Subgroups — C₂²×C₃×#474 #10679080

Order 768.

2790 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#474	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃×#107	C₂	1	2	yes	maximal, complemented
5	C₂²×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃×#102	C₂	1	2	yes	maximal
7	C₂²×#474	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₃×Q₈	—	2	1	no	—
9	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic
10	C₂⁴×C₃×C₄	—	2	1	no	—
11	C₂³×C₃×C₈	C₂²	1	2	yes	—
12	C₂×C₃×C₄⋊Q₈	—	2	6	no	—
13	C₂×C₃×C₄⋊Q₈	C₂²	1	12	yes	complemented
14	C₂×C₃×C₄×D₄	—	2	6	no	—
15	C₂×C₃×C₄×D₄	C₂²	1	12	yes	—
16	C₂²×C₃×C₄⋊C₄	—	2	1	no	—
17	C₂²×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂²×C₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂²×C₃×C₂²⋊C₄	—	2	1	no	—
20	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
21	C₃×#474	C₂²	1	64	yes	direct product factor
22	C₂×C₃×#107	C₂²	1	24	yes	complemented
23	C₂×C₃×C₄⋊C₈	C₂²	1	12	yes	complemented
24	C₂×C₃×#102	C₂²	1	24	yes	—
25	C₂²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂²×C₄×D₄	C₂×C₃	1	1	yes	characteristic
27	C₂×#474	C₂×C₃	1	24	yes	direct product factor
28	C₂²×#107	C₂×C₃	1	2	yes	complemented
29	C₂²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂²×#102	C₂×C₃	1	2	yes	—
31	C₂⁵×C₃	—	2	1	no	—
32	C₂²×C₃×Q₈	—	2	12	no	—
33	C₂²×C₃×Q₈	—	4	8	no	—
34	C₂²×C₃×D₄	—	2	3	no	—
35	C₂²×C₃×D₄	—	2	4	no	—
36	C₂²×C₃×D₄	Q₈	1	8	yes	complemented
37	C₂²×C₃×D₄	C₂³	1	6	yes	—
38	C₂³×C₃×C₄	—	2	1	no	—
39	C₂³×C₃×C₄	—	2	2	no	—
40	C₂³×C₃×C₄	—	2	12	no	—
41	C₂³×C₃×C₄	—	4	8	no	—
42	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
43	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
44	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
45	C₂²×C₃×C₈	—	2	8	no	—
46	C₂²×C₃×C₈	C₂³	1	12	yes	—
47	C₃×C₄⋊Q₈	—	2	24	no	—
48	C₃×C₄⋊Q₈	C₂³	1	16	yes	complemented
49	C₃×C₄×D₄	—	2	24	no	—
50	C₃×C₄×D₄	C₂³	1	16	yes	—
51	C₂×C₃×C₄⋊C₄	—	2	12	no	—
52	C₂×C₃×C₄⋊C₄	—	2	3	no	—
53	C₂×C₃×C₄⋊C₄	—	2	6	no	—
54	C₂×C₃×C₄⋊C₄	C₂³	1	12	yes	—
55	C₂×C₃×C₄⋊C₄	C₂³	1	6	yes	—
56	C₂×C₃×C₂²⋊C₄	—	2	12	no	—
57	C₂×C₃×C₄²	D₄	1	6	yes	—
58	C₂×C₃×C₄²	C₂³	1	6	yes	—
59	C₃×#107	C₂³	1	32	yes	complemented
60	C₃×C₄⋊C₈	C₂³	1	16	yes	complemented
61	C₃×#102	C₂³	1	32	yes	—
62	C₂³×Q₈	—	2	1	no	—
63	C₂³×D₄	C₂²×C₃	1	1	yes	characteristic
64	C₂⁴×C₄	—	2	1	no	—
65	C₂³×C₈	C₂²×C₃	1	2	yes	—
66	C₂×C₄⋊Q₈	—	2	6	no	—
67	C₂×C₄⋊Q₈	C₂²×C₃	1	12	yes	complemented
68	C₂×C₄×D₄	—	2	6	no	—
69	C₂×C₄×D₄	C₂²×C₃	1	12	yes	—
70	C₂²×C₄⋊C₄	—	2	1	no	—
71	C₂²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
72	C₂²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
