Finite Group Databasebeta

← #10692820

Subgroups — #10692820

Order 768.

456 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 456 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#78529C211yesmaximal, characteristic
2#73394C211yesmaximal, characteristic, complemented
3#70832C211yesmaximal, characteristic
4#70814C211yesmaximal, characteristic, complemented
5#69432C211yesmaximal, characteristic
6C3×#1300C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#9442—31nomaximal, Sylow
9#5477C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×#376C2211yescharacteristic
14#5133—21no—
15#5133—21no—
16#4933—21no—
17#4932C2211yescharacteristic
18C4×Dic3⋊C4—21no—
19#4926C2211yescharacteristic
20#4713C412yescomplemented
21#4703C412yescomplemented
22#4697—22no—
23#4654—21no—
24#4653—21no—
25#4652—21no—
26#4649—21no—
27#4470—21no—
28#4470—21no—
29#4465—22no—
30#4451—21no—
31#4451—21no—
32#3142—31no—
33#2371—31no—
34#1534—31no—
35#1505—31no—
36#1505—31no—
37#1300—31no—
38#1300D311yescharacteristic, complemented
39C2×#235—21no—
40C2×C3×M16—21no—
41C2×C3×C42—21no—
42C2×C3×C42C2311yescharacteristic
43#930—21no—
44#930—21no—
45C2×C4⋊Dic3D411yescharacteristic
46C2×C4⋊Dic3D411yescharacteristic
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3—21no—
52C2×C4×Dic3—21no—
53C2×C4×Dic3—21no—
54#875—21no—
55#874C2×C411yescharacteristic
56C4×D12—21no—
57C4×D12—21no—
58#870—21no—
59#870—21no—
60#870C2×C412yes—
61#870D412yes—
62#869C2×C411yescharacteristic
63C3×C4⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#97—21no—
66C3×#97C2×C412yes—
67C3×C4×C8—21no—
68C3×#95—21no—
69C3×#95D411yescharacteristic
70C3×#95D411yescharacteristic
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95D411yescharacteristic
77C3⋊#95D411yescharacteristic
78#830—21no—
79#821—22no—
80#821—22no—
81#816—22no—
82#816—22no—
83C4⋊(C3⋊C8)—21no—
84#803—21no—
85#532—31no—
86C2×#125—31no—
87#491—32no—
88#490—32no—
89#488—62no—
90#432—31no—
91#432C2×D311yescharacteristic
92#417—61no—
93#417—61no—
94C4×M16C2×D311yescharacteristic
95#387—61no—
96#387—61no—
97#387—61no—
98#387—61no—
99C4×C4⋊C4—61no—
100C4×C4⋊C4—61no—
101#376—31no—
102#376C2×D311yescharacteristic
103#330—62no—
104#327—61no—
105#327—61no—
106#327—61no—
107#327—61no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C2×#67—41no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118#235—42no—
119#235—42no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×C4×D3—41no—
124C2×Dic6—21no—
125C3×M16—41no—
126C3×M16—41no—
127C2×C3×C8—22no—
128C2×C3×C8—41no—
129C3×C42—42no—
130C3×C42C22×C411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133C3×C42#4111yescharacteristic
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3C22⋊C412yes—
141C4⋊Dic3C22⋊C412yes—
142Dic3⋊C4—22no—
143Dic3⋊C4—22no—
144Dic3⋊C4—22no—
145Dic3⋊C4—22no—
146Dic3⋊C4—42no—
147Dic3⋊C4—42no—
148C4×Dic3—42no—
149C4×Dic3—42no—
150C4×Dic3—42no—
151C2×C3⋊C8—41no—
152C2×#41—61no—
153C2×M16—21no—
154C4⋊Q8—31no—
155#127—31no—
156#125—32no—
157#125—32no—
158#125—61no—
159#125—61no—
160#124—61no—
161#121—61no—
162#121—61no—
163C4×D4—61no—
164C4×D4—61no—
165C2×C4⋊C4—31no—
166C2×C4⋊C4—31no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C42—21no—
172C2×C42—61no—
173C2×C42—61no—
174C2×C42—61no—
175C2×C42C22×D311yescharacteristic
176C4⋊C8—21no—
177C4⋊C8—61no—
178#103—62no—
179#103—62no—
180#102—62no—
181#102—62no—
182C22⋊C8—21no—
183C22⋊C8—61no—
184#97—21no—
185#97—61no—
186#97C4×D312yes—
