Subgroups — C₂²×C₃×#461 #10679011

Order 768.

3222 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×SD₁₆	C₂	1	2	yes	maximal
2	C₂²×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#461	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃×#103	C₂	1	2	yes	maximal, complemented
7	C₂²×#461	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₃×Q₈	C₂²	1	2	yes	—
9	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic
10	C₂⁴×C₃×C₄	—	2	1	no	—
11	C₂²×C₃×SD₁₆	—	2	16	no	—
12	C₂²×C₃×SD₁₆	C₂²	1	24	yes	—
13	C₂³×C₃×C₈	C₂²	1	2	yes	—
14	C₂×C₃×C₄⋊Q₈	—	2	6	no	—
15	C₂×C₃×C₄⋊Q₈	C₂²	1	12	yes	complemented
16	C₂×C₃×C₄×D₄	—	2	6	no	—
17	C₂×C₃×C₄×D₄	C₂²	1	12	yes	—
18	C₂²×C₃×C₄⋊C₄	—	2	1	no	—
19	C₂²×C₃×C₄⋊C₄	—	2	1	no	—
20	C₂²×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂²×C₃×C₂²⋊C₄	—	2	1	no	—
22	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
23	C₃×#461	C₂²	1	64	yes	direct product factor
24	C₂×C₃×C₄⋊C₈	C₂²	1	12	yes	complemented
25	C₂×C₃×#103	C₂²	1	24	yes	complemented
26	C₂³×SD₁₆	C₂×C₃	1	2	yes	—
27	C₂²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂²×C₄×D₄	C₂×C₃	1	1	yes	characteristic
29	C₂×#461	C₂×C₃	1	24	yes	direct product factor
30	C₂²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂²×#103	C₂×C₃	1	2	yes	complemented
32	C₂⁵×C₃	—	2	1	no	—
33	C₂²×C₃×Q₈	—	2	6	no	—
34	C₂²×C₃×Q₈	—	2	8	no	—
35	C₂²×C₃×Q₈	—	2	8	no	—
36	C₂²×C₃×Q₈	C₂³	1	12	yes	—
37	C₂²×C₃×D₄	—	2	3	no	—
38	C₂²×C₃×D₄	—	2	4	no	—
39	C₂²×C₃×D₄	D₄	1	8	yes	—
40	C₂²×C₃×D₄	C₂³	1	6	yes	—
41	C₂³×C₃×C₄	—	2	1	no	—
42	C₂³×C₃×C₄	—	2	2	no	—
43	C₂³×C₃×C₄	—	2	12	no	—
44	C₂³×C₃×C₄	—	4	8	no	—
45	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
46	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
47	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
48	C₂×C₃×SD₁₆	—	2	96	no	—
49	C₂×C₃×SD₁₆	C₂³	1	32	yes	—
50	C₂²×C₃×C₈	—	2	8	no	—
51	C₂²×C₃×C₈	C₂³	1	12	yes	—
52	C₃×C₄⋊Q₈	—	2	24	no	—
53	C₃×C₄⋊Q₈	C₂³	1	16	yes	complemented
54	C₃×C₄×D₄	—	2	24	no	—
55	C₃×C₄×D₄	C₂³	1	16	yes	—
56	C₂×C₃×C₄⋊C₄	—	2	12	no	—
57	C₂×C₃×C₄⋊C₄	—	2	12	no	—
58	C₂×C₃×C₄⋊C₄	—	2	3	no	—
59	C₂×C₃×C₄⋊C₄	C₂³	1	6	yes	—
60	C₂×C₃×C₂²⋊C₄	—	2	12	no	—
61	C₂×C₃×C₄²	D₄	1	6	yes	—
62	C₂×C₃×C₄²	C₂³	1	6	yes	—
63	C₃×C₄⋊C₈	C₂³	1	16	yes	complemented
64	C₃×#103	C₂³	1	32	yes	complemented
65	C₂³×Q₈	C₂²×C₃	1	2	yes	—
66	C₂³×D₄	C₂²×C₃	1	1	yes	characteristic
67	C₂⁴×C₄	—	2	1	no	—
68	C₂²×SD₁₆	—	2	16	no	—
69	C₂²×SD₁₆	C₂²×C₃	1	24	yes	—
70	C₂³×C₈	C₂²×C₃	1	2	yes	—
71	C₂×C₄⋊Q₈	—	2	6	no	—
72	C₂×C₄⋊Q₈	C₂²×C₃	1	12	yes	complemented
73	C₂×C₄×D₄	—	2	6	no	—
74	C₂×C₄×D₄	C₂²×C₃	1	12	yes	—
75	C₂²×C₄⋊C₄	—	2	1	no	—
76	C₂²×C₄⋊C₄	—	2	1	no	—
77	C₂²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
78	C₂²×C₂²⋊C₄	—	2	1	no	—
79	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
