Subgroups — #33311

Order 256.

415 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499	C₂	1	4	yes	maximal
2	#2859	C₂	1	2	yes	maximal
3	#1846	C₂	1	4	yes	maximal
4	#1845	C₂	1	2	yes	maximal
5	#1844	C₂	1	2	yes	maximal
6	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
7	C₂×#125	C₂²	1	4	yes	—
8	#499	—	2	16	no	—
9	C₂×#107	C₂²	1	8	yes	—
10	C₂×#106	C₂²	1	8	yes	—
11	C₂×C₄×C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	2	yes	—
15	#389	—	2	8	no	—
16	#384	C₂²	1	4	yes	—
17	#382	C₂²	1	4	yes	—
18	C₄³	C₂²	1	1	yes	characteristic
19	C₂²×C₈	C₂³	1	2	yes	—
20	C₂²×C₈	C₂³	1	2	yes	—
21	#125	—	2	16	no	—
22	C₂×C₄⋊C₄	—	2	8	no	—
23	C₂×C₄⋊C₄	—	2	8	no	—
24	C₂×C₄⋊C₄	C₂³	1	8	yes	—
25	C₂×C₄²	D₄	1	4	yes	—
26	C₂×C₄²	C₂³	1	1	yes	characteristic
27	C₂×C₄²	C₂³	1	1	yes	characteristic
28	C₂×C₄²	C₂³	1	1	yes	characteristic
29	#107	—	2	16	no	—
30	#106	—	2	16	no	—
31	C₄×C₈	D₄	1	8	yes	—
32	C₄×C₈	Q₈	1	4	yes	—
33	C₄×C₈	Q₈	1	4	yes	—
34	C₄×C₈	Q₈	1	8	yes	—
35	#95	—	2	8	no	—
36	C₂²×C₄	—	2	8	no	—
37	C₂²×C₄	C₂×D₄	1	2	yes	—
38	C₂²×C₄	C₂×D₄	1	2	yes	—
39	C₂²×C₄	C₂×D₄	1	2	yes	—
40	C₂²×C₄	C₂⁴	1	1	yes	characteristic
41	C₂×C₈	C₂×D₄	1	4	yes	—
42	C₂×C₈	C₂×D₄	1	4	yes	—
43	C₂×C₈	C₂×Q₈	1	4	yes	—
44	C₂×C₈	C₂×Q₈	1	4	yes	—
45	C₂×C₈	C₂×Q₈	1	4	yes	—
46	C₂×C₈	C₂×Q₈	1	4	yes	—
47	C₄⋊C₄	—	2	16	no	—
48	C₄⋊C₄	—	4	16	no	—
49	C₄⋊C₄	—	4	16	no	—
50	C₄²	—	2	8	no	—
51	C₄²	C₂×D₄	1	4	yes	—
52	C₄²	C₂×Q₈	1	4	yes	—
53	C₄²	C₂×Q₈	1	2	yes	—
54	C₄²	C₂×Q₈	1	2	yes	—
55	C₂³	C₂²×D₄	1	1	yes	characteristic
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	—	2	4	no	—
58	C₂×C₄	—	4	8	no	—
59	C₂×C₄	—	4	16	no	—
60	C₂×C₄	#127	1	4	yes	—
61	C₂×C₄	C₄⋊Q₈	1	4	yes	—
62	C₂×C₄	C₄⋊Q₈	1	4	yes	—
63	C₂×C₄	#135	1	8	yes	—
64	C₂×C₄	C₂²×D₄	1	2	yes	—
65	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
66	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
67	C₂×C₄	C₂²×Q₈	1	2	yes	—
68	C₈	C₄⋊Q₈	1	8	yes	—
69	C₈	C₄⋊Q₈	1	8	yes	—
70	C₂²	C₂×#127	1	1	yes	characteristic
71	C₂²	C₂×C₄⋊Q₈	1	1	yes	characteristic
72	C₂²	C₂×C₄⋊Q₈	1	1	yes	characteristic
73	C₂²	C₂×#135	1	4	yes	—
74	C₄	—	2	4	no	—
75	C₄	—	2	4	no	—
76	C₄	—	8	8	no	—
77	C₄	#499	1	8	yes	complemented
78	C₄	C₂×C₄⋊Q₈	1	2	yes	—
79	C₄	C₂×C₄⋊Q₈	1	2	yes	—
80	C₂	#2868	1	1	yes	characteristic
81	C₂	#3147	1	2	yes	—
82	C₂	C₂×#499	1	2	yes	—
83	C₂	#3160	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
77. C₄	#499	semidirect

Maximal subgroups

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

subgroup	index
C₂×#499	2
#2859	2
#1846	2
#1845	2
#1844	2
C₄²×C₈	2