187C4×C8—21no—
188#95—21no—
189#95—31no—
190#95—31no—
191#95—61no—
192#95—61no—
193#95—61no—
194#95—61no—
195#95—61no—
196#95D1211yescharacteristic
197#95D1211yescharacteristic
198C23×C3#12011yescharacteristic
199C22×D3—41no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—22no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4C2×C22⋊C411yescharacteristic
211C2×C3×C4C4×D411yescharacteristic
212C2×C3×C4C4×D411yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12311yescharacteristic
216#67—81no—
217#67—81no—
218C2×Dic3—22no—
219C2×Dic3—22no—
220C2×Dic3—22no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231D12—42no—
232D12—81no—
233D12—81no—
234C4×D3—82no—
235Dic6—42no—
236C3×C8—42no—
237C3×C8—81no—
238C3⋊C8—81no—
239#41—122no—
240#41—122no—
241C2×Q8—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—121no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—121no—
253C22×C4C2×D1211yescharacteristic
254C22×C4D3×D411yescharacteristic
255C22×C4D3×D411yescharacteristic
256M16—41no—
257M16—41no—
258C2×C8—22no—
259C2×C8—41no—
260C2×C8—121no—
261C4⋊C4—32no—
262C4⋊C4—32no—
263C4⋊C4—62no—
264C4⋊C4—62no—
265C4⋊C4—62no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—62no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—122no—
272C4⋊C4—122no—
273C22⋊C4—121no—
274C22⋊C4—121no—
275C42—42no—
276C42—122no—
277C42—122no—
278C42—122no—
279C42C2×C4×D311yescharacteristic
280C42#23511yescharacteristic
281C42D3×D411yescharacteristic
282C42#23911yescharacteristic
283C22×C3—21no—
284C22×C3—21no—
285C22×C3#38611yescharacteristic
286C22×C3#45411yescharacteristic
287C22×C3#45411yescharacteristic
288C2×D3—81no—
289C2×D3—81no—
290C2×D3—81no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—42no—
298C3×C4—81no—
299Dic3—42no—
300Dic3—42no—
301Dic3—42no—
302Dic3—81no—
303Dic3—81no—
304C23—121no—
305C23#88211yescharacteristic
306Q8—122no—
307D4—122no—
308D4—241no—
309D4—241no—
310D4—241no—
311D4—241no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—62no—
323C2×C4—62no—
324C2×C4—62no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—242no—
336C2×C4C4×D1211yescharacteristic
337C2×C4D3×C22⋊C411yescharacteristic
338C2×C4#88411yescharacteristic
339C2×C4#89311yescharacteristic
340C2×C4#89411yescharacteristic
341C2×C4#89511yescharacteristic
342C8—42no—
343C8—81no—
344C8—241no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3#188811yescharacteristic
348C2×C3#190611yescharacteristic
349C2×C3#190611yescharacteristic
350D3—161no—
351C22—21no—
352C22—21no—
353C22—241no—
354C22—241no—
355C22—241no—
356C22#466511yescharacteristic
357C22#482711yescharacteristic
358C22#482711yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—81no—
367C4—122no—
368C4—122no—
369C4—122no—
370C4—241no—
371C4—241no—
372C3#944211yescharacteristic, complemented, Sylow
373C2—21no—
374C2—21no—
375C2—481no—
376C2#6876611yescharacteristic
377C2#6877811yescharacteristic
378C2#6903911yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1300C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1300D3
hypocenterC3#9442
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
2. #73394C2semidirect
4. #70814C2semidirect
6. C3×#1300C2semidirect
7. #67665C2semidirect
20. #4713C4semidirect
21. #4703C4semidirect
38. #1300D3semidirect
372. C3#9442semidirect

Maximal subgroups

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

subgroupindex
#785292
#733942
#708322
#708142
#694322
C3×#13002
#676652
#94423