Subgroups — C₃×#60450 #10679057

Order 768.

2722 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#3449	C₂	1	2	yes	maximal, complemented
4	C₃×#3061	C₂	1	4	yes	maximal, complemented
5	C₃×#3060	C₂	1	4	yes	maximal, complemented
6	C₃×#3059	C₂	1	2	yes	maximal, complemented
7	C₃×#3058	C₂	1	2	yes	maximal, complemented
8	C₃×#3057	C₂	1	4	yes	maximal, complemented
9	C₃×#3054	C₂	1	4	yes	maximal, complemented
10	C₃×#3050	C₂	1	4	yes	maximal, complemented
11	C₂×C₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#2896	C₂	1	2	yes	maximal, complemented
13	#60450	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂²×C₃×#41	C₂²	1	2	yes	—
15	C₂²×C₃×#41	C₂²	1	1	yes	characteristic
16	C₂×C₃×#135	—	2	8	no	—
17	C₂×C₃×#135	C₂²	1	2	yes	—
18	C₂×C₃×#135	C₂²	1	2	yes	—
19	C₂×C₃×#135	C₂²	1	2	yes	complemented
20	C₂×C₃×#135	C₂²	1	2	yes	complemented
21	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
22	C₂²×C₃×SD₁₆	C₂²	1	2	yes	complemented
23	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₃×#131	—	2	8	no	—
25	C₂×C₃×#131	C₂²	1	2	yes	—
26	C₂×C₃×#131	C₂²	1	2	yes	—
27	C₂×C₃×#131	C₂²	1	2	yes	complemented
28	C₂×C₃×#131	C₂²	1	2	yes	complemented
29	C₂²×C₃×M₁₆	C₂²	1	1	yes	characteristic
30	C₂²×C₃×M₁₆	C₂²	1	2	yes	complemented
31	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
32	C₂³×C₃×C₈	C₂²	1	2	yes	complemented
33	C₃×#537	C₂²	1	8	yes	complemented
34	C₃×#536	—	2	2	no	—
35	C₃×#536	C₂²	1	4	yes	—
36	C₃×#535	—	2	2	no	—
37	C₃×#535	C₂²	1	4	yes	—
38	C₃×#534	—	2	4	no	—
39	C₃×#532	—	2	4	no	—
40	C₃×#525	—	2	4	no	—
41	C₃×#525	C₂²	1	8	yes	complemented
42	C₂×C₃×#117	C₂²	1	2	yes	—
43	C₃×#470	C₂²	1	8	yes	complemented
44	C₃×#469	C₂²	1	8	yes	complemented
45	C₃×#468	C₂²	1	16	yes	complemented
46	C₃×#467	C₂²	1	8	yes	complemented
47	C₃×#466	C₂²	1	8	yes	complemented
48	C₃×#465	C₂²	1	16	yes	complemented
49	C₃×#427	C₂²	1	2	yes	complemented
50	C₃×#427	C₂²	1	2	yes	complemented
51	C₃×#427	C₂²	1	2	yes	complemented
52	C₃×#427	C₂²	1	2	yes	complemented
53	C₂×C₃×#107	C₂²	1	2	yes	complemented
54	C₂×C₃×#106	C₂²	1	2	yes	complemented
55	C₃×#419	C₂²	1	4	yes	complemented
56	C₃×#418	C₂²	1	4	yes	complemented
57	C₃×#417	C₂²	1	8	yes	complemented
58	C₃×#416	C₂²	1	8	yes	complemented
59	C₂²×#135	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂²×#131	C₂×C₃	1	1	yes	characteristic, complemented
61	#3449	C₂×C₃	1	2	yes	complemented
62	#3061	C₂×C₃	1	4	yes	complemented
63	#3060	C₂×C₃	1	4	yes	complemented
64	#3059	C₂×C₃	1	2	yes	complemented
65	#3058	C₂×C₃	1	2	yes	complemented
66	#3057	C₂×C₃	1	4	yes	complemented
67	#3054	C₂×C₃	1	4	yes	complemented
68	#3050	C₂×C₃	1	4	yes	complemented
69	C₂×#427	C₂×C₃	1	1	yes	characteristic, complemented
70	#2896	C₂×C₃	1	2	yes	complemented
71	C₂×C₃×#41	—	2	4	no	—
72	C₂×C₃×#41	—	2	2	no	—
73	C₂×C₃×#41	—	2	2	no	—
74	C₂×C₃×#41	—	2	4	no	—
