Finite Group Databasebeta

← #10764137

Subgroups — #10764137

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85850C211yesmaximal, characteristic
2C3×#2991C211yesmaximal, characteristic, complemented
3#82522C211yesmaximal, characteristic, complemented
4#70780C211yesmaximal, characteristic, complemented
5#70776C212yesmaximal, complemented
6#70775C212yesmaximal, complemented
7#70773C211yesmaximal, characteristic, complemented
8#70510C212yesmaximal, complemented
9#70506C211yesmaximal, characteristic, complemented
10#70505C211yesmaximal, characteristic, complemented
11#70503C212yesmaximal, complemented
12#17781—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14#5833C412yes—
15#5584—21no—
16#5579—21no—
17#5565—22no—
18#5563—21no—
19#5552—22no—
20#5551—21no—
21#5534C414yes—
22#5523C412yes—
23#5477C2211yescharacteristic
24#5477C2212yes—
25#5470C2212yes—
26C2×#869C2211yescharacteristic
27C3×C8×Q8—21no—
28C3×#436—21no—
29C3×#436—22no—
30C3×#431C2211yescharacteristic
31C3×#431C2212yes—
32C3×#423C2212yes—
33C3×#423C2211yescharacteristic
34#5024—21no—
35#5011—22no—
36D4×C3⋊C8—21no—
37#4926C2212yes—
38#4925C2211yescharacteristic
39#4924C2211yescharacteristic
40C2×C4⋊(C3⋊C8)C2212yes—
41#4485C2211yescharacteristic
42#4485C2212yes—
43#4485C2211yescharacteristic
44#4482C2212yes—
45#4482C2212yes—
46#4482C2212yes—
47#4482C2212yes—
48#4481C2212yes—
49#4481C2211yescharacteristic
50#4481C2211yescharacteristic
51#3511—31no—
52#2991D311yescharacteristic, complemented
53#2990—31no—
54#1560—31no—
55#1559—32no—
56#1553—32no—
57#1549—31no—
58#1547—32no—
59#1546—31no—
60#1543—31no—
61#1542—32no—
62C2×C3×#41C2×C411yescharacteristic
63C2×#235—22no—
64C2×#235C2×C412yes—
65C22×Dic6—21no—
66C2×C3×M16—22no—
67C2×C3×M16—21no—
68C3×C4×Q8D411yescharacteristic
69C3×C4×Q8D411yescharacteristic
70C3×C4×D4D411yescharacteristic
71C3×C4×D4D412yes—
72C3×C4×D4D411yescharacteristic
73C3×C4×D4D412yes—
74C3×#117C2×C411yescharacteristic
75C3×#117C2×C412yes—
76C2×C3×C42C2312yes—
77C2×C3×C42C2311yescharacteristic
78#946—22no—
79#939—24no—
80#938—22no—
81#930—21no—
82#930—22no—
83#930—21no—
84#930C2×C414yes—
85#926—22no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3C2×C412yes—
88#924—22no—
89#924C2×C414yes—
90C2×#208—21no—
91C22×C3⋊C8—22no—
92#898—24no—
93#896—24no—
94#892—22no—
95D3×C4⋊C4—22no—
96#885—24no—
97#883—24no—
98#67⋊C4—24no—
99#875—22no—
100#875C2×C412yes—
101#874C2311yescharacteristic
102C4×D12—21no—
103C4×D12—21no—
104C4×D12—22no—
105C4×D12C2×C414yes—
106#870—22no—
107#869—21no—
108#869—21no—
109#869C2×C412yes—
110#869C2311yescharacteristic
111#869C2312yes—
112C4×Dic6—22no—
113C4×Dic6C2×C414yes—
114C3×C4⋊C8—22no—
115C3×C4⋊C8—21no—
116C3×C4⋊C8C2312yes—
117C3×C4⋊C8C2311yescharacteristic
118C3×C4⋊C8C2311yescharacteristic
119C3×C22⋊C8—22no—
120C3×C22⋊C8—21no—
121C3×C4×C8—22no—
122C3×C4×C8—21no—
123#830—21no—
124#830—22no—
125C4⋊(C3⋊C8)—22no—
126C4⋊(C3⋊C8)—21no—
127C4⋊(C3⋊C8)C2311yescharacteristic
128C4⋊(C3⋊C8)C2312yes—
129C4⋊(C3⋊C8)C2311yescharacteristic
130#803—22no—
131C4×C3⋊C8—21no—
132#556—61no—
133#555—61no—
134#554—61no—
135#553—62no—
136#548—62no—
137#547—61no—
138#542—34no—
139#537—32no—
140#533—32no—
141#532—32no—
142#532—31no—
143C2×C4⋊Q8—31no—
144#520—32no—
145C4×#41C2×D311yescharacteristic
146#446—61no—
147C8×Q8—21no—
148#436—21no—
149#436—22no—
150#435—62no—
151C8×D4—61no—
152#432—32no—
153#431—31no—
154#431C2×D312yes—
155#431C2×D311yescharacteristic
156#423—31no—
157#423C2×D311yescharacteristic
158#423C2×D312yes—
159C2×C4⋊C8—32no—
160#333—31no—
161#333—32no—
162#333—31no—
163#332—32no—
164#332—32no—
165#332—32no—
166#332—32no—
167#331—31no—
168#331—31no—
169#331—32no—
170C3×#41—42no—
171C2×C3×Q8C22⋊C411yescharacteristic
