Subgroups — C₂²×C₃×#462 #10679014

Order 768.

2790 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×Q₁₆	C₂	1	2	yes	maximal
2	C₂²×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#462	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₃×#103	C₂	1	2	yes	maximal
7	C₂²×#462	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₃×Q₈	C₂²	1	2	yes	—
9	C₂³×C₃×Q₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×Q₁₆	—	2	16	no	—
11	C₂²×C₃×Q₁₆	C₂²	1	24	yes	—
12	C₂³×C₃×C₈	C₂²	1	2	yes	—
13	C₂×C₃×C₄⋊Q₈	—	2	6	no	—
14	C₂×C₃×C₄⋊Q₈	C₂²	1	12	yes	—
15	C₂×C₃×C₄×Q₈	—	2	6	no	—
16	C₂×C₃×C₄×Q₈	C₂²	1	12	yes	—
17	C₂²×C₃×C₄⋊C₄	—	2	1	no	—
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×C₄²	—	2	1	no	—
22	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
23	C₃×#462	C₂²	1	64	yes	direct product factor
24	C₂×C₃×C₄⋊C₈	C₂²	1	12	yes	—
25	C₂×C₃×#103	C₂²	1	24	yes	—
26	C₂³×Q₁₆	C₂×C₃	1	2	yes	—
27	C₂²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
28	C₂²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
29	C₂×#462	C₂×C₃	1	24	yes	direct product factor
30	C₂²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
31	C₂²×#103	C₂×C₃	1	2	yes	—
32	C₂²×C₃×Q₈	—	2	3	no	—
33	C₂²×C₃×Q₈	—	2	6	no	—
34	C₂²×C₃×Q₈	—	2	4	no	—
35	C₂²×C₃×Q₈	—	2	8	no	—
36	C₂²×C₃×Q₈	—	2	8	no	—
37	C₂²×C₃×Q₈	D₄	1	8	yes	—
38	C₂²×C₃×Q₈	C₂³	1	12	yes	—
39	C₂²×C₃×Q₈	C₂³	1	6	yes	—
40	C₂³×C₃×C₄	—	2	1	no	—
41	C₂³×C₃×C₄	—	2	2	no	—
42	C₂³×C₃×C₄	—	2	1	no	—
43	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
44	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
45	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
46	C₂×C₃×Q₁₆	—	2	96	no	—
47	C₂×C₃×Q₁₆	C₂³	1	32	yes	—
48	C₂²×C₃×C₈	—	2	8	no	—
49	C₂²×C₃×C₈	C₂³	1	12	yes	—
50	C₃×C₄⋊Q₈	—	2	24	no	—
51	C₃×C₄⋊Q₈	C₂³	1	16	yes	—
52	C₃×C₄×Q₈	—	2	24	no	—
53	C₃×C₄×Q₈	C₂³	1	16	yes	—
54	C₂×C₃×C₄⋊C₄	—	2	12	no	—
55	C₂×C₃×C₄⋊C₄	—	2	12	no	—
56	C₂×C₃×C₄⋊C₄	—	2	12	no	—
57	C₂×C₃×C₄⋊C₄	—	2	3	no	—
58	C₂×C₃×C₄⋊C₄	C₂³	1	6	yes	—
59	C₂×C₃×C₄²	—	2	12	no	—
60	C₂×C₃×C₄²	D₄	1	6	yes	—
61	C₂×C₃×C₄²	C₂³	1	6	yes	—
62	C₃×C₄⋊C₈	C₂³	1	16	yes	—
63	C₃×#103	C₂³	1	32	yes	—
64	C₂³×Q₈	C₂²×C₃	1	2	yes	—
65	C₂³×Q₈	C₂²×C₃	1	1	yes	characteristic
66	C₂²×Q₁₆	—	2	16	no	—
67	C₂²×Q₁₆	C₂²×C₃	1	24	yes	—
68	C₂³×C₈	C₂²×C₃	1	2	yes	—
69	C₂×C₄⋊Q₈	—	2	6	no	—
70	C₂×C₄⋊Q₈	C₂²×C₃	1	12	yes	—
71	C₂×C₄×Q₈	—	2	6	no	—
72	C₂×C₄×Q₈	C₂²×C₃	1	12	yes	—
73	C₂²×C₄⋊C₄	—	2	1	no	—
74	C₂²×C₄⋊C₄	—	2	1	no	—
