Finite Group Databasebeta

← #10705808

Subgroups — #10705808

Order 768.

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

normal subgroups only

This group has 516 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×#2542C211yesmaximal, characteristic
2C4×#834C211yesmaximal, characteristic
3#71602C211yesmaximal, characteristic
4#67651C214yesmaximal, complemented
5#11788—31nomaximal, Sylow
6C2×C3×C4⋊Q8C411yescharacteristic, complemented
7C2×C3×C22⋊Q8—21no—
8C2×C3×C22⋊Q8—21no—
9C2×C3×C4×D4C2211yescharacteristic
10C3×#399C412yescomplemented
11C3×#399C2212yes—
12C3×#390—21no—
13C3×#384—21no—
14C3×C4×C4⋊C4C411yescharacteristic, complemented
15C2×#834—21no—
16C4×#217—21no—
17#5207—21no—
18#4528—21no—
19#4526C2212yes—
20#4525C2212yes—
21#2542D311yescharacteristic
22C4×#99—31no—
23#1465—31no—
24#1352—34no—
25C22×C3×Q8—21no—
26C22×C3×D4D411yescharacteristic
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C3×C4⋊Q8—41no—
30C3×C4⋊Q8—41no—
31C3×C22⋊Q8—44no—
32C3×C22⋊Q8—44no—
33C3×C4×D4—41no—
34C3×C4×D4—41no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C4⋊C4C2×C412yes—
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4C2×C411yescharacteristic
44C2×C3×C22⋊C4C2311yescharacteristic
45C2×C3×C42—21no—
46C2×C3×C42C2×C411yescharacteristic
47C2×#217—22no—
48C2×C4×Dic3—41no—
49C2×#208—41no—
50C3×#95C2×C412yes—
51#834—24no—
52C3⋊#95—41no—
53#830—22no—
54#830—41no—
55C2×C4⋊Q8Dic311yescharacteristic, complemented
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C4×D4C2×D311yescharacteristic
59C2×#99—61no—
60#407—61no—
61#399Dic312yescomplemented
62#399C2×D312yes—
63#390—21no—
64#384—21no—
65C4×C4⋊C4Dic311yescharacteristic, complemented
66C4×C22⋊C4—61no—
67#342—32no—
68#324—61no—
69#323—32no—
70C24×C3C22⋊C411yescharacteristic
71C24×C3C2×D411yescharacteristic
72C2×C3×Q8—42no—
73C2×C3×Q8—42no—
74C2×C3×Q8—41no—
75C2×C3×Q8—41no—
76C2×C3×D4—22no—
77C2×C3×D4—22no—
78C2×C3×D4—41no—
79C22×C3×C4—22no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—22no—
85C22×C3×C4—42no—
86C22×C3×C4—42no—
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×Dic3—42no—
93C3×C4⋊C4—24no—
94C3×C4⋊C4—41no—
95C3×C4⋊C4—42no—
96C3×C4⋊C4—44no—
97C3×C4⋊C4—44no—
98C3×C4⋊C4—42no—
99C3×C22⋊C4—41no—
100C3×C22⋊C4—41no—
101C3×C22⋊C4—44no—
102C3×C22⋊C4—44no—
103C3×C42—21no—
104C3×C42—21no—
105C3×C42—42no—
106#217—24no—
107#217—42no—
108C4×Dic3—82no—
109#208—82no—
110C2×C3⋊C8—42no—
111C22×Q8—21no—
112C22×D4#6711yescharacteristic
113C23×C4#6711yescharacteristic
114C23×C4#6711yescharacteristic
115C2×M16—121no—
116C4⋊Q8—41no—
117C4⋊Q8—41no—
118C22⋊Q8—44no—
119C22⋊Q8—44no—
120C4×D4—41no—
121C4×D4—41no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4C2×Dic312yes—
127C2×C4⋊C4#6711yescharacteristic
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—62no—
131C2×C22⋊C4C2×Dic311yescharacteristic
132C2×C22⋊C4C22×D311yescharacteristic
133C2×C42—21no—
134C2×C42—121no—
135C2×C42C2×Dic311yescharacteristic
136#99—64no—
137C22⋊C8—62no—
138C22⋊C8—121no—
139#95—121no—
140#95C2×Dic312yes—
141C23×C3—21no—
142C23×C3—21no—
