Finite Group Databasebeta

← #10692471

Subgroups — #10692471

Order 768.

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

normal subgroups only

This group has 492 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#83357C211yesmaximal, characteristic
2#74262C211yesmaximal, characteristic
3C3×#1515C211yesmaximal, characteristic, complemented
4#71841C211yesmaximal, characteristic, complemented
5#70804C211yesmaximal, characteristic
6#67750C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#9379—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#875C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#383C2211yescharacteristic
13#5089C412yescomplemented
14#5086C412yescomplemented
15#5050—21no—
16#5046—21no—
17#5028—22no—
18C4×C4⋊Dic3C2211yescharacteristic
19#4926C2211yescharacteristic
20C4×#208C2211yescharacteristic
21#4670—21no—
22#4669—21no—
23#4668—21no—
24#4666—21no—
25#4665—21no—
26#4646—21no—
27C3×#328—21no—
28C3×#328—21no—
29#4532—21no—
30#4532—21no—
31#4464—22no—
32#3215—31no—
33#2337—31no—
34#1986—31no—
35#1527—31no—
36#1515D311yescharacteristic, complemented
37#1288—31no—
38#1285—31no—
39C22×D12D411yescharacteristic
40C22×C4×D3—21no—
41C22×Dic6D411yescharacteristic
42C3×C4⋊Q8C2×C411yescharacteristic, complemented
43C3×C4⋊Q8C2×C411yescharacteristic, complemented
44C3×#125—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×#117—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#216—22no—
52C2×#215—22no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×C4⋊Dic3—21no—
59C2×Dic3⋊C4—21no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3D411yescharacteristic
62C2×C4×Dic3D411yescharacteristic
63C2×#208—21no—
64#875—21no—
65#875—21no—
66#875C2×C412yes—
67#875D412yes—
68C3×C4⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71C3×#95—21no—
72C3×#95—21no—
73C3×#95D411yescharacteristic
74C3×#95D411yescharacteristic
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78#830—21no—
79#810—22no—
80#809—22no—
81C4⋊(C3⋊C8)—21no—
82#803—21no—
83#803C2×C412yescomplemented
84C4×C3⋊C8—21no—
85#533C2×D311yescharacteristic
86C2×#124—31no—
87#497—32no—
88#489—32no—
89#464—62no—
90#451—61no—
91#450—61no—
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#383C2×D311yescharacteristic
101C4×C4⋊C4—31no—
102C4×C22⋊C4—61no—
103#339—61no—
104#339—61no—
105#329—62no—
106#328—21no—
107#328—21no—
108C23×D3—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C2×D12—21no—
120C2×D12—81no—
121C2×D12C22⋊C412yes—
122C2×C4×D3—42no—
123C2×C4×D3—81no—
124C2×C4×D3—81no—
125C2×Dic6—21no—
126C2×Dic6—81no—
127C2×Dic6C22⋊C412yes—
128C2×C3×C8—41no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42—41no—
137C3×C42C22×C411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140C3×C42#4111yescharacteristic
141#216—82no—
142#215—82no—
143(C2×D3)⋊C4—22no—
144(C2×D3)⋊C4—22no—
145(C2×D3)⋊C4—22no—
146(C2×D3)⋊C4—22no—
147(C2×D3)⋊C4—42no—
148(C2×D3)⋊C4—81no—
149(C2×D3)⋊C4—81no—
150C4⋊Dic3—42no—
151Dic3⋊C4—81no—
152C4×Dic3—41no—
153C4×Dic3—41no—
154C4×Dic3—81no—
155#208—22no—
156#208—41no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—41no—
159C22×Q8—31no—
160C22×D4—31no—
161C23×C4—61no—
162C2×Q16—62no—
163C2×SD16—62no—
164C2×M16—61no—
165C4⋊Q8C4×D311yescharacteristic, complemented
166C4⋊Q8C4×D311yescharacteristic, complemented
167#125—21no—
168#124—32no—
169#124—32no—
170#124—61no—
171#124—61no—
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—61no—
180C2×C22⋊C4—61no—
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—62no—
191#102—62no—
192C22⋊C8—21no—
193C22⋊C8—61no—
194#97—21no—
195#97—32no—
196#97—61no—
197C4×C8—61no—
198#95—21no—
