Subgroups — #13496

Order 256.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#367	C₂	1	1	yes	maximal, characteristic, complemented
2	#2113	C₂	1	1	yes	maximal, characteristic
3	#1816	C₂	1	2	yes	maximal
4	#1387	C₂	1	2	yes	maximal, complemented
5	#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#131	C₂²	1	1	yes	characteristic
7	C₂×M₃₂	C₂²	1	1	yes	characteristic
8	C₂²×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×#108	C₂²	1	2	yes	—
10	C₂×#107	C₂²	1	2	yes	—
11	#417	—	2	2	no	—
12	#367	—	2	2	no	—
13	C₂²⋊C₁₆	—	2	2	no	—
14	#343	—	2	2	no	—
15	C₂×#41	D₄	1	1	yes	characteristic
16	#131	—	2	2	no	—
17	#131	—	2	2	no	—
18	C₂×M₁₆	—	2	1	no	—
19	C₂×M₁₆	—	2	2	no	—
20	C₂×M₁₆	D₄	1	1	yes	characteristic
21	C₂²×C₈	—	2	1	no	—
22	C₂²×C₈	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄	—	2	2	no	—
24	M₃₂	—	2	2	no	—
25	C₂×C₁₆	—	2	2	no	—
26	C₂×C₁₆	—	2	1	no	—
27	C₂×C₁₆	D₄	1	2	yes	—
28	#108	—	2	2	no	—
29	#108	—	2	2	no	—
30	#107	—	2	2	no	—
31	#107	—	2	2	no	—
32	#103	—	4	2	no	—
33	#102	—	4	2	no	—
34	#41	—	2	2	no	—
35	#41	—	2	2	no	—
36	C₂×Q₈	D₈	1	1	yes	characteristic
37	C₂×D₄	—	2	1	no	—
38	C₂×D₄	D₈	1	1	yes	characteristic
39	C₂²×C₄	—	2	1	no	—
40	C₂²×C₄	—	4	2	no	—
41	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
42	M₁₆	—	2	2	no	—
43	M₁₆	—	4	2	no	—
44	M₁₆	—	4	2	no	—
45	M₁₆	—	4	2	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	4	1	no	—
48	C₂×C₈	—	4	1	no	—
49	C₂×C₈	—	4	2	no	—
50	C₂×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈	#41	1	2	yes	—
52	C₂×C₈	#41	1	2	yes	—
53	C₂×C₈	#41	1	1	yes	characteristic
54	C₄⋊C₄	—	4	2	no	—
55	C₄⋊C₄	—	4	2	no	—
56	C₁₆	—	2	2	no	—
57	C₁₆	—	4	1	no	—
58	C₂³	—	2	1	no	—
59	C₂³	#135	1	1	yes	characteristic
60	Q₈	—	2	2	no	—
61	D₄	—	2	2	no	—
62	D₄	—	4	2	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	4	1	no	—
65	C₂×C₄	—	4	1	no	—
66	C₂×C₄	—	8	2	no	—
67	C₂×C₄	—	16	2	no	—
68	C₂×C₄	#121	1	1	yes	characteristic
69	C₂×C₄	#123	1	2	yes	—
70	C₂×C₄	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄	#137	1	2	yes	—
72	C₈	—	4	1	no	—
73	C₈	—	8	2	no	—
74	C₈	#123	1	1	yes	characteristic
75	C₈	#124	1	1	yes	characteristic
76	C₈	#126	1	2	yes	—
77	C₂²	—	2	2	no	—
78	C₂²	—	4	1	no	—
79	C₂²	—	4	2	no	—
80	C₂²	#466	1	1	yes	characteristic
81	C₂²	#485	1	2	yes	—
82	C₄	—	4	1	no	—
83	C₄	—	16	2	no	—
84	C₄	#397	1	1	yes	characteristic
85	C₄	#470	1	1	yes	characteristic
86	C₄	#480	1	2	yes	—
87	C₂	—	2	2	no	—
88	C₂	—	8	1	no	—
89	C₂	#2075	1	1	yes	characteristic
90	C₂	#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#367	C₂	semidirect
4. #1387	C₂	semidirect
5. #1381	C₂	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
#2113	2
#1816	2
#1387	2
#1381	2