Finite Group Databasebeta

← #10692642

Subgroups — #10692642

Order 768.

606 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2312C211yesmaximal, characteristic
2#82116C211yesmaximal, characteristic
3#71539C211yesmaximal, characteristic
4#69445C212yesmaximal, complemented
5#67618C212yesmaximal, complemented
6#9411—31nomaximal, Sylow
7C22×C3×C22⋊C4C2211yescharacteristic
8C2×#924C2211yescharacteristic
9C3×#388—24no—
10C3×#380—21no—
11C3×#379—21no—
12C3×C4×C22⋊C4C2212yes—
13#5247—21no—
14#5211—21no—
15C4×#217C2211yescharacteristic
16#5207C2212yes—
17#4953C412yescomplemented
18#4947C412yescomplemented
19#4945—22no—
20#4525—22no—
21#4452—24no—
22#2561—31no—
23#2312D311yescharacteristic
24#1508—32no—
25#1462—31no—
26#1297—32no—
27C25×C3D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×Dic6C2×C411yescharacteristic
31C2×C3×C4⋊C4—22no—
32C2×C3×C22⋊C4—22no—
33C2×C3×C22⋊C4—22no—
34C2×C3×C22⋊C4—22no—
35C2×C3×C22⋊C4—22no—
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4C2×C412yes—
38C2×C3×C22⋊C4D412yes—
39C2×C3×C42—22no—
40C2×#217—22no—
41C2×#217—21no—
42C2×#217—21no—
43C2×#217—22no—
44C2×C4⋊Dic3C2×C411yescharacteristic
45#924—22no—
46#924—22no—
47C2×Dic3⋊C4—21no—
48C2×C4×Dic3—21no—
49C2×#208—22no—
50C3×#95—22no—
51C3×#95D414yes—
52C3⋊#95—22no—
53C3⋊#95—22no—
54C3⋊#95C2×C412yes—
55#830—22no—
56#830—22no—
57C2×C22⋊Q8—31no—
58C22×C22⋊C4C2×D311yescharacteristic
59#407—32no—
60#397—32no—
61#393—32no—
62#388—24no—
63#388—62no—
64#381—61no—
65#380—21no—
66#379—21no—
67#379—61no—
68C4×C22⋊C4—31no—
69C4×C22⋊C4C2×D312yes—
70#328—64no—
71#323—62no—
72C24×C3—21no—
73C24×C3—21no—
74C24×C3—42no—
75C24×C3—42no—
76C24×C3—81no—
77C24×C3C22⋊C412yes—
78C24×C3C2×D411yescharacteristic
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—24no—
82C22×C3×C4—42no—
83C22×C3×C4—81no—
84C22×C3×C4C22⋊C412yes—
85C22×C3×C4C22×C411yescharacteristic
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4C2×D412yes—
88C22×C3×C4#4112yes—
89C22×Dic3—22no—
90C22×Dic3—21no—
91C22×Dic3—21no—
92C2×Dic6—22no—
93C2×Dic6—81no—
94C3×C4⋊C4—82no—
95C3×C22⋊C4—24no—
96C3×C22⋊C4—24no—
97C3×C22⋊C4—44no—
98C3×C22⋊C4—82no—
99C3×C22⋊C4—82no—
100C3×C42—82no—
101#217—42no—
102#217—44no—
103#217—41no—
104#217—41no—
105#217—82no—
106C4⋊Dic3—22no—
107Dic3⋊C4—42no—
108C4×Dic3—81no—
109#208—82no—
110C2×C3⋊C8—42no—
111C25#6711yescharacteristic
112C22×Q8—31no—
113C23×C4#6711yescharacteristic
114C23×C4C22×D311yescharacteristic
115C2×M16—62no—
116C22⋊Q8—62no—
117C22⋊Q8—62no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—61no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—62no—
127C2×C22⋊C4—61no—
128C2×C22⋊C4—61no—
129C2×C22⋊C4—62no—
130C2×C22⋊C4C4×D312yes—
131C2×C22⋊C4D1212yes—
132C2×C42—22no—
133C2×C42—61no—
134C22⋊C8—62no—
135C22⋊C8—62no—
136#95—22no—
137#95—32no—
138#95—62no—
139#95—62no—
140#95#6714yes—
141C23×C3—22no—
142C23×C3—22no—
143C23×C3—22no—
144C23×C3—22no—
145C23×C3—22no—
146C23×C3—41no—
147C23×C3—42no—
148C23×C3—42no—
149C23×C3—42no—
150C23×C3—42no—
151C23×C3—42no—
152C23×C3—42no—
153C23×C3—81no—
154C23×C3—81no—
155C23×C3—88no—
156C23×C3C2×C22⋊C411yescharacteristic
