Subgroups — Q₈×#13093960 #26235171

Order 12032.

658 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#10625329	C₂	1	2	yes	maximal
2	#20244164	C₂	1	6	yes	maximal
3	#20243560	C₂	1	3	yes	maximal
4	C₄×#13093960	C₂	1	3	yes	maximal
5	C₂×C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15689254	—	2	6	no	—
7	Q₈×C₄₇⋊C₈	—	2	2	no	—
8	C₂×#13093960	C₂²	1	3	yes	—
9	C₂×#13093960	C₂²	1	6	yes	—
10	C₂×#13093960	C₂²	1	1	yes	characteristic
11	#15689147	C₂²	1	6	yes	—
12	C₄×#10625329	C₂²	1	6	yes	—
13	#15688888	C₂²	1	6	yes	—
14	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₄₇	C₄	1	6	yes	—
16	C₈×Q₈×C₄₇	C₄	1	2	yes	—
17	C₂×C₄⋊C₈×C₄₇	C₂²	1	3	yes	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
19	C₂×#10625329	C₂³	1	6	yes	—
20	C₂×#10625329	C₂³	1	2	yes	—
21	#13093960	—	2	6	no	—
22	#13093960	—	4	6	no	—
23	#13093960	Q₈	1	4	yes	direct product factor
24	C₄⋊(C₄₇⋊C₈)	—	2	6	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	6	no	—
26	#13093935	—	2	6	no	—
27	C₄×C₄₇⋊C₈	—	2	6	no	—
28	C₂²×Q₈×C₄₇	Q₈	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	3	yes	—
32	C₄×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
34	C₄×Q₈×C₄₇	D₄	1	3	yes	—
35	C₂×C₄⋊C₄×C₄₇	Q₈	1	3	yes	—
36	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
37	C₄⋊C₈×C₄₇	C₂×C₄	1	6	yes	—
38	C₄⋊C₈×C₄₇	C₂×C₄	1	6	yes	—
39	C₄×C₈×C₄₇	C₂×C₄	1	6	yes	—
40	C₄×C₈×C₄₇	C₂×C₄	1	6	yes	—
41	C₂×Q₈×C₄₇	—	2	4	no	—
42	C₂×Q₈×C₄₇	C₄⋊C₄	1	3	yes	—
43	C₂×Q₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂×Q₈	1	3	yes	—
45	C₂²×C₄×C₄₇	C₂×Q₈	1	3	yes	—
46	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
47	C₂×C₈×C₄₇	—	2	3	no	—
48	C₂×C₈×C₄₇	—	2	3	no	—
49	C₂×C₈×C₄₇	C₂²×C₄	1	6	yes	—
50	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
51	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₄₇	#41	1	2	yes	—
53	C₄⋊C₄×C₄₇	C₄⋊C₄	1	6	yes	—
54	C₄⋊C₄×C₄₇	C₄⋊C₄	1	3	yes	—
55	C₄⋊C₄×C₄₇	C₄⋊C₄	1	3	yes	—
56	C₄²×C₄₇	C₂²×C₄	1	3	yes	—
57	C₄²×C₄₇	C₂²×C₄	1	3	yes	—
58	C₄²×C₄₇	C₂×D₄	1	3	yes	—
59	C₄²×C₄₇	C₂×D₄	1	3	yes	—
60	#10625329	—	2	2	no	—
61	#10625329	—	2	6	no	—
62	#10625329	—	2	6	no	—
63	#10625329	C₂×Q₈	1	4	yes	—
64	C₂×C₄₇⋊C₈	—	2	2	no	—
65	C₂×C₄₇⋊C₈	—	2	6	no	—
66	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
67	Q₈×C₄₇	—	2	6	no	—
68	Q₈×C₄₇	#108	1	4	yes	complemented
69	C₂×C₄×C₄₇	—	2	6	no	—
70	C₂×C₄×C₄₇	—	4	3	no	—
71	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
72	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
73	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
79	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
80	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
81	C₈×C₄₇	—	2	6	no	—
82	C₈×C₄₇	C₄×Q₈	1	4	yes	—
83	C₄₇⋊C₈	—	2	4	no	—
84	C₄₇⋊C₈	—	4	6	no	—
85	Q₈×#108	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	—	2	1	no	—
87	C₂²×C₄₇	—	2	1	no	—
88	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄₇	#557	1	1	yes	characteristic
90	C₂²×C₄₇	Q₈²	1	1	yes	characteristic
91	C₄×C₄₇	—	2	3	no	—
92	C₄×C₄₇	—	2	3	no	—
93	C₄×C₄₇	—	2	3	no	—
94	C₄×C₄₇	C₂×#108	1	6	yes	—
95	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
96	C₄×C₄₇	#520	1	1	yes	characteristic
97	C₄×C₄₇	D₄×Q₈	1	1	yes	characteristic
98	C₄×C₄₇	#550	1	1	yes	characteristic
99	Q₈×M₁₆	—	47	2	no	—
100	C₂×C₈×Q₈	D₄₇	1	1	yes	characteristic
101	#1950	—	47	6	no	—
102	#1848	—	47	3	no	—
103	C₄×#108	—	47	3	no	—
104	C₂×C₄₇	—	2	1	no	—
105	C₂×C₄₇	—	2	1	no	—
106	C₂×C₄₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄₇	C₂²×#108	1	1	yes	characteristic
108	C₂×C₄₇	#2916	1	1	yes	characteristic
109	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
110	#446	—	94	6	no	—
111	C₈×Q₈	—	94	2	no	—
112	C₈×Q₈	Dic₄₇	1	6	yes	—
113	C₈×Q₈	Dic₄₇	1	2	yes	—
