Subgroups — #94694

Order 512.

93 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13697	C₂	1	1	yes	maximal, characteristic
2	#7513	C₂	1	1	yes	maximal, characteristic, complemented
3	#7475	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₃₂	C₄	1	1	yes	characteristic, complemented
5	C₂×M₆₄	C₂²	1	1	yes	characteristic
6	#2215	—	2	1	no	—
7	#2158	—	2	1	no	—
8	M₁₂₈	—	2	1	no	—
9	#1424	C₄	1	1	yes	characteristic, complemented
10	#1415	—	2	1	no	—
11	C₂×D₁₆	—	2	1	no	—
12	C₂²×C₁₆	D₄	1	1	yes	characteristic
13	#466	—	4	1	no	—
14	#427	—	4	1	no	—
15	D₃₂	—	4	1	no	—
16	M₆₄	—	4	1	no	—
17	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃₂	D₄	1	1	yes	characteristic
19	#367	—	2	2	no	—
20	#366	—	2	1	no	—
21	#366	—	4	1	no	—
22	#357	—	4	1	no	—
23	C₆₄	—	4	1	no	—
24	C₂×D₈	—	4	1	no	—
25	C₂²×C₈	D₈	1	1	yes	characteristic
26	#121	—	8	1	no	—
27	#117	—	8	1	no	—
28	D₁₆	—	4	1	no	—
29	D₁₆	—	4	1	no	—
30	C₂×C₁₆	—	2	1	no	—
31	C₂×C₁₆	—	2	1	no	—
32	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₁₆	SD₁₆	1	1	yes	characteristic
34	#107	—	4	2	no	—
35	#107	—	4	1	no	—
36	#106	—	8	1	no	—
37	#102	—	8	1	no	—
38	C₃₂	—	2	1	no	—
39	C₃₂	—	2	1	no	—
40	C₂×D₄	—	8	1	no	—
41	C₂×D₄	—	16	1	no	—
42	C₂²×C₄	D₁₆	1	1	yes	characteristic
43	D₈	—	8	1	no	—
44	D₈	—	8	1	no	—
45	C₂×C₈	—	2	1	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	#102	1	1	yes	characteristic
48	C₂×C₈	SD₃₂	1	1	yes	characteristic
49	C₄⋊C₄	—	8	2	no	—
50	C₄⋊C₄	—	8	1	no	—
51	C₂²⋊C₄	—	16	1	no	—
52	C₂²⋊C₄	—	16	1	no	—
53	C₄²	—	16	1	no	—
54	C₁₆	—	2	1	no	—
55	C₁₆	#99	1	1	yes	characteristic
56	C₁₆	#104	1	1	yes	characteristic
57	C₂³	—	16	1	no	—
58	C₂³	D₃₂	1	1	yes	characteristic
59	D₄	—	16	1	no	—
60	D₄	—	16	1	no	—
61	D₄	—	32	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	16	2	no	—
65	C₂×C₄	—	16	1	no	—
66	C₂×C₄	#357	1	1	yes	characteristic
67	C₂×C₄	SD₆₄	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	#327	1	1	yes	characteristic
70	C₈	#360	1	1	yes	characteristic
71	C₂²	—	2	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	32	1	no	—
74	C₂²	—	32	1	no	—
75	C₂²	—	32	1	no	—
76	C₂²	#1415	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	32	1	no	—
79	C₄	—	32	2	no	—
80	C₄	#1347	1	1	yes	characteristic
81	C₄	#1418	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	4	1	no	—
84	C₂	—	64	1	no	—
85	C₂	#7418	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#7418
commutator	C₂×C₃₂	C₂×C₄
Frattini, cosocle	C₂×M₆₄	C₂²
prime radical	#13697	C₂
LCS	G ▷ C₂×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄ ◁ G
LECS	G ▷ C₂×M₆₄ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #7513	C₂	semidirect
3. #7475	C₂	semidirect
4. C₂×D₃₂	C₄	semidirect
9. #1424	C₄	semidirect

Maximal subgroups

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

subgroup	index
#13697	2
#7513	2
#7475	2