Finite Group Databasebeta

← #10696770

Subgroups — #10696770

Order 768.

476 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 81 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#79350C211yesmaximal, characteristic
2#79318C211yesmaximal, characteristic, complemented
3#78529C211yesmaximal, characteristic, complemented
4#70831C211yesmaximal, characteristic, complemented
5#70801C211yesmaximal, characteristic, complemented
6#69853C211yesmaximal, characteristic, complemented
7C3×#1291C211yesmaximal, characteristic, complemented
8#10122—31nomaximal, Sylow
9#5477C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5133—21no—
16#5133—21no—
17#4925C2211yescharacteristic
18#4847—22no—
19#4843—21no—
20#4841—21no—
21#4736—21no—
22#4735—21no—
23#4734—22no—
24#4714—21no—
25#4713—21no—
26#4713—21no—
27#4712—21no—
28#4703—21no—
29#4702—21no—
30#4697—22no—
31#4483—22no—
32#4469—22no—
33#4459—22no—
34#3234—31no—
35#3214—31no—
36#3142—31no—
37#1496—31no—
38#1484—31no—
39#1470—31no—
40#1291D311yescharacteristic, complemented
41C2×#235—21no—
42C2×#235—21no—
43C2×C3×M16—21no—
44C2×C3×C42C2311yescharacteristic
45#930—21no—
46#930—21no—
47#930—21no—
48#930—21no—
49C2×C4⋊Dic3D411yescharacteristic
50C2×C4⋊Dic3D411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×D24—22no—
54C2×#204—22no—
55#875—21no—
56#875—21no—
57#874D411yescharacteristic
58#874D411yescharacteristic
59C4×D12—21no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12—21no—
63#870—21no—
64#870—21no—
65#870—21no—
66#870—21no—
67#869D411yescharacteristic
68#869D411yescharacteristic
69C3×C4⋊C8—21no—
70C3×C4⋊C8D412yes—
71C3×C22⋊C8—21no—
72C3×C22⋊C8D412yes—
73C3×#97—21no—
74C3×#97—21no—
75C3×#97—21no—
76C3×C4×C8D412yes—
77#830—21no—
78#821—22no—
79#821—22no—
80#821—22no—
81#821—22no—
82#821—22no—
83#818—22no—
84#817—22no—
85#816—22no—
86#816—22no—
87#816—22no—
88C4⋊(C3⋊C8)—21no—
89C4×C3⋊C8—21no—
90#532—31no—
91#532—31no—
92C2×#125—31no—
93#501—61no—
94#496—61no—
95#491—61no—
96#490—61no—
97#490—61no—
98#489—61no—
99#488—62no—
100#481—61no—
101#480—61no—
102#478—62no—
103#461—61no—
104#459—61no—
105#449—62no—
106#432C2×D311yescharacteristic
107#432C2×D311yescharacteristic
108#431—31no—
109#417—61no—
110#417—61no—
111C4×M16C2×D311yescharacteristic
112D4⋊C8—62no—
113D4⋊C8—62no—
114#324—62no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C2×#67—41no—
119C2×#67—41no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122#235—44no—
123#235—82no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×C4×D3—41no—
131C2×C4×D3—41no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C3×M16—42no—
135C2×C3×C8—22no—
136C2×C3×C8—22no—
137C2×C3×C8—22no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—41no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158C2×C3⋊C8—41no—
159D24—82no—
160#204—82no—
161C2×#41—61no—
162C2×#41—61no—
163C2×SD16—62no—
164C2×D8—62no—
165C2×M16—21no—
166C4⋊Q8—31no—
167C4⋊Q8—31no—
168#127—31no—
169#127—31no—
170#125—61no—
171#125—61no—
172#125—61no—
173#125—61no—
174#124—61no—
175#124—61no—
176#121—61no—
177#121—61no—
178#121—61no—
179#121—61no—
180C4×D4—61no—
181C4×D4—61no—
182C4×D4—61no—
183C4×D4—61no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C42C22×D311yescharacteristic
189#107—62no—
190#106—62no—
191C4⋊C8—21no—
192C4⋊C8—61no—
193C4⋊C8D1212yes—
194#103—62no—
195#103—62no—
196#103—62no—
197#102—62no—
198#102—62no—
199#102—62no—
