Finite Group Databasebeta

Subgroups — #125063

Order 512.

278 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 90 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×#499	C₂	1	1	yes	maximal, characteristic
2	#12883	C₂	1	1	yes	maximal, characteristic
3	#12883	C₂	1	1	yes	maximal, characteristic
4	#7135	C₂	1	4	yes	maximal
5	C₂×#499	C₄	1	1	yes	characteristic
6	C₄×#125	C₄	1	2	yes	—
7	#2150	—	2	1	no	—
8	#2150	—	2	1	no	—
9	#1991	C₂²	1	2	yes	—
10	#1942	—	2	1	no	—
11	#1941	—	2	1	no	—
12	#1846	C₄	1	1	yes	characteristic
13	C₄×#107	—	2	1	no	—
14	C₄×#106	—	2	1	no	—
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	#1376	—	2	1	no	—
17	#1376	—	2	1	no	—
18	#1376	C₂²	1	2	yes	—
19	#1376	C₂²	1	2	yes	—
20	C₂×#125	C₂×C₄	1	2	yes	—
21	C₂×M₃₂	—	2	2	no	—
22	C₂×M₃₂	—	2	2	no	—
23	#499	—	2	4	no	—
24	C₂×#107	—	2	1	no	—
25	C₂×#107	—	2	1	no	—
26	C₂×#106	—	2	1	no	—
27	C₂×#106	—	2	1	no	—
28	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈	D₄	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	#387	—	4	2	no	—
36	#382	C₂×C₄	1	2	yes	—
37	C₄×C₄⋊C₄	—	2	2	no	—
38	C₄×C₄⋊C₄	—	4	2	no	—
39	C₄³	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₁₆	—	4	1	no	—
41	C₄⋊C₁₆	—	4	1	no	—
42	C₄⋊C₁₆	—	4	1	no	—
43	C₄⋊C₁₆	—	4	1	no	—
44	#340	—	2	2	no	—
45	#340	—	2	2	no	—
46	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
50	#125	—	2	2	no	—
51	#125	—	4	2	no	—
52	#125	M₁₆	1	4	yes	—
53	C₂×C₄⋊C₄	—	2	2	no	—
54	C₂×C₄⋊C₄	—	2	2	no	—
55	C₂×C₄⋊C₄	—	4	2	no	—
56	C₂×C₄⋊C₄	—	4	2	no	—
57	C₂×C₄²	—	2	1	no	—
58	C₂×C₄²	—	2	1	no	—
59	C₂×C₄²	—	4	2	no	—
60	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
63	M₃₂	—	4	2	no	—
64	M₃₂	—	4	2	no	—
65	M₃₂	—	4	2	no	—
66	M₃₂	—	4	2	no	—
67	C₂×C₁₆	—	4	2	no	—
68	C₂×C₁₆	—	4	2	no	—
69	#107	—	2	4	no	—
70	#107	—	4	2	no	—
71	#106	—	2	4	no	—
72	#106	—	4	2	no	—
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	—	4	1	no	—
75	C₄×C₈	—	4	1	no	—
76	C₄×C₈	—	4	1	no	—
77	C₄×C₈	SD₁₆	1	1	yes	characteristic
78	C₄×C₈	SD₁₆	1	1	yes	characteristic
79	C₄×C₈	SD₁₆	1	1	yes	characteristic
80	C₄×C₈	SD₁₆	1	1	yes	characteristic
81	C₄×C₈	Q₁₆	1	1	yes	characteristic
82	C₄×C₈	Q₁₆	1	1	yes	characteristic
83	C₄×C₈	Q₁₆	1	1	yes	characteristic
84	C₄×C₈	Q₁₆	1	1	yes	characteristic
85	#95	—	4	2	no	—
86	#95	—	4	2	no	—
87	#95	—	4	2	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	—	4	2	no	—
91	C₂²×C₄	—	4	2	no	—
92	C₂²×C₄	#100	1	1	yes	characteristic
93	C₂²×C₄	#100	1	1	yes	characteristic
94	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈	—	2	1	no	—
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	#103	1	1	yes	characteristic
104	C₂×C₈	#103	1	1	yes	characteristic
105	C₂×C₈	#103	1	1	yes	characteristic
106	C₂×C₈	#103	1	1	yes	characteristic
107	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
110	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
111	C₄⋊C₄	—	2	4	no	—
112	C₄⋊C₄	—	4	4	no	—
113	C₄⋊C₄	—	4	4	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄⋊C₄	—	8	4	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	2	1	no	—
121	C₄²	—	4	2	no	—
122	C₄²	—	4	2	no	—
123	C₄²	—	8	4	no	—
124	C₄²	#103	1	1	yes	characteristic
125	C₄²	#103	1	1	yes	characteristic
126	C₄²	#103	1	1	yes	characteristic
127	C₄²	#103	1	1	yes	characteristic
128	C₄²	C₂×M₁₆	1	2	yes	—
129	C₁₆	—	8	2	no	—
130	C₁₆	—	8	2	no	—
131	C₂³	C₂×#100	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	4	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	#333	1	1	yes	characteristic
147	C₂×C₄	#333	1	1	yes	characteristic
148	C₂×C₄	#333	1	1	yes	characteristic
149	C₂×C₄	#333	1	1	yes	characteristic
150	C₂×C₄	#407	1	1	yes	characteristic
151	C₂×C₄	#410	1	1	yes	characteristic
152	C₂×C₄	C₂×#103	1	1	yes	characteristic
153	C₂×C₄	C₂×#103	1	1	yes	characteristic
154	C₂×C₄	#421	1	1	yes	characteristic
155	C₂×C₄	#421	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#1466	1	1	yes	characteristic
167	C₂²	#1536	1	1	yes	characteristic
168	C₂²	C₂×#333	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	4	2	no	—
173	C₄	—	4	2	no	—
174	C₄	—	8	4	no	—
175	C₄	—	16	2	no	—
176	C₄	#1366	1	2	yes	—
177	C₄	#1487	1	1	yes	characteristic
178	C₄	#1487	1	1	yes	characteristic
179	C₄	#1518	1	1	yes	characteristic
180	C₄	#1518	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	#7653	1	1	yes	characteristic
184	C₂	C₂×#1366	1	1	yes	characteristic
185	C₂	#11956	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#333
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ 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
C₄×#499	2
#12883	2
#7135	2