75	C₂×C₃×#41	—	2	8	no	—
76	C₂×C₃×#41	—	2	8	no	—
77	C₂×C₃×#41	D₄	1	2	yes	—
78	C₂×C₃×#41	D₄	1	2	yes	—
79	C₂×C₃×#41	C₂³	1	4	yes	—
80	C₂×C₃×#41	C₂³	1	4	yes	—
81	C₂×C₃×#41	C₂³	1	2	yes	—
82	C₂×C₃×#41	C₂³	1	2	yes	—
83	C₂²×C₃×Q₈	D₄	1	1	yes	characteristic
84	C₂²×C₃×Q₈	C₂³	1	2	yes	—
85	C₂²×C₃×D₄	—	2	2	no	—
86	C₂²×C₃×D₄	D₄	1	2	yes	—
87	C₂²×C₃×D₄	D₄	1	1	yes	characteristic
88	C₂²×C₃×D₄	C₂³	1	2	yes	—
89	C₂³×C₃×C₄	—	2	2	no	—
90	C₂³×C₃×C₄	C₂³	1	2	yes	—
91	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
92	C₃×#135	—	2	8	no	—
93	C₃×#135	—	2	8	no	—
94	C₃×#135	—	2	8	no	—
95	C₃×#135	—	2	8	no	—
96	C₂×C₃×Q₁₆	—	2	4	no	—
97	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
98	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
99	C₂×C₃×SD₁₆	—	2	8	no	—
100	C₂×C₃×SD₁₆	C₂³	1	4	yes	—
101	C₂×C₃×SD₁₆	C₂³	1	4	yes	—
102	C₂×C₃×D₈	—	2	4	no	—
103	C₂×C₃×D₈	C₂³	1	2	yes	—
104	C₂×C₃×D₈	C₂³	1	2	yes	—
105	C₃×#131	—	2	8	no	—
106	C₃×#131	—	2	8	no	—
107	C₃×#131	—	2	8	no	—
108	C₃×#131	—	2	8	no	—
109	C₂×C₃×M₁₆	—	2	4	no	—
110	C₂×C₃×M₁₆	—	2	8	no	—
111	C₂×C₃×M₁₆	C₂³	1	2	yes	—
112	C₂×C₃×M₁₆	C₂³	1	2	yes	—
113	C₂×C₃×M₁₆	C₂³	1	4	yes	—
114	C₂×C₃×M₁₆	C₂³	1	4	yes	—
115	C₂²×C₃×C₈	—	2	8	no	—
116	C₂²×C₃×C₈	D₄	1	8	yes	complemented
117	C₂²×C₃×C₈	C₂³	1	2	yes	—
118	C₂²×C₃×C₈	C₂³	1	4	yes	—
119	C₂²×C₃×C₈	C₂³	1	4	yes	—
120	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
121	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
122	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
123	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
124	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
125	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
126	C₃×C₄⋊Q₈	—	2	4	no	—
127	C₃×#127	—	2	4	no	—
128	C₃×#124	—	2	4	no	—
129	C₃×#124	—	2	4	no	—
130	C₃×#123	—	2	8	no	—
131	C₃×#123	—	2	8	no	—
132	C₃×C₂²⋊Q₈	—	2	8	no	—
133	C₃×C₂²⋊Q₈	C₂³	1	16	yes	—
134	C₃×#121	—	2	8	no	—
135	C₃×#121	—	2	16	no	—
136	C₃×#121	C₂³	1	16	yes	—
137	C₃×#120	—	2	8	no	—
138	C₃×#120	—	2	8	no	—
139	C₃×C₄×D₄	—	2	16	no	—
140	C₃×#117	—	2	2	no	—
141	C₃×#117	—	2	2	no	—
142	C₃×#117	C₂³	1	4	yes	—
143	C₃×#117	C₂³	1	4	yes	—
144	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
145	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
146	C₂×C₃×C₄²	—	2	2	no	—
147	C₃×#107	C₂³	1	4	yes	—
148	C₃×#107	C₂³	1	4	yes	—
149	C₃×#106	C₂³	1	4	yes	—
150	C₃×#106	C₂³	1	4	yes	—
151	C₃×#103	C₂³	1	16	yes	—
152	C₃×#102	C₂³	1	16	yes	—
153	C₂²×#41	C₂²×C₃	1	2	yes	—
154	C₂²×#41	C₂²×C₃	1	1	yes	characteristic
155	C₂×#135	—	2	8	no	—
156	C₂×#135	C₂²×C₃	1	2	yes	—
157	C₂×#135	C₂²×C₃	1	2	yes	—
158	C₂×#135	C₂²×C₃	1	2	yes	complemented
159	C₂×#135	C₂²×C₃	1	2	yes	complemented
160	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
161	C₂²×SD₁₆	C₂²×C₃	1	2	yes	complemented
162	C₂²×D₈	C₂²×C₃	1	1	yes	characteristic, complemented
