Subgroups — #128221

Order 512.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12592	C₂	1	1	yes	maximal, characteristic
2	#12541	C₂	1	1	yes	maximal, characteristic
3	#8137	C₂	1	1	yes	maximal, characteristic
4	#7141	C₂	1	2	yes	maximal
5	#7140	C₂	1	2	yes	maximal
6	C₂×#347	C₄	1	4	yes	—
7	C₄×M₃₂	C₂²	1	2	yes	—
8	C₂×#346	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	#1726	C₂²	1	1	yes	characteristic
11	#1450	C₂²	1	2	yes	—
12	#1374	C₄	1	4	yes	—
13	#1275	C₄	1	2	yes	—
14	#1274	C₄	1	2	yes	—
15	C₂×M₃₂	C₂×C₄	1	4	yes	—
16	C₂²×C₁₆	C₂×C₄	1	2	yes	—
17	C₂²×C₁₆	C₂×C₄	1	2	yes	—
18	#431	C₂×C₄	1	4	yes	—
19	C₂×#97	C₂×C₄	1	2	yes	—
20	C₂×#97	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
23	#376	C₂×C₄	1	1	yes	characteristic
24	#347	—	2	8	no	—
25	#346	—	2	2	no	—
26	#346	—	2	2	no	—
27	C₄×C₁₆	—	2	2	no	—
28	C₄×C₁₆	—	2	2	no	—
29	#322	—	2	2	no	—
30	C₈²	—	2	2	no	—
31	C₂²×C₈	C₄²	1	2	yes	—
32	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄²	C₄²	1	2	yes	—
35	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
36	M₃₂	—	2	4	no	—
37	M₃₂	M₁₆	1	8	yes	—
38	C₂×C₁₆	—	2	4	no	—
39	C₂×C₁₆	—	2	2	no	—
40	C₂×C₁₆	—	2	2	no	—
41	C₂×C₁₆	—	2	4	no	—
42	C₂×C₁₆	—	2	4	no	—
43	C₄⋊C₈	—	2	4	no	—
44	C₂²⋊C₈	—	2	4	no	—
45	#97	—	2	4	no	—
46	#97	—	2	2	no	—
47	C₄×C₈	—	2	4	no	—
48	C₄×C₈	—	2	4	no	—
49	C₄×C₈	#41	1	2	yes	—
50	C₄×C₈	#41	1	2	yes	—
51	#95	—	2	2	no	—
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
54	C₂²×C₄	#131	1	2	yes	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	4	4	no	—
59	C₂×C₈	#117	1	2	yes	—
60	C₂×C₈	#117	1	1	yes	characteristic
61	C₂×C₈	#117	1	1	yes	characteristic
62	C₂×C₈	C₂×M₁₆	1	4	yes	—
63	C₂×C₈	#131	1	4	yes	—
64	C₄²	—	2	2	no	—
65	C₄²	—	2	2	no	—
66	C₄²	#117	1	2	yes	—
67	C₄²	#131	1	2	yes	—
68	C₁₆	—	2	8	no	—
69	C₁₆	—	4	2	no	—
70	C₁₆	—	4	2	no	—
71	C₂³	#405	1	1	yes	characteristic
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	4	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	#376	1	1	yes	characteristic
76	C₂×C₄	C₄×M₁₆	1	2	yes	—
77	C₂×C₄	#405	1	1	yes	characteristic
78	C₂×C₄	#405	1	2	yes	—
79	C₂×C₄	#433	1	4	yes	—
80	C₈	—	4	2	no	—
81	C₈	—	8	4	no	—
82	C₈	#432	1	4	yes	—
83	C₈	#433	1	2	yes	—
84	C₈	#433	1	2	yes	—
85	C₂²	—	2	2	no	—
86	C₂²	#1453	1	2	yes	—
87	C₂²	#1746	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	—	8	2	no	—
90	C₄	#1453	1	2	yes	—
91	C₄	#1745	1	1	yes	characteristic
92	C₄	#1746	1	1	yes	characteristic
93	C₂	—	2	2	no	—
94	C₂	#8134	1	1	yes	characteristic
95	C₂	#12597	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1746
prime radical	C₂³	#405
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#12592	2
#12541	2
#8137	2
#7141	2
#7140	2