Finite Group Databasebeta

← #10692665

Subgroups — #10692665

Order 768.

420 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 85 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#83347C211yesmaximal, characteristic
2#79337C211yesmaximal, characteristic
3C3×#1537C211yesmaximal, characteristic, complemented
4#71837C211yesmaximal, characteristic, complemented
5#70835C211yesmaximal, characteristic
6#67753C211yesmaximal, characteristic, complemented
7#67752C211yesmaximal, characteristic, complemented
8#9415—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13#5088C411yescharacteristic, complemented
14#5085C411yescharacteristic, complemented
15#5079C411yescharacteristic, complemented
16#5078C411yescharacteristic, complemented
17#5049—22no—
18#5026—22no—
19#4937C2211yescharacteristic
20#4926C2211yescharacteristic
21C4×#208C2211yescharacteristic
22#4836—22no—
23#4727—22no—
24#4706—22no—
25C3×#330—22no—
26#4538—22no—
27#4470—22no—
28#3182—31no—
29#2980—31no—
30#1537D311yescharacteristic, complemented
31#1515—31no—
32#1514—31no—
33#1291—31no—
34#1288—31no—
35C2×#235—21no—
36C3×C4⋊Q8C2×C411yescharacteristic
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×#125—21no—
39C3×C22⋊Q8—21no—
40C3×C22⋊Q8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C3×#117—21no—
44C2×C3×C42C2311yescharacteristic
45C2×#215—22no—
46#930—21no—
47#930D412yes—
48C2×#208—21no—
49C2×ΓL(1,𝔽25)—22no—
50C2×D3×C8—22no—
51#875—21no—
52#874C2×C411yescharacteristic
53C4×D12—21no—
54C4×D12D412yes—
55#869C2×C411yescharacteristic
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C22⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×#97—21no—
61C3×#97D412yes—
62#835—22no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66#830D412yes—
67(C2×D3)⋊C8—24no—
68#809—22no—
69#807—22no—
70C4⋊(C3⋊C8)—21no—
71#803—21no—
72#803—21no—
73C4×C3⋊C8C2×C411yescharacteristic
74C4×C3⋊C8C2×C411yescharacteristic
75#533C2×D311yescharacteristic
76#532—31no—
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#432C2×D311yescharacteristic
87#432C2×D311yescharacteristic
88#408—62no—
89C4×M16—31no—
90#383—31no—
91#330—22no—
92#328—62no—
93#327—62no—
94C2×C3×Q8—21no—
95C2×C3×Q8—21no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4#4111yescharacteristic
99C2×#67—22no—
100C22×Dic3—41no—
101#235—81no—
102#235—81no—
103C2×D12—21no—
104C2×D12—21no—
105C2×D12C22⋊C412yes—
106C2×C4×D3—22no—
107C2×Dic6—21no—
108C2×C3×C8—22no—
109C2×C3×C8—41no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C22⋊C4—41no—
117C3×C42—41no—
118C3×C42C22×C411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42#4111yescharacteristic
122#215—44no—
123(C2×D3)⋊C4—22no—
124(C2×D3)⋊C4—22no—
125C4⋊Dic3—21no—
126C4⋊Dic3C22⋊C412yes—
127#208—42no—
128C2×C3⋊C8—21no—
129C2×C3⋊C8—21no—
130C2×C3⋊C8—22no—
131ΓL(1,𝔽25)—82no—
132D3×C8—82no—
133C2×#41—61no—
134C2×SD16—62no—
135C2×M16—62no—
136C2×M16—61no—
137C22×C8—62no—
138C4⋊Q8—31no—
139C4⋊Q8C4×D311yescharacteristic
140C4⋊Q8C4×D311yescharacteristic
141#127—31no—
142#125—21no—
143#124—61no—
144C22⋊Q8—21no—
145C22⋊Q8—21no—
146#121—32no—
147#121—61no—
148C4×Q8—21no—
149C4×Q8—21no—
150C4×D4—32no—
151C4×D4—61no—
152#117—21no—
153C2×C42C22×D311yescharacteristic
154#107—62no—
155C4⋊C8—21no—
156C4⋊C8—21no—
157C4⋊C8—61no—
158#103—62no—
159#102—62no—
160C22⋊C8—21no—
161C22⋊C8—21no—
162C22⋊C8—32no—
163C22⋊C8—64no—
