Subgroups — #12903

Order 256.

167 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#365	C₂	1	4	yes	maximal
2	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
3	#1848	C₂	1	1	yes	maximal, characteristic
4	#1844	C₂	1	1	yes	maximal, characteristic
5	C₂×M₃₂	C₂²	1	2	yes	—
6	C₂×#108	—	2	1	no	—
7	C₂×#108	C₂²	1	2	yes	—
8	C₂×#106	—	2	1	no	—
9	C₂×#106	C₂²	1	2	yes	—
10	#423	—	2	1	no	—
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	#384	—	2	1	no	—
13	#365	—	2	8	no	—
14	#346	C₄	1	2	yes	complemented
15	C₄×C₁₆	C₄	1	2	yes	complemented
16	C₂×M₁₆	—	2	2	no	—
17	C₂²×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₈	C₂³	1	1	yes	characteristic
19	C₂×C₄⋊C₄	—	2	2	no	—
20	C₂×C₄⋊C₄	—	4	2	no	—
21	C₂×C₄²	D₄	1	1	yes	characteristic
22	M₃₂	D₄	1	4	yes	—
23	M₃₂	Q₈	1	4	yes	—
24	C₂×C₁₆	C₂×C₄	1	4	yes	—
25	#108	—	2	4	no	—
26	#108	—	4	2	no	—
27	#106	—	2	4	no	—
28	#106	—	4	2	no	—
29	C₄⋊C₈	—	4	2	no	—
30	C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈	Q₈	1	1	yes	characteristic
33	C₄×C₈	Q₈	1	1	yes	characteristic
34	C₂²×C₄	—	4	2	no	—
35	C₂²×C₄	SD₁₆	1	2	yes	—
36	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
37	M₁₆	—	4	2	no	—
38	M₁₆	—	4	2	no	—
39	C₂×C₈	—	4	2	no	—
40	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈	SD₁₆	1	2	yes	—
43	C₂×C₈	SD₁₆	1	2	yes	—
44	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈	C₂×D₄	1	2	yes	—
46	C₂×C₈	C₂×Q₈	1	2	yes	—
47	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
48	C₄⋊C₄	—	4	2	no	—
49	C₄⋊C₄	—	4	2	no	—
50	C₄⋊C₄	—	8	4	no	—
51	C₄²	C₄⋊C₄	1	1	yes	characteristic
52	C₄²	C₄⋊C₄	1	1	yes	characteristic
53	C₄²	SD₁₆	1	2	yes	—
54	C₁₆	C₄⋊C₄	1	8	yes	complemented
55	C₂³	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	—	8	4	no	—
58	C₂×C₄	—	8	2	no	—
59	C₂×C₄	#106	1	2	yes	—
60	C₂×C₄	#106	1	2	yes	—
61	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄	#127	1	1	yes	characteristic
63	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
67	C₈	—	8	2	no	—
68	C₈	#106	1	2	yes	—
69	C₈	#106	1	2	yes	—
70	C₈	C₂×C₄⋊C₄	1	2	yes	—
71	C₈	C₄⋊Q₈	1	2	yes	—
72	C₂²	—	2	2	no	—
73	C₂²	C₂×#106	1	1	yes	characteristic
74	C₂²	#492	1	1	yes	characteristic
75	C₂²	#498	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	16	2	no	—
78	C₄	#365	1	4	yes	complemented
79	C₄	#384	1	1	yes	characteristic
80	C₄	C₂×#106	1	1	yes	characteristic
81	C₄	#492	1	1	yes	characteristic
82	C₄	#498	1	1	yes	characteristic
83	C₂	—	2	1	no	—
84	C₂	—	2	1	no	—
85	C₂	#1844	1	1	yes	characteristic
86	C₂	C₂×#365	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×#106
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×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
14. #346	C₄	semidirect
15. C₄×C₁₆	C₄	semidirect
54. C₁₆	C₄⋊C₄	semidirect
78. C₄	#365	semidirect

Maximal subgroups

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

subgroup	index
C₂×#365	2
C₄×M₃₂	2
#1848	2
#1844	2