Finite Group Databasebeta

← #10691782

Subgroups — #10691782

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#79316C211yesmaximal, characteristic
2#79273C211yesmaximal, characteristic
3#70836C211yesmaximal, characteristic
4#70836C211yesmaximal, characteristic
5#70803C211yesmaximal, characteristic
6C3×#1287C211yesmaximal, characteristic
7#67737C211yesmaximal, characteristic
8#9280—31nomaximal, Sylow
9#5470C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#4932C2211yescharacteristic
15#4925C2211yescharacteristic
16#4831—21no—
17#4830—21no—
18#4829—22no—
19#4723—22no—
20#4720—22no—
21#4701—22no—
22#4682C411yescharacteristic, complemented
23#4681C411yescharacteristic, complemented
24#4681C411yescharacteristic, complemented
25#4680C411yescharacteristic, complemented
26#4484—21no—
27#4484—21no—
28#4467—22no—
29#4467—22no—
30#3231—31no—
31#2985—31no—
32#1506—31no—
33#1506—31no—
34#1497—31no—
35#1287D311yescharacteristic
36#1274—31no—
37C2×C3×C42C2311yescharacteristic
38#926—21no—
39#926—21no—
40#924—21no—
41#924—21no—
42#924—21no—
43#924D412yes—
44C2×C4×Dic3—21no—
45#870—21no—
46#870—21no—
47#869C2×C411yescharacteristic
48#869C2×C411yescharacteristic
49#869C2×C411yescharacteristic
50#869C2×C411yescharacteristic
51C4×Dic6—21no—
52C4×Dic6—21no—
53C4×Dic6—21no—
54C4×Dic6D412yes—
55C3×C4⋊C8—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C22⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×C22⋊C8D412yes—
61C3×#97—21no—
62C3×#97—21no—
63C3×C4×C8C2×C411yescharacteristic
64C3×C4×C8C2×C411yescharacteristic
65C3⋊#95—21no—
66C3⋊#95—21no—
67#830—21no—
68#818—22no—
69#817—22no—
70#816—22no—
71#816—22no—
72#815—22no—
73Dic3⋊C8—22no—
74Dic3⋊C8—22no—
75C8×Dic3—22no—
76C4⋊(C3⋊C8)—21no—
77C4×C3⋊C8D411yescharacteristic
78C4×C3⋊C8D411yescharacteristic
79#533—31no—
80#533—31no—
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#432C2×D311yescharacteristic
92#432C2×D311yescharacteristic
93#431—31no—
94#431C2×D311yescharacteristic
95#376—31no—
96#328—62no—
97#328—62no—
98Q8⋊C8—61no—
99Q8⋊C8—61no—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C22×Dic3—41no—
104C2×Dic6—21no—
105C2×Dic6—21no—
106C2×Dic6—21no—
107C2×Dic6C22⋊C412yes—
108C2×C3×C8—21no—
109C2×C3×C8—21no—
110C2×C3×C8—22no—
111C2×C3×C8—41no—
112C3×C42C22×C411yescharacteristic
113C3×C42C2×D411yescharacteristic
114C3×C42C2×D411yescharacteristic
115C3×C42#4111yescharacteristic
116#217—22no—
117#217—41no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3C22⋊C412yes—
126Dic3⋊C4—22no—
127Dic3⋊C4—22no—
128Dic3⋊C4—41no—
129Dic3⋊C4—41no—
130C4×Dic3—22no—
131C4×Dic3—22no—
132C4×Dic3—41no—
133C4×Dic3—41no—
134C2×C3⋊C8—21no—
135C2×C3⋊C8—21no—
136C4⋊Q8—31no—
137C4⋊Q8—31no—
138C4⋊Q8—31no—
139C4⋊Q8—31no—
140#125—61no—
141#125—61no—
142C22⋊Q8—32no—
143C22⋊Q8—61no—
144C22⋊Q8—61no—
145C22⋊Q8—61no—
146C4×Q8—32no—
147C4×Q8—61no—
148C4×Q8—61no—
149C4×Q8—61no—
150#117—61no—
151#117—61no—
152C2×C42—61no—
153C2×C42C22×D311yescharacteristic
154#107—62no—
155#106—62no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—62no—
160C4⋊C8—62no—
161C4⋊C8—61no—
162#103—62no—
163#103—62no—
164C22⋊C8—21no—
165C22⋊C8—21no—