143C23×C3—21no—
144C23×C3—21no—
145C23×C3—21no—
146C23×C3—41no—
147C23×C3—41no—
148C23×C3—41no—
149C23×C3—41no—
150C23×C3C2×C22⋊C411yescharacteristic
151C23×C3#13712yes—
152C3×Q8—82no—
153C3×Q8—82no—
154C3×D4—44no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—22no—
158C2×C3×C4—24no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—81no—
173C2×C3×C4—81no—
174C2×C3×C4#9911yescharacteristic
175C2×C3×C4#9911yescharacteristic
176C2×C3×C4#10412yes—
177C2×Dic3—44no—
178C2×Dic3—82no—
179C2×Dic3—82no—
180C3⋊C8—82no—
181C24#21711yescharacteristic
182C24C2×#6711yescharacteristic
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—41no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—122no—
199C22×C4#21711yescharacteristic
200C22×C4#21711yescharacteristic
201C22×C4#21711yescharacteristic
202C22×C4C22×Dic311yescharacteristic
203C22×C4C2×#6711yescharacteristic
204M16—242no—
205C2×C8—122no—
206C4⋊C4—24no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—42no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—44no—
215C22⋊C4—44no—
216C22⋊C4—64no—
217C22⋊C4—122no—
218C42—21no—
219C42—21no—
220C42—42no—
221C42—242no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—22no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—41no—
232C22×C3—81no—
233C22×C3—81no—
234C22×C3C2×#9911yescharacteristic
235C22×C3#41911yescharacteristic
236C22×C3C2×#10411yescharacteristic
237C3×C4—21no—
238C3×C4—22no—
239C3×C4—44no—
240C3×C4—41no—
241C3×C4—41no—
242C3×C4—41no—
243C3×C4—82no—
244C3×C4#35312yes—
245Dic3—84no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23#94212yes—
256C23C2×#21711yescharacteristic
257Q8—82no—
258Q8—82no—
259D4—44no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—124no—
280C2×C4—242no—
281C2×C4—242no—
282C2×C4#83411yescharacteristic
283C2×C4#83411yescharacteristic
284C2×C4#83712yes—
285C8—242no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—42no—
289C2×C3—81no—
290C2×C3#150811yescharacteristic
291C2×C3#212311yescharacteristic
292C2×C3C2×#35311yescharacteristic
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22—81no—
305C22C2×#83411yescharacteristic
306C22#522511yescharacteristic
307C22C2×#83711yescharacteristic
308C4—22no—
309C4—21no—
310C4—44no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—82no—
315C4—244no—
316C4#454512yes—
317C3#1178811yescharacteristic, complemented, Sylow
318C2—21no—
319C2—21no—
320C2—42no—
321C2—81no—
322C2#7168811yescharacteristic
323C2C2×#454511yescharacteristic
324C2#7187011yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#2542C2
socleC22×C3C2×#99
prime radicalC22×C3×D4D4
cosocleC2×C3×C22⋊C4C23
hypercenter#2542D3
hypocenterC3#11788
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
4. #67651C2semidirect
6. C2×C3×C4⋊Q8C4semidirect
10. C3×#399C4semidirect
14. C3×C4×C4⋊C4C4semidirect
55. C2×C4⋊Q8Dic3semidirect
61. #399Dic3semidirect
65. C4×C4⋊C4Dic3semidirect
317. C3#11788semidirect

Maximal subgroups

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

subgroupindex
C3×#25422
C4×#8342
#716022
#676512
#117883