75	C₂²×C₄⋊C₄	—	2	1	no	—
76	C₂²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
77	C₂²×C₄²	—	2	1	no	—
78	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
79	#462	C₂²×C₃	1	64	yes	direct product factor
80	C₂×C₄⋊C₈	C₂²×C₃	1	12	yes	—
81	C₂×#103	C₂²×C₃	1	24	yes	—
82	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×Q₈	—	2	24	no	—
84	C₂×C₃×Q₈	—	2	6	no	—
85	C₂×C₃×Q₈	—	2	12	no	—
86	C₂×C₃×Q₈	—	2	12	no	—
87	C₂×C₃×Q₈	—	2	24	no	—
88	C₂×C₃×Q₈	—	2	24	no	—
89	C₂×C₃×Q₈	—	4	24	no	—
90	C₂×C₃×Q₈	C₂×D₄	1	24	yes	—
91	C₂×C₃×Q₈	C₂⁴	1	8	yes	—
92	C₂×C₃×Q₈	C₂⁴	1	4	yes	—
93	C₂²×C₃×C₄	—	2	8	no	—
94	C₂²×C₃×C₄	—	2	4	no	—
95	C₂²×C₃×C₄	—	2	6	no	—
96	C₂²×C₃×C₄	—	2	6	no	—
97	C₂²×C₃×C₄	—	2	12	no	—
98	C₂²×C₃×C₄	—	4	8	no	—
99	C₂²×C₃×C₄	—	4	4	no	—
100	C₂²×C₃×C₄	Q₁₆	1	8	yes	complemented
101	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
102	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
103	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
104	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
105	C₂²×C₃×C₄	#41	1	4	yes	—
106	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
107	C₃×Q₁₆	—	2	64	no	—
108	C₂×C₃×C₈	—	2	24	no	—
109	C₂×C₃×C₈	C₂⁴	1	8	yes	—
110	C₃×C₄⋊C₄	—	2	16	no	—
111	C₃×C₄⋊C₄	—	2	16	no	—
112	C₃×C₄⋊C₄	—	2	16	no	—
113	C₃×C₄⋊C₄	—	2	6	no	—
114	C₃×C₄⋊C₄	C₂⁴	1	4	yes	—
115	C₃×C₄²	—	2	16	no	—
116	C₃×C₄²	C₂×D₄	1	12	yes	—
117	C₃×C₄²	C₂⁴	1	4	yes	—
118	C₂²×Q₈	—	2	3	no	—
119	C₂²×Q₈	—	2	6	no	—
120	C₂²×Q₈	—	2	4	no	—
121	C₂²×Q₈	—	2	8	no	—
122	C₂²×Q₈	—	2	8	no	—
123	C₂²×Q₈	C₃×D₄	1	8	yes	—
124	C₂²×Q₈	C₂³×C₃	1	12	yes	—
125	C₂²×Q₈	C₂³×C₃	1	6	yes	—
126	C₂³×C₄	—	2	1	no	—
127	C₂³×C₄	—	2	2	no	—
128	C₂³×C₄	—	2	1	no	—
129	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
130	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
131	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
132	C₂×Q₁₆	—	2	96	no	—
133	C₂×Q₁₆	C₂³×C₃	1	32	yes	—
134	C₂²×C₈	—	2	8	no	—
135	C₂²×C₈	C₂³×C₃	1	12	yes	—
136	C₄⋊Q₈	—	2	24	no	—
137	C₄⋊Q₈	C₂³×C₃	1	16	yes	—
138	C₄×Q₈	—	2	24	no	—
139	C₄×Q₈	C₂³×C₃	1	16	yes	—
140	C₂×C₄⋊C₄	—	2	12	no	—
141	C₂×C₄⋊C₄	—	2	12	no	—
142	C₂×C₄⋊C₄	—	2	12	no	—
143	C₂×C₄⋊C₄	—	2	3	no	—
144	C₂×C₄⋊C₄	C₂³×C₃	1	6	yes	—
145	C₂×C₄²	—	2	12	no	—
146	C₂×C₄²	C₃×D₄	1	6	yes	—
147	C₂×C₄²	C₂³×C₃	1	6	yes	—
148	C₄⋊C₈	C₂³×C₃	1	16	yes	—
149	#103	C₂³×C₃	1	32	yes	—
150	C₂³×C₃	#121	1	4	yes	—
151	C₂³×C₃	C₂×Q₁₆	1	4	yes	—
152	C₂³×C₃	#137	1	4	yes	—
