Subgroups — C₂³×#911 #11708977

Order 768.

13310 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₃×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×#237	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#214	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#213	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#911	C₂	1	56	yes	maximal, direct product factor
7	C₂³×#204	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×AGL(1,ℤ/8)	—	3	1	no	maximal, Sylow
10	C₂⁵×D₃	—	2	1	no	—
11	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic
12	C₂³×C₃×D₄	C₂²	1	1	yes	characteristic, complemented
13	C₂³×#67	—	2	1	no	—
14	C₂³×#67	—	2	1	no	—
15	C₂⁴×Dic₃	—	2	1	no	—
16	C₂²×#237	—	2	14	no	—
17	C₂²×#237	C₂²	1	28	yes	complemented
18	C₂²×D₃×D₄	—	2	14	no	—
19	C₂²×D₃×D₄	C₂²	1	28	yes	complemented
20	C₂³×D₁₂	C₂²	1	1	yes	characteristic
21	C₂³×C₄×D₃	C₂²	1	1	yes	characteristic
22	C₂³×Dic₆	C₂²	1	1	yes	characteristic, complemented
23	C₂²×C₃×D₈	C₂²	1	28	yes	complemented
24	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic, complemented
25	C₂²×#214	C₂²	1	28	yes	complemented
26	C₂²×#213	C₂²	1	28	yes	complemented
27	C₂³×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
28	C₂×#911	C₂²	1	448	yes	direct product factor
29	C₂²×#204	C₂²	1	28	yes	complemented
30	C₂²×ΓL(1,𝔽₂₅)	C₂²	1	28	yes	complemented
31	C₂³×#41	—	3	1	no	—
32	C₂⁴×D₄	—	3	1	no	—
33	C₂²×AGL(1,ℤ/8)	—	3	56	no	—
34	C₂³×SD₁₆	—	3	1	no	—
35	C₂³×SD₁₆	—	3	1	no	—
36	C₂³×D₈	—	3	1	no	—
37	C₂³×D₈	D₃	1	1	yes	characteristic, complemented
38	C₂³×M₁₆	—	3	1	no	—
39	C₂⁵×C₃	—	2	1	no	—
40	C₂⁵×C₃	—	2	1	no	—
41	C₂⁴×D₃	—	2	1	no	—
42	C₂⁴×D₃	—	2	28	no	—
43	C₂⁴×D₃	—	4	16	no	—
44	C₂⁴×D₃	D₄	1	1	yes	characteristic
45	C₂²×C₃×D₄	—	2	7	no	—
46	C₂²×C₃×D₄	—	2	7	no	—
47	C₂²×C₃×D₄	C₂³	1	14	yes	—
48	C₂²×C₃×D₄	C₂³	1	14	yes	complemented
49	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
50	C₂²×#67	—	2	28	no	—
51	C₂²×#67	—	2	28	no	—
52	C₂³×Dic₃	—	2	1	no	—
53	C₂³×Dic₃	—	2	28	no	—
54	C₂³×Dic₃	D₄	1	1	yes	characteristic
55	C₂×#237	—	2	168	no	—
56	C₂×#237	C₂³	1	112	yes	complemented
57	C₂×D₃×D₄	—	2	168	no	—
58	C₂×D₃×D₄	C₂³	1	112	yes	complemented
59	C₂²×D₁₂	—	2	7	no	—
60	C₂²×D₁₂	C₂³	1	14	yes	—
61	C₂²×C₄×D₃	D₄	1	14	yes	—
62	C₂²×C₄×D₃	C₂³	1	14	yes	—
63	C₂²×Dic₆	—	2	7	no	—
