Subgroups — C₂×Q₈×D₃₇₆ #27096693

Order 12032.

2286 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₁₈₈	C₂	1	2	yes	maximal, complemented
2	Q₈×D₃₇₆	C₂	1	16	yes	maximal, direct product factor
3	C₂×#15689528	C₂	1	3	yes	maximal, complemented
4	C₂×#15689404	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×D₃₇₆	C₂	1	3	yes	maximal
6	C₂×C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×D₄₇	—	2	2	no	—
8	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
9	Q₈×D₁₈₈	—	2	8	no	—
10	Q₈×D₁₈₈	C₂²	1	16	yes	complemented
11	C₂×#13094024	C₂²	1	6	yes	—
12	C₂×#13094004	—	2	6	no	—
13	C₂×C₄×D₁₈₈	C₂²	1	6	yes	—
14	#15689528	C₂²	1	24	yes	complemented
15	#15689404	C₂²	1	48	yes	complemented
16	C₄×D₃₇₆	C₂²	1	24	yes	—
17	C₂×#13093958	C₂²	1	3	yes	—
18	C₂×#13093958	C₂²	1	6	yes	—
19	C₂×#13093946	C₂²	1	6	yes	—
20	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₄₇	C₂²	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₄₇	C₂²	1	3	yes	—
23	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
24	C₂²×Dic₉₄	—	2	6	no	—
25	C₂×Q₈×D₄₇	—	2	16	no	—
26	C₂×Q₈×D₄₇	—	4	16	no	—
27	C₂²×D₁₈₈	C₂³	1	2	yes	—
28	C₂²×C₄×D₄₇	—	2	6	no	—
29	C₂×D₃₇₆	Q₈	1	8	yes	direct product factor
30	C₂×D₃₇₆	C₂³	1	4	yes	—
31	#13094024	—	2	12	no	—
32	#13094024	C₂³	1	24	yes	—
33	#13094004	—	2	48	no	—
34	C₄×D₁₈₈	—	2	12	no	—
35	C₄×D₁₈₈	C₂³	1	24	yes	—
36	C₂×C₄⋊Dic₄₇	—	2	6	no	—
37	C₂×C₄⋊Dic₄₇	C₂³	1	6	yes	—
38	C₂×(C₂×D₄₇)⋊C₄	—	2	6	no	—
39	#13093958	C₂³	1	12	yes	—
40	#13093958	C₂³	1	24	yes	—
41	#13093946	C₂³	1	24	yes	—
42	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
44	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₄₇	D₄	1	4	yes	—
46	C₄×Q₈×C₄₇	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₄₇	D₄	1	3	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
49	C₄⋊C₈×C₄₇	C₂³	1	12	yes	—
50	C₄×C₈×C₄₇	C₂³	1	12	yes	—
51	C₂³×D₄₇	—	2	2	no	—
52	C₂×Q₈×C₄₇	D₈	1	8	yes	complemented
53	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
57	C₂²×Dic₄₇	—	2	6	no	—
58	Q₈×D₄₇	—	4	64	no	—
59	C₂×D₁₈₈	—	2	2	no	—
60	C₂×D₁₈₈	—	2	4	no	—
61	C₂×D₁₈₈	C₂×Q₈	1	8	yes	complemented
62	C₂×D₁₈₈	C₂⁴	1	4	yes	—
63	C₂×C₄×D₄₇	—	2	24	no	—
64	C₂×C₄×D₄₇	—	4	24	no	—
65	C₂×Dic₉₄	—	2	24	no	—
66	C₂×Dic₉₄	—	4	24	no	—
67	C₂×C₈×C₄₇	—	2	6	no	—
68	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	complemented
69	C₂×C₈×C₄₇	C₂⁴	1	6	yes	—
70	C₂×C₈×C₄₇	C₂⁴	1	2	yes	—
71	C₄⋊C₄×C₄₇	C₂×D₄	1	12	yes	—
72	C₄²×C₄₇	C₂×D₄	1	6	yes	—
73	C₄²×C₄₇	C₂⁴	1	6	yes	—
74	(C₂×D₄₇)⋊C₄	—	2	24	no	—
75	C₄⋊Dic₄₇	—	2	24	no	—
76	C₄⋊Dic₄₇	—	2	6	no	—
77	C₄⋊Dic₄₇	C₂⁴	1	12	yes	—
78	D₃₇₆	C₂×Q₈	1	16	yes	direct product factor
79	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
80	C₂²×D₄₇	—	2	8	no	—
81	C₂²×D₄₇	—	2	4	no	—
82	C₂²×D₄₇	—	4	4	no	—
83	C₂²×D₄₇	—	4	4	no	—
84	Q₈×C₄₇	C₂×D₈	1	16	yes	complemented
85	C₂×C₄×C₄₇	—	2	6	no	—
86	C₂×C₄×C₄₇	C₂×D₈	1	12	yes	—
87	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
88	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂²×D₄	1	6	yes	—
