Subgroups — #12668

Order 256.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2155	C₂	1	2	yes	maximal
2	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
3	#1772	C₂	1	2	yes	maximal
4	#1385	C₂	1	1	yes	maximal, characteristic
5	#1384	C₂	1	1	yes	maximal, characteristic
6	C₂×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×#108	C₂²	1	2	yes	—
9	#427	C₂²	1	2	yes	—
10	#405	—	2	2	no	—
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	#365	C₄	1	8	yes	complemented
13	#346	C₄	1	2	yes	complemented
14	C₄×C₁₆	C₄	1	2	yes	complemented
15	#339	—	2	2	no	—
16	C₂×M₁₆	—	2	2	no	—
17	C₂²×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₈	C₂³	1	1	yes	characteristic
19	#117	—	2	2	no	—
20	C₂×C₄²	D₄	1	1	yes	characteristic
21	M₃₂	—	2	2	no	—
22	M₃₂	C₂×C₄	1	4	yes	—
23	C₂×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₆	C₂×C₄	1	2	yes	—
25	#108	—	2	2	no	—
26	#108	C₂×C₄	1	4	yes	—
27	#107	—	2	2	no	—
28	#106	C₂×C₄	1	4	yes	—
29	#97	—	4	2	no	—
30	C₄×C₈	—	4	2	no	—
31	C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈	Q₈	1	1	yes	characteristic
34	C₄×C₈	Q₈	1	1	yes	characteristic
35	C₂²×C₄	—	2	1	no	—
36	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
37	M₁₆	—	2	4	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	C₂²×C₄	1	2	yes	—
46	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈	#41	1	2	yes	—
48	C₄⋊C₄	—	2	4	no	—
49	C₂²⋊C₄	—	4	2	no	—
50	C₄²	—	2	1	no	—
51	C₄²	—	4	2	no	—
52	C₄²	C₄⋊C₄	1	1	yes	characteristic
53	C₄²	C₄⋊C₄	1	1	yes	characteristic
54	C₁₆	—	2	2	no	—
55	C₁₆	C₄²	1	4	yes	—
56	C₂³	#135	1	1	yes	characteristic
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	—	4	4	no	—
61	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄	C₄×D₄	1	2	yes	—
63	C₂×C₄	C₂×SD₁₆	1	2	yes	—
64	C₂×C₄	#135	1	1	yes	characteristic
65	C₈	—	4	4	no	—
66	C₈	#106	1	2	yes	—
67	C₈	#106	1	1	yes	characteristic
68	C₈	#106	1	1	yes	characteristic
69	C₈	C₂×C₄²	1	1	yes	characteristic
70	C₈	#117	1	1	yes	characteristic
71	C₈	C₄×Q₈	1	2	yes	—
72	C₂²	—	4	1	no	—
73	C₂²	#427	1	1	yes	characteristic
74	C₂²	C₄×SD₁₆	1	2	yes	—
75	C₄	—	2	2	no	—
76	C₄	—	4	1	no	—
77	C₄	—	8	4	no	—
78	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
79	C₄	C₂×#106	1	1	yes	characteristic
80	C₄	C₄×SD₁₆	1	2	yes	—
81	C₂	—	2	1	no	—
82	C₂	—	2	2	no	—
83	C₂	C₄×#106	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#106
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	C₄×#106
prime radical	C₂³	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ 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
12. #365	C₄	semidirect
13. #346	C₄	semidirect
14. C₄×C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
#2155	2
C₄×M₃₂	2
#1772	2
#1385	2
#1384	2