80	#461	C₂²×C₃	1	64	yes	direct product factor
81	C₂×C₄⋊C₈	C₂²×C₃	1	12	yes	complemented
82	C₂×#103	C₂²×C₃	1	24	yes	complemented
83	C₂⁴×C₃	—	2	8	no	—
84	C₂⁴×C₃	—	2	6	no	—
85	C₂⁴×C₃	—	4	4	no	—
86	C₂⁴×C₃	—	4	4	no	—
87	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×Q₈	—	2	12	no	—
89	C₂×C₃×Q₈	—	2	24	no	—
90	C₂×C₃×Q₈	—	2	24	no	—
91	C₂×C₃×Q₈	—	4	24	no	—
92	C₂×C₃×Q₈	C₂⁴	1	8	yes	—
93	C₂×C₃×D₄	—	2	24	no	—
94	C₂×C₃×D₄	—	2	6	no	—
95	C₂×C₃×D₄	—	2	12	no	—
96	C₂×C₃×D₄	C₂×D₄	1	24	yes	—
97	C₂×C₃×D₄	C₂⁴	1	4	yes	—
98	C₂²×C₃×C₄	—	2	4	no	—
99	C₂²×C₃×C₄	—	2	6	no	—
100	C₂²×C₃×C₄	—	2	12	no	—
101	C₂²×C₃×C₄	—	2	16	no	—
102	C₂²×C₃×C₄	—	4	8	no	—
103	C₂²×C₃×C₄	—	4	4	no	—
104	C₂²×C₃×C₄	—	4	48	no	—
105	C₂²×C₃×C₄	SD₁₆	1	8	yes	complemented
106	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
107	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
108	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
109	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
110	C₂²×C₃×C₄	#41	1	4	yes	—
111	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
112	C₃×SD₁₆	—	2	64	no	—
113	C₂×C₃×C₈	—	2	24	no	—
114	C₂×C₃×C₈	C₂⁴	1	8	yes	—
115	C₃×C₄⋊C₄	—	2	16	no	—
116	C₃×C₄⋊C₄	—	2	16	no	—
117	C₃×C₄⋊C₄	—	2	6	no	—
118	C₃×C₄⋊C₄	C₂⁴	1	4	yes	—
119	C₃×C₂²⋊C₄	—	2	16	no	—
120	C₃×C₄²	C₂×D₄	1	12	yes	—
121	C₃×C₄²	C₂⁴	1	4	yes	—
122	C₂⁵	—	2	1	no	—
123	C₂²×Q₈	—	2	6	no	—
124	C₂²×Q₈	—	2	8	no	—
125	C₂²×Q₈	—	2	8	no	—
126	C₂²×Q₈	C₂³×C₃	1	12	yes	—
127	C₂²×D₄	—	2	3	no	—
128	C₂²×D₄	—	2	4	no	—
129	C₂²×D₄	C₃×D₄	1	8	yes	—
130	C₂²×D₄	C₂³×C₃	1	6	yes	—
131	C₂³×C₄	—	2	1	no	—
132	C₂³×C₄	—	2	2	no	—
133	C₂³×C₄	—	2	12	no	—
134	C₂³×C₄	—	4	8	no	—
135	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
136	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
137	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
138	C₂×SD₁₆	—	2	96	no	—
139	C₂×SD₁₆	C₂³×C₃	1	32	yes	—
140	C₂²×C₈	—	2	8	no	—
141	C₂²×C₈	C₂³×C₃	1	12	yes	—
142	C₄⋊Q₈	—	2	24	no	—
143	C₄⋊Q₈	C₂³×C₃	1	16	yes	complemented
144	C₄×D₄	—	2	24	no	—
145	C₄×D₄	C₂³×C₃	1	16	yes	—
146	C₂×C₄⋊C₄	—	2	12	no	—
147	C₂×C₄⋊C₄	—	2	12	no	—
148	C₂×C₄⋊C₄	—	2	3	no	—
149	C₂×C₄⋊C₄	C₂³×C₃	1	6	yes	—
150	C₂×C₂²⋊C₄	—	2	12	no	—
151	C₂×C₄²	C₃×D₄	1	6	yes	—
152	C₂×C₄²	C₂³×C₃	1	6	yes	—
153	C₄⋊C₈	C₂³×C₃	1	16	yes	complemented
154	#103	C₂³×C₃	1	32	yes	complemented
155	C₂³×C₃	—	2	24	no	—
156	C₂³×C₃	—	2	4	no	—
157	C₂³×C₃	—	4	12	no	—
158	C₂³×C₃	—	4	12	no	—
159	C₂³×C₃	—	4	32	no	—
160	C₂³×C₃	#121	1	4	yes	—
161	C₂³×C₃	C₂×SD₁₆	1	4	yes	—
162	C₂³×C₃	#137	1	4	yes	—
163	C₂³×C₃	C₂²×D₄	1	3	yes	—
164	C₃×Q₈	—	2	8	no	—
165	C₃×Q₈	—	2	8	no	—
166	C₃×Q₈	—	4	24	no	—
167	C₃×D₄	—	2	24	no	—
168	C₃×D₄	—	2	4	no	—
169	C₃×D₄	C₂²×D₄	1	8	yes	—
170	C₂×C₃×C₄	—	2	12	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	—	2	8	no	—
