Finite Group Databasebeta

← #10692647

Subgroups — #10692647

Order 768.

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

normal subgroups only

This group has 526 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×#2313C211yesmaximal, characteristic
2#82181C211yesmaximal, characteristic
3#71552C211yesmaximal, characteristic
4#69445C212yesmaximal
5#67618C212yesmaximal
6#9412—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C2×#924C2211yescharacteristic
9C3×#387—24no—
10C3×#381—21no—
11C3×#380—21no—
12C3×C4×C22⋊C4C2212yes—
13#5210—21no—
14#5210—21no—
15C4×#217C2211yescharacteristic
16#5207C2212yes—
17#4981C412yescomplemented
18#4979C412yescomplemented
19#4976—22no—
20#4528—22no—
21#4452—24no—
22#2583—31no—
23#2313D311yescharacteristic
24#1508—32no—
25#1463—31no—
26#1297—32no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×Dic6C2×C411yescharacteristic
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4C2×C412yes—
35C2×C3×C4⋊C4D412yes—
36C2×C3×C22⋊C4—22no—
37C2×C3×C42—22no—
38C2×#217—21no—
39C2×#217—21no—
40C2×C4⋊Dic3C2×C411yescharacteristic
41#924—22no—
42#924—22no—
43C2×Dic3⋊C4—22no—
44C2×Dic3⋊C4—21no—
45C2×Dic3⋊C4—22no—
46C2×C4×Dic3—21no—
47C2×#208—22no—
48C3×#95—22no—
49C3×#95—24no—
50C3×#95D414yes—
51C3⋊#95—22no—
52C3⋊#95—22no—
53C3⋊#95C2×C412yes—
54#830—22no—
55#830—22no—
56C2×C22⋊Q8—31no—
57C22×C4⋊C4C2×D311yescharacteristic
58#407—32no—
59#398—32no—
60#395—32no—
61#389—62no—
62#387—24no—
63#385—61no—
64#385—61no—
65#381—21no—
66#380—21no—
67C4×C22⋊C4—31no—
68C4×C22⋊C4C2×D312yes—
69#328—64no—
70#324—62no—
71C24×C3C2×D411yescharacteristic
72C22×C3×C4—21no—
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—24no—
77C22×C3×C4—42no—
78C22×C3×C4—81no—
79C22×C3×C4—81no—
80C22×C3×C4C22⋊C412yes—
81C22×C3×C4C22⋊C412yes—
82C22×C3×C4C22×C411yescharacteristic
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4C2×D412yes—
85C22×C3×C4#4112yes—
86C22×Dic3—22no—
87C22×Dic3—21no—
88C22×Dic3—21no—
89C2×Dic6—22no—
90C2×Dic6—81no—
91C3×C4⋊C4—24no—
92C3×C4⋊C4—24no—
93C3×C4⋊C4—82no—
94C3×C22⋊C4—44no—
95C3×C42—82no—
96#217—24no—
97#217—41no—
98#217—41no—
99C4⋊Dic3—22no—
100Dic3⋊C4—44no—
101Dic3⋊C4—42no—
102Dic3⋊C4—82no—
103C4×Dic3—81no—
104#208—82no—
105C2×C3⋊C8—42no—
106C22×Q8—31no—
107C23×C4#6711yescharacteristic
108C23×C4#6711yescharacteristic
109C23×C4C22×D311yescharacteristic
110C2×M16—62no—
111C22⋊Q8—62no—
112C22⋊Q8—62no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—62no—
118C2×C4⋊C4—61no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4C4×D312yes—
121C2×C4⋊C4D1212yes—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—61no—
124C2×C22⋊C4—61no—
125C2×C42—22no—
126C2×C42—61no—
127C22⋊C8—62no—
128C22⋊C8—62no—
129#95—22no—
130#95—24no—
131#95—32no—
132#95—62no—
133#95—62no—
134#95#6714yes—
135C23×C3—22no—
136C23×C3—22no—
137C23×C3—41no—
138C23×C3C2×C22⋊C411yescharacteristic
139C23×C3#12011yescharacteristic
140C23×C3#12311yescharacteristic
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—42no—
146C2×C3×C4—41no—
147C2×C3×C4—42no—
148C2×C3×C4—41no—
149C2×C3×C4—42no—
150C2×C3×C4—42no—
151C2×C3×C4—42no—
152C2×C3×C4—42no—
153C2×C3×C4—44no—
154C2×C3×C4—44no—
155C2×C3×C4—44no—
156C2×C3×C4—81no—
157C2×C3×C4C4×D412yes—
158C2×C3×C4#12112yes—
159C2×Dic3—24no—
160C2×Dic3—42no—
161C2×Dic3—42no—
162C2×Dic3—42no—
163C2×Dic3—41no—
164C2×Dic3—41no—
165C2×Dic3—42no—
166Dic6—82no—
167C3⋊C8—82no—
168C24C2×#6711yescharacteristic
169C2×Q8—62no—
170C2×Q8—241no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—24no—
176C22×C4—42no—
177C22×C4—62no—
178C22×C4—61no—
179C22×C4—61no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4(C2×D3)⋊C412yes—
183C22×C4(C2×D3)⋊C412yes—
184C22×C4C2×C4×D311yescharacteristic
185C22×C4C2×D1211yescharacteristic
186C22×C4#23512yes—
187C22×C4C2×#6712yes—
188M16—242no—
189C2×C8—122no—
190C4⋊C4—24no—
191C4⋊C4—24no—
192C4⋊C4—62no—
193C4⋊C4—82no—
194C4⋊C4—124no—
195C4⋊C4—122no—
196C4⋊C4—242no—
197C22⋊C4—44no—
198C22⋊C4—64no—
199C22⋊C4—121no—
200C22⋊C4—121no—
201C42—82no—
202C42—241no—
203C22×C3—21no—
204C22×C3—21no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—44no—
211C22×C3#38611yescharacteristic
212C22×C3#42112yes—
213C22×C3#45612yes—
214C3×C4—22no—
215C3×C4—22no—
216C3×C4—42no—
217C3×C4—42no—
218C3×C4—84no—
219Dic3—44no—
220Dic3—81no—
221Dic3—81no—
222C23—22no—
223C23—22no—
224C23—41no—
225C23C2×(C2×D3)⋊C411yescharacteristic
226C23#92911yescharacteristic
227C23#95311yescharacteristic
228Q8—242no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—64no—
245C2×C4—81no—
246C2×C4—122no—
247C2×C4—122no—
248C2×C4—122no—
249C2×C4—121no—
250C2×C4—121no—
251C2×C4—122no—
252C2×C4C4×#6712yes—
253C2×C4#93012yes—
254C8—242no—
255C2×C3—22no—
256C2×C3—22no—
257C2×C3—41no—
258C2×C3#188611yescharacteristic
259C2×C3#189711yescharacteristic
260C2×C3#191311yescharacteristic
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—44no—
269C22#500512yes—
270C22#521311yescharacteristic
271C22#524612yes—
272C4—22no—
273C4—22no—
274C4—42no—
275C4—42no—
276C4—84no—
277C4—124no—
278C4—241no—
279C4—241no—
280C3#941211yescharacteristic, complemented, Sylow
281C2—22no—
282C2—22no—
283C2—41no—
284C2#7159111yescharacteristic
285C2#7165611yescharacteristic
286C2#7166011yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2313C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2313D3
hypocenterC3#9412
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
17. #4981C4semidirect
18. #4979C4semidirect
280. C3#9412semidirect

Maximal subgroups

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

subgroupindex
C3×#23132
#821812
#715522
#694452
#676182
#94123