Subgroups — #12835

Order 256.

219 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#367	C₂	1	2	yes	maximal, complemented
3	#1808	C₂	1	2	yes	maximal
4	#1380	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆	—	2	4	no	—
7	C₂²×C₁₆	C₄	1	4	yes	complemented
8	C₂²×C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×#107	C₂²	1	4	yes	—
11	#380	—	2	2	no	—
12	#367	—	2	4	no	—
13	#340	—	2	2	no	—
14	C₂³×C₄	D₄	1	1	yes	characteristic
15	C₂²×C₈	—	2	2	no	—
16	C₂²×C₈	—	2	2	no	—
17	C₂²×C₈	C₂×C₄	1	2	yes	—
18	C₂²×C₈	D₄	1	1	yes	characteristic
19	C₂²×C₈	Q₈	1	2	yes	—
20	C₂²×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊C₄	—	2	4	no	—
22	C₂×C₂²⋊C₄	—	4	2	no	—
23	C₂×C₁₆	—	2	2	no	—
24	C₂×C₁₆	—	2	8	no	—
25	C₂×C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₆	D₄	1	4	yes	—
28	#107	—	2	4	no	—
29	#107	—	2	4	no	—
30	#95	—	4	2	no	—
31	C₂⁴	D₈	1	1	yes	characteristic
32	C₂²×C₄	—	2	2	no	—
33	C₂²×C₄	—	2	2	no	—
34	C₂²×C₄	—	4	4	no	—
35	C₂²×C₄	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄	D₈	1	1	yes	characteristic
37	C₂²×C₄	Q₁₆	1	2	yes	—
38	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈	—	2	2	no	—
40	C₂×C₈	—	2	2	no	—
41	C₂×C₈	—	2	2	no	—
42	C₂×C₈	—	2	4	no	—
43	C₂×C₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈	C₂×D₄	1	2	yes	—
46	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈	#41	1	2	yes	—
48	C₄⋊C₄	—	4	4	no	—
49	C₄⋊C₄	—	4	4	no	—
50	C₂²⋊C₄	—	8	4	no	—
51	C₁₆	—	2	2	no	—
52	C₁₆	C₂²⋊C₄	1	4	yes	complemented
53	C₂³	—	2	2	no	—
54	C₂³	—	2	2	no	—
55	C₂³	#107	1	2	yes	—
56	C₂³	SD₃₂	1	4	yes	—
57	C₂³	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	—	8	4	no	—
63	C₂×C₄	—	8	4	no	—
64	C₂×C₄	—	8	4	no	—
65	C₂×C₄	#107	1	2	yes	—
66	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄	#121	1	2	yes	—
68	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄	#135	1	2	yes	—
70	C₈	—	2	2	no	—
71	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₈	#117	1	1	yes	characteristic
73	C₈	C₂²⋊Q₈	1	2	yes	—
74	C₂²	—	2	2	no	—
75	C₂²	—	2	2	no	—
76	C₂²	—	2	8	no	—
77	C₂²	#367	1	4	yes	complemented
78	C₂²	C₂×#107	1	1	yes	characteristic
79	C₂²	#466	1	2	yes	—
80	C₂²	C₂×SD₃₂	1	2	yes	—
81	C₂²	#508	1	2	yes	—
82	C₄	—	2	2	no	—
83	C₄	—	16	4	no	—
84	C₄	#380	1	1	yes	characteristic
85	C₄	#427	1	1	yes	characteristic
86	C₄	#467	1	2	yes	—
87	C₂	—	2	4	no	—
88	C₂	#1808	1	1	yes	characteristic
89	C₂	C₂×#367	1	1	yes	characteristic
90	C₂	#2158	1	1	yes	characteristic
91	C₂	#2217	1	4	yes	—

	subgroup	G/H
center, socle	C₂³	C₂×D₈
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂⁴	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×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
2. C₂×#367	C₂	semidirect
4. #1380	C₂	semidirect
7. C₂²×C₁₆	C₄	semidirect
52. C₁₆	C₂²⋊C₄	semidirect
77. C₂²	#367	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₆	2
C₂×#367	2
#1808	2
#1380	2