Subgroups — C₂²×C₃×#460 #10679013

Order 768.

3478 conjugacy classes of subgroups.

The subgroup lattice has 304 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, complemented
2	C₂²×C₃×#127	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#460	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃×#102	C₂	1	2	yes	maximal, complemented
7	C₂²×#460	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₃×Q₈	C₂²	1	1	yes	characteristic
9	C₂³×C₃×D₄	—	2	1	no	—
10	C₂³×C₃×D₄	—	2	1	no	—
11	C₂³×C₃×D₄	C₂²	1	2	yes	—
12	C₂²×C₃×SD₁₆	—	2	16	no	—
13	C₂²×C₃×SD₁₆	C₂²	1	24	yes	complemented
14	C₂³×C₃×C₈	C₂²	1	2	yes	—
15	C₂×C₃×#127	—	2	6	no	—
16	C₂×C₃×#127	C₂²	1	12	yes	—
17	C₂×C₃×C₄×Q₈	—	2	6	no	—
18	C₂×C₃×C₄×Q₈	C₂²	1	12	yes	complemented
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×#460	C₂²	1	64	yes	direct product factor
24	C₂×C₃×C₄⋊C₈	C₂²	1	12	yes	complemented
25	C₂×C₃×#102	C₂²	1	24	yes	complemented
26	C₂³×SD₁₆	C₂×C₃	1	2	yes	complemented
27	C₂²×#127	C₂×C₃	1	1	yes	characteristic
28	C₂²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×#460	C₂×C₃	1	24	yes	direct product factor
30	C₂²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂²×#102	C₂×C₃	1	2	yes	complemented
32	C₂⁵×C₃	—	2	2	no	—
33	C₂²×C₃×Q₈	—	2	3	no	—
34	C₂²×C₃×Q₈	—	2	4	no	—
35	C₂²×C₃×Q₈	D₄	1	8	yes	complemented
36	C₂²×C₃×Q₈	C₂³	1	6	yes	—
37	C₂²×C₃×D₄	—	2	8	no	—
38	C₂²×C₃×D₄	—	2	12	no	—
39	C₂²×C₃×D₄	—	2	12	no	—
40	C₂²×C₃×D₄	—	2	6	no	—
41	C₂²×C₃×D₄	—	2	8	no	—
42	C₂²×C₃×D₄	—	4	8	no	—
43	C₂²×C₃×D₄	—	4	8	no	—
44	C₂²×C₃×D₄	C₂³	1	12	yes	—
45	C₂³×C₃×C₄	—	2	1	no	—
46	C₂³×C₃×C₄	—	2	1	no	—
47	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
48	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
49	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
50	C₂×C₃×SD₁₆	—	2	96	no	—
51	C₂×C₃×SD₁₆	C₂³	1	32	yes	complemented
52	C₂²×C₃×C₈	—	2	8	no	—
53	C₂²×C₃×C₈	C₂³	1	12	yes	—
54	C₃×#127	—	2	24	no	—
55	C₃×#127	C₂³	1	16	yes	—
56	C₃×C₄×Q₈	—	2	24	no	—
57	C₃×C₄×Q₈	C₂³	1	16	yes	complemented
58	C₂×C₃×C₄⋊C₄	—	2	12	no	—
59	C₂×C₃×C₄⋊C₄	—	2	3	no	—
60	C₂×C₃×C₄⋊C₄	C₂³	1	6	yes	—
61	C₂×C₃×C₄²	—	2	12	no	—
62	C₂×C₃×C₄²	D₄	1	6	yes	—
63	C₂×C₃×C₄²	C₂³	1	6	yes	—
64	C₃×C₄⋊C₈	C₂³	1	16	yes	complemented
65	C₃×#102	C₂³	1	32	yes	complemented
