Subgroups — #13499

Order 256.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2161	C₂	1	1	yes	maximal, characteristic, complemented
2	#2113	C₂	1	1	yes	maximal, characteristic
3	#1814	C₂	1	2	yes	maximal
4	#1384	C₂	1	1	yes	maximal, characteristic, complemented
5	#1382	C₂	1	2	yes	maximal, complemented
6	C₂×#131	C₂²	1	1	yes	characteristic
7	C₂×M₃₂	C₂²	1	1	yes	characteristic
8	C₂²×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×#108	C₂²	1	2	yes	—
10	#427	C₂²	1	2	yes	—
11	#421	—	2	2	no	—
12	#368	—	2	2	no	—
13	C₂²⋊C₁₆	—	2	2	no	—
14	#344	—	2	2	no	—
15	C₂×#41	D₄	1	1	yes	characteristic
16	#131	—	2	2	no	—
17	#131	—	2	2	no	—
18	C₂×M₁₆	—	2	1	no	—
19	C₂×M₁₆	—	2	2	no	—
20	C₂×M₁₆	D₄	1	1	yes	characteristic
21	C₂²×C₈	—	2	1	no	—
22	C₂²×C₈	C₂³	1	1	yes	characteristic
23	#117	—	2	2	no	—
24	M₃₂	—	2	2	no	—
25	C₂×C₁₆	—	2	1	no	—
26	C₂×C₁₆	—	2	2	no	—
27	C₂×C₁₆	D₄	1	2	yes	—
28	#108	—	2	2	no	—
29	#108	—	2	2	no	—
30	#107	—	2	2	no	—
31	#106	—	2	2	no	—
32	#104	—	4	4	no	—
33	#41	—	2	2	no	—
34	#41	—	2	2	no	—
35	C₂×Q₈	Q₁₆	1	1	yes	characteristic
36	C₂×D₄	—	2	1	no	—
37	C₂×D₄	Q₁₆	1	1	yes	characteristic
38	C₂²×C₄	—	2	1	no	—
39	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
40	M₁₆	—	2	2	no	—
41	M₁₆	—	4	2	no	—
42	M₁₆	—	4	2	no	—
43	M₁₆	—	4	2	no	—
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	4	1	no	—
46	C₂×C₈	—	4	1	no	—
47	C₂×C₈	—	4	2	no	—
48	C₂×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈	#41	1	2	yes	—
50	C₂×C₈	#41	1	2	yes	—
51	C₂×C₈	#41	1	1	yes	characteristic
52	C₄⋊C₄	—	4	2	no	—
53	C₂²⋊C₄	—	4	2	no	—
54	C₄²	—	4	2	no	—
55	C₁₆	—	2	2	no	—
56	C₁₆	—	4	1	no	—
57	C₂³	—	2	1	no	—
58	C₂³	#135	1	1	yes	characteristic
59	Q₈	—	2	2	no	—
60	D₄	—	2	2	no	—
61	D₄	—	4	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	—	4	1	no	—
65	C₂×C₄	—	8	2	no	—
66	C₂×C₄	#121	1	1	yes	characteristic
67	C₂×C₄	#123	1	2	yes	—
68	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
70	C₈	—	4	1	no	—
71	C₈	—	8	2	no	—
72	C₈	#123	1	1	yes	characteristic
73	C₈	#124	1	1	yes	characteristic
74	C₈	#126	1	2	yes	—
75	C₂²	—	4	1	no	—
76	C₂²	—	4	2	no	—
77	C₂²	—	4	1	no	—
78	C₂²	#467	1	1	yes	characteristic
79	C₂²	#482	1	2	yes	—
80	C₄	—	4	1	no	—
81	C₄	—	16	2	no	—
82	C₄	#397	1	1	yes	characteristic
83	C₄	#469	1	1	yes	characteristic
84	C₄	#484	1	2	yes	—
85	C₂	—	2	2	no	—
86	C₂	—	2	1	no	—
87	C₂	—	8	1	no	—
88	C₂	#2076	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#2076
prime radical	C₂×#41	D₄
LCS	G ▷ C₂²×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
1. #2161	C₂	semidirect
4. #1384	C₂	semidirect
5. #1382	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2161	2
#2113	2
#1814	2
#1384	2
#1382	2