163	C₂×#131	—	2	8	no	—
164	C₂×#131	C₂²×C₃	1	2	yes	—
165	C₂×#131	C₂²×C₃	1	2	yes	—
166	C₂×#131	C₂²×C₃	1	2	yes	complemented
167	C₂×#131	C₂²×C₃	1	2	yes	complemented
168	C₂²×M₁₆	C₂²×C₃	1	1	yes	characteristic
169	C₂²×M₁₆	C₂²×C₃	1	2	yes	complemented
170	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
171	C₂³×C₈	C₂²×C₃	1	2	yes	complemented
172	#537	C₂²×C₃	1	8	yes	complemented
173	#536	—	2	2	no	—
174	#536	C₂²×C₃	1	4	yes	—
175	#535	—	2	2	no	—
176	#535	C₂²×C₃	1	4	yes	—
177	#534	—	2	4	no	—
178	#532	—	2	4	no	—
179	#525	—	2	4	no	—
180	#525	C₂²×C₃	1	8	yes	complemented
181	C₂×#117	C₂²×C₃	1	2	yes	—
182	#470	C₂²×C₃	1	8	yes	complemented
183	#469	C₂²×C₃	1	8	yes	complemented
184	#468	C₂²×C₃	1	16	yes	complemented
185	#467	C₂²×C₃	1	8	yes	complemented
186	#466	C₂²×C₃	1	8	yes	complemented
187	#465	C₂²×C₃	1	16	yes	complemented
188	#427	C₂²×C₃	1	2	yes	complemented
189	#427	C₂²×C₃	1	2	yes	complemented
190	#427	C₂²×C₃	1	2	yes	complemented
191	#427	C₂²×C₃	1	2	yes	complemented
192	C₂×#107	C₂²×C₃	1	2	yes	complemented
193	C₂×#106	C₂²×C₃	1	2	yes	complemented
194	#419	C₂²×C₃	1	4	yes	complemented
195	#418	C₂²×C₃	1	4	yes	complemented
196	#417	C₂²×C₃	1	8	yes	complemented
197	#416	C₂²×C₃	1	8	yes	complemented
198	C₂⁴×C₃	—	2	2	no	—
199	C₂⁴×C₃	C₂×D₄	1	2	yes	—
200	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
201	C₃×#41	—	2	8	no	—
202	C₃×#41	—	2	8	no	—
203	C₃×#41	—	2	4	no	—
204	C₃×#41	—	2	4	no	—
205	C₃×#41	—	2	4	no	—
206	C₃×#41	—	2	4	no	—
207	C₃×#41	—	2	8	no	—
208	C₃×#41	—	2	8	no	—
209	C₃×#41	—	2	8	no	—
210	C₃×#41	—	2	8	no	—
211	C₃×#41	—	4	8	no	—
212	C₃×#41	—	4	8	no	—
213	C₂×C₃×Q₈	—	2	4	no	—
214	C₂×C₃×Q₈	—	2	2	no	—
215	C₂×C₃×Q₈	—	2	4	no	—
216	C₂×C₃×Q₈	—	2	4	no	—
217	C₂×C₃×Q₈	C₂×D₄	1	2	yes	—
218	C₂×C₃×Q₈	C₂×D₄	1	2	yes	—
219	C₂×C₃×Q₈	C₂⁴	1	4	yes	—
220	C₂×C₃×D₄	—	2	8	no	—
221	C₂×C₃×D₄	—	2	4	no	—
222	C₂×C₃×D₄	—	2	4	no	—
223	C₂×C₃×D₄	—	2	16	no	—
224	C₂×C₃×D₄	—	2	4	no	—
225	C₂×C₃×D₄	—	2	4	no	—
226	C₂×C₃×D₄	—	2	2	no	—
227	C₂×C₃×D₄	—	2	4	no	—
228	C₂×C₃×D₄	—	4	8	no	—
229	C₂×C₃×D₄	C₂×D₄	1	4	yes	—
230	C₂×C₃×D₄	C₂×D₄	1	4	yes	—
231	C₂×C₃×D₄	C₂×D₄	1	2	yes	—
232	C₂×C₃×D₄	C₂×D₄	1	2	yes	—
233	C₂×C₃×D₄	C₂⁴	1	4	yes	—
234	C₂²×C₃×C₄	—	2	4	no	—
235	C₂²×C₃×C₄	—	2	4	no	—
236	C₂²×C₃×C₄	—	2	2	no	—
237	C₂²×C₃×C₄	—	2	4	no	—
238	C₂²×C₃×C₄	—	2	4	no	—
239	C₂²×C₃×C₄	—	2	4	no	—
240	C₂²×C₃×C₄	—	4	8	no	—
241	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
242	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
243	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
244	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
245	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
246	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
247	C₂²×C₃×C₄	#41	1	4	yes	—
248	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
249	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
