Finite Group Databasebeta

← #10692660

Subgroups — #10692660

Order 768.

420 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 91 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#83294C211yesmaximal, characteristic
2#83279C211yesmaximal, characteristic
3C3×#1514C211yesmaximal, characteristic
4#70815C211yesmaximal, characteristic, complemented
5#69566C211yesmaximal, characteristic, complemented
6#67753C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#9415—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#383C2211yescharacteristic
13#5234—22no—
14#5093C411yescharacteristic, complemented
15#5078C411yescharacteristic, complemented
16#5076C411yescharacteristic, complemented
17#5070C411yescharacteristic, complemented
18#5042—22no—
19#5018—22no—
20#5011—22no—
21#5011—22no—
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24C4×#208C2211yescharacteristic
25C3×#327—22no—
26#4465—22no—
27#4452—22no—
28#3182—31no—
29#2980—31no—
30#1537—31no—
31#1515—31no—
32#1514D311yescharacteristic
33#1291—31no—
34#1288—31no—
35C2×C3×#41—21no—
36C3×C4⋊Q8C2×C411yescharacteristic
37C3×#127C2×C411yescharacteristic
38C3×#124—21no—
39C3×#121—21no—
40C3×#121D412yes—
41C3×C4×D4—21no—
42C3×C4×D4D412yes—
43C2×C3×C42C2311yescharacteristic
44C2×#214—22no—
45#926—21no—
46#924—21no—
47#924—21no—
48C2×#208—21no—
49C2×#208—22no—
50C22×C3⋊C8—22no—
51#870—21no—
52#869C2×C411yescharacteristic
53#869C2×C411yescharacteristic
54C4×Dic6—21no—
55C4×Dic6—21no—
56C3×C4⋊C8—21no—
57C3×C22⋊C8—21no—
58C3×#97—21no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95—21no—
62#832—22no—
63#830—24no—
64#830—21no—
65#830D412yes—
66#810—22no—
67#807—22no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70#803—21no—
71#803—21no—
72#803D412yes—
73C4×C3⋊C8C2×C411yescharacteristic
74C4×C3⋊C8C2×C411yescharacteristic
75#533—31no—
76#532C2×D311yescharacteristic
77#500—31no—
78#492—31no—
79#487—31no—
80#486—31no—
81#468—62no—
82#440—62no—
83#435—62no—
84#435—62no—
85#432—31no—
86#432—31no—
87#432C2×D311yescharacteristic
88#408—62no—
89C4×M16—31no—
90#383C2×D311yescharacteristic
91#330—62no—
92#328—62no—
93#327—22no—
94C3×#41—81no—
95C3×#41—81no—
96C2×C3×Q8—21no—
97C2×C3×D4—22no—
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4C22⋊C412yes—
101C22×C3×C4—22no—
102C22×C3×C4—41no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C2×Dic6—21no—
107C2×Dic6—21no—
108C2×C3×C8—41no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4C22⋊C412yes—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—22no—
113C3×C42C22×C411yescharacteristic
114C3×C42C2×D411yescharacteristic
115C3×C42C2×D411yescharacteristic
116C3×C42#4111yescharacteristic
117#217—41no—
118#214—44no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123Dic3⋊C4—41no—
124Dic3⋊C4—41no—
125C4×Dic3—41no—
126#208—42no—
127#208—82no—
128C2×C3⋊C8—21no—
129C2×C3⋊C8—21no—
130C2×C3⋊C8—22no—
131C2×C3⋊C8—22no—
132C2×C3⋊C8—82no—
133C2×#41—21no—
134C2×SD16—62no—
135C2×M16—61no—
136C2×M16—62no—
137C22×C8—62no—
138C4⋊Q8—31no—
139C4⋊Q8—31no—
140C4⋊Q8C4×D311yescharacteristic
141#127C4×D311yescharacteristic
142#125—61no—
143#124—21no—
144C22⋊Q8—61no—
145C22⋊Q8—61no—
146#121—21no—
147#121#6712yes—
148C4×Q8—61no—