90	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
91	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
92	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
93	C₂×Dic₄₇	—	2	12	no	—
94	C₂×Dic₄₇	—	4	12	no	—
95	D₁₈₈	—	2	8	no	—
96	D₁₈₈	—	2	4	no	—
97	D₁₈₈	C₂²×Q₈	1	8	yes	complemented
98	C₄×D₄₇	—	4	48	no	—
99	Dic₉₄	—	4	48	no	—
100	C₈×C₄₇	—	2	6	no	—
101	C₈×C₄₇	C₂²×Q₈	1	4	yes	complemented
102	C₂×Q₈×D₈	—	47	1	no	maximal, Sylow
103	C₂²×C₄₇	D₄×Q₈	1	2	yes	—
104	C₂²×C₄₇	C₂²×D₈	1	2	yes	—
105	C₂²×C₄₇	#578	1	2	yes	—
106	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
107	C₂×D₄₇	—	2	8	no	—
108	C₂×D₄₇	—	4	16	no	—
109	C₂×D₄₇	—	4	4	no	—
110	C₂×D₄₇	—	4	4	no	—
111	C₄×C₄₇	—	2	6	no	—
112	C₄×C₄₇	D₄×Q₈	1	2	yes	—
113	C₄×C₄₇	C₂²×D₈	1	12	yes	—
114	C₄×C₄₇	C₂³×Q₈	1	1	yes	characteristic
115	C₄×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
116	Dic₄₇	—	4	12	no	—
117	C₂×D₄×Q₈	—	47	2	no	—
118	Q₈×D₈	—	47	16	no	—
119	C₂×#500	—	47	3	no	—
120	C₂×#474	—	47	6	no	—
121	C₂×C₈×Q₈	D₄₇	1	1	yes	characteristic, complemented
122	C₂×C₄×D₈	—	47	3	no	—
123	C₂×C₄₇	Q₈×D₈	1	4	yes	complemented
124	C₂×C₄₇	C₂×D₄×Q₈	1	1	yes	characteristic
125	C₂×C₄₇	C₂³×D₈	1	1	yes	characteristic
126	C₂×C₄₇	C₂×#578	1	1	yes	characteristic
127	D₄₇	—	4	8	no	—
128	C₂³×Q₈	—	94	2	no	—
129	C₂²×D₈	—	47	1	no	—
130	D₄×Q₈	—	47	16	no	—
131	D₄×Q₈	—	94	8	no	—
132	C₂×C₄⋊Q₈	—	47	6	no	—
133	C₂×C₂²⋊Q₈	—	94	6	no	—
134	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
135	C₂×C₄×D₄	—	47	6	no	—
136	#500	—	47	24	no	—
137	#474	—	47	48	no	—
138	C₈×Q₈	C₂×D₄₇	1	8	yes	complemented
139	C₄×D₈	—	47	24	no	—
140	C₂×#107	—	47	3	no	—
141	C₂×#107	—	47	6	no	—
142	C₂×C₄⋊C₈	C₂×D₄₇	1	3	yes	—
143	C₂×#102	—	47	6	no	—
144	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
145	C₄₇	C₂×Q₈×D₈	1	1	yes	characteristic, complemented, Sylow
146	C₂²×Q₈	—	94	16	no	—
147	C₂²×Q₈	—	94	6	no	—
148	C₂²×Q₈	—	188	16	no	—
149	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
150	C₂²×D₄	—	47	2	no	—
151	C₂³×C₄	—	94	6	no	—
152	C₂×D₈	—	47	8	no	—
153	C₂×D₈	—	47	4	no	—
154	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
155	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
156	C₄⋊Q₈	—	47	24	no	—
157	C₄⋊Q₈	—	94	12	no	—
158	C₂²⋊Q₈	—	94	48	no	—
159	C₄×Q₈	D₁₈₈	1	4	yes	—
160	C₄×Q₈	C₂²×D₄₇	1	4	yes	—
161	C₄×D₄	—	47	24	no	—
162	C₄×D₄	—	94	12	no	—
163	C₂×C₄⋊C₄	—	47	6	no	—
164	C₂×C₄⋊C₄	—	94	6	no	—
165	C₂×C₄⋊C₄	D₁₈₈	1	3	yes	—
166	C₂×C₂²⋊C₄	—	94	6	no	—
167	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
168	#107	—	47	12	no	—
169	#107	—	47	24	no	—
170	C₄⋊C₈	C₂²×D₄₇	1	12	yes	—
171	#102	—	47	24	no	—
172	C₄×C₈	C₂²×D₄₇	1	12	yes	—
173	C₂⁴	—	94	2	no	—
174	C₂×Q₈	—	94	24	no	—
175	C₂×Q₈	—	188	64	no	—
176	C₂×Q₈	—	188	24	no	—
177	C₂×Q₈	D₃₇₆	1	8	yes	direct product factor
178	C₂×Q₈	C₂×D₁₈₈	1	4	yes	—
179	C₂×D₄	—	47	8	no	—
180	C₂×D₄	—	47	4	no	—
181	C₂×D₄	—	94	2	no	—
182	C₂×D₄	—	94	4	no	—
183	C₂²×C₄	—	94	6	no	—
184	C₂²×C₄	—	94	24	no	—
185	C₂²×C₄	—	188	24	no	—
