Finite Group Databasebeta

← #10692663

Subgroups — #10692663

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#79337C211yesmaximal, characteristic
2#78533C211yesmaximal, characteristic
3#70837C211yesmaximal, characteristic, complemented
4#70835C211yesmaximal, characteristic
5#70815C211yesmaximal, characteristic, complemented
6C3×#1291C211yesmaximal, characteristic, complemented
7#67666C211yesmaximal, characteristic, complemented
8#9415—31nomaximal, Sylow
9#5477C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5134—22no—
15#4937C2211yescharacteristic
16#4926C2211yescharacteristic
17#4836—22no—
18#4727—22no—
19#4710—22no—
20#4706—22no—
21#4694C411yescharacteristic, complemented
22#4693C411yescharacteristic, complemented
23#4682C411yescharacteristic, complemented
24#4681C411yescharacteristic, complemented
25#4470—22no—
26#4467—22no—
27#4465—22no—
28#3182—31no—
29#2980—31no—
30#1537—31no—
31#1515—31no—
32#1514—31no—
33#1291D311yescharacteristic, complemented
34#1288—31no—
35C2×#235—21no—
36C2×C3×M16—21no—
37C2×C3×C42C2311yescharacteristic
38#930—21no—
39#930D412yes—
40#926—21no—
41#924—21no—
42#924—21no—
43C2×#204—22no—
44C2×ΓL(1,𝔽25)—22no—
45C2×D3×C8—22no—
46#875—21no—
47#874C2×C411yescharacteristic
48C4×D12—21no—
49C4×D12D412yes—
50#870—21no—
51#869C2×C411yescharacteristic
52#869C2×C411yescharacteristic
53#869C2×C411yescharacteristic
54C4×Dic6—21no—
55C4×Dic6—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C22⋊C8—21no—
59C3×C22⋊C8D412yes—
60C3×#97—21no—
61C3×#97—21no—
62C3×#97—21no—
63C3×C4×C8C2×C411yescharacteristic
64C3×C4×C8C2×C411yescharacteristic
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68#830—21no—
69#821—22no—
70(C2×D3)⋊C8—24no—
71#818—22no—
72#816—22no—
73C4⋊(C3⋊C8)—21no—
74#803D412yes—
75#533—31no—
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×M16C2×D311yescharacteristic
90#383—31no—
91#330—62no—
92#328—62no—
93#327—62no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4#4111yescharacteristic
97C2×#67—22no—
98C22×Dic3—41no—
99#235—81no—
100#235—81no—
101C2×D12—21no—
102C2×D12—21no—
103C2×D12C22⋊C412yes—
104C2×C4×D3—22no—
105C2×Dic6—21no—
106C2×Dic6—21no—
107C2×Dic6—21no—
108C3×M16—42no—
109C2×C3×C8—21no—
110C2×C3×C8—21no—
111C2×C3×C8—22no—
112C2×C3×C8—41no—
113C3×C42C22×C411yescharacteristic
114C3×C42C2×D411yescharacteristic
115C3×C42C2×D411yescharacteristic
116C3×C42#4111yescharacteristic
117#217—41no—
118(C2×D3)⋊C4—22no—
119(C2×D3)⋊C4—22no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3C22⋊C412yes—
126Dic3⋊C4—41no—
127Dic3⋊C4—41no—
128C4×Dic3—41no—
129C2×C3⋊C8—22no—
130#204—44no—
131ΓL(1,𝔽25)—82no—
132D3×C8—82no—
133C2×#41—61no—
134C2×SD16—62no—
135C2×M16—21no—
136C2×M16—62no—
137C22×C8—62no—
138C4⋊Q8—31no—
139C4⋊Q8—31no—
140C4⋊Q8—31no—
141#127—31no—
142#125—61no—
143#124—61no—
144C22⋊Q8—61no—
145C22⋊Q8—61no—
146#121—32no—
147#121—61no—
148C4×Q8—61no—
149C4×Q8—61no—
150C4×D4—32no—
151C4×D4—61no—
152#117—61no—
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—64no—
162C22⋊C8—61no—
163C22⋊C8D1212yes—
164#97—21no—
165#97—21no—
166#97—21no—
167#97—32no—