157C23×C3#12011yescharacteristic
158C23×C3#12311yescharacteristic
159C2×C3×C4—22no—
160C2×C3×C4—42no—
161C2×C3×C4—41no—
162C2×C3×C4—42no—
163C2×C3×C4—41no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—44no—
167C2×C3×C4—44no—
168C2×C3×C4—44no—
169C2×C3×C4—81no—
170C2×C3×C4C4×D412yes—
171C2×C3×C4#12112yes—
172C2×Dic3—42no—
173C2×Dic3—42no—
174C2×Dic3—42no—
175C2×Dic3—41no—
176C2×Dic3—41no—
177C2×Dic3—42no—
178C2×Dic3—42no—
179Dic6—82no—
180C3⋊C8—82no—
181C24—21no—
182C24—21no—
183C24—42no—
184C24—42no—
185C24—81no—
186C24(C2×D3)⋊C412yes—
187C24C2×#6711yescharacteristic
188C2×Q8—62no—
189C2×Q8—241no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—24no—
193C22×C4—42no—
194C22×C4—62no—
195C22×C4—61no—
196C22×C4—61no—
197C22×C4—81no—
198C22×C4(C2×D3)⋊C412yes—
199C22×C4C2×C4×D311yescharacteristic
200C22×C4C2×D1211yescharacteristic
201C22×C4#23512yes—
202C22×C4C2×#6712yes—
203M16—242no—
204C2×C8—122no—
205C4⋊C4—62no—
206C4⋊C4—82no—
207C4⋊C4—122no—
208C22⋊C4—24no—
209C22⋊C4—24no—
210C22⋊C4—44no—
211C22⋊C4—82no—
212C22⋊C4—82no—
213C22⋊C4—122no—
214C22⋊C4—124no—
215C22⋊C4—121no—
216C22⋊C4—121no—
217C22⋊C4—242no—
218C42—82no—
219C42—241no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—44no—
228C22×C3—44no—
229C22×C3—42no—
230C22×C3—42no—
231C22×C3—42no—
232C22×C3—82no—
233C22×C3—84no—
234C22×C3—84no—
235C22×C3#38611yescharacteristic
236C22×C3#42112yes—
237C22×C3#45512yes—
238C3×C4—22no—
239C3×C4—22no—
240C3×C4—42no—
241C3×C4—84no—
242Dic3—82no—
243Dic3—81no—
244Dic3—81no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—41no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—81no—
258C23—81no—
259C23—88no—
260C23C2×(C2×D3)⋊C411yescharacteristic
261C23#92911yescharacteristic
262C23#95311yescharacteristic
263Q8—242no—
264C2×C4—22no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—81no—
275C2×C4—122no—
276C2×C4—122no—
277C2×C4—122no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4—122no—
281C2×C4—122no—
282C2×C4C4×#6712yes—
283C2×C4#93012yes—
284C8—242no—
285C2×C3—22no—
286C2×C3—22no—
287C2×C3—41no—
288C2×C3—82no—
289C2×C3#188511yescharacteristic
290C2×C3#189711yescharacteristic
291C2×C3#191311yescharacteristic
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—44no—
300C22—44no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—82no—
305C22—84no—
306C22—84no—
307C22#500512yes—
308C22#521311yescharacteristic
309C22#524212yes—
310C4—22no—
311C4—22no—
312C4—42no—
313C4—84no—
314C4—242no—
315C4—241no—
316C4—241no—
317C3#941111yescharacteristic, complemented, Sylow
318C2—22no—
319C2—22no—
320C2—41no—
321C2—82no—
322C2#7159111yescharacteristic
323C2#7164911yescharacteristic
324C2#7165611yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2312C2
socleC22×C3#386
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenter#2312D3
hypocenterC3#9411
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
4. #69445C2semidirect
5. #67618C2semidirect
17. #4953C4semidirect
18. #4947C4semidirect
317. C3#9411semidirect

Maximal subgroups

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

subgroupindex
C3×#23122
#821162
#715392
#694452
#676182
#94113