Subgroups — #8790

Order 256.

147 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1848	C₂	1	1	yes	maximal, characteristic
2	#1846	C₂	1	1	yes	maximal, characteristic
3	#1450	C₂	1	1	yes	maximal, characteristic
4	#1286	C₂	1	4	yes	maximal
5	#431	C₂²	1	2	yes	—
6	C₂×#108	—	2	1	no	—
7	C₂×#108	—	2	1	no	—
8	C₂×#107	—	2	1	no	—
9	C₂×#106	—	2	1	no	—
10	#423	C₂²	1	2	yes	—
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	#382	C₂²	1	2	yes	—
13	#322	C₄	1	2	yes	complemented
14	C₈²	C₄	1	2	yes	complemented
15	C₂×M₁₆	—	2	2	no	—
16	C₂²×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₈	D₄	1	1	yes	characteristic
18	C₂×C₄⋊C₄	—	2	2	no	—
19	C₂×C₄²	C₂³	1	1	yes	characteristic
20	#108	—	4	2	no	—
21	#108	—	4	2	no	—
22	#107	—	4	2	no	—
23	#106	—	4	2	no	—
24	C₄⋊C₈	—	2	2	no	—
25	C₄⋊C₈	—	2	4	no	—
26	C₂²⋊C₈	—	2	2	no	—
27	C₄×C₈	C₂×C₄	1	4	yes	—
28	C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈	Q₈	1	1	yes	characteristic
31	C₄×C₈	Q₈	1	1	yes	characteristic
32	#95	—	2	4	no	—
33	C₂²×C₄	—	4	2	no	—
34	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄	#41	1	2	yes	—
36	M₁₆	—	4	2	no	—
37	M₁₆	—	4	2	no	—
38	C₂×C₈	—	2	4	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	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈	D₈	1	2	yes	—
45	C₂×C₈	SD₁₆	1	2	yes	—
46	C₂×C₈	SD₁₆	1	2	yes	—
47	C₂×C₈	Q₁₆	1	2	yes	—
48	C₄⋊C₄	—	4	2	no	—
49	C₄⋊C₄	—	4	2	no	—
50	C₄²	C₂²×C₄	1	1	yes	characteristic
51	C₄²	C₂×Q₈	1	1	yes	characteristic
52	C₄²	#41	1	2	yes	—
53	C₂³	#124	1	1	yes	characteristic
54	C₂×C₄	—	4	2	no	—
55	C₂×C₄	—	8	4	no	—
56	C₂×C₄	—	8	2	no	—
57	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄	#117	1	2	yes	—
59	C₂×C₄	#124	1	1	yes	characteristic
60	C₂×C₄	#125	1	2	yes	—
61	C₂×C₄	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
65	C₈	—	4	4	no	—
66	C₈	—	8	2	no	—
67	C₈	#106	1	2	yes	—
68	C₈	#106	1	2	yes	—
69	C₈	#107	1	2	yes	—
70	C₈	#107	1	2	yes	—
71	C₂²	—	2	2	no	—
72	C₂²	#382	1	1	yes	characteristic
73	C₂²	#486	1	1	yes	characteristic
74	C₂²	#487	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	16	2	no	—
78	C₄	C₂×#106	1	1	yes	characteristic
79	C₄	C₂×#107	1	1	yes	characteristic
80	C₄	#486	1	1	yes	characteristic
81	C₄	#487	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	2	1	no	—
84	C₂	#1833	1	2	yes	—
85	C₂	#1834	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#382
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
13. #322	C₄	semidirect
14. C₈²	C₄	semidirect

Maximal subgroups

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

subgroup	index
#1848	2
#1846	2
#1450	2
#1286	2