168C4×C8C4×D311yescharacteristic
169C4×C8C4×D311yescharacteristic
170#95—61no—
171#95—61no—
172#95—61no—
173C23×C3#12111yescharacteristic
174C22×D3—22no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4C2×C22⋊C411yescharacteristic
179C2×C3×C4C4×D411yescharacteristic
180C2×C3×C4C4×D411yescharacteristic
181C2×C3×C4#12011yescharacteristic
182C2×C3×C4#12111yescharacteristic
183C2×C3×C4#12311yescharacteristic
184#67—82no—
185C2×Dic3—22no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C2×Dic3—81no—
189C2×Dic3—81no—
190C2×Dic3—81no—
191D12—42no—
192D12—81no—
193D12—81no—
194C4×D3—42no—
195C4×D3—42no—
196Dic6—42no—
197Dic6—81no—
198Dic6—81no—
199C3×C8—22no—
200C3×C8—42no—
201C3×C8—41no—
202C3×C8—81no—
203C3⋊C8—42no—
204#41—241no—
205#41—241no—
206C2×Q8—61no—
207C2×Q8—61no—
208C2×Q8—61no—
209C2×D4—32no—
210C2×D4—62no—
211C2×D4—61no—
212C2×D4—61no—
213C22×C4—62no—
214C22×C4—121no—
215C22×C4C2×D1211yescharacteristic
216C22×C4D3×D411yescharacteristic
217C22×C4#23911yescharacteristic
218SD16—124no—
219M16—42no—
220M16—242no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—22no—
224C2×C8—41no—
225C2×C8—62no—
226C2×C8—242no—
227C4⋊C4—32no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—121no—
234C4⋊C4—121no—
235C22⋊C4—62no—
236C22⋊C4—62no—
237C22⋊C4—121no—
238C42—121no—
239C42C2×C4×D311yescharacteristic
240C42#23511yescharacteristic
241C42D3×D411yescharacteristic
242C42D3×D411yescharacteristic
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—
256Dic3—42no—
257Dic3—81no—
258Dic3—81no—
259Dic3—161no—
260C23—62no—
261C23#89511yescharacteristic
262Q8—122no—
263Q8—241no—
264Q8—241no—
265D4—122no—
266D4—241no—
267D4—241no—
268D4—242no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—62no—
273C2×C4—122no—
274C2×C4—122no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4—241no—
278C2×C4—241no—
279C2×C4—241no—
280C2×C4C4×D1211yescharacteristic
281C2×C4D3×C22⋊C411yescharacteristic
282C2×C4#88211yescharacteristic
283C2×C4#88411yescharacteristic
284C2×C4#89311yescharacteristic
285C2×C4#89411yescharacteristic
286C8—22no—
287C8—42no—
288C8—41no—
289C8—81no—
290C8—122no—
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#466511yescharacteristic
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—122no—
310C4—241no—
311C4—241no—
312C4—481no—
313C3#941511yescharacteristic, complemented, Sylow
314C2—41no—
315C2—242no—
316C2#6872511yescharacteristic
317C2#6877711yescharacteristic
318C2#6902711yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1291C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1291D3
hypocenterC3#9415
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1291

Complements

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

NH ≅ G/Ntype
3. #70837C2semidirect
5. #70815C2semidirect
6. C3×#1291C2semidirect
7. #67666C2semidirect
21. #4694C4semidirect
22. #4693C4semidirect
23. #4682C4semidirect
24. #4681C4semidirect
33. #1291D3semidirect
313. C3#9415semidirect

Maximal subgroups

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

subgroupindex
#793372
#785332
#708372
#708352
#708152
C3×#12912
#676662
#94153