Subgroups — #8820

Order 256.

219 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1845	C₂	1	1	yes	maximal, characteristic
2	#1845	C₂	1	2	yes	maximal
3	#1844	C₂	1	1	yes	maximal, characteristic
4	#1834	C₂	1	2	yes	maximal
5	C₂×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#107	—	2	4	no	—
7	C₂×#107	—	2	2	no	—
8	C₂×#106	—	2	2	no	—
9	C₂×C₄×C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	#384	C₂²	1	1	yes	characteristic
12	#384	C₂²	1	2	yes	—
13	#384	C₂²	1	1	yes	characteristic
14	#340	—	2	4	no	—
15	#322	C₄	1	4	yes	complemented
16	C₂²×C₈	D₄	1	4	yes	—
17	C₂²×C₈	D₄	1	2	yes	—
18	C₂×C₄⋊C₄	—	2	2	no	—
19	C₂×C₄⋊C₄	—	2	2	no	—
20	C₂×C₄⋊C₄	—	2	1	no	—
21	C₂×C₄⋊C₄	—	2	1	no	—
22	C₂×C₄⋊C₄	—	2	2	no	—
23	C₂×C₄⋊C₄	—	2	2	no	—
24	C₂×C₄⋊C₄	—	2	2	no	—
25	C₂×C₄²	C₂³	1	1	yes	characteristic
26	#107	—	4	8	no	—
27	#107	—	4	4	no	—
28	#106	—	4	4	no	—
29	C₄×C₈	C₂×C₄	1	4	yes	—
30	C₄×C₈	C₂×C₄	1	2	yes	—
31	C₄×C₈	Q₈	1	4	yes	—
32	C₄×C₈	Q₈	1	2	yes	—
33	C₂²×C₄	—	4	2	no	—
34	C₂²×C₄	—	4	1	no	—
35	C₂²×C₄	—	4	1	no	—
36	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄	C₂×D₄	1	2	yes	—
38	C₂×C₈	—	2	4	no	—
39	C₂×C₈	—	2	4	no	—
40	C₂×C₈	C₄⋊C₄	1	8	yes	—
41	C₂×C₈	C₄⋊C₄	1	4	yes	—
42	C₂×C₈	D₈	1	4	yes	—
43	C₂×C₈	Q₁₆	1	4	yes	—
44	C₄⋊C₄	—	4	2	no	—
45	C₄⋊C₄	—	4	2	no	—
46	C₄⋊C₄	—	4	4	no	—
47	C₄⋊C₄	—	4	4	no	—
48	C₄⋊C₄	—	4	4	no	—
49	C₄⋊C₄	—	4	4	no	—
50	C₄⋊C₄	—	4	4	no	—
51	C₄²	C₂²×C₄	1	1	yes	characteristic
52	C₄²	C₂×Q₈	1	1	yes	characteristic
53	C₄²	C₂×Q₈	1	2	yes	—
54	C₂³	#127	1	1	yes	characteristic
55	C₂×C₄	—	8	1	no	—
56	C₂×C₄	—	8	2	no	—
57	C₂×C₄	—	8	1	no	—
58	C₂×C₄	—	8	2	no	—
59	C₂×C₄	—	8	2	no	—
60	C₂×C₄	—	8	2	no	—
61	C₂×C₄	—	8	2	no	—
62	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
64	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄	C₂×D₈	1	2	yes	—
67	C₂×C₄	C₂×Q₁₆	1	2	yes	—
68	C₂×C₄	AGL(1,ℤ/8)	1	4	yes	—
69	C₂×C₄	#137	1	4	yes	—
70	C₈	—	2	8	no	—
71	C₈	#107	1	8	yes	complemented
72	C₂²	#384	1	1	yes	characteristic
73	C₂²	#493	1	1	yes	characteristic
74	C₂²	#494	1	1	yes	characteristic
75	C₂²	#496	1	2	yes	—
76	C₂²	#497	1	2	yes	—
77	C₄	—	16	1	no	—
78	C₄	—	16	2	no	—
79	C₄	—	16	1	no	—
80	C₄	C₂×#107	1	2	yes	—
81	C₄	#428	1	4	yes	—
82	C₄	#500	1	2	yes	—
83	C₄	#501	1	4	yes	—
84	C₂	#1721	1	1	yes	characteristic
85	C₂	#1722	1	1	yes	characteristic
86	C₂	#1723	1	2	yes	—
87	C₂	#1845	1	1	yes	characteristic
88	C₂	#1847	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³	#127
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ 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
15. #322	C₄	semidirect
71. C₈	#107	semidirect

Maximal subgroups

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

subgroup	index
#1845	2
#1844	2
#1834	2
C₂×#322	2