66	C₂³×Q₈	C₂²×C₃	1	1	yes	characteristic
67	C₂³×D₄	—	2	1	no	—
68	C₂³×D₄	—	2	1	no	—
69	C₂³×D₄	C₂²×C₃	1	2	yes	—
70	C₂²×SD₁₆	—	2	16	no	—
71	C₂²×SD₁₆	C₂²×C₃	1	24	yes	complemented
72	C₂³×C₈	C₂²×C₃	1	2	yes	—
73	C₂×#127	—	2	6	no	—
74	C₂×#127	C₂²×C₃	1	12	yes	—
75	C₂×C₄×Q₈	—	2	6	no	—
76	C₂×C₄×Q₈	C₂²×C₃	1	12	yes	complemented
77	C₂²×C₄⋊C₄	—	2	1	no	—
78	C₂²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
79	C₂²×C₄²	—	2	1	no	—
80	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
81	#460	C₂²×C₃	1	64	yes	direct product factor
82	C₂×C₄⋊C₈	C₂²×C₃	1	12	yes	complemented
83	C₂×#102	C₂²×C₃	1	24	yes	complemented
84	C₂⁴×C₃	—	2	12	no	—
85	C₂⁴×C₃	—	4	8	no	—
86	C₂⁴×C₃	—	4	8	no	—
87	C₂⁴×C₃	—	4	8	no	—
88	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×Q₈	—	2	24	no	—
90	C₂×C₃×Q₈	—	2	6	no	—
91	C₂×C₃×Q₈	—	2	12	no	—
92	C₂×C₃×Q₈	C₂×D₄	1	24	yes	complemented
93	C₂×C₃×Q₈	C₂⁴	1	4	yes	—
94	C₂×C₃×D₄	—	2	24	no	—
95	C₂×C₃×D₄	—	2	12	no	—
96	C₂×C₃×D₄	—	2	16	no	—
97	C₂×C₃×D₄	—	2	16	no	—
98	C₂×C₃×D₄	—	2	24	no	—
99	C₂×C₃×D₄	—	4	48	no	—
100	C₂×C₃×D₄	—	4	24	no	—
101	C₂×C₃×D₄	—	4	48	no	—
102	C₂×C₃×D₄	C₂⁴	1	8	yes	—
103	C₂²×C₃×C₄	—	2	8	no	—
104	C₂²×C₃×C₄	—	2	4	no	—
105	C₂²×C₃×C₄	—	2	6	no	—
106	C₂²×C₃×C₄	—	2	6	no	—
107	C₂²×C₃×C₄	—	4	4	no	—
108	C₂²×C₃×C₄	SD₁₆	1	8	yes	complemented
109	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
110	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
111	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
112	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
113	C₂²×C₃×C₄	#41	1	4	yes	—
114	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
115	C₃×SD₁₆	—	2	64	no	—
116	C₂×C₃×C₈	—	2	24	no	—
117	C₂×C₃×C₈	C₂⁴	1	8	yes	—
118	C₃×C₄⋊C₄	—	2	16	no	—
119	C₃×C₄⋊C₄	—	2	6	no	—
120	C₃×C₄⋊C₄	C₂⁴	1	4	yes	—
121	C₃×C₄²	—	2	16	no	—
122	C₃×C₄²	C₂×D₄	1	12	yes	—
123	C₃×C₄²	C₂⁴	1	4	yes	—
124	C₂⁵	—	2	2	no	—
125	C₂²×Q₈	—	2	3	no	—
126	C₂²×Q₈	—	2	4	no	—
127	C₂²×Q₈	C₃×D₄	1	8	yes	complemented
128	C₂²×Q₈	C₂³×C₃	1	6	yes	—
129	C₂²×D₄	—	2	8	no	—
130	C₂²×D₄	—	2	6	no	—
131	C₂²×D₄	—	2	12	no	—
132	C₂²×D₄	—	2	12	no	—
133	C₂²×D₄	—	2	8	no	—
134	C₂²×D₄	—	4	8	no	—
135	C₂²×D₄	—	4	8	no	—
136	C₂²×D₄	C₂³×C₃	1	12	yes	—
137	C₂³×C₄	—	2	1	no	—
138	C₂³×C₄	—	2	1	no	—
139	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
