Finite Group Databasebeta

← #10691781

Subgroups — #10691781

Order 768.

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

normal subgroups only

This group has 382 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×#2985C211yesmaximal, characteristic
2#83325C211yesmaximal, characteristic
3#71836C211yesmaximal, characteristic
4#70836C211yesmaximal, characteristic
5#70771C211yesmaximal, characteristic
6#67757C211yesmaximal, characteristic
7#67737C211yesmaximal, characteristic
8#9280—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#446—22no—
12C3×#446—21no—
13C3×C8×Q8—21no—
14C3×#433—22no—
15C3×#432C2211yescharacteristic
16C3×#431C2211yescharacteristic
17#5080C411yescharacteristic, complemented
18#5079C411yescharacteristic, complemented
19#5079C411yescharacteristic, complemented
20#5078C411yescharacteristic, complemented
21#5045—22no—
22#5022—22no—
23#4932C2211yescharacteristic
24#4926C2211yescharacteristic
25#4925C2211yescharacteristic
26#4538—22no—
27#4467—22no—
28#4456—21no—
29#4456—21no—
30#3231—31no—
31#2985D311yescharacteristic
32#1506—31no—
33#1506—31no—
34#1497—31no—
35#1287—31no—
36#1274—31no—
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×#125—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8D412yes—
42C3×C4×Q8—21no—
43C3×C4×Q8D412yes—
44C3×#117—21no—
45C2×C3×C42C2311yescharacteristic
46#926—21no—
47#924—21no—
48#924—21no—
49C2×C4×Dic3—21no—
50#870—21no—
51#869C2×C411yescharacteristic
52#869C2×C411yescharacteristic
53C4×Dic6—21no—
54C4×Dic6—21no—
55C3×C4⋊C8—22no—
56C3×C4⋊C8—22no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8D412yes—
61C3×#97—22no—
62C3×#97—21no—
63C3×C4×C8—22no—
64C3×C4×C8D411yescharacteristic
65C3×C4×C8D411yescharacteristic
66#835—22no—
67#835—22no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70#830—21no—
71#830—21no—
72#808—22no—
73#807—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76#803—21no—
77C4×C3⋊C8C2×C411yescharacteristic
78C4×C3⋊C8C2×C411yescharacteristic
79#533—31no—
80#533C2×D311yescharacteristic
81#500—31no—
82#498—31no—
83#487—31no—
84#487—31no—
85#485—62no—
86#479—62no—
87#446—22no—
88#446—21no—
89C8×Q8—21no—
90#433—22no—
91#432—31no—
92#432C2×D311yescharacteristic
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×Q8C22⋊C412yes—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×Dic3—41no—
106C2×Dic6—21no—
107C2×Dic6—21no—
108C2×C3×C8—21no—
109C2×C3×C8—22no—
110C2×C3×C8—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4C22⋊C412yes—
117C3×C22⋊C4—22no—
118C3×C42—22no—
119C3×C42C22×C411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42#4111yescharacteristic
123#217—41no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128Dic3⋊C4—41no—
129Dic3⋊C4—41no—
130C4×Dic3—22no—
131C4×Dic3—41no—
132C4×Dic3—41no—
133C2×C3⋊C8—21no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—41no—
136C4⋊Q8—31no—
137C4⋊Q8—31no—
138C4⋊Q8C4×D311yescharacteristic
139C4⋊Q8C4×D311yescharacteristic
140#125—21no—
141#125—61no—
142C22⋊Q8—21no—
143C22⋊Q8—61no—
144C22⋊Q8—61no—
145C22⋊Q8D1212yes—
146C4×Q8—21no—
147C4×Q8—61no—
148C4×Q8—61no—
149C4×Q8#6712yes—
150#117—21no—
151#117—61no—
152C2×C42—61no—
153C2×C42C22×D311yescharacteristic
154#107—62no—
155#106—62no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8—61no—
161C4⋊C8—61no—
162#103—62no—
163#103—62no—
164C22⋊C8—21no—
165C22⋊C8—61no—
166C22⋊C8—61no—
167C22⋊C8#6712yes—
168#97—22no—
169#97—21no—
170#97—61no—
171C4×C8—22no—
172C4×C8—31no—
173C4×C8—31no—
174C4×C8#6711yescharacteristic
175C4×C8#6711yescharacteristic
176#95—61no—
177#95—61no—
178C23×C3#12111yescharacteristic
179C3×Q8—42no—
180C3×Q8—41no—
181C3×Q8—41no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4C2×C22⋊C411yescharacteristic
189C2×C3×C4C4×D411yescharacteristic
190C2×C3×C4C4×D411yescharacteristic
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12311yescharacteristic
194C2×Dic3—42no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—81no—
198C2×Dic3—81no—
199Dic6—41no—
200Dic6—41no—
201Dic6—81no—
202C3×C8—41no—
203C3×C8—42no—
204C3×C8—41no—
205C3⋊C8—41no—
206C3⋊C8—41no—
207C3⋊C8—81no—
208C2×Q8—21no—
209C2×Q8—61no—
210C2×Q8—61no—
211C2×Q8(C2×D3)⋊C412yes—
212C22×C4—121no—
213C22×C4C2×D1211yescharacteristic
214C22×C4#23511yescharacteristic
215C22×C4C2×#6711yescharacteristic
216C2×C8—21no—
217C2×C8—22no—
218C2×C8—21no—
219C2×C8—61no—
220C2×C8—61no—
221C2×C8—121no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—121no—
232C4⋊C4—121no—
233C4⋊C4(C2×D3)⋊C412yes—
234C22⋊C4—22no—
235C22⋊C4—121no—
236C42—22no—
237C42—62no—
238C42—121no—
239C42—121no—
240C42C2×C4×D311yescharacteristic
241C42#23511yescharacteristic
242C42C2×#6711yescharacteristic
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#44311yescharacteristic
256C3×C4#44411yescharacteristic
257Dic3—82no—
258Dic3—81no—
259Dic3—81no—
260C23#93011yescharacteristic
261Q8—42no—
262Q8—41no—
263Q8—41no—
264Q8—121no—
265Q8—121no—
266Q8—241no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—122no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4—241no—
277C2×C4—241no—
278C2×C4C2×(C2×D3)⋊C411yescharacteristic
279C2×C4C4×#6711yescharacteristic
280C2×C4C4×#6711yescharacteristic
281C2×C4#92911yescharacteristic
282C2×C4#93011yescharacteristic
283C2×C4#95311yescharacteristic
284C8—41no—
285C8—42no—
286C8—41no—
287C8—121no—
288C8—121no—
289C8—241no—
290C2×C3—41no—
291C2×C3#187611yescharacteristic
292C2×C3#190511yescharacteristic
293C2×C3#190911yescharacteristic
294C22—21no—
295C22—21no—
296C22—21no—
297C22#521311yescharacteristic
298C4—21no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—42no—
304C4—42no—
305C4—242no—
306C4—241no—
307C4—241no—
308C4#514411yescharacteristic
309C4#514511yescharacteristic
310C3#928011yescharacteristic, complemented, Sylow
311C2—41no—
312C2#7166211yescharacteristic
313C2#7201011yescharacteristic
314C2#7202411yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2985C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#2985D3
hypocenterC3#9280
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2985

Complements

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

NH ≅ G/Ntype
17. #5080C4semidirect
18. #5079C4semidirect
19. #5079C4semidirect
20. #5078C4semidirect
310. C3#9280semidirect

Maximal subgroups

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

subgroupindex
C3×#29852
#833252
#718362
#708362
#707712
#677572
#677372
#92803