173	C₂×C₃×C₄	—	4	24	no	—
174	C₂×C₃×C₄	—	4	12	no	—
175	C₂×C₃×C₄	—	4	32	no	—
176	C₂×C₃×C₄	#121	1	12	yes	—
177	C₂×C₃×C₄	C₂×SD₁₆	1	24	yes	complemented
178	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
179	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
180	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
181	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
182	C₂×C₃×C₄	C₂×#41	1	6	yes	—
183	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
184	C₃×C₈	—	2	8	no	—
185	C₂⁴	—	2	8	no	—
186	C₂⁴	—	2	6	no	—
187	C₂⁴	—	4	4	no	—
188	C₂⁴	—	4	4	no	—
189	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
190	C₂×Q₈	—	2	12	no	—
191	C₂×Q₈	—	2	24	no	—
192	C₂×Q₈	—	2	24	no	—
193	C₂×Q₈	—	4	24	no	—
194	C₂×Q₈	C₂⁴×C₃	1	8	yes	—
195	C₂×D₄	—	2	24	no	—
196	C₂×D₄	—	2	6	no	—
197	C₂×D₄	—	2	12	no	—
198	C₂×D₄	C₂×C₃×D₄	1	24	yes	—
199	C₂×D₄	C₂⁴×C₃	1	4	yes	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	6	no	—
202	C₂²×C₄	—	2	12	no	—
203	C₂²×C₄	—	2	16	no	—
204	C₂²×C₄	—	4	8	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	—	4	48	no	—
207	C₂²×C₄	C₃×SD₁₆	1	8	yes	complemented
208	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
209	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
210	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
211	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
212	C₂²×C₄	C₃×#41	1	4	yes	—
213	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
214	SD₁₆	—	2	64	no	—
215	C₂×C₈	—	2	24	no	—
216	C₂×C₈	C₂⁴×C₃	1	8	yes	—
217	C₄⋊C₄	—	2	16	no	—
218	C₄⋊C₄	—	2	16	no	—
219	C₄⋊C₄	—	2	6	no	—
220	C₄⋊C₄	C₂⁴×C₃	1	4	yes	—
221	C₂²⋊C₄	—	2	16	no	—
222	C₄²	C₂×C₃×D₄	1	12	yes	—
223	C₄²	C₂⁴×C₃	1	4	yes	—
224	C₂²×C₃	—	2	8	no	—
225	C₂²×C₃	—	4	48	no	—
226	C₂²×C₃	—	4	4	no	—
227	C₂²×C₃	—	4	4	no	—
228	C₂²×C₃	#461	1	16	yes	direct product factor
229	C₂²×C₃	C₂×#121	1	6	yes	—
230	C₂²×C₃	C₂²×SD₁₆	1	6	yes	—
231	C₂²×C₃	C₂×#137	1	6	yes	—
232	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
233	C₃×C₄	—	2	4	no	—
234	C₃×C₄	—	4	8	no	—
235	C₃×C₄	—	4	4	no	—
236	C₃×C₄	C₂×#121	1	6	yes	—
237	C₃×C₄	C₂²×SD₁₆	1	8	yes	complemented
238	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
239	C₃×C₄	C₂²×#41	1	1	yes	characteristic
240	C₂³	—	2	4	no	—
241	C₂³	—	2	24	no	—
242	C₂³	—	4	12	no	—
243	C₂³	—	4	12	no	—
244	C₂³	—	4	32	no	—
245	C₂³	C₃×#121	1	4	yes	—
246	C₂³	C₂×C₃×SD₁₆	1	4	yes	—
247	C₂³	C₃×#137	1	4	yes	—
248	C₂³	C₂²×C₃×D₄	1	3	yes	—
249	Q₈	—	2	8	no	—
250	Q₈	—	2	8	no	—
251	Q₈	—	4	24	no	—
252	D₄	—	2	4	no	—
253	D₄	—	2	24	no	—
254	D₄	C₂²×C₃×D₄	1	8	yes	—
255	C₂×C₄	—	2	12	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	4	12	no	—
259	C₂×C₄	—	4	24	no	—
260	C₂×C₄	—	4	32	no	—
261	C₂×C₄	C₃×#121	1	12	yes	—
