Finite Group Databasebeta

Subgroups — #99289

Order 512.

232 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 94 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	#8807	C₂	1	1	yes	maximal, characteristic
2	#8785	C₂	1	1	yes	maximal, characteristic
3	#8133	C₂	1	1	yes	maximal, characteristic
4	#7018	C₂	1	1	yes	maximal, characteristic
5	#7018	C₂	1	1	yes	maximal, characteristic
6	#7018	C₂	1	1	yes	maximal, characteristic
7	#7018	C₂	1	1	yes	maximal, characteristic
8	#1831	C₂²	1	1	yes	characteristic
9	#1831	C₂²	1	1	yes	characteristic
10	#1831	C₂²	1	1	yes	characteristic
11	#1831	C₂²	1	1	yes	characteristic
12	#1744	C₂²	1	1	yes	characteristic
13	#1744	C₂²	1	1	yes	characteristic
14	C₄²×C₈	C₂²	1	1	yes	characteristic
15	C₂×#335	—	2	1	no	—
16	C₂×#334	—	2	1	no	—
17	#1277	—	2	1	no	—
18	#1277	—	2	1	no	—
19	#1273	C₄	1	2	yes	—
20	#1273	C₄	1	2	yes	—
21	C₂×C₄⋊C₈	—	2	1	no	—
22	C₂×C₄⋊C₈	—	2	1	no	—
23	C₂×C₄⋊C₈	—	2	1	no	—
24	C₂×C₄⋊C₈	—	2	1	no	—
25	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
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	Q₈	1	1	yes	characteristic
32	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
33	C₄³	C₂³	1	1	yes	characteristic
34	#336	—	2	1	no	—
35	#336	—	2	1	no	—
36	#336	—	2	1	no	—
37	#336	—	2	1	no	—
38	#336	—	2	1	no	—
39	#336	—	2	1	no	—
40	#336	—	2	1	no	—
41	#336	—	2	1	no	—
42	#336	—	2	1	no	—
43	#336	—	2	1	no	—
44	#336	—	2	1	no	—
45	#336	—	2	1	no	—
46	#335	—	4	2	no	—
47	#334	—	4	2	no	—
48	C₂²×C₈	—	2	2	no	—
49	C₂²×C₈	—	2	2	no	—
50	C₂²×C₈	—	4	1	no	—
51	C₂²×C₈	—	4	1	no	—
52	C₂²×C₈	—	4	1	no	—
53	C₂²×C₈	—	4	1	no	—
54	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄²	#41	1	1	yes	characteristic
62	C₂×C₄²	#41	1	1	yes	characteristic
63	C₂×C₄²	#41	1	1	yes	characteristic
64	C₂×C₄²	#41	1	1	yes	characteristic
65	C₄⋊C₈	—	4	1	no	—
66	C₄⋊C₈	—	4	1	no	—
67	C₄⋊C₈	—	4	1	no	—
68	C₄⋊C₈	—	4	1	no	—
69	C₄⋊C₈	—	4	1	no	—
70	C₄⋊C₈	—	4	1	no	—
71	C₄⋊C₈	—	4	1	no	—
72	C₄⋊C₈	—	4	1	no	—
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	—	2	2	no	—
76	C₄×C₈	—	2	2	no	—
77	C₄×C₈	—	2	2	no	—
78	C₄×C₈	—	2	2	no	—
79	C₄×C₈	—	2	2	no	—
80	C₄×C₈	—	2	2	no	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	—	2	1	no	—
83	C₄×C₈	—	2	1	no	—
84	C₄×C₈	—	2	1	no	—
85	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄	#117	1	1	yes	characteristic
87	C₂²×C₄	#117	1	1	yes	characteristic
88	C₂²×C₄	#124	1	1	yes	characteristic
89	C₂²×C₄	#124	1	1	yes	characteristic
90	C₂²×C₄	#125	1	1	yes	characteristic
91	C₂²×C₄	#125	1	1	yes	characteristic
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
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	—	4	2	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	8	1	no	—
109	C₂×C₈	—	8	1	no	—
110	C₂×C₈	—	8	1	no	—
111	C₂×C₈	—	8	1	no	—
112	C₂×C₈	—	8	1	no	—
113	C₂×C₈	—	8	1	no	—
114	C₂×C₈	—	8	1	no	—
115	C₂×C₈	—	8	1	no	—
116	C₂×C₈	—	8	1	no	—
117	C₂×C₈	—	8	1	no	—
118	C₂×C₈	—	8	1	no	—
119	C₂×C₈	—	8	1	no	—
120	C₂×C₈	#108	1	2	yes	—
121	C₂×C₈	#108	1	2	yes	—
122	C₄²	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	4	1	no	—
125	C₄²	—	4	1	no	—
126	C₄²	—	4	1	no	—
127	C₄²	—	4	1	no	—
128	C₄²	#131	1	2	yes	—
129	C₄²	#131	1	2	yes	—
130	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₄²	#137	1	1	yes	characteristic
133	C₄²	#137	1	1	yes	characteristic
134	C₂³	#382	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	—	2	1	no	—
149	C₂×C₄	#424	1	1	yes	characteristic
150	C₂×C₄	#424	1	1	yes	characteristic
151	C₂×C₄	#428	1	1	yes	characteristic
152	C₂×C₄	#428	1	1	yes	characteristic
153	C₂×C₄	C₂×#108	1	1	yes	characteristic
154	C₂×C₄	C₂×#108	1	1	yes	characteristic
155	C₂×C₄	#433	1	2	yes	—
156	C₂×C₄	#433	1	2	yes	—
157	C₂×C₄	#489	1	1	yes	characteristic
158	C₂×C₄	#489	1	1	yes	characteristic
159	C₂×C₄	#490	1	1	yes	characteristic
160	C₂×C₄	#491	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	—	4	1	no	—
164	C₈	—	4	1	no	—
165	C₈	—	8	2	no	—
166	C₈	—	8	2	no	—
167	C₈	—	16	1	no	—
168	C₈	—	16	1	no	—
169	C₈	—	16	1	no	—
170	C₈	—	16	1	no	—
171	C₂²	#1832	1	1	yes	characteristic
172	C₂²	#1833	1	1	yes	characteristic
173	C₂²	#1833	1	1	yes	characteristic
174	C₂²	#1833	1	1	yes	characteristic
175	C₂²	#1833	1	1	yes	characteristic
176	C₂²	#1836	1	1	yes	characteristic
177	C₂²	#1837	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	2	2	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	1	no	—
186	C₄	#1573	1	1	yes	characteristic
187	C₄	#1573	1	1	yes	characteristic
188	C₄	#1574	1	1	yes	characteristic
189	C₄	#1574	1	1	yes	characteristic
190	C₂	#8782	1	1	yes	characteristic
191	C₂	#8783	1	1	yes	characteristic
192	C₂	#8789	1	1	yes	characteristic
193	C₂	#8794	1	1	yes	characteristic
194	C₂	#8795	1	1	yes	characteristic
195	C₂	#8796	1	1	yes	characteristic
196	C₂	#8797	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#382
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Maximal subgroups

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

subgroup	index
#8807	2
#8785	2
#8133	2
#7018	2