64	C₂²×Dic₆	C₂³	1	14	yes	complemented
65	C₂×C₃×D₈	C₂³	1	112	yes	complemented
66	C₂²×C₃×C₈	C₂³	1	14	yes	complemented
67	C₂×#214	C₂³	1	112	yes	complemented
68	C₂×#213	C₂³	1	112	yes	complemented
69	C₂²×C₃⋊C₈	C₂³	1	14	yes	complemented
70	#911	C₂³	1	512	yes	direct product factor
71	C₂×#204	C₂³	1	112	yes	complemented
72	C₂×ΓL(1,𝔽₂₅)	C₂³	1	112	yes	complemented
73	C₂⁶	—	6	1	no	—
74	C₂²×#41	—	3	28	no	—
75	C₂²×#41	—	6	14	no	—
76	C₂³×Q₈	—	3	1	no	—
77	C₂³×D₄	—	3	28	no	—
78	C₂³×D₄	—	3	1	no	—
79	C₂³×D₄	—	6	14	no	—
80	C₂³×D₄	—	6	1	no	—
81	C₂³×D₄	—	6	1	no	—
82	C₂³×D₄	C₂×D₃	1	1	yes	characteristic
83	C₂³×D₄	C₂×D₃	1	1	yes	characteristic, complemented
84	C₂⁴×C₄	—	3	1	no	—
85	C₂⁴×C₄	—	6	1	no	—
86	C₂×AGL(1,ℤ/8)	—	3	448	no	—
87	C₂²×SD₁₆	—	3	28	no	—
88	C₂²×SD₁₆	—	3	28	no	—
89	C₂²×D₈	—	3	28	no	—
90	C₂²×D₈	C₂×D₃	1	28	yes	complemented
91	C₂²×M₁₆	—	3	28	no	—
92	C₂³×C₈	—	3	1	no	—
93	C₂³×C₈	C₂×D₃	1	1	yes	characteristic, complemented
94	C₂⁴×C₃	—	2	14	no	—
95	C₂⁴×C₃	—	2	14	no	—
96	C₂⁴×C₃	—	4	8	no	—
97	C₂⁴×C₃	—	4	8	no	—
98	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
99	C₂³×D₃	—	2	8	no	—
100	C₂³×D₃	—	2	14	no	—
101	C₂³×D₃	—	2	112	no	—
102	C₂³×D₃	—	4	224	no	—
103	C₂³×D₃	—	4	8	no	—
104	C₂³×D₃	C₂×D₄	1	14	yes	—
105	C₂×C₃×D₄	—	2	42	no	—
106	C₂×C₃×D₄	—	2	42	no	—
107	C₂×C₃×D₄	C₂⁴	1	28	yes	—
108	C₂×C₃×D₄	C₂⁴	1	28	yes	complemented
109	C₂²×C₃×C₄	C₂×D₄	1	7	yes	—
110	C₂²×C₃×C₄	C₂⁴	1	7	yes	—
111	C₂×#67	—	2	112	no	—
112	C₂×#67	—	2	112	no	—
113	C₂²×Dic₃	—	2	14	no	—
114	C₂²×Dic₃	—	2	112	no	—
115	C₂²×Dic₃	C₂×D₄	1	14	yes	—
116	#237	—	2	224	no	—
117	#237	C₂⁴	1	64	yes	complemented
118	D₃×D₄	—	2	224	no	—
119	D₃×D₄	C₂⁴	1	64	yes	complemented
120	C₂×D₁₂	—	2	42	no	—
121	C₂×D₁₂	C₂⁴	1	28	yes	—
122	C₂×C₄×D₃	C₂×D₄	1	84	yes	—
123	C₂×C₄×D₃	C₂⁴	1	28	yes	—
124	C₂×Dic₆	—	2	42	no	—
125	C₂×Dic₆	C₂⁴	1	28	yes	complemented
126	C₃×D₈	C₂⁴	1	64	yes	complemented
127	C₂×C₃×C₈	C₂⁴	1	28	yes	complemented
128	#214	C₂⁴	1	64	yes	complemented
129	#213	C₂⁴	1	64	yes	complemented
130	C₂×C₃⋊C₈	C₂⁴	1	28	yes	complemented
131	#204	C₂⁴	1	64	yes	complemented
132	ΓL(1,𝔽₂₅)	C₂⁴	1	64	yes	complemented
133	C₂⁵	—	2	1	no	—
134	C₂⁵	—	2	1	no	—
135	C₂⁵	—	3	1	no	—
136	C₂⁵	—	6	28	no	—
137	C₂⁵	—	6	1	no	—
138	C₂⁵	—	12	16	no	—
139	C₂×#41	—	3	112	no	—
140	C₂×#41	—	6	168	no	—
