Subgroups — #12908

Order 256.

175 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#367	C₂	1	2	yes	maximal
2	C₂×#366	C₂	1	2	yes	maximal
3	C₂×#346	C₂	1	1	yes	maximal, characteristic
4	#1846	C₂	1	1	yes	maximal, characteristic
5	#1845	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₆	C₂²	1	2	yes	—
7	C₂×#107	—	2	1	no	—
8	C₂×#107	C₂²	1	2	yes	—
9	C₂×#107	C₂²	1	1	yes	characteristic
10	C₂×#107	C₂²	1	1	yes	characteristic
11	C₂×#106	—	2	1	no	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#384	—	2	1	no	—
14	#382	—	2	1	no	—
15	#367	—	2	4	no	—
16	#366	—	2	4	no	—
17	#346	C₄	1	4	yes	complemented
18	C₂²×C₈	D₄	1	1	yes	characteristic
19	C₂²×C₈	C₂³	1	1	yes	characteristic
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	—	4	2	no	—
24	C₂×C₄²	D₄	1	1	yes	characteristic
25	C₂×C₁₆	C₂×C₄	1	4	yes	—
26	C₂×C₁₆	D₄	1	4	yes	—
27	C₂×C₁₆	Q₈	1	4	yes	—
28	#107	—	2	2	no	—
29	#107	—	2	2	no	—
30	#107	—	2	2	no	—
31	#107	—	2	2	no	—
32	#107	—	4	2	no	—
33	#106	—	4	2	no	—
34	C₄×C₈	C₂×C₄	1	2	yes	—
35	C₄×C₈	Q₈	1	2	yes	—
36	#95	—	4	2	no	—
37	C₂²×C₄	—	4	1	no	—
38	C₂²×C₄	—	4	1	no	—
39	C₂²×C₄	—	4	2	no	—
40	C₂²×C₄	D₈	1	2	yes	—
41	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈	C₄⋊C₄	1	2	yes	—
43	C₂×C₈	D₈	1	2	yes	—
44	C₂×C₈	Q₁₆	1	2	yes	—
45	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈	C₂×D₄	1	2	yes	—
47	C₂×C₈	C₂×Q₈	1	2	yes	—
48	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₄⋊C₄	—	4	1	no	—
50	C₄⋊C₄	—	4	2	no	—
51	C₄⋊C₄	—	4	1	no	—
52	C₄⋊C₄	—	4	1	no	—
53	C₄⋊C₄	—	4	1	no	—
54	C₄⋊C₄	—	4	2	no	—
55	C₄⋊C₄	—	8	4	no	—
56	C₄²	C₄⋊C₄	1	2	yes	—
57	C₄²	Q₁₆	1	2	yes	—
58	C₁₆	C₄⋊C₄	1	8	yes	complemented
59	C₂³	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	8	1	no	—
63	C₂×C₄	—	8	2	no	—
64	C₂×C₄	—	8	1	no	—
65	C₂×C₄	—	8	4	no	—
66	C₂×C₄	—	8	2	no	—
67	C₂×C₄	—	8	2	no	—
68	C₂×C₄	#107	1	4	yes	—
69	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄	#127	1	1	yes	characteristic
71	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
75	C₈	#107	1	4	yes	—
76	C₈	C₂×C₄⋊C₄	1	2	yes	—
77	C₈	C₄⋊Q₈	1	2	yes	—
78	C₂²	C₂×#107	1	1	yes	characteristic
79	C₂²	#493	1	1	yes	characteristic
80	C₂²	#500	1	1	yes	characteristic
81	C₂²	#509	1	2	yes	—
82	C₂²	#510	1	2	yes	—
83	C₄	—	2	4	no	—
84	C₄	—	16	2	no	—
85	C₄	—	16	1	no	—
86	C₄	—	16	1	no	—
87	C₄	#384	1	1	yes	characteristic
88	C₄	C₂×#107	1	1	yes	characteristic
89	C₄	#494	1	1	yes	characteristic
90	C₄	#500	1	1	yes	characteristic
91	C₂	#1845	1	1	yes	characteristic
92	C₂	#2159	1	2	yes	—
93	C₂	#2250	1	1	yes	characteristic
94	C₂	#2251	1	1	yes	characteristic
95	C₂	#2255	1	2	yes	—

	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
17. #346	C₄	semidirect
58. C₁₆	C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#367	2
C₂×#366	2
C₂×#346	2
#1846	2
#1845	2