Finite Group Databasebeta

← #10692818

Subgroups — #10692818

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#83284C211yesmaximal, characteristic
2#73394C211yesmaximal, characteristic, complemented
3C3×#1505C211yesmaximal, characteristic
4#71812C211yesmaximal, characteristic
5#70814C211yesmaximal, characteristic, complemented
6#67748C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#9442—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#376C2211yescharacteristic
13#5232—21no—
14#5232—21no—
15#5082C412yescomplemented
16#5071C412yescomplemented
17#5056—22no—
18#4933—21no—
19#4932C2211yescharacteristic
20C4×Dic3⋊C4—21no—
21#4926C2211yescharacteristic
22C4×#208C2211yescharacteristic
23#4654—21no—
24#4653—21no—
25#4652—21no—
26#4649—21no—
27C3×#327—21no—
28C3×#327—21no—
29#4537—21no—
30#4537—21no—
31#4465—22no—
32#3142—31no—
33#2371—31no—
34#1534—31no—
35#1505—31no—
36#1505D311yescharacteristic
37#1300—31no—
38#1300—31no—
39C2×C3×#41—21no—
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#127C2×C411yescharacteristic
42C3×#124—21no—
43C3×#121—21no—
44C3×#121—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C2×C3×C42—21no—
48C2×C3×C42C2311yescharacteristic
49C2×C4⋊Dic3D411yescharacteristic
50C2×C4⋊Dic3D411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3—21no—
57C2×C4×Dic3—21no—
58C2×#208—21no—
59#870—21no—
60#870—21no—
61#870C2×C412yes—
62#870D412yes—
63C3×C4⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#97—21no—
66C3×#95—21no—
67C3×#95D411yescharacteristic
68C3×#95D411yescharacteristic
69#835—22no—
70#835—22no—
71#832—22no—
72#832—22no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95D411yescharacteristic
79C3⋊#95D411yescharacteristic
80#830—21no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83#803C2×C412yes—
84C4×C3⋊C8—21no—
85#532C2×D311yescharacteristic
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×M16—31no—
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—21no—
105#327—21no—
106#327—61no—
107#327—61no—
108C3×#41—42no—
109C3×#41—42no—
110C2×C3×Q8—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—41no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C2×C3×C8—41no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C22⋊C4—41no—
129C3×C22⋊C4—41no—
130C3×C42—42no—
131C3×C42C22×C411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42#4111yescharacteristic
134C3×C42#4111yescharacteristic
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3C22⋊C412yes—
138C4⋊Dic3C22⋊C412yes—
139Dic3⋊C4—22no—
140Dic3⋊C4—22no—
141Dic3⋊C4—22no—
142Dic3⋊C4—22no—
143Dic3⋊C4—42no—
144Dic3⋊C4—42no—
145C4×Dic3—42no—
146C4×Dic3—42no—
147C4×Dic3—42no—
148#208—41no—
149#208—41no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—41no—
152C2×#41—21no—
153C2×M16—61no—
154C4⋊Q8C4×D311yescharacteristic
155#127C4×D311yescharacteristic
156#125—32no—
157#125—32no—
158#125—61no—
159#125—61no—
160#124—21no—
161#121—21no—
162#121—21no—
163C4×D4—21no—
164C4×D4—21no—
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—32no—
186#97—61no—
187C4×C8—61no—
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#95#6711yescharacteristic
197#95#6711yescharacteristic
198C23×C3—41no—
199C23×C3#12011yescharacteristic
200C3×Q8—42no—
201C3×D4—42no—
202C3×D4—81no—
203C3×D4—81no—
204C3×D4—81no—
205C3×D4—81no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—22no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—82no—
218C2×C3×C4C2×C22⋊C411yescharacteristic
219C2×C3×C4C4×D411yescharacteristic
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12311yescharacteristic
224C2×Dic3—22no—
225C2×Dic3—22no—
226C2×Dic3—22no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C3×C8—81no—
237C3⋊C8—42no—
238C3⋊C8—81no—
239#41—42no—
240#41—42no—
241C2×Q8—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—41no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4D3×D411yescharacteristic
254C22×C4D3×D411yescharacteristic
255C22×C4C2×#6711yescharacteristic
256M16—121no—
257M16—121no—
258C2×C8—41no—
259C2×C8—62no—
260C2×C8—121no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—32no—
264C4⋊C4—32no—
265C4⋊C4—62no—
266C4⋊C4—61no—
267C4⋊C4—61no—
268C4⋊C4—62no—
269C4⋊C4—62no—
270C4⋊C4—62no—
271C4⋊C4—122no—
272C4⋊C4—122no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C42—42no—
276C42—122no—
277C42—122no—
278C42—122no—
279C42C2×C4×D311yescharacteristic
280C42#23511yescharacteristic
281C42D3×D411yescharacteristic
282C42#23711yescharacteristic
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—81no—
286C22×C3—81no—
287C22×C3—81no—
288C22×C3#38611yescharacteristic
289C22×C3#45411yescharacteristic
290C22×C3#45411yescharacteristic
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—42no—
298C3×C4—81no—
299C3×C4—81no—
300Dic3—42no—
301Dic3—42no—
302Dic3—42no—
303Dic3—81no—
304C23—41no—
305C23#93711yescharacteristic
306Q8—42no—
307D4—42no—
308D4—81no—
309D4—81no—
310D4—81no—
311D4—81no—
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—41no—
323C2×C4—62no—
324C2×C4—62no—
325C2×C4—62no—
326C2×C4—82no—
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—121no—
336C2×C4D3×C22⋊C411yescharacteristic
337C2×C4#67⋊C411yescharacteristic
338C2×C4#88311yescharacteristic
339C2×C4#88411yescharacteristic
340C2×C4C4×#6711yescharacteristic
341C2×C4#93811yescharacteristic
342C8—81no—
343C8—122no—
344C8—241no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—161no—
348C2×C3#188811yescharacteristic
349C2×C3#190611yescharacteristic
350C2×C3#190611yescharacteristic
351C22—21no—
352C22—21no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#495611yescharacteristic
357C22#520111yescharacteristic
358C22#520111yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—81no—
367C4—81no—
368C4—122no—
369C4—122no—
370C4—122no—
371C4—241no—
372C3#944211yescharacteristic, complemented, Sylow
373C2—21no—
374C2—21no—
375C2—161no—
376C2#7005511yescharacteristic
377C2#7032511yescharacteristic
378C2#7039111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1505C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1505D3
hypocenterC3#9442
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1505

Complements

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

NH ≅ G/Ntype
2. #73394C2semidirect
5. #70814C2semidirect
6. #67748C2semidirect
7. #67615C2semidirect
15. #5082C4semidirect
16. #5071C4semidirect
372. C3#9442semidirect

Maximal subgroups

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

subgroupindex
#832842
#733942
C3×#15052
#718122
#708142
#677482
#676152
#94423