141	C₂²×Q₈	—	3	14	no	—
142	C₂²×Q₈	—	6	7	no	—
143	C₂²×D₄	—	2	7	no	—
144	C₂²×D₄	—	2	7	no	—
145	C₂²×D₄	—	3	112	no	—
146	C₂²×D₄	—	3	14	no	—
147	C₂²×D₄	—	6	168	no	—
148	C₂²×D₄	—	6	28	no	—
149	C₂²×D₄	—	6	28	no	—
150	C₂²×D₄	—	6	7	no	—
151	C₂²×D₄	C₂²×D₃	1	14	yes	—
152	C₂²×D₄	C₂²×D₃	1	14	yes	complemented
153	C₂³×C₄	—	3	1	no	—
154	C₂³×C₄	—	3	14	no	—
155	C₂³×C₄	—	3	14	no	—
156	C₂³×C₄	—	6	28	no	—
157	C₂³×C₄	—	6	1	no	—
158	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
159	AGL(1,ℤ/8)	—	3	512	no	—
160	C₂×SD₁₆	—	3	112	no	—
161	C₂×SD₁₆	—	3	112	no	—
162	C₂×D₈	—	3	112	no	—
163	C₂×D₈	C₂²×D₃	1	112	yes	complemented
164	C₂×M₁₆	—	3	112	no	—
165	C₂²×C₈	—	3	14	no	—
166	C₂²×C₈	C₂²×D₃	1	14	yes	complemented
167	C₂³×C₃	—	2	28	no	—
168	C₂³×C₃	—	2	28	no	—
169	C₂³×C₃	—	4	56	no	—
170	C₂³×C₃	—	4	56	no	—
171	C₂³×C₃	AGL(1,ℤ/8)	1	8	yes	complemented
172	C₂³×C₃	C₂²×D₄	1	7	yes	—
173	C₂²×D₃	—	2	56	no	—
174	C₂²×D₃	—	2	28	no	—
175	C₂²×D₃	—	2	64	no	—
176	C₂²×D₃	—	4	448	no	—
177	C₂²×D₃	—	4	56	no	—
178	C₂²×D₃	C₂²×D₄	1	28	yes	—
179	C₃×D₄	—	2	28	no	—
180	C₃×D₄	—	2	28	no	—
181	C₃×D₄	C₂⁵	1	8	yes	—
182	C₃×D₄	C₂⁵	1	8	yes	complemented
183	C₂×C₃×C₄	C₂²×D₄	1	21	yes	—
184	C₂×C₃×C₄	C₂⁵	1	7	yes	—
185	#67	—	2	64	no	—
186	#67	—	2	64	no	—
187	C₂×Dic₃	—	2	28	no	—
188	C₂×Dic₃	—	2	64	no	—
189	C₂×Dic₃	C₂²×D₄	1	28	yes	—
190	D₁₂	—	2	28	no	—
191	D₁₂	C₂⁵	1	8	yes	—
192	C₄×D₃	C₂²×D₄	1	56	yes	—
193	C₄×D₃	C₂⁵	1	8	yes	—
194	Dic₆	—	2	28	no	—
195	Dic₆	C₂⁵	1	8	yes	complemented
196	C₃×C₈	C₂⁵	1	8	yes	complemented
197	C₃⋊C₈	C₂⁵	1	8	yes	complemented
198	C₂⁴	—	2	14	no	—
199	C₂⁴	—	2	14	no	—
200	C₂⁴	—	3	14	no	—
201	C₂⁴	—	4	8	no	—
202	C₂⁴	—	4	8	no	—
203	C₂⁴	—	6	112	no	—
204	C₂⁴	—	6	8	no	—
205	C₂⁴	—	6	14	no	—
206	C₂⁴	—	12	224	no	—
207	C₂⁴	—	12	8	no	—
208	C₂⁴	D₃×D₄	1	1	yes	characteristic
209	#41	—	3	64	no	—
210	#41	—	6	224	no	—
211	C₂×Q₈	—	3	28	no	—
212	C₂×Q₈	—	6	42	no	—
213	C₂×D₄	—	2	42	no	—
214	C₂×D₄	—	2	42	no	—
215	C₂×D₄	—	3	64	no	—
216	C₂×D₄	—	3	28	no	—
217	C₂×D₄	—	6	224	no	—
218	C₂×D₄	—	6	112	no	—
219	C₂×D₄	—	6	112	no	—
220	C₂×D₄	—	6	42	no	—
221	C₂×D₄	C₂³×D₃	1	28	yes	—
222	C₂×D₄	C₂³×D₃	1	28	yes	complemented
223	C₂²×C₄	—	3	14	no	—
224	C₂²×C₄	—	3	84	no	—
225	C₂²×C₄	—	3	28	no	—
226	C₂²×C₄	—	6	112	no	—
227	C₂²×C₄	—	6	14	no	—
228	C₂²×C₄	D₃×D₄	1	7	yes	—