73	C₂²×C₂²⋊C₄	—	2	1	no	—
74	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
75	#474	C₂²×C₃	1	64	yes	direct product factor
76	C₂×#107	C₂²×C₃	1	24	yes	complemented
77	C₂×C₄⋊C₈	C₂²×C₃	1	12	yes	complemented
78	C₂×#102	C₂²×C₃	1	24	yes	—
79	C₂⁴×C₃	—	2	8	no	—
80	C₂⁴×C₃	—	2	6	no	—
81	C₂⁴×C₃	—	4	4	no	—
82	C₂⁴×C₃	—	4	4	no	—
83	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×Q₈	—	2	16	no	—
85	C₂×C₃×Q₈	—	4	48	no	—
86	C₂×C₃×D₄	—	2	24	no	—
87	C₂×C₃×D₄	—	2	6	no	—
88	C₂×C₃×D₄	—	2	12	no	—
89	C₂×C₃×D₄	C₂×Q₈	1	24	yes	complemented
90	C₂×C₃×D₄	C₂⁴	1	4	yes	—
91	C₂²×C₃×C₄	—	2	4	no	—
92	C₂²×C₃×C₄	—	2	6	no	—
93	C₂²×C₃×C₄	—	2	12	no	—
94	C₂²×C₃×C₄	—	2	16	no	—
95	C₂²×C₃×C₄	—	4	4	no	—
96	C₂²×C₃×C₄	—	4	8	no	—
97	C₂²×C₃×C₄	—	4	48	no	—
98	C₂²×C₃×C₄	D₈	1	8	yes	—
99	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
100	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
101	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
102	C₂²×C₃×C₄	C₂×Q₈	1	4	yes	—
103	C₂²×C₃×C₄	#41	1	4	yes	—
104	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
105	C₂×C₃×C₈	—	2	24	no	—
106	C₂×C₃×C₈	C₂⁴	1	8	yes	—
107	C₃×C₄⋊C₄	—	2	16	no	—
108	C₃×C₄⋊C₄	—	2	6	no	—
109	C₃×C₄⋊C₄	—	2	12	no	—
110	C₃×C₄⋊C₄	C₂⁴	1	8	yes	—
111	C₃×C₄⋊C₄	C₂⁴	1	4	yes	—
112	C₃×C₂²⋊C₄	—	2	16	no	—
113	C₃×C₄²	C₂×D₄	1	12	yes	—
114	C₃×C₄²	C₂⁴	1	4	yes	—
115	C₂⁵	—	2	1	no	—
116	C₂²×Q₈	—	2	12	no	—
117	C₂²×Q₈	—	4	8	no	—
118	C₂²×D₄	—	2	3	no	—
119	C₂²×D₄	—	2	4	no	—
120	C₂²×D₄	C₃×Q₈	1	8	yes	complemented
121	C₂²×D₄	C₂³×C₃	1	6	yes	—
122	C₂³×C₄	—	2	1	no	—
123	C₂³×C₄	—	2	2	no	—
124	C₂³×C₄	—	2	12	no	—
125	C₂³×C₄	—	4	8	no	—
126	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
127	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
128	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
129	C₂²×C₈	—	2	8	no	—
130	C₂²×C₈	C₂³×C₃	1	12	yes	—
131	C₄⋊Q₈	—	2	24	no	—
132	C₄⋊Q₈	C₂³×C₃	1	16	yes	complemented
133	C₄×D₄	—	2	24	no	—
134	C₄×D₄	C₂³×C₃	1	16	yes	—
135	C₂×C₄⋊C₄	—	2	12	no	—
136	C₂×C₄⋊C₄	—	2	3	no	—
137	C₂×C₄⋊C₄	—	2	6	no	—
138	C₂×C₄⋊C₄	C₂³×C₃	1	6	yes	—
139	C₂×C₄⋊C₄	C₂³×C₃	1	12	yes	—
140	C₂×C₂²⋊C₄	—	2	12	no	—
141	C₂×C₄²	C₃×D₄	1	6	yes	—
142	C₂×C₄²	C₂³×C₃	1	6	yes	—
143	#107	C₂³×C₃	1	32	yes	complemented
144	C₄⋊C₈	C₂³×C₃	1	16	yes	complemented
145	#102	C₂³×C₃	1	32	yes	—
146	C₂³×C₃	—	2	24	no	—
147	C₂³×C₃	—	2	4	no	—
148	C₂³×C₃	—	4	12	no	—
149	C₂³×C₃	—	4	12	no	—
150	C₂³×C₃	—	4	32	no	—
151	C₂³×C₃	C₂²⋊Q₈	1	4	yes	—
152	C₂³×C₃	C₂×D₈	1	4	yes	—
153	C₂³×C₃	#137	1	4	yes	—
154	C₂³×C₃	C₂²×D₄	1	3	yes	—
155	C₃×Q₈	—	4	32	no	—
156	C₃×D₄	—	2	24	no	—
157	C₃×D₄	—	2	4	no	—
158	C₃×D₄	C₂²×Q₈	1	8	yes	complemented
