Finite Group Databasebeta

← #303040

Subgroups — #303040

Order 512.

695 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 695 conjugacy classes of subgroups — too many to draw. Only its 172 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#59869C211yesmaximal, characteristic
2#29320C212yesmaximal
3#29144C211yesmaximal, characteristic
4#10578C214yesmaximal
5#9868C212yesmaximal
6#8927C214yesmaximal
7#8232C211yesmaximal, characteristic
8#2859C2211yescharacteristic
9#2809—22no—
10#2571C2212yes—
11#2568—22no—
12#2546—22no—
13#2508—24no—
14#2369C2212yes—
15#2355—22no—
16C4×C4⋊Q8C2212yes—
17C4×C4⋊Q8C2211yescharacteristic
18#2083—22no—
19#2058—22no—
20#2026—22no—
21#1956C2212yes—
22#1948C2214yes—
23#1947C2214yes—
24#1944—24no—
25#1834C2212yes—
26#1805—22no—
27#1767C2212yes—
28C4×C4⋊C8C2211yescharacteristic
29C2×#333C2212yes—
30#1296C2212yes—
31#1279C2218yes—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8D412yes—
34C2×C4⋊Q8C2311yescharacteristic
35C2×#125—21no—
36C2×C4×Q8—22no—
37C2×C4×Q8D412yes—
38C2×#107—24no—
39C2×#106—24no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8C2314yes—
42C2×#103—24no—
43C2×#97—22no—
44C2×C4×C8C2312yes—
45#400—24no—
46#400—24no—
47#398—28no—
48#391—24no—
49#391—24no—
50#391—22no—
51#391D412yes—
52#389—24no—
53#389—24no—
54#389—22no—
55#389—22no—
56#389—22no—
57#389—22no—
58#389D414yes—
59#387—28no—
60#384D412yes—
61#384C2312yes—
62#384C2311yescharacteristic
63#382—24no—
64#382—24no—
65#382—21no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—24no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4C2314yes—
71C43C2311yescharacteristic
72#340—24no—
73#340—24no—
74#340—24no—
75#336—22no—
76#336—21no—
77#333Q818yescomplemented
78C22×Q8—22no—
79C22×Q8C2×D412yes—
80C22×C8—24no—
81C22×C8—22no—
82C4⋊Q8—22no—
83C4⋊Q8—44no—
84C4⋊Q8—84no—
85C4⋊Q8C2×Q814yes—
86#125—82no—
87C4×Q8—48no—
88C4×Q8—44no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4C2×D414yes—
103C2×C4⋊C4C2×D414yes—
104C2×C4⋊C4C2×D412yes—
105C2×C42—22no—
106C2×C42—24no—
107C2×C42—24no—
108C2×C42C2×D412yes—
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D412yes—
111C2×C42C2×D411yescharacteristic
112C2×C42C2411yescharacteristic
113#107—84no—
114#106—48no—
115C4⋊C8—24no—
116C4⋊C8—42no—
117C4⋊C8C2×Q818yes—
118#103—84no—
119#97—82no—
120C4×C8—24no—
121#95—24no—
122#95—28no—
123#95—28no—
124#95—24no—
125#95—24no—
126#95—24no—
127#95—41no—
128#95—41no—
129C2×Q8—22no—
130C2×Q8—44no—
131C2×Q8—44no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—44no—
135C2×Q8C4⋊Q814yes—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—42no—
141C22×C4#12011yescharacteristic
142C22×C4#12012yes—
143C22×C4#12011yescharacteristic
144C22×C4C22×D411yescharacteristic
145C22×C4C22×D412yes—
146C2×C8—28no—
147C2×C8—44no—
148C2×C8—44no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—42no—
152C4⋊C4—28no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—24no—
157C4⋊C4—42no—
158C4⋊C4—44no—
159C4⋊C4—48no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—44no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C4⋊C4—84no—
170C4⋊C4—84no—
171C4⋊C4—84no—
172C42—24no—
173C42—22no—
174C42—22no—
175C42—22no—
176C42—41no—
177C42—48no—
178C42—44no—
179C42—84no—
180C42C22×Q812yes—
181C422−1+412yes—
182C23C2×#12011yescharacteristic
183Q8—44no—
184Q8—82no—
185Q8—88no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—24no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4#45512yes—
209C2×C4C2×C4⋊Q811yescharacteristic
210C2×C4#53611yescharacteristic
211C2×C4D4×Q814yes—
212C2×C4#57514yes—
213C2×C4#57712yes—
214C2×C4#57812yes—
215C8—48no—
216C8—82no—
217C22#243411yescharacteristic
218C22#300512yes—
219C22#301312yes—
220C22C2×#45511yescharacteristic
221C22#302111yescharacteristic
222C4—22no—
223C4—21no—
224C4—21no—
225C4—41no—
226C4—42no—
227C4—44no—
228C4—82no—
229C4—84no—
230C4—84no—
231C4—162no—
232C4#338814yes—
233C4#339014yes—
234C2#1855211yescharacteristic
235C2#1869311yescharacteristic
236C2#1884112yes—
237C2#2582311yescharacteristic
238C2#2584612yes—
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
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
77. #333Q8semidirect

Maximal subgroups

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

subgroupindex
#598692
#293202
#291442
#105782
#98682
#89272
#82322