Finite Group Databasebeta

← #10698080

Subgroups — #10698080

Order 768.

1174 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 89 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#78641C211yesmaximal, characteristic, complemented
2#73587C212yesmaximal, complemented
3#69988C211yesmaximal, characteristic, complemented
4#69444C212yesmaximal, complemented
5C3×#1297C211yesmaximal, characteristic, complemented
6#10326—31nomaximal, Sylow
7C23×D12C2211yescharacteristic
8C2×#930C2212yes—
9C2×#882—24no—
10C2×#874—22no—
11C3×#407C2211yescharacteristic
12C3×C4×C22⋊C4C2212yes—
13#5207C2211yescharacteristic
14#5135—22no—
15C2×#823—22no—
16#4939—22no—
17#4735—22no—
18#4732—22no—
19#4671—24no—
20#4451—24no—
21#3192—31no—
22#2433—32no—
23#1783—31no—
24#1507—32no—
25#1297D311yescharacteristic, complemented
26C24×D3D412yes—
27C23×C3×C4C2311yescharacteristic
28C22×#67—22no—
29C22×D12—21no—
30C22×D12—22no—
31C22×D12—22no—
32C22×D12—21no—
33C22×D12—28no—
34C22×D12—44no—
35C22×D12D412yescomplemented
36C2×C3×M16D412yescomplemented
37C2×C3×C22⋊C4—22no—
38C2×C3×C42—22no—
39#930—22no—
40#930—22no—
41#930—22no—
42#930—22no—
43C2×(C2×D3)⋊C4—28no—
44C2×(C2×D3)⋊C4—22no—
45C2×C4⋊Dic3D412yescomplemented
46C2×#208—21no—
47C2×D24—22no—
48#882—416no—
49#874—48no—
50C3×C22⋊C8—22no—
51C3×#95D414yescomplemented
52#830—22no—
53#823—44no—
54#821—24no—
55#817—22no—
56C23×D4—31no—
57C2×#127—62no—
58C2×#121—32no—
59C2×#120—64no—
60#481—62no—
61#469—62no—
62#447—62no—
63C2×#100—62no—
64#407—31no—
65#407C2×D311yescharacteristic
66#392—64no—
67#390—62no—
68C4×C22⋊C4C2×D312yes—
69#327—64no—
70C24×C3C2×D411yescharacteristic
71C23×D3—24no—
72C23×D3—22no—
73C23×D3—22no—
74C23×D3—44no—
75C23×D3—48no—
76C22×C3×C4—24no—
77C22×C3×C4—42no—
78C22×C3×C4C2×D411yescharacteristic
79C22×C3×C4C2×D412yes—
80C22×C3×C4C2×D412yes—
81C22×C3×C4#4111yescharacteristic
82C2×#67—48no—
83C2×#67—42no—
84C2×#67—42no—
85C22×Dic3—22no—
86C2×D12—24no—
87C2×D12—28no—
88C2×D12—22no—
89C2×D12—22no—
90C2×D12—42no—
91C2×D12—44no—
92C2×D12—48no—
93C2×D12—44no—
94C2×D12—41no—
95C2×D12—42no—
96C2×D12—41no—
97C2×D12—44no—
98C2×D12—416no—
99C2×D12—416no—
100C2×D12—88no—
101C3×M16—42no—
102C2×C3×C8—22no—
103C3×C22⋊C4—28no—
104C3×C42—44no—
105(C2×D3)⋊C4—416no—
106(C2×D3)⋊C4—42no—
107(C2×D3)⋊C4—42no—
108C4⋊Dic3—22no—
109C4⋊Dic3—22no—
110#208—42no—
111C2×C3⋊C8—41no—
112D24—82no—
113C25—32no—
114C22×D4—32no—
115C22×D4—68no—
116C22×D4—62no—
117C22×D4—61no—
118C22×D4—62no—
119C22×D4—62no—
120C22×D4—61no—
121C22×D4—124no—
122C23×C4C22×D311yescharacteristic
123C2×D8—62no—
124C2×M16—61no—
125C2×M16D1212yescomplemented
126#127—128no—
127#121—62no—
128#121—62no—
129#121—62no—
130#121—62no—
131#120—1216no—
132C2×C4⋊C4—32no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—68no—
135C2×C22⋊C4—62no—
136C2×C42—22no—
137#106—62no—
138#102—64no—
139#100—124no—
140C22⋊C8—22no—
141C22⋊C8—62no—
142#95D1214yescomplemented
143C23×C3—24no—
144C23×C3—41no—
145C23×C3#12012yes—
146C23×C3#12111yescharacteristic
147C22×D3—28no—
148C22×D3—24no—
149C22×D3—48no—
150C22×D3—44no—
151C22×D3—42no—
152C22×D3—42no—
153C22×D3—48no—
154C22×D3—44no—
155C22×D3—44no—
156C22×D3—42no—
157C22×D3—44no—
158C22×D3—816no—
159C22×D3—84no—
160C2×C3×C4—22no—
161C2×C3×C4—24no—
162C2×C3×C4—28no—
163C2×C3×C4—42no—
164C2×C3×C4—41no—
