Finite Group Databasebeta

← #10692628

Subgroups — #10692628

Order 768.

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

normal subgroups only

This group has 618 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×#2583C211yesmaximal, characteristic
2#82144C211yesmaximal, characteristic
3#71688C212yesmaximal
4#71552C211yesmaximal, characteristic
5#67618C212yesmaximal
6#9408—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C2×#926—21no—
10C3×#398C412yescomplemented
11C3×#395C412yescomplemented
12C3×#389—22no—
13C3×#385—21no—
14C3×#385—21no—
15C3×C4×C22⋊C4C2211yescharacteristic
16C4×#217—21no—
17C4×#217C2212yes—
18#5207C2212yes—
19Dic3×C4⋊C4—24no—
20#4538—24no—
21#4528—22no—
22#2583D311yescharacteristic
23#2274—31no—
24#1508—32no—
25#1463—31no—
26#1297—32no—
27C22×C3×Q8C2×C411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4C2311yescharacteristic
31C3×C22⋊Q8—22no—
32C3×C22⋊Q8—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4C2×C412yes—
40C2×C3×C4⋊C4D412yes—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C42—21no—
44C2×#217—22no—
45#926—28no—
46C2×C4⋊Dic3—22no—
47C2×C4×Dic3—22no—
48C2×C4×Dic3—22no—
49C2×C4×Dic3—22no—
50C2×#208—22no—
51C3×#95—22no—
52C3×#95—22no—
53C3×#95C2×C412yes—
54C3⋊#95—22no—
55C3⋊#95D414yes—
56#830—22no—
57#830—22no—
58C2×C22⋊Q8C2×D311yescharacteristic
59C2×#117—61no—
60C22×C4⋊C4C2×D311yescharacteristic
61#407—32no—
62#398Dic312yescomplemented
63#395Dic312yescomplemented
64#389—22no—
65#385—21no—
66#385—21no—
67C4×C4⋊C4—64no—
68C4×C22⋊C4—32no—
69C4×C22⋊C4—61no—
70C4×C22⋊C4C2×D311yescharacteristic
71#328—64no—
72#324—62no—
73C24×C3C2×D411yescharacteristic
74C2×C3×Q8—22no—
75C2×C3×Q8—81no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—24no—
79C22×C3×C4—22no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—81no—
85C22×C3×C4—81no—
86C22×C3×C4C22⋊C412yes—
87C22×C3×C4C22⋊C412yes—
88C22×C3×C4C22×C411yescharacteristic
89C22×C3×C4C2×D412yes—
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4#4112yes—
92C22×Dic3—24no—
93C3×C4⋊C4—24no—
94C3×C4⋊C4—24no—
95C3×C4⋊C4—22no—
96C3×C4⋊C4—44no—
97C3×C4⋊C4—42no—
98C3×C4⋊C4—82no—
99C3×C22⋊C4—24no—
100C3×C22⋊C4—41no—
101C3×C22⋊C4—41no—
102C3×C42—81no—
103#217—28no—
104C4⋊Dic3—28no—
105C4×Dic3—28no—
106C4×Dic3—82no—
107C4×Dic3—82no—
108#208—82no—
109C2×C3⋊C8—42no—
110C22×Q8C2×Dic311yescharacteristic
111C23×C4#6711yescharacteristic
112C23×C4#6711yescharacteristic
113C23×C4C22×D311yescharacteristic
114C2×M16—62no—
115C22⋊Q8—22no—
116C22⋊Q8—22no—
117#117—68no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—62no—
124C2×C4⋊C4C2×Dic311yescharacteristic
125C2×C4⋊C4C2×Dic312yes—
126C2×C4⋊C4#6712yes—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—62no—
130C2×C42—21no—
131C2×C42—62no—
132C2×C42—62no—
133C2×C42—62no—
134C22⋊C8—62no—
135C22⋊C8—62no—
136#95—22no—
137#95—22no—
138#95—34no—
139#95—62no—
140#95C2×Dic312yes—
141C23×C3—22no—
142C23×C3—22no—
143C23×C3—41no—
144C23×C3C2×C22⋊C411yescharacteristic
145C23×C3#12011yescharacteristic
146C23×C3#12311yescharacteristic
147C3×Q8—82no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—24no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—24no—
154C2×C3×C4—22no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—41no—
158C2×C3×C4—41no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—41no—
165C2×C3×C4—41no—
166C2×C3×C4—42no—
167C2×C3×C4—81no—
168C2×C3×C4#10414yes—
169C2×C3×C4C4×D412yes—
170C2×C3×C4#12112yes—
171C2×Dic3—28no—
172C2×Dic3—44no—
173C2×Dic3—44no—
174C3⋊C8—82no—
175C24C2×#6711yescharacteristic
176C2×Q8—22no—
177C2×Q8—81no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—24no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—64no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4#21712yes—
190C22×C4#21712yes—
191C22×C4D3×D412yes—
192C22×C4#23712yes—
193C22×C4C22×Dic311yescharacteristic
194C22×C4C2×#6711yescharacteristic
195M16—242no—
196C2×C8—122no—
197C4⋊C4—24no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—44no—
201C4⋊C4—42no—
202C4⋊C4—68no—
203C4⋊C4—82no—
204C22⋊C4—24no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—68no—
208C42—68no—
209C42—81no—
210C42—242no—
211C42—242no—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—21no—
217C22×C3—21no—
218C22×C3—21no—
219C22×C3—44no—
220C22×C3#38611yescharacteristic
221C22×C3C2×#10412yes—
222C22×C3#45612yes—
223C3×C4—24no—
224C3×C4—22no—
225C3×C4—22no—
226C3×C4—44no—
227C3×C4—42no—
228C3×C4—81no—
229C3×C4—81no—
230Dic3—48no—
231C23—22no—
232C23—22no—
233C23—41no—
234C23#93511yescharacteristic
235C23#93711yescharacteristic
236C23C2×#21711yescharacteristic
237Q8—82no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—68no—
258C2×C4—81no—
259C2×C4—124no—
260C2×C4—124no—
261C2×C4#83714yes—
262C2×C4D4×Dic312yes—
263C2×C4#93912yes—
264C8—242no—
265C2×C3—22no—
266C2×C3—22no—
267C2×C3—41no—
268C2×C3#188611yescharacteristic
269C2×C3#189611yescharacteristic
270C2×C3#191311yescharacteristic
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—44no—
279C22#509512yes—
280C22#522011yescharacteristic
281C22C2×#83712yes—
282C4—24no—
283C4—22no—
284C4—22no—
285C4—44no—
286C4—42no—
287C4—81no—
288C4—81no—
289C4—128no—
290C3#940811yescharacteristic, complemented, Sylow
291C2—22no—
292C2—22no—
293C2—41no—
294C2#7171111yescharacteristic
295C2#7178411yescharacteristic
296C2#7184811yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2583C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2583D3
hypocenterC3#9408
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2583

Complements

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

NH ≅ G/Ntype
10. C3×#398C4semidirect
11. C3×#395C4semidirect
62. #398Dic3semidirect
63. #395Dic3semidirect
290. C3#9408semidirect

Maximal subgroups

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

subgroupindex
C3×#25832
#821442
#716882
#715522
#676182
#94083