Subgroups — Q₈×#13093958 #26235118

Order 12032.

838 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₄⋊Dic₄₇	C₂	1	2	yes	maximal
2	#20244147	C₂	1	6	yes	maximal
3	#20243548	C₂	1	3	yes	maximal
4	C₄×#13093958	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×Dic₄₇	—	2	2	no	—
7	C₂×#13093958	C₂²	1	3	yes	—
8	C₂×#13093958	C₂²	1	6	yes	—
9	C₂×#13093958	C₂²	1	1	yes	characteristic
10	#15689022	—	2	6	no	—
11	#15688993	C₂²	1	6	yes	—
12	C₄×C₄⋊Dic₄₇	C₂²	1	6	yes	—
13	#15688886	C₂²	1	6	yes	—
14	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₄₇	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₈×C₄₇	C₂²	1	3	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
18	Q₈×Dic₄₇	—	4	8	no	—
19	C₂×C₄⋊Dic₄₇	—	2	6	no	—
20	C₂×C₄⋊Dic₄₇	—	2	6	no	—
21	C₂×C₄⋊Dic₄₇	C₂³	1	6	yes	—
22	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
23	C₂×C₄×Dic₄₇	—	2	6	no	—
24	#13093958	—	2	6	no	—
25	#13093958	—	4	6	no	—
26	#13093958	Q₈	1	4	yes	direct product factor
27	C₄₇⋊#95	—	2	6	no	—
28	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
30	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	C₄×Q₈×C₄₇	C₂×C₄	1	4	yes	—
32	C₄×Q₈×C₄₇	Q₈	1	4	yes	—
33	C₂×C₄⋊C₄×C₄₇	D₄	1	3	yes	—
34	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
35	C₄⋊C₈×C₄₇	C₂×C₄	1	12	yes	—
36	C₄×C₈×C₄₇	C₂×C₄	1	12	yes	—
37	C₂×Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
38	C₂×Q₈×C₄₇	D₈	1	4	yes	—
39	C₂×Q₈×C₄₇	Q₁₆	1	4	yes	—
40	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
41	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
42	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
43	C₂²×Dic₄₇	—	2	2	no	—
44	C₂²×Dic₄₇	—	2	6	no	—
45	C₂×C₈×C₄₇	—	2	3	no	—
46	C₂×C₈×C₄₇	—	2	3	no	—
47	C₂×C₈×C₄₇	C₂²×C₄	1	6	yes	—
48	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₄₇	#41	1	2	yes	—
51	C₄⋊C₄×C₄₇	C₄⋊C₄	1	12	yes	—
52	C₄²×C₄₇	C₂²×C₄	1	6	yes	—
53	C₄²×C₄₇	C₂×Q₈	1	6	yes	—
54	C₄⋊Dic₄₇	—	2	2	no	—
55	C₄⋊Dic₄₇	—	2	6	no	—
56	C₄⋊Dic₄₇	—	2	6	no	—
57	C₄⋊Dic₄₇	—	4	12	no	—
58	C₄⋊Dic₄₇	—	8	6	no	—
59	C₄⋊Dic₄₇	C₂×Q₈	1	4	yes	—
60	C₄×Dic₄₇	—	4	12	no	—
61	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
62	Q₈×C₄₇	#107	1	16	yes	complemented
63	C₂×C₄×C₄₇	—	2	3	no	—
64	C₂×C₄×C₄₇	—	2	3	no	—
65	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
66	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
67	C₂×C₄×C₄₇	C₂×D₈	1	6	yes	—
68	C₂×C₄×C₄₇	C₂×Q₁₆	1	6	yes	—
69	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
73	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
74	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
75	C₂×Dic₄₇	—	2	4	no	—
76	C₂×Dic₄₇	—	4	2	no	—
