Subgroups — C₃×#60144 #10678751

Order 768.

2778 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#568	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×#532	C₂	1	1	yes	maximal, characteristic
3	C₃×#2935	C₂	1	8	yes	maximal, complemented
4	C₃×D₄×M₁₆	C₂	1	8	yes	maximal, complemented
5	C₃×#2932	C₂	1	4	yes	maximal, complemented
6	C₃×#2931	C₂	1	4	yes	maximal, complemented
7	C₂×C₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2879	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#60144	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×#41	C₂²	1	2	yes	—
12	C₃×#568	—	2	16	no	—
13	C₂×C₃×#131	C₂²	1	8	yes	complemented
14	C₂×C₃×#131	C₂²	1	8	yes	complemented
15	C₂²×C₃×M₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₃×M₁₆	C₂²	1	2	yes	—
17	C₂²×C₃×M₁₆	C₂²	1	8	yes	—
18	C₂²×C₃×M₁₆	C₂²	1	1	yes	characteristic
19	C₃×#532	C₄	1	2	yes	—
20	C₃×#532	C₄	1	4	yes	—
21	C₃×#532	C₄	1	2	yes	—
22	C₃×#532	C₂²	1	2	yes	—
23	C₃×#532	C₂²	1	2	yes	—
24	C₃×#532	C₂²	1	4	yes	—
25	C₂×C₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
26	C₂×C₃×#127	C₄	1	1	yes	characteristic
27	C₂×C₃×#124	C₄	1	2	yes	—
28	C₂×C₃×#121	C₄	1	4	yes	—
29	C₂×C₃×C₄×D₄	C₂²	1	4	yes	—
30	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
31	C₃×#436	C₂²	1	16	yes	complemented
32	C₃×#435	C₂²	1	32	yes	complemented
33	C₃×C₈×D₄	C₂²	1	16	yes	complemented
34	C₃×#423	C₂²	1	4	yes	—
35	C₃×#423	C₂²	1	2	yes	complemented
36	C₃×#423	C₂²	1	2	yes	complemented
37	C₂×C₃×C₄⋊C₈	C₂²	1	4	yes	complemented
38	C₃×#408	C₂²	1	8	yes	complemented
39	C₃×#408	C₂²	1	8	yes	complemented
40	C₃×#407	C₂²	1	8	yes	—
41	C₃×C₄×M₁₆	C₂²	1	4	yes	—
42	C₃×C₄×M₁₆	C₂²	1	4	yes	complemented
43	C₂×C₃×#97	C₂²	1	2	yes	complemented
44	C₂×C₃×C₄×C₈	C₂²	1	2	yes	complemented
45	C₂×#568	C₂×C₃	1	2	yes	complemented
46	C₂×#532	C₂×C₃	1	1	yes	characteristic
47	#2935	C₂×C₃	1	8	yes	complemented
48	D₄×M₁₆	C₂×C₃	1	8	yes	complemented
49	#2932	C₂×C₃	1	4	yes	complemented
50	#2931	C₂×C₃	1	4	yes	complemented
51	C₂×#423	C₂×C₃	1	1	yes	characteristic, complemented
52	#2879	C₂×C₃	1	2	yes	complemented
53	C₂×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
54	C₂×C₃×#41	—	2	8	no	—
55	C₂×C₃×#41	—	2	8	no	—
56	C₂×C₃×#41	C₂×C₄	1	4	yes	—
57	C₂×C₃×#41	C₂×C₄	1	4	yes	—
58	C₂×C₃×#41	C₂³	1	4	yes	—
59	C₂×C₃×#41	C₂³	1	4	yes	—
60	C₂²×C₃×Q₈	C₂×C₄	1	2	yes	—
61	C₂²×C₃×D₄	C₂×C₄	1	4	yes	—
62	C₂²×C₃×D₄	C₂×C₄	1	2	yes	—
63	C₂²×C₃×D₄	C₂×C₄	1	4	yes	—
64	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
65	C₂³×C₃×C₄	C₂³	1	4	yes	—
66	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
67	C₃×#131	—	2	32	no	—
68	C₃×#131	—	2	32	no	—
69	C₂×C₃×M₁₆	—	2	8	no	—
70	C₂×C₃×M₁₆	—	2	4	no	—
71	C₂×C₃×M₁₆	—	2	32	no	—
72	C₂×C₃×M₁₆	D₄	1	8	yes	complemented
73	C₂×C₃×M₁₆	C₂³	1	4	yes	—