250	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
251	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
252	C₃×Q₁₆	—	2	4	no	—
253	C₃×Q₁₆	—	2	4	no	—
254	C₃×SD₁₆	—	2	8	no	—
255	C₃×SD₁₆	—	2	8	no	—
256	C₃×D₈	—	2	4	no	—
257	C₃×D₈	—	2	4	no	—
258	C₃×M₁₆	—	2	4	no	—
259	C₃×M₁₆	—	2	4	no	—
260	C₃×M₁₆	—	2	8	no	—
261	C₃×M₁₆	—	2	8	no	—
262	C₂×C₃×C₈	—	2	4	no	—
263	C₂×C₃×C₈	—	2	8	no	—
264	C₂×C₃×C₈	—	2	8	no	—
265	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
266	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
267	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
268	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
269	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
270	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
271	C₂×C₃×C₈	C₂⁴	1	4	yes	—
272	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
273	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
274	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
275	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
276	C₃×C₄⋊C₄	—	2	4	no	—
277	C₃×C₄⋊C₄	—	2	16	no	—
278	C₃×C₄⋊C₄	C₂⁴	1	8	yes	—
279	C₃×C₂²⋊C₄	—	2	4	no	—
280	C₃×C₂²⋊C₄	—	2	4	no	—
281	C₃×C₂²⋊C₄	—	2	16	no	—
282	C₃×C₂²⋊C₄	—	2	16	no	—
283	C₃×C₄²	—	2	4	no	—
284	C₃×C₄²	—	2	4	no	—
285	C₂×#41	—	2	4	no	—
286	C₂×#41	—	2	2	no	—
287	C₂×#41	—	2	2	no	—
288	C₂×#41	—	2	4	no	—
289	C₂×#41	—	2	8	no	—
290	C₂×#41	—	2	8	no	—
291	C₂×#41	C₃×D₄	1	2	yes	—
292	C₂×#41	C₃×D₄	1	2	yes	—
293	C₂×#41	C₂³×C₃	1	4	yes	—
294	C₂×#41	C₂³×C₃	1	4	yes	—
295	C₂×#41	C₂³×C₃	1	2	yes	—
296	C₂×#41	C₂³×C₃	1	2	yes	—
297	C₂²×Q₈	C₃×D₄	1	1	yes	characteristic
298	C₂²×Q₈	C₂³×C₃	1	2	yes	—
299	C₂²×D₄	—	2	2	no	—
300	C₂²×D₄	C₃×D₄	1	2	yes	—
301	C₂²×D₄	C₃×D₄	1	1	yes	characteristic
302	C₂²×D₄	C₂³×C₃	1	2	yes	—
303	C₂³×C₄	—	2	2	no	—
304	C₂³×C₄	C₂³×C₃	1	2	yes	—
305	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
306	#135	—	2	8	no	—
307	#135	—	2	8	no	—
308	#135	—	2	8	no	—
309	#135	—	2	8	no	—
310	C₂×Q₁₆	—	2	4	no	—
311	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
312	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
313	C₂×SD₁₆	—	2	8	no	—
314	C₂×SD₁₆	C₂³×C₃	1	4	yes	—
315	C₂×SD₁₆	C₂³×C₃	1	4	yes	—
316	C₂×D₈	—	2	4	no	—
317	C₂×D₈	C₂³×C₃	1	2	yes	—
318	C₂×D₈	C₂³×C₃	1	2	yes	—
319	#131	—	2	8	no	—
320	#131	—	2	8	no	—
321	#131	—	2	8	no	—
322	#131	—	2	8	no	—
323	C₂×M₁₆	—	2	4	no	—
324	C₂×M₁₆	—	2	8	no	—
325	C₂×M₁₆	C₂³×C₃	1	2	yes	—
326	C₂×M₁₆	C₂³×C₃	1	2	yes	—
327	C₂×M₁₆	C₂³×C₃	1	4	yes	—
328	C₂×M₁₆	C₂³×C₃	1	4	yes	—
329	C₂²×C₈	—	2	8	no	—
330	C₂²×C₈	C₃×D₄	1	8	yes	complemented
331	C₂²×C₈	C₂³×C₃	1	2	yes	—
332	C₂²×C₈	C₂³×C₃	1	4	yes	—
333	C₂²×C₈	C₂³×C₃	1	4	yes	—
334	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
335	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
336	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
