Subgroups — #8260

Order 256.

251 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#322	C₂	1	2	yes	maximal
4	#1273	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈	C₄	1	4	yes	—
6	C₂×C₄×C₈	C₄	1	2	yes	—
7	C₂×C₄×C₈	C₄	1	2	yes	—
8	C₂×C₄×C₈	C₂²	1	4	yes	—
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₄³	C₂²	1	1	yes	characteristic
12	#336	C₄	1	4	yes	—
13	#322	—	2	4	no	—
14	C₂²×C₈	C₂×C₄	1	8	yes	—
15	C₂²×C₈	C₂×C₄	1	2	yes	—
16	C₂²×C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄²	C₂×C₄	1	2	yes	—
18	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²	C₂×C₄	1	2	yes	—
21	C₂×C₄²	C₂³	1	1	yes	characteristic
22	C₄⋊C₈	—	2	8	no	—
23	C₄×C₈	—	2	4	no	—
24	C₄×C₈	—	2	4	no	—
25	C₄×C₈	—	2	2	no	—
26	C₄×C₈	—	2	4	no	—
27	C₄×C₈	C₈	1	8	yes	complemented
28	C₄×C₈	D₄	1	2	yes	—
29	C₄×C₈	Q₈	1	2	yes	—
30	C₂²×C₄	C₄²	1	1	yes	characteristic
31	C₂²×C₄	C₄²	1	2	yes	—
32	C₂²×C₄	C₄²	1	1	yes	characteristic
33	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄	C₂²×C₄	1	2	yes	—
35	C₂×C₈	—	2	2	no	—
36	C₂×C₈	—	2	2	no	—
37	C₂×C₈	—	2	8	no	—
38	C₂×C₈	—	2	8	no	—
39	C₂×C₈	—	2	8	no	—
40	C₂×C₈	C₄⋊C₄	1	4	yes	—
41	C₂×C₈	C₂×C₈	1	4	yes	—
42	C₂×C₈	C₂×C₈	1	4	yes	—
43	C₂×C₈	M₁₆	1	4	yes	—
44	C₄²	—	2	2	no	—
45	C₄²	—	2	4	no	—
46	C₄²	C₂×C₈	1	4	yes	—
47	C₄²	M₁₆	1	8	yes	—
48	C₄²	C₂×D₄	1	1	yes	characteristic
49	C₄²	C₂×Q₈	1	1	yes	characteristic
50	C₄²	#41	1	2	yes	—
51	C₂³	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₄×C₈	1	2	yes	—
56	C₂×C₄	C₄×C₈	1	2	yes	—
57	C₂×C₄	#97	1	4	yes	—
58	C₂×C₄	#97	1	4	yes	—
59	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄	#117	1	1	yes	characteristic
61	C₂×C₄	C₄×D₄	1	2	yes	—
62	C₂×C₄	C₄×Q₈	1	2	yes	—
63	C₂×C₄	C₂²×C₈	1	2	yes	—
64	C₂×C₄	C₂×M₁₆	1	2	yes	—
65	C₂×C₄	C₂×M₁₆	1	4	yes	—
66	C₂×C₄	#131	1	4	yes	—
67	C₈	—	2	4	no	—
68	C₈	—	4	8	no	—
69	C₈	C₄⋊C₈	1	8	yes	complemented
70	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
71	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
72	C₂²	C₂×#97	1	2	yes	—
73	C₂²	C₄×M₁₆	1	1	yes	characteristic
74	C₂²	#405	1	2	yes	—
75	C₄	—	2	2	no	—
76	C₄	—	2	4	no	—
77	C₄	#322	1	4	yes	complemented
78	C₄	C₂×C₄⋊C₈	1	2	yes	—
79	C₄	#431	1	2	yes	—
80	C₄	#436	1	4	yes	—
81	C₄	#446	1	4	yes	—
82	C₂	C₂×#322	1	1	yes	characteristic
83	C₂	#1450	1	1	yes	characteristic
84	C₂	#1451	1	2	yes	—
85	C₂	C₄×C₄⋊C₈	1	1	yes	characteristic
86	C₂	#1770	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
27. C₄×C₈	C₈	semidirect
69. C₈	C₄⋊C₈	semidirect
77. C₄	#322	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊C₈	2
C₄²×C₈	2
C₂×#322	2
#1273	2