Finite Group Databasebeta

← #10692664

Subgroups — #10692664

Order 768.

420 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 99 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×#3182C211yesmaximal, characteristic
2#83279C211yesmaximal, characteristic
3#71837C211yesmaximal, characteristic, complemented
4#71823C211yesmaximal, characteristic, complemented
5#71813C211yesmaximal, characteristic
6#67670C211yesmaximal, characteristic, complemented
7#67666C211yesmaximal, characteristic, complemented
8#9415—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#532C2211yescharacteristic
11C3×#500C411yescharacteristic, complemented
12C3×#492C411yescharacteristic, complemented
13C3×#487C411yescharacteristic, complemented
14C3×#486C411yescharacteristic, complemented
15C3×#468—22no—
16C3×#440—22no—
17C3×C4×M16C2211yescharacteristic
18#5234—22no—
19#5011—22no—
20#5011—22no—
21#4937C2211yescharacteristic
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24#4926C2211yescharacteristic
25#4543—22no—
26#4538—22no—
27#4537—22no—
28#3182D311yescharacteristic
29#2980—31no—
30#1537—31no—
31#1515—31no—
32#1514—31no—
33#1291—31no—
34#1288—31no—
35C2×C3×#41—21no—
36C2×C3×SD16—22no—
37C2×C3×M16—21no—
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#127C2×C411yescharacteristic
42C3×#125—21no—
43C3×#124—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×#121—21no—
47C3×#121D412yes—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×D4—21no—
51C3×C4×D4D412yes—
52C3×#117—21no—
53C2×C3×C42C2311yescharacteristic
54C2×#208—22no—
55C22×C3⋊C8—22no—
56C3×#107—22no—
57C3×#103—22no—
58C3×#102—22no—
59C3×#97—21no—
60C3×C4×C8C2×C411yescharacteristic
61C3×C4×C8C2×C411yescharacteristic
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65#830—24no—
66#830—21no—
67#830—21no—
68#830D412yes—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72#803—21no—
73#803—21no—
74#803D412yes—
75#533C2×D311yescharacteristic
76#532C2×D311yescharacteristic
77#500Dic311yescharacteristic, complemented
78#492Dic311yescharacteristic, complemented
79#487Dic311yescharacteristic, complemented
80#486Dic311yescharacteristic, complemented
81#468—22no—
82#440—22no—
83#435—62no—
84#435—62no—
85#432—31no—
86#432—31no—
87#432—31no—
88#408—62no—
89C4×M16C2×D311yescharacteristic
90#383—31no—
91#330—62no—
92#328—62no—
93#327—62no—
94C3×#41—81no—
95C3×#41—81no—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×Q8—21no—
99C2×C3×D4—22no—
100C2×C3×D4—21no—
101C2×C3×D4—21no—
102C2×C3×D4C22⋊C412yes—
103C22×C3×C4—22no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×Dic3—41no—
108C3×SD16—44no—
109C3×M16—42no—
110C2×C3×C8—21no—
111C2×C3×C8—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4C22⋊C412yes—
120C3×C22⋊C4—22no—
121C3×C22⋊C4—22no—
122C3×C22⋊C4—41no—
123C3×C42—41no—
124C3×C42C22×C411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42#4111yescharacteristic
128#208—82no—
129C2×C3⋊C8—22no—
130C2×C3⋊C8—22no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8—82no—
133C2×#41—21no—
134C2×SD16—22no—
135C2×M16—21no—
136C2×M16—62no—
137C22×C8—62no—
138C4⋊Q8C2×Dic311yescharacteristic
139C4⋊Q8C2×Dic311yescharacteristic
140C4⋊Q8C2×Dic311yescharacteristic
141#127C2×Dic311yescharacteristic
142#125—21no—
143#124—21no—
144C22⋊Q8—21no—
145C22⋊Q8—21no—
146#121—21no—
147#121#6712yes—
148C4×Q8—21no—
149C4×Q8—21no—
150C4×D4—21no—
151C4×D4#6712yes—
152#117—21no—
153C2×C42C22×D311yescharacteristic
154#107—22no—
155C4⋊C8—61no—