153	C₂³×C₃	C₂²×D₄	1	3	yes	—
154	C₃×Q₈	—	2	24	no	—
155	C₃×Q₈	—	2	4	no	—
156	C₃×Q₈	—	2	8	no	—
157	C₃×Q₈	—	2	8	no	—
158	C₃×Q₈	—	4	24	no	—
159	C₃×Q₈	C₂²×D₄	1	8	yes	—
160	C₂×C₃×C₄	—	2	24	no	—
161	C₂×C₃×C₄	—	2	12	no	—
162	C₂×C₃×C₄	—	2	4	no	—
163	C₂×C₃×C₄	—	2	4	no	—
164	C₂×C₃×C₄	—	2	8	no	—
165	C₂×C₃×C₄	—	4	24	no	—
166	C₂×C₃×C₄	—	4	12	no	—
167	C₂×C₃×C₄	#121	1	12	yes	—
168	C₂×C₃×C₄	C₂×Q₁₆	1	24	yes	complemented
169	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
170	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
171	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
172	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
173	C₂×C₃×C₄	C₂×#41	1	6	yes	—
174	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
175	C₃×C₈	—	2	8	no	—
176	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
177	C₂×Q₈	—	2	24	no	—
178	C₂×Q₈	—	2	6	no	—
179	C₂×Q₈	—	2	12	no	—
180	C₂×Q₈	—	2	12	no	—
181	C₂×Q₈	—	2	24	no	—
182	C₂×Q₈	—	2	24	no	—
183	C₂×Q₈	—	4	24	no	—
184	C₂×Q₈	C₂×C₃×D₄	1	24	yes	—
185	C₂×Q₈	C₂⁴×C₃	1	8	yes	—
186	C₂×Q₈	C₂⁴×C₃	1	4	yes	—
187	C₂²×C₄	—	2	8	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	6	no	—
190	C₂²×C₄	—	2	6	no	—
191	C₂²×C₄	—	2	12	no	—
192	C₂²×C₄	—	4	8	no	—
193	C₂²×C₄	—	4	4	no	—
194	C₂²×C₄	C₃×Q₁₆	1	8	yes	complemented
195	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
196	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
197	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
198	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
199	C₂²×C₄	C₃×#41	1	4	yes	—
200	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
201	Q₁₆	—	2	64	no	—
202	C₂×C₈	—	2	24	no	—
203	C₂×C₈	C₂⁴×C₃	1	8	yes	—
204	C₄⋊C₄	—	2	16	no	—
205	C₄⋊C₄	—	2	16	no	—
206	C₄⋊C₄	—	2	16	no	—
207	C₄⋊C₄	—	2	6	no	—
208	C₄⋊C₄	C₂⁴×C₃	1	4	yes	—
209	C₄²	—	2	16	no	—
210	C₄²	C₂×C₃×D₄	1	12	yes	—
211	C₄²	C₂⁴×C₃	1	4	yes	—
212	C₂²×C₃	#462	1	16	yes	direct product factor
213	C₂²×C₃	C₂×#121	1	6	yes	—
214	C₂²×C₃	C₂²×Q₁₆	1	6	yes	—
215	C₂²×C₃	C₂×#137	1	6	yes	—
216	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
217	C₃×C₄	—	2	8	no	—
218	C₃×C₄	—	2	4	no	—
219	C₃×C₄	—	4	8	no	—
220	C₃×C₄	—	4	4	no	—
221	C₃×C₄	C₂×#121	1	6	yes	—
222	C₃×C₄	C₂²×Q₁₆	1	8	yes	complemented
223	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
224	C₃×C₄	C₂²×#41	1	1	yes	characteristic
225	C₂³	C₃×#121	1	4	yes	—
226	C₂³	C₂×C₃×Q₁₆	1	4	yes	—
227	C₂³	C₃×#137	1	4	yes	—
