Finite Group Databasebeta

Subgroups — #152809

Order 512.

207 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 92 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	#13630	C₂	1	1	yes	maximal, characteristic, complemented
2	#13558	C₂	1	1	yes	maximal, characteristic
3	C₂×#1401	C₂	1	1	yes	maximal, characteristic
4	#13549	C₂	1	1	yes	maximal, characteristic, complemented
5	#12613	C₂	1	1	yes	maximal, characteristic
6	#7470	C₂	1	1	yes	maximal, characteristic, complemented
7	#7469	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#504	C₂²	1	1	yes	characteristic
9	C₂×M₆₄	C₂²	1	1	yes	characteristic
10	C₂×M₆₄	C₂²	1	1	yes	characteristic
11	C₂×M₆₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃₂	C₂²	1	1	yes	characteristic
13	#2115	C₄	1	1	yes	characteristic
14	#2108	C₂²	1	1	yes	characteristic
15	#2108	C₂²	1	1	yes	characteristic
16	#1757	C₄	1	1	yes	characteristic
17	#1422	—	2	2	no	—
18	#1401	—	2	2	no	—
19	#1401	—	2	1	no	—
20	#1401	—	2	1	no	—
21	#1397	—	2	1	no	—
22	C₄×C₃₂	—	2	1	no	—
23	#1378	C₄	1	1	yes	characteristic
24	#1378	C₄	1	1	yes	characteristic
25	C₂×#131	C₂×C₄	1	1	yes	characteristic
26	#504	—	2	1	no	—
27	#504	—	2	1	no	—
28	#504	—	2	1	no	—
29	#504	—	2	1	no	—
30	C₂×M₃₂	—	2	1	no	—
31	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
32	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
33	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₁₆	—	2	1	no	—
35	C₂²×C₁₆	C₂³	1	1	yes	characteristic
36	#413	C₈	1	1	yes	characteristic
37	#410	C₈	1	1	yes	characteristic
38	#405	C₂×C₄	1	1	yes	characteristic
39	#405	C₂×C₄	1	1	yes	characteristic
40	M₆₄	—	2	1	no	—
41	M₆₄	—	2	1	no	—
42	M₆₄	—	2	1	no	—
43	M₆₄	—	2	1	no	—
44	M₆₄	—	2	1	no	—
45	M₆₄	D₄	1	2	yes	—
46	C₂×C₃₂	—	2	1	no	—
47	C₂×C₃₂	—	2	1	no	—
48	C₂×C₃₂	—	2	1	no	—
49	C₂×C₃₂	—	2	1	no	—
50	C₂×C₃₂	—	2	1	no	—
51	C₂×C₃₂	D₄	1	1	yes	characteristic
52	C₂×C₃₂	D₄	1	1	yes	characteristic
53	#349	—	2	2	no	—
54	#346	—	2	1	no	—
55	#346	—	2	1	no	—
56	C₄×C₁₆	—	2	1	no	—
57	C₄×C₁₆	—	2	1	no	—
58	#344	C₈	1	1	yes	characteristic
59	#344	C₈	1	1	yes	characteristic
60	C₂×#41	C₂×C₈	1	1	yes	characteristic
61	#131	—	2	1	no	—
62	#131	—	2	1	no	—
63	#131	—	2	1	no	—
64	#131	—	2	1	no	—
65	C₂×M₁₆	—	2	1	no	—
66	C₂×M₁₆	C₂×C₈	1	1	yes	characteristic
67	C₂×M₁₆	C₂×C₈	1	1	yes	characteristic
68	C₂×M₁₆	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₈	—	2	1	no	—
70	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
71	#117	C₂×C₈	1	1	yes	characteristic
72	#117	C₂×C₈	1	1	yes	characteristic
73	M₃₂	—	2	1	no	—
74	M₃₂	—	2	1	no	—
75	M₃₂	—	2	1	no	—
76	M₃₂	—	2	1	no	—
77	M₃₂	—	2	1	no	—
78	M₃₂	—	4	1	no	—
79	M₃₂	—	4	1	no	—
80	M₃₂	M₁₆	1	2	yes	—
81	C₂×C₁₆	—	2	1	no	—
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	—	2	1	no	—
84	C₂×C₁₆	—	4	1	no	—
85	C₂×C₁₆	—	4	1	no	—
86	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
87	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
88	C₂×C₁₆	#41	1	1	yes	characteristic
89	C₂×C₁₆	#41	1	1	yes	characteristic