114	C₂×#108	—	47	1	no	—
115	C₂×#108	—	47	3	no	—
116	C₂×#108	—	47	6	no	—
117	#423	—	47	6	no	—
118	C₂×C₄⋊C₈	C₂×D₄₇	1	3	yes	—
119	C₄×M₁₆	—	47	6	no	—
120	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
121	#341	—	47	6	no	—
122	C₄₇	Q₈×#108	1	1	yes	characteristic, complemented, Sylow
123	C₂²×Q₈	Dic₉₄	1	1	yes	characteristic
124	C₂×M₁₆	—	47	2	no	—
125	C₂×M₁₆	—	47	6	no	—
126	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
127	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
128	C₄×Q₈	D₁₈₈	1	1	yes	characteristic
129	C₄×Q₈	D₁₈₈	1	3	yes	—
130	C₄×Q₈	C₂×Dic₄₇	1	3	yes	—
131	C₄×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
132	C₂×C₄⋊C₄	Dic₉₄	1	3	yes	—
133	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
134	#108	—	47	4	no	—
135	#108	—	94	6	no	—
136	#108	—	188	6	no	—
137	C₄⋊C₈	—	94	6	no	—
138	C₄⋊C₈	—	94	6	no	—
139	C₄⋊C₈	C₂×Dic₄₇	1	6	yes	—
140	C₄⋊C₈	C₂×Dic₄₇	1	6	yes	—
141	#97	—	94	6	no	—
142	C₄×C₈	—	94	6	no	—
143	C₄×C₈	C₂×Dic₄₇	1	6	yes	—
144	C₄×C₈	C₂×Dic₄₇	1	6	yes	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	C₄⋊Dic₄₇	1	3	yes	—
147	C₂×Q₈	C₄⋊Dic₄₇	1	1	yes	characteristic
148	C₂²×C₄	C₂×Dic₉₄	1	3	yes	—
149	C₂²×C₄	C₂×Dic₉₄	1	3	yes	—
150	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
151	M₁₆	—	47	4	no	—
152	M₁₆	—	94	2	no	—
153	M₁₆	—	94	6	no	—
154	M₁₆	—	94	6	no	—
155	C₂×C₈	—	2	3	no	—
156	C₂×C₈	—	2	3	no	—
157	C₂×C₈	—	94	2	no	—
158	C₂×C₈	—	94	6	no	—
159	C₂×C₈	Q₈×D₄₇	1	2	yes	—
160	C₂×C₈	#10625354	1	2	yes	—
161	C₂×C₈	C₂²×Dic₄₇	1	6	yes	—
162	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
163	C₄⋊C₄	C₄⋊Dic₄₇	1	6	yes	—
164	C₄⋊C₄	C₄⋊Dic₄₇	1	3	yes	—
165	C₄⋊C₄	C₄⋊Dic₄₇	1	3	yes	—
166	C₄²	C₂×D₁₈₈	1	3	yes	—
167	C₄²	C₂×D₁₈₈	1	3	yes	—
168	C₄²	C₂²×Dic₄₇	1	3	yes	—
169	C₄²	C₂²×Dic₄₇	1	3	yes	—
170	C₂³	C₂²×Dic₉₄	1	1	yes	characteristic
171	Q₈	—	2	6	no	—
172	Q₈	#13093960	1	4	yes	direct product factor
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	4	3	no	—
175	C₂×C₄	C₂×C₄⋊Dic₄₇	1	3	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₄₇	1	3	yes	—
177	C₂×C₄	C₂×C₄⋊Dic₄₇	1	3	yes	—
178	C₂×C₄	C₂×C₄⋊Dic₄₇	1	3	yes	—
179	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
180	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
181	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
182	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
183	C₂×C₄	#13094070	1	1	yes	characteristic
184	C₂×C₄	#13094071	1	1	yes	characteristic
185	C₈	—	2	6	no	—
186	C₈	—	94	4	no	—
187	C₈	—	188	6	no	—
188	C₈	Q₈×Dic₄₇	1	4	yes	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
192	C₂²	#15689852	1	1	yes	characteristic
193	C₂²	Q₈×Dic₉₄	1	1	yes	characteristic
194	C₄	—	2	3	no	—
195	C₄	—	2	3	no	—
196	C₄	—	2	3	no	—
197	C₄	C₂×#13093960	1	6	yes	—
198	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
199	C₄	#15689606	1	1	yes	characteristic
200	C₄	Q₈×D₁₈₈	1	1	yes	characteristic
201	C₄	#15689795	1	1	yes	characteristic
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	Q₈×C₄⋊Dic₄₇	1	1	yes	characteristic
205	C₂	C₂²×#13093960	1	1	yes	characteristic
206	C₂	#20251999	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₈×Q₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂³×C₄₇	C₂²×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₄₇
hypocenter	C₄₇	Q₈×#108
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
23. #13093960	Q₈	direct (N × H)
68. Q₈×C₄₇	#108	semidirect
122. C₄₇	Q₈×#108	semidirect
172. Q₈	#13093960	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#10625329	2
#20244164	2
#20243560	2
C₄×#13093960	2
C₂×C₈×Q₈×C₄₇	2
Q₈×#108	47

Subgroups — Q8×#13093960 #26235171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13093960 #26235171