337	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
338	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
339	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
340	C₄⋊Q₈	—	2	4	no	—
341	#127	—	2	4	no	—
342	#124	—	2	4	no	—
343	#124	—	2	4	no	—
344	#123	—	2	8	no	—
345	#123	—	2	8	no	—
346	C₂²⋊Q₈	—	2	8	no	—
347	C₂²⋊Q₈	C₂³×C₃	1	16	yes	—
348	#121	—	2	8	no	—
349	#121	—	2	16	no	—
350	#121	C₂³×C₃	1	16	yes	—
351	#120	—	2	8	no	—
352	#120	—	2	8	no	—
353	C₄×D₄	—	2	16	no	—
354	#117	—	2	2	no	—
355	#117	—	2	2	no	—
356	#117	C₂³×C₃	1	4	yes	—
357	#117	C₂³×C₃	1	4	yes	—
358	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
359	C₂×C₂²⋊C₄	—	2	2	no	—
360	C₂×C₄²	—	2	2	no	—
361	#107	C₂³×C₃	1	4	yes	—
362	#107	C₂³×C₃	1	4	yes	—
363	#106	C₂³×C₃	1	4	yes	—
364	#106	C₂³×C₃	1	4	yes	—
365	#103	C₂³×C₃	1	16	yes	—
366	#102	C₂³×C₃	1	16	yes	—
367	C₂³×C₃	—	2	4	no	—
368	C₂³×C₃	—	2	4	no	—
369	C₂³×C₃	—	4	2	no	—
370	C₂³×C₃	—	4	4	no	—
371	C₂³×C₃	—	4	4	no	—
372	C₂³×C₃	—	8	4	no	—
373	C₂³×C₃	#121	1	4	yes	—
374	C₂³×C₃	C₂²×D₄	1	4	yes	—
375	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
376	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
377	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
378	C₃×Q₈	—	2	4	no	—
379	C₃×Q₈	—	2	4	no	—
380	C₃×Q₈	—	2	4	no	—
381	C₃×Q₈	—	2	4	no	—
382	C₃×Q₈	—	4	4	no	—
383	C₃×D₄	—	2	8	no	—
384	C₃×D₄	—	2	8	no	—
385	C₃×D₄	—	2	4	no	—
386	C₃×D₄	—	2	4	no	—
387	C₃×D₄	—	2	4	no	—
388	C₃×D₄	—	2	4	no	—
389	C₃×D₄	—	4	8	no	—
390	C₃×D₄	—	4	8	no	—
391	C₃×D₄	—	4	4	no	—
392	C₃×D₄	—	8	16	no	—
393	C₂×C₃×C₄	—	2	4	no	—
394	C₂×C₃×C₄	—	2	8	no	—
395	C₂×C₃×C₄	—	2	4	no	—
396	C₂×C₃×C₄	—	2	4	no	—
397	C₂×C₃×C₄	—	2	8	no	—
398	C₂×C₃×C₄	—	2	4	no	—
399	C₂×C₃×C₄	—	4	4	no	—
400	C₂×C₃×C₄	—	4	8	no	—
401	C₂×C₃×C₄	—	4	8	no	—
402	C₂×C₃×C₄	—	8	4	no	—
403	C₂×C₃×C₄	#121	1	4	yes	—
404	C₂×C₃×C₄	#121	1	4	yes	—
405	C₂×C₃×C₄	#121	1	4	yes	—
406	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
407	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
408	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
409	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
410	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
411	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
412	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
413	C₂×C₃×C₄	C₂×#41	1	2	yes	—
414	C₂×C₃×C₄	C₂×#41	1	2	yes	—
415	C₂×C₃×C₄	C₂×#41	1	2	yes	—
416	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
417	C₃×C₈	—	2	4	no	—
418	C₃×C₈	C₂²×D₄	1	2	yes	—
419	C₃×C₈	C₂²×D₄	1	2	yes	—
420	C₃×C₈	C₂²×D₄	1	2	yes	—
421	C₃×C₈	C₂²×D₄	1	2	yes	—
422	C₂⁴	—	2	2	no	—
423	C₂⁴	C₂×C₃×D₄	1	2	yes	—
424	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
425	#41	—	2	8	no	—
426	#41	—	2	8	no	—
427	#41	—	2	4	no	—
428	#41	—	2	4	no	—
429	#41	—	2	4	no	—
430	#41	—	2	4	no	—
