Finite Group Databasebeta

← #10822589

Subgroups — #10822589

Order 768.

1378 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 1378 conjugacy classes of subgroups — too many to draw. Only its 209 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#86571C211yesmaximal, characteristic, complemented
2#73817C213yesmaximal
3C3×#2029C211yesmaximal, characteristic
4#71894C211yesmaximal, characteristic, complemented
5#69497C213yesmaximal
6#68956C213yesmaximal, complemented
7#68951C213yesmaximal, complemented
8#23467—31nomaximal, Sylow
9C24×Dic3C2211yescharacteristic
10C3×#543C2211yescharacteristic
11#5623—26no—
12#5622—26no—
13#5483C2213yes—
14C2×#877C2213yes—
15C2×#877C2213yes—
16C3×#379C2213yes—
17#5247C2213yes—
18#4943C2213yes—
19#4943C2213yes—
20#4942—26no—
21Dic3×C22⋊C4—26no—
22C3×#342C2213yes—
23#4527C2216yes—
24#4526C2213yes—
25#3458—31no—
26#2280—33no—
27#2030—33no—
28#2030—31no—
29#2029—33no—
30#2029D311yescharacteristic
31#1740—33no—
32C25×C3C2311yescharacteristic
33C23×Dic3—23no—
34C23×Dic3—23no—
35C23×Dic3—26no—
36C23×Dic3Q814yescomplemented
37C23×Dic3C2311yescharacteristic
38C23×Dic3C2311yescharacteristic
39C3×C22⋊Q8—26no—
40C2×C3×C22⋊C4—26no—
41C2×C3×C22⋊C4C2313yes—
42C2×C3×C22⋊C4C2313yes—
43C2×#217—26no—
44C2×#217C2313yes—
45C2×#217C2313yes—
46Q8×Dic3—22no—
47#944—26no—
48#924—26no—
49C2×Dic3⋊C4—23no—
50C2×Dic3⋊C4D416yes—
51C2×C4×Dic3—23no—
52C2×C4×Dic3D416yes—
53#889—212no—
54#888—26no—
55#887—26no—
56#878—212no—
57#877—26no—
58#877—26no—
59#877—212no—
60C3⋊#95—26no—
61C3⋊#95—26no—
62C24×C4—31no—
63#544—66no—
64#543—33no—
65#543C2×D311yescharacteristic
66#533—66no—
67C2×#117—33no—
68C2×#117—33no—
69#381—33no—
70#381—33no—
71#380—66no—
72#379—33no—
73#379C2×D313yes—
74C4×C22⋊C4—66no—
75#342—33no—
76#342—36no—
77#342C2×D313yes—
78C24×C3—23no—
79C24×C3—23no—
80C24×C3—44no—
81C24×C3C2×Q812yes—
82C24×C3C2411yescharacteristic
83C2×C3×Q8—22no—
84C22×C3×C4—23no—
85C22×C3×C4C2×D416yes—
86C22×Dic3—23no—
87C22×Dic3—23no—
88C22×Dic3—212no—
89C22×Dic3—212no—
90C22×Dic3—23no—
91C22×Dic3—41no—
92C22×Dic3—43no—
93C22×Dic3—46no—
94C22×Dic3—46no—
95C22×Dic3C2×D416yes—
96C22×Dic3C2×D413yes—
97C22×Dic3C2×D413yes—
98C22×Dic3C2×Q812yes—
99C22×Dic3C2×Q812yes—
100C22×Dic3#4116yes—
101C2×Dic6—26no—
102C3×C4⋊C4—26no—
103C3×C22⋊C4—26no—
104C3×C22⋊C4—26no—
105C3×C22⋊C4—412no—
106#217—26no—
107#217—26no—
108#217—412no—
109C4⋊Dic3—26no—
110Dic3⋊C4—212no—
111Dic3⋊C4—212no—
112Dic3⋊C4—46no—
113C4×Dic3—26no—
114C4×Dic3—26no—
115C4×Dic3—46no—
116C25C22×D311yescharacteristic
117C23×C4—31no—
118C23×C4—34no—
119C23×C4—31no—
120C23×C4—63no—
121C23×C4—63no—
122C23×C4—66no—
123C4⋊Q8—66no—
124C4⋊Q8—66no—
125#125—612no—
126C22⋊Q8—26no—
127C22⋊Q8—612no—
128C22⋊Q8—66no—
129C4×Q8—66no—
130C4×Q8—62no—
131#117—66no—
132#117—66no—
133#117—612no—
134C2×C4⋊C4—36no—
135C2×C4⋊C4—63no—
136C2×C22⋊C4—26no—
137C2×C22⋊C4—33no—
138C2×C22⋊C4—33no—
139C2×C22⋊C4—66no—
140C2×C22⋊C4C22×D313yes—
141C2×C22⋊C4C22×D313yes—
142C2×C42—36no—
143C2×C42—63no—
144#95—66no—
145#95—66no—
146C23×C3—212no—
147C23×C3—26no—
148C23×C3—44no—
149C23×C3—42no—
150C23×C3—46no—
