Finite Group Databasebeta

← #511955

Subgroups — #511955

Order 512.

1032 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1032 conjugacy classes of subgroups — too many to draw. Only its 202 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
1C23×M32C211yesmaximal, characteristic
2C2×#2219C212yesmaximal, complemented
3C2×#2216C211yesmaximal, characteristic, complemented
4#32931C212yesmaximal
5#13062C214yesmaximal, complemented
6#13060C212yesmaximal, complemented
7#12836C212yesmaximal, complemented
8#12835C211yesmaximal, characteristic, complemented
9C24×C8C2211yescharacteristic
10C22×Q32C2211yescharacteristic, complemented
11C22×M32—28no—
12C22×M32—24no—
13C22×M32C2212yes—
14C22×M32C2212yes—
15C23×C16C2211yescharacteristic
16C23×C16C2211yescharacteristic
17C2×#467C2212yes—
18C2×#467C2214yes—
19#2848—22no—
20#2219—28no—
21#2216—24no—
22C2×#367C2212yescomplemented
23C2×#366C2211yescharacteristic, complemented
24C2×#358C2214yescomplemented
25C2×#358C2212yescomplemented
26#1892—22no—
27#1892—24no—
28#1808C2212yes—
29#1808C2214yes—
30#1380C2214yescomplemented
31#1380C2212yescomplemented
32C24×C4D411yescharacteristic
33C22×Q16C2312yes—
34C23×C8—24no—
35C23×C8—22no—
36C23×C8—24no—
37C23×C8—22no—
38C23×C8D412yes—
39C23×C8D411yescharacteristic
40C23×C8C2311yescharacteristic
41C23×C8C2312yes—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—24no—
44C2×Q32—22no—
45C2×Q32—24no—
46C2×M32—232no—
47C2×M32—24no—
48C2×M32—24no—
49C2×M32—24no—
50C2×M32—24no—
51C2×M32—24no—
52C2×M32D418yes—
53C22×C16—22no—
54C22×C16—24no—
55C22×C16—24no—
56C22×C16D414yes—
57C22×C16C2312yes—
58C22×C16C2311yescharacteristic
59C22×C16C2311yescharacteristic
60#467—24no—
61#467—28no—
62#467—24no—
63#467—28no—
64C2×#107C2312yes—
65C2×#107C2314yes—
66C2×#103—22no—
67C2×#103—24no—
68#385—44no—
69#385—42no—
70#380—22no—
71#380—24no—
72#379—42no—
73#367—24no—
74#366—22no—
75#358—24no—
76#358—28no—
77#340—22no—
78#340—24no—
79C25D811yescharacteristic
80C22×Q8—22no—
81C23×C4—24no—
82C23×C4—22no—
83C23×C4—24no—
84C23×C4—22no—
85C23×C4D812yes—
86C23×C4D811yescharacteristic
87C23×C4C2×D412yes—
88C23×C4C2×D411yescharacteristic
89C2×Q16—22no—
90C2×Q16—24no—
91C2×Q16—22no—
92C2×Q16—24no—
93C22×C8—28no—
94C22×C8—216no—
95C22×C8—28no—
96C22×C8—24no—
97C22×C8—22no—
98C22×C8—24no—
99C22×C8—24no—
100C22×C8—24no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D412yes—
109C22×C8C2×D412yes—
110C22×C8C2×D414yes—
111C22×C8#4112yes—
112C22×C8#4114yes—
113C22×C8C2411yescharacteristic
114C22⋊Q8—44no—
115C22⋊Q8—48no—
116C22⋊Q8—48no—
117C22⋊Q8—44no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—42no—
122C2×C22⋊C4—44no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—44no—
125C2×C22⋊C4—42no—
126Q32—44no—
127M32—216no—
128M32—216no—
129C2×C16—28no—
130C2×C16—22no—
131C2×C16—22no—
132C2×C16—22no—
133C2×C16—28no—
134C2×C16C2×D414yes—
135C2×C16C2×D412yes—
136C2×C16C2×D412yes—
137C2×C16C2×D414yes—
138#107—22no—
