Subgroups — #93321

Order 512.

175 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#347	C₂	1	1	yes	maximal, characteristic
2	#8800	C₂	1	2	yes	maximal
3	C₂×#347	—	2	2	no	—
4	C₄×M₃₂	C₄	1	1	yes	characteristic
5	C₄×M₃₂	C₄	1	1	yes	characteristic
6	C₄×#97	C₂²	1	1	yes	characteristic
7	#1281	C₄	1	4	yes	—
8	#1276	—	2	2	no	—
9	#1275	—	2	2	no	—
10	C₂×M₃₂	—	2	1	no	—
11	C₂×M₃₂	—	2	1	no	—
12	#431	—	2	4	no	—
13	C₂×#97	C₂×C₄	1	2	yes	—
14	C₂×#97	D₄	1	2	yes	—
15	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
17	C₄³	D₄	1	1	yes	characteristic
18	#347	—	2	8	no	—
19	#346	—	2	1	no	—
20	#346	C₈	1	2	yes	complemented
21	C₄×C₁₆	—	2	1	no	—
22	C₄×C₁₆	C₈	1	2	yes	complemented
23	C₂²×C₈	C₄²	1	1	yes	characteristic
24	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₈	C₄⋊C₄	1	2	yes	—
26	C₂×C₄²	—	4	1	no	—
27	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
28	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
29	M₃₂	—	2	4	no	—
30	M₃₂	—	2	4	no	—
31	C₂×C₁₆	—	2	2	no	—
32	C₂×C₁₆	—	2	2	no	—
33	C₄⋊C₈	—	4	4	no	—
34	C₂²⋊C₈	—	4	4	no	—
35	#97	—	2	4	no	—
36	#97	—	4	2	no	—
37	C₄×C₈	—	4	1	no	—
38	C₄×C₈	—	4	4	no	—
39	C₄×C₈	C₂×C₈	1	1	yes	characteristic
40	C₄×C₈	C₂×C₈	1	1	yes	characteristic
41	C₄×C₈	M₁₆	1	1	yes	characteristic
42	C₄×C₈	M₁₆	1	1	yes	characteristic
43	C₂²×C₄	—	2	1	no	—
44	C₂²×C₄	—	2	2	no	—
45	C₂²×C₄	#95	1	1	yes	characteristic
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	—	8	4	no	—
52	C₂×C₈	C₄×C₈	1	1	yes	characteristic
53	C₂×C₈	#97	1	1	yes	characteristic
54	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
55	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
56	C₄²	—	2	1	no	—
57	C₄²	—	2	2	no	—
58	C₄²	—	8	2	no	—
59	C₄²	#104	1	4	yes	—
60	C₄²	C₄⋊C₈	1	1	yes	characteristic
61	C₄²	C₄⋊C₈	1	1	yes	characteristic
62	C₁₆	—	2	4	no	—
63	C₁₆	—	2	4	no	—
64	C₂³	H(3,ℤ/4)	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	8	2	no	—
70	C₂×C₄	#336	1	1	yes	characteristic
71	C₂×C₄	#338	1	1	yes	characteristic
72	C₂×C₄	#339	1	2	yes	—
73	C₂×C₄	#344	1	2	yes	—
74	C₈	—	2	1	no	—
75	C₈	—	2	1	no	—
76	C₈	—	2	1	no	—
77	C₈	—	2	1	no	—
78	C₈	—	4	2	no	—
79	C₈	—	16	4	no	—
80	C₂²	—	4	1	no	—
81	C₂²	#1275	1	1	yes	characteristic
82	C₂²	#1288	1	2	yes	—
83	C₄	—	2	2	no	—
84	C₄	—	4	4	no	—
85	C₄	—	4	1	no	—
86	C₄	#1276	1	1	yes	characteristic
87	C₄	#1281	1	1	yes	characteristic
88	C₄	#1291	1	2	yes	—
89	C₂	—	2	1	no	—
90	C₂	—	2	2	no	—
91	C₂	#7019	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#1281
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₄×#97	C₂²
socle	C₂	#7019
prime radical	C₂³	H(3,ℤ/4)
LCS	G ▷ C₂×C₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#97 ◁ G
LECS	G ▷ C₄×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
20. #346	C₈	semidirect
22. C₄×C₁₆	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#347	2
#8800	2