Subgroups — C₂×C₃×#3223 #10679402

Order 768.

2022 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#536	C₂	1	1	yes	maximal, characteristic
3	C₃×#3223	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#491	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#488	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃×#464	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×#462	C₂	1	2	yes	maximal, complemented
10	C₂×C₃×#452	C₂	1	2	yes	maximal
11	C₂×C₃×#451	C₂	1	2	yes	maximal, complemented
12	C₂×C₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#3223	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂²×C₃×#41	C₂²	1	1	yes	characteristic
15	C₂³×C₃×Q₈	C₂²	1	1	yes	characteristic
16	C₂²×C₃×Q₁₆	C₂²	1	2	yes	—
17	C₂²×C₃×Q₁₆	C₂²	1	2	yes	—
18	C₂²×C₃×Q₁₆	C₂²	1	2	yes	—
19	C₂²×C₃×SD₁₆	C₂²	1	2	yes	—
20	C₃×#541	—	2	4	no	—
21	C₃×#541	C₂²	1	8	yes	complemented
22	C₃×#536	—	2	4	no	—
23	C₃×#536	C₂²	1	8	yes	—
24	C₂×C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂×C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
26	C₂×C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
27	C₂×C₃×#126	—	2	2	no	—
28	C₂×C₃×#125	—	2	2	no	—
29	C₂×C₃×#125	C₂²	1	1	yes	characteristic
30	C₂×C₃×#124	C₂²	1	1	yes	characteristic
31	C₂×C₃×#124	C₂²	1	1	yes	characteristic
32	C₂×C₃×#123	—	2	2	no	—
33	C₂×C₃×C₂²⋊Q₈	—	2	2	no	—
34	C₂×C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
35	C₂×C₃×C₄×Q₈	C₂²	1	2	yes	—
36	C₂×C₃×#117	C₂²	1	1	yes	characteristic
37	C₃×#497	C₂²	1	8	yes	complemented
38	C₃×#494	C₂²	1	8	yes	complemented
39	C₃×#491	C₂²	1	8	yes	complemented
40	C₃×#488	C₂²	1	8	yes	complemented
41	C₃×#464	C₂²	1	16	yes	complemented
42	C₃×#462	C₂²	1	16	yes	complemented
43	C₃×#452	C₂²	1	16	yes	—
44	C₃×#451	C₂²	1	16	yes	complemented
45	C₃×#433	C₂²	1	8	yes	complemented
46	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
47	C₂×C₃×#103	C₂²	1	2	yes	—
48	C₂×C₃×#103	C₂²	1	2	yes	—
49	C₂×C₃×#103	C₂²	1	2	yes	—
50	C₂×C₃×#102	C₂²	1	2	yes	—
51	C₂×C₃×C₂²⋊C₈	C₂²	1	2	yes	—
52	C₂×C₃×#97	C₂²	1	1	yes	characteristic
53	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
54	C₂×#541	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂×#536	C₂×C₃	1	1	yes	characteristic
56	#3223	C₂×C₃	1	16	yes	direct product factor
57	C₂×#497	C₂×C₃	1	1	yes	characteristic, complemented
58	C₂×#494	C₂×C₃	1	1	yes	characteristic, complemented
59	C₂×#491	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂×#488	C₂×C₃	1	1	yes	characteristic, complemented
