Finite Group Databasebeta

← #10691778

Subgroups — #10691778

Order 768.

382 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 89 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
1#83371C211yesmaximal, characteristic
2#79316C211yesmaximal, characteristic
3C3×#1506C211yesmaximal, characteristic
4#71827C211yesmaximal, characteristic
5#69433C211yesmaximal, characteristic
6#67757C211yesmaximal, characteristic
7#67614C211yesmaximal, characteristic
8#9280—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#376C2211yescharacteristic
13#5093C411yescharacteristic, complemented
14#5093C411yescharacteristic, complemented
15#5080C411yescharacteristic, complemented
16#5078C411yescharacteristic, complemented
17#5063—22no—
18#4996—22no—
19#4926C2211yescharacteristic
20#4925C2211yescharacteristic
21#4925C2211yescharacteristic
22#4831—21no—
23#4830—21no—
24#4720—22no—
25#4701—22no—
26C3×#328—22no—
27#4535—21no—
28#4535—21no—
29#4452—22no—
30#3231—31no—
31#2985—31no—
32#1506—31no—
33#1506D311yescharacteristic
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⋊Q8—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×#117—21no—
45C2×C3×C42—21no—
46C2×C3×C42C2311yescharacteristic
47#926—21no—
48#924—21no—
49#924D412yes—
50#870—21no—
51#869C2×C411yescharacteristic
52#869C2×C411yescharacteristic
53C4×Dic6—21no—
54C4×Dic6D412yes—
55C3×C4⋊C8—21no—
56C3×C22⋊C8D412yes—
57C3×#97—21no—
58C3×#95—21no—
59C3×#95—21no—
60#830—21no—
61#830—21no—
62#830—21no—
63#815—22no—
64Dic3⋊C8—22no—
65Dic3⋊C8—22no—
66C8×Dic3—22no—
67#810—22no—
68#810—22no—
69#808—22no—
70#807—22no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74#803—21no—
75C4×C3⋊C8C2×C411yescharacteristic
76C4×C3⋊C8C2×C411yescharacteristic
77C4×C3⋊C8D411yescharacteristic
78C4×C3⋊C8D411yescharacteristic
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—61no—
88#446—62no—
89C8×Q8—61no—
90#433—62no—
91#432—31no—
92#432C2×D311yescharacteristic
93#431—31no—
94#431—31no—
95#376C2×D311yescharacteristic
96#328—22no—
97#328—62no—
98Q8⋊C8—61no—
99Q8⋊C8—61no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C22×C3×C4—41no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C2×Dic6—21no—
107C2×Dic6C22⋊C412yes—
108C2×C3×C8—22no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C22⋊C4—41no—
116C3×C42—22no—
117C3×C42—41no—
118C3×C42—41no—
119C3×C42C22×C411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42#4111yescharacteristic
123#217—22no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3C22⋊C412yes—
128Dic3⋊C4—22no—
129Dic3⋊C4—22no—
130C4×Dic3—22no—
131C2×C3⋊C8—21no—
132C2×C3⋊C8—21no—
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—21no—
144C22⋊Q8—32no—
145C22⋊Q8—61no—
146C4×Q8—21no—
147C4×Q8—21no—
148C4×Q8—32no—
149C4×Q8—61no—
150#117—21no—
151#117—61no—
152C2×C42—21no—
153C2×C42C22×D311yescharacteristic
154#107—62no—
155#106—62no—
156C4⋊C8—21no—
157C4⋊C8—62no—
158C4⋊C8—62no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161C4⋊C8—61no—
162#103—62no—
163#103—62no—
164C22⋊C8—61no—
165C22⋊C8—61no—
166C22⋊C8—61no—
167C22⋊C8#6712yes—
168#97—21no—
169#97—62no—
170#97—61no—
171C4×C8—31no—
172C4×C8—31no—
173C4×C8—31no—
174C4×C8—31no—
175C4×C8—62no—
176#95—21no—
177#95—21no—
178C23×C3#12111yescharacteristic
179C3×Q8—41no—
180C3×Q8—41no—
181C3×Q8—81no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—81no—
190C2×C3×C4—81no—
191C2×C3×C4C2×C22⋊C411yescharacteristic
192C2×C3×C4C4×D411yescharacteristic
193C2×C3×C4C4×D411yescharacteristic
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12311yescharacteristic
197C2×Dic3—22no—
198C2×Dic3—22no—
199Dic6—41no—
200Dic6—41no—
201Dic6—42no—
202C3×C8—42no—
203C3⋊C8—41no—
204C3⋊C8—41no—
205C3⋊C8—41no—
206C3⋊C8—41no—
207C3⋊C8—81no—
208C2×Q8—21no—
209C2×Q8—21no—
210C2×Q8—32no—
211C2×Q8—61no—
212C22×C4—41no—
213C22×C4D3×D411yescharacteristic
214C22×C4#23711yescharacteristic
215C22×C4C2×#6711yescharacteristic
216C2×C8—22no—
217C2×C8—61no—
218C2×C8—61no—
219C2×C8—61no—
220C2×C8—61no—
221C2×C8—121no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—32no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—61no—
230C4⋊C4—62no—
231C4⋊C4—62no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C22⋊C4—41no—
235C22⋊C4—62no—
236C42—22no—
237C42—41no—
238C42—41no—
239C42—62no—
240C42C2×C4×D311yescharacteristic
241C42#23511yescharacteristic
242C42D3×D411yescharacteristic
243C42D3×D411yescharacteristic
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—82no—
254C3×C4—81no—
255C3×C4—81no—
256C3×C4#44311yescharacteristic
257C3×C4#44411yescharacteristic
258Dic3—42no—
259Dic3—42no—
260C23#93811yescharacteristic
261Q8—41no—
262Q8—41no—
263Q8—81no—
264Q8—122no—
265Q8—121no—
266Q8—121no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—62no—
275C2×C4—62no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4D3×C22⋊C411yescharacteristic
279C2×C4#67⋊C411yescharacteristic
280C2×C4#88311yescharacteristic
281C2×C4#88411yescharacteristic
282C2×C4C4×#6711yescharacteristic
283C2×C4#93711yescharacteristic
284C8—42no—
285C8—121no—
286C8—121no—
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#495611yescharacteristic
298C4—21no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—82no—
304C4—81no—
305C4—81no—
306C4—122no—
307C4—122no—
308C4#482311yescharacteristic
309C4#482411yescharacteristic
310C3#928011yescharacteristic, complemented, Sylow
311C2—41no—
312C2#7005211yescharacteristic
313C2#7035011yescharacteristic
314C2#7039011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1506C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1506D3
hypocenterC3#9280
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1506

Complements

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

NH ≅ G/Ntype
13. #5093C4semidirect
14. #5093C4semidirect
15. #5080C4semidirect
16. #5078C4semidirect
310. C3#9280semidirect

Maximal subgroups

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

subgroupindex
#833712
#793162
C3×#15062
#718272
#694332
#677572
#676142
#92803