Subgroups — #93992

Order 512.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₆₄	C₂	1	2	yes	maximal
2	C₂×#1311	C₂	1	1	yes	maximal, characteristic
3	C₂×M₆₄	C₄	1	4	yes	—
4	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
5	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₈²	C₄	1	1	yes	characteristic
7	#1397	—	2	2	no	—
8	C₄×C₃₂	—	2	2	no	—
9	#1311	—	2	2	no	—
10	C₂²×C₁₆	C₈	1	2	yes	—
11	C₂²×C₁₆	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₈	C₈	1	2	yes	—
13	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
14	M₆₄	—	2	4	no	—
15	M₆₄	C₈	1	8	yes	complemented
16	C₂×C₃₂	—	2	4	no	—
17	C₄×C₁₆	—	2	1	no	—
18	C₄×C₁₆	—	2	1	no	—
19	C₄×C₁₆	D₄	1	2	yes	—
20	C₄×C₁₆	D₄	1	1	yes	characteristic
21	C₄×C₁₆	Q₈	1	1	yes	characteristic
22	C₈²	—	2	1	no	—
23	C₈²	—	2	1	no	—
24	C₂²×C₈	C₁₆	1	4	yes	—
25	C₂²×C₈	C₄²	1	1	yes	characteristic
26	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
28	C₂×C₁₆	—	2	2	no	—
29	C₂×C₁₆	—	2	2	no	—
30	C₂×C₁₆	C₁₆	1	4	yes	—
31	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
32	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
33	C₂×C₁₆	C₂×C₈	1	4	yes	—
34	C₂×C₁₆	M₁₆	1	4	yes	—
35	C₄×C₈	—	2	2	no	—
36	C₄×C₈	—	2	2	no	—
37	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
39	C₄×C₈	M₁₆	1	1	yes	characteristic
40	C₄×C₈	M₁₆	1	1	yes	characteristic
41	C₃₂	—	2	8	no	—
42	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
43	C₂²×C₄	C₂×C₁₆	1	2	yes	—
44	C₂×C₈	—	2	4	no	—
45	C₂×C₈	—	4	4	no	—
46	C₂×C₈	#95	1	1	yes	characteristic
47	C₂×C₈	C₄×C₈	1	2	yes	—
48	C₂×C₈	#97	1	1	yes	characteristic
49	C₂×C₈	#97	1	2	yes	—
50	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₈	C₂×C₁₆	1	2	yes	—
53	C₂×C₈	M₃₂	1	2	yes	—
54	C₄²	C₂²⋊C₈	1	1	yes	characteristic
55	C₄²	C₄⋊C₈	1	1	yes	characteristic
56	C₄²	M₃₂	1	2	yes	—
57	C₁₆	—	2	4	no	—
58	C₁₆	C₂²⋊C₈	1	4	yes	—
59	C₁₆	C₄⋊C₈	1	4	yes	—
60	C₂³	C₄×C₁₆	1	1	yes	characteristic
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	C₈²	1	1	yes	characteristic
63	C₂×C₄	#322	1	1	yes	characteristic
64	C₂×C₄	#336	1	1	yes	characteristic
65	C₂×C₄	C₄×C₁₆	1	1	yes	characteristic
66	C₂×C₄	#346	1	1	yes	characteristic
67	C₂×C₄	#346	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊C₁₆	1	2	yes	—
69	C₂×C₄	C₄⋊C₁₆	1	2	yes	—
70	C₈	—	4	4	no	—
71	C₈	#322	1	2	yes	—
72	C₈	#336	1	2	yes	—
73	C₈	C₂²⋊C₁₆	1	2	yes	—
74	C₈	C₄⋊C₁₆	1	2	yes	—
75	C₂²	—	2	2	no	—
76	C₂²	C₈×C₁₆	1	1	yes	characteristic
77	C₂²	#1312	1	1	yes	characteristic
78	C₂²	#1374	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	#1273	1	1	yes	characteristic
81	C₄	#1311	1	1	yes	characteristic
82	C₄	#1312	1	1	yes	characteristic
83	C₄	#1374	1	1	yes	characteristic
84	C₄	#1400	1	4	yes	—
85	C₂	—	2	1	no	—
86	C₂	—	2	1	no	—
87	C₂	#7136	1	1	yes	characteristic
88	C₂	#7468	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²⋊C₄
commutator	C₂²	C₈×C₁₆
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	#1374
prime radical	C₂³	C₄×C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
15. M₆₄	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₆₄	2
C₂×#1311	2