Subgroups — #12890

Order 256.

159 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#346	C₂	1	1	yes	maximal, characteristic
2	#1846	C₂	1	1	yes	maximal, characteristic
3	#1845	C₂	1	1	yes	maximal, characteristic
4	#1380	C₂	1	4	yes	maximal
5	C₂²×C₁₆	C₂²	1	2	yes	—
6	C₂×#107	—	2	1	no	—
7	C₂×#107	C₂²	1	2	yes	—
8	C₂×#107	C₂²	1	2	yes	—
9	C₂×#106	—	2	1	no	—
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	#384	—	2	1	no	—
12	#382	—	2	1	no	—
13	#346	C₄	1	4	yes	complemented
14	C₂²×C₈	D₄	1	1	yes	characteristic
15	C₂²×C₈	C₂³	1	1	yes	characteristic
16	C₂×C₄⋊C₄	—	2	2	no	—
17	C₂×C₄⋊C₄	—	2	2	no	—
18	C₂×C₄⋊C₄	—	4	2	no	—
19	C₂×C₄²	D₄	1	1	yes	characteristic
20	C₂×C₁₆	—	2	2	no	—
21	C₂×C₁₆	—	2	2	no	—
22	C₂×C₁₆	C₂×C₄	1	4	yes	—
23	#107	—	2	4	no	—
24	#107	—	2	4	no	—
25	#107	—	4	2	no	—
26	#106	—	4	2	no	—
27	C₄×C₈	C₂×C₄	1	2	yes	—
28	C₄×C₈	Q₈	1	2	yes	—
29	#95	—	4	2	no	—
30	C₂²×C₄	—	4	2	no	—
31	C₂²×C₄	—	4	2	no	—
32	C₂²×C₄	D₈	1	2	yes	—
33	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈	C₄⋊C₄	1	2	yes	—
35	C₂×C₈	SD₁₆	1	2	yes	—
36	C₂×C₈	SD₁₆	1	2	yes	—
37	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈	#41	1	2	yes	—
40	C₂×C₈	#41	1	2	yes	—
41	C₄⋊C₄	—	4	2	no	—
42	C₄⋊C₄	—	4	2	no	—
43	C₄⋊C₄	—	4	2	no	—
44	C₄⋊C₄	—	4	2	no	—
45	C₄⋊C₄	—	8	4	no	—
46	C₄²	C₄⋊C₄	1	2	yes	—
47	C₄²	Q₁₆	1	2	yes	—
48	C₁₆	—	2	4	no	—
49	C₂³	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	8	2	no	—
53	C₂×C₄	—	8	2	no	—
54	C₂×C₄	—	8	4	no	—
55	C₂×C₄	—	8	2	no	—
56	C₂×C₄	—	8	2	no	—
57	C₂×C₄	#107	1	4	yes	—
58	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄	#124	1	1	yes	characteristic
60	C₂×C₄	#124	1	1	yes	characteristic
61	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
64	C₈	#106	1	4	yes	—
65	C₈	#117	1	2	yes	—
66	C₈	#125	1	2	yes	—
67	C₂²	C₂×#107	1	1	yes	characteristic
68	C₂²	#486	1	1	yes	characteristic
69	C₂²	#486	1	1	yes	characteristic
70	C₂²	#509	1	2	yes	—
71	C₂²	#510	1	2	yes	—
72	C₄	—	2	4	no	—
73	C₄	—	16	2	no	—
74	C₄	—	16	2	no	—
75	C₄	#382	1	1	yes	characteristic
76	C₄	C₂×#106	1	1	yes	characteristic
77	C₄	#487	1	1	yes	characteristic
78	C₄	#487	1	1	yes	characteristic
79	C₂	#1834	1	1	yes	characteristic
80	C₂	#2159	1	2	yes	—
81	C₂	#2243	1	2	yes	—
82	C₂	#2244	1	1	yes	characteristic
83	C₂	#2245	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₈
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ 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
13. #346	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#346	2
#1846	2
#1845	2
#1380	2