Finite Group Databasebeta

← #10691783

Subgroups — #10691783

Order 768.

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

normal subgroups only

This group has 382 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×#3231C211yesmaximal, characteristic
2#83331C211yesmaximal, characteristic
3#71836C211yesmaximal, characteristic
4#71836C211yesmaximal, characteristic
5#71827C211yesmaximal, characteristic
6#67747C211yesmaximal, characteristic
7#67737C211yesmaximal, characteristic
8#9280—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#533C2211yescharacteristic
11C3×#500C411yescharacteristic, complemented
12C3×#498C411yescharacteristic, complemented
13C3×#487C411yescharacteristic, complemented
14C3×#487C411yescharacteristic, complemented
15C3×#485—22no—
16C3×#479—22no—
17C3×#431C2211yescharacteristic
18#5024—22no—
19#5024—21no—
20Q8×C3⋊C8—21no—
21#5000—22no—
22#4932C2211yescharacteristic
23#4926C2211yescharacteristic
24#4926C2211yescharacteristic
25#4925C2211yescharacteristic
26#4538—22no—
27#4538—22no—
28#4535—21no—
29#4535—21no—
30#3231D311yescharacteristic
31#2985—31no—
32#1506—31no—
33#1506—31no—
34#1497—31no—
35#1287—31no—
36#1274—31no—
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#125—21no—
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8D412yes—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8D412yes—
51C3×#117—21no—
52C3×#117—21no—
53C2×C3×C42C2311yescharacteristic
54C2×C4×Dic3—21no—
55C3×#107—22no—
56C3×#106—22no—
57C3×C4⋊C8—21no—
58C3×#103—22no—
59C3×#103—22no—
60C3×C22⋊C8—21no—
61C3×C4×C8C2×C411yescharacteristic
62C3×C4×C8C2×C411yescharacteristic
63C3⋊#95—21no—
64C3⋊#95—21no—
65#830—21no—
66#830—21no—
67#830D412yes—
68C4⋊(C3⋊C8)—22no—
69C4⋊(C3⋊C8)—22no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73#803—22no—
74#803—21no—
75#803—21no—
76C4×C3⋊C8—22no—
77C4×C3⋊C8D411yescharacteristic
78C4×C3⋊C8D411yescharacteristic
79#533C2×D311yescharacteristic
80#533C2×D311yescharacteristic
81#500Dic311yescharacteristic, complemented
82#498Dic311yescharacteristic, complemented
83#487Dic311yescharacteristic, complemented
84#487Dic311yescharacteristic, complemented
85#485—22no—
86#479—22no—
87#446—61no—
88#446—62no—
89C8×Q8—61no—
90#433—62no—
91#432—31no—
92#432—31no—
93#431—31no—
94#431C2×D311yescharacteristic
95#376—31no—
96#328—62no—
97#328—62no—
98Q8⋊C8—61no—
99Q8⋊C8—61no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8C22⋊C412yes—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×Dic3—41no—
108C2×C3×C8—21no—
109C2×C3×C8—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4C22⋊C412yes—
122C3×C22⋊C4—22no—
123C3×C22⋊C4—41no—
124C3×C42—22no—
125C3×C42—41no—
126C3×C42C22×C411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130C4×Dic3—22no—
131C4×Dic3—41no—
132C2×C3⋊C8—21no—
133C2×C3⋊C8—22no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—41no—
136C4⋊Q8C2×Dic311yescharacteristic
137C4⋊Q8C2×Dic311yescharacteristic
138C4⋊Q8C2×Dic311yescharacteristic
139C4⋊Q8C2×Dic311yescharacteristic
140#125—21no—
141#125—21no—
142C22⋊Q8—21no—
143C22⋊Q8—21no—
144C22⋊Q8—21no—
145C22⋊Q8#6712yes—
146C4×Q8—21no—
147C4×Q8—21no—
148C4×Q8—21no—
149C4×Q8#6712yes—
150#117—21no—
151#117—21no—
152C2×C42—61no—
153C2×C42C22×D311yescharacteristic
154#107—22no—
155#106—22no—
156C4⋊C8—21no—
157C4⋊C8—62no—
