Subgroups — #13615

Order 256.

149 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#2158	C₂	1	1	yes	maximal, characteristic, complemented
4	#1416	C₂	1	2	yes	maximal, complemented
5	#1415	C₂	1	2	yes	maximal, complemented
6	C₂×#135	—	2	1	no	—
7	#508	—	2	2	no	—
8	#508	C₄	1	4	yes	complemented
9	C₂×Q₃₂	C₂²	1	1	yes	characteristic
10	C₂×SD₃₂	—	2	1	no	—
11	C₂×D₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆	C₂²	1	1	yes	characteristic
13	#427	—	2	1	no	—
14	C₂×C₃₂	—	2	2	no	—
15	C₂×C₃₂	C₂²	1	2	yes	—
16	#367	—	2	1	no	—
17	#366	C₂²	1	2	yes	—
18	C₂×#41	—	4	1	no	—
19	#135	—	2	4	no	—
20	#135	—	4	2	no	—
21	C₂×Q₁₆	—	2	1	no	—
22	C₂×SD₁₆	—	4	1	no	—
23	C₂×D₈	—	2	1	no	—
24	C₂²×C₈	D₄	1	1	yes	characteristic
25	#117	—	4	1	no	—
26	Q₃₂	—	2	1	no	—
27	Q₃₂	C₂×C₄	1	2	yes	—
28	SD₃₂	—	2	4	no	—
29	D₁₆	—	2	1	no	—
30	D₁₆	C₂×C₄	1	2	yes	—
31	C₂×C₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₁₆	D₄	1	1	yes	characteristic
33	C₂×C₁₆	D₄	1	2	yes	—
34	C₂×C₁₆	C₂³	1	1	yes	characteristic
35	#107	—	2	2	no	—
36	#106	—	4	1	no	—
37	C₃₂	—	2	2	no	—
38	#41	—	4	4	no	—
39	#41	—	8	2	no	—
40	C₂×Q₈	—	4	1	no	—
41	C₂×D₄	—	4	1	no	—
42	C₂×D₄	—	8	1	no	—
43	C₂²×C₄	—	8	1	no	—
44	C₂²×C₄	D₈	1	1	yes	characteristic
45	Q₁₆	—	2	2	no	—
46	Q₁₆	—	4	1	no	—
47	SD₁₆	—	4	4	no	—
48	D₈	—	2	2	no	—
49	D₈	—	4	1	no	—
50	C₂×C₈	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈	D₈	1	1	yes	characteristic
52	C₂×C₈	SD₁₆	1	2	yes	—
53	C₂×C₈	C₂×D₄	1	1	yes	characteristic
54	C₄⋊C₄	—	4	2	no	—
55	C₂²⋊C₄	—	8	1	no	—
56	C₄²	—	8	1	no	—
57	C₁₆	C₂²⋊C₄	1	2	yes	—
58	C₁₆	C₂²×C₄	1	1	yes	characteristic
59	C₁₆	C₂×D₄	1	1	yes	characteristic
60	C₂³	—	8	1	no	—
61	C₂³	D₁₆	1	1	yes	characteristic
62	Q₈	—	4	2	no	—
63	Q₈	—	8	1	no	—
64	D₄	—	4	2	no	—
65	D₄	—	8	4	no	—
66	D₄	—	8	1	no	—
67	C₂×C₄	—	8	1	no	—
68	C₂×C₄	—	8	4	no	—
69	C₂×C₄	—	8	2	no	—
70	C₂×C₄	#102	1	2	yes	—
71	C₂×C₄	D₁₆	1	1	yes	characteristic
72	C₂×C₄	SD₃₂	1	2	yes	—
73	C₂×C₄	C₂×D₈	1	1	yes	characteristic
74	C₈	#102	1	2	yes	—
75	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₈	C₂×SD₁₆	1	1	yes	characteristic
77	C₂²	—	2	2	no	—
78	C₂²	—	8	2	no	—
79	C₂²	—	16	2	no	—
80	C₂²	#357	1	2	yes	—
81	C₂²	C₂×D₁₆	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	16	2	no	—
84	C₄	#357	1	2	yes	—
85	C₄	C₂×#102	1	1	yes	characteristic
86	C₄	C₂×SD₃₂	1	1	yes	characteristic
87	C₂	—	2	2	no	—
88	C₂	—	16	2	no	—
89	C₂	C₂×#357	1	1	yes	characteristic
90	C₂	#2262	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₁₆
commutator	C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₁₆	C₂³
socle	C₂²	C₂×D₁₆
prime radical	C₂×#508	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₂	C₂	semidirect
3. #2158	C₂	semidirect
4. #1416	C₂	semidirect
5. #1415	C₂	semidirect
8. #508	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508	2
C₂²×C₃₂	2
#2158	2
#1416	2
#1415	2