Subgroups — C₂²×C₃×C₈×D₄ #10678737

Order 768.

3694 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂³×C₃×C₈	C₂²
commutator	C₂	C₂⁴×C₃×C₈
Frattini, cosocle	C₂×C₄	C₂⁵×C₃
socle	C₂⁴×C₃	C₂²×C₄
prime radical	C₂³×C₃×D₄	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃×C₈	C₂	semidirect
3. C₂×C₃×C₈×D₄	C₂	direct (N × H)
4. C₂²×C₃×C₄⋊C₈	C₂	semidirect
5. C₂²×C₃×C₂²⋊C₈	C₂	semidirect
6. C₂²×C₃×C₄×C₈	C₂	semidirect
7. C₂²×C₈×D₄	C₃	direct (N × H)
14. C₂³×C₃×C₈	C₂²	semidirect
20. C₃×C₈×D₄	C₂²	direct (N × H)
21. C₂×C₃×C₄⋊C₈	C₂²	semidirect
22. C₂×C₃×C₂²⋊C₈	C₂²	semidirect
23. C₂×C₃×C₄×C₈	C₂²	semidirect
24. C₂⁴×C₈	C₂×C₃	semidirect
26. C₂×C₈×D₄	C₂×C₃	direct (N × H)
27. C₂²×C₄⋊C₈	C₂×C₃	semidirect
28. C₂²×C₂²⋊C₈	C₂×C₃	semidirect
29. C₂²×C₄×C₈	C₂×C₃	semidirect
31. C₂²×C₃×D₄	C₈	direct (N × H)
44. C₂²×C₃×C₈	D₄	direct (N × H)
48. C₂²×C₃×C₈	C₂³	semidirect
55. C₃×C₄⋊C₈	C₂³	semidirect
56. C₃×C₂²⋊C₈	C₂³	semidirect
57. C₃×C₄×C₈	C₂³	semidirect
64. C₂³×C₈	C₂²×C₃	semidirect
70. C₈×D₄	C₂²×C₃	direct (N × H)
71. C₂×C₄⋊C₈	C₂²×C₃	semidirect
72. C₂×C₂²⋊C₈	C₂²×C₃	semidirect
73. C₂×C₄×C₈	C₂²×C₃	semidirect
76. C₂⁴×C₃	C₂×C₈	semidirect
79. C₂×C₃×D₄	C₂×C₈	direct (N × H)
86. C₂²×C₃×C₄	C₂×C₈	semidirect
99. C₂×C₃×C₈	C₂×D₄	direct (N × H)
108. C₂²×D₄	C₃×C₈	direct (N × H)
121. C₂²×C₈	C₃×D₄	direct (N × H)
125. C₂²×C₈	C₂³×C₃	semidirect
132. C₄⋊C₈	C₂³×C₃	semidirect
133. C₂²⋊C₈	C₂³×C₃	semidirect
134. C₄×C₈	C₂³×C₃	semidirect
140. C₂³×C₃	C₂²×C₈	semidirect
144. C₃×D₄	C₂²×C₈	direct (N × H)
151. C₂×C₃×C₄	C₂²×C₈	semidirect
161. C₃×C₈	C₂²×D₄	direct (N × H)
164. C₂⁴	C₂×C₃×C₈	semidirect
167. C₂×D₄	C₂×C₃×C₈	direct (N × H)
174. C₂²×C₄	C₂×C₃×C₈	semidirect
187. C₂×C₈	C₂×C₃×D₄	direct (N × H)
198. C₂²×C₃	C₈×D₄	direct (N × H)
201. C₂²×C₃	C₂³×C₈	semidirect
207. C₃×C₄	C₂³×C₈	semidirect
215. C₂³	C₂²×C₃×C₈	semidirect
219. D₄	C₂²×C₃×C₈	direct (N × H)
226. C₂×C₄	C₂²×C₃×C₈	semidirect
236. C₈	C₂²×C₃×D₄	direct (N × H)
238. C₂×C₃	C₂×C₈×D₄	direct (N × H)
245. C₂²	C₃×C₈×D₄	direct (N × H)
248. C₂²	C₂³×C₃×C₈	semidirect
254. C₄	C₂³×C₃×C₈	semidirect
257. C₃	C₂²×C₈×D₄	direct (N × H)
259. C₂	C₂×C₃×C₈×D₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×C₈	2
C₂²×C₃×C₄×D₄	2
C₂×C₃×C₈×D₄	2
C₂²×C₃×C₄⋊C₈	2
C₂²×C₃×C₂²⋊C₈	2
C₂²×C₃×C₄×C₈	2
C₂²×C₈×D₄	3

Subgroups — C22×C3×C8×D4 #10678737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₈×D₄ #10678737