186	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
187	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
188	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
189	D₈	—	47	16	no	—
190	C₂×C₈	—	2	6	no	—
191	C₂×C₈	Q₈×D₄₇	1	4	yes	complemented
192	C₂×C₈	C₂³×D₄₇	1	6	yes	—
193	C₂×C₈	C₂³×D₄₇	1	2	yes	—
194	C₄⋊C₄	—	47	12	no	—
195	C₄⋊C₄	—	94	6	no	—
196	C₄⋊C₄	—	94	24	no	—
197	C₄⋊C₄	C₂×D₁₈₈	1	12	yes	—
198	C₂²⋊C₄	—	94	24	no	—
199	C₄²	C₂×D₁₈₈	1	6	yes	—
200	C₄²	C₂³×D₄₇	1	6	yes	—
201	C₂³	—	94	8	no	—
202	C₂³	—	94	4	no	—
203	C₂³	—	188	4	no	—
204	C₂³	—	188	4	no	—
205	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
206	Q₈	—	188	48	no	—
207	Q₈	C₂×D₃₇₆	1	16	yes	direct product factor
208	D₄	—	47	8	no	—
209	D₄	—	94	8	no	—
210	D₄	—	94	4	no	—
211	C₂×C₄	—	2	6	no	—
212	C₂×C₄	—	94	12	no	—
213	C₂×C₄	—	188	12	no	—
214	C₂×C₄	—	188	48	no	—
215	C₂×C₄	C₂×D₃₇₆	1	12	yes	—
216	C₂×C₄	C₂²×D₁₈₈	1	6	yes	—
217	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
218	C₂×C₄	C₂²×D₁₈₈	1	6	yes	—
219	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
220	C₂×C₄	#13094071	1	2	yes	—
221	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
222	C₈	—	2	6	no	—
223	C₈	C₂×Q₈×D₄₇	1	4	yes	complemented
224	C₂²	—	94	8	no	—
225	C₂²	—	188	4	no	—
226	C₂²	—	188	4	no	—
227	C₂²	—	188	16	no	—
228	C₂²	Q₈×D₁₈₈	1	2	yes	—
229	C₂²	C₂²×D₃₇₆	1	2	yes	—
230	C₂²	#15689940	1	2	yes	—
231	C₂²	C₂³×D₁₈₈	1	1	yes	characteristic
232	C₄	—	2	6	no	—
233	C₄	—	188	12	no	—
234	C₄	Q₈×D₁₈₈	1	2	yes	—
235	C₄	C₂²×D₃₇₆	1	12	yes	—
236	C₄	C₂²×Q₈×D₄₇	1	1	yes	characteristic
237	C₄	C₂×#13094071	1	1	yes	characteristic
238	C₂	—	188	8	no	—
239	C₂	Q₈×D₃₇₆	1	4	yes	direct product factor
240	C₂	C₂×Q₈×D₁₈₈	1	1	yes	characteristic
241	C₂	C₂³×D₃₇₆	1	1	yes	characteristic
242	C₂	C₂×#15689940	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂×C₈×Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂²×D₃₇₆	C₂²
hypercenter	C₂×C₈×Q₈	D₄₇
hypocenter	C₄₇	C₂×Q₈×D₈
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₁₈₈	C₂	semidirect
2. Q₈×D₃₇₆	C₂	direct (N × H)
3. C₂×#15689528	C₂	semidirect
4. C₂×#15689404	C₂	semidirect
6. C₂×C₈×Q₈×C₄₇	C₂	semidirect
10. Q₈×D₁₈₈	C₂²	semidirect
14. #15689528	C₂²	semidirect
15. #15689404	C₂²	semidirect
21. C₈×Q₈×C₄₇	C₂²	semidirect
29. C₂×D₃₇₆	Q₈	direct (N × H)
52. C₂×Q₈×C₄₇	D₈	semidirect
61. C₂×D₁₈₈	C₂×Q₈	semidirect
68. C₂×C₈×C₄₇	C₂×Q₈	semidirect
78. D₃₇₆	C₂×Q₈	direct (N × H)
84. Q₈×C₄₇	C₂×D₈	semidirect
97. D₁₈₈	C₂²×Q₈	semidirect
101. C₈×C₄₇	C₂²×Q₈	semidirect
121. C₂×C₈×Q₈	D₄₇	semidirect
123. C₂×C₄₇	Q₈×D₈	semidirect
138. C₈×Q₈	C₂×D₄₇	semidirect
145. C₄₇	C₂×Q₈×D₈	semidirect
177. C₂×Q₈	D₃₇₆	direct (N × H)
191. C₂×C₈	Q₈×D₄₇	semidirect
207. Q₈	C₂×D₃₇₆	direct (N × H)
223. C₈	C₂×Q₈×D₄₇	semidirect
239. C₂	Q₈×D₃₇₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₁₈₈	2
Q₈×D₃₇₆	2
C₂×#15689528	2
C₂×#15689404	2
C₂×C₄×D₃₇₆	2
C₂×C₈×Q₈×C₄₇	2
C₂×Q₈×D₈	47

Subgroups — C2×Q8×D376 #27096693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×D₃₇₆ #27096693