Subgroups — #12013

Order 256.

125 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2163	C₂	1	2	yes	maximal
2	C₂×C₈²	C₂	1	1	yes	maximal, characteristic
3	#1370	C₂	1	2	yes	maximal, complemented
4	#1367	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆	C₂²	1	4	yes	—
9	C₂²⋊C₁₆	—	2	2	no	—
10	#346	—	2	2	no	—
11	C₈²	C₄	1	1	yes	characteristic
12	C₈²	C₄	1	1	yes	characteristic
13	C₈²	C₂²	1	1	yes	characteristic
14	C₈²	C₂²	1	1	yes	characteristic
15	C₂²×C₈	—	2	1	no	—
16	C₂²×C₈	C₈	1	1	yes	characteristic
17	C₂²×C₈	C₈	1	1	yes	characteristic
18	C₂²×C₈	D₄	1	1	yes	characteristic
19	C₂²×C₈	Q₈	1	1	yes	characteristic
20	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆	—	2	4	no	—
22	C₄×C₈	—	2	1	no	—
23	C₄×C₈	C₈	1	1	yes	characteristic
24	C₄×C₈	C₈	1	1	yes	characteristic
25	C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈	D₄	1	1	yes	characteristic
31	C₄×C₈	Q₈	1	1	yes	characteristic
32	C₄×C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄	M₁₆	1	1	yes	characteristic
36	C₂×C₈	—	2	4	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	2	2	no	—
40	C₂×C₈	—	2	2	no	—
41	C₂×C₈	—	2	2	no	—
42	C₂×C₈	—	2	2	no	—
43	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈	C₂×C₈	1	1	yes	characteristic
47	C₂×C₈	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
51	C₄²	C₄⋊C₄	1	1	yes	characteristic
52	C₄²	C₂×C₈	1	1	yes	characteristic
53	C₄²	M₁₆	1	1	yes	characteristic
54	C₄²	C₂²×C₄	1	1	yes	characteristic
55	C₁₆	—	4	4	no	—
56	C₂³	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
68	C₈	—	2	2	no	—
69	C₈	—	2	2	no	—
70	C₈	—	2	1	no	—
71	C₈	—	2	1	no	—
72	C₈	—	2	1	no	—
73	C₈	—	2	1	no	—
74	C₈	#108	1	2	yes	—
75	C₈	M₃₂	1	2	yes	—
76	C₈	M₃₂	1	2	yes	—
77	C₈	#135	1	2	yes	—
78	C₂²	—	2	1	no	—
79	C₂²	—	2	1	no	—
80	C₂²	#364	1	2	yes	—
81	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	2	1	no	—
86	C₄	#427	1	1	yes	characteristic
87	C₄	C₂×#108	1	1	yes	characteristic
88	C₄	C₂×M₃₂	1	1	yes	characteristic
89	C₄	C₂×M₃₂	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	#1564	1	1	yes	characteristic
92	C₂	#2150	1	1	yes	characteristic
93	C₂	C₂×#364	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂²	C₂×C₄⋊C₈
prime radical	C₂³	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ 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
3. #1370	C₂	semidirect
4. #1367	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2163	2
C₂×C₈²	2
#1370	2
#1367	2