Finite Group Databasebeta

← #10692476

Subgroups — #10692476

Order 768.

492 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C3×#3215C211yesmaximal, characteristic
2#82954C211yesmaximal, characteristic
3#71841C211yesmaximal, characteristic, complemented
4#71837C211yesmaximal, characteristic, complemented
5#71718C211yesmaximal, characteristic
6#67752C211yesmaximal, characteristic, complemented
7#67749C211yesmaximal, characteristic, complemented
8#9379—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C3×#497C412yescomplemented
12C3×#489C412yescomplemented
13C3×#464—22no—
14C3×#451—21no—
15C3×#450—21no—
16C3×#432C2211yescharacteristic
17#5229—21no—
18#5220—21no—
19#4951—21no—
20#4951—21no—
21#4950—21no—
22Dic3×C22⋊C4—21no—
23#4937C2211yescharacteristic
24C4×C4⋊Dic3C2211yescharacteristic
25#4926C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4540—22no—
28#4538—21no—
29#4538—21no—
30#4532—21no—
31#4532—21no—
32#3215D311yescharacteristic
33#2337—31no—
34#1986—31no—
35#1527—31no—
36#1515—31no—
37#1288—31no—
38#1285—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4D411yescharacteristic
41C23×Dic3—21no—
42C2×C3×Q16—22no—
43C2×C3×SD16—22no—
44C3×C4⋊Q8C2×C411yescharacteristic, complemented
45C3×C4⋊Q8C2×C411yescharacteristic, complemented
46C3×#125—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124C2×C412yes—
50C3×#124D412yes—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×#117—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#217—21no—
60C2×#217—21no—
61C2×#217—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4×Dic3—21no—
65C2×C4×Dic3D411yescharacteristic
66C2×C4×Dic3D411yescharacteristic
67C2×#208—21no—
68C3×C4⋊C8—21no—
69C3×#103—22no—
70C3×#102—22no—
71C3×C22⋊C8—21no—
72C3×#97C2×C412yescomplemented
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#803—21no—
84C4×C3⋊C8—21no—
85#533C2×D311yescharacteristic
86C2×#124C2×D311yescharacteristic
87#497Dic312yescomplemented
88#489Dic312yescomplemented
89#464—22no—
90#451—21no—
91#450—21no—
92#432—31no—
93#432C2×D311yescharacteristic
94C4×M16—31no—
95#391—61no—
96#388—61no—
97#388—61no—
98#386—61no—
99#385—61no—
100#383—31no—
101C4×C4⋊C4—31no—
102C4×C22⋊C4—61no—
103#339—61no—
104#339—61no—
105#329—62no—
106#328—61no—
107#328—61no—
108C24×C3—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—81no—
113C2×C3×Q8C22⋊C412yes—
114C2×C3×D4—21no—
115C2×C3×D4—81no—
116C2×C3×D4C22⋊C412yes—
117C22×C3×C4—21no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C22×Dic3—42no—
126C22×Dic3—81no—
127C22×Dic3—81no—
128C3×Q16—82no—
129C3×SD16—82no—
130C2×C3×C8—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C22⋊C4—22no—
138C3×C22⋊C4—22no—
139C3×C22⋊C4—22no—
140C3×C22⋊C4—22no—
141C3×C22⋊C4—41no—
142C3×C42—41no—
143C3×C42C22×C411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146C3×C42#4111yescharacteristic
147#217—42no—
148#217—81no—
149#217—81no—
150C4⋊Dic3—42no—
151C4⋊Dic3—81no—
152C4×Dic3—41no—
153C4×Dic3—41no—
154C4×Dic3—81no—
155#208—22no—
156#208—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C22×Q8#6711yescharacteristic
160C22×D4#6711yescharacteristic
161C23×C4—61no—
162C2×Q16—22no—
163C2×SD16—22no—
164C2×M16—61no—
165C4⋊Q8C2×Dic311yescharacteristic, complemented
166C4⋊Q8C2×Dic311yescharacteristic, complemented
167#125—21no—
168#124—21no—
169#124—21no—
170#124C2×Dic312yes—
171#124#6712yes—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176#117—21no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C42—31no—
185C2×C42—31no—
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188C4⋊C8—21no—
189C4⋊C8—61no—
190#103—22no—
191#102—22no—
192C22⋊C8—21no—
193C22⋊C8—61no—
194#97—61no—
195#97—61no—
196#97C2×Dic312yescomplemented
197C4×C8—61no—
198#95—31no—