159	C₂×C₃×C₄	—	2	12	no	—
160	C₂×C₃×C₄	—	2	4	no	—
161	C₂×C₃×C₄	—	2	8	no	—
162	C₂×C₃×C₄	—	4	12	no	—
163	C₂×C₃×C₄	—	4	24	no	—
164	C₂×C₃×C₄	—	4	32	no	—
165	C₂×C₃×C₄	C₂²⋊Q₈	1	12	yes	—
166	C₂×C₃×C₄	C₂×D₈	1	24	yes	—
167	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
168	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
169	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
170	C₂×C₃×C₄	C₂²×Q₈	1	6	yes	—
171	C₂×C₃×C₄	C₂×#41	1	6	yes	—
172	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
173	C₃×C₈	—	2	8	no	—
174	C₂⁴	—	2	8	no	—
175	C₂⁴	—	2	6	no	—
176	C₂⁴	—	4	4	no	—
177	C₂⁴	—	4	4	no	—
178	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
179	C₂×Q₈	—	2	16	no	—
180	C₂×Q₈	—	4	48	no	—
181	C₂×D₄	—	2	24	no	—
182	C₂×D₄	—	2	6	no	—
183	C₂×D₄	—	2	12	no	—
184	C₂×D₄	C₂×C₃×Q₈	1	24	yes	complemented
185	C₂×D₄	C₂⁴×C₃	1	4	yes	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	6	no	—
188	C₂²×C₄	—	2	12	no	—
189	C₂²×C₄	—	2	16	no	—
190	C₂²×C₄	—	4	4	no	—
191	C₂²×C₄	—	4	8	no	—
192	C₂²×C₄	—	4	48	no	—
193	C₂²×C₄	C₃×D₈	1	8	yes	—
194	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
195	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
196	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
197	C₂²×C₄	C₂×C₃×Q₈	1	4	yes	—
198	C₂²×C₄	C₃×#41	1	4	yes	—
199	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
200	C₂×C₈	—	2	24	no	—
201	C₂×C₈	C₂⁴×C₃	1	8	yes	—
202	C₄⋊C₄	—	2	16	no	—
203	C₄⋊C₄	—	2	6	no	—
204	C₄⋊C₄	—	2	12	no	—
205	C₄⋊C₄	C₂⁴×C₃	1	8	yes	—
206	C₄⋊C₄	C₂⁴×C₃	1	4	yes	—
207	C₂²⋊C₄	—	2	16	no	—
208	C₄²	C₂×C₃×D₄	1	12	yes	—
209	C₄²	C₂⁴×C₃	1	4	yes	—
210	C₂²×C₃	—	2	8	no	—
211	C₂²×C₃	—	4	4	no	—
212	C₂²×C₃	—	4	4	no	—
213	C₂²×C₃	—	4	48	no	—
214	C₂²×C₃	#474	1	16	yes	direct product factor
215	C₂²×C₃	C₂×C₂²⋊Q₈	1	6	yes	—
216	C₂²×C₃	C₂²×D₈	1	6	yes	—
217	C₂²×C₃	C₂×#137	1	6	yes	—
218	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
219	C₃×C₄	—	2	4	no	—
220	C₃×C₄	—	4	4	no	—
221	C₃×C₄	—	4	8	no	—
222	C₃×C₄	C₂×C₂²⋊Q₈	1	6	yes	—
223	C₃×C₄	C₂²×D₈	1	8	yes	—
224	C₃×C₄	C₂³×Q₈	1	1	yes	characteristic
225	C₃×C₄	C₂²×#41	1	1	yes	characteristic
226	C₂³	—	2	4	no	—
227	C₂³	—	2	24	no	—
228	C₂³	—	4	12	no	—
229	C₂³	—	4	12	no	—
230	C₂³	—	4	32	no	—
231	C₂³	C₃×C₂²⋊Q₈	1	4	yes	—
232	C₂³	C₂×C₃×D₈	1	4	yes	—
233	C₂³	C₃×#137	1	4	yes	—
234	C₂³	C₂²×C₃×D₄	1	3	yes	—
235	Q₈	—	4	32	no	—
236	D₄	—	2	4	no	—
237	D₄	—	2	24	no	—
238	D₄	C₂²×C₃×Q₈	1	8	yes	complemented
239	C₂×C₄	—	2	12	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	4	12	no	—
243	C₂×C₄	—	4	24	no	—
244	C₂×C₄	—	4	32	no	—
245	C₂×C₄	C₃×C₂²⋊Q₈	1	12	yes	—
246	C₂×C₄	C₂×C₃×D₈	1	24	yes	—
