Finite Group Databasebeta

← #325597

Subgroups — #325597

Order 512.

600 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 178 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#33310C211yesmaximal, characteristic
2#29226C211yesmaximal, characteristic
3#28755C211yesmaximal, characteristic
4C8×#125C211yesmaximal, characteristic
5#10544C214yesmaximal
6#9729C214yesmaximal
7#9705C211yesmaximal, characteristic
8#9702C211yesmaximal, characteristic
9#8273C211yesmaximal, characteristic
10C2×#498C2212yes—
11C2×#498C2212yes—
12C2×#491—21no—
13#2868C2211yescharacteristic
14#2868C2211yescharacteristic
15C4×#125C2211yescharacteristic
16#2099—22no—
17#2043—22no—
18#1992—21no—
19#1987C2214yes—
20#1981C2214yes—
21#1960—21no—
22#1939C2212yes—
23#1916—22no—
24#1844—22no—
25#1844—22no—
26#1836—21no—
27#1831C2211yescharacteristic
28C8×C4⋊C4—22no—
29C8×C4⋊C4C2212yes—
30#1767C2211yescharacteristic
31#1767C2211yescharacteristic
32C42×C8C2211yescharacteristic
33C2×#334C2212yes—
34C2×#333C2214yes—
35C2×#332C2214yes—
36#1277C2212yes—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8C2312yes—
41C2×C4⋊Q8C2312yes—
42C2×#125D411yescharacteristic
43#498—28no—
44#498—44no—
45#491—44no—
46C2×#106—24no—
47C2×#106—22no—
48C2×#106—22no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8C2312yes—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×#103—24no—
54C2×#103—24no—
55C2×#97—21no—
56C2×#97—21no—
57C2×C4×C8—22no—
58C2×C4×C8D412yes—
59C2×C4×C8D412yes—
60C2×C4×C8C2311yescharacteristic
61C2×C4×C8C2311yescharacteristic
62#389D412yes—
63#387—22no—
64#384—21no—
65#384—21no—
66#384—22no—
67#384—22no—
68#384—22no—
69#382D411yescharacteristic
70C4×C4⋊C4—22no—
71C4×C4⋊C4C2312yes—
72C43C2311yescharacteristic
73#340—24no—
74#340—24no—
75#336—22no—
76#336—22no—
77#336—22no—
78#336—21no—
79#336—21no—
80#334—24no—
81#333—44no—
82#332—44no—
83C22×Q8—22no—
84C22×Q8—22no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D412yes—
90C22×C8C2×D411yescharacteristic
91C4⋊Q8—24no—
92C4⋊Q8—24no—
93C4⋊Q8—24no—
94C4⋊Q8—24no—
95C4⋊Q8—42no—
96C4⋊Q8—42no—
97C4⋊Q8—82no—
98C4⋊Q8—84no—
99#125—22no—
100#125SD1614yes—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4C2×D412yes—
112C2×C4⋊C4C2×D412yes—
113C2×C42—22no—
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D412yes—
116C2×C42C2×D411yescharacteristic
117C2×C42#4112yes—
118C2×C42C2411yescharacteristic
119#106—48no—
120#106—44no—
121#106—44no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—24no—
126C4⋊C8—22no—
127C4⋊C8—44no—
128#103—48no—
129#103—84no—
130#97—42no—
131#97—42no—
132C4×C8—24no—
133C4×C8—24no—
134C4×C8—22no—
135C4×C8—44no—
136C4×C8#4114yes—
137#95—22no—
138#95—22no—
139#95—22no—
140C2×Q8—44no—
141C2×Q8—44no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—44no—
145C2×Q8—84no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4#12112yes—
151C22×C4#12112yes—
152C22×C4C22×D411yescharacteristic
153C22×C4C22×D411yescharacteristic
154C22×C4C2×#4111yescharacteristic
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8#12412yes—
168C2×C8#12412yes—
169C2×C8#13714yes—
170C2×C8#13714yes—
171C4⋊C4—22no—
172C4⋊C4—28no—
173C4⋊C4—24no—
174C4⋊C4—44no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—84no—
182C4⋊C4—84no—
183C4⋊C4—84no—
184C4⋊C4—84no—
185C4⋊C4C2×SD1614yes—
186C42—22no—
187C42—22no—
188C42—22no—
189C42—24no—
190C42—42no—
191C42—42no—
192C42C2×SD1612yes—
193C42#13512yes—
194C42C2×#4112yes—
195C42H(5,𝔽2)12yes—
196C23C2×#12111yescharacteristic
197Q8—84no—
198Q8—84no—
199Q8—164no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—84no—
216C2×C4#46514yes—
217C2×C4#47114yes—
218C2×C4C2×#12411yescharacteristic
219C2×C4#53511yescharacteristic
220C2×C4#55012yes—
221C2×C4#55912yes—
222C2×C4C22×SD1611yescharacteristic
223C2×C4C2×#13511yescharacteristic
224C2×C4C2×#13712yes—
225C2×C4C2×#13712yes—
226C2×C4#57712yes—
227C8—22no—
228C8—24no—
229C8—82no—
230C8—82no—
231C8—82no—
232C8#49114yescomplemented
233C22#254811yescharacteristic
234C22#303912yes—
235C22C2×#46511yescharacteristic
236C22#305711yescharacteristic
237C22C2×#47112yes—
238C4—24no—
239C4—22no—
240C4—24no—
241C4—44no—
242C4—82no—
243C4—162no—
244C4—162no—
245C4C2×#49111yescharacteristic
246C4#313911yescharacteristic
247C4#321812yes—
248C4#339212yes—
249C4#339412yes—
250C2#1927012yes—
251C2#1935712yes—
252C2#2873811yescharacteristic
253C2#2875511yescharacteristic
254C2#2876411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
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
232. C8#491semidirect

Maximal subgroups

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

subgroupindex
#333102
#292262
#287552
C8×#1252
#105442
#97292
#97052
#97022
#82732