Subgroups — #12459

Order 256.

99 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2254	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#1676	C₂	1	1	yes	maximal, characteristic
4	#1674	C₂	1	1	yes	maximal, characteristic
5	#1372	C₂	1	1	yes	maximal, characteristic
6	#1365	C₂	1	1	yes	maximal, characteristic
7	#1364	C₂	1	1	yes	maximal, characteristic
8	#500	C₂²	1	1	yes	characteristic
9	#500	C₂²	1	1	yes	characteristic
10	#477	—	2	1	no	—
11	#462	—	2	1	no	—
12	C₈×Q₈	C₂²	1	1	yes	characteristic
13	#367	—	2	1	no	—
14	#367	—	2	1	no	—
15	C₄⋊C₁₆	—	2	1	no	—
16	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₄×C₁₆	—	2	1	no	—
18	C₄×C₁₆	C₂²	1	1	yes	characteristic
19	#335	C₂²	1	1	yes	characteristic
20	#335	C₂²	1	1	yes	characteristic
21	#333	—	2	1	no	—
22	#332	—	2	1	no	—
23	C₂×Q₁₆	—	4	1	no	—
24	C₄⋊Q₈	—	2	1	no	—
25	C₄⋊Q₈	—	2	1	no	—
26	C₄×Q₈	D₄	1	1	yes	characteristic
27	C₂×C₁₆	—	2	1	no	—
28	C₂×C₁₆	D₄	1	1	yes	characteristic
29	C₂×C₁₆	D₄	1	1	yes	characteristic
30	#107	—	2	1	no	—
31	#107	—	2	1	no	—
32	#107	—	2	1	no	—
33	#107	—	2	1	no	—
34	#107	—	4	1	no	—
35	C₄⋊C₈	—	2	1	no	—
36	C₄⋊C₈	—	2	1	no	—
37	C₄⋊C₈	—	2	1	no	—
38	C₄⋊C₈	D₄	1	1	yes	characteristic
39	#103	—	4	1	no	—
40	#103	—	4	1	no	—
41	C₄×C₈	—	2	1	no	—
42	C₄×C₈	C₂³	1	1	yes	characteristic
43	C₂×Q₈	—	4	1	no	—
44	C₂×Q₈	—	4	1	no	—
45	C₂×Q₈	D₈	1	1	yes	characteristic
46	Q₁₆	—	8	2	no	—
47	C₂×C₈	—	2	1	no	—
48	C₂×C₈	—	4	1	no	—
49	C₂×C₈	—	4	1	no	—
50	C₂×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈	#41	1	1	yes	characteristic
52	C₄⋊C₄	—	2	1	no	—
53	C₄⋊C₄	—	4	1	no	—
54	C₄⋊C₄	—	4	1	no	—
55	C₄⋊C₄	—	4	1	no	—
56	C₄⋊C₄	—	4	1	no	—
57	C₄⋊C₄	D₈	1	1	yes	characteristic
58	C₄²	—	2	1	no	—
59	C₄²	C₂×D₄	1	1	yes	characteristic
60	C₁₆	—	2	1	no	—
61	C₁₆	—	4	1	no	—
62	C₁₆	Q₁₆	1	2	yes	complemented
63	Q₈	—	2	1	no	—
64	Q₈	—	8	1	no	—
65	Q₈	—	8	1	no	—
66	Q₈	—	16	1	no	—
67	Q₈	SD₃₂	1	2	yes	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	8	1	no	—
71	C₂×C₄	—	8	1	no	—
72	C₂×C₄	#121	1	1	yes	characteristic
73	C₂×C₄	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄	#135	1	1	yes	characteristic
75	C₈	—	2	1	no	—
76	C₈	—	4	1	no	—
77	C₈	—	8	1	no	—
78	C₈	—	8	1	no	—
79	C₈	C₂×Q₁₆	1	1	yes	characteristic
80	C₈	#137	1	1	yes	characteristic
81	C₂²	#466	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	16	1	no	—
86	C₄	—	16	1	no	—
87	C₄	#462	1	1	yes	characteristic
88	C₄	#473	1	1	yes	characteristic
89	C₄	C₂×SD₃₂	1	1	yes	characteristic
90	C₄	#508	1	1	yes	characteristic
91	C₂	#1674	1	1	yes	characteristic
92	C₂	#2217	1	1	yes	characteristic
93	C₂	#2223	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
62. C₁₆	Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#2254	2
Q₈×C₁₆	2
#1676	2
#1674	2
#1372	2
#1365	2
#1364	2