Subgroups — Q₈×#6386 #13507310

Order 1792.

530 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#11728703	C₂	1	6	yes	maximal, complemented
3	Q₈×C₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	#11727497	C₂	1	3	yes	maximal
5	C₄×#6386	C₂	1	3	yes	maximal
6	C₂×C₄×C₇×Q₈	C₄	1	1	yes	characteristic
7	C₇×C₈×Q₈	C₄	1	3	yes	—
8	C₇×C₈×Q₈	C₄	1	1	yes	characteristic
9	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
10	C₇×C₈×Q₈	C₂²	1	3	yes	—
11	C₂×C₇×C₄⋊C₈	C₄	1	3	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	3	yes	—
13	C₂×#6386	C₂²	1	3	yes	—
14	C₂×#6386	C₂²	1	1	yes	characteristic
15	C₄⋊(C₇⋊C₁₆)	C₂²	1	6	yes	—
16	C₄⋊(C₇⋊C₁₆)	C₂²	1	6	yes	—
17	#89256	C₂²	1	6	yes	—
18	C₄×C₇⋊C₁₆	C₂²	1	6	yes	—
19	Q₈×M₃₂	—	7	1	no	maximal, Sylow
20	C₂²×C₇×Q₈	C₈	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂×C₄	1	3	yes	—
22	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
23	C₄×C₇×Q₈	C₈	1	3	yes	—
24	C₄×C₇×Q₈	C₈	1	1	yes	characteristic
25	C₄×C₇×Q₈	C₂×C₄	1	3	yes	—
26	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₇×C₄⋊C₄	C₈	1	3	yes	—
28	C₂×C₄²×C₇	C₂×C₄	1	3	yes	—
29	C₇×C₄⋊C₈	C₂×C₄	1	3	yes	—
30	C₇×C₄⋊C₈	C₂×C₄	1	6	yes	—
31	C₇×C₄⋊C₈	C₂×C₄	1	3	yes	—
32	C₄×C₇×C₈	C₂×C₄	1	3	yes	—
33	C₄×C₇×C₈	C₂×C₄	1	3	yes	—
34	C₄×C₇×C₈	C₂³	1	3	yes	—
35	C₄×C₇×C₈	C₂³	1	3	yes	—
36	#6386	—	2	6	no	—
37	#6386	Q₈	1	4	yes	direct product factor
38	C₂×C₇⋊C₁₆	C₂³	1	6	yes	—
39	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
40	C₂×C₈×Q₈	D₇	1	1	yes	characteristic
41	#2183	—	7	6	no	—
42	Q₈×C₁₆	—	7	2	no	—
43	#2150	—	7	3	no	—
44	C₄×M₃₂	—	7	3	no	—
45	C₂×C₇×Q₈	—	2	4	no	—
46	C₂×C₇×Q₈	C₂×C₈	1	3	yes	—
47	C₂×C₇×Q₈	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂×C₈	1	3	yes	—
49	C₂²×C₄×C₇	C₂×C₈	1	3	yes	—
50	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₇×C₈	—	2	6	no	—
52	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
53	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
54	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
56	C₂×C₇×C₈	#41	1	2	yes	—
57	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
58	C₇×C₄⋊C₄	C₂×C₈	1	6	yes	—
59	C₇×C₄⋊C₄	C₂×C₈	1	3	yes	—
60	C₇×C₄⋊C₄	C₂×C₈	1	3	yes	—
61	C₄²×C₇	C₂×C₈	1	3	yes	—
62	C₄²×C₇	C₂×C₈	1	3	yes	—
63	C₄²×C₇	C₂²×C₄	1	3	yes	—
64	C₄²×C₇	C₂²×C₄	1	3	yes	—
65	C₇⋊C₁₆	—	2	6	no	—
