Subgroups — #392557

Order 512.

1842 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1844	C₂	1	12	yes	maximal
2	C₂×C₄²×C₈	C₂	1	1	yes	maximal, characteristic
3	#14692	C₂	1	2	yes	maximal
4	C₂²×#106	C₂²	1	16	yes	—
5	C₂²×C₄×C₈	C₂²	1	6	yes	—
6	C₂×#384	—	2	8	no	—
7	C₂×#384	C₂²	1	12	yes	—
8	C₂×C₄³	C₂²	1	1	yes	characteristic
9	#1844	—	2	48	no	—
10	C₄²×C₈	C₄	1	8	yes	complemented
11	C₂³×C₈	C₂³	1	4	yes	—
12	C₂²×C₄⋊C₄	—	2	24	no	—
13	C₂²×C₄⋊C₄	C₂³	1	8	yes	—
14	C₂²×C₄²	D₄	1	4	yes	—
15	C₂²×C₄²	C₂³	1	3	yes	—
16	C₂×#106	—	2	96	no	—
17	C₂×C₄×C₈	C₂×C₄	1	24	yes	—
18	C₂×C₄×C₈	D₄	1	24	yes	—
19	C₂×C₄×C₈	Q₈	1	24	yes	—
20	#384	—	2	48	no	—
21	#384	—	4	32	no	—
22	C₄³	C₂×C₄	1	4	yes	—
23	C₄³	Q₈	1	4	yes	—
24	C₂³×C₄	—	2	8	no	—
25	C₂³×C₄	C₂×D₄	1	6	yes	—
26	C₂³×C₄	C₂⁴	1	1	yes	characteristic
27	C₂²×C₈	C₂²×C₄	1	8	yes	—
28	C₂²×C₈	C₂×D₄	1	24	yes	—
29	C₂²×C₈	C₂×Q₈	1	24	yes	—
30	C₂×C₄⋊C₄	—	2	48	no	—
31	C₂×C₄⋊C₄	—	4	48	no	—
32	C₂×C₄⋊C₄	—	4	96	no	—
33	C₂×C₄²	C₄⋊C₄	1	16	yes	—
34	C₂×C₄²	SD₁₆	1	32	yes	—
35	C₂×C₄²	C₂²×C₄	1	6	yes	—
36	C₂×C₄²	C₂×D₄	1	12	yes	—
37	C₂×C₄²	C₂×Q₈	1	12	yes	—
38	C₂×C₄²	C₂×Q₈	1	6	yes	—
39	#106	—	4	64	no	—
40	C₄×C₈	C₄⋊C₄	1	96	yes	complemented
41	C₂⁴	C₂²×D₄	1	1	yes	characteristic
42	C₂²×C₄	—	4	24	no	—
43	C₂²×C₄	—	4	32	no	—
44	C₂²×C₄	C₂×C₄⋊C₄	1	12	yes	—
45	C₂²×C₄	#127	1	12	yes	—
46	C₂²×C₄	C₄⋊Q₈	1	12	yes	—
47	C₂²×C₄	C₂×SD₁₆	1	48	yes	—
48	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
49	C₂²×C₄	C₂²×D₄	1	6	yes	—
50	C₂²×C₄	C₂²×Q₈	1	6	yes	—
51	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
52	C₂×C₈	C₂×C₄⋊C₄	1	48	yes	—
53	C₂×C₈	#127	1	16	yes	—
54	C₂×C₈	C₄⋊Q₈	1	48	yes	—
55	C₄⋊C₄	—	4	32	no	—
56	C₄⋊C₄	—	8	96	no	—
57	C₄²	#106	1	64	yes	complemented
58	C₄²	C₂×C₄⋊C₄	1	24	yes	—
59	C₄²	C₄⋊Q₈	1	24	yes	—
60	C₂³	C₂²×C₄⋊C₄	1	1	yes	characteristic
61	C₂³	C₂×#127	1	3	yes	—
62	C₂³	C₂×C₄⋊Q₈	1	3	yes	—
63	C₂³	C₂²×SD₁₆	1	8	yes	—
64	C₂×C₄	—	8	8	no	—
65	C₂×C₄	—	8	48	no	—
66	C₂×C₄	#384	1	24	yes	—
67	C₂×C₄	C₂×#106	1	48	yes	—
68	C₂×C₄	#492	1	48	yes	—
69	C₂×C₄	#498	1	48	yes	—
70	C₂×C₄	C₂²×C₄⋊C₄	1	6	yes	—
71	C₂×C₄	C₂×#127	1	4	yes	—
72	C₂×C₄	C₂×C₄⋊Q₈	1	6	yes	—
73	C₂×C₄	C₂×C₄⋊Q₈	1	12	yes	—
74	C₈	#384	1	32	yes	complemented
75	C₂²	C₂×#384	1	3	yes	—
76	C₂²	#2867	1	1	yes	characteristic
77	C₂²	#2868	1	3	yes	—
78	C₂²	C₂²×#106	1	4	yes	—
79	C₂²	C₂×#492	1	12	yes	—
80	C₂²	C₂×#498	1	12	yes	—
81	C₄	—	16	8	no	—
82	C₄	#1844	1	48	yes	complemented
83	C₄	C₂×#384	1	4	yes	—
84	C₄	#2868	1	4	yes	—
85	C₂	#14692	1	1	yes	characteristic
86	C₂	C₂×#1844	1	6	yes	—
87	C₂	#33300	1	2	yes	—
88	C₂	#33310	1	6	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
10. C₄²×C₈	C₄	semidirect
40. C₄×C₈	C₄⋊C₄	semidirect
57. C₄²	#106	semidirect
74. C₈	#384	semidirect
82. C₄	#1844	semidirect

Maximal subgroups

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

subgroup	index
C₂×#1844	2
C₂×C₄²×C₈	2
#14692	2