200#102—62no—
201#102—62no—
202C22⋊C8—21no—
203C22⋊C8—61no—
204C22⋊C8D1212yes—
205#97—21no—
206#97—21no—
207#97—21no—
208C4×C8—61no—
209C4×C8D1212yes—
210C23×C3#12111yescharacteristic
211C22×D3—41no—
212C22×D3—41no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12711yescharacteristic
222#67—82no—
223#67—82no—
224C2×Dic3—42no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—81no—
230D12—41no—
231D12—41no—
232D12—42no—
233D12—42no—
234D12—42no—
235D12—42no—
236D12—42no—
237C4×D3—82no—
238C4×D3—82no—
239Dic6—42no—
240Dic6—42no—
241C3×C8—42no—
242C3×C8—42no—
243C3×C8—42no—
244C3⋊C8—81no—
245#41—124no—
246#41—242no—
247C2×Q8—61no—
248C2×Q8—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—121no—
256C2×D4—121no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—121no—
260C22×C4—121no—
261C22×C4C2×D1211yescharacteristic
262C22×C4D3×D411yescharacteristic
263C22×C4#23911yescharacteristic
264SD16—242no—
265D8—242no—
266M16—42no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—121no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—121no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C22⋊C4—121no—
284C22⋊C4—121no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C42C2×D1211yescharacteristic
288C42C2×D1211yescharacteristic
289C42D3×D411yescharacteristic
290C42D3×D411yescharacteristic
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3#39211yescharacteristic
295C2×D3—81no—
296C2×D3—81no—
297C2×D3—81no—
298C2×D3—81no—
299C2×D3—81no—
300C2×D3—81no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4#45412yes—
307Dic3—81no—
308Dic3—81no—
309Dic3—81no—
310Dic3—81no—
311C23—121no—
312C23—121no—
313C23#89511yescharacteristic
314Q8—122no—
315Q8—122no—
316D4—121no—
317D4—122no—
318D4—121no—
319D4—122no—
320D4—122no—
321D4—122no—
322D4—122no—
323D4—242no—
324D4—242no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—122no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—242no—
334C2×C4—242no—
335C2×C4—241no—
336C2×C4#87411yescharacteristic
337C2×C4#88211yescharacteristic
338C2×C4#88211yescharacteristic
339C2×C4#88211yescharacteristic
340C2×C4#89511yescharacteristic
341C2×C4#89511yescharacteristic
342C8—42no—
343C8—42no—
344C8—42no—
345C8—241no—
346C2×C3—41no—
347C2×C3#200311yescharacteristic
348C2×C3#200811yescharacteristic
349C2×C3#201711yescharacteristic
350D3—161no—
351D3—161no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—241no—
356C22—241no—
357C22—241no—
358C22—241no—
359C22—241no—
360C22—241no—
361C22#467111yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—241no—
368C4—241no—
369C4—241no—
370C4—241no—
371C4#482712yes—
372C3#1012211yescharacteristic, complemented, Sylow
373C2—41no—
374C2—481no—
375C2—481no—
376C2#6872711yescharacteristic
377C2#6876711yescharacteristic
378C2#6902811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1291C2
socleC22×C3#392
prime radical#79350C2
hypercenter#1291D3
hypocenterC3#10122
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1291

Complements

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

NH ≅ G/Ntype
2. #79318C2semidirect
3. #78529C2semidirect
4. #70831C2semidirect
5. #70801C2semidirect
6. #69853C2semidirect
7. C3×#1291C2semidirect
40. #1291D3semidirect
372. C3#10122semidirect

Maximal subgroups

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

subgroupindex
#793502
#793182
#785292
#708312
#708012
#698532
C3×#12912
#101223