Subgroups — #13498

Order 256.

146 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	#2112	C₂	1	1	yes	maximal, characteristic
3	#1810	C₂	1	2	yes	maximal
4	#1388	C₂	1	2	yes	maximal, complemented
5	#1385	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₈	C₂²	1	1	yes	characteristic
7	C₂×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×#108	C₂²	1	2	yes	—
10	C₂×#108	C₂²	1	2	yes	—
11	#407	—	2	2	no	—
12	#368	—	2	2	no	—
13	C₂²⋊C₁₆	—	2	2	no	—
14	#341	—	2	2	no	—
15	C₂³×C₄	D₄	1	1	yes	characteristic
16	C₂×M₁₆	—	2	2	no	—
17	C₂×M₁₆	—	2	2	no	—
18	C₂²×C₈	—	2	1	no	—
19	C₂²×C₈	—	2	1	no	—
20	C₂²×C₈	—	2	2	no	—
21	C₂²×C₈	—	2	2	no	—
22	C₂²×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₈	C₂³	1	1	yes	characteristic
24	M₃₂	—	2	2	no	—
25	M₃₂	—	2	2	no	—
26	C₂×C₁₆	—	2	1	no	—
27	C₂×C₁₆	D₄	1	2	yes	—
28	#108	—	2	2	no	—
29	#108	—	2	2	no	—
30	#108	—	2	2	no	—
31	#108	—	2	2	no	—
32	C₂²⋊C₈	—	4	4	no	—
33	C₂⁴	Q₁₆	1	1	yes	characteristic
34	C₂²×C₄	—	2	1	no	—
35	C₂²×C₄	—	2	2	no	—
36	C₂²×C₄	—	2	1	no	—
37	C₂²×C₄	—	2	2	no	—
38	C₂²×C₄	Q₁₆	1	1	yes	characteristic
39	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
40	M₁₆	—	4	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	—	2	2	no	—
46	C₂×C₈	—	4	1	no	—
47	C₂×C₈	—	4	1	no	—
48	C₂×C₈	—	4	2	no	—
49	C₂×C₈	—	4	2	no	—
50	C₂×C₈	—	4	2	no	—
51	C₂×C₈	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈	#41	1	2	yes	—
53	C₂×C₈	#41	1	2	yes	—
54	C₂×C₈	#41	1	1	yes	characteristic
55	C₁₆	—	2	2	no	—
56	C₁₆	—	4	1	no	—
57	C₂³	—	2	1	no	—
58	C₂³	—	2	2	no	—
59	C₂³	—	4	2	no	—
60	C₂³	C₂×Q₁₆	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	—	4	1	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	#121	1	1	yes	characteristic
67	C₂×C₄	#123	1	2	yes	—
68	C₂×C₄	#135	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	—	8	2	no	—
73	C₈	#123	1	1	yes	characteristic
74	C₈	#124	1	1	yes	characteristic
75	C₈	#126	1	2	yes	—
76	C₂²	—	2	2	no	—
77	C₂²	—	2	4	no	—
78	C₂²	—	4	1	no	—
79	C₂²	—	4	1	no	—
80	C₂²	—	4	2	no	—
81	C₂²	—	4	1	no	—
82	C₂²	#467	1	1	yes	characteristic
83	C₂²	#484	1	2	yes	—
84	C₄	—	4	1	no	—
85	C₄	#397	1	1	yes	characteristic
86	C₄	#469	1	1	yes	characteristic
87	C₄	#482	1	2	yes	—
88	C₂	—	2	1	no	—
89	C₂	—	2	2	no	—
90	C₂	—	4	2	no	—
91	C₂	#2076	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#2076
prime radical	C₂⁴	Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ 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. #1388	C₂	semidirect
5. #1385	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
#2112	2
#1810	2
#1388	2
#1385	2