151C23×C3—412no—
152C23×C3—41no—
153C23×C3—43no—
154C23×C3C22×D413yes—
155C23×C3C22×Q811yescharacteristic
156C23×C3C2×#4113yes—
157C3×Q8—82no—
158C2×C3×C4—26no—
159C2×C3×C4—46no—
160C2×C3×C4—46no—
161C2×C3×C4—43no—
162C2×Dic3—26no—
163C2×Dic3—26no—
164C2×Dic3—41no—
165C2×Dic3—43no—
166C2×Dic3—44no—
167C2×Dic3—412no—
168C2×Dic3—46no—
169C2×Dic3—46no—
170C2×Dic3—43no—
171C2×Dic3#12011yescharacteristic
172C2×Dic3#12013yes—
173C2×Dic3C22⋊Q816yes—
174C2×Dic3C22⋊Q816yes—
175C2×Dic3#12416yes—
176C2×Dic3#12416yes—
177Dic6—86no—
178C24—23no—
179C24—23no—
180C24—44no—
181C24D3×Q812yes—
182C24C23×D311yescharacteristic
183C2×Q8—22no—
184C2×Q8—66no—
185C22×C4—23no—
186C22×C4—33no—
187C22×C4—33no—
188C22×C4—32no—
189C22×C4—36no—
190C22×C4—32no—
191C22×C4—36no—
192C22×C4—612no—
193C22×C4—612no—
194C22×C4—63no—
195C22×C4—63no—
196C22×C4—63no—
197C22×C4—126no—
198C22×C4—126no—
199C22×C4—121no—
200C22×C4—123no—
201C22×C4D3×D416yes—
202C4⋊C4—26no—
203C4⋊C4—612no—
204C4⋊C4—612no—
205C4⋊C4—66no—
206C4⋊C4—126no—
207C22⋊C4—26no—
208C22⋊C4—26no—
209C22⋊C4—412no—
210C22⋊C4—66no—
211C22⋊C4—66no—
212C22⋊C4—1212no—
213C42—66no—
214C42—66no—
215C42—126no—
216C22×C3—26no—
217C22×C3—212no—
218C22×C3—41no—
219C22×C3—43no—
220C22×C3—412no—
221C22×C3—46no—
222C22×C3—42no—
223C22×C3—44no—
224C22×C3C2×#12011yescharacteristic
225C22×C3C2×C22⋊Q813yes—
226C22×C3C2×#12413yes—
227C3×C4—46no—
228C3×C4—83no—
229Dic3—42no—
230Dic3—46no—
231Dic3—83no—
232Dic3#39312yes—
233Dic3#39316yes—
234C23—212no—
235C23—26no—
236C23—44no—
237C23—42no—
238C23—46no—
239C23—412no—
240C23—43no—
241C23—41no—
242C23C2×D3×D413yes—
243C23C2×D3×Q811yescharacteristic
244C23D3×#4113yes—
245Q8—82no—
246Q8—246no—
247C2×C4—26no—
248C2×C4—36no—
249C2×C4—36no—
250C2×C4—31no—
251C2×C4—33no—
252C2×C4—36no—
253C2×C4—36no—
254C2×C4—46no—
255C2×C4—46no—
256C2×C4—43no—
257C2×C4—66no—
258C2×C4—66no—
259C2×C4—124no—
260C2×C4—1212no—
261C2×C4—123no—
262C2×C4—121no—
263C2×C4—126no—
264C2×C4—126no—
265C2×C4—123no—
266C2×C3—23no—
267C2×C3—23no—
268C2×C3—44no—
269C2×C3C2×#39311yescharacteristic
270C2×C3#302512yes—
271C22—26no—
272C22—212no—
273C22—41no—
274C22—43no—
275C22—412no—
276C22—46no—
277C22—42no—
278C22—44no—
279C22D3×#12011yescharacteristic
280C22D3×C22⋊Q813yes—
281C22D3×#12413yes—
282C4—32no—
283C4—36no—
284C4—46no—
285C4—83no—
286C4—122no—
287C4—126no—
288C4—243no—
289C3#2346711yescharacteristic, complemented, Sylow
290C2—23no—
291C2—23no—
292C2—44no—
293C2D3×#39311yescharacteristic
294C2#8030412yes—
subgroupG/H
centerC22D3×#120
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#2029C2
socleC22×C3C2×#120
prime radicalC25×C3C23
hypercenter#2029D3
hypocenterC3#23467
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ #2029

Complements

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

NH ≅ G/Ntype
1. #86571C2semidirect
4. #71894C2semidirect
6. #68956C2semidirect
7. #68951C2semidirect
36. C23×Dic3Q8semidirect
289. C3#23467semidirect

Maximal subgroups

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

subgroupindex
#865712
#738172
C3×#20292
#718942
#694972
#689562
#689512
#234673