61	C₂×#464	C₂×C₃	1	2	yes	complemented
62	C₂×#462	C₂×C₃	1	2	yes	complemented
63	C₂×#452	C₂×C₃	1	2	yes	—
64	C₂×#451	C₂×C₃	1	2	yes	complemented
65	C₂×#433	C₂×C₃	1	1	yes	characteristic, complemented
66	C₂×C₃×#41	—	2	2	no	—
67	C₂×C₃×#41	—	4	4	no	—
68	C₂×C₃×#41	C₂³	1	4	yes	—
69	C₂²×C₃×Q₈	—	2	1	no	—
70	C₂²×C₃×Q₈	—	2	2	no	—
71	C₂²×C₃×Q₈	—	4	4	no	—
72	C₂²×C₃×Q₈	C₂³	1	2	yes	—
73	C₂²×C₃×Q₈	C₂³	1	1	yes	characteristic
74	C₂²×C₃×Q₈	C₂³	1	1	yes	characteristic
75	C₂²×C₃×Q₈	C₂³	1	1	yes	characteristic
76	C₂²×C₃×Q₈	C₂³	1	4	yes	—
77	C₂²×C₃×D₄	—	2	1	no	—
78	C₂²×C₃×D₄	C₂³	1	1	yes	characteristic
79	C₂³×C₃×C₄	—	2	1	no	—
80	C₂³×C₃×C₄	—	2	1	no	—
81	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
82	C₂×C₃×Q₁₆	—	4	4	no	—
83	C₂×C₃×Q₁₆	—	4	4	no	—
84	C₂×C₃×Q₁₆	—	4	4	no	—
85	C₂×C₃×Q₁₆	C₂³	1	8	yes	—
86	C₂×C₃×Q₁₆	C₂³	1	8	yes	—
87	C₂×C₃×Q₁₆	C₂³	1	8	yes	—
88	C₂×C₃×SD₁₆	—	4	4	no	—
89	C₂×C₃×SD₁₆	C₂³	1	8	yes	—
90	C₂²×C₃×C₈	C₂³	1	2	yes	—
91	C₂²×C₃×C₈	C₂³	1	2	yes	—
92	C₃×C₄⋊Q₈	—	2	2	no	—
93	C₃×C₄⋊Q₈	—	2	2	no	—
94	C₃×C₄⋊Q₈	—	2	2	no	—
95	C₃×C₄⋊Q₈	C₂³	1	4	yes	—
96	C₃×C₄⋊Q₈	C₂³	1	4	yes	—
97	C₃×C₄⋊Q₈	C₂³	1	4	yes	—
98	C₃×#126	—	2	16	no	—
99	C₃×#125	—	2	16	no	—
100	C₃×#125	—	2	2	no	—
101	C₃×#125	C₂³	1	4	yes	—
102	C₃×#124	—	2	2	no	—
103	C₃×#124	—	2	2	no	—
104	C₃×#124	C₂³	1	4	yes	—
105	C₃×#124	C₂³	1	4	yes	—
106	C₃×#123	—	2	16	no	—
107	C₃×C₂²⋊Q₈	—	2	4	no	—
108	C₃×C₂²⋊Q₈	—	2	16	no	—
109	C₃×C₂²⋊Q₈	C₂³	1	8	yes	—
110	C₃×C₄×Q₈	—	2	4	no	—
111	C₃×C₄×Q₈	C₂³	1	8	yes	—
112	C₃×#117	D₄	1	4	yes	—
113	C₃×#117	C₂³	1	4	yes	—
114	C₂×C₃×C₄⋊C₄	—	2	2	no	—
115	C₂×C₃×C₄⋊C₄	—	2	2	no	—
116	C₂×C₃×C₄⋊C₄	—	2	2	no	—
117	C₂×C₃×C₄⋊C₄	—	2	2	no	—
118	C₂×C₃×C₄⋊C₄	—	2	2	no	—
119	C₂×C₃×C₄⋊C₄	—	2	2	no	—
120	C₂×C₃×C₄⋊C₄	D₄	1	2	yes	—
121	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
122	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
123	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
124	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
125	C₂×C₃×C₂²⋊C₄	D₄	1	2	yes	—
126	C₂×C₃×C₄²	—	2	2	no	—
127	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
128	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
129	C₃×C₄⋊C₈	C₂³	1	8	yes	—
130	C₃×#103	C₂³	1	8	yes	—
131	C₃×#103	C₂³	1	8	yes	—
132	C₃×#103	C₂³	1	8	yes	—
133	C₃×#102	C₂³	1	8	yes	—
134	C₃×C₂²⋊C₈	C₂³	1	8	yes	—
135	C₃×#97	C₂³	1	4	yes	—
136	C₃×C₄×C₈	C₂³	1	4	yes	—
137	C₂²×#41	C₂²×C₃	1	1	yes	characteristic
138	C₂³×Q₈	C₂²×C₃	1	1	yes	characteristic