74	C₂×C₃×M₁₆	C₂³	1	4	yes	—
75	C₂×C₃×M₁₆	C₂³	1	16	yes	—
76	C₂×C₃×M₁₆	C₂³	1	2	yes	—
77	C₂×C₃×M₁₆	C₂³	1	2	yes	—
78	C₂×C₃×M₁₆	C₂³	1	2	yes	—
79	C₂×C₃×M₁₆	C₂³	1	2	yes	—
80	C₂²×C₃×C₈	C₂³	1	4	yes	—
81	C₂²×C₃×C₈	C₂³	1	16	yes	—
82	C₂²×C₃×C₈	C₂³	1	2	yes	—
83	C₂²×C₃×C₈	C₂³	1	2	yes	—
84	C₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
85	C₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
86	C₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
87	C₃×#127	C₂×C₄	1	2	yes	—
88	C₃×#127	C₂×C₄	1	4	yes	—
89	C₃×#127	C₂×C₄	1	2	yes	—
90	C₃×#124	C₂×C₄	1	4	yes	—
91	C₃×#124	C₂×C₄	1	4	yes	—
92	C₃×#124	C₂×C₄	1	8	yes	—
93	C₃×#121	C₂×C₄	1	8	yes	—
94	C₃×#121	C₂×C₄	1	8	yes	—
95	C₃×#121	C₂×C₄	1	8	yes	—
96	C₃×#121	C₂×C₄	1	8	yes	—
97	C₃×C₄×D₄	C₂×C₄	1	8	yes	—
98	C₃×C₄×D₄	C₂×C₄	1	8	yes	—
99	C₃×C₄×D₄	C₂³	1	8	yes	—
100	C₃×C₄×D₄	C₂³	1	8	yes	—
101	C₂×C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
102	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
103	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
104	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
105	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
106	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
107	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
108	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
109	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
110	C₂×C₃×C₄²	C₂³	1	2	yes	—
111	C₂×C₃×C₄²	C₂³	1	2	yes	—
112	C₃×C₄⋊C₈	C₂³	1	8	yes	—
113	C₃×C₄⋊C₈	C₂³	1	8	yes	—
114	C₃×C₂²⋊C₈	C₂³	1	32	yes	—
115	C₃×#97	C₂³	1	8	yes	—
116	C₃×C₄×C₈	C₂³	1	8	yes	—
117	C₂²×#41	C₂²×C₃	1	2	yes	—
118	#568	—	2	16	no	—
119	C₂×#131	C₂²×C₃	1	8	yes	complemented
120	C₂×#131	C₂²×C₃	1	8	yes	complemented
121	C₂²×M₁₆	C₂²×C₃	1	1	yes	characteristic
122	C₂²×M₁₆	C₂²×C₃	1	2	yes	—
123	C₂²×M₁₆	C₂²×C₃	1	8	yes	—
124	C₂²×M₁₆	C₂²×C₃	1	1	yes	characteristic
125	#532	C₃×C₄	1	2	yes	—
126	#532	C₃×C₄	1	4	yes	—
127	#532	C₃×C₄	1	2	yes	—
128	#532	C₂²×C₃	1	2	yes	—
129	#532	C₂²×C₃	1	2	yes	—
130	#532	C₂²×C₃	1	4	yes	—
131	C₂×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
132	C₂×#127	C₃×C₄	1	1	yes	characteristic
133	C₂×#124	C₃×C₄	1	2	yes	—
134	C₂×#121	C₃×C₄	1	4	yes	—
135	C₂×C₄×D₄	C₂²×C₃	1	4	yes	—
136	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
137	#436	C₂²×C₃	1	16	yes	complemented
138	#435	C₂²×C₃	1	32	yes	complemented
139	C₈×D₄	C₂²×C₃	1	16	yes	complemented
140	#423	C₂²×C₃	1	4	yes	—
141	#423	C₂²×C₃	1	2	yes	complemented
142	#423	C₂²×C₃	1	2	yes	complemented
143	C₂×C₄⋊C₈	C₂²×C₃	1	4	yes	complemented
144	#408	C₂²×C₃	1	8	yes	complemented
145	#408	C₂²×C₃	1	8	yes	complemented
146	#407	C₂²×C₃	1	8	yes	—
147	C₄×M₁₆	C₂²×C₃	1	4	yes	—
148	C₄×M₁₆	C₂²×C₃	1	4	yes	complemented
149	C₂×#97	C₂²×C₃	1	2	yes	complemented
