Finite Group Databasebeta

← #6434718

Subgroups — #6434718

Order 512.

3733 conjugacy classes of subgroups; only the 614 normal subgroups are shown.

normal subgroups only

This group has 3733 conjugacy classes of subgroups — too many to draw. Only its 614 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
1C22×#543C211yesmaximal, characteristic, complemented
2#34992C213yesmaximal
3#29657C218yesmaximal, complemented
4#29656C218yesmaximal, complemented
5#23565C214yesmaximal, complemented
6#23468C214yesmaximal, complemented
7C2×#1819C213yesmaximal, complemented
8C2×#543—26no—
9C2×#543—212no—
10C2×#543C2214yes—
11C2×#543C2218yes—
12C22×C22⋊Q8—26no—
13C22×#120C2211yescharacteristic
14C23×C22⋊C4C2213yes—
15#2849—26no—
16#2622—26no—
17#2483C22112yes—
18#2412—212no—
19#2405C22112yes—
20#2315C2216yes—
21#2308C2216yes—
22C2×#379C2213yes—
23#2062C22132yescomplemented
24#2030C22116yescomplemented
25#2029C22116yescomplemented
26#1819C22124yes—
27C2×#342C22112yes—
28C26C2311yescharacteristic
29C23×Q8—22no—
30C23×D4C2311yescharacteristic
31C23×D4C2312yes—
32C24×C4D416yes—
33#543—248no—
34#543—224no—
35#543—248no—
36#543C23116yes—
37C2×#123—212no—
38C2×C22⋊Q8—224no—
39C2×C22⋊Q8—224no—
40C2×C22⋊Q8—224no—
41C2×C22⋊Q8—224no—
42C2×C22⋊Q8—224no—
43C2×C22⋊Q8—224no—
44C2×#120—23no—
45C2×#120—23no—
46C2×#120—26no—
47C22×C4⋊C4—26no—
48C22×C22⋊C4—26no—
49C22×C22⋊C4—212no—
50C22×C22⋊C4—26no—
51C22×C22⋊C4—26no—
52C22×C22⋊C4D416yes—
53C22×C22⋊C4D4112yes—
54C22×C22⋊C4C2311yescharacteristic
55C22×C22⋊C4C2312yes—
56C22×C22⋊C4C2316yes—
57C22×C22⋊C4C2316yes—
58C22×C22⋊C4C23112yes—
59#397—224no—
60#394—224no—
61#386—212no—
62#386—212no—
63#386—212no—
64#385—212no—
65#385—212no—
66#385C23124yes—
67#381—212no—
68#381C23112yes—
69#380—212no—
70#379—23no—
71#379—23no—
72#379C23112yes—
73#379C23112yes—
74#342C23148yes—
75C25—28no—
76C25—84no—
77C25C2×D413yes—
78C25C2×D416yes—
79C25C2×D413yes—
80C25C2411yescharacteristic
81C25C2412yes—
82C25C2411yescharacteristic
83C22×Q8—28no—
84C22×Q8—28no—
85C22×Q8—28no—
86C22×Q8—88no—
87C22×D4—26no—
88C22×D4—23no—
89C22×D4—23no—
90C22×D4—28no—
91C22×D4—26no—
92C22×D4—84no—
93C23×C4—26no—
94C23×C4—26no—
95C23×C4—26no—
96C23×C4—26no—
97C23×C4—26no—
98C23×C4—812no—
99C23×C4C2×D4112yes—
100C23×C4C2×D4112yes—
101C23×C4C2×D4112yes—
102C23×C4C2411yescharacteristic
103C23×C4C2412yes—
104#123—824no—
105C22⋊Q8—296no—
106C22⋊Q8—296no—
107C22⋊Q8—296no—
108C22⋊Q8—296no—
109#120—41no—
110#120—46no—
111#120—46no—
112#120—43no—
113C2×C4⋊C4—224no—
114C2×C4⋊C4—224no—
115C2×C4⋊C4—224no—
116C2×C4⋊C4—224no—
117C2×C22⋊C4—224no—
118C2×C22⋊C4—212no—
119C2×C22⋊C4—212no—
120C2×C22⋊C4—224no—
121C2×C22⋊C4—224no—
122C2×C22⋊C4—212no—
123C2×C22⋊C4—224no—
124C2×C22⋊C4—212no—
125C2×C22⋊C4—224no—
126C2×C22⋊C4—212no—
127C2×C22⋊C4—23no—
128C2×C22⋊C4—26no—
129C2×C22⋊C4—23no—
130C2×C22⋊C4—26no—
131C2×C22⋊C4—26no—
132C2×C22⋊C4—212no—
133C2×C22⋊C4—224no—
134C2×C22⋊C4—812no—
135C2×C22⋊C4C2×D4112yes—
136C2×C22⋊C4C2×D4124yes—
137C2×C22⋊C4C24112yes—
138C2×C22⋊C4C24112yes—
139#95—224no—
140#95—224no—
141C24—212no—
