Finite Group Databasebeta

← #10705804

Subgroups — #10705804

Order 768.

460 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2563C211yesmaximal, characteristic
2C4×#834C211yesmaximal, characteristic
3#71602C211yesmaximal, characteristic
4#67646C214yesmaximal, complemented
5#11787—31nomaximal, Sylow
6C2×C3×C4⋊Q8C411yescharacteristic, complemented
7C2×C3×C4×D4C2211yescharacteristic
8C3×#393C412yescomplemented
9C3×#393C2212yes—
10C3×#390—21no—
11C3×#382—21no—
12C3×#380—21no—
13C3×#380—21no—
14C3×C4×C4⋊C4C411yescharacteristic, complemented
15C2×#834—21no—
16C4×#217—21no—
17#5207—21no—
18#4528—21no—
19#4526C2212yes—
20#4525C2212yes—
21#2563D311yescharacteristic
22C4×#99—31no—
23#1465—31no—
24#1351—34no—
25C22×C3×Q8C2×C412yes—
26C22×C3×D4D411yescharacteristic
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C3×C4⋊Q8—41no—
30C3×C4⋊Q8—41no—
31C3×C4×D4—41no—
32C3×C4×D4—41no—
33C2×C3×C4⋊C4—21no—
34C2×C3×C4⋊C4—21no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4D411yescharacteristic
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4C2×C411yescharacteristic
40C2×C3×C22⋊C4C2311yescharacteristic
41C2×C3×C42—21no—
42C2×C3×C42C2×C411yescharacteristic
43C2×#217—22no—
44C2×C4×Dic3—41no—
45C2×#208—41no—
46C3×#95—22no—
47C3×#95—22no—
48C3×#95C2×C412yes—
49#834—24no—
50C3⋊#95—41no—
51#830—22no—
52#830—41no—
53C2×C4⋊Q8Dic311yescharacteristic, complemented
54C2×C4×D4C2×D311yescharacteristic
55C2×#99—61no—
56#407—61no—
57#393Dic312yescomplemented
58#393C2×D312yes—
59#390—21no—
60#382—21no—
61#380—21no—
62#380—21no—
63C4×C4⋊C4Dic311yescharacteristic, complemented
64C4×C22⋊C4—61no—
65#342—32no—
66#324—61no—
67#323—32no—
68C24×C3C22⋊C411yescharacteristic
69C24×C3C2×D411yescharacteristic
70C2×C3×Q8—42no—
71C2×C3×Q8—82no—
72C2×C3×D4—22no—
73C2×C3×D4—22no—
74C2×C3×D4—41no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—22no—
80C22×C3×C4—22no—
81C22×C3×C4—42no—
82C22×C3×C4—42no—
83C22×C3×C4C22⋊C411yescharacteristic
84C22×C3×C4C22⋊C411yescharacteristic
85C22×C3×C4C22⋊C411yescharacteristic
86C22×C3×C4C22×C411yescharacteristic
87C22×C3×C4C2×D411yescharacteristic
88C22×Dic3—42no—
89C3×C4⋊C4—41no—
90C3×C4⋊C4—42no—
91C3×C4⋊C4—42no—
92C3×C4⋊C4—41no—
93C3×C4⋊C4—41no—
94C3×C22⋊C4—41no—
95C3×C22⋊C4—41no—
96C3×C22⋊C4—82no—
97C3×C22⋊C4—82no—
98C3×C42—21no—
99C3×C42—21no—
100C3×C42—42no—
101#217—24no—
102#217—42no—
103C4×Dic3—82no—
104#208—82no—
105C2×C3⋊C8—42no—
106C22×Q8C2×Dic312yes—
107C22×D4#6711yescharacteristic
108C23×C4#6711yescharacteristic
109C23×C4#6711yescharacteristic
110C2×M16—121no—
111C4⋊Q8—41no—
112C4⋊Q8—41no—
113C4×D4—41no—
114C4×D4—41no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4#6711yescharacteristic
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—62no—
122C2×C22⋊C4C2×Dic311yescharacteristic
123C2×C22⋊C4C22×D311yescharacteristic
124C2×C42—21no—
125C2×C42—121no—
126C2×C42C2×Dic311yescharacteristic
127#99—64no—
128C22⋊C8—62no—
129C22⋊C8—121no—
130#95—22no—
131#95—22no—
132#95—121no—
133#95C2×Dic312yes—
134C23×C3—21no—
135C23×C3—21no—
136C23×C3—21no—
137C23×C3—21no—
138C23×C3—21no—
139C23×C3—41no—
140C23×C3—41no—
141C23×C3—41no—