150	C₂×C₄×C₈	C₂²×C₃	1	2	yes	complemented
151	C₂⁴×C₃	C₂²×C₄	1	4	yes	—
152	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
153	C₃×#41	—	2	16	no	—
154	C₃×#41	—	2	16	no	—
155	C₃×#41	—	2	8	no	—
156	C₃×#41	—	2	8	no	—
157	C₃×#41	—	2	8	no	—
158	C₃×#41	—	2	8	no	—
159	C₂×C₃×Q₈	—	2	8	no	—
160	C₂×C₃×Q₈	C₂²×C₄	1	4	yes	—
161	C₂×C₃×Q₈	C₂²×C₄	1	4	yes	—
162	C₂×C₃×D₄	—	2	16	no	—
163	C₂×C₃×D₄	—	2	8	no	—
164	C₂×C₃×D₄	—	4	8	no	—
165	C₂×C₃×D₄	C₂²×C₄	1	8	yes	—
166	C₂×C₃×D₄	C₂²×C₄	1	8	yes	—
167	C₂×C₃×D₄	C₂²×C₄	1	8	yes	—
168	C₂×C₃×D₄	C₂²×C₄	1	4	yes	—
169	C₂×C₃×D₄	C₂²×C₄	1	4	yes	—
170	C₂×C₃×D₄	C₂²×C₄	1	8	yes	—
171	C₂²×C₃×C₄	—	2	8	no	—
172	C₂²×C₃×C₄	—	2	8	no	—
173	C₂²×C₃×C₄	—	4	4	no	—
174	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
175	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
176	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
177	C₂²×C₃×C₄	C₂²×C₄	1	4	yes	—
178	C₂²×C₃×C₄	C₂²×C₄	1	8	yes	—
179	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
180	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
181	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
182	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
183	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
184	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
185	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
186	C₂²×C₃×C₄	#41	1	4	yes	—
187	C₂²×C₃×C₄	C₂⁴	1	8	yes	—
188	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
189	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
190	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
191	C₃×M₁₆	—	2	8	no	—
192	C₃×M₁₆	—	2	8	no	—
193	C₃×M₁₆	—	2	4	no	—
194	C₃×M₁₆	—	2	4	no	—
195	C₃×M₁₆	—	2	32	no	—
196	C₃×M₁₆	C₂×D₄	1	8	yes	—
197	C₃×M₁₆	C₂×D₄	1	8	yes	complemented
198	C₂×C₃×C₈	—	2	8	no	—
199	C₂×C₃×C₈	—	2	4	no	—
200	C₂×C₃×C₈	—	2	32	no	—
201	C₂×C₃×C₈	C₂×D₄	1	8	yes	complemented
202	C₂×C₃×C₈	C₂⁴	1	4	yes	—
203	C₂×C₃×C₈	C₂⁴	1	8	yes	—
204	C₂×C₃×C₈	C₂⁴	1	4	yes	—
205	C₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
206	C₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
207	C₃×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
208	C₃×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
209	C₃×C₄²	C₂²×C₄	1	2	yes	—
210	C₃×C₄²	C₂²×C₄	1	2	yes	—
211	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
212	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
213	C₃×C₄²	C₂²×C₄	1	2	yes	—
214	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
215	C₃×C₄²	C₂²×C₄	1	2	yes	—
216	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
217	C₃×C₄²	C₂⁴	1	2	yes	—
218	C₃×C₄²	C₂⁴	1	2	yes	—
219	C₂×#41	—	2	8	no	—
220	C₂×#41	—	2	8	no	—
221	C₂×#41	C₂×C₃×C₄	1	4	yes	—
222	C₂×#41	C₂×C₃×C₄	1	4	yes	—
223	C₂×#41	C₂³×C₃	1	4	yes	—