158C4⋊C8—62no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161C4⋊C8—61no—
162#103—22no—
163#103—22no—
164C22⋊C8—21no—
165C22⋊C8—32no—
166C22⋊C8—61no—
167C22⋊C8—61no—
168#97—62no—
169#97—61no—
170#97—61no—
171C4×C8—31no—
172C4×C8—31no—
173C4×C8—62no—
174C4×C8C2×Dic311yescharacteristic
175C4×C8C2×Dic311yescharacteristic
176#95—61no—
177#95—61no—
178C23×C3#12111yescharacteristic
179C3×Q8—42no—
180C3×Q8—41no—
181C3×Q8—41no—
182C3×Q8—41no—
183C3×Q8—41no—
184C3×Q8—81no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4C2×C22⋊C411yescharacteristic
194C2×C3×C4C4×D411yescharacteristic
195C2×C3×C4C4×D411yescharacteristic
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12311yescharacteristic
199C2×Dic3—42no—
200C2×Dic3—81no—
201C2×Dic3—81no—
202C3×C8—41no—
203C3×C8—41no—
204C3⋊C8—41no—
205C3⋊C8—42no—
206C3⋊C8—41no—
207C3⋊C8—81no—
208C2×Q8—21no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8#21712yes—
212C22×C4—121no—
213C22×C4D3×D411yescharacteristic
214C22×C4#23711yescharacteristic
215C22×C4C2×#6711yescharacteristic
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—61no—
219C2×C8—61no—
220C2×C8—62no—
221C2×C8—121no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4#21712yes—
234C22⋊C4—22no—
235C22⋊C4—41no—
236C42—22no—
237C42—41no—
238C42—62no—
239C42—121no—
240C42D3×D411yescharacteristic
241C42#23711yescharacteristic
242C42C22×Dic311yescharacteristic
243C42C2×#6711yescharacteristic
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3#38611yescharacteristic
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—42no—
254C3×C4—42no—
255C3×C4—81no—
256C3×C4—81no—
257C3×C4#44311yescharacteristic
258C3×C4#44411yescharacteristic
259Dic3—82no—
260C23#93911yescharacteristic
261Q8—42no—
262Q8—41no—
263Q8—41no—
264Q8—41no—
265Q8—41no—
266Q8—81no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—122no—
276C2×C4—241no—
277C2×C4—241no—
278C2×C4D4×Dic311yescharacteristic
279C2×C4D4×Dic311yescharacteristic
280C2×C4#93511yescharacteristic
281C2×C4#93711yescharacteristic
282C2×C4#93911yescharacteristic
283C2×C4C2×#21711yescharacteristic
284C8—41no—
285C8—41no—
286C8—121no—
287C8—121no—
288C8—122no—
289C8—241no—
290C2×C3—41no—
291C2×C3#187611yescharacteristic
292C2×C3#190511yescharacteristic
293C2×C3#190911yescharacteristic
294C22—21no—
295C22—21no—
296C22—21no—
297C22#522011yescharacteristic
298C4—21no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—42no—
304C4—42no—
305C4—81no—
306C4—81no—
307C4—242no—
308C4#519611yescharacteristic
309C4#519711yescharacteristic
310C3#928011yescharacteristic, complemented, Sylow
311C2—41no—
312C2#7186511yescharacteristic
313C2#7208711yescharacteristic
314C2#7210111yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3231C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#3231D3
hypocenterC3#9280
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3231

Complements

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

NH ≅ G/Ntype
11. C3×#500C4semidirect
12. C3×#498C4semidirect
13. C3×#487C4semidirect
14. C3×#487C4semidirect
81. #500Dic3semidirect
82. #498Dic3semidirect
83. #487Dic3semidirect
84. #487Dic3semidirect
310. C3#9280semidirect

Maximal subgroups

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

subgroupindex
C3×#32312
#833312
#718362
#718272
#677472
#677372
#92803