Subgroups — #9609

Order 256.

147 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#108	C₂	1	1	yes	maximal, characteristic
2	#1565	C₂	1	2	yes	maximal
3	C₂×#334	C₂	1	1	yes	maximal, characteristic
4	#1452	C₂	1	1	yes	maximal, characteristic
5	#1283	C₂	1	2	yes	maximal
6	#431	C₂²	1	1	yes	characteristic
7	#431	C₂²	1	1	yes	characteristic
8	C₂×#108	C₄	1	2	yes	—
9	C₂×C₄⋊C₈	C₂²	1	2	yes	—
10	C₄×M₁₆	C₂²	1	2	yes	—
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	#341	C₄	1	1	yes	characteristic
13	#341	C₄	1	1	yes	characteristic
14	#335	—	2	2	no	—
15	#334	—	2	2	no	—
16	#334	—	2	2	no	—
17	#322	—	2	2	no	—
18	C₂×M₁₆	C₂×C₄	1	2	yes	—
19	C₂×M₁₆	C₂×C₄	1	2	yes	—
20	C₂²×C₈	—	2	2	no	—
21	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²	C₂³	1	1	yes	characteristic
24	#108	—	2	4	no	—
25	C₄⋊C₈	—	2	2	no	—
26	C₄⋊C₈	—	2	2	no	—
27	C₄⋊C₈	—	2	1	no	—
28	C₄⋊C₈	D₄	1	2	yes	—
29	C₂²⋊C₈	—	2	1	no	—
30	C₂²⋊C₈	Q₈	1	2	yes	—
31	#97	—	2	2	no	—
32	C₄×C₈	—	2	1	no	—
33	C₄×C₈	—	2	1	no	—
34	C₄×C₈	—	2	2	no	—
35	C₄×C₈	D₄	1	2	yes	—
36	C₄×C₈	Q₈	1	2	yes	—
37	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄	#41	1	1	yes	characteristic
40	M₁₆	—	2	2	no	—
41	M₁₆	—	2	2	no	—
42	M₁₆	—	2	2	no	—
43	M₁₆	M₁₆	1	4	yes	—
44	C₂×C₈	—	2	2	no	—
45	C₂×C₈	—	2	2	no	—
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	—	4	2	no	—
51	C₂×C₈	—	8	2	no	—
52	C₂×C₈	C₄⋊C₄	1	2	yes	—
53	C₂×C₈	C₄⋊C₄	1	2	yes	—
54	C₄²	C₂×D₄	1	1	yes	characteristic
55	C₄²	C₂×D₄	1	1	yes	characteristic
56	C₄²	C₂×Q₈	1	1	yes	characteristic
57	C₄²	#41	1	1	yes	characteristic
58	C₂³	C₄×Q₈	1	1	yes	characteristic
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄	#121	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂×C₄	#125	1	1	yes	characteristic
66	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄	C₂×M₁₆	1	2	yes	—
68	C₂×C₄	#131	1	2	yes	—
69	C₈	—	2	2	no	—
70	C₈	—	2	2	no	—
71	C₈	—	2	4	no	—
72	C₈	—	4	2	no	—
73	C₈	—	4	2	no	—
74	C₈	—	4	2	no	—
75	C₈	—	8	2	no	—
76	C₂²	—	2	1	no	—
77	C₂²	—	2	1	no	—
78	C₂²	#389	1	1	yes	characteristic
79	C₂²	#446	1	2	yes	—
80	C₄	—	2	1	no	—
81	C₄	—	2	1	no	—
82	C₄	—	2	1	no	—
83	C₄	#423	1	1	yes	characteristic
84	C₄	#424	1	1	yes	characteristic
85	C₄	#435	1	2	yes	—
86	C₄	#471	1	2	yes	—
87	C₂	—	2	2	no	—
88	C₂	#1940	1	1	yes	characteristic
89	C₂	#1951	1	1	yes	characteristic
90	C₂	#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#389
prime radical	C₂³	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
C₄×#108	2
#1565	2
C₂×#334	2
#1452	2
#1283	2