224	C₂×#41	C₂³×C₃	1	4	yes	—
225	C₂²×Q₈	C₂×C₃×C₄	1	2	yes	—
226	C₂²×D₄	C₂×C₃×C₄	1	4	yes	—
227	C₂²×D₄	C₂×C₃×C₄	1	2	yes	—
228	C₂²×D₄	C₂×C₃×C₄	1	4	yes	—
229	C₂³×C₄	C₂×C₃×C₄	1	2	yes	—
230	C₂³×C₄	C₂³×C₃	1	4	yes	—
231	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
232	#131	—	2	32	no	—
233	#131	—	2	32	no	—
234	C₂×M₁₆	—	2	8	no	—
235	C₂×M₁₆	—	2	4	no	—
236	C₂×M₁₆	—	2	32	no	—
237	C₂×M₁₆	C₃×D₄	1	8	yes	complemented
238	C₂×M₁₆	C₂³×C₃	1	4	yes	—
239	C₂×M₁₆	C₂³×C₃	1	4	yes	—
240	C₂×M₁₆	C₂³×C₃	1	16	yes	—
241	C₂×M₁₆	C₂³×C₃	1	2	yes	—
242	C₂×M₁₆	C₂³×C₃	1	2	yes	—
243	C₂×M₁₆	C₂³×C₃	1	2	yes	—
244	C₂×M₁₆	C₂³×C₃	1	2	yes	—
245	C₂²×C₈	C₂³×C₃	1	4	yes	—
246	C₂²×C₈	C₂³×C₃	1	16	yes	—
247	C₂²×C₈	C₂³×C₃	1	2	yes	—
248	C₂²×C₈	C₂³×C₃	1	2	yes	—
249	C₄⋊Q₈	C₂×C₃×C₄	1	2	yes	—
250	C₄⋊Q₈	C₂×C₃×C₄	1	2	yes	—
251	C₄⋊Q₈	C₂×C₃×C₄	1	4	yes	—
252	#127	C₂×C₃×C₄	1	2	yes	—
253	#127	C₂×C₃×C₄	1	4	yes	—
254	#127	C₂×C₃×C₄	1	2	yes	—
255	#124	C₂×C₃×C₄	1	4	yes	—
256	#124	C₂×C₃×C₄	1	4	yes	—
257	#124	C₂×C₃×C₄	1	8	yes	—
258	#121	C₂×C₃×C₄	1	8	yes	—
259	#121	C₂×C₃×C₄	1	8	yes	—
260	#121	C₂×C₃×C₄	1	8	yes	—
261	#121	C₂×C₃×C₄	1	8	yes	—
262	C₄×D₄	C₂×C₃×C₄	1	8	yes	—
263	C₄×D₄	C₂×C₃×C₄	1	8	yes	—
264	C₄×D₄	C₂³×C₃	1	8	yes	—
265	C₄×D₄	C₂³×C₃	1	8	yes	—
266	C₂×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
267	C₂×C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
268	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
269	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
270	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
271	C₂×C₄²	C₂×C₃×C₄	1	2	yes	—
272	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
273	C₂×C₄²	C₂³×C₃	1	2	yes	—
274	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
275	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
276	C₂×C₄²	C₂³×C₃	1	2	yes	—
277	C₄⋊C₈	C₂³×C₃	1	8	yes	—
278	C₄⋊C₈	C₂³×C₃	1	8	yes	—
279	C₂²⋊C₈	C₂³×C₃	1	32	yes	—
280	#97	C₂³×C₃	1	8	yes	—
281	C₄×C₈	C₂³×C₃	1	8	yes	—
282	C₂³×C₃	—	2	8	no	—
283	C₂³×C₃	—	4	2	no	—
284	C₂³×C₃	—	4	8	no	—
285	C₂³×C₃	C₄×D₄	1	4	yes	—
286	C₂³×C₃	C₂³×C₄	1	8	yes	—
287	C₂³×C₃	C₂³×C₄	1	1	yes	characteristic
288	C₂³×C₃	C₂³×C₄	1	1	yes	characteristic
289	C₂³×C₃	C₂³×C₄	1	1	yes	characteristic
290	C₃×Q₈	—	2	8	no	—
291	C₃×Q₈	—	2	8	no	—
292	C₃×D₄	—	2	16	no	—
293	C₃×D₄	—	2	16	no	—
294	C₃×D₄	—	2	8	no	—
295	C₃×D₄	—	2	8	no	—
296	C₃×D₄	—	2	16	no	—
297	C₃×D₄	—	4	8	no	—
298	C₂×C₃×C₄	—	2	8	no	—
299	C₂×C₃×C₄	—	2	16	no	—
300	C₂×C₃×C₄	—	2	4	no	—
301	C₂×C₃×C₄	—	2	4	no	—
302	C₂×C₃×C₄	—	2	4	no	—
303	C₂×C₃×C₄	—	2	8	no	—
304	C₂×C₃×C₄	—	2	8	no	—
305	C₂×C₃×C₄	—	4	2	no	—
306	C₂×C₃×C₄	—	4	2	no	—
307	C₂×C₃×C₄	—	4	2	no	—
308	C₂×C₃×C₄	C₄×D₄	1	4	yes	—
309	C₂×C₃×C₄	C₄×D₄	1	4	yes	—
310	C₂×C₃×C₄	C₄×D₄	1	4	yes	—
311	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