139	C₂²×Q₁₆	C₂²×C₃	1	2	yes	—
140	C₂²×Q₁₆	C₂²×C₃	1	2	yes	—
141	C₂²×Q₁₆	C₂²×C₃	1	2	yes	—
142	C₂²×SD₁₆	C₂²×C₃	1	2	yes	—
143	#541	—	2	4	no	—
144	#541	C₂²×C₃	1	8	yes	complemented
145	#536	—	2	4	no	—
146	#536	C₂²×C₃	1	8	yes	—
147	C₂×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
148	C₂×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
149	C₂×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
150	C₂×#126	—	2	2	no	—
151	C₂×#125	—	2	2	no	—
152	C₂×#125	C₂²×C₃	1	1	yes	characteristic
153	C₂×#124	C₂²×C₃	1	1	yes	characteristic
154	C₂×#124	C₂²×C₃	1	1	yes	characteristic
155	C₂×#123	—	2	2	no	—
156	C₂×C₂²⋊Q₈	—	2	2	no	—
157	C₂×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
158	C₂×C₄×Q₈	C₂²×C₃	1	2	yes	—
159	C₂×#117	C₂²×C₃	1	1	yes	characteristic
160	#497	C₂²×C₃	1	8	yes	complemented
161	#494	C₂²×C₃	1	8	yes	complemented
162	#491	C₂²×C₃	1	8	yes	complemented
163	#488	C₂²×C₃	1	8	yes	complemented
164	#464	C₂²×C₃	1	16	yes	complemented
165	#462	C₂²×C₃	1	16	yes	complemented
166	#452	C₂²×C₃	1	16	yes	—
167	#451	C₂²×C₃	1	16	yes	complemented
168	#433	C₂²×C₃	1	8	yes	complemented
169	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
170	C₂×#103	C₂²×C₃	1	2	yes	—
171	C₂×#103	C₂²×C₃	1	2	yes	—
172	C₂×#103	C₂²×C₃	1	2	yes	—
173	C₂×#102	C₂²×C₃	1	2	yes	—
174	C₂×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
175	C₂×#97	C₂²×C₃	1	1	yes	characteristic
176	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
177	C₂⁴×C₃	—	2	1	no	—
178	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
179	C₃×#41	—	4	8	no	—
180	C₃×#41	—	4	8	no	—
181	C₂×C₃×Q₈	—	2	2	no	—
182	C₂×C₃×Q₈	—	2	4	no	—
183	C₂×C₃×Q₈	—	2	1	no	—
184	C₂×C₃×Q₈	—	2	1	no	—
185	C₂×C₃×Q₈	—	2	1	no	—
186	C₂×C₃×Q₈	—	2	4	no	—
187	C₂×C₃×Q₈	—	2	4	no	—
188	C₂×C₃×Q₈	—	2	4	no	—
189	C₂×C₃×Q₈	—	2	4	no	—
190	C₂×C₃×Q₈	—	2	4	no	—
191	C₂×C₃×Q₈	—	4	8	no	—
192	C₂×C₃×Q₈	—	4	4	no	—
193	C₂×C₃×Q₈	—	4	8	no	—
194	C₂×C₃×Q₈	C₂⁴	1	2	yes	—
195	C₂×C₃×Q₈	C₂⁴	1	2	yes	—
196	C₂×C₃×Q₈	C₂⁴	1	2	yes	—
197	C₂×C₃×Q₈	C₂⁴	1	4	yes	—
198	C₂×C₃×D₄	—	2	4	no	—
199	C₂×C₃×D₄	—	2	1	no	—
200	C₂×C₃×D₄	—	2	4	no	—
201	C₂×C₃×D₄	—	4	4	no	—
202	C₂×C₃×D₄	C₂⁴	1	2	yes	—
203	C₂²×C₃×C₄	—	2	4	no	—
204	C₂²×C₃×C₄	—	2	4	no	—
205	C₂²×C₃×C₄	—	2	2	no	—
206	C₂²×C₃×C₄	—	2	2	no	—
207	C₂²×C₃×C₄	—	2	1	no	—
208	C₂²×C₃×C₄	—	2	1	no	—
209	C₂²×C₃×C₄	—	2	1	no	—
210	C₂²×C₃×C₄	—	2	4	no	—
211	C₂²×C₃×C₄	—	4	4	no	—
212	C₂²×C₃×C₄	—	4	4	no	—
213	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