66	C₇⋊C₁₆	C₂×Q₈	1	4	yes	complemented
67	C₂×C₄×Q₈	Dic₇	1	1	yes	characteristic
68	C₂×M₃₂	—	7	1	no	—
69	C₂×M₃₂	—	7	3	no	—
70	C₈×Q₈	Dic₇	1	3	yes	—
71	C₈×Q₈	Dic₇	1	1	yes	characteristic
72	C₈×Q₈	C₂×D₇	1	1	yes	characteristic
73	C₈×Q₈	C₂×D₇	1	3	yes	—
74	C₂×C₄⋊C₈	Dic₇	1	3	yes	—
75	C₂×C₄×C₈	C₂×D₇	1	3	yes	—
76	C₄⋊C₁₆	—	7	6	no	—
77	C₄⋊C₁₆	—	7	6	no	—
78	#346	—	7	6	no	—
79	C₄×C₁₆	—	7	6	no	—
80	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
81	C₇×Q₈	—	2	6	no	—
82	C₇×Q₈	M₃₂	1	4	yes	complemented
83	C₂×C₄×C₇	—	2	6	no	—
84	C₂×C₄×C₇	—	4	3	no	—
85	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
86	C₂×C₄×C₇	C₂²×C₈	1	3	yes	—
87	C₂×C₄×C₇	C₂²×C₈	1	3	yes	—
88	C₂×C₄×C₇	C₂²×C₈	1	3	yes	—
89	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂²×C₈	1	3	yes	—
91	C₂×C₄×C₇	#131	1	2	yes	—
92	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
93	C₇×C₈	—	2	3	no	—
94	C₇×C₈	—	2	3	no	—
95	C₇×C₈	C₄×Q₈	1	2	yes	—
96	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
97	C₇×C₈	C₂×#41	1	1	yes	characteristic
98	C₂²×Q₈	C₇⋊C₈	1	1	yes	characteristic
99	C₂²×C₈	C₂×Dic₇	1	3	yes	—
100	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
101	C₄×Q₈	C₇⋊C₈	1	3	yes	—
102	C₄×Q₈	C₇⋊C₈	1	1	yes	characteristic
103	C₄×Q₈	C₂×Dic₇	1	3	yes	—
104	C₄×Q₈	C₂×Dic₇	1	1	yes	characteristic
105	C₂×C₄⋊C₄	C₇⋊C₈	1	3	yes	—
106	C₂×C₄²	C₂×Dic₇	1	3	yes	—
107	M₃₂	—	7	4	no	—
108	M₃₂	—	14	6	no	—
109	C₂×C₁₆	—	7	2	no	—
110	C₂×C₁₆	—	7	6	no	—
111	C₄⋊C₈	C₂×Dic₇	1	3	yes	—
112	C₄⋊C₈	C₂×Dic₇	1	6	yes	—
113	C₄⋊C₈	C₂×Dic₇	1	3	yes	—
114	C₄×C₈	C₂×Dic₇	1	3	yes	—
115	C₄×C₈	C₂×Dic₇	1	3	yes	—
116	C₄×C₈	C₂²×D₇	1	3	yes	—
117	C₄×C₈	C₂²×D₇	1	3	yes	—
118	C₂²×C₇	—	2	1	no	—
119	C₂²×C₇	—	2	1	no	—
120	C₂²×C₇	C₈×Q₈	1	2	yes	—
121	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
122	C₄×C₇	—	2	3	no	—
123	C₄×C₇	—	2	3	no	—
124	C₄×C₇	—	2	3	no	—
125	C₄×C₇	C₈×Q₈	1	2	yes	—
126	C₄×C₇	C₂×M₃₂	1	6	yes	—
127	C₄×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
128	C₄×C₇	C₂×#131	1	1	yes	characteristic
129	C₂×Q₈	—	2	4	no	—
130	C₂×Q₈	C₂×C₇⋊C₈	1	3	yes	—
131	C₂×Q₈	C₂×C₇⋊C₈	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₇⋊C₈	1	3	yes	—
133	C₂²×C₄	C₂×C₇⋊C₈	1	3	yes	—
134	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
135	C₂×C₈	—	2	6	no	—
136	C₂×C₈	Q₈×D₇	1	2	yes	—
137	C₂×C₈	#1161	1	2	yes	—
