Subgroups — #7128

Order 256.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1984	C₂	1	1	yes	maximal, characteristic
2	#1374	C₂	1	2	yes	maximal
3	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
4	C₂²×C₁₆	—	2	2	no	—
5	C₂×C₄⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
7	#336	—	2	2	no	—
8	C₂²×Q₈	C₈	1	1	yes	characteristic
9	C₂²×C₈	—	2	1	no	—
10	C₂²×C₈	D₄	1	2	yes	—
11	C₄×Q₈	—	4	1	no	—
12	C₄×Q₈	—	4	1	no	—
13	C₂×C₄⋊C₄	—	2	1	no	—
14	C₂×C₄⋊C₄	C₈	1	1	yes	characteristic
15	C₂×C₄²	—	2	1	no	—
16	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆	—	2	4	no	—
18	C₂×C₁₆	—	4	2	no	—
19	C₂×C₁₆	—	4	2	no	—
20	C₄⋊C₈	—	4	1	no	—
21	C₄×C₈	—	4	1	no	—
22	C₂×Q₈	—	2	1	no	—
23	C₂×Q₈	—	4	1	no	—
24	C₂×Q₈	—	4	1	no	—
25	C₂×Q₈	C₁₆	1	2	yes	complemented
26	C₂²×C₄	—	2	1	no	—
27	C₂²×C₄	—	2	1	no	—
28	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄	M₁₆	1	1	yes	characteristic
31	C₂×C₈	—	2	2	no	—
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	—	4	1	no	—
34	C₂×C₈	—	4	2	no	—
35	C₂×C₈	SD₁₆	1	2	yes	—
36	C₂×C₈	Q₁₆	1	2	yes	—
37	C₄⋊C₄	—	4	1	no	—
38	C₄⋊C₄	—	4	2	no	—
39	C₄²	—	2	1	no	—
40	C₄²	—	2	1	no	—
41	C₄²	—	4	1	no	—
42	C₄²	—	4	1	no	—
43	C₁₆	—	4	4	no	—
44	C₂³	C₂²⋊C₈	1	1	yes	characteristic
45	Q₈	—	4	2	no	—
46	Q₈	—	4	2	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	4	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	#99	1	1	yes	characteristic
57	C₂×C₄	#100	1	1	yes	characteristic
58	C₂×C₄	#103	1	2	yes	—
59	C₂×C₄	#104	1	2	yes	—
60	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
61	C₂×C₄	M₃₂	1	1	yes	characteristic
62	C₈	—	2	2	no	—
63	C₈	—	2	2	no	—
64	C₈	—	8	1	no	—
65	C₂²	#323	1	1	yes	characteristic
66	C₂²	Q₈⋊C₈	1	2	yes	—
67	C₂²	C₂²⋊C₁₆	1	1	yes	characteristic
68	C₂²	#349	1	1	yes	characteristic
69	C₂²	#350	1	2	yes	—
70	C₄	—	2	1	no	—
71	C₄	—	2	1	no	—
72	C₄	—	4	2	no	—
73	C₄	—	4	1	no	—
74	C₄	—	8	1	no	—
75	C₄	#328	1	2	yes	—
76	C₄	#329	1	1	yes	characteristic
77	C₄	#333	1	1	yes	characteristic
78	C₂	#1272	1	1	yes	characteristic
79	C₂	#1314	1	1	yes	characteristic
80	C₂	#1315	1	1	yes	characteristic
81	C₂	Q₈⋊C₁₆	1	2	yes	—
82	C₂	#1338	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂²
socle, prime radical	C₂³	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
25. C₂×Q₈	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#1984	2
#1374	2