140	C₂³×C₄	C₃×D₄	1	1	yes	characteristic
141	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
142	C₂×SD₁₆	—	2	96	no	—
143	C₂×SD₁₆	C₂³×C₃	1	32	yes	complemented
144	C₂²×C₈	—	2	8	no	—
145	C₂²×C₈	C₂³×C₃	1	12	yes	—
146	#127	—	2	24	no	—
147	#127	C₂³×C₃	1	16	yes	—
148	C₄×Q₈	—	2	24	no	—
149	C₄×Q₈	C₂³×C₃	1	16	yes	complemented
150	C₂×C₄⋊C₄	—	2	12	no	—
151	C₂×C₄⋊C₄	—	2	3	no	—
152	C₂×C₄⋊C₄	C₂³×C₃	1	6	yes	—
153	C₂×C₄²	—	2	12	no	—
154	C₂×C₄²	C₃×D₄	1	6	yes	—
155	C₂×C₄²	C₂³×C₃	1	6	yes	—
156	C₄⋊C₈	C₂³×C₃	1	16	yes	complemented
157	#102	C₂³×C₃	1	32	yes	complemented
158	C₂³×C₃	—	2	8	no	—
159	C₂³×C₃	—	4	24	no	—
160	C₂³×C₃	—	4	24	no	—
161	C₂³×C₃	—	4	24	no	—
162	C₂³×C₃	—	8	32	no	—
163	C₂³×C₃	#121	1	4	yes	—
164	C₂³×C₃	C₂×SD₁₆	1	4	yes	—
165	C₂³×C₃	AGL(1,ℤ/8)	1	4	yes	—
166	C₂³×C₃	C₂²×D₄	1	3	yes	—
167	C₃×Q₈	—	2	24	no	—
168	C₃×Q₈	—	2	4	no	—
169	C₃×Q₈	C₂²×D₄	1	8	yes	complemented
170	C₃×D₄	—	2	8	no	—
171	C₃×D₄	—	2	8	no	—
172	C₃×D₄	—	4	32	no	—
173	C₃×D₄	—	4	24	no	—
174	C₃×D₄	—	4	32	no	—
175	C₂×C₃×C₄	—	2	24	no	—
176	C₂×C₃×C₄	—	2	12	no	—
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	—	2	4	no	—
179	C₂×C₃×C₄	—	4	12	no	—
180	C₂×C₃×C₄	#121	1	12	yes	—
181	C₂×C₃×C₄	C₂×SD₁₆	1	24	yes	complemented
182	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
183	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
184	C₂×C₃×C₄	C₂²×D₄	1	6	yes	—
185	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
186	C₂×C₃×C₄	C₂×#41	1	6	yes	—
187	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
188	C₃×C₈	—	2	8	no	—
189	C₂⁴	—	2	12	no	—
190	C₂⁴	—	4	8	no	—
191	C₂⁴	—	4	8	no	—
192	C₂⁴	—	4	8	no	—
193	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
194	C₂×Q₈	—	2	24	no	—
195	C₂×Q₈	—	2	6	no	—
196	C₂×Q₈	—	2	12	no	—
197	C₂×Q₈	C₂×C₃×D₄	1	24	yes	complemented
198	C₂×Q₈	C₂⁴×C₃	1	4	yes	—
199	C₂×D₄	—	2	24	no	—
200	C₂×D₄	—	2	12	no	—
201	C₂×D₄	—	2	16	no	—
202	C₂×D₄	—	2	16	no	—
203	C₂×D₄	—	2	24	no	—
204	C₂×D₄	—	4	48	no	—
205	C₂×D₄	—	4	24	no	—
206	C₂×D₄	—	4	48	no	—
207	C₂×D₄	C₂⁴×C₃	1	8	yes	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	6	no	—
211	C₂²×C₄	—	2	6	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	C₃×SD₁₆	1	8	yes	complemented
214	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
215	C₂²×C₄	C₂×C₃×D₄	1	3	yes	—
