Finite Group Databasebeta

← #10692623

Subgroups — #10692623

Order 768.

726 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C3×#2561C211yesmaximal, characteristic
2#82083C211yesmaximal, characteristic
3#71688C212yesmaximal, complemented
4#71539C211yesmaximal, characteristic
5#67618C212yesmaximal, complemented
6#9407—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2211yescharacteristic
8C22×C3×C22⋊C4C2211yescharacteristic
9C22×#217—21no—
10C3×#397C412yescomplemented
11C3×#393C412yescomplemented
12C3×#388—22no—
13C3×#381—21no—
14C3×#379—21no—
15C3×C4×C22⋊C4C2211yescharacteristic
16C4×#217—21no—
17C4×#217C2212yes—
18#5207C2212yes—
19Dic3×C22⋊C4—24no—
20#4538—24no—
21#4525—22no—
22#2561D311yescharacteristic
23#2273—31no—
24#1508—32no—
25#1462—31no—
26#1297—32no—
27C25×C3D411yescharacteristic
28C22×C3×Q8C2×C411yescharacteristic, complemented
29C23×C3×C4D411yescharacteristic
30C23×C3×C4C2311yescharacteristic
31C23×Dic3—22no—
32C3×C22⋊Q8—22no—
33C3×C22⋊Q8—22no—
34C2×C3×C4⋊C4—21no—
35C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4—21no—
40C2×C3×C22⋊C4—21no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4C2×C412yes—
43C2×C3×C22⋊C4D412yes—
44C2×C3×C42—21no—
45C2×#217—22no—
46C2×#217—28no—
47C2×#217—22no—
48C2×C4×Dic3—22no—
49C2×C4×Dic3—22no—
50C2×#208—22no—
51C3×#95—22no—
52C3×#95—22no—
53C3×#95C2×C412yescomplemented
54C3⋊#95—22no—
55C3⋊#95D414yes—
56#830—22no—
57#830—22no—
58C2×C22⋊Q8C2×D311yescharacteristic
59C22×C22⋊C4—61no—
60C22×C22⋊C4C2×D311yescharacteristic
61#407—32no—
62#397Dic312yescomplemented
63#393Dic312yescomplemented
64#388—22no—
65#381—21no—
66#379—21no—
67C4×C22⋊C4—32no—
68C4×C22⋊C4—64no—
69C4×C22⋊C4—61no—
70C4×C22⋊C4C2×D311yescharacteristic
71#328—64no—
72#323—62no—
73C24×C3—21no—
74C24×C3—24no—
75C24×C3—21no—
76C24×C3—42no—
77C24×C3—81no—
78C24×C3C22⋊C412yes—
79C24×C3C2×D411yescharacteristic
80C2×C3×Q8—22no—
81C2×C3×Q8—81no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—22no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—42no—
88C22×C3×C4—81no—
89C22×C3×C4C22⋊C412yes—
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D412yes—
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4#4112yes—
94C22×Dic3—24no—
95C22×Dic3—28no—
96C22×Dic3—82no—
97C22×Dic3—82no—
98C3×C4⋊C4—22no—
99C3×C4⋊C4—42no—
100C3×C22⋊C4—24no—
101C3×C22⋊C4—24no—
102C3×C22⋊C4—42no—
103C3×C22⋊C4—44no—
104C3×C22⋊C4—41no—
105C3×C22⋊C4—41no—
106C3×C22⋊C4—82no—
107C3×C42—81no—
108#217—28no—
109#217—28no—
110C4×Dic3—82no—
111C4×Dic3—82no—
112#208—82no—
113C2×C3⋊C8—42no—
114C25#6711yescharacteristic
115C22×Q8C2×Dic311yescharacteristic, complemented
116C23×C4—62no—
117C23×C4#6711yescharacteristic
118C23×C4C22×D311yescharacteristic
119C2×M16—62no—
120C22⋊Q8—22no—
121C22⋊Q8—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4C2×Dic311yescharacteristic, complemented
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—62no—
131C2×C22⋊C4—68no—
132C2×C22⋊C4—62no—
133C2×C22⋊C4C2×Dic312yes—
134C2×C22⋊C4#6712yes—
135C2×C42—21no—
136C2×C42—62no—
137C2×C42—62no—
138C22⋊C8—62no—
139C22⋊C8—62no—
140#95—22no—
141#95—22no—
142#95—34no—
143#95—62no—
144#95C2×Dic312yescomplemented
145C23×C3—22no—
146C23×C3—22no—
147C23×C3—24no—
148C23×C3—22no—
149C23×C3—22no—
150C23×C3—22no—
151C23×C3—22no—
152C23×C3—41no—
153C23×C3—42no—
154C23×C3—42no—
155C23×C3—42no—
156C23×C3—42no—
157C23×C3—81no—
158C23×C3—81no—
159C23×C3—88no—
160C23×C3#10414yes—