142C24—224no—
143C24—212no—
144C24—28no—
145C24—23no—
146C24—24no—
147C24—412no—
148C24—412no—
149C24—44no—
150C24—86no—
151C24—86no—
152C24—82no—
153C24—82no—
154C24—82no—
155C24—832no—
156C24#12011yescharacteristic
157C24#12016yes—
158C24#12016yes—
159C24#12013yes—
160C24C22×D416yes—
161C24C22×D416yes—
162C24C22×D413yes—
163C24C22×D413yes—
164C242−1+418yes—
165C24C2511yescharacteristic
166C2×Q8—232no—
167C2×Q8—816no—
168C2×Q8—816no—
169C2×Q8—816no—
170C2×D4—424no—
171C2×D4—46no—
172C2×D4—43no—
173C2×D4—41no—
174C2×D4—42no—
175C2×D4—412no—
176C2×D4—412no—
177C2×D4—812no—
178C22×C4—212no—
179C22×C4—212no—
180C22×C4—212no—
181C22×C4—212no—
182C22×C4—212no—
183C22×C4—212no—
184C22×C4—23no—
185C22×C4—212no—
186C22×C4—26no—
187C22×C4—24no—
188C22×C4—424no—
189C22×C4—424no—
190C22×C4—412no—
191C22×C4—82no—
192C22×C4—812no—
193C22×C4—812no—
194C22×C4—812no—
195C22×C4—812no—
196C22×C4—812no—
197C22×C4—812no—
198C22×C4C22×D4124yes—
199C4⋊C4—296no—
200C4⋊C4—824no—
201C22⋊C4—248no—
202C22⋊C4—224no—
203C22⋊C4—224no—
204C22⋊C4—224no—
205C22⋊C4—424no—
206C22⋊C4—46no—
207C22⋊C4—43no—
208C22⋊C4—41no—
209C22⋊C4—42no—
210C22⋊C4—412no—
211C22⋊C4—812no—
212C22⋊C4—824no—
213C23—22no—
214C23—24no—
215C23—212no—
216C23—26no—
217C23—212no—
218C23—212no—
219C23—212no—
220C23—212no—
221C23—212no—
222C23—46no—
223C23—46no—
224C23—46no—
225C23—46no—
226C23—46no—
227C23—42no—
228C23—46no—
229C23—46no—
230C23—412no—
231C23—448no—
232C23—412no—
233C23—41no—
234C23—46no—
235C23—88no—
236C23—848no—
237C23—816no—
238C23—824no—
239C23—164no—
240C23C2×#12016yes—
241C23C2×#12013yes—
242C23C2×#12011yescharacteristic
243C23C2×#12012yes—
244C23#536112yes—
245C23#536112yes—
246C23C23×D413yes—
247C23C2×2−1+414yes—
248Q8—832no—
249D4—812no—
250D4—82no—
251D4—86no—
252D4—84no—
253C2×C4—224no—
254C2×C4—424no—
255C2×C4—424no—
256C2×C4—424no—
257C2×C4—424no—
258C2×C4—424no—
259C2×C4—412no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—412no—
263C2×C4—46no—
264C2×C4—812no—
265C2×C4—812no—
266C2×C4—812no—
267C2×C4—1612no—
268C22—28no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—26no—
273C22—26no—
274C22—26no—
275C22—26no—
276C22—26no—
277C22—412no—
278C22—412no—
279C22—44no—
280C22—424no—
281C22—424no—
282C22—412no—
283C22—812no—
284C22—812no—
285C22—812no—
286C22—81no—
287C22—82no—
288C22#261412yes—
289C22#261416yes—
290C22#302614yes—
291C22C22×#12011yescharacteristic
292C22C2×#53616yes—
293C4—424no—
294C4—812no—
295C4—81no—
296C4—82no—
297C2—26no—
298C2—22no—
299C2—24no—
300C2—26no—
301C2—46no—
302C2—161no—
303C2C2×#261411yescharacteristic
304C2C2×#302612yes—
subgroupG/H
center, socleC22C22×#120
commutator, Frattini, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C26 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#543C2semidirect
3. #29657C2semidirect
4. #29656C2semidirect
5. #23565C2semidirect
6. #23468C2semidirect
7. C2×#1819C2semidirect
23. #2062C22semidirect
24. #2030C22semidirect
25. #2029C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5432
#349922
#296572
#296562
#235652
#234682
C2×#18192