Subgroups — #135692

Order 512.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11412	C₂	1	1	yes	maximal, characteristic
2	#8797	C₂	1	1	yes	maximal, characteristic
3	#8791	C₂	1	1	yes	maximal, characteristic
4	#7220	C₂	1	2	yes	maximal
5	#7218	C₂	1	2	yes	maximal
6	#2162	C₂²	1	2	yes	—
7	#1837	C₂²	1	1	yes	characteristic
8	#1835	C₂²	1	1	yes	characteristic
9	#1568	C₂²	1	2	yes	—
10	#1450	C₂²	1	1	yes	characteristic
11	#1312	C₄	1	2	yes	complemented
12	#1311	C₄	1	1	yes	characteristic, complemented
13	C₈×C₁₆	C₄	1	1	yes	characteristic, complemented
14	#1287	—	2	2	no	—
15	#1286	—	2	2	no	—
16	#432	—	2	2	no	—
17	#431	D₄	1	2	yes	—
18	#423	—	2	1	no	—
19	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
20	#382	—	2	1	no	—
21	C₄⋊C₁₆	—	2	2	no	—
22	C₂²⋊C₁₆	—	2	2	no	—
23	C₄×C₁₆	C₂×C₄	1	2	yes	—
24	C₄×C₁₆	C₂×C₄	1	2	yes	—
25	#341	—	2	2	no	—
26	#340	—	2	2	no	—
27	#335	—	2	2	no	—
28	#334	—	2	2	no	—
29	#322	Q₈	1	2	yes	—
30	C₈²	C₂×C₄	1	1	yes	characteristic
31	C₈²	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆	—	4	1	no	—
33	C₂²×C₈	#41	1	2	yes	—
34	C₂×C₄⋊C₄	—	4	1	no	—
35	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
36	C₂×C₁₆	—	2	2	no	—
37	C₂×C₁₆	—	2	2	no	—
38	C₄⋊C₈	—	2	2	no	—
39	C₄⋊C₈	—	4	2	no	—
40	C₄⋊C₈	—	4	2	no	—
41	C₂²⋊C₈	—	2	2	no	—
42	C₂²⋊C₈	—	4	2	no	—
43	#97	—	4	2	no	—
44	C₄×C₈	C₄⋊C₄	1	2	yes	—
45	C₄×C₈	C₄⋊C₄	1	2	yes	—
46	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
48	C₄×C₈	#41	1	2	yes	—
49	#95	—	4	2	no	—
50	C₂²×C₄	—	8	1	no	—
51	C₂²×C₄	#124	1	1	yes	characteristic
52	C₂²×C₄	#135	1	2	yes	—
53	M₁₆	—	8	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	4	2	no	—
57	C₂×C₈	—	8	2	no	—
58	C₂×C₈	—	8	1	no	—
59	C₂×C₈	#117	1	2	yes	—
60	C₂×C₈	#125	1	2	yes	—
61	C₄⋊C₄	—	8	2	no	—
62	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₄²	#124	1	1	yes	characteristic
64	C₄²	#135	1	2	yes	—
65	C₁₆	—	4	2	no	—
66	C₁₆	—	4	2	no	—
67	C₂³	#488	1	1	yes	characteristic
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	4	2	no	—
70	C₂×C₄	—	16	1	no	—
71	C₂×C₄	—	16	2	no	—
72	C₂×C₄	#382	1	1	yes	characteristic
73	C₂×C₄	#427	1	2	yes	—
74	C₂×C₄	#488	1	2	yes	—
75	C₂×C₄	#488	1	1	yes	characteristic
76	C₈	—	2	2	no	—
77	C₈	—	2	2	no	—
78	C₈	—	4	2	no	—
79	C₈	—	4	2	no	—
80	C₈	—	16	2	no	—
81	C₈	—	16	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	2	2	no	—
84	C₂²	#1835	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	2	2	no	—
87	C₄	—	2	2	no	—
88	C₄	—	32	1	no	—
89	C₄	#1833	1	2	yes	—
90	C₂	—	4	1	no	—
91	C₂	#8791	1	1	yes	characteristic
92	C₂	#12893	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#1835
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#488
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
11. #1312	C₄	semidirect
12. #1311	C₄	semidirect
13. C₈×C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
#11412	2
#8797	2
#8791	2
#7220	2
#7218	2