228	C₂³	C₂²×C₃×D₄	1	3	yes	—
229	Q₈	—	2	4	no	—
230	Q₈	—	2	24	no	—
231	Q₈	—	2	8	no	—
232	Q₈	—	2	8	no	—
233	Q₈	—	4	24	no	—
234	Q₈	C₂²×C₃×D₄	1	8	yes	—
235	C₂×C₄	—	2	12	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	24	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	4	12	no	—
241	C₂×C₄	—	4	24	no	—
242	C₂×C₄	C₃×#121	1	12	yes	—
243	C₂×C₄	C₂×C₃×Q₁₆	1	24	yes	complemented
244	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
245	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
246	C₂×C₄	C₂²×C₃×D₄	1	6	yes	—
247	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
248	C₂×C₄	C₂×C₃×#41	1	6	yes	—
249	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
250	C₈	—	2	8	no	—
251	C₂×C₃	C₂×#462	1	12	yes	direct product factor
252	C₂×C₃	C₂²×#121	1	1	yes	characteristic
253	C₂×C₃	C₂³×Q₁₆	1	1	yes	characteristic
254	C₂×C₃	C₂²×#137	1	1	yes	characteristic
255	C₂²	C₃×#462	1	16	yes	direct product factor
256	C₂²	C₂×C₃×#121	1	6	yes	—
257	C₂²	C₂²×C₃×Q₁₆	1	6	yes	—
258	C₂²	C₂×C₃×#137	1	6	yes	—
259	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
260	C₄	—	2	8	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	8	no	—
263	C₄	—	4	4	no	—
264	C₄	C₂×C₃×#121	1	6	yes	—
265	C₄	C₂²×C₃×Q₁₆	1	8	yes	complemented
266	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
267	C₄	C₂²×C₃×#41	1	1	yes	characteristic
268	C₃	C₂²×#462	1	1	yes	characteristic, direct product factor, Sylow
269	C₂	C₂×C₃×#462	1	12	yes	direct product factor
270	C₂	C₂²×C₃×#121	1	1	yes	characteristic
271	C₂	C₂³×C₃×Q₁₆	1	1	yes	characteristic
272	C₂	C₂²×C₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
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
4. C₂×C₃×#462	C₂	direct (N × H)
7. C₂²×#462	C₃	direct (N × H)
23. C₃×#462	C₂²	direct (N × H)
29. C₂×#462	C₂×C₃	direct (N × H)
79. #462	C₂²×C₃	direct (N × H)
100. C₂²×C₃×C₄	Q₁₆	semidirect
168. C₂×C₃×C₄	C₂×Q₁₆	semidirect
194. C₂²×C₄	C₃×Q₁₆	semidirect
212. C₂²×C₃	#462	direct (N × H)
222. C₃×C₄	C₂²×Q₁₆	semidirect
243. C₂×C₄	C₂×C₃×Q₁₆	semidirect
251. C₂×C₃	C₂×#462	direct (N × H)
255. C₂²	C₃×#462	direct (N × H)
265. C₄	C₂²×C₃×Q₁₆	semidirect
268. C₃	C₂²×#462	direct (N × H)
269. C₂	C₂×C₃×#462	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×Q₁₆	2
C₂²×C₃×C₄⋊Q₈	2
C₂²×C₃×C₄×Q₈	2
C₂×C₃×#462	2
C₂²×C₃×C₄⋊C₈	2
C₂²×C₃×#103	2
C₂²×#462	3

Subgroups — C22×C3×#462 #10679014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#462 #10679014