262	C₂×C₄	C₂×C₃×SD₁₆	1	24	yes	complemented
263	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
264	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
265	C₂×C₄	C₂²×C₃×D₄	1	6	yes	—
266	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
267	C₂×C₄	C₂×C₃×#41	1	6	yes	—
268	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
269	C₈	—	2	8	no	—
270	C₂×C₃	—	4	8	no	—
271	C₂×C₃	C₂×#461	1	12	yes	direct product factor
272	C₂×C₃	C₂²×#121	1	1	yes	characteristic
273	C₂×C₃	C₂³×SD₁₆	1	1	yes	characteristic
274	C₂×C₃	C₂²×#137	1	1	yes	characteristic
275	C₂²	—	2	8	no	—
276	C₂²	—	4	48	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	C₃×#461	1	16	yes	direct product factor
280	C₂²	C₂×C₃×#121	1	6	yes	—
281	C₂²	C₂²×C₃×SD₁₆	1	6	yes	—
282	C₂²	C₂×C₃×#137	1	6	yes	—
283	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	4	8	no	—
286	C₄	—	4	4	no	—
287	C₄	C₂×C₃×#121	1	6	yes	—
288	C₄	C₂²×C₃×SD₁₆	1	8	yes	complemented
289	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
290	C₄	C₂²×C₃×#41	1	1	yes	characteristic
291	C₃	C₂²×#461	1	1	yes	characteristic, direct product factor, Sylow
292	C₂	—	4	8	no	—
293	C₂	C₂×C₃×#461	1	12	yes	direct product factor
294	C₂	C₂²×C₃×#121	1	1	yes	characteristic
295	C₂	C₂³×C₃×SD₁₆	1	1	yes	characteristic
296	C₂	C₂²×C₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
prime radical	C₂³×C₃×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂²×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₄⋊Q₈	C₂	semidirect
4. C₂×C₃×#461	C₂	direct (N × H)
5. C₂²×C₃×C₄⋊C₈	C₂	semidirect
6. C₂²×C₃×#103	C₂	semidirect
7. C₂²×#461	C₃	direct (N × H)
15. C₂×C₃×C₄⋊Q₈	C₂²	semidirect
23. C₃×#461	C₂²	direct (N × H)
24. C₂×C₃×C₄⋊C₈	C₂²	semidirect
25. C₂×C₃×#103	C₂²	semidirect
27. C₂²×C₄⋊Q₈	C₂×C₃	semidirect
29. C₂×#461	C₂×C₃	direct (N × H)
30. C₂²×C₄⋊C₈	C₂×C₃	semidirect
31. C₂²×#103	C₂×C₃	semidirect
53. C₃×C₄⋊Q₈	C₂³	semidirect
63. C₃×C₄⋊C₈	C₂³	semidirect
64. C₃×#103	C₂³	semidirect
72. C₂×C₄⋊Q₈	C₂²×C₃	semidirect
80. #461	C₂²×C₃	direct (N × H)
81. C₂×C₄⋊C₈	C₂²×C₃	semidirect
82. C₂×#103	C₂²×C₃	semidirect
105. C₂²×C₃×C₄	SD₁₆	semidirect
143. C₄⋊Q₈	C₂³×C₃	semidirect
153. C₄⋊C₈	C₂³×C₃	semidirect
154. #103	C₂³×C₃	semidirect
177. C₂×C₃×C₄	C₂×SD₁₆	semidirect
207. C₂²×C₄	C₃×SD₁₆	semidirect
228. C₂²×C₃	#461	direct (N × H)
237. C₃×C₄	C₂²×SD₁₆	semidirect
262. C₂×C₄	C₂×C₃×SD₁₆	semidirect
271. C₂×C₃	C₂×#461	direct (N × H)
279. C₂²	C₃×#461	direct (N × H)
288. C₄	C₂²×C₃×SD₁₆	semidirect
291. C₃	C₂²×#461	direct (N × H)
293. C₂	C₂×C₃×#461	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×SD₁₆	2
C₂²×C₃×C₄⋊Q₈	2
C₂²×C₃×C₄×D₄	2
C₂×C₃×#461	2
C₂²×C₃×C₄⋊C₈	2
C₂²×C₃×#103	2
C₂²×#461	3

Subgroups — C22×C3×#461 #10679011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#461 #10679011