229	C₂²×C₄	C₂³×D₃	1	7	yes	—
230	SD₁₆	—	3	64	no	—
231	SD₁₆	—	3	64	no	—
232	D₈	—	3	64	no	—
233	D₈	C₂³×D₃	1	64	yes	complemented
234	M₁₆	—	3	64	no	—
235	C₂×C₈	—	3	28	no	—
236	C₂×C₈	C₂³×D₃	1	28	yes	complemented
237	C₂²×C₃	—	2	8	no	—
238	C₂²×C₃	—	2	8	no	—
239	C₂²×C₃	—	4	56	no	—
240	C₂²×C₃	—	4	56	no	—
241	C₂²×C₃	C₂×AGL(1,ℤ/8)	1	28	yes	complemented
242	C₂²×C₃	C₂³×D₄	1	7	yes	—
243	C₂×D₃	—	2	56	no	—
244	C₂×D₃	—	2	8	no	—
245	C₂×D₃	—	4	128	no	—
246	C₂×D₃	—	4	56	no	—
247	C₂×D₃	C₂³×D₄	1	8	yes	—
248	C₃×C₄	C₂³×D₄	1	7	yes	—
249	C₃×C₄	C₂⁶	1	1	yes	characteristic
250	Dic₃	—	2	8	no	—
251	Dic₃	C₂³×D₄	1	8	yes	—
252	C₂³	—	2	28	no	—
253	C₂³	—	2	28	no	—
254	C₂³	—	3	28	no	—
255	C₂³	—	4	56	no	—
256	C₂³	—	4	56	no	—
257	C₂³	—	6	64	no	—
258	C₂³	—	6	56	no	—
259	C₂³	—	6	28	no	—
260	C₂³	—	12	448	no	—
261	C₂³	—	12	56	no	—
262	C₂³	#911	1	8	yes	direct product factor
263	C₂³	C₂×D₃×D₄	1	7	yes	—
264	Q₈	—	3	8	no	—
265	Q₈	—	6	28	no	—
266	D₄	—	2	28	no	—
267	D₄	—	2	28	no	—
268	D₄	—	3	8	no	—
269	D₄	—	6	64	no	—
270	D₄	—	6	64	no	—
271	D₄	—	6	28	no	—
272	D₄	C₂⁴×D₃	1	8	yes	—
273	D₄	C₂⁴×D₃	1	8	yes	complemented
274	C₂×C₄	—	3	28	no	—
275	C₂×C₄	—	3	56	no	—
276	C₂×C₄	—	3	8	no	—
277	C₂×C₄	—	6	64	no	—
278	C₂×C₄	—	6	28	no	—
279	C₂×C₄	C₂×D₃×D₄	1	21	yes	—
280	C₂×C₄	C₂⁴×D₃	1	7	yes	—
281	C₈	—	3	8	no	—
282	C₈	C₂⁴×D₃	1	8	yes	complemented
283	C₂×C₃	—	4	8	no	—
284	C₂×C₃	—	4	8	no	—
285	C₂×C₃	C₂²×AGL(1,ℤ/8)	1	14	yes	complemented
286	C₂×C₃	C₂⁴×D₄	1	1	yes	characteristic
287	D₃	—	2	8	no	—
288	D₃	—	4	8	no	—
289	C₂²	—	2	8	no	—
290	C₂²	—	2	8	no	—
291	C₂²	—	3	8	no	—
292	C₂²	—	4	56	no	—
293	C₂²	—	4	56	no	—
294	C₂²	—	6	56	no	—
295	C₂²	—	6	8	no	—
296	C₂²	—	12	128	no	—
297	C₂²	—	12	56	no	—
298	C₂²	C₂×#911	1	28	yes	direct product factor
299	C₂²	C₂²×D₃×D₄	1	7	yes	—
300	C₄	—	3	8	no	—
301	C₄	—	6	8	no	—
302	C₄	C₂²×D₃×D₄	1	7	yes	—
303	C₄	C₂⁵×D₃	1	1	yes	characteristic
304	C₃	C₂³×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
305	C₂	—	4	8	no	—
306	C₂	—	4	8	no	—
307	C₂	—	6	8	no	—
308	C₂	—	12	8	no	—
309	C₂	C₂²×#911	1	14	yes	direct product factor
310	C₂	C₂³×D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁶
Frattini	C₄	C₂⁵×D₃
Fitting	C₂³×C₃×D₈	C₂
socle	C₂⁴×C₃	C₂×D₄
hypercenter	C₂³×D₈	D₃
hypocenter	C₃	C₂³×AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×D₈