164#97—21no—
165#97—61no—
166#97—61no—
167#97#6712yes—
168C4×C8—31no—
169C4×C8—31no—
170#95—61no—
171#95—61no—
172#95—61no—
173C23×C3#12111yescharacteristic
174C22×D3—22no—
175C3×Q8—42no—
176C3×Q8—81no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4C2×C22⋊C411yescharacteristic
183C2×C3×C4C4×D411yescharacteristic
184C2×C3×C4C4×D411yescharacteristic
185C2×C3×C4#12011yescharacteristic
186C2×C3×C4#12111yescharacteristic
187C2×C3×C4#12311yescharacteristic
188#67—82no—
189C2×Dic3—22no—
190C2×Dic3—81no—
191C2×Dic3—81no—
192C2×Dic3—81no—
193D12—42no—
194D12—81no—
195D12—81no—
196C4×D3—42no—
197C4×D3—42no—
198Dic6—81no—
199C3×C8—42no—
200C3×C8—81no—
201C3⋊C8—22no—
202C3⋊C8—42no—
203C3⋊C8—41no—
204#41—241no—
205#41—241no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—61no—
209C2×D4—32no—
210C2×D4—62no—
211C2×D4—61no—
212C2×D4—61no—
213C22×C4—62no—
214C22×C4—121no—
215C22×C4#23511yescharacteristic
216C22×C4D3×D411yescharacteristic
217C22×C4D3×D411yescharacteristic
218SD16—124no—
219M16—122no—
220M16—242no—
221C2×C8—22no—
222C2×C8—41no—
223C2×C8—61no—
224C2×C8—61no—
225C2×C8—62no—
226C2×C8—242no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—32no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—61no—
235C22⋊C4—41no—
236C22⋊C4—62no—
237C22⋊C4—62no—
238C42—41no—
239C42C2×C4×D311yescharacteristic
240C42D3×D411yescharacteristic
241C42#23711yescharacteristic
242C42C2×#6711yescharacteristic
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3#38611yescharacteristic
247C2×D3—42no—
248C2×D3—42no—
249C2×D3—42no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—81no—
257C3×C4—81no—
258Dic3—42no—
259Dic3—161no—
260C23—62no—
261C23#88411yescharacteristic
262Q8—42no—
263Q8—81no—
264Q8—241no—
265D4—122no—
266D4—241no—
267D4—241no—
268D4—242no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—62no—
275C2×C4—122no—
276C2×C4—122no—
277C2×C4—241no—
278C2×C4—241no—
279C2×C4—241no—
280C2×C4D3×C22⋊C411yescharacteristic
281C2×C4#67⋊C411yescharacteristic
282C2×C4#88311yescharacteristic
283C2×C4C4×#6711yescharacteristic
284C2×C4#93711yescharacteristic
285C2×C4#93811yescharacteristic
286C8—42no—
287C8—62no—
288C8—81no—
289C8—122no—
290C8—121no—
291C2×C3—41no—
292C2×C3#188411yescharacteristic
293C2×C3#190211yescharacteristic
294C2×C3#190411yescharacteristic
295D3—82no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—122no—
300C22—122no—
301C22—122no—
302C22#495611yescharacteristic
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—81no—
310C4—81no—
311C4—122no—
312C4—481no—
313C3#941511yescharacteristic, complemented, Sylow
314C2—41no—
315C2—242no—
316C2#7012211yescharacteristic
317C2#7040711yescharacteristic
318C2#7041711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1537C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1537D3
hypocenterC3#9415
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1537

Complements

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

NH ≅ G/Ntype
3. C3×#1537C2semidirect
4. #71837C2semidirect
6. #67753C2semidirect
7. #67752C2semidirect
13. #5088C4semidirect
14. #5085C4semidirect
15. #5079C4semidirect
16. #5078C4semidirect
30. #1537D3semidirect
313. C3#9415semidirect

Maximal subgroups

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

subgroupindex
#833472
#793372
C3×#15372
#718372
#708352
#677532
#677522
#94153