156C4⋊C8—61no—
157C4⋊C8—61no—
158#103—22no—
159#102—22no—
160C22⋊C8—32no—
161C22⋊C8—64no—
162C22⋊C8—61no—
163C22⋊C8—61no—
164#97—21no—
165#97—32no—
166#97—61no—
167#97—61no—
168C4×C8C2×Dic311yescharacteristic
169C4×C8C2×Dic311yescharacteristic
170#95—61no—
171#95—61no—
172#95—61no—
173C23×C3—22no—
174C23×C3#12111yescharacteristic
175C3×Q8—42no—
176C3×Q8—81no—
177C3×Q8—81no—
178C3×D4—42no—
179C3×D4—81no—
180C3×D4—81no—
181C3×D4—82no—
182C2×C3×C4—22no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4C2×C22⋊C411yescharacteristic
191C2×C3×C4C4×D411yescharacteristic
192C2×C3×C4C4×D411yescharacteristic
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12311yescharacteristic
196C2×Dic3—81no—
197C2×Dic3—81no—
198C2×Dic3—81no—
199C3×C8—22no—
200C3×C8—41no—
201C3⋊C8—42no—
202C3⋊C8—42no—
203C3⋊C8—81no—
204#41—81no—
205#41—81no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×D4—22no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4#21712yes—
213C22×C4—22no—
214C22×C4—121no—
215C22×C4D3×D411yescharacteristic
216C22×C4#23711yescharacteristic
217C22×C4C2×#6711yescharacteristic
218SD16—44no—
219M16—42no—
220M16—242no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—62no—
224C2×C8—62no—
225C2×C8—121no—
226C2×C8—242no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4#21712yes—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—41no—
238C42—41no—
239C42D3×D411yescharacteristic
240C42#23711yescharacteristic
241C42C22×Dic311yescharacteristic
242C42C2×#6711yescharacteristic
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—42no—
247C22×C3—42no—
248C22×C3—42no—
249C22×C3#38611yescharacteristic
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—42no—
257C3×C4—81no—
258C3×C4—81no—
259Dic3—161no—
260C23—22no—
261C23#93911yescharacteristic
262Q8—42no—
263Q8—81no—
264Q8—81no—
265D4—42no—
266D4—81no—
267D4—81no—
268D4—82no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—241no—
278C2×C4—241no—
279C2×C4—241no—
280C2×C4D4×Dic311yescharacteristic
281C2×C4D4×Dic311yescharacteristic
282C2×C4#93511yescharacteristic
283C2×C4#93711yescharacteristic
284C2×C4#93911yescharacteristic
285C2×C4C2×#21711yescharacteristic
286C8—22no—
287C8—41no—
288C8—122no—
289C8—122no—
290C8—241no—
291C2×C3—41no—
292C2×C3—82no—
293C2×C3#188411yescharacteristic
294C2×C3#190211yescharacteristic
295C2×C3#190411yescharacteristic
296C22—21no—
297C22—21no—
298C22—21no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22#522011yescharacteristic
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—42no—
310C4—81no—
311C4—81no—
312C4—481no—
313C3#941511yescharacteristic, complemented, Sylow
314C2—41no—
315C2—82no—
316C2#7181711yescharacteristic
317C2#7208911yescharacteristic
318C2#7210211yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3182C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3182D3
hypocenterC3#9415
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3182

Complements

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

NH ≅ G/Ntype
3. #71837C2semidirect
4. #71823C2semidirect
6. #67670C2semidirect
7. #67666C2semidirect
11. C3×#500C4semidirect
12. C3×#492C4semidirect
13. C3×#487C4semidirect
14. C3×#486C4semidirect
77. #500Dic3semidirect
78. #492Dic3semidirect
79. #487Dic3semidirect
80. #486Dic3semidirect
313. C3#9415semidirect

Maximal subgroups

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

subgroupindex
C3×#31822
#832792
#718372
#718232
#718132
#676702
#676662
#94153