214	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
215	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
216	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
217	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
218	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
219	C₃×Q₁₆	—	4	8	no	—
220	C₃×Q₁₆	—	4	8	no	—
221	C₃×Q₁₆	—	4	8	no	—
222	C₃×SD₁₆	—	4	8	no	—
223	C₂×C₃×C₈	—	2	4	no	—
224	C₂×C₃×C₈	—	2	4	no	—
225	C₂×C₃×C₈	C₂⁴	1	4	yes	—
226	C₂×C₃×C₈	C₂⁴	1	4	yes	—
227	C₃×C₄⋊C₄	—	2	8	no	—
228	C₃×C₄⋊C₄	—	2	8	no	—
229	C₃×C₄⋊C₄	—	2	8	no	—
230	C₃×C₄⋊C₄	—	2	8	no	—
231	C₃×C₄⋊C₄	—	2	8	no	—
232	C₃×C₄⋊C₄	—	2	8	no	—
233	C₃×C₄⋊C₄	—	2	2	no	—
234	C₃×C₄⋊C₄	C₂×D₄	1	8	yes	—
235	C₃×C₄⋊C₄	C₂⁴	1	4	yes	—
236	C₃×C₂²⋊C₄	—	2	8	no	—
237	C₃×C₂²⋊C₄	—	2	8	no	—
238	C₃×C₂²⋊C₄	—	2	8	no	—
239	C₃×C₂²⋊C₄	C₂×D₄	1	8	yes	—
240	C₃×C₄²	—	2	8	no	—
241	C₃×C₄²	C₂×D₄	1	2	yes	—
242	C₃×C₄²	C₂×D₄	1	2	yes	—
243	C₃×C₄²	C₂⁴	1	2	yes	—
244	C₃×C₄²	C₂⁴	1	2	yes	—
245	C₂×#41	—	2	2	no	—
246	C₂×#41	—	4	4	no	—
247	C₂×#41	C₂³×C₃	1	4	yes	—
248	C₂²×Q₈	—	2	1	no	—
249	C₂²×Q₈	—	2	2	no	—
250	C₂²×Q₈	—	4	4	no	—
251	C₂²×Q₈	C₂³×C₃	1	2	yes	—
252	C₂²×Q₈	C₂³×C₃	1	1	yes	characteristic
253	C₂²×Q₈	C₂³×C₃	1	1	yes	characteristic
254	C₂²×Q₈	C₂³×C₃	1	1	yes	characteristic
255	C₂²×Q₈	C₂³×C₃	1	4	yes	—
256	C₂²×D₄	—	2	1	no	—
257	C₂²×D₄	C₂³×C₃	1	1	yes	characteristic
258	C₂³×C₄	—	2	1	no	—
259	C₂³×C₄	—	2	1	no	—
260	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
261	C₂×Q₁₆	—	4	4	no	—
262	C₂×Q₁₆	—	4	4	no	—
263	C₂×Q₁₆	—	4	4	no	—
264	C₂×Q₁₆	C₂³×C₃	1	8	yes	—
265	C₂×Q₁₆	C₂³×C₃	1	8	yes	—
266	C₂×Q₁₆	C₂³×C₃	1	8	yes	—
267	C₂×SD₁₆	—	4	4	no	—
268	C₂×SD₁₆	C₂³×C₃	1	8	yes	—
269	C₂²×C₈	C₂³×C₃	1	2	yes	—
270	C₂²×C₈	C₂³×C₃	1	2	yes	—
271	C₄⋊Q₈	—	2	2	no	—
272	C₄⋊Q₈	—	2	2	no	—
273	C₄⋊Q₈	—	2	2	no	—
274	C₄⋊Q₈	C₂³×C₃	1	4	yes	—
275	C₄⋊Q₈	C₂³×C₃	1	4	yes	—
276	C₄⋊Q₈	C₂³×C₃	1	4	yes	—
277	#126	—	2	16	no	—
278	#125	—	2	16	no	—
279	#125	—	2	2	no	—
280	#125	C₂³×C₃	1	4	yes	—
281	#124	—	2	2	no	—
282	#124	—	2	2	no	—
283	#124	C₂³×C₃	1	4	yes	—
284	#124	C₂³×C₃	1	4	yes	—
285	#123	—	2	16	no	—
286	C₂²⋊Q₈	—	2	4	no	—
287	C₂²⋊Q₈	—	2	16	no	—
288	C₂²⋊Q₈	C₂³×C₃	1	8	yes	—
289	C₄×Q₈	—	2	4	no	—
290	C₄×Q₈	C₂³×C₃	1	8	yes	—
291	#117	C₃×D₄	1	4	yes	—
292	#117	C₂³×C₃	1	4	yes	—
293	C₂×C₄⋊C₄	—	2	2	no	—
294	C₂×C₄⋊C₄	—	2	2	no	—
295	C₂×C₄⋊C₄	—	2	2	no	—
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	—	2	2	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	C₃×D₄	1	2	yes	—
