Finite Group Databasebeta

Subgroups — #145889

Order 512.

265 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 82 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	#28830	C₂	1	1	yes	maximal, characteristic
2	#13185	C₂	1	1	yes	maximal, characteristic
3	#12903	C₂	1	1	yes	maximal, characteristic
4	#9627	C₂	1	1	yes	maximal, characteristic
5	#7278	C₂	1	1	yes	maximal, characteristic
6	#7175	C₂	1	1	yes	maximal, characteristic
7	#7169	C₂	1	1	yes	maximal, characteristic
8	#2571	—	2	1	no	—
9	C₂×#365	—	2	2	no	—
10	C₂×#365	—	2	1	no	—
11	#2150	C₂²	1	1	yes	characteristic
12	C₄×M₃₂	C₂²	1	1	yes	characteristic
13	#2085	—	2	2	no	—
14	#1947	—	2	1	no	—
15	#1941	—	2	1	no	—
16	#1848	C₂²	1	1	yes	characteristic
17	#1846	—	2	1	no	—
18	#1844	C₂²	1	1	yes	characteristic
19	C₄×#106	C₂²	1	1	yes	characteristic
20	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	#1571	—	2	1	no	—
22	#1564	C₂²	1	1	yes	characteristic
23	#1383	—	2	1	no	—
24	#1303	—	2	1	no	—
25	C₂×M₃₂	—	2	1	no	—
26	C₂×M₃₂	—	2	1	no	—
27	#431	—	2	1	no	—
28	C₂×#108	—	2	1	no	—
29	C₂×#108	—	2	1	no	—
30	C₂×#107	—	2	2	no	—
31	C₂×#106	—	2	1	no	—
32	C₂×#106	D₄	1	1	yes	characteristic
33	C₂×#106	D₄	1	1	yes	characteristic
34	C₂×#106	D₄	1	2	yes	—
35	#423	—	2	1	no	—
36	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	—	2	1	no	—
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	#400	—	4	2	no	—
40	#389	—	2	2	no	—
41	#389	—	4	1	no	—
42	#384	—	2	1	no	—
43	#382	—	2	2	no	—
44	#382	—	4	1	no	—
45	C₄×C₄⋊C₄	—	2	1	no	—
46	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
47	#365	—	4	2	no	—
48	#365	—	4	4	no	—
49	C₄⋊C₁₆	—	2	1	no	—
50	C₄⋊C₁₆	Q₈	1	2	yes	—
51	#346	Q₈	1	2	yes	—
52	C₄×C₁₆	—	2	1	no	—
53	#340	—	2	2	no	—
54	#340	—	2	1	no	—
55	#336	—	2	1	no	—
56	#335	—	4	2	no	—
57	#335	Q₈	1	2	yes	—
58	#334	—	2	1	no	—
59	C₂×M₁₆	—	4	1	no	—
60	C₂²×C₈	—	2	1	no	—
61	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄	—	2	1	no	—
64	C₂×C₄⋊C₄	—	2	1	no	—
65	C₂×C₄⋊C₄	—	2	1	no	—
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	4	1	no	—
68	C₂×C₄⋊C₄	—	4	2	no	—
69	C₂×C₄⋊C₄	—	4	1	no	—
70	C₂×C₄²	—	2	1	no	—
71	C₂×C₄²	—	4	1	no	—
72	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
73	M₃₂	—	2	2	no	—
74	M₃₂	—	2	2	no	—
75	M₃₂	—	4	1	no	—
76	M₃₂	—	4	1	no	—
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	—	2	2	no	—
79	#108	—	4	2	no	—
80	#108	—	4	1	no	—
81	#108	—	4	1	no	—
82	#107	—	8	2	no	—
83	#106	—	2	1	no	—
84	#106	—	2	1	no	—
85	#106	—	2	1	no	—
86	#106	—	2	1	no	—
87	#106	—	4	1	no	—
88	#106	—	4	1	no	—
89	#106	—	4	2	no	—
90	C₄⋊C₈	—	2	2	no	—
91	C₄⋊C₈	—	2	1	no	—
92	C₄⋊C₈	—	4	1	no	—
93	C₄⋊C₈	—	4	1	no	—
94	C₄⋊C₈	SD₁₆	1	2	yes	—
95	C₂²⋊C₈	—	4	1	no	—
96	C₄×C₈	—	4	1	no	—
97	C₄×C₈	—	4	1	no	—
98	C₄×C₈	—	4	1	no	—
99	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
100	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
101	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