161C23×C3C2×C22⋊C411yescharacteristic
162C23×C3#12011yescharacteristic
163C23×C3#12311yescharacteristic
164C3×Q8—82no—
165C2×C3×C4—22no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—42no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—81no—
180C2×C3×C4C4×D412yes—
181C2×C3×C4#12112yes—
182C2×Dic3—28no—
183C2×Dic3—44no—
184C2×Dic3—44no—
185C2×Dic3—88no—
186C3⋊C8—82no—
187C24—21no—
188C24—24no—
189C24—21no—
190C24—42no—
191C24—81no—
192C24#21712yes—
193C24C2×#6711yescharacteristic
194C2×Q8—22no—
195C2×Q8—81no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—42no—
202C22×C4—64no—
203C22×C4—68no—
204C22×C4—81no—
205C22×C4—242no—
206C22×C4—242no—
207C22×C4#21712yes—
208C22×C4D3×D412yes—
209C22×C4#23712yes—
210C22×C4C22×Dic311yescharacteristic
211C22×C4C2×#6711yescharacteristic
212M16—242no—
213C2×C8—122no—
214C4⋊C4—22no—
215C4⋊C4—42no—
216C22⋊C4—24no—
217C22⋊C4—24no—
218C22⋊C4—42no—
219C22⋊C4—44no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—68no—
223C22⋊C4—68no—
224C22⋊C4—82no—
225C42—81no—
226C42—242no—
227C42—242no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—24no—
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—44no—
237C22×C3—44no—
238C22×C3—44no—
239C22×C3—44no—
240C22×C3—42no—
241C22×C3—42no—
242C22×C3—82no—
243C22×C3—84no—
244C22×C3#38611yescharacteristic
245C22×C3C2×#10412yes—
246C22×C3#45512yes—
247C3×C4—22no—
248C3×C4—22no—
249C3×C4—42no—
250C3×C4—82no—
251C3×C4—81no—
252C3×C4—81no—
253Dic3—48no—
254C23—22no—
255C23—22no—
256C23—24no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—41no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—81no—
267C23—81no—
268C23—88no—
269C23#83714yes—
270C23#93511yescharacteristic
271C23#93711yescharacteristic
272C23C2×#21711yescharacteristic
273Q8—82no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—68no—
289C2×C4—81no—
290C2×C4—124no—
291C2×C4—124no—
292C2×C4—248no—
293C2×C4D4×Dic312yes—
294C2×C4#93912yes—
295C8—242no—
296C2×C3—22no—
297C2×C3—22no—
298C2×C3—41no—
299C2×C3—44no—
300C2×C3#188511yescharacteristic
301C2×C3#189611yescharacteristic
302C2×C3#191311yescharacteristic
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—24no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—44no—
312C22—44no—
313C22—44no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—82no—
318C22—84no—
319C22#506112yes—
320C22#522011yescharacteristic
321C22C2×#83712yes—
322C4—22no—
323C4—22no—
324C4—42no—
325C4—82no—
326C4—81no—
327C4—81no—
328C4—128no—
329C3#940711yescharacteristic, complemented, Sylow
330C2—22no—
331C2—22no—
332C2—41no—
333C2—44no—
334C2#7171111yescharacteristic
335C2#7172311yescharacteristic
336C2#7184811yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2561C2
socleC22×C3#386
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenter#2561D3
hypocenterC3#9407
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2561

Complements

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

NH ≅ G/Ntype
3. #71688C2semidirect
5. #67618C2semidirect
10. C3×#397C4semidirect
11. C3×#393C4semidirect
28. C22×C3×Q8C2×C4semidirect
35. C2×C3×C4⋊C4C2×C4semidirect
53. C3×#95C2×C4semidirect
62. #397Dic3semidirect
63. #393Dic3semidirect
115. C22×Q8C2×Dic3semidirect
123. C2×C4⋊C4C2×Dic3semidirect
144. #95C2×Dic3semidirect
329. C3#9407semidirect

Maximal subgroups

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

subgroupindex
C3×#25612
#820832
#716882
#715392
#676182
#94073