Subgroups — #94690

Order 512.

99 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13694	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#7471	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₃₂	C₄	1	1	yes	characteristic, complemented
5	C₂²×C₃₂	C₂²	1	1	yes	characteristic
6	#2215	—	2	1	no	—
7	C₂×#366	—	2	1	no	—
8	C₂×C₆₄	—	2	1	no	—
9	#1423	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	C₂×#107	—	4	1	no	—
15	D₃₂	—	4	1	no	—
16	C₂×C₃₂	—	4	1	no	—
17	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃₂	D₄	1	1	yes	characteristic
19	#366	—	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	C₂×C₄⋊C₄	—	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	#107	—	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	—	16	1	no	—
43	C₂²×C₄	D₁₆	1	1	yes	characteristic
44	D₈	—	8	1	no	—
45	D₈	—	8	1	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	2	1	no	—
48	C₂×C₈	#102	1	1	yes	characteristic
49	C₂×C₈	SD₃₂	1	1	yes	characteristic
50	C₄⋊C₄	—	8	2	no	—
51	C₄⋊C₄	—	8	1	no	—
52	C₄⋊C₄	—	16	1	no	—
53	C₂²⋊C₄	—	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	1	no	—
62	D₄	—	32	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	16	2	no	—
66	C₂×C₄	—	16	1	no	—
67	C₂×C₄	—	32	1	no	—
68	C₂×C₄	—	32	1	no	—
69	C₂×C₄	#357	1	1	yes	characteristic
70	C₂×C₄	SD₆₄	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	#327	1	1	yes	characteristic
73	C₈	#360	1	1	yes	characteristic
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	32	1	no	—
77	C₂²	—	32	1	no	—
78	C₂²	—	32	1	no	—
79	C₂²	#1415	1	1	yes	characteristic
80	C₂²	D₆₄	1	1	yes	characteristic
81	C₂²	SD₁₂₈	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	32	2	no	—
84	C₄	—	32	1	no	—
85	C₄	#1347	1	1	yes	characteristic
86	C₄	#1418	1	1	yes	characteristic
87	C₂	—	2	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	64	1	no	—
90	C₂	#7418	1	1	yes	characteristic
91	C₂	#7537	1	1	yes	characteristic
92	C₂	#7540	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1415
commutator	C₂×C₃₂	C₂×C₄
Frattini, cosocle	C₂²×C₃₂	C₂²
prime radical	#13694	C₂
LCS	G ▷ C₂×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²⋊C₆₄	C₂	semidirect
3. #7471	C₂	semidirect
4. C₂×D₃₂	C₄	semidirect
9. #1423	C₄	semidirect

Maximal subgroups

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

subgroup	index
#13694	2
C₂²⋊C₆₄	2
#7471	2