199#95—31no—
200#95—61no—
201#95—61no—
202#95—61no—
203#95—61no—
204#95—61no—
205C23×C3—22no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3#12011yescharacteristic
213C3×Q8—42no—
214C3×Q8—42no—
215C3×Q8—81no—
216C3×Q8—81no—
217C3×D4—42no—
218C3×D4—81no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—22no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4C2×C22⋊C411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12311yescharacteristic
234C2×Dic3—22no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—84no—
247C3×C8—42no—
248C3⋊C8—42no—
249C3⋊C8—81no—
250C24—21no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—81no—
255C2×Q8#21712yes—
256C2×D4—21no—
257C2×D4—81no—
258C2×D4#21712yes—
259C22×C4—21no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—122no—
265C22×C4—241no—
266C22×C4—241no—
267C22×C4D3×D411yescharacteristic
268C22×C4D3×D411yescharacteristic
269C22×C4C2×#6711yescharacteristic
270Q16—82no—
271SD16—82no—
272M16—62no—
273M16—121no—
274C2×C8—22no—
275C2×C8—121no—
276C2×C8—121no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—122no—
284C4⋊C4—241no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—41no—
290C22⋊C4—122no—
291C22⋊C4—241no—
292C22⋊C4—241no—
293C42—41no—
294C42—121no—
295C42—121no—
296C42—241no—
297C42#23711yescharacteristic
298C42#23711yescharacteristic
299C42C22×Dic311yescharacteristic
300C42C2×#6711yescharacteristic
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—42no—
304C22×C3—42no—
305C22×C3—42no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3—81no—
309C22×C3—81no—
310C22×C3#38611yescharacteristic
311C22×C3#45511yescharacteristic
312C22×C3#45611yescharacteristic
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—42no—
320C3×C4—81no—
321C3×C4—81no—
322Dic3—42no—
323Dic3—81no—
324Dic3—81no—
325Dic3—81no—
326C23—22no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23#93711yescharacteristic
334Q8—42no—
335Q8—42no—
336Q8—81no—
337Q8—81no—
338D4—42no—
339D4—81no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—62no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—244no—
362C2×C4D4×Dic311yescharacteristic
363C2×C4D4×Dic311yescharacteristic
364C2×C4#93511yescharacteristic
365C2×C4#93911yescharacteristic
366C2×C4#93911yescharacteristic
367C2×C4C2×#21711yescharacteristic
368C8—42no—
369C8—122no—
370C8—241no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—82no—
374C2×C3#188211yescharacteristic
375C2×C3#190611yescharacteristic
376C2×C3#191111yescharacteristic
377C22—21no—
378C22—21no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#520211yescharacteristic
387C22#520511yescharacteristic
388C22#522011yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—42no—
396C4—81no—
397C4—81no—
398C4—122no—
399C4—241no—
400C4—241no—
401C4—241no—
402C3#937911yescharacteristic, complemented, Sylow
403C2—21no—
404C2—21no—
405C2—82no—
406C2#7185311yescharacteristic
407C2#7210011yescharacteristic
408C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3215C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#3215D3
hypocenterC3#9379
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3215

Complements

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

NH ≅ G/Ntype
3. #71841C2semidirect
4. #71837C2semidirect
6. #67752C2semidirect
7. #67749C2semidirect
11. C3×#497C4semidirect
12. C3×#489C4semidirect
44. C3×C4⋊Q8C2×C4semidirect
45. C3×C4⋊Q8C2×C4semidirect
72. C3×#97C2×C4semidirect
87. #497Dic3semidirect
88. #489Dic3semidirect
165. C4⋊Q8C2×Dic3semidirect
166. C4⋊Q8C2×Dic3semidirect
196. #97C2×Dic3semidirect
402. C3#9379semidirect

Maximal subgroups

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

subgroupindex
C3×#32152
#829542
#718412
#718372
#717182
#677522
#677492
#93793