300	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
301	C₂×C₂²⋊C₄	—	2	2	no	—
302	C₂×C₂²⋊C₄	—	2	2	no	—
303	C₂×C₂²⋊C₄	—	2	2	no	—
304	C₂×C₂²⋊C₄	C₃×D₄	1	2	yes	—
305	C₂×C₄²	—	2	2	no	—
306	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
307	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
308	C₄⋊C₈	C₂³×C₃	1	8	yes	—
309	#103	C₂³×C₃	1	8	yes	—
310	#103	C₂³×C₃	1	8	yes	—
311	#103	C₂³×C₃	1	8	yes	—
312	#102	C₂³×C₃	1	8	yes	—
313	C₂²⋊C₈	C₂³×C₃	1	8	yes	—
314	#97	C₂³×C₃	1	4	yes	—
315	C₄×C₈	C₂³×C₃	1	4	yes	—
316	C₂³×C₃	—	2	2	no	—
317	C₂³×C₃	—	2	4	no	—
318	C₂³×C₃	—	2	2	no	—
319	C₂³×C₃	—	4	2	no	—
320	C₂³×C₃	—	4	4	no	—
321	C₂³×C₃	C₂²×D₄	1	2	yes	—
322	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
323	C₃×Q₈	—	2	4	no	—
324	C₃×Q₈	—	2	4	no	—
325	C₃×Q₈	—	2	4	no	—
326	C₃×Q₈	—	2	4	no	—
327	C₃×Q₈	—	2	4	no	—
328	C₃×Q₈	—	4	8	no	—
329	C₃×Q₈	—	4	4	no	—
330	C₃×Q₈	—	4	2	no	—
331	C₃×Q₈	—	4	2	no	—
332	C₃×Q₈	—	4	2	no	—
333	C₃×D₄	—	2	4	no	—
334	C₃×D₄	—	4	8	no	—
335	C₃×D₄	—	4	2	no	—
336	C₂×C₃×C₄	—	2	2	no	—
337	C₂×C₃×C₄	—	2	4	no	—
338	C₂×C₃×C₄	—	2	4	no	—
339	C₂×C₃×C₄	—	2	4	no	—
340	C₂×C₃×C₄	—	2	4	no	—
341	C₂×C₃×C₄	—	2	4	no	—
342	C₂×C₃×C₄	—	2	4	no	—
343	C₂×C₃×C₄	—	2	2	no	—
344	C₂×C₃×C₄	—	2	2	no	—
345	C₂×C₃×C₄	—	2	2	no	—
346	C₂×C₃×C₄	—	4	8	no	—
347	C₂×C₃×C₄	—	4	8	no	—
348	C₂×C₃×C₄	—	4	2	no	—
349	C₂×C₃×C₄	—	4	2	no	—
350	C₂×C₃×C₄	—	4	2	no	—
351	C₂×C₃×C₄	—	4	4	no	—
352	C₂×C₃×C₄	—	4	4	no	—
353	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
354	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
355	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
356	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
357	C₂×C₃×C₄	H(5,𝔽₂)	1	4	yes	—
358	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
359	C₃×C₈	—	2	4	no	—
360	C₃×C₈	—	2	4	no	—
361	C₂⁴	—	2	1	no	—
362	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
363	#41	—	4	8	no	—
364	#41	—	4	8	no	—
365	C₂×Q₈	—	2	2	no	—
366	C₂×Q₈	—	2	4	no	—
367	C₂×Q₈	—	2	1	no	—
368	C₂×Q₈	—	2	1	no	—
369	C₂×Q₈	—	2	1	no	—
370	C₂×Q₈	—	2	4	no	—
371	C₂×Q₈	—	2	4	no	—
372	C₂×Q₈	—	2	4	no	—
373	C₂×Q₈	—	2	4	no	—
374	C₂×Q₈	—	2	4	no	—
375	C₂×Q₈	—	4	8	no	—
376	C₂×Q₈	—	4	4	no	—
377	C₂×Q₈	—	4	8	no	—
378	C₂×Q₈	C₂⁴×C₃	1	2	yes	—
379	C₂×Q₈	C₂⁴×C₃	1	2	yes	—
380	C₂×Q₈	C₂⁴×C₃	1	2	yes	—
381	C₂×Q₈	C₂⁴×C₃	1	4	yes	—
382	C₂×D₄	—	2	4	no	—
383	C₂×D₄	—	2	1	no	—
384	C₂×D₄	—	2	4	no	—
385	C₂×D₄	—	4	4	no	—
386	C₂×D₄	C₂⁴×C₃	1	2	yes	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	4	4	no	—
