Subgroups — #12660

Order 256.

165 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#364	C₂	1	2	yes	maximal
2	C₂×#347	C₂	1	2	yes	maximal
3	#1726	C₂	1	1	yes	maximal, characteristic
4	#1377	C₂	1	2	yes	maximal
5	C₂×M₃₂	C₂²	1	4	yes	—
6	C₂×#97	C₄	1	1	yes	characteristic
7	C₂×#97	C₂²	1	1	yes	characteristic
8	C₂×#97	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	#376	C₄	1	1	yes	characteristic
13	#364	C₄	1	8	yes	complemented
14	#347	—	2	4	no	—
15	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈	C₂³	1	1	yes	characteristic
19	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
22	M₃₂	—	2	4	no	—
23	M₃₂	C₂×C₄	1	8	yes	—
24	C₂×C₁₆	—	2	4	no	—
25	#97	—	2	2	no	—
26	#97	—	2	2	no	—
27	#97	D₄	1	2	yes	—
28	#97	Q₈	1	2	yes	—
29	C₄×C₈	—	2	2	no	—
30	C₄×C₈	—	2	2	no	—
31	C₄×C₈	C₂×C₄	1	4	yes	—
32	#95	—	2	2	no	—
33	C₂²×C₄	—	2	1	no	—
34	C₂²×C₄	—	2	1	no	—
35	C₂²×C₄	M₁₆	1	2	yes	—
36	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	C₄²	1	2	yes	—
42	C₂×C₈	C₄⋊C₄	1	2	yes	—
43	C₂×C₈	M₁₆	1	2	yes	—
44	C₂×C₈	M₁₆	1	2	yes	—
45	C₂×C₈	M₁₆	1	2	yes	—
46	C₂×C₈	C₂²×C₄	1	2	yes	—
47	C₂×C₈	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈	#41	1	2	yes	—
50	C₄²	—	2	1	no	—
51	C₄²	—	2	1	no	—
52	C₄²	—	2	1	no	—
53	C₄²	—	2	1	no	—
54	C₄²	C₄²	1	2	yes	—
55	C₄²	C₄⋊C₄	1	2	yes	—
56	C₁₆	—	2	8	no	—
57	C₂³	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	4	2	no	—
62	C₂×C₄	—	4	2	no	—
63	C₂×C₄	—	4	2	no	—
64	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄	#117	1	1	yes	characteristic
67	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	2	2	no	—
72	C₈	#97	1	4	yes	—
73	C₈	C₄×D₄	1	2	yes	—
74	C₈	C₄×Q₈	1	2	yes	—
75	C₂²	—	2	2	no	—
76	C₂²	C₄×M₁₆	1	1	yes	characteristic
77	C₂²	#423	1	1	yes	characteristic
78	C₂²	#432	1	1	yes	characteristic
79	C₄	—	4	2	no	—
80	C₄	—	4	2	no	—
81	C₄	—	4	2	no	—
82	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
83	C₄	C₂×#97	1	1	yes	characteristic
84	C₄	#423	1	1	yes	characteristic
85	C₄	#432	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2	1	no	—
88	C₂	#1767	1	1	yes	characteristic
89	C₂	#2166	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#117
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	#432
prime radical	C₂³	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
13. #364	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#364	2
C₂×#347	2
#1726	2
#1377	2