Subgroups — #12877

Order 256.

183 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2916	C₂	1	1	yes	maximal, characteristic
2	#2113	C₂	1	1	yes	maximal, characteristic
3	#1812	C₂	1	1	yes	maximal, characteristic
4	#1389	C₂	1	2	yes	maximal, complemented
5	#1382	C₂	1	2	yes	maximal, complemented
6	#568	—	2	1	no	—
7	C₂×#131	C₂²	1	1	yes	characteristic
8	C₂×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆	C₂²	1	1	yes	characteristic
10	#430	—	2	1	no	—
11	#430	C₄	1	4	yes	complemented
12	C₂×#108	—	2	2	no	—
13	C₂×#108	C₂²	1	2	yes	—
14	#427	C₂²	1	2	yes	—
15	#410	—	2	1	no	—
16	C₂²⋊C₁₆	—	2	2	no	—
17	H(3,ℤ/4)	—	2	1	no	—
18	C₂×#41	D₄	1	1	yes	characteristic
19	#131	—	2	2	no	—
20	#131	—	2	2	no	—
21	#131	—	4	4	no	—
22	C₂×M₁₆	—	2	4	no	—
23	C₂×M₁₆	—	2	2	no	—
24	C₂×M₁₆	C₂×C₄	1	2	yes	—
25	C₂×M₁₆	D₄	1	1	yes	characteristic
26	C₂²×C₈	D₄	1	2	yes	—
27	C₂²×C₈	C₂³	1	1	yes	characteristic
28	#117	—	2	2	no	—
29	M₃₂	—	2	2	no	—
30	C₂×C₁₆	—	2	1	no	—
31	C₂×C₁₆	—	2	1	no	—
32	C₂×C₁₆	—	2	2	no	—
33	#108	—	2	4	no	—
34	#108	—	2	2	no	—
35	#108	C₂×C₄	1	4	yes	—
36	#107	—	2	2	no	—
37	#106	—	2	2	no	—
38	#99	—	4	2	no	—
39	#41	—	2	2	no	—
40	#41	—	2	2	no	—
41	C₂×Q₈	Q₁₆	1	1	yes	characteristic
42	C₂×D₄	SD₁₆	1	2	yes	—
43	C₂×D₄	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
46	M₁₆	—	2	2	no	—
47	M₁₆	—	2	4	no	—
48	M₁₆	—	2	4	no	—
49	M₁₆	—	4	4	no	—
50	M₁₆	—	4	2	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	2	2	no	—
53	C₂×C₈	—	4	4	no	—
54	C₂×C₈	—	4	2	no	—
55	C₂×C₈	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈	#41	1	2	yes	—
59	C₂×C₈	#41	1	2	yes	—
60	C₄⋊C₄	—	4	2	no	—
61	C₂²⋊C₄	—	4	2	no	—
62	C₄²	—	4	2	no	—
63	C₁₆	—	4	1	no	—
64	C₁₆	—	4	1	no	—
65	C₂³	#103	1	2	yes	—
66	C₂³	C₂×SD₁₆	1	1	yes	characteristic
67	Q₈	—	2	2	no	—
68	D₄	—	2	4	no	—
69	D₄	—	2	2	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	8	2	no	—
73	C₂×C₄	#103	1	2	yes	—
74	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄	#123	1	2	yes	—
76	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
78	C₈	—	2	2	no	—
79	C₈	—	4	4	no	—
80	C₈	#117	1	2	yes	—
81	C₈	#123	1	1	yes	characteristic
82	C₈	#123	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	—	4	1	no	—
85	C₂²	—	4	2	no	—
86	C₂²	C₂×#103	1	1	yes	characteristic
87	C₂²	#481	1	2	yes	—
88	C₄	—	2	2	no	—
89	C₄	—	16	2	no	—
90	C₄	#381	1	1	yes	characteristic
91	C₄	#418	1	1	yes	characteristic
92	C₄	#484	1	2	yes	—
93	C₂	—	2	2	no	—
94	C₂	—	2	1	no	—
95	C₂	—	4	2	no	—
96	C₂	#1824	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#418
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1824
prime radical	C₂×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1389	C₂	semidirect
5. #1382	C₂	semidirect
11. #430	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2916	2
#2113	2
#1812	2
#1389	2
#1382	2