247	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
248	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
249	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
250	C₂×C₄	C₂²×C₃×Q₈	1	6	yes	—
251	C₂×C₄	C₂×C₃×#41	1	6	yes	—
252	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
253	C₈	—	2	8	no	—
254	C₂×C₃	—	4	8	no	—
255	C₂×C₃	C₂×#474	1	12	yes	direct product factor
256	C₂×C₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
257	C₂×C₃	C₂³×D₈	1	1	yes	characteristic
258	C₂×C₃	C₂²×#137	1	1	yes	characteristic
259	C₂²	—	2	8	no	—
260	C₂²	—	4	4	no	—
261	C₂²	—	4	4	no	—
262	C₂²	—	4	48	no	—
263	C₂²	C₃×#474	1	16	yes	direct product factor
264	C₂²	C₂×C₃×C₂²⋊Q₈	1	6	yes	—
265	C₂²	C₂²×C₃×D₈	1	6	yes	—
266	C₂²	C₂×C₃×#137	1	6	yes	—
267	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
268	C₄	—	2	4	no	—
269	C₄	—	4	4	no	—
270	C₄	—	4	8	no	—
271	C₄	C₂×C₃×C₂²⋊Q₈	1	6	yes	—
272	C₄	C₂²×C₃×D₈	1	8	yes	—
273	C₄	C₂³×C₃×Q₈	1	1	yes	characteristic
274	C₄	C₂²×C₃×#41	1	1	yes	characteristic
275	C₃	C₂²×#474	1	1	yes	characteristic, direct product factor, Sylow
276	C₂	—	4	8	no	—
277	C₂	C₂×C₃×#474	1	12	yes	direct product factor
278	C₂	C₂²×C₃×C₂²⋊Q₈	1	1	yes	characteristic
279	C₂	C₂³×C₃×D₈	1	1	yes	characteristic
280	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
1. C₂²×C₃×C₄⋊Q₈	C₂	semidirect
3. C₂×C₃×#474	C₂	direct (N × H)
4. C₂²×C₃×#107	C₂	semidirect
5. C₂²×C₃×C₄⋊C₈	C₂	semidirect
7. C₂²×#474	C₃	direct (N × H)
13. C₂×C₃×C₄⋊Q₈	C₂²	semidirect
21. C₃×#474	C₂²	direct (N × H)
22. C₂×C₃×#107	C₂²	semidirect
23. C₂×C₃×C₄⋊C₈	C₂²	semidirect
25. C₂²×C₄⋊Q₈	C₂×C₃	semidirect
27. C₂×#474	C₂×C₃	direct (N × H)
28. C₂²×#107	C₂×C₃	semidirect
29. C₂²×C₄⋊C₈	C₂×C₃	semidirect
36. C₂²×C₃×D₄	Q₈	semidirect
48. C₃×C₄⋊Q₈	C₂³	semidirect
59. C₃×#107	C₂³	semidirect
60. C₃×C₄⋊C₈	C₂³	semidirect
67. C₂×C₄⋊Q₈	C₂²×C₃	semidirect
75. #474	C₂²×C₃	direct (N × H)
76. C₂×#107	C₂²×C₃	semidirect
77. C₂×C₄⋊C₈	C₂²×C₃	semidirect
89. C₂×C₃×D₄	C₂×Q₈	semidirect
120. C₂²×D₄	C₃×Q₈	semidirect
132. C₄⋊Q₈	C₂³×C₃	semidirect
143. #107	C₂³×C₃	semidirect
144. C₄⋊C₈	C₂³×C₃	semidirect
158. C₃×D₄	C₂²×Q₈	semidirect
184. C₂×D₄	C₂×C₃×Q₈	semidirect
214. C₂²×C₃	#474	direct (N × H)
238. D₄	C₂²×C₃×Q₈	semidirect
255. C₂×C₃	C₂×#474	direct (N × H)
263. C₂²	C₃×#474	direct (N × H)
275. C₃	C₂²×#474	direct (N × H)
277. C₂	C₂×C₃×#474	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×C₄⋊Q₈	2
C₂²×C₃×C₄×D₄	2
C₂×C₃×#474	2
C₂²×C₃×#107	2
C₂²×C₃×C₄⋊C₈	2
C₂²×C₃×#102	2
C₂²×#474	3

Subgroups — C22×C3×#474 #10679080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#474 #10679080