Finite Group Databasebeta

← #324472

Subgroups — #324472

Order 512.

652 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30230C211yesmaximal, characteristic, complemented
2#28724C211yesmaximal, characteristic
3#28509C211yesmaximal, characteristic
4#22260C211yesmaximal, characteristic
5#10480C214yesmaximal, complemented
6#9899C211yesmaximal, characteristic, complemented
7#9702C211yesmaximal, characteristic, complemented
8#9215C214yesmaximal, complemented
9#8277C211yesmaximal, characteristic, complemented
10C2×#490—21no—
11C2×#476C2214yes—
12C2×C8×D4C2212yes—
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15C4×#127C2211yescharacteristic
16#2100—24no—
17#2094—22no—
18#2073—22no—
19#1992—21no—
20#1990C2212yes—
21#1990C2212yes—
22#1957—21no—
23#1954C2212yes—
24#1869—24no—
25#1844—21no—
26#1831C2211yescharacteristic
27#1802C2214yes—
28#1767C2211yescharacteristic
29C4×C4⋊C8C2211yescharacteristic
30#1744C2211yescharacteristic
31C2×#334C2212yes—
32C2×#333C2214yes—
33#1296C2214yes—
34#1277C2212yes—
35C23×C8—22no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8C2312yes—
38C2×#127D411yescharacteristic
39C2×#125—21no—
40C2×C4×D4—22no—
41C2×C4×D4C2312yes—
42#490—44no—
43#476—28no—
44#476—44no—
45C8×D4—44no—
46C2×#106—24no—
47C2×C4⋊C8C2312yes—
48C2×C4⋊C8C2312yes—
49C2×C4⋊C8C2312yes—
50C2×#102—24no—
51C2×#102—24no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—24no—
54C2×#97—21no—
55C2×C4×C8—21no—
56C2×C4×C8C2311yescharacteristic
57C2×C4×C8C2311yescharacteristic
58#390—22no—
59#390D412yes—
60#389—22no—
61#389—22no—
62#384—21no—
63#384—21no—
64#384—22no—
65#384D411yescharacteristic
66#384C2312yes—
67#382—21no—
68C43C2311yescharacteristic
69#340—24no—
70#340—24no—
71#340—24no—
72#336—21no—
73#336—22no—
74#336—21no—
75#336D414yes—
76#334—24no—
77#333—44no—
78C22×Q8—22no—
79C22×D4—22no—
80C22×D4C2×D412yes—
81C23×C4—22no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—44no—
86C22×C8—84no—
87C22×C8C2×D412yes—
88C22×C8C2×D412yes—
89C4⋊Q8—24no—
90C4⋊Q8—24no—
91C4⋊Q8—84no—
92#127—22no—
93#127SD1614yes—
94#125—42no—
95#125—42no—
96C4×D4—24no—
97C4×D4—24no—
98C4×D4—84no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4C2×D412yes—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—24no—
111C2×C42C2×D412yes—
112C2×C42C2×D411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42#4112yes—
115C2×C42C2411yescharacteristic
116#106—48no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120C4⋊C8—24no—
121C4⋊C8#4114yes—
122#102—48no—
123#102—48no—
124C22⋊C8—44no—
125C22⋊C8—84no—
126#97—42no—
127C4×C8—22no—
128C4×C8—22no—
129C4×C8—42no—
130#95—42no—
131C24—22no—
132C2×Q8—44no—
133C2×Q8—84no—
134C2×D4—24no—
135C2×D4—22no—
136C2×D4—24no—
137C2×D4—28no—
138C2×D4—82no—
139C2×D4—82no—
140C2×D4C2×SD1614yes—
141C22×C4—22no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4#12112yes—
149C22×C4#12112yes—
150C22×C4C22×D411yescharacteristic
151C22×C4C22×D411yescharacteristic
152C22×C4C2×#4111yescharacteristic
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—24no—
156C2×C8—24no—
157C2×C8—24no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—44no—
164C2×C8—42no—
165C2×C8—88no—
166C4⋊C4—24no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—44no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—44no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C4⋊C4—84no—
176C4⋊C4—84no—
177C4⋊C4—84no—
178C22⋊C4—44no—
179C22⋊C4—84no—
180C42—24no—
181C42—22no—
182C42—42no—
183C42C2×SD1612yes—
184C42#13512yes—
185C42C2×#4112yes—
186C42H(5,𝔽2)12yes—
187C23—24no—
188C23—44no—
189C23—42no—
190C23—42no—
191C23—44no—
192C23C2×#12111yescharacteristic
193Q8—164no—
194D4—28no—
195D4—44no—
196D4—88no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—84no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—84no—
212C2×C4—82no—
213C2×C4#46514yes—
214C2×C4#53211yescharacteristic
215C2×C4#53511yescharacteristic
216C2×C4#54612yes—
217C2×C4#55912yes—
218C2×C4C22×SD1611yescharacteristic
219C2×C4C2×#13511yescharacteristic
220C2×C4#57714yes—
221C2×C4#57812yes—
222C8—42no—
223C8—42no—
224C8—44no—
225C8—82no—
226C8—82no—
227C22—44no—
228C22—48no—
229C22—84no—
230C22—82no—
231C22—82no—
232C22#254411yescharacteristic
233C22#304212yes—
234C22C2×#46511yescharacteristic
235C22#305911yescharacteristic
236C22#306812yes—
237C4—24no—
238C4—22no—
239C4—42no—
240C4—82no—
241C4—162no—
242C4—162no—
243C4#317412yes—
244C4#322212yes—
245C4#329812yes—
246C4#330212yes—
247C2—84no—
248C2#1939112yes—
249C2#1951512yes—
250C2#2860011yescharacteristic
251C2#2862211yescharacteristic
252C2#2863511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #30230C2semidirect
5. #10480C2semidirect
6. #9899C2semidirect
7. #9702C2semidirect
8. #9215C2semidirect
9. #8277C2semidirect

Maximal subgroups

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

subgroupindex
#302302
#287242
#285092
#222602
#104802
#98992
#97022
#92152
#82772