138	C₂×C₈	C₂²×Dic₇	1	3	yes	—
139	C₂×C₈	C₂²×Dic₇	1	3	yes	—
140	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
141	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
142	C₄⋊C₄	C₂×C₇⋊C₈	1	6	yes	—
143	C₄⋊C₄	C₂×C₇⋊C₈	1	3	yes	—
144	C₄⋊C₄	C₂×C₇⋊C₈	1	3	yes	—
145	C₄²	C₂×C₇⋊C₈	1	3	yes	—
146	C₄²	C₂×C₇⋊C₈	1	3	yes	—
147	C₄²	C₂²×Dic₇	1	3	yes	—
148	C₄²	C₂²×Dic₇	1	3	yes	—
149	C₁₆	—	7	4	no	—
150	C₁₆	—	14	6	no	—
151	C₂×C₇	—	2	2	no	—
152	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
153	C₂×C₇	C₂²×M₃₂	1	1	yes	characteristic
154	C₂×C₇	#3407	1	1	yes	characteristic
155	C₂³	C₂²×C₇⋊C₈	1	1	yes	characteristic
156	Q₈	—	2	6	no	—
157	Q₈	#6386	1	4	yes	direct product factor
158	C₂×C₄	—	2	6	no	—
159	C₂×C₄	—	4	3	no	—
160	C₂×C₄	C₂²×C₇⋊C₈	1	3	yes	—
161	C₂×C₄	C₂²×C₇⋊C₈	1	3	yes	—
162	C₂×C₄	C₂²×C₇⋊C₈	1	3	yes	—
163	C₂×C₄	C₂²×C₇⋊C₈	1	3	yes	—
164	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
165	C₂×C₄	Q₈×Dic₇	1	2	yes	—
166	C₂×C₄	#6511	1	2	yes	—
167	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
168	C₈	—	2	3	no	—
169	C₈	—	2	3	no	—
170	C₈	Q₈×Dic₇	1	2	yes	—
171	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
172	C₈	C₂×#1161	1	1	yes	characteristic
173	C₇	Q₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	Q₈×C₇⋊C₈	1	2	yes	—
177	C₂²	C₂³×C₇⋊C₈	1	1	yes	characteristic
178	C₄	—	2	3	no	—
179	C₄	—	2	3	no	—
180	C₄	—	2	3	no	—
181	C₄	Q₈×C₇⋊C₈	1	2	yes	—
182	C₄	C₂×#6386	1	6	yes	—
183	C₄	C₂×Q₈×Dic₇	1	1	yes	characteristic
184	C₄	C₂×#6511	1	1	yes	characteristic
185	C₂	—	2	2	no	—
186	C₂	C₂×Q₈×C₇⋊C₈	1	1	yes	characteristic
187	C₂	C₂²×#6386	1	1	yes	characteristic
188	C₂	#11741225	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₈
Fitting	C₂×C₇×C₈×Q₈	C₂
prime radical	C₂³×C₇	C₂²×C₈
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂×C₈×Q₈	D₇
hypocenter	C₇	Q₈×M₃₂
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #11728703	C₂	semidirect
3. Q₈×C₇⋊C₁₆	C₂	semidirect
37. #6386	Q₈	direct (N × H)
66. C₇⋊C₁₆	C₂×Q₈	semidirect
82. C₇×Q₈	M₃₂	semidirect
157. Q₈	#6386	direct (N × H)
173. C₇	Q₈×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₈×Q₈	2
#11728703	2
Q₈×C₇⋊C₁₆	2
#11727497	2
C₄×#6386	2
Q₈×M₃₂	7

Subgroups — Q8×#6386 #13507310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#6386 #13507310