77	C₂×Dic₄₇	—	4	2	no	—
78	C₂×Dic₄₇	—	4	6	no	—
79	C₂×Dic₄₇	—	4	6	no	—
80	C₂×Dic₄₇	—	4	6	no	—
81	C₈×C₄₇	—	2	6	no	—
82	C₈×C₄₇	C₄×Q₈	1	4	yes	complemented
83	Q₈×#107	—	47	1	no	maximal, Sylow
84	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄₇	D₄×Q₈	1	1	yes	characteristic
86	C₂²×C₄₇	#550	1	1	yes	characteristic
87	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
88	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
89	C₂²×C₄₇	#576	1	1	yes	characteristic
90	C₂²×C₄₇	#578	1	1	yes	characteristic
91	C₄×C₄₇	—	2	6	no	—
92	C₄×C₄₇	C₂×#107	1	12	yes	—
93	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
94	C₄×C₄₇	#520	1	1	yes	characteristic
95	C₄×C₄₇	#557	1	1	yes	characteristic
96	C₄×C₄₇	Q₈²	1	1	yes	characteristic
97	Dic₄₇	—	4	4	no	—
98	Dic₄₇	—	8	6	no	—
99	C₂×C₈×Q₈	D₄₇	1	1	yes	characteristic
100	Q₈×C₄⋊C₄	—	47	2	no	—
101	#1947	—	47	6	no	—
102	#1845	—	47	3	no	—
103	C₄×#107	—	47	3	no	—
104	C₂×C₄₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄₇	C₂²×#107	1	1	yes	characteristic
106	C₂×C₄₇	#2913	1	1	yes	characteristic
107	C₂×C₄₇	Q₈×D₈	1	1	yes	characteristic
108	C₂×C₄₇	Q₈×Q₁₆	1	1	yes	characteristic
109	C₂×C₄₇	#3391	1	1	yes	characteristic
110	C₂×C₄₇	#3395	1	1	yes	characteristic
111	C₂×C₄×Q₈	—	94	2	no	—
112	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
113	C₈×Q₈	Dic₄₇	1	8	yes	complemented
114	C₂×#107	—	47	1	no	—
115	C₂×#107	—	47	3	no	—
116	C₂×#107	—	47	6	no	—
117	C₂×C₄⋊C₈	C₂×D₄₇	1	3	yes	—
118	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
119	#389	—	94	6	no	—
120	#384	—	47	6	no	—
121	C₄×C₄⋊C₄	—	47	6	no	—
122	#340	—	47	6	no	—
123	C₄₇	Q₈×#107	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
126	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
127	C₄×Q₈	—	188	8	no	—
128	C₄×Q₈	Dic₉₄	1	4	yes	—
129	C₄×Q₈	C₂×Dic₄₇	1	4	yes	—
130	C₂×C₄⋊C₄	—	47	2	no	—
131	C₂×C₄⋊C₄	—	47	6	no	—
132	C₂×C₄⋊C₄	—	94	6	no	—
133	C₂×C₄⋊C₄	—	94	6	no	—
134	C₂×C₄⋊C₄	D₁₈₈	1	3	yes	—
135	C₂×C₄²	—	94	6	no	—
136	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
137	#107	—	47	4	no	—
138	#107	—	94	6	no	—
139	#107	—	188	6	no	—
140	C₄⋊C₈	C₂×Dic₄₇	1	12	yes	—
141	C₄×C₈	C₂×Dic₄₇	1	12	yes	—
142	#95	—	94	6	no	—
143	C₂×Q₈	D₃₇₆	1	4	yes	—
144	C₂×Q₈	Dic₁₈₈	1	4	yes	—
145	C₂×Q₈	C₄⋊Dic₄₇	1	4	yes	—
146	C₂²×C₄	—	94	2	no	—
147	C₂²×C₄	—	94	6	no	—
148	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
149	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
150	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
151	C₂×C₈	—	2	3	no	—
152	C₂×C₈	—	2	3	no	—
153	C₂×C₈	Q₈×D₄₇	1	2	yes	—
154	C₂×C₈	#10625354	1	2	yes	—
155	C₂×C₈	C₂²×Dic₄₇	1	6	yes	—
