Finite Group Databasebeta

Subgroups — #129120

Order 512.

263 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#347	C₂	1	1	yes	maximal, characteristic
2	#12009	C₂	1	1	yes	maximal, characteristic
3	#12000	C₂	1	1	yes	maximal, characteristic
4	#8275	C₂	1	1	yes	maximal, characteristic
5	#8268	C₂	1	1	yes	maximal, characteristic
6	#7159	C₂	1	1	yes	maximal, characteristic
7	#7158	C₂	1	1	yes	maximal, characteristic
8	C₂×#347	C₄	1	2	yes	—
9	C₂×#347	C₄	1	2	yes	—
10	C₄×M₃₂	C₂²	1	1	yes	characteristic
11	C₄×M₃₂	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₈	—	2	1	no	—
13	C₄×#97	C₂²	1	1	yes	characteristic
14	#1564	C₂²	1	1	yes	characteristic
15	#1564	C₂²	1	1	yes	characteristic
16	C₂×#335	C₂²	1	1	yes	characteristic
17	C₂×#334	C₂²	1	1	yes	characteristic
18	#1450	—	2	1	no	—
19	#1372	C₄	1	2	yes	complemented
20	#1371	C₄	1	2	yes	complemented
21	#1368	C₄	1	4	yes	complemented
22	#1277	—	2	1	no	—
23	#1276	—	2	1	no	—
24	C₂×M₃₂	C₂×C₄	1	2	yes	—
25	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
26	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
27	#431	—	2	1	no	—
28	#431	—	2	1	no	—
29	C₂×C₄⋊C₈	—	2	1	no	—
30	C₂×C₄⋊C₈	—	2	1	no	—
31	C₂×#97	C₂×C₄	1	2	yes	—
32	C₂×#97	Q₈	1	2	yes	—
33	C₂×C₄×C₈	—	4	1	no	—
34	C₂×C₄×C₈	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
36	C₄³	D₄	1	1	yes	characteristic
37	#347	—	2	4	no	—
38	#347	C₈	1	8	yes	complemented
39	#346	—	2	1	no	—
40	#346	C₂×C₄	1	2	yes	—
41	C₄×C₁₆	—	2	1	no	—
42	C₄×C₁₆	C₂×C₄	1	2	yes	—
43	#336	—	4	1	no	—
44	#335	—	2	1	no	—
45	#335	—	2	1	no	—
46	#335	C₂×C₄	1	2	yes	—
47	#335	C₂×C₄	1	1	yes	characteristic
48	#335	C₂×C₄	1	1	yes	characteristic
49	#334	—	2	1	no	—
50	#334	—	2	1	no	—
51	#334	C₂×C₄	1	2	yes	—
52	#334	C₂×C₄	1	2	yes	—
53	#322	—	4	1	no	—
54	C₈²	—	4	1	no	—
55	C₂²×C₈	—	4	1	no	—
56	C₂²×C₈	—	4	1	no	—
57	C₂²×C₈	C₄⋊C₄	1	2	yes	—
58	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄²	—	2	1	no	—
61	C₂×C₄²	—	2	1	no	—
62	C₂×C₄²	C₄⋊C₄	1	2	yes	—
63	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
64	M₃₂	—	2	2	no	—
65	M₃₂	—	2	2	no	—
66	M₃₂	C₂×C₈	1	4	yes	—
67	M₃₂	C₂×C₈	1	4	yes	—
68	C₂×C₁₆	—	2	2	no	—
69	C₂×C₁₆	C₄²	1	2	yes	—
70	C₂×C₁₆	C₄²	1	2	yes	—
71	C₄⋊C₈	—	2	2	no	—
72	C₄⋊C₈	—	2	1	no	—
73	C₄⋊C₈	—	2	1	no	—
74	C₄⋊C₈	—	2	2	no	—
75	C₄⋊C₈	—	2	2	no	—
76	C₄⋊C₈	—	4	1	no	—
77	C₄⋊C₈	—	4	1	no	—
78	C₂²⋊C₈	—	4	1	no	—
79	C₂²⋊C₈	—	4	1	no	—
80	#97	—	4	2	no	—
81	#97	C₂×C₈	1	4	yes	—
82	#97	M₁₆	1	4	yes	—
83	C₄×C₈	—	2	1	no	—
84	C₄×C₈	—	2	1	no	—
85	C₄×C₈	—	4	1	no	—
86	C₄×C₈	—	4	1	no	—
87	C₄×C₈	—	8	2	no	—
88	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₄×C₈	#41	1	1	yes	characteristic
91	C₄×C₈	#41	1	1	yes	characteristic
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂²×C₄	#137	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	—	4	2	no	—
101	C₂×C₈	—	4	1	no	—
102	C₂×C₈	—	4	1	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	8	1	no	—
105	C₂×C₈	—	8	1	no	—
106	C₂×C₈	—	8	1	no	—
107	C₂×C₈	—	8	1	no	—
108	C₂×C₈	C₄⋊C₈	1	4	yes	—
109	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₈	#117	1	1	yes	characteristic
111	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₈	C₂²×C₈	1	2	yes	—
114	C₂×C₈	C₂×M₁₆	1	2	yes	—
115	C₄²	—	2	2	no	—
116	C₄²	—	4	1	no	—
117	C₄²	—	4	1	no	—
118	C₄²	—	8	1	no	—
119	C₄²	C₄⋊C₈	1	4	yes	—
120	C₄²	C₄×D₄	1	1	yes	characteristic
121	C₄²	C₄×D₄	1	1	yes	characteristic
122	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₄²	#137	1	1	yes	characteristic
124	C₁₆	—	2	2	no	—
125	C₁₆	—	2	4	no	—
126	C₁₆	C₄×C₈	1	4	yes	complemented
127	C₂³	#428	1	1	yes	characteristic
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
136	C₂×C₄	#428	1	1	yes	characteristic
137	C₂×C₄	#430	1	2	yes	—
138	C₂×C₄	#440	1	1	yes	characteristic
139	C₂×C₄	#440	1	1	yes	characteristic
140	C₂×C₄	#441	1	1	yes	characteristic
141	C₂×C₄	#442	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	8	2	no	—
146	C₈	—	8	2	no	—
147	C₈	—	8	1	no	—
148	C₈	—	8	1	no	—
149	C₈	—	8	2	no	—
150	C₈	C₂×C₄×C₈	1	1	yes	characteristic
151	C₈	C₄×M₁₆	1	1	yes	characteristic
152	C₈	#431	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#1565	1	2	yes	—
156	C₂²	#1776	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	4	4	no	—
161	C₄	#1565	1	2	yes	—
162	C₄	C₄×C₄⋊C₈	1	1	yes	characteristic
163	C₄	#1778	1	1	yes	characteristic
164	C₄	#2180	1	1	yes	characteristic
165	C₄	AGL(1,ℤ/16)	1	1	yes	characteristic
166	C₂	—	2	2	no	—
167	C₂	#8275	1	1	yes	characteristic
168	C₂	#12677	1	1	yes	characteristic
169	C₂	#12678	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#442
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1776
prime radical	C₂³	#428
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#97 ◁ 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
19. #1372	C₄	semidirect
20. #1371	C₄	semidirect
21. #1368	C₄	semidirect
38. #347	C₈	semidirect
126. C₁₆	C₄×C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#347	2
#12009	2
#12000	2
#8275	2
#8268	2
#7159	2
#7158	2