Complements

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

N	H ≅ G/N	type
1. C₂³×D₃×D₄	C₂	semidirect
2. C₂³×#237	C₂	semidirect
3. C₂³×C₃×D₈	C₂	semidirect
4. C₂³×#214	C₂	semidirect
5. C₂³×#213	C₂	semidirect
6. C₂²×#911	C₂	direct (N × H)
7. C₂³×#204	C₂	semidirect
8. C₂³×ΓL(1,𝔽₂₅)	C₂	semidirect
12. C₂³×C₃×D₄	C₂²	semidirect
17. C₂²×#237	C₂²	semidirect
19. C₂²×D₃×D₄	C₂²	semidirect
22. C₂³×Dic₆	C₂²	semidirect
23. C₂²×C₃×D₈	C₂²	semidirect
24. C₂³×C₃×C₈	C₂²	semidirect
25. C₂²×#214	C₂²	semidirect
26. C₂²×#213	C₂²	semidirect
27. C₂³×C₃⋊C₈	C₂²	semidirect
28. C₂×#911	C₂²	direct (N × H)
29. C₂²×#204	C₂²	semidirect
30. C₂²×ΓL(1,𝔽₂₅)	C₂²	semidirect
37. C₂³×D₈	D₃	semidirect
48. C₂²×C₃×D₄	C₂³	semidirect
56. C₂×#237	C₂³	semidirect
58. C₂×D₃×D₄	C₂³	semidirect
64. C₂²×Dic₆	C₂³	semidirect
65. C₂×C₃×D₈	C₂³	semidirect
66. C₂²×C₃×C₈	C₂³	semidirect
67. C₂×#214	C₂³	semidirect
68. C₂×#213	C₂³	semidirect
69. C₂²×C₃⋊C₈	C₂³	semidirect
70. #911	C₂³	direct (N × H)
71. C₂×#204	C₂³	semidirect
72. C₂×ΓL(1,𝔽₂₅)	C₂³	semidirect
83. C₂³×D₄	C₂×D₃	semidirect
90. C₂²×D₈	C₂×D₃	semidirect
93. C₂³×C₈	C₂×D₃	semidirect
108. C₂×C₃×D₄	C₂⁴	semidirect
117. #237	C₂⁴	semidirect
119. D₃×D₄	C₂⁴	semidirect
125. C₂×Dic₆	C₂⁴	semidirect
126. C₃×D₈	C₂⁴	semidirect
127. C₂×C₃×C₈	C₂⁴	semidirect
128. #214	C₂⁴	semidirect
129. #213	C₂⁴	semidirect
130. C₂×C₃⋊C₈	C₂⁴	semidirect
131. #204	C₂⁴	semidirect
132. ΓL(1,𝔽₂₅)	C₂⁴	semidirect
152. C₂²×D₄	C₂²×D₃	semidirect
163. C₂×D₈	C₂²×D₃	semidirect
166. C₂²×C₈	C₂²×D₃	semidirect
171. C₂³×C₃	AGL(1,ℤ/8)	semidirect
182. C₃×D₄	C₂⁵	semidirect
195. Dic₆	C₂⁵	semidirect
196. C₃×C₈	C₂⁵	semidirect
197. C₃⋊C₈	C₂⁵	semidirect
222. C₂×D₄	C₂³×D₃	semidirect
233. D₈	C₂³×D₃	semidirect
236. C₂×C₈	C₂³×D₃	semidirect
241. C₂²×C₃	C₂×AGL(1,ℤ/8)	semidirect
262. C₂³	#911	direct (N × H)
273. D₄	C₂⁴×D₃	semidirect
282. C₈	C₂⁴×D₃	semidirect
285. C₂×C₃	C₂²×AGL(1,ℤ/8)	semidirect
298. C₂²	C₂×#911	direct (N × H)
304. C₃	C₂³×AGL(1,ℤ/8)	semidirect
309. C₂	C₂²×#911	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×D₃×D₄	2
C₂³×#237	2
C₂³×C₃×D₈	2
C₂³×#214	2
C₂³×#213	2
C₂²×#911	2
C₂³×#204	2
C₂³×ΓL(1,𝔽₂₅)	2
C₂³×AGL(1,ℤ/8)	3

Subgroups — C23×#911 #11708977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#911 #11708977