Finite Group Databasebeta

← #10822591

Subgroups — #10822591

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#74050C213yesmaximal
3C3×#2030C211yesmaximal, characteristic, complemented
4#71897C211yesmaximal, characteristic
5#69536C213yesmaximal, complemented
6#68971C213yesmaximal
7#68951C213yesmaximal, complemented
8#23467—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C3×#543C2211yescharacteristic
11#5623—26no—
12#5622—26no—
13#5483C2213yes—
14C2×#877C2213yes—
15C2×#877C2213yes—
16C3×#381C2213yes—
17#5211C2213yes—
18#4955C2213yes—
19#4955C2213yes—
20#4667—26no—
21#4666—26no—
22C3×#342C2213yes—
23#4527C2216yes—
24#4526C2213yes—
25#3458—31no—
26#2280—33no—
27#2030—33no—
28#2030D311yescharacteristic, complemented
29#2029—33no—
30#2029—31no—
31#1740—33no—
32C24×D3C2311yescharacteristic
33C23×C3×C4C2311yescharacteristic
34C23×Dic3C2311yescharacteristic
35C22×C4×D3—23no—
36C22×C4×D3—23no—
37C22×C4×D3—26no—
38C22×C4×D3Q814yescomplemented
39C3×C22⋊Q8—26no—
40C2×C3×C22⋊C4C2313yes—
41C2×C3×C22⋊C4C2313yes—
42C2×#217C2313yes—
43C2×#217C2313yes—
44Q8×Dic3—22no—
45#944—26no—
46C2×(C2×D3)⋊C4—26no—
47C2×(C2×D3)⋊C4—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#381C2×D313yes—
71#380—66no—
72#379—33no—
73#379—33no—
74C4×C22⋊C4—66no—
75#342—33no—
76#342—36no—
77#342C2×D313yes—
78C24×C3C2411yescharacteristic
79C23×D3—23no—
80C23×D3—23no—
81C23×D3—44no—
82C23×D3C2×Q812yes—
83C2×C3×Q8—22no—
84C22×C3×C4—23no—
85C22×C3×C4—23no—
86C22×C3×C4—23no—
87C22×C3×C4C2×D416yes—
88C22×C3×C4C2×Q812yes—
89C22×Dic3—23no—
90C22×Dic3C2×D416yes—
91C22×Dic3C2×D413yes—
92C22×Dic3C2×D413yes—
93C22×Dic3C2×Q812yes—
94C22×Dic3#4116yes—
95C2×C4×D3—212no—
96C2×C4×D3—212no—
97C2×C4×D3—41no—
98C2×C4×D3—43no—
99C2×C4×D3—46no—
100C2×C4×D3—46no—
101C2×Dic6—26no—
102C3×C4⋊C4—26no—
103C3×C22⋊C4—26no—
104C3×C22⋊C4—26no—
105#217—26no—
106#217—26no—
107(C2×D3)⋊C4—412no—
108(C2×D3)⋊C4—412no—
109C4⋊Dic3—26no—
110Dic3⋊C4—212no—
111Dic3⋊C4—212no—
112Dic3⋊C4—46no—
113C4×Dic3—26no—
114C4×Dic3—26no—
115C4×Dic3—46no—
116C25—31no—
117C23×C4—31no—
118C23×C4—34no—
119C23×C4—63no—
120C23×C4—63no—
121C23×C4—66no—
122C23×C4C22×D311yescharacteristic
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—33no—
137C2×C22⋊C4—33no—
138C2×C22⋊C4—66no—
139C2×C22⋊C4—66no—
140C2×C22⋊C4C22×D313yes—
141C2×C22⋊C4C22×D313yes—
142C2×C42—36no—
143C2×C42—63no—
144#95—26no—
145#95—66no—
146C23×C3—42no—
147C23×C3C22×D413yes—
148C23×C3C22×Q811yescharacteristic
149C23×C3C2×#4113yes—
150C22×D3—212no—
151C22×D3—26no—
152C22×D3—44no—
153C22×D3—412no—
154C22×D3—46no—
155C22×D3—41no—
156C22×D3—43no—
157C3×Q8—82no—
158C2×C3×C4—26no—
159C2×C3×C4—26no—
160C2×C3×C4—41no—
161C2×C3×C4—43no—
162C2×C3×C4—46no—
163C2×C3×C4—46no—
164C2×C3×C4—43no—
165C2×Dic3—26no—
166C2×Dic3—46no—
167C2×Dic3—46no—
168C2×Dic3—43no—