312	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
313	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
314	C₂×C₃×C₄	C₂³×C₄	1	8	yes	—
315	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
316	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
317	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
318	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
319	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
320	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
321	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
322	C₂×C₃×C₄	C₂×#41	1	2	yes	—
323	C₂×C₃×C₄	C₂×#41	1	2	yes	—
324	C₂×C₃×C₄	C₂×#41	1	2	yes	—
325	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
326	C₃×C₈	—	2	8	no	—
327	C₃×C₈	—	2	4	no	—
328	C₃×C₈	C₂²×D₄	1	8	yes	—
329	C₂⁴	C₂²×C₃×C₄	1	4	yes	—
330	C₂⁴	C₂²×C₃×C₄	1	1	yes	characteristic
331	#41	—	2	16	no	—
332	#41	—	2	16	no	—
333	#41	—	2	8	no	—
334	#41	—	2	8	no	—
335	#41	—	2	8	no	—
336	#41	—	2	8	no	—
337	C₂×Q₈	—	2	8	no	—
338	C₂×Q₈	C₂²×C₃×C₄	1	4	yes	—
339	C₂×Q₈	C₂²×C₃×C₄	1	4	yes	—
340	C₂×D₄	—	2	16	no	—
341	C₂×D₄	—	2	8	no	—
342	C₂×D₄	—	4	8	no	—
343	C₂×D₄	C₂²×C₃×C₄	1	8	yes	—
344	C₂×D₄	C₂²×C₃×C₄	1	8	yes	—
345	C₂×D₄	C₂²×C₃×C₄	1	8	yes	—
346	C₂×D₄	C₂²×C₃×C₄	1	4	yes	—
347	C₂×D₄	C₂²×C₃×C₄	1	4	yes	—
348	C₂×D₄	C₂²×C₃×C₄	1	8	yes	—
349	C₂²×C₄	—	2	8	no	—
350	C₂²×C₄	—	2	8	no	—
351	C₂²×C₄	—	4	4	no	—
352	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
353	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
354	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
355	C₂²×C₄	C₂²×C₃×C₄	1	4	yes	—
356	C₂²×C₄	C₂²×C₃×C₄	1	8	yes	—
357	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
358	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
359	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
360	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
361	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
362	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
363	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
364	C₂²×C₄	C₃×#41	1	4	yes	—
365	C₂²×C₄	C₂⁴×C₃	1	8	yes	—
366	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
367	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
368	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
369	M₁₆	—	2	8	no	—
370	M₁₆	—	2	8	no	—
371	M₁₆	—	2	4	no	—
372	M₁₆	—	2	4	no	—
373	M₁₆	—	2	32	no	—
374	M₁₆	C₂×C₃×D₄	1	8	yes	—
375	M₁₆	C₂×C₃×D₄	1	8	yes	complemented
376	C₂×C₈	—	2	8	no	—
377	C₂×C₈	—	2	4	no	—
378	C₂×C₈	—	2	32	no	—
379	C₂×C₈	C₂×C₃×D₄	1	8	yes	complemented
380	C₂×C₈	C₂⁴×C₃	1	4	yes	—
381	C₂×C₈	C₂⁴×C₃	1	8	yes	—
382	C₂×C₈	C₂⁴×C₃	1	4	yes	—
383	C₄⋊C₄	C₂²×C₃×C₄	1	8	yes	—
384	C₄⋊C₄	C₂²×C₃×C₄	1	8	yes	—
385	C₂²⋊C₄	C₂²×C₃×C₄	1	16	yes	—