166C22⋊C8—61no—
167C22⋊C8D1212yes—
168#97—21no—
169#97—21no—
170#97—62no—
171C4×C8—31no—
172C4×C8—31no—
173C4×C8—62no—
174C4×C8C4×D311yescharacteristic
175C4×C8C4×D311yescharacteristic
176#95—61no—
177#95—61no—
178C23×C3#12111yescharacteristic
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4C2×C22⋊C411yescharacteristic
184C2×C3×C4C4×D411yescharacteristic
185C2×C3×C4C4×D411yescharacteristic
186C2×C3×C4#12011yescharacteristic
187C2×C3×C4#12111yescharacteristic
188C2×C3×C4#12311yescharacteristic
189C2×Dic3—22no—
190C2×Dic3—22no—
191C2×Dic3—42no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—81no—
195C2×Dic3—81no—
196Dic6—41no—
197Dic6—41no—
198Dic6—42no—
199Dic6—41no—
200Dic6—41no—
201Dic6—81no—
202C3×C8—42no—
203C3×C8—41no—
204C3×C8—41no—
205C3×C8—81no—
206C3⋊C8—41no—
207C3⋊C8—41no—
208C2×Q8—32no—
209C2×Q8—61no—
210C2×Q8—61no—
211C2×Q8—61no—
212C22×C4—121no—
213C22×C4C2×D1211yescharacteristic
214C22×C4D3×D411yescharacteristic
215C22×C4#23911yescharacteristic
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—22no—
219C2×C8—41no—
220C2×C8—61no—
221C2×C8—61no—
222C4⋊C4—32no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—62no—
226C4⋊C4—61no—
227C4⋊C4—62no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—121no—
233C4⋊C4—121no—
234C22⋊C4—62no—
235C22⋊C4—121no—
236C42—62no—
237C42—62no—
238C42—121no—
239C42—121no—
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#44311yescharacteristic
254C3×C4#44411yescharacteristic
255Dic3—42no—
256Dic3—42no—
257Dic3—82no—
258Dic3—81no—
259Dic3—81no—
260C23#89511yescharacteristic
261Q8—122no—
262Q8—121no—
263Q8—121no—
264Q8—121no—
265Q8—121no—
266Q8—241no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—62no—
272C2×C4—62no—
273C2×C4—122no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4—241no—
277C2×C4—241no—
278C2×C4C4×D1211yescharacteristic
279C2×C4D3×C22⋊C411yescharacteristic
280C2×C4#88211yescharacteristic
281C2×C4#88411yescharacteristic
282C2×C4#89311yescharacteristic
283C2×C4#89411yescharacteristic
284C8—41no—
285C8—41no—
286C8—42no—
287C8—81no—
288C8—121no—
289C8—121no—
290C2×C3—41no—
291C2×C3#187611yescharacteristic
292C2×C3#190511yescharacteristic
293C2×C3#190911yescharacteristic
294C22—21no—
295C22—21no—
296C22—21no—
297C22#466511yescharacteristic
298C4—21no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—122no—
304C4—122no—
305C4—242no—
306C4—241no—
307C4—241no—
308C4#489411yescharacteristic
309C4#489511yescharacteristic
310C3#928011yescharacteristic, complemented, Sylow
311C2—41no—
312C2#6871211yescharacteristic
313C2#6894611yescharacteristic
314C2#6903011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1287C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1287D3
hypocenterC3#9280
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1287

Complements

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

NH ≅ G/Ntype
22. #4682C4semidirect
23. #4681C4semidirect
24. #4681C4semidirect
25. #4680C4semidirect
310. C3#9280semidirect

Maximal subgroups

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

subgroupindex
#793162
#792732
#708362
#708032
C3×#12872
#677372
#92803