90	C₂×C₁₆	#41	1	1	yes	characteristic
91	C₂×C₁₆	#41	1	1	yes	characteristic
92	#101	C₁₆	1	2	yes	—
93	#100	C₁₆	1	2	yes	—
94	#99	—	2	2	no	—
95	#97	—	2	1	no	—
96	#97	—	2	1	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	2	1	no	—
99	C₃₂	—	2	2	no	—
100	C₃₂	—	2	1	no	—
101	C₃₂	—	2	1	no	—
102	C₃₂	—	4	1	no	—
103	C₃₂	—	4	1	no	—
104	#41	—	2	1	no	—
105	#41	—	2	1	no	—
106	#41	—	2	1	no	—
107	#41	—	2	1	no	—
108	C₂×Q₈	C₂×C₁₆	1	1	yes	characteristic
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	C₂×C₁₆	1	1	yes	characteristic
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
113	M₁₆	—	2	1	no	—
114	M₁₆	—	2	1	no	—
115	M₁₆	—	2	1	no	—
116	M₁₆	—	2	1	no	—
117	M₁₆	—	4	1	no	—
118	M₁₆	—	4	1	no	—
119	M₁₆	C₂×C₁₆	1	2	yes	—
120	M₁₆	C₂×C₁₆	1	2	yes	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	4	1	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	#117	1	1	yes	characteristic
127	C₂×C₈	C₄×D₄	1	1	yes	characteristic
128	C₂×C₈	C₄×D₄	1	1	yes	characteristic
129	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₈	#131	1	1	yes	characteristic
131	C₂×C₈	#131	1	1	yes	characteristic
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₂²⋊C₄	—	2	1	no	—
135	C₂²⋊C₄	M₃₂	1	2	yes	—
136	C₄²	—	2	1	no	—
137	C₄²	—	2	1	no	—
138	C₁₆	—	2	2	no	—
139	C₁₆	—	4	1	no	—
140	C₁₆	—	4	1	no	—
141	C₁₆	#121	1	1	yes	characteristic
142	C₁₆	#123	1	1	yes	characteristic
143	C₁₆	#124	1	1	yes	characteristic
144	C₁₆	#126	1	1	yes	characteristic
145	C₂³	—	2	1	no	—
146	C₂³	C₂×M₃₂	1	1	yes	characteristic
147	Q₈	—	2	1	no	—
148	Q₈	—	2	1	no	—
149	D₄	—	2	1	no	—
150	D₄	—	2	1	no	—
151	D₄	—	4	1	no	—
152	D₄	—	4	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	#431	1	1	yes	characteristic
159	C₂×C₄	C₈×D₄	1	1	yes	characteristic
160	C₂×C₄	C₈×D₄	1	1	yes	characteristic
161	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
162	C₂×C₄	#504	1	1	yes	characteristic
163	C₂×C₄	#504	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	2	2	no	—
166	C₈	—	4	1	no	—
167	C₈	#388	1	1	yes	characteristic
168	C₈	#433	1	1	yes	characteristic
169	C₈	#435	1	1	yes	characteristic
170	C₈	#436	1	1	yes	characteristic
171	C₂²	—	4	1	no	—
172	C₂²	—	4	1	no	—
173	C₂²	—	8	1	no	—
174	C₂²	#2162	1	1	yes	characteristic
175	C₂²	#2168	1	1	yes	characteristic
176	C₂²	#2169	1	1	yes	characteristic
177	C₄	—	4	2	no	—
178	C₄	—	4	1	no	—
179	C₄	—	8	1	no	—
180	C₄	#1923	1	1	yes	characteristic
181	C₄	#2164	1	1	yes	characteristic
182	C₄	D₄×C₁₆	1	1	yes	characteristic
183	C₄	D₄×C₁₆	1	1	yes	characteristic
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	—	8	1	no	—
188	C₂	#13120	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#121
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13120
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #13630	C₂	semidirect
4. #13549	C₂	semidirect
6. #7470	C₂	semidirect
7. #7469	C₂	semidirect

Maximal subgroups

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

subgroup	index
#13630	2
#13558	2
C₂×#1401	2
#13549	2
#12613	2
#7470	2
#7469	2