156	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
157	C₄⋊C₄	—	47	4	no	—
158	C₄⋊C₄	—	94	2	no	—
159	C₄⋊C₄	—	94	6	no	—
160	C₄⋊C₄	—	94	6	no	—
161	C₄⋊C₄	—	376	6	no	—
162	C₄⋊C₄	—	188	12	no	—
163	C₄⋊C₄	C₄⋊Dic₄₇	1	12	yes	—
164	C₄²	—	188	12	no	—
165	C₄²	C₂×Dic₉₄	1	6	yes	—
166	C₄²	C₂²×Dic₄₇	1	6	yes	—
167	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
168	Q₈	#13093958	1	16	yes	direct product factor
169	C₂×C₄	—	2	3	no	—
170	C₂×C₄	—	2	3	no	—
171	C₂×C₄	—	94	4	no	—
172	C₂×C₄	—	188	6	no	—
173	C₂×C₄	—	188	6	no	—
174	C₂×C₄	—	188	6	no	—
175	C₂×C₄	—	188	2	no	—
176	C₂×C₄	—	188	2	no	—
177	C₂×C₄	C₂×C₄⋊Dic₄₇	1	6	yes	—
178	C₂×C₄	C₂×C₄⋊Dic₄₇	1	6	yes	—
179	C₂×C₄	C₂×D₃₇₆	1	6	yes	—
180	C₂×C₄	C₂×Dic₁₈₈	1	6	yes	—
181	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
182	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
183	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
184	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
185	C₂×C₄	#13094070	1	1	yes	characteristic
186	C₂×C₄	#13094071	1	1	yes	characteristic
187	C₈	—	2	6	no	—
188	C₈	Q₈×Dic₄₇	1	4	yes	complemented
189	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
190	C₂²	Q₈×D₁₈₈	1	1	yes	characteristic
191	C₂²	#15689795	1	1	yes	characteristic
192	C₂²	C₂²×D₃₇₆	1	1	yes	characteristic
193	C₂²	C₂²×Dic₁₈₈	1	1	yes	characteristic
194	C₂²	#15689928	1	1	yes	characteristic
195	C₂²	#15689940	1	1	yes	characteristic
196	C₄	—	2	6	no	—
197	C₄	—	376	6	no	—
198	C₄	—	188	4	no	—
199	C₄	C₂×#13093958	1	12	yes	—
200	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
201	C₄	#15689606	1	1	yes	characteristic
202	C₄	#15689852	1	1	yes	characteristic
203	C₄	Q₈×Dic₉₄	1	1	yes	characteristic
204	C₂	Q₈×C₄⋊Dic₄₇	1	1	yes	characteristic
205	C₂	C₂²×#13093958	1	1	yes	characteristic
206	C₂	#20251988	1	1	yes	characteristic
207	C₂	Q₈×D₃₇₆	1	1	yes	characteristic
208	C₂	Q₈×Dic₁₈₈	1	1	yes	characteristic
209	C₂	#20256363	1	1	yes	characteristic
210	C₂	#20256394	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₈×Q₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₄₇
hypocenter	C₄₇	Q₈×#107
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
15. C₈×Q₈×C₄₇	C₄	semidirect
26. #13093958	Q₈	direct (N × H)
62. Q₈×C₄₇	#107	semidirect
82. C₈×C₄₇	C₄×Q₈	semidirect
113. C₈×Q₈	Dic₄₇	semidirect
123. C₄₇	Q₈×#107	semidirect
168. Q₈	#13093958	direct (N × H)
188. C₈	Q₈×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₄⋊Dic₄₇	2
#20244147	2
#20243548	2
C₄×#13093958	2
C₂×C₈×Q₈×C₄₇	2
Q₈×#107	47

Subgroups — Q8×#13093958 #26235118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13093958 #26235118