431	#41	—	2	8	no	—
432	#41	—	2	8	no	—
433	#41	—	2	8	no	—
434	#41	—	2	8	no	—
435	#41	—	4	8	no	—
436	#41	—	4	8	no	—
437	C₂×Q₈	—	2	4	no	—
438	C₂×Q₈	—	2	2	no	—
439	C₂×Q₈	—	2	4	no	—
440	C₂×Q₈	—	2	4	no	—
441	C₂×Q₈	C₂×C₃×D₄	1	2	yes	—
442	C₂×Q₈	C₂×C₃×D₄	1	2	yes	—
443	C₂×Q₈	C₂⁴×C₃	1	4	yes	—
444	C₂×D₄	—	2	8	no	—
445	C₂×D₄	—	2	4	no	—
446	C₂×D₄	—	2	4	no	—
447	C₂×D₄	—	2	16	no	—
448	C₂×D₄	—	2	4	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	4	no	—
452	C₂×D₄	—	4	8	no	—
453	C₂×D₄	C₂×C₃×D₄	1	4	yes	—
454	C₂×D₄	C₂×C₃×D₄	1	4	yes	—
455	C₂×D₄	C₂×C₃×D₄	1	2	yes	—
456	C₂×D₄	C₂×C₃×D₄	1	2	yes	—
457	C₂×D₄	C₂⁴×C₃	1	4	yes	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	4	8	no	—
465	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
466	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
467	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
468	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
469	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
470	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
471	C₂²×C₄	C₃×#41	1	4	yes	—
472	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
473	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
474	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
475	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
476	Q₁₆	—	2	4	no	—
477	Q₁₆	—	2	4	no	—
478	SD₁₆	—	2	8	no	—
479	SD₁₆	—	2	8	no	—
480	D₈	—	2	4	no	—
481	D₈	—	2	4	no	—
482	M₁₆	—	2	4	no	—
483	M₁₆	—	2	4	no	—
484	M₁₆	—	2	8	no	—
485	M₁₆	—	2	8	no	—
486	C₂×C₈	—	2	4	no	—
487	C₂×C₈	—	2	8	no	—
488	C₂×C₈	—	2	8	no	—
489	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
490	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
491	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
492	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
493	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
494	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
495	C₂×C₈	C₂⁴×C₃	1	4	yes	—
496	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
497	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
498	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
499	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
500	C₄⋊C₄	—	2	4	no	—
501	C₄⋊C₄	—	2	16	no	—
502	C₄⋊C₄	C₂⁴×C₃	1	8	yes	—
503	C₂²⋊C₄	—	2	4	no	—
504	C₂²⋊C₄	—	2	4	no	—
505	C₂²⋊C₄	—	2	16	no	—
506	C₂²⋊C₄	—	2	16	no	—
507	C₄²	—	2	4	no	—
508	C₄²	—	2	4	no	—
509	C₂²×C₃	—	2	2	no	—
510	C₂²×C₃	—	2	2	no	—
511	C₂²×C₃	—	2	2	no	—
512	C₂²×C₃	—	2	8	no	—
513	C₂²×C₃	—	2	4	no	—
514	C₂²×C₃	—	4	4	no	—
515	C₂²×C₃	—	4	8	no	—
516	C₂²×C₃	—	4	8	no	—
517	C₂²×C₃	—	4	4	no	—
518	C₂²×C₃	—	8	8	no	—
519	C₂²×C₃	C₂×#121	1	2	yes	—
520	C₂²×C₃	C₂×#121	1	2	yes	—
521	C₂²×C₃	C₂×#121	1	2	yes	—
522	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
523	C₃×C₄	—	2	4	no	—
524	C₃×C₄	—	4	8	no	—