216	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
217	C₂²×C₄	C₂×C₃×D₄	1	6	yes	—
218	C₂²×C₄	C₃×#41	1	4	yes	—
219	C₂²×C₄	C₂⁴×C₃	1	3	yes	—
220	SD₁₆	—	2	64	no	—
221	C₂×C₈	—	2	24	no	—
222	C₂×C₈	C₂⁴×C₃	1	8	yes	—
223	C₄⋊C₄	—	2	16	no	—
224	C₄⋊C₄	—	2	6	no	—
225	C₄⋊C₄	C₂⁴×C₃	1	4	yes	—
226	C₄²	—	2	16	no	—
227	C₄²	C₂×C₃×D₄	1	12	yes	—
228	C₄²	C₂⁴×C₃	1	4	yes	—
229	C₂²×C₃	—	4	8	no	—
230	C₂²×C₃	—	4	8	no	—
231	C₂²×C₃	—	4	8	no	—
232	C₂²×C₃	—	8	48	no	—
233	C₂²×C₃	#460	1	16	yes	direct product factor
234	C₂²×C₃	C₂×#121	1	6	yes	—
235	C₂²×C₃	C₂²×SD₁₆	1	6	yes	—
236	C₂²×C₃	C₂×AGL(1,ℤ/8)	1	6	yes	—
237	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
238	C₃×C₄	—	2	8	no	—
239	C₃×C₄	—	2	4	no	—
240	C₃×C₄	—	4	4	no	—
241	C₃×C₄	C₂×#121	1	6	yes	—
242	C₃×C₄	C₂²×SD₁₆	1	8	yes	complemented
243	C₃×C₄	C₂³×D₄	1	1	yes	characteristic
244	C₃×C₄	C₂²×#41	1	1	yes	characteristic
245	C₂³	—	2	8	no	—
246	C₂³	—	4	24	no	—
247	C₂³	—	4	24	no	—
248	C₂³	—	4	24	no	—
249	C₂³	—	8	32	no	—
250	C₂³	C₃×#121	1	4	yes	—
251	C₂³	C₂×C₃×SD₁₆	1	4	yes	—
252	C₂³	C₃×AGL(1,ℤ/8)	1	4	yes	—
253	C₂³	C₂²×C₃×D₄	1	3	yes	—
254	Q₈	—	2	4	no	—
255	Q₈	—	2	24	no	—
256	Q₈	C₂²×C₃×D₄	1	8	yes	complemented
257	D₄	—	2	8	no	—
258	D₄	—	2	8	no	—
259	D₄	—	4	32	no	—
260	D₄	—	4	32	no	—
261	D₄	—	4	24	no	—
262	C₂×C₄	—	2	12	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	24	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	12	no	—
267	C₂×C₄	C₃×#121	1	12	yes	—
268	C₂×C₄	C₂×C₃×SD₁₆	1	24	yes	complemented
269	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
270	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
271	C₂×C₄	C₂²×C₃×D₄	1	6	yes	—
272	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
273	C₂×C₄	C₂×C₃×#41	1	6	yes	—
274	C₂×C₄	C₂⁵×C₃	1	1	yes	characteristic
275	C₈	—	2	8	no	—
276	C₂×C₃	—	8	8	no	—
277	C₂×C₃	C₂×#460	1	12	yes	direct product factor
278	C₂×C₃	C₂²×#121	1	1	yes	characteristic
279	C₂×C₃	C₂³×SD₁₆	1	1	yes	characteristic
280	C₂×C₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
281	C₂²	—	4	8	no	—
282	C₂²	—	4	8	no	—
283	C₂²	—	4	8	no	—
284	C₂²	—	8	48	no	—
285	C₂²	C₃×#460	1	16	yes	direct product factor
286	C₂²	C₂×C₃×#121	1	6	yes	—
287	C₂²	C₂²×C₃×SD₁₆	1	6	yes	—
288	C₂²	C₂×C₃×AGL(1,ℤ/8)	1	6	yes	—
289	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
