Subgroups — #92907

Order 512.

167 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11411	C₂	1	1	yes	maximal, characteristic
2	#11410	C₂	1	1	yes	maximal, characteristic
3	#11410	C₂	1	1	yes	maximal, characteristic
4	#2162	C₄	1	2	yes	—
5	#2162	C₄	1	2	yes	—
6	#2162	C₄	1	2	yes	—
7	C₂×C₈²	C₂²	1	1	yes	characteristic
8	#1312	—	2	1	no	—
9	#1312	—	2	1	no	—
10	#1312	—	2	1	no	—
11	#1311	—	2	1	no	—
12	C₈×C₁₆	—	2	1	no	—
13	C₈×C₁₆	—	2	1	no	—
14	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
17	C₄⋊C₁₆	—	2	2	no	—
18	C₄⋊C₁₆	C₈	1	4	yes	—
19	C₄⋊C₁₆	C₈	1	4	yes	—
20	C₂²⋊C₁₆	—	2	2	no	—
21	C₂²⋊C₁₆	—	2	2	no	—
22	C₂²⋊C₁₆	C₈	1	4	yes	—
23	C₄×C₁₆	—	2	2	no	—
24	C₄×C₁₆	—	2	2	no	—
25	C₄×C₁₆	—	2	2	no	—
26	C₈²	D₄	1	1	yes	characteristic
27	C₈²	D₄	1	1	yes	characteristic
28	C₈²	D₄	1	1	yes	characteristic
29	C₈²	Q₈	1	1	yes	characteristic
30	C₂²×C₈	C₂×C₈	1	2	yes	—
31	C₂²×C₈	M₁₆	1	2	yes	—
32	C₂²×C₈	M₁₆	1	2	yes	—
33	C₂×C₄²	C₄²	1	1	yes	characteristic
34	C₂×C₁₆	—	2	4	no	—
35	C₂×C₁₆	—	2	4	no	—
36	C₂×C₁₆	—	2	4	no	—
37	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
39	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₄×C₈	C₂×C₈	1	2	yes	—
44	C₄×C₈	C₂×C₈	1	2	yes	—
45	C₄×C₈	M₁₆	1	2	yes	—
46	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
47	C₂²×C₄	#97	1	1	yes	characteristic
48	C₂²×C₄	#97	1	1	yes	characteristic
49	C₂×C₈	—	2	4	no	—
50	C₂×C₈	—	2	4	no	—
51	C₂×C₈	—	4	2	no	—
52	C₂×C₈	C₂²⋊C₈	1	2	yes	—
53	C₂×C₈	C₂²⋊C₈	1	2	yes	—
54	C₂×C₈	C₂²⋊C₈	1	2	yes	—
55	C₂×C₈	C₄⋊C₈	1	2	yes	—
56	C₂×C₈	C₄⋊C₈	1	2	yes	—
57	C₂×C₈	C₄⋊C₈	1	2	yes	—
58	C₄²	#95	1	1	yes	characteristic
59	C₄²	C₄×C₈	1	1	yes	characteristic
60	C₄²	C₄×C₈	1	1	yes	characteristic
61	C₄²	#97	1	1	yes	characteristic
62	C₁₆	—	4	4	no	—
63	C₁₆	—	4	4	no	—
64	C₁₆	—	4	4	no	—
65	C₂³	#322	1	1	yes	characteristic
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	C₈²	1	1	yes	characteristic
70	C₂×C₄	#322	1	1	yes	characteristic
71	C₂×C₄	#322	1	1	yes	characteristic
72	C₂×C₄	#336	1	1	yes	characteristic
73	C₂×C₄	#336	1	1	yes	characteristic
74	C₂×C₄	#336	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₈	—	2	2	no	—
77	C₈	—	2	2	no	—
78	C₈	—	2	2	no	—
79	C₈	#350	1	4	yes	—
80	C₈	#350	1	4	yes	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	2	1	no	—
84	C₂²	#1273	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	#1378	1	2	yes	—
89	C₄	#1379	1	2	yes	—
90	C₄	#1379	1	2	yes	—
91	C₂	—	4	1	no	—
92	C₂	#7140	1	1	yes	characteristic
93	C₂	#7141	1	1	yes	characteristic
94	C₂	#7141	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₈²
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1273
prime radical	C₂³	#322
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
#11411	2
#11410	2