396	C₂²×C₄	—	4	4	no	—
397	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
398	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
399	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
400	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
401	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
402	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
403	Q₁₆	—	4	8	no	—
404	Q₁₆	—	4	8	no	—
405	Q₁₆	—	4	8	no	—
406	SD₁₆	—	4	8	no	—
407	C₂×C₈	—	2	4	no	—
408	C₂×C₈	—	2	4	no	—
409	C₂×C₈	C₂⁴×C₃	1	4	yes	—
410	C₂×C₈	C₂⁴×C₃	1	4	yes	—
411	C₄⋊C₄	—	2	8	no	—
412	C₄⋊C₄	—	2	8	no	—
413	C₄⋊C₄	—	2	8	no	—
414	C₄⋊C₄	—	2	8	no	—
415	C₄⋊C₄	—	2	8	no	—
416	C₄⋊C₄	—	2	8	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	C₂×C₃×D₄	1	8	yes	—
419	C₄⋊C₄	C₂⁴×C₃	1	4	yes	—
420	C₂²⋊C₄	—	2	8	no	—
421	C₂²⋊C₄	—	2	8	no	—
422	C₂²⋊C₄	—	2	8	no	—
423	C₂²⋊C₄	C₂×C₃×D₄	1	8	yes	—
424	C₄²	—	2	8	no	—
425	C₄²	C₂×C₃×D₄	1	2	yes	—
426	C₄²	C₂×C₃×D₄	1	2	yes	—
427	C₄²	C₂⁴×C₃	1	2	yes	—
428	C₄²	C₂⁴×C₃	1	2	yes	—
429	C₂²×C₃	—	2	2	no	—
430	C₂²×C₃	—	2	4	no	—
431	C₂²×C₃	—	4	4	no	—
432	C₂²×C₃	—	4	2	no	—
433	C₂²×C₃	—	4	4	no	—
434	C₂²×C₃	—	8	4	no	—
435	C₂²×C₃	#535	1	2	yes	—
436	C₂²×C₃	#578	1	4	yes	—
437	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
438	C₃×C₄	—	2	2	no	—
439	C₃×C₄	—	2	4	no	—
440	C₃×C₄	—	2	4	no	—
441	C₃×C₄	—	4	2	no	—
442	C₃×C₄	—	4	2	no	—
443	C₃×C₄	—	4	2	no	—
444	C₃×C₄	—	4	4	no	—
445	C₃×C₄	—	4	4	no	—
446	C₃×C₄	#535	1	2	yes	—
447	C₃×C₄	C₂×H(5,𝔽₂)	1	2	yes	—
448	C₂³	—	2	4	no	—
449	C₂³	—	2	2	no	—
450	C₂³	—	2	2	no	—
451	C₂³	—	4	4	no	—
452	C₂³	—	4	2	no	—
453	C₂³	C₂²×C₃×D₄	1	2	yes	—
454	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
455	Q₈	—	2	4	no	—
456	Q₈	—	2	4	no	—
457	Q₈	—	2	4	no	—
458	Q₈	—	2	4	no	—
459	Q₈	—	2	4	no	—
460	Q₈	—	4	8	no	—
461	Q₈	—	4	2	no	—
462	Q₈	—	4	2	no	—
463	Q₈	—	4	2	no	—
464	Q₈	—	4	4	no	—
465	D₄	—	2	4	no	—
466	D₄	—	4	8	no	—
467	D₄	—	4	2	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	4	8	no	—
479	C₂×C₄	—	4	8	no	—
480	C₂×C₄	—	4	4	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	4	no	—
485	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
486	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
487	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
488	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
489	C₂×C₄	C₃×H(5,𝔽₂)	1	4	yes	—
490	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
491	C₈	—	2	4	no	—
492	C₈	—	2	4	no	—
