Subgroups — #12915

Order 256.

187 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3407	C₂	1	1	yes	maximal, characteristic
2	#2151	C₂	1	2	yes	maximal, complemented
3	#2113	C₂	1	1	yes	maximal, characteristic
4	#1757	C₂	1	1	yes	maximal, characteristic
5	#1378	C₂	1	2	yes	maximal, complemented
6	C₂×#131	C₂²	1	1	yes	characteristic
7	#504	—	2	4	no	—
8	C₂×M₃₂	—	2	2	no	—
9	C₂×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×M₃₂	C₂²	1	2	yes	—
11	C₂²×C₁₆	C₂²	1	1	yes	characteristic
12	#413	C₄	1	1	yes	characteristic
13	#410	C₄	1	1	yes	characteristic
14	#405	C₂²	1	2	yes	—
15	C₄⋊C₁₆	—	2	4	no	—
16	C₂²⋊C₁₆	—	2	2	no	—
17	#344	C₄	1	2	yes	—
18	C₂×#41	C₂×C₄	1	1	yes	characteristic
19	#131	—	2	2	no	—
20	#131	—	2	2	no	—
21	C₂×M₁₆	C₂×C₄	1	2	yes	—
22	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆	Q₈	1	2	yes	—
24	C₂²×C₈	D₄	1	2	yes	—
25	C₂²×C₈	C₂³	1	1	yes	characteristic
26	#117	C₂×C₄	1	2	yes	—
27	M₃₂	—	2	4	no	—
28	M₃₂	—	2	2	no	—
29	M₃₂	—	2	2	no	—
30	M₃₂	—	2	2	no	—
31	C₂×C₁₆	—	2	4	no	—
32	C₂×C₁₆	—	2	1	no	—
33	C₂×C₁₆	—	2	1	no	—
34	C₂×C₁₆	—	2	2	no	—
35	C₂×C₁₆	D₄	1	4	yes	—
36	#101	—	2	1	no	—
37	#100	—	2	1	no	—
38	#99	C₈	1	4	yes	—
39	#97	—	2	2	no	—
40	C₄×C₈	—	2	2	no	—
41	#41	—	2	2	no	—
42	#41	—	2	2	no	—
43	C₂×Q₈	M₁₆	1	1	yes	characteristic
44	C₂×D₄	C₂×C₈	1	2	yes	—
45	C₂×D₄	M₁₆	1	1	yes	characteristic
46	C₂²×C₄	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
48	M₁₆	—	2	2	no	—
49	M₁₆	—	2	4	no	—
50	M₁₆	—	2	2	no	—
51	M₁₆	—	2	2	no	—
52	C₂×C₈	—	2	2	no	—
53	C₂×C₈	—	2	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	C₄⋊C₄	1	2	yes	—
56	C₂×C₈	C₂×D₄	1	2	yes	—
57	C₂×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈	#41	1	2	yes	—
60	C₄⋊C₄	—	2	2	no	—
61	C₂²⋊C₄	C₂×C₈	1	4	yes	—
62	C₄²	—	2	2	no	—
63	C₁₆	—	2	4	no	—
64	C₁₆	—	4	1	no	—
65	C₁₆	—	4	1	no	—
66	C₂³	C₄⋊C₈	1	2	yes	—
67	C₂³	C₂²×C₈	1	1	yes	characteristic
68	Q₈	—	2	2	no	—
69	D₄	—	2	4	no	—
70	D₄	—	2	2	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	4	no	—
74	C₂×C₄	C₄⋊C₈	1	2	yes	—
75	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄	C₄×D₄	1	2	yes	—
77	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄	#131	1	2	yes	—
79	C₈	—	2	2	no	—
80	C₈	—	4	2	no	—
81	C₈	#120	1	1	yes	characteristic
82	C₈	C₂²⋊Q₈	1	2	yes	—
83	C₈	#123	1	1	yes	characteristic
84	C₂²	—	2	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	2	no	—
87	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₂²	C₈×D₄	1	2	yes	—
89	C₄	—	2	2	no	—
90	C₄	—	4	4	no	—
91	C₄	#385	1	1	yes	characteristic
92	C₄	#424	1	1	yes	characteristic
93	C₄	#435	1	2	yes	—
94	C₂	—	2	2	no	—
95	C₂	—	2	1	no	—
96	C₂	—	4	2	no	—
97	C₂	#1850	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#120
commutator	C₂³	C₂²×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1850
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×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. #2151	C₂	semidirect
5. #1378	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3407	2
#2151	2
#2113	2
#1757	2
#1378	2