199#95—21no—
200#95—61no—
201#95—61no—
202#95—61no—
203#95#6711yescharacteristic
204#95#6711yescharacteristic
205C23×C3#12011yescharacteristic
206C22×D3—22no—
207C22×D3—41no—
208C22×D3—41no—
209C22×D3—41no—
210C22×D3—41no—
211C22×D3—41no—
212C22×D3—41no—
213C3×Q8—42no—
214C3×Q8—81no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4C2×C22⋊C411yescharacteristic
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12311yescharacteristic
233C2×Dic3—22no—
234C2×Dic3—22no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242D12—42no—
243D12—81no—
244C4×D3—84no—
245Dic6—42no—
246Dic6—81no—
247C3×C8—81no—
248C3⋊C8—42no—
249C3⋊C8—42no—
250C24—61no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—32no—
254C2×Q8—61no—
255C2×Q8—241no—
256C2×D4—32no—
257C2×D4—61no—
258C2×D4—241no—
259C22×C4—21no—
260C22×C4—21no—
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—242no—
271SD16—242no—
272M16—62no—
273M16—121no—
274C2×C8—41no—
275C2×C8—62no—
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—41no—
286C22⋊C4—62no—
287C22⋊C4—62no—
288C22⋊C4—62no—
289C22⋊C4—62no—
290C22⋊C4—122no—
291C22⋊C4—241no—
292C22⋊C4—241no—
293C42—41no—
294C42—121no—
295C42—121no—
296C42—241no—
297C42C2×C4×D311yescharacteristic
298C42#23511yescharacteristic
299C42D3×D411yescharacteristic
300C42#23711yescharacteristic
301C22×C3—21no—
302C22×C3—21no—
303C22×C3#38611yescharacteristic
304C22×C3#45511yescharacteristic
305C22×C3#45611yescharacteristic
306C2×D3—42no—
307C2×D3—42no—
308C2×D3—42no—
309C2×D3—81no—
310C2×D3—81no—
311C2×D3—81no—
312C2×D3—81no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—81no—
320C3×C4—81no—
321C3×C4—81no—
322C3×C4—81no—
323Dic3—42no—
324Dic3—42no—
325Dic3—81no—
326C23—62no—
327C23—121no—
328C23—121no—
329C23—121no—
330C23—121no—
331C23—121no—
332C23—121no—
333C23#93711yescharacteristic
334Q8—42no—
335Q8—81no—
336Q8—122no—
337Q8—241no—
338D4—122no—
339D4—241no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—62no—
353C2×C4—62no—
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×C4D3×C22⋊C411yescharacteristic
363C2×C4#67⋊C411yescharacteristic
364C2×C4#88311yescharacteristic
365C2×C4#88411yescharacteristic
366C2×C4C4×#6711yescharacteristic
367C2×C4#93811yescharacteristic
368C8—81no—
369C8—122no—
370C8—122no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3#188211yescharacteristic
374C2×C3#190611yescharacteristic
375C2×C3#191111yescharacteristic
376D3—82no—
377C22—21no—
378C22—21no—
379C22—122no—
380C22—122no—
381C22—122no—
382C22—241no—
383C22—241no—
384C22—241no—
385C22—241no—
386C22#495611yescharacteristic
387C22#520211yescharacteristic
388C22#520511yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—122no—
400C4—122no—
401C4—241no—
402C3#937911yescharacteristic, complemented, Sylow
403C2—21no—
404C2—21no—
405C2—242no—
406C2#7005511yescharacteristic
407C2#7034811yescharacteristic
408C2#7040811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1515C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1515D3
hypocenterC3#9379
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1515

Complements

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

NH ≅ G/Ntype
3. C3×#1515C2semidirect
4. #71841C2semidirect
6. #67750C2semidirect
7. #67633C2semidirect
13. #5089C4semidirect
14. #5086C4semidirect
36. #1515D3semidirect
42. C3×C4⋊Q8C2×C4semidirect
43. C3×C4⋊Q8C2×C4semidirect
83. #803C2×C4semidirect
165. C4⋊Q8C4×D3semidirect
166. C4⋊Q8C4×D3semidirect
402. C3#9379semidirect

Maximal subgroups

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

subgroupindex
#833572
#742622
C3×#15152
#718412
#708042
#677502
#676332
#93793