149C4×Q8—61no—
150C4×D4—21no—
151C4×D4D1212yes—
152#117—61no—
153C2×C42C22×D311yescharacteristic
154#107—62no—
155C4⋊C8—21no—
156C4⋊C8—61no—
157C4⋊C8—61no—
158#103—62no—
159#102—62no—
160C22⋊C8—21no—
161C22⋊C8—32no—
162C22⋊C8—64no—
163C22⋊C8—61no—
164#97—21no—
165#97—32no—
166#97—61no—
167#97—61no—
168C4×C8—31no—
169C4×C8—31no—
170#95—21no—
171#95—21no—
172#95—21no—
173C23×C3—22no—
174C23×C3#12111yescharacteristic
175C3×Q8—81no—
176C3×D4—42no—
177C3×D4—81no—
178C3×D4—81no—
179C3×D4—82no—
180C2×C3×C4—22no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—81no—
187C2×C3×C4—81no—
188C2×C3×C4—81no—
189C2×C3×C4C2×C22⋊C411yescharacteristic
190C2×C3×C4C4×D411yescharacteristic
191C2×C3×C4C4×D411yescharacteristic
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12311yescharacteristic
195C2×Dic3—41no—
196C2×Dic3—41no—
197Dic6—42no—
198Dic6—81no—
199C3×C8—81no—
200C3⋊C8—22no—
201C3⋊C8—42no—
202C3⋊C8—42no—
203C3⋊C8—41no—
204#41—81no—
205#41—81no—
206C2×Q8—21no—
207C2×Q8—61no—
208C2×Q8—61no—
209C2×D4—22no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4(C2×D3)⋊C412yes—
213C22×C4—22no—
214C22×C4—41no—
215C22×C4D3×D411yescharacteristic
216C22×C4#23711yescharacteristic
217C22×C4C2×#6711yescharacteristic
218SD16—124no—
219M16—122no—
220M16—242no—
221C2×C8—41no—
222C2×C8—61no—
223C2×C8—62no—
224C2×C8—62no—
225C2×C8—61no—
226C2×C8—242no—
227C4⋊C4—21no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—121no—
233C4⋊C4—121no—
234C4⋊C4(C2×D3)⋊C412yes—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—121no—
238C42—121no—
239C42C2×C4×D311yescharacteristic
240C42C2×D1211yescharacteristic
241C42D3×D411yescharacteristic
242C42#23711yescharacteristic
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—161no—
258Dic3—81no—
259Dic3—81no—
260C23—22no—
261C23#93911yescharacteristic
262Q8—81no—
263Q8—122no—
264Q8—241no—
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—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4#67⋊C411yescharacteristic
281C2×C4#67⋊C411yescharacteristic
282C2×C4#88211yescharacteristic
283C2×C4#88611yescharacteristic
284C2×C4C2×(C2×D3)⋊C411yescharacteristic
285C2×C4#93911yescharacteristic
286C8—62no—
287C8—81no—
288C8—122no—
289C8—122no—
290C8—121no—
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#495811yescharacteristic
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—42no—
310C4—161no—
311C4—241no—
312C4—241no—
313C3#941511yescharacteristic, complemented, Sylow
314C2—41no—
315C2—82no—
316C2#7004511yescharacteristic
317C2#7031411yescharacteristic
318C2#7039911yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1514C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1514D3
hypocenterC3#9415
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1514

Complements

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

NH ≅ G/Ntype
4. #70815C2semidirect
5. #69566C2semidirect
6. #67753C2semidirect
7. #67633C2semidirect
14. #5093C4semidirect
15. #5078C4semidirect
16. #5076C4semidirect
17. #5070C4semidirect
313. C3#9415semidirect

Maximal subgroups

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

subgroupindex
#832942
#832792
C3×#15142
#708152
#695662
#677532
#676332
#94153