386	C₂²⋊C₄	C₂²×C₃×C₄	1	16	yes	—
387	C₄²	C₂²×C₃×C₄	1	2	yes	—
388	C₄²	C₂²×C₃×C₄	1	2	yes	—
389	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
390	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
391	C₄²	C₂²×C₃×C₄	1	2	yes	—
392	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
393	C₄²	C₂²×C₃×C₄	1	2	yes	—
394	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
395	C₄²	C₂⁴×C₃	1	2	yes	—
396	C₄²	C₂⁴×C₃	1	2	yes	—
397	C₂²×C₃	—	2	2	no	—
398	C₂²×C₃	—	2	2	no	—
399	C₂²×C₃	—	2	2	no	—
400	C₂²×C₃	—	2	8	no	—
401	C₂²×C₃	—	2	16	no	—
402	C₂²×C₃	—	4	4	no	—
403	C₂²×C₃	—	8	8	no	—
404	C₂²×C₃	C₂×C₄×D₄	1	2	yes	—
405	C₂²×C₃	C₂×C₄×D₄	1	2	yes	—
406	C₂²×C₃	C₂×C₄×D₄	1	2	yes	—
407	C₂²×C₃	C₂⁴×C₄	1	1	yes	characteristic
408	C₃×C₄	—	2	8	no	—
409	C₃×C₄	—	2	2	no	—
410	C₃×C₄	—	2	2	no	—
411	C₃×C₄	—	2	2	no	—
412	C₃×C₄	C₂×C₄×D₄	1	2	yes	—
413	C₃×C₄	C₂×C₄×D₄	1	2	yes	—
414	C₃×C₄	C₂×C₄×D₄	1	2	yes	—
415	C₃×C₄	#568	1	4	yes	complemented
416	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
417	C₃×C₄	C₂²×#41	1	1	yes	characteristic
418	C₂³	—	2	8	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	8	no	—
421	C₂³	C₃×C₄×D₄	1	4	yes	—
422	C₂³	C₂³×C₃×C₄	1	8	yes	—
423	C₂³	C₂³×C₃×C₄	1	1	yes	characteristic
424	C₂³	C₂³×C₃×C₄	1	1	yes	characteristic
425	C₂³	C₂³×C₃×C₄	1	1	yes	characteristic
426	Q₈	—	2	8	no	—
427	Q₈	—	2	8	no	—
428	D₄	—	2	16	no	—
429	D₄	—	2	16	no	—
430	D₄	—	2	16	no	—
431	D₄	—	2	8	no	—
432	D₄	—	2	8	no	—
433	D₄	—	4	8	no	—
434	C₂×C₄	—	2	8	no	—
435	C₂×C₄	—	2	4	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	2	8	no	—
439	C₂×C₄	—	2	8	no	—
440	C₂×C₄	—	2	16	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	C₃×C₄×D₄	1	4	yes	—
445	C₂×C₄	C₃×C₄×D₄	1	4	yes	—
446	C₂×C₄	C₃×C₄×D₄	1	4	yes	—
447	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
448	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
449	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
450	C₂×C₄	C₂³×C₃×C₄	1	8	yes	—
451	C₂×C₄	C₂³×C₃×C₄	1	2	yes	—
452	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
453	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
454	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
455	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
456	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
457	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
458	C₂×C₄	C₂×C₃×#41	1	2	yes	—
459	C₂×C₄	C₂×C₃×#41	1	2	yes	—
460	C₂×C₄	C₂×C₃×#41	1	2	yes	—
461	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
462	C₈	—	2	8	no	—
463	C₈	—	2	4	no	—
464	C₈	C₂²×C₃×D₄	1	8	yes	—
465	C₂×C₃	—	2	2	no	—
466	C₂×C₃	—	2	2	no	—
467	C₂×C₃	—	2	2	no	—
468	C₂×C₃	—	4	8	no	—
469	C₂×C₃	C₂²×C₄×D₄	1	1	yes	characteristic
470	C₂×C₃	C₂×#568	1	2	yes	—