525	C₃×C₄	—	4	4	no	—
526	C₃×C₄	C₂×#121	1	2	yes	—
527	C₃×C₄	C₂×#121	1	2	yes	—
528	C₃×C₄	C₂×#121	1	2	yes	—
529	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
530	C₃×C₄	C₂²×#41	1	1	yes	characteristic
531	C₂³	—	2	4	no	—
532	C₂³	—	2	4	no	—
533	C₂³	—	4	2	no	—
534	C₂³	—	4	4	no	—
535	C₂³	—	4	4	no	—
536	C₂³	—	8	4	no	—
537	C₂³	C₃×#121	1	4	yes	—
538	C₂³	C₂²×C₃×D₄	1	4	yes	—
539	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
540	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
541	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
542	Q₈	—	2	4	no	—
543	Q₈	—	2	4	no	—
544	Q₈	—	2	4	no	—
545	Q₈	—	2	4	no	—
546	Q₈	—	4	4	no	—
547	D₄	—	2	8	no	—
548	D₄	—	2	8	no	—
549	D₄	—	2	4	no	—
550	D₄	—	2	4	no	—
551	D₄	—	2	4	no	—
552	D₄	—	2	4	no	—
553	D₄	—	4	8	no	—
554	D₄	—	4	8	no	—
555	D₄	—	4	4	no	—
556	D₄	—	8	16	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	8	no	—
561	C₂×C₄	—	2	8	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	4	4	no	—
564	C₂×C₄	—	4	8	no	—
565	C₂×C₄	—	4	8	no	—
566	C₂×C₄	—	8	4	no	—
567	C₂×C₄	C₃×#121	1	4	yes	—
568	C₂×C₄	C₃×#121	1	4	yes	—
569	C₂×C₄	C₃×#121	1	4	yes	—
570	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
571	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
572	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
573	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
574	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
575	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
576	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
577	C₂×C₄	C₂×C₃×#41	1	2	yes	—
578	C₂×C₄	C₂×C₃×#41	1	2	yes	—
579	C₂×C₄	C₂×C₃×#41	1	2	yes	—
580	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
581	C₈	—	2	4	no	—
582	C₈	C₂²×C₃×D₄	1	2	yes	—
583	C₈	C₂²×C₃×D₄	1	2	yes	—
584	C₈	C₂²×C₃×D₄	1	2	yes	—
585	C₈	C₂²×C₃×D₄	1	2	yes	—
586	C₂×C₃	—	2	2	no	—
587	C₂×C₃	—	2	2	no	—
588	C₂×C₃	—	2	2	no	—
589	C₂×C₃	—	4	4	no	—
590	C₂×C₃	—	8	4	no	—
591	C₂×C₃	C₂²×#121	1	1	yes	characteristic
592	C₂×C₃	#3584	1	2	yes	—
593	C₂²	—	2	2	no	—
594	C₂²	—	2	2	no	—
595	C₂²	—	2	2	no	—
596	C₂²	—	2	4	no	—
597	C₂²	—	2	8	no	—
598	C₂²	—	4	4	no	—
599	C₂²	—	4	8	no	—
600	C₂²	—	4	4	no	—
601	C₂²	—	4	8	no	—
602	C₂²	—	8	8	no	—
603	C₂²	C₂×C₃×#121	1	2	yes	—
604	C₂²	C₂×C₃×#121	1	2	yes	—
605	C₂²	C₂×C₃×#121	1	2	yes	—
606	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
607	C₄	—	2	4	no	—
608	C₄	—	4	4	no	—
609	C₄	—	4	8	no	—
610	C₄	C₂×C₃×#121	1	2	yes	—
611	C₄	C₂×C₃×#121	1	2	yes	—
612	C₄	C₂×C₃×#121	1	2	yes	—
613	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
614	C₄	C₂²×C₃×#41	1	1	yes	characteristic
615	C₃	#60450	1	1	yes	characteristic, direct product factor, Sylow
616	C₂	—	2	2	no	—
617	C₂	—	2	2	no	—
618	C₂	—	2	2	no	—
619	C₂	—	4	4	no	—
620	C₂	—	8	4	no	—
621	C₂	C₂²×C₃×#121	1	1	yes	characteristic
622	C₂	C₃×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
