Subgroups — #11990

Order 256.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3177	C₂	1	1	yes	maximal, characteristic
2	#2166	C₂	1	1	yes	maximal, characteristic
3	#2153	C₂	1	1	yes	maximal, characteristic
4	#1366	C₂	1	2	yes	maximal, complemented
5	#1358	C₂	1	2	yes	maximal, complemented
6	#541	C₄	1	2	yes	—
7	C₂×M₃₂	—	2	1	no	—
8	#499	C₂²	1	1	yes	characteristic
9	#494	C₂²	1	1	yes	characteristic
10	#491	C₄	1	2	yes	—
11	#470	—	2	1	no	—
12	#467	—	2	1	no	—
13	#441	—	2	1	no	—
14	C₄×Q₁₆	—	2	1	no	—
15	#405	C₂²	1	1	yes	characteristic
16	#364	—	2	1	no	—
17	#364	C₂²	1	2	yes	—
18	#364	C₂²	1	2	yes	—
19	#349	—	2	1	no	—
20	#347	—	2	1	no	—
21	C₂×Q₁₆	—	2	1	no	—
22	C₂×Q₁₆	—	2	1	no	—
23	C₂×M₁₆	D₄	1	1	yes	characteristic
24	C₂²×C₈	D₄	1	1	yes	characteristic
25	C₄⋊Q₈	C₂×C₄	1	2	yes	—
26	#126	—	4	2	no	—
27	#125	—	4	2	no	—
28	#125	C₂×C₄	1	2	yes	—
29	C₂²⋊Q₈	—	2	2	no	—
30	C₄×Q₈	—	2	2	no	—
31	#117	C₂×C₄	1	1	yes	characteristic
32	M₃₂	—	2	2	no	—
33	M₃₂	—	2	2	no	—
34	M₃₂	—	4	1	no	—
35	C₂×C₁₆	—	4	1	no	—
36	#107	—	2	1	no	—
37	#106	—	2	1	no	—
38	#103	—	2	2	no	—
39	#103	—	2	2	no	—
40	#97	C₂×C₄	1	1	yes	characteristic
41	#97	D₄	1	1	yes	characteristic
42	C₄×C₈	D₄	1	1	yes	characteristic
43	C₄×C₈	C₂³	1	1	yes	characteristic
44	C₂×Q₈	—	2	2	no	—
45	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
46	Q₁₆	—	4	2	no	—
47	Q₁₆	—	4	1	no	—
48	Q₁₆	—	4	1	no	—
49	M₁₆	—	2	2	no	—
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	—	2	1	no	—
52	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈	C₂×D₄	1	1	yes	characteristic
56	C₄⋊C₄	—	2	1	no	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄⋊C₄	—	4	2	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	4	2	no	—
61	C₄⋊C₄	—	4	2	no	—
62	C₂²⋊C₄	—	2	1	no	—
63	C₂²⋊C₄	—	4	2	no	—
64	C₄²	—	4	2	no	—
65	C₄²	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²	C₂²×C₄	1	1	yes	characteristic
67	C₁₆	—	4	2	no	—
68	C₁₆	—	4	2	no	—
69	C₂³	#100	1	1	yes	characteristic
70	Q₈	—	4	2	no	—
71	Q₈	—	4	2	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	#100	1	1	yes	characteristic
79	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	2	1	no	—
82	C₈	#135	1	2	yes	—
83	C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₈	#137	1	1	yes	characteristic
85	C₂²	—	2	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	C₂×#100	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	4	1	no	—
90	C₄	—	4	1	no	—
91	C₄	—	8	2	no	—
92	C₄	—	8	2	no	—
93	C₄	#416	1	1	yes	characteristic
94	C₄	#417	1	1	yes	characteristic
95	C₂	—	2	1	no	—
96	C₂	—	4	1	no	—
97	C₂	#1551	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#1551
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₄×C₈	C₂³
prime radical	C₂³	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ 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
4. #1366	C₂	semidirect
5. #1358	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3177	2
#2166	2
#2153	2
#1366	2
#1358	2