102	C₄×C₈	#41	1	1	yes	characteristic
103	#95	—	4	2	no	—
104	#95	—	4	2	no	—
105	#95	—	4	2	no	—
106	#95	—	4	1	no	—
107	#95	SD₁₆	1	2	yes	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	4	1	no	—
110	C₂²×C₄	—	4	1	no	—
111	C₂²×C₄	—	4	2	no	—
112	C₂²×C₄	#120	1	1	yes	characteristic
113	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
114	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
115	M₁₆	—	8	1	no	—
116	M₁₆	—	8	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	8	1	no	—
124	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
127	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₈	#135	1	1	yes	characteristic
129	C₂×C₈	#135	1	1	yes	characteristic
130	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₈	#137	1	1	yes	characteristic
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	1	no	—
134	C₄⋊C₄	—	4	1	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄⋊C₄	—	4	1	no	—
139	C₄⋊C₄	—	8	2	no	—
140	C₄⋊C₄	—	16	1	no	—
141	C₄⋊C₄	—	16	2	no	—
142	C₄²	—	4	1	no	—
143	C₄²	—	4	1	no	—
144	C₄²	—	8	2	no	—
145	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
146	C₄²	C₄⋊Q₈	1	1	yes	characteristic
147	C₄²	C₂×SD₁₆	1	1	yes	characteristic
148	C₄²	#137	1	1	yes	characteristic
149	C₁₆	—	2	4	no	—
150	C₁₆	—	4	2	no	—
151	C₂³	#450	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	8	1	no	—
158	C₂×C₄	—	8	1	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	8	1	no	—
166	C₂×C₄	#395	1	1	yes	characteristic
167	C₂×C₄	#448	1	1	yes	characteristic
168	C₂×C₄	#449	1	1	yes	characteristic
169	C₂×C₄	#452	1	1	yes	characteristic
170	C₂×C₄	#453	1	1	yes	characteristic
171	C₂×C₄	#456	1	1	yes	characteristic
172	C₂×C₄	#475	1	1	yes	characteristic
173	C₂×C₄	#476	1	1	yes	characteristic
174	C₂×C₄	#498	1	1	yes	characteristic
175	C₂×C₄	#501	1	1	yes	characteristic
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	4	2	no	—
179	C₈	—	8	2	no	—
180	C₈	—	16	1	no	—
181	C₈	#478	1	1	yes	characteristic
182	C₈	#479	1	1	yes	characteristic
183	C₈	#499	1	1	yes	characteristic
184	C₈	#501	1	1	yes	characteristic
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	#1631	1	1	yes	characteristic
188	C₂²	#1654	1	1	yes	characteristic
189	C₂²	#2058	1	1	yes	characteristic
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	8	2	no	—
194	C₄	—	16	1	no	—
195	C₄	—	16	1	no	—
196	C₄	—	16	2	no	—
197	C₄	#1622	1	1	yes	characteristic
198	C₄	#1653	1	1	yes	characteristic
199	C₄	#2059	1	1	yes	characteristic
200	C₄	#2060	1	1	yes	characteristic
201	C₄	#2231	1	2	yes	—
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	#10580	1	1	yes	characteristic
205	C₂	#13458	1	1	yes	characteristic
206	C₂	#13460	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#2058
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#450
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#28830	2
#13185	2
#12903	2
#9627	2
#7278	2
#7175	2
#7169	2