Subgroups — #94580

Order 512.

95 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#8791	C₂	1	1	yes	maximal, characteristic
2	#7366	C₂	1	2	yes	maximal
3	#1837	C₄	1	1	yes	characteristic
4	#1835	C₂²	1	1	yes	characteristic
5	#1450	C₄	1	1	yes	characteristic
6	#1324	—	2	2	no	—
7	#1286	—	2	2	no	—
8	#431	—	2	1	no	—
9	#423	—	2	1	no	—
10	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
11	#382	D₄	1	2	yes	—
12	#341	—	4	1	no	—
13	#340	—	4	1	no	—
14	#324	—	4	2	no	—
15	#322	C₈	1	2	yes	complemented
16	C₈²	—	2	1	no	—
17	C₂×M₁₆	—	4	1	no	—
18	C₂²×C₈	—	2	1	no	—
19	C₂×C₄⋊C₄	—	2	2	no	—
20	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
21	C₄⋊C₈	—	4	1	no	—
22	C₄⋊C₈	—	4	2	no	—
23	C₂²⋊C₈	—	4	1	no	—
24	C₂²⋊C₈	—	8	2	no	—
25	C₄×C₈	—	2	2	no	—
26	C₄×C₈	—	2	1	no	—
27	C₄×C₈	C₂×C₈	1	1	yes	characteristic
28	C₄×C₈	M₁₆	1	1	yes	characteristic
29	#95	—	4	2	no	—
30	C₂²×C₄	—	2	1	no	—
31	C₂²×C₄	—	4	2	no	—
32	C₂²×C₄	#99	1	1	yes	characteristic
33	M₁₆	—	8	2	no	—
34	C₂×C₈	—	2	1	no	—
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	—	4	2	no	—
37	C₂×C₈	—	4	2	no	—
38	C₂×C₈	—	8	1	no	—
39	C₂×C₈	—	16	2	no	—
40	C₄⋊C₄	—	8	2	no	—
41	C₄²	—	2	1	no	—
42	C₄²	C₂²⋊C₈	1	1	yes	characteristic
43	C₄²	#101	1	1	yes	characteristic
44	C₂³	#354	1	1	yes	characteristic
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	8	1	no	—
49	C₂×C₄	—	8	2	no	—
50	C₂×C₄	—	16	2	no	—
51	C₂×C₄	#324	1	1	yes	characteristic
52	C₂×C₄	#355	1	1	yes	characteristic
53	C₈	—	4	1	no	—
54	C₈	—	4	1	no	—
55	C₈	—	8	2	no	—
56	C₈	—	16	1	no	—
57	C₈	—	32	2	no	—
58	C₂²	—	4	1	no	—
59	C₂²	#1324	1	1	yes	characteristic
60	C₂²	#1414	1	2	yes	—
61	C₄	—	2	1	no	—
62	C₄	—	2	1	no	—
63	C₄	—	4	1	no	—
64	C₄	—	4	1	no	—
65	C₄	—	16	2	no	—
66	C₂	—	2	1	no	—
67	C₂	—	2	2	no	—
68	C₂	#7366	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#7366
commutator	C₄×C₈	C₂×C₈
Frattini, cosocle	#1835	C₂²
prime radical	C₂³	#354
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1835 ◁ G
LECS	G ▷ #1835 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
15. #322	C₈	semidirect

Maximal subgroups

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

subgroup	index
#8791	2
#7366	2