Subgroups — #11914

Order 256.

136 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	#2110	C₂	1	1	yes	maximal, characteristic
4	#1356	C₂	1	4	yes	maximal, complemented
5	#541	C₄	1	2	yes	—
6	C₂×M₃₂	—	2	1	no	—
7	#499	C₄	1	1	yes	characteristic
8	#494	C₄	1	1	yes	characteristic
9	#491	C₂²	1	2	yes	—
10	#470	—	2	1	no	—
11	#467	—	2	1	no	—
12	#441	—	2	1	no	—
13	C₄×Q₁₆	—	2	1	no	—
14	#405	C₂²	1	1	yes	characteristic
15	#364	—	2	1	no	—
16	#349	—	2	1	no	—
17	#347	—	2	1	no	—
18	#347	C₂²	1	2	yes	—
19	#347	C₂²	1	2	yes	—
20	C₂×Q₁₆	—	2	1	no	—
21	C₂×Q₁₆	—	2	1	no	—
22	C₂×M₁₆	D₄	1	1	yes	characteristic
23	C₂²×C₈	D₄	1	1	yes	characteristic
24	C₄⋊Q₈	C₂×C₄	1	2	yes	—
25	#126	—	4	2	no	—
26	#125	—	4	2	no	—
27	#125	C₂×C₄	1	2	yes	—
28	C₂²⋊Q₈	—	2	2	no	—
29	C₄×Q₈	—	2	2	no	—
30	#117	C₂×C₄	1	1	yes	characteristic
31	M₃₂	—	2	2	no	—
32	M₃₂	—	2	2	no	—
33	M₃₂	—	2	2	no	—
34	C₂×C₁₆	—	2	2	no	—
35	#107	—	2	1	no	—
36	#106	—	2	1	no	—
37	#103	—	2	2	no	—
38	#103	—	2	2	no	—
39	#97	D₄	1	1	yes	characteristic
40	#97	C₂³	1	1	yes	characteristic
41	C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×Q₈	—	2	2	no	—
44	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
45	Q₁₆	—	4	2	no	—
46	Q₁₆	—	4	1	no	—
47	Q₁₆	—	4	1	no	—
48	M₁₆	—	4	1	no	—
49	C₂×C₈	—	2	1	no	—
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈	C₂×D₄	1	1	yes	characteristic
55	C₄⋊C₄	—	2	1	no	—
56	C₄⋊C₄	—	2	2	no	—
57	C₄⋊C₄	—	4	2	no	—
58	C₄⋊C₄	—	4	2	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	4	2	no	—
61	C₂²⋊C₄	—	2	1	no	—
62	C₂²⋊C₄	—	4	2	no	—
63	C₄²	—	4	2	no	—
64	C₄²	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²	C₂²×C₄	1	1	yes	characteristic
66	C₁₆	—	2	4	no	—
67	C₁₆	—	4	2	no	—
68	C₂³	#100	1	1	yes	characteristic
69	Q₈	—	4	2	no	—
70	Q₈	—	4	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	#100	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₈	—	2	1	no	—
80	C₈	—	2	1	no	—
81	C₈	—	2	1	no	—
82	C₈	#104	1	1	yes	characteristic
83	C₈	#104	1	1	yes	characteristic
84	C₂²	—	2	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	C₂×#100	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	4	1	no	—
89	C₄	—	4	1	no	—
90	C₄	—	8	2	no	—
91	C₄	—	8	2	no	—
92	C₄	C₂×#104	1	1	yes	characteristic
93	C₄	#421	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	4	1	no	—
96	C₂	#1530	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#1530
commutator	C₄²	C₂²×C₄
Frattini, cosocle	#97	C₂³
prime radical	C₂³	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ #97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1356	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
#2110	2
#1356	2