139#107—24no—
140#107—22no—
141#107—24no—
142#103—44no—
143#103—48no—
144#95—42no—
145#95—44no—
146C24—24no—
147C24—22no—
148C24—24no—
149C24—42no—
150C24C2×D812yes—
151C24C2×D811yescharacteristic
152C2×Q8—42no—
153C2×Q8—44no—
154C2×Q8—42no—
155C2×Q8—44no—
156C22×C4—28no—
157C22×C4—28no—
158C22×C4—216no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—44no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4#12114yes—
174C22×C4#12112yes—
175C22×C4C2×D812yes—
176C22×C4C2×D811yescharacteristic
177C22×C4#13514yes—
178C22×C4#13512yes—
179C22×C4C22×D411yescharacteristic
180Q16—42no—
181Q16—42no—
182Q16—44no—
183C2×C8—216no—
184C2×C8—28no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—24no—
190C2×C8—22no—
191C2×C8—24no—
192C2×C8—24no—
193C2×C8—24no—
194C2×C8—24no—
195C2×C8#12112yes—
196C2×C8#12114yes—
197C2×C8C22×D411yescharacteristic
198C2×C8C22×D412yes—
199C2×C8C2×#4111yescharacteristic
200C2×C8C2×#4112yes—
201C4⋊C4—42no—
202C4⋊C4—44no—
203C4⋊C4—42no—
204C4⋊C4—44no—
205C4⋊C4—84no—
206C4⋊C4—88no—
207C22⋊C4—88no—
208C22⋊C4—88no—
209C22⋊C4—84no—
210C22⋊C4—84no—
211C16—24no—
212C16—22no—
213C16#12014yes—
214C23—22no—
215C23—22no—
216C23—28no—
217C23—24no—
218C23—24no—
219C23—22no—
220C23—22no—
221C23—416no—
222C23—44no—
223C23—44no—
224C23#46614yes—
225C23#46612yes—
226C23#51014yes—
227C23C22×D811yescharacteristic
228Q8—82no—
229Q8—82no—
230Q8—84no—
231C2×C4—28no—
232C2×C4—216no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—84no—
244C2×C4—84no—
245C2×C4—84no—
246C2×C4—82no—
247C2×C4—84no—
248C2×C4—82no—
249C2×C4—84no—
250C2×C4#46612yes—
251C2×C4#46614yes—
252C2×C4C2×#12111yescharacteristic
253C2×C4C2×#12112yes—
254C2×C4C2×#13511yescharacteristic
255C2×C4C2×#13512yes—
256C8—22no—
257C8—24no—
258C8C2×#12011yescharacteristic
259C8#52511yescharacteristic
260C8#52512yes—
261C22—22no—
262C22—28no—
263C22—22no—
264C22—28no—
265C22—24no—
266C22—416no—
267C22—42no—
268C22—42no—
269C22—44no—
270C22#221914yescomplemented
271C22C2×#46611yescharacteristic
272C22C2×#46612yes—
273C22C2×#51012yes—
274C22#341412yes—
275C4—24no—
276C4—22no—
277C4—162no—
278C4—164no—
279C4—162no—
280C4#284711yescharacteristic
281C4#305011yescharacteristic
282C4#305012yes—
283C2—24no—
284C2—44no—
285C2#3293011yescharacteristic
286C2C2×#221911yescharacteristic
287C2#3360411yescharacteristic
288C2#3360614yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC25D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2219C2semidirect
3. C2×#2216C2semidirect
5. #13062C2semidirect
6. #13060C2semidirect
7. #12836C2semidirect
8. #12835C2semidirect
10. C22×Q32C22semidirect
22. C2×#367C22semidirect
23. C2×#366C22semidirect
24. C2×#358C22semidirect
25. C2×#358C22semidirect
30. #1380C22semidirect
31. #1380C22semidirect
270. C22#2219semidirect

Maximal subgroups

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

subgroupindex
C23×M322
C2×#22192
C2×#22162
#329312
#130622
#130602
#128362
#128352