142C23×C3—41no—
143C23×C3C2×C22⋊C411yescharacteristic
144C23×C3#13712yes—
145C3×Q8—84no—
146C3×D4—44no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—22no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—41no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—81no—
165C2×C3×C4—81no—
166C2×C3×C4#9911yescharacteristic
167C2×C3×C4#9911yescharacteristic
168C2×C3×C4#10412yes—
169C2×Dic3—44no—
170C2×Dic3—82no—
171C2×Dic3—82no—
172C3⋊C8—82no—
173C24#21711yescharacteristic
174C24C2×#6711yescharacteristic
175C2×Q8—42no—
176C2×Q8—82no—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—41no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—122no—
189C22×C4#21711yescharacteristic
190C22×C4#21711yescharacteristic
191C22×C4#21711yescharacteristic
192C22×C4C22×Dic311yescharacteristic
193C22×C4C2×#6711yescharacteristic
194M16—242no—
195C2×C8—122no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—64no—
204C22⋊C4—82no—
205C22⋊C4—82no—
206C22⋊C4—122no—
207C42—21no—
208C42—21no—
209C42—42no—
210C42—242no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—22no—
214C22×C3—42no—
215C22×C3—42no—
216C22×C3—41no—
217C22×C3—41no—
218C22×C3—41no—
219C22×C3—41no—
220C22×C3—41no—
221C22×C3—81no—
222C22×C3—81no—
223C22×C3C2×#9911yescharacteristic
224C22×C3#41911yescharacteristic
225C22×C3C2×#10411yescharacteristic
226C3×C4—21no—
227C3×C4—22no—
228C3×C4—41no—
229C3×C4—41no—
230C3×C4—41no—
231C3×C4—82no—
232C3×C4—82no—
233C3×C4#35412yes—
234Dic3—84no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23#94212yes—
245C23C2×#21711yescharacteristic
246Q8—84no—
247D4—44no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—124no—
268C2×C4—242no—
269C2×C4—242no—
270C2×C4#83411yescharacteristic
271C2×C4#83411yescharacteristic
272C2×C4#83712yes—
273C8—242no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—42no—
277C2×C3—81no—
278C2×C3#150811yescharacteristic
279C2×C3#212211yescharacteristic
280C2×C3C2×#35411yescharacteristic
281C22—21no—
282C22—21no—
283C22—22no—
284C22—42no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22—81no—
293C22C2×#83411yescharacteristic
294C22#522511yescharacteristic
295C22C2×#83711yescharacteristic
296C4—22no—
297C4—21no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—82no—
302C4—82no—
303C4—244no—
304C4#454112yes—
305C3#1178711yescharacteristic, complemented, Sylow
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2#7168811yescharacteristic
311C2C2×#454111yescharacteristic
312C2#7185911yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#2563C2
socleC22×C3C2×#99
prime radicalC22×C3×D4D4
cosocleC2×C3×C22⋊C4C23
hypercenter#2563D3
hypocenterC3#11787
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2563

Complements

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

NH ≅ G/Ntype
4. #67646C2semidirect
6. C2×C3×C4⋊Q8C4semidirect
8. C3×#393C4semidirect
14. C3×C4×C4⋊C4C4semidirect
53. C2×C4⋊Q8Dic3semidirect
57. #393Dic3semidirect
63. C4×C4⋊C4Dic3semidirect
305. C3#11787semidirect

Maximal subgroups

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

subgroupindex
C3×#25632
C4×#8342
#716022
#676462
#117873