Subgroups — #392555

Order 512.

2070 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1842	C₂	1	14	yes	maximal
2	C₄⁴	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₈	C₂²	1	28	yes	—
4	C₂×C₄³	C₄	1	7	yes	—
5	C₂×C₄³	C₄	1	1	yes	characteristic
6	C₂×C₄³	C₂²	1	7	yes	—
7	#1842	—	2	56	no	—
8	C₂³×C₈	C₂³	1	8	yes	—
9	C₂²×C₄²	C₂×C₄	1	21	yes	—
10	C₂²×C₄²	C₂×C₄	1	7	yes	—
11	C₂²×C₄²	C₂³	1	7	yes	—
12	C₂×C₄⋊C₈	—	2	168	no	—
13	C₄³	—	2	28	no	—
14	C₄³	C₈	1	8	yes	complemented
15	C₄³	D₄	1	28	yes	—
16	C₄³	Q₈	1	28	yes	—
17	C₂³×C₄	C₂²×C₄	1	7	yes	—
18	C₂³×C₄	C₂²×C₄	1	7	yes	—
19	C₂³×C₄	C₂⁴	1	1	yes	characteristic
20	C₂²×C₈	—	2	56	no	—
21	C₂×C₄²	—	2	84	no	—
22	C₂×C₄²	C₄⋊C₄	1	84	yes	—
23	C₂×C₄²	C₄⋊C₄	1	28	yes	—
24	C₂×C₄²	C₂×C₈	1	28	yes	—
25	C₂×C₄²	M₁₆	1	28	yes	—
26	C₂×C₄²	C₂×D₄	1	42	yes	—
27	C₂×C₄²	C₂×Q₈	1	42	yes	—
28	C₄⋊C₈	—	4	112	no	—
29	C₂⁴	C₂³×C₄	1	1	yes	characteristic
30	C₂²×C₄	—	2	28	no	—
31	C₂²×C₄	C₂×C₄⋊C₄	1	42	yes	—
32	C₂²×C₄	C₂×C₄⋊C₄	1	42	yes	—
33	C₂²×C₄	C₂²×C₈	1	14	yes	—
34	C₂²×C₄	C₂×M₁₆	1	42	yes	—
35	C₂²×C₄	C₂²×D₄	1	7	yes	—
36	C₂²×C₄	C₂²×Q₈	1	7	yes	—
37	C₂×C₈	—	4	56	no	—
38	C₄²	—	2	168	no	—
39	C₄²	C₄⋊C₈	1	112	yes	complemented
40	C₄²	#127	1	28	yes	—
41	C₄²	C₄⋊Q₈	1	84	yes	—
42	C₂³	C₂²×C₄⋊C₄	1	7	yes	—
43	C₂³	C₂³×C₈	1	1	yes	characteristic
44	C₂³	C₂²×M₁₆	1	7	yes	—
45	C₂×C₄	—	2	84	no	—
46	C₂×C₄	#384	1	28	yes	—
47	C₂×C₄	#384	1	28	yes	—
48	C₂×C₄	C₂×C₄⋊C₈	1	84	yes	—
49	C₂×C₄	#423	1	84	yes	—
50	C₂×C₄	C₂×#127	1	7	yes	—
51	C₂×C₄	C₂×C₄⋊Q₈	1	21	yes	—
52	C₈	—	8	8	no	—
53	C₂²	C₂×#384	1	7	yes	—
54	C₂²	C₂²×C₄⋊C₈	1	7	yes	—
55	C₂²	C₂×#423	1	21	yes	—
56	C₄	—	2	28	no	—
57	C₄	#1842	1	56	yes	complemented
58	C₄	#2867	1	1	yes	characteristic
59	C₄	#2868	1	7	yes	—
60	C₂	#14692	1	1	yes	characteristic
61	C₂	C₂×#1842	1	7	yes	—
62	C₂	#20791	1	7	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂³×C₈
socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
14. C₄³	C₈	semidirect
39. C₄²	C₄⋊C₈	semidirect
57. C₄	#1842	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1842	2
C₄⁴	2