Finite Group Databasebeta

← #10692808

Subgroups — #10692808

Order 768.

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

normal subgroups only

This group has 434 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#83342C211yesmaximal, characteristic
2#73394C211yesmaximal, characteristic
3C3×#1506C211yesmaximal, characteristic
4#71836C211yesmaximal, characteristic
5#70814C211yesmaximal, characteristic
6#67748C211yesmaximal, characteristic
7#67615C211yesmaximal, characteristic
8#9440—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#376C2211yescharacteristic
13#5081C412yescomplemented
14#5081C412yescomplemented
15#5065—22no—
16#4995—21no—
17#4995—21no—
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×#328—21no—
28C3×#328—21no—
29#4538—21no—
30#4538—21no—
31#4465—22no—
32#3166—31no—
33#2371—31no—
34#1534—31no—
35#1506—31no—
36#1506D311yescharacteristic
37#1300—31no—
38#1300—31no—
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#125—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×#117—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
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95D411yescharacteristic
75C3⋊#95D411yescharacteristic
76#830—21no—
77#808—22no—
78#808—22no—
79#807—22no—
80#807—22no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83#803C2×C412yes—
84C4×C3⋊C8—21no—
85#533C2×D311yescharacteristic
86C2×#125—31no—
87#501—32no—
88#501—32no—
89#499—62no—
90#432—31no—
91#432C2×D311yescharacteristic
92#428—61no—
93#428—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#328—21no—
105#328—21no—
106#328—61no—
107#328—61no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C2×C3×C8—41no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C42—42no—
128C3×C42—41no—
129C3×C42C22×C411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132C3×C42#4111yescharacteristic
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3C22⋊C412yes—
136C4⋊Dic3C22⋊C412yes—
137Dic3⋊C4—22no—
138Dic3⋊C4—22no—
139Dic3⋊C4—22no—
140Dic3⋊C4—22no—
141Dic3⋊C4—42no—
142Dic3⋊C4—42no—
143C4×Dic3—42no—
144C4×Dic3—42no—
145C4×Dic3—42no—
146#208—22no—
147#208—22no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—41no—
150C2×M16—61no—
151C4⋊Q8C4×D311yescharacteristic
152C4⋊Q8C4×D311yescharacteristic
153#125—21no—
154#125—32no—
155#125—32no—
156#125—61no—
157#125—61no—
158C22⋊Q8—21no—
159C22⋊Q8—21no—
160C4×Q8—21no—
161C4×Q8—21no—
162#117—21no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C42—21no—
170C2×C42—61no—
171C2×C42—61no—
172C2×C42—61no—
173C2×C42C22×D311yescharacteristic
174#107—62no—
175#107—62no—
176#106—62no—
177#106—62no—
178C4⋊C8—21no—
179C4⋊C8—61no—
180C22⋊C8—21no—
181C22⋊C8—61no—
182#97—21no—
183#97—32no—
184#97—61no—
185C4×C8—61no—
186#95—21no—
187#95—31no—
188#95—31no—
189#95—61no—
190#95—61no—
191#95—61no—
192#95—61no—
193#95—61no—
194#95#6711yescharacteristic
195#95#6711yescharacteristic
196C23×C3#12011yescharacteristic
197C3×Q8—81no—
198C3×Q8—81no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—22no—
202C2×C3×C4—41no—
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×C4—41no—
211C2×C3×C4C2×C22⋊C411yescharacteristic
212C2×C3×C4C4×D411yescharacteristic
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12311yescharacteristic
217C2×Dic3—22no—
218C2×Dic3—22no—
219C2×Dic3—22no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C3×C8—81no—
230C3⋊C8—24no—
231C3⋊C8—81no—
232C2×Q8—21no—
233C2×Q8—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4D3×D411yescharacteristic
241C22×C4D3×D411yescharacteristic
242C22×C4C2×#6711yescharacteristic
243M16—62no—
244M16—62no—
245C2×C8—41no—
246C2×C8—62no—
247C2×C8—121no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—32no—
253C4⋊C4—32no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—62no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—62no—
260C4⋊C4—62no—
261C4⋊C4—62no—
262C4⋊C4—122no—
263C4⋊C4—122no—
264C22⋊C4—41no—
265C42—42no—
266C42—41no—
267C42—122no—
268C42—122no—
269C42—122no—
270C42C2×C4×D311yescharacteristic
271C42#23511yescharacteristic
272C42D3×D411yescharacteristic
273C42#23711yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3#38611yescharacteristic
277C22×C3#45611yescharacteristic
278C22×C3#45611yescharacteristic
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—42no—
286C3×C4—81no—
287C3×C4—81no—
288C3×C4—81no—
289Dic3—42no—
290Dic3—42no—
291Dic3—42no—
292Dic3—81no—
293C23#93711yescharacteristic
294Q8—81no—
295Q8—81no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—62no—
309C2×C4—62no—
310C2×C4—62no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4D3×C22⋊C411yescharacteristic
321C2×C4#67⋊C411yescharacteristic
322C2×C4#88311yescharacteristic
323C2×C4#88411yescharacteristic
324C2×C4C4×#6711yescharacteristic
325C2×C4#93811yescharacteristic
326C8—64no—
327C8—81no—
328C8—241no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3#188611yescharacteristic
332C2×C3#190611yescharacteristic
333C2×C3#190611yescharacteristic
334C22—21no—
335C22—21no—
336C22#495611yescharacteristic
337C22#520511yescharacteristic
338C22#520511yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—42no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—122no—
350C4—122no—
351C4—122no—
352C4—241no—
353C3#944011yescharacteristic, complemented, Sylow
354C2—21no—
355C2—21no—
356C2#7005511yescharacteristic
357C2#7034511yescharacteristic
358C2#7039111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1506C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1506D3
hypocenterC3#9440
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1506

Complements

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

NH ≅ G/Ntype
13. #5081C4semidirect
14. #5081C4semidirect
353. C3#9440semidirect

Maximal subgroups

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

subgroupindex
#833422
#733942
C3×#15062
#718362
#708142
#677482
#676152
#94403