Subgroups — #33258

Order 256.

359 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2859	C₂	1	2	yes	maximal
2	#1942	C₂	1	8	yes	maximal
3	#1845	C₂	1	2	yes	maximal
4	#1834	C₂	1	2	yes	maximal
5	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×#125	—	2	2	no	—
7	C₂×#107	C₂²	1	8	yes	—
8	C₂×C₄×C₈	C₂²	1	4	yes	—
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	#389	—	2	4	no	—
12	#389	C₂²	1	8	yes	—
13	#384	C₂²	1	4	yes	—
14	#382	—	2	2	no	—
15	C₄³	C₂²	1	1	yes	characteristic
16	#340	C₂²	1	8	yes	—
17	C₂²×C₈	C₂³	1	2	yes	—
18	C₂²×C₈	C₂³	1	2	yes	—
19	#125	—	4	8	no	—
20	C₂×C₄⋊C₄	—	2	4	no	—
21	C₂×C₄⋊C₄	—	2	8	no	—
22	C₂×C₄⋊C₄	—	2	4	no	—
23	C₂×C₄⋊C₄	C₂³	1	8	yes	—
24	C₂×C₄²	D₄	1	2	yes	—
25	C₂×C₄²	D₄	1	1	yes	characteristic
26	C₂×C₄²	D₄	1	1	yes	characteristic
27	C₂×C₄²	C₂³	1	2	yes	—
28	C₂×C₄²	C₂³	1	1	yes	characteristic
29	#107	—	2	16	no	—
30	C₄×C₈	—	2	8	no	—
31	C₄×C₈	Q₈	1	4	yes	—
32	C₄×C₈	Q₈	1	4	yes	—
33	#95	—	2	8	no	—
34	C₂²×C₄	—	2	8	no	—
35	C₂²×C₄	C₂×D₄	1	2	yes	—
36	C₂²×C₄	C₂×D₄	1	2	yes	—
37	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄	C₂⁴	1	1	yes	characteristic
40	C₂×C₈	—	2	4	no	—
41	C₂×C₈	C₂×Q₈	1	4	yes	—
42	C₂×C₈	C₂×Q₈	1	4	yes	—
43	C₂×C₈	#41	1	8	yes	—
44	C₄⋊C₄	—	2	16	no	—
45	C₄⋊C₄	—	4	16	no	—
46	C₄⋊C₄	—	4	8	no	—
47	C₄⋊C₄	—	4	8	no	—
48	C₄²	—	2	4	no	—
49	C₄²	—	2	4	no	—
50	C₄²	D₈	1	4	yes	—
51	C₄²	Q₁₆	1	4	yes	—
52	C₄²	C₂×Q₈	1	2	yes	—
53	C₄²	C₂×Q₈	1	2	yes	—
54	C₂³	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	4	8	no	—
60	C₂×C₄	—	4	8	no	—
61	C₂×C₄	—	4	8	no	—
62	C₂×C₄	C₄⋊Q₈	1	2	yes	—
63	C₂×C₄	C₄⋊Q₈	1	2	yes	—
64	C₂×C₄	C₂×D₈	1	4	yes	—
65	C₂×C₄	C₂×D₈	1	2	yes	—
66	C₂×C₄	C₂×Q₁₆	1	4	yes	—
67	C₂×C₄	C₂×Q₁₆	1	2	yes	—
68	C₂×C₄	#135	1	4	yes	—
69	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
70	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄	C₂×#41	1	4	yes	—
72	C₈	—	2	4	no	—
73	C₈	#125	1	8	yes	complemented
74	C₂²	C₂×C₄⋊Q₈	1	1	yes	characteristic
75	C₂²	#532	1	2	yes	—
76	C₂²	C₂²×D₈	1	1	yes	characteristic
77	C₂²	C₂²×Q₁₆	1	1	yes	characteristic
78	C₂²	C₂×#135	1	2	yes	—
79	C₄	—	2	4	no	—
80	C₄	—	2	4	no	—
81	C₄	—	8	8	no	—
82	C₄	#499	1	4	yes	—
83	C₄	#500	1	4	yes	—
84	C₄	C₂×#125	1	2	yes	—
85	C₄	#533	1	2	yes	—
86	C₂	#2859	1	1	yes	characteristic
87	C₂	C₂×#499	1	1	yes	characteristic
88	C₂	C₂×#500	1	1	yes	characteristic
89	C₂	#3169	1	2	yes	—
90	C₂	#3170	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
73. C₈	#125	semidirect

Maximal subgroups

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

subgroup	index
#2859	2
#1942	2
#1845	2
#1834	2
C₄²×C₈	2