471	C₂²	—	2	2	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	8	no	—
475	C₂²	—	2	16	no	—
476	C₂²	—	4	4	no	—
477	C₂²	—	8	8	no	—
478	C₂²	C₂×C₃×C₄×D₄	1	2	yes	—
479	C₂²	C₂×C₃×C₄×D₄	1	2	yes	—
480	C₂²	C₂×C₃×C₄×D₄	1	2	yes	—
481	C₂²	C₂⁴×C₃×C₄	1	1	yes	characteristic
482	C₄	—	2	8	no	—
483	C₄	—	2	2	no	—
484	C₄	—	2	2	no	—
485	C₄	—	2	2	no	—
486	C₄	C₂×C₃×C₄×D₄	1	2	yes	—
487	C₄	C₂×C₃×C₄×D₄	1	2	yes	—
488	C₄	C₂×C₃×C₄×D₄	1	2	yes	—
489	C₄	C₃×#568	1	4	yes	complemented
490	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
491	C₄	C₂²×C₃×#41	1	1	yes	characteristic
492	C₃	#60144	1	1	yes	characteristic, direct product factor, Sylow
493	C₂	—	2	2	no	—
494	C₂	—	2	2	no	—
495	C₂	—	2	2	no	—
496	C₂	—	4	8	no	—
497	C₂	C₂²×C₃×C₄×D₄	1	1	yes	characteristic
498	C₂	C₂×C₃×#568	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂⁵
commutator	C₂²	C₂⁴×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁵×C₃
socle	C₂²×C₃	C₂⁴×C₄
prime radical	C₂×C₃×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#568	C₂	semidirect
3. C₃×#2935	C₂	semidirect
4. C₃×D₄×M₁₆	C₂	semidirect
5. C₃×#2932	C₂	semidirect
6. C₃×#2931	C₂	semidirect
7. C₂×C₃×#423	C₂	semidirect
8. C₃×#2879	C₂	semidirect
9. C₂×C₃×C₄×M₁₆	C₂	semidirect
10. #60144	C₃	direct (N × H)
13. C₂×C₃×#131	C₂²	semidirect
14. C₂×C₃×#131	C₂²	semidirect
31. C₃×#436	C₂²	semidirect
32. C₃×#435	C₂²	semidirect
33. C₃×C₈×D₄	C₂²	semidirect
35. C₃×#423	C₂²	semidirect
36. C₃×#423	C₂²	semidirect
37. C₂×C₃×C₄⋊C₈	C₂²	semidirect
38. C₃×#408	C₂²	semidirect
39. C₃×#408	C₂²	semidirect
42. C₃×C₄×M₁₆	C₂²	semidirect
43. C₂×C₃×#97	C₂²	semidirect
44. C₂×C₃×C₄×C₈	C₂²	semidirect
45. C₂×#568	C₂×C₃	semidirect
47. #2935	C₂×C₃	semidirect
48. D₄×M₁₆	C₂×C₃	semidirect
49. #2932	C₂×C₃	semidirect
50. #2931	C₂×C₃	semidirect
51. C₂×#423	C₂×C₃	semidirect
52. #2879	C₂×C₃	semidirect
53. C₂×C₄×M₁₆	C₂×C₃	semidirect
72. C₂×C₃×M₁₆	D₄	semidirect
119. C₂×#131	C₂²×C₃	semidirect
120. C₂×#131	C₂²×C₃	semidirect
137. #436	C₂²×C₃	semidirect
138. #435	C₂²×C₃	semidirect
139. C₈×D₄	C₂²×C₃	semidirect
141. #423	C₂²×C₃	semidirect
142. #423	C₂²×C₃	semidirect
143. C₂×C₄⋊C₈	C₂²×C₃	semidirect
144. #408	C₂²×C₃	semidirect
145. #408	C₂²×C₃	semidirect
148. C₄×M₁₆	C₂²×C₃	semidirect
149. C₂×#97	C₂²×C₃	semidirect
150. C₂×C₄×C₈	C₂²×C₃	semidirect
197. C₃×M₁₆	C₂×D₄	semidirect
201. C₂×C₃×C₈	C₂×D₄	semidirect
237. C₂×M₁₆	C₃×D₄	semidirect
375. M₁₆	C₂×C₃×D₄	semidirect
379. C₂×C₈	C₂×C₃×D₄	semidirect
415. C₃×C₄	#568	semidirect
489. C₄	C₃×#568	semidirect
492. C₃	#60144	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#568	2
C₂×C₃×#532	2
C₃×#2935	2
C₃×D₄×M₁₆	2
C₃×#2932	2
C₃×#2931	2
C₂×C₃×#423	2
C₃×#2879	2
C₂×C₃×C₄×M₁₆	2
#60144	3

Subgroups — C3×#60144 #10678751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#60144 #10678751