Finite Group Databasebeta

Subgroups — #125303

Order 512.

306 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 102 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	#33258	C₂	1	1	yes	maximal, characteristic
2	#12590	C₂	1	1	yes	maximal, characteristic
3	#8274	C₂	1	1	yes	maximal, characteristic
4	#7120	C₂	1	4	yes	maximal
5	#2859	—	2	1	no	—
6	C₂×C₄×C₁₆	—	2	1	no	—
7	#1942	—	2	1	no	—
8	#1942	—	2	1	no	—
9	#1942	C₄	1	2	yes	—
10	#1942	C₂²	1	2	yes	—
11	#1845	C₄	1	1	yes	characteristic
12	#1845	C₄	1	1	yes	characteristic
13	#1834	—	2	1	no	—
14	#1767	—	2	1	no	—
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	C₂×#335	—	2	1	no	—
17	#1374	—	2	1	no	—
18	#1374	C₂²	1	2	yes	—
19	#1277	C₂²	1	2	yes	—
20	C₂×#125	—	4	1	no	—
21	C₂²×C₁₆	—	2	2	no	—
22	C₂×#107	—	2	1	no	—
23	C₂×#107	—	2	1	no	—
24	C₂×#107	C₂×C₄	1	2	yes	—
25	C₂×#107	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊C₈	—	2	2	no	—
27	C₂×#97	—	4	1	no	—
28	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₈	D₄	1	1	yes	characteristic
31	C₂×C₄×C₈	D₄	1	1	yes	characteristic
32	C₂×C₄×C₈	D₄	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
34	#389	—	2	2	no	—
35	#389	—	2	2	no	—
36	#389	—	4	1	no	—
37	#389	—	4	1	no	—
38	#384	—	2	1	no	—
39	#384	—	2	1	no	—
40	#382	—	4	1	no	—
41	C₄³	D₄	1	1	yes	characteristic
42	C₄×C₁₆	—	2	4	no	—
43	#340	—	2	2	no	—
44	#340	—	2	2	no	—
45	#336	—	4	1	no	—
46	#335	—	2	4	no	—
47	C₂²×C₈	—	4	2	no	—
48	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
52	#125	—	8	2	no	—
53	#125	—	8	2	no	—
54	C₂×C₄⋊C₄	—	2	2	no	—
55	C₂×C₄⋊C₄	—	2	2	no	—
56	C₂×C₄⋊C₄	—	4	1	no	—
57	C₂×C₄⋊C₄	—	4	1	no	—
58	C₂×C₄⋊C₄	—	4	2	no	—
59	C₂×C₄⋊C₄	—	4	1	no	—
60	C₂×C₄⋊C₄	—	4	1	no	—
61	C₂×C₄⋊C₄	—	4	2	no	—
62	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄²	D₈	1	1	yes	characteristic
65	C₂×C₄²	D₈	1	1	yes	characteristic
66	C₂×C₄²	SD₁₆	1	1	yes	characteristic
67	C₂×C₄²	SD₁₆	1	1	yes	characteristic
68	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
69	C₂×C₁₆	—	2	4	no	—
70	C₂×C₁₆	—	4	2	no	—
71	C₂×C₁₆	—	4	2	no	—
72	#107	—	2	4	no	—
73	#107	—	4	2	no	—
74	#107	—	4	2	no	—
75	#107	—	4	2	no	—
76	C₄⋊C₈	—	2	4	no	—
77	C₄⋊C₈	—	4	2	no	—
78	#97	—	8	2	no	—
79	C₄×C₈	—	2	2	no	—
80	C₄×C₈	—	2	2	no	—
81	C₄×C₈	—	2	2	no	—
82	C₄×C₈	—	2	2	no	—
83	C₄×C₈	—	2	2	no	—
84	C₄×C₈	—	4	1	no	—
85	#95	—	4	2	no	—
86	#95	—	4	2	no	—
87	C₂²×C₄	—	4	2	no	—
88	C₂²×C₄	—	4	2	no	—
89	C₂²×C₄	#102	1	1	yes	characteristic
90	C₂²×C₄	#102	1	1	yes	characteristic
91	C₂²×C₄	#102	1	1	yes	characteristic
92	C₂²×C₄	#102	1	1	yes	characteristic
93	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
95	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	4	4	no	—
104	C₂×C₈	—	8	2	no	—
105	C₂×C₈	—	8	2	no	—
106	C₂×C₈	#101	1	2	yes	—
107	C₂×C₈	#104	1	2	yes	—
108	C₂×C₈	#104	1	2	yes	—
109	C₂×C₈	#131	1	2	yes	—
110	C₂×C₈	#137	1	2	yes	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	8	2	no	—
116	C₄⋊C₄	—	8	2	no	—
117	C₄⋊C₄	—	8	2	no	—
118	C₄⋊C₄	—	8	4	no	—
119	C₄⋊C₄	—	8	2	no	—
120	C₄⋊C₄	—	8	2	no	—
121	C₄⋊C₄	—	8	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	2	1	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	4	1	no	—
131	C₄²	D₁₆	1	2	yes	—
132	C₄²	SD₃₂	1	2	yes	—
133	C₁₆	—	4	4	no	—
134	C₂³	C₂×#102	1	1	yes	characteristic
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	#357	1	2	yes	—
155	C₂×C₄	#357	1	2	yes	—
156	C₂×C₄	#359	1	2	yes	—
157	C₂×C₄	#362	1	2	yes	—
158	C₂×C₄	#408	1	1	yes	characteristic
159	C₂×C₄	#410	1	1	yes	characteristic
160	C₂×C₄	C₂×#101	1	1	yes	characteristic
161	C₂×C₄	#419	1	1	yes	characteristic
162	C₂×C₄	C₂×#104	1	1	yes	characteristic
163	C₂×C₄	C₂×#104	1	1	yes	characteristic
164	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
165	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	—	2	2	no	—
168	C₈	—	4	1	no	—
169	C₈	—	8	4	no	—
170	C₈	#330	1	4	yes	—
171	C₂²	#1492	1	1	yes	characteristic
172	C₂²	#1502	1	1	yes	characteristic
173	C₂²	#1546	1	1	yes	characteristic
174	C₂²	C₂×#357	1	1	yes	characteristic
175	C₂²	C₂×#359	1	1	yes	characteristic
176	C₂²	#2145	1	1	yes	characteristic
177	C₂²	C₂×#362	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	2	2	no	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	1	no	—
185	C₄	—	16	2	no	—
186	C₄	—	16	2	no	—
187	C₄	#1470	1	1	yes	characteristic
188	C₄	#1485	1	1	yes	characteristic
189	C₄	#1506	1	1	yes	characteristic
190	C₄	C₂×#330	1	1	yes	characteristic
191	C₂	#7680	1	1	yes	characteristic
192	C₂	#11671	1	1	yes	characteristic
193	C₂	#11679	1	1	yes	characteristic
194	C₂	#11742	1	1	yes	characteristic
195	C₂	#11748	1	1	yes	characteristic
196	C₂	#11967	1	1	yes	characteristic
197	C₂	#11971	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#33258	2
#12590	2
#8274	2
#7120	2