socle	C₂²×C₃	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×#41 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×C₃×#135	C₂	semidirect
2. C₂²×C₃×#131	C₂	semidirect
3. C₃×#3449	C₂	semidirect
4. C₃×#3061	C₂	semidirect
5. C₃×#3060	C₂	semidirect
6. C₃×#3059	C₂	semidirect
7. C₃×#3058	C₂	semidirect
8. C₃×#3057	C₂	semidirect
9. C₃×#3054	C₂	semidirect
10. C₃×#3050	C₂	semidirect
11. C₂×C₃×#427	C₂	semidirect
12. C₃×#2896	C₂	semidirect
13. #60450	C₃	direct (N × H)
19. C₂×C₃×#135	C₂²	semidirect
20. C₂×C₃×#135	C₂²	semidirect
21. C₂²×C₃×Q₁₆	C₂²	semidirect
22. C₂²×C₃×SD₁₆	C₂²	semidirect
23. C₂²×C₃×D₈	C₂²	semidirect
27. C₂×C₃×#131	C₂²	semidirect
28. C₂×C₃×#131	C₂²	semidirect
30. C₂²×C₃×M₁₆	C₂²	semidirect
32. C₂³×C₃×C₈	C₂²	semidirect
33. C₃×#537	C₂²	semidirect
41. C₃×#525	C₂²	semidirect
43. C₃×#470	C₂²	semidirect
44. C₃×#469	C₂²	semidirect
45. C₃×#468	C₂²	semidirect
46. C₃×#467	C₂²	semidirect
47. C₃×#466	C₂²	semidirect
48. C₃×#465	C₂²	semidirect
49. C₃×#427	C₂²	semidirect
50. C₃×#427	C₂²	semidirect
51. C₃×#427	C₂²	semidirect
52. C₃×#427	C₂²	semidirect
53. C₂×C₃×#107	C₂²	semidirect
54. C₂×C₃×#106	C₂²	semidirect
55. C₃×#419	C₂²	semidirect
56. C₃×#418	C₂²	semidirect
57. C₃×#417	C₂²	semidirect
58. C₃×#416	C₂²	semidirect
59. C₂²×#135	C₂×C₃	semidirect
60. C₂²×#131	C₂×C₃	semidirect
61. #3449	C₂×C₃	semidirect
62. #3061	C₂×C₃	semidirect
63. #3060	C₂×C₃	semidirect
64. #3059	C₂×C₃	semidirect
65. #3058	C₂×C₃	semidirect
66. #3057	C₂×C₃	semidirect
67. #3054	C₂×C₃	semidirect
68. #3050	C₂×C₃	semidirect
69. C₂×#427	C₂×C₃	semidirect
70. #2896	C₂×C₃	semidirect
116. C₂²×C₃×C₈	D₄	semidirect
158. C₂×#135	C₂²×C₃	semidirect
159. C₂×#135	C₂²×C₃	semidirect
160. C₂²×Q₁₆	C₂²×C₃	semidirect
161. C₂²×SD₁₆	C₂²×C₃	semidirect
162. C₂²×D₈	C₂²×C₃	semidirect
166. C₂×#131	C₂²×C₃	semidirect
167. C₂×#131	C₂²×C₃	semidirect
169. C₂²×M₁₆	C₂²×C₃	semidirect
171. C₂³×C₈	C₂²×C₃	semidirect
172. #537	C₂²×C₃	semidirect
180. #525	C₂²×C₃	semidirect
182. #470	C₂²×C₃	semidirect
183. #469	C₂²×C₃	semidirect
184. #468	C₂²×C₃	semidirect
185. #467	C₂²×C₃	semidirect
186. #466	C₂²×C₃	semidirect
187. #465	C₂²×C₃	semidirect
188. #427	C₂²×C₃	semidirect
189. #427	C₂²×C₃	semidirect
190. #427	C₂²×C₃	semidirect
191. #427	C₂²×C₃	semidirect
192. C₂×#107	C₂²×C₃	semidirect
193. C₂×#106	C₂²×C₃	semidirect
194. #419	C₂²×C₃	semidirect
195. #418	C₂²×C₃	semidirect
196. #417	C₂²×C₃	semidirect
197. #416	C₂²×C₃	semidirect
330. C₂²×C₈	C₃×D₄	semidirect
615. C₃	#60450	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×#135	2
C₂²×C₃×#131	2
C₃×#3449	2
C₃×#3061	2
C₃×#3060	2
C₃×#3059	2
C₃×#3058	2
C₃×#3057	2
C₃×#3054	2
C₃×#3050	2
C₂×C₃×#427	2
C₃×#2896	2
#60450	3

Subgroups — C3×#60450 #10679057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#60450 #10679057