169C2×Dic3#12011yescharacteristic
170C2×Dic3#12013yes—
171C2×Dic3C22⋊Q816yes—
172C2×Dic3C22⋊Q816yes—
173C2×Dic3#12416yes—
174C2×Dic3#12416yes—
175C4×D3—44no—
176C4×D3—412no—
177Dic6—86no—
178C24—32no—
179C24—63no—
180C24—63no—
181C24—124no—
182C24C23×D311yescharacteristic
183C2×Q8—22no—
184C2×Q8—66no—
185C22×C4—23no—
186C22×C4—23no—
187C22×C4—23no—
188C22×C4—33no—
189C22×C4—33no—
190C22×C4—32no—
191C22×C4—36no—
192C22×C4—36no—
193C22×C4—612no—
194C22×C4—612no—
195C22×C4—63no—
196C22×C4—126no—
197C22×C4—126no—
198C22×C4—121no—
199C22×C4—123no—
200C22×C4D3×D416yes—
201C22×C4D3×Q812yes—
202C4⋊C4—26no—
203C4⋊C4—612no—
204C4⋊C4—612no—
205C4⋊C4—66no—
206C4⋊C4—126no—
207C22⋊C4—26no—
208C22⋊C4—26no—
209C22⋊C4—66no—
210C22⋊C4—66no—
211C22⋊C4—1212no—
212C22⋊C4—1212no—
213C42—66no—
214C42—66no—
215C42—126no—
216C22×C3—26no—
217C22×C3—41no—
218C22×C3—43no—
219C22×C3C2×#12011yescharacteristic
220C22×C3C2×C22⋊Q813yes—
221C22×C3C2×#12413yes—
222C2×D3—212no—
223C2×D3—412no—
224C2×D3—46no—
225C2×D3—42no—
226C2×D3—44no—
227C3×C4—42no—
228C3×C4—46no—
229C3×C4—83no—
230Dic3—46no—
231Dic3—83no—
232Dic3#39312yes—
233Dic3#39316yes—
234C23—42no—
235C23—612no—
236C23—66no—
237C23—126no—
238C23—124no—
239C23—1212no—
240C23—123no—
241C23—121no—
242C23C2×D3×D413yes—
243C23C2×D3×Q811yescharacteristic
244C23D3×#4113yes—
245Q8—82no—
246Q8—246no—
247C2×C4—26no—
248C2×C4—26no—
249C2×C4—36no—
250C2×C4—36no—
251C2×C4—31no—
252C2×C4—33no—
253C2×C4—36no—
254C2×C4—36no—
255C2×C4—43no—
256C2×C4—41no—
257C2×C4—46no—
258C2×C4—46no—
259C2×C4—43no—
260C2×C4—66no—
261C2×C4—124no—
262C2×C4—1212no—
263C2×C4—126no—
264C2×C4—126no—
265C2×C4—123no—
266C2×C3—23no—
267C2×C3—23no—
268C2×C3C2×#39311yescharacteristic
269C2×C3#302512yes—
270D3—44no—
271C22—26no—
272C22—41no—
273C22—43no—
274C22—612no—
275C22—1212no—
276C22—126no—
277C22—122no—
278C22—124no—
279C22D3×#12011yescharacteristic
280C22D3×C22⋊Q813yes—
281C22D3×#12413yes—
282C4—32no—
283C4—36no—
284C4—42no—
285C4—46no—
286C4—83no—
287C4—126no—
288C4—243no—
289C3#2346711yescharacteristic, complemented, Sylow
290C2—23no—
291C2—23no—
292C2—124no—
293C2D3×#39311yescharacteristic
294C2#8030812yes—
subgroupG/H
centerC22D3×#120
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#2030C2
socleC22×C3C2×#120
prime radicalC24×D3C23
hypercenter#2030D3
hypocenterC3#23467
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ #2030

Complements

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

NH ≅ G/Ntype
1. #86571C2semidirect
3. C3×#2030C2semidirect
5. #69536C2semidirect
7. #68951C2semidirect
28. #2030D3semidirect
38. C22×C4×D3Q8semidirect
289. C3#23467semidirect

Maximal subgroups

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

subgroupindex
#865712
#740502
C3×#20302
#718972
#695362
#689712
#689512
#234673