165C2×C3×C4—44no—
166C2×C3×C4—44no—
167C2×C3×C4#12011yescharacteristic
168C2×C3×C4#12112yes—
169C2×C3×C4#12711yescharacteristic
170#67—84no—
171#67—84no—
172C2×Dic3—44no—
173C2×Dic3—42no—
174D12—44no—
175D12—48no—
176D12—44no—
177D12—44no—
178D12—82no—
179D12—832no—
180C3×C8—42no—
181C3⋊C8—81no—
182C24—64no—
183C24—62no—
184C24—62no—
185C24—128no—
186C24—124no—
187C24D3×D411yescharacteristic
188C2×D4—64no—
189C2×D4—68no—
190C2×D4—62no—
191C2×D4—62no—
192C2×D4—124no—
193C2×D4—122no—
194C2×D4—124no—
195C2×D4—1216no—
196C2×D4—128no—
197C2×D4—124no—
198C2×D4—1216no—
199C2×D4—128no—
200C2×D4—122no—
201C2×D4—122no—
202C2×D4—121no—
203C2×D4—121no—
204C2×D4—122no—
205C2×D4—248no—
206C22×C4—24no—
207C22×C4—42no—
208C22×C4—62no—
209C22×C4C2×D1211yescharacteristic
210C22×C4C2×D1212yes—
211C22×C4D3×D412yes—
212C22×C4#23911yescharacteristic
213D8—242no—
214M16—42no—
215M16—122no—
216C2×C8—22no—
217C2×C8—121no—
218C4⋊C4—62no—
219C4⋊C4—62no—
220C22⋊C4—28no—
221C22⋊C4—1216no—
222C22⋊C4—122no—
223C22⋊C4—122no—
224C42—44no—
225C22×C3—22no—
226C22×C3—22no—
227C22×C3—22no—
228C22×C3—21no—
229C22×C3—44no—
230C22×C3#39211yescharacteristic
231C22×C3#45314yes—
232C2×D3—416no—
233C2×D3—44no—
234C2×D3—88no—
235C2×D3—84no—
236C2×D3—84no—
237C2×D3—84no—
238C2×D3—88no—
239C2×D3—82no—
240C2×D3—82no—
241C2×D3—82no—
242C3×C4—21no—
243C3×C4—22no—
244C3×C4—48no—
245C3×C4B(4,𝔽2)12yes—
246Dic3—82no—
247C23—24no—
248C23—41no—
249C23—68no—
250C23—64no—
251C23—128no—
252C23—124no—
253C23—122no—
254C23—124no—
255C23—124no—
256C23—122no—
257C23—128no—
258C23—122no—
259C23—124no—
260C23—244no—
261C23—2416no—
262C23#88212yes—
263C23#89511yescharacteristic
264D4—124no—
265D4—128no—
266D4—124no—
267D4—124no—
268D4—242no—
269D4—2432no—
270D4—244no—
271D4—244no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—28no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—124no—
280C2×C4—122no—
281C2×C4#87411yescharacteristic
282C2×C4#88211yescharacteristic
283C2×C4#89512yes—
284C8—42no—
285C8—241no—
286C2×C3—24no—
287C2×C3—41no—
288C2×C3#200212yes—
289C2×C3#202111yescharacteristic
290D3—84no—
291D3—162no—
292C22—22no—
293C22—22no—
294C22—21no—
295C22—22no—
296C22—44no—
297C22—1216no—
298C22—124no—
299C22—248no—
300C22—248no—
301C22—244no—
302C22—244no—
303C22—244no—
304C22—242no—
305C22—242no—
306C22—242no—
307C22#467111yescharacteristic
308C22#482214yes—
309C4—21no—
310C4—22no—
311C4—48no—
312C4—242no—
313C4#474112yes—
314C3#1032611yescharacteristic, complemented, Sylow
315C2—24no—
316C2—41no—
317C2—244no—
318C2—482no—
319C2#6876112yes—
320C2#6896511yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1297C2
socleC22×C3#392
hypercenter#1297D3
hypocenterC3#10326
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1297

Complements

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

NH ≅ G/Ntype
1. #78641C2semidirect
2. #73587C2semidirect
3. #69988C2semidirect
4. #69444C2semidirect
5. C3×#1297C2semidirect
25. #1297D3semidirect
35. C22×D12D4semidirect
36. C2×C3×M16D4semidirect
45. C2×C4⋊Dic3D4semidirect
51. C3×#95D4semidirect
125. C2×M16D12semidirect
142. #95D12semidirect
314. C3#10326semidirect

Maximal subgroups

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

subgroupindex
#786412
#735872
#699882
#694442
C3×#12972
#103263