290	C₄	—	2	8	no	—
291	C₄	—	2	4	no	—
292	C₄	—	4	4	no	—
293	C₄	C₂×C₃×#121	1	6	yes	—
294	C₄	C₂²×C₃×SD₁₆	1	8	yes	complemented
295	C₄	C₂³×C₃×D₄	1	1	yes	characteristic
296	C₄	C₂²×C₃×#41	1	1	yes	characteristic
297	C₃	C₂²×#460	1	1	yes	characteristic, direct product factor, Sylow
298	C₂	—	8	8	no	—
299	C₂	C₂×C₃×#460	1	12	yes	direct product factor
300	C₂	C₂²×C₃×#121	1	1	yes	characteristic
301	C₂	C₂³×C₃×SD₁₆	1	1	yes	characteristic
302	C₂	C₂²×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃
prime radical	C₂²×C₃×#127	C₂
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. C₂³×C₃×SD₁₆	C₂	semidirect
3. C₂²×C₃×C₄×Q₈	C₂	semidirect
4. C₂×C₃×#460	C₂	direct (N × H)
5. C₂²×C₃×C₄⋊C₈	C₂	semidirect
6. C₂²×C₃×#102	C₂	semidirect
7. C₂²×#460	C₃	direct (N × H)
13. C₂²×C₃×SD₁₆	C₂²	semidirect
18. C₂×C₃×C₄×Q₈	C₂²	semidirect
23. C₃×#460	C₂²	direct (N × H)
24. C₂×C₃×C₄⋊C₈	C₂²	semidirect
25. C₂×C₃×#102	C₂²	semidirect
26. C₂³×SD₁₆	C₂×C₃	semidirect
28. C₂²×C₄×Q₈	C₂×C₃	semidirect
29. C₂×#460	C₂×C₃	direct (N × H)
30. C₂²×C₄⋊C₈	C₂×C₃	semidirect
31. C₂²×#102	C₂×C₃	semidirect
35. C₂²×C₃×Q₈	D₄	semidirect
51. C₂×C₃×SD₁₆	C₂³	semidirect
57. C₃×C₄×Q₈	C₂³	semidirect
64. C₃×C₄⋊C₈	C₂³	semidirect
65. C₃×#102	C₂³	semidirect
71. C₂²×SD₁₆	C₂²×C₃	semidirect
76. C₂×C₄×Q₈	C₂²×C₃	semidirect
81. #460	C₂²×C₃	direct (N × H)
82. C₂×C₄⋊C₈	C₂²×C₃	semidirect
83. C₂×#102	C₂²×C₃	semidirect
92. C₂×C₃×Q₈	C₂×D₄	semidirect
108. C₂²×C₃×C₄	SD₁₆	semidirect
127. C₂²×Q₈	C₃×D₄	semidirect
143. C₂×SD₁₆	C₂³×C₃	semidirect
149. C₄×Q₈	C₂³×C₃	semidirect
156. C₄⋊C₈	C₂³×C₃	semidirect
157. #102	C₂³×C₃	semidirect
169. C₃×Q₈	C₂²×D₄	semidirect
181. C₂×C₃×C₄	C₂×SD₁₆	semidirect
197. C₂×Q₈	C₂×C₃×D₄	semidirect
213. C₂²×C₄	C₃×SD₁₆	semidirect
233. C₂²×C₃	#460	direct (N × H)
242. C₃×C₄	C₂²×SD₁₆	semidirect
256. Q₈	C₂²×C₃×D₄	semidirect
268. C₂×C₄	C₂×C₃×SD₁₆	semidirect
277. C₂×C₃	C₂×#460	direct (N × H)
285. C₂²	C₃×#460	direct (N × H)
294. C₄	C₂²×C₃×SD₁₆	semidirect
297. C₃	C₂²×#460	direct (N × H)
299. C₂	C₂×C₃×#460	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×#127	2
C₂²×C₃×C₄×Q₈	2
C₂×C₃×#460	2
C₂²×C₃×C₄⋊C₈	2
C₂²×C₃×#102	2
C₂²×#460	3

Subgroups — C22×C3×#460 #10679013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#460 #10679013