493	C₂×C₃	—	4	2	no	—
494	C₂×C₃	—	8	2	no	—
495	C₂×C₃	#3223	1	4	yes	direct product factor
496	C₂×C₃	C₂×#535	1	1	yes	characteristic
497	C₂×C₃	C₂×#578	1	2	yes	—
498	C₂²	—	2	2	no	—
499	C₂²	—	2	4	no	—
500	C₂²	—	4	4	no	—
501	C₂²	—	4	2	no	—
502	C₂²	—	4	4	no	—
503	C₂²	—	8	4	no	—
504	C₂²	C₃×#535	1	2	yes	—
505	C₂²	C₃×#578	1	4	yes	—
506	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
507	C₄	—	2	2	no	—
508	C₄	—	2	4	no	—
509	C₄	—	2	4	no	—
510	C₄	—	4	2	no	—
511	C₄	—	4	2	no	—
512	C₄	—	4	2	no	—
513	C₄	—	4	4	no	—
514	C₄	—	4	4	no	—
515	C₄	C₃×#535	1	2	yes	—
516	C₄	C₂×C₃×H(5,𝔽₂)	1	2	yes	—
517	C₃	C₂×#3223	1	1	yes	characteristic, direct product factor, Sylow
518	C₂	—	4	2	no	—
519	C₂	—	8	2	no	—
520	C₂	C₃×#3223	1	4	yes	direct product factor
521	C₂	C₂×C₃×#535	1	1	yes	characteristic
522	C₂	C₂×C₃×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
prime radical	C₂²×C₃×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#541	C₂	semidirect
3. C₃×#3223	C₂	direct (N × H)
4. C₂×C₃×#497	C₂	semidirect
5. C₂×C₃×#494	C₂	semidirect
6. C₂×C₃×#491	C₂	semidirect
7. C₂×C₃×#488	C₂	semidirect
8. C₂×C₃×#464	C₂	semidirect
9. C₂×C₃×#462	C₂	semidirect
11. C₂×C₃×#451	C₂	semidirect
12. C₂×C₃×#433	C₂	semidirect
13. C₂×#3223	C₃	direct (N × H)
21. C₃×#541	C₂²	semidirect
37. C₃×#497	C₂²	semidirect
38. C₃×#494	C₂²	semidirect
39. C₃×#491	C₂²	semidirect
40. C₃×#488	C₂²	semidirect
41. C₃×#464	C₂²	semidirect
42. C₃×#462	C₂²	semidirect
44. C₃×#451	C₂²	semidirect
45. C₃×#433	C₂²	semidirect
54. C₂×#541	C₂×C₃	semidirect
56. #3223	C₂×C₃	direct (N × H)
57. C₂×#497	C₂×C₃	semidirect
58. C₂×#494	C₂×C₃	semidirect
59. C₂×#491	C₂×C₃	semidirect
60. C₂×#488	C₂×C₃	semidirect
61. C₂×#464	C₂×C₃	semidirect
62. C₂×#462	C₂×C₃	semidirect
64. C₂×#451	C₂×C₃	semidirect
65. C₂×#433	C₂×C₃	semidirect
144. #541	C₂²×C₃	semidirect
160. #497	C₂²×C₃	semidirect
161. #494	C₂²×C₃	semidirect
162. #491	C₂²×C₃	semidirect
163. #488	C₂²×C₃	semidirect
164. #464	C₂²×C₃	semidirect
165. #462	C₂²×C₃	semidirect
167. #451	C₂²×C₃	semidirect
168. #433	C₂²×C₃	semidirect
495. C₂×C₃	#3223	direct (N × H)
517. C₃	C₂×#3223	direct (N × H)
520. C₂	C₃×#3223	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×#541	2
C₂×C₃×#536	2
C₃×#3223	2
C₂×C₃×#497	2
C₂×C₃×#494	2
C₂×C₃×#491	2
C₂×C₃×#488	2
C₂×C₃×#464	2
C₂×C₃×#462	2
C₂×C₃×#452	2
C₂×C₃×#451	2
C₂×C₃×#433	2
C₂×#3223	3

Subgroups — C2×C3×#3223 #10679402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#3223 #10679402