172C2×C3×D4C22⋊C411yescharacteristic
173C2×C3×D4C22⋊C412yes—
174C22×C3×C4C22×C411yescharacteristic
175C22×C3×C4C22×C412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C2×#67—22no—
181C2×#67—24no—
182C22×Dic3—22no—
183#235—44no—
184#235—82no—
185#235—82no—
186C2×D12—21no—
187C2×D12—21no—
188C2×D12C22×C412yes—
189C2×C4×D3—22no—
190C2×C4×D3—24no—
191C2×Dic6—22no—
192C2×Dic6—22no—
193C2×Dic6—81no—
194C2×Dic6—81no—
195C2×Dic6C22×C412yes—
196C3×M16—81no—
197C3×M16—82no—
198C2×C3×C8—21no—
199C2×C3×C8—22no—
200C2×C3×C8—21no—
201C3×C4⋊C4C22⋊C411yescharacteristic
202C3×C4⋊C4C22⋊C411yescharacteristic
203C3×C4⋊C4C22⋊C412yes—
204C3×C4⋊C4C22⋊C412yes—
205C3×C22⋊C4C22⋊C412yes—
206C3×C22⋊C4C22⋊C412yes—
207C3×C22⋊C4C22⋊C412yes—
208C3×C42C22×C412yes—
209C3×C42C22×C411yescharacteristic
210C3×C42C2×D411yescharacteristic
211C3×C42C2×D412yes—
212C3×C42C2×D412yes—
213C3×C42C2×D411yescharacteristic
214C3×C42C2411yescharacteristic
215#217—24no—
216(C2×D3)⋊C4—24no—
217(C2×D3)⋊C4—22no—
218(C2×D3)⋊C4—22no—
219(C2×D3)⋊C4—24no—
220C4⋊Dic3—22no—
221C4⋊Dic3—22no—
222C4⋊Dic3—21no—
223C4⋊Dic3—22no—
224C4⋊Dic3—21no—
225C4⋊Dic3C22×C414yes—
226Dic3⋊C4—24no—
227Dic3⋊C4—24no—
228C4×Dic3—24no—
229#208—81no—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—82no—
234C2×#41—32no—
235C2×#41—62no—
236C2×#41C4×D311yescharacteristic
237C22×Q8—61no—
238C2×M16—21no—
239C2×M16—22no—
240C2×M16—61no—
241C22×C8—62no—
242C4⋊Q8—32no—
243C4⋊Q8—32no—
244C4⋊Q8—31no—
245C4⋊Q8—61no—
246C4⋊Q8—61no—
247#127—31no—
248#126—64no—
249#125—62no—
250#124—32no—
251#124—64no—
252#124—62no—
253#123—64no—
254C22⋊Q8—34no—
255C22⋊Q8—62no—
256C22⋊Q8—62no—
257C22⋊Q8—64no—
258#121—34no—
259#121—62no—
260#121—64no—
261#121—61no—
262#121—62no—
263#121—61no—
264C4×Q8—34no—
265C4×Q8—62no—
266C4×Q8D1211yescharacteristic
267C4×Q8#6711yescharacteristic
268C4×D4—34no—
269C4×D4—64no—
270C4×D4—62no—
271C4×D4—61no—
272C4×D4—61no—
273C4×D4D1212yes—
274C4×D4D1211yescharacteristic
275C4×D4#6711yescharacteristic
276C4×D4#6712yes—
277#117—62no—
278#117—62no—
279#117C4×D311yescharacteristic
280#117C4×D312yes—
281C2×C4⋊C4—32no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—62no—
284C2×C42C22×D311yescharacteristic
285C2×C42C22×D312yes—
286C4⋊C8—21no—
287C4⋊C8—22no—
288C4⋊C8—31no—
289C4⋊C8—32no—
290C4⋊C8—31no—
291C4⋊C8—62no—
292C4⋊C8—61no—
293C4⋊C8C22×D311yescharacteristic
294C4⋊C8C22×D312yes—
295C4⋊C8C22×D311yescharacteristic
296C22⋊C8—21no—
297C22⋊C8—22no—
298C22⋊C8—61no—
299C22⋊C8—62no—
300#97—62no—
301C4×C8—21no—
302C4×C8—22no—
303C4×C8—61no—
304C23×C3C2×C22⋊C412yes—
305C23×C3C2×C22⋊C411yescharacteristic
306C22×D3—22no—
307C3×Q8—22no—
308C3×D4—22no—
309C3×D4—42no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—42no—
320C2×C3×C4C2×C22⋊C412yes—
321C2×C3×C4C2×C22⋊C412yes—
322C2×C3×C4C2×C22⋊C412yes—
323C2×C3×C4C2×C22⋊C411yescharacteristic
324C2×C3×C4C2×C22⋊C411yescharacteristic
325C2×C3×C4C2×C22⋊C411yescharacteristic
326C2×C3×C4C23×C411yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C22×D411yescharacteristic
329#67—44no—
330#67—84no—
331C2×Dic3—22no—
332C2×Dic3—24no—
333C2×Dic3—82no—
334D12—42no—
335D12—41no—
336D12—41no—
337D12—41no—
338D12—41no—
339C4×D3—44no—
340C4×D3—84no—
341Dic6—42no—
342Dic6—42no—
343Dic6—42no—
344Dic6—82no—
345C3×C8—42no—
346C3×C8—41no—
347C3×C8—41no—
348C3⋊C8—41no—
349C3⋊C8—42no—
350C3⋊C8—41no—
351#41—42no—
352#41—124no—
353#41—242no—
354#41—242no—
355C2×Q8—32no—
356C2×Q8—62no—
357C2×Q8—62no—
358C2×Q8—241no—
359C2×Q8—241no—
360C2×Q8(C2×D3)⋊C411yescharacteristic
361C2×D4—32no—
362C2×D4—62no—
363C2×D4—64no—
364C2×D4—61no—
365C2×D4—61no—
366C2×D4(C2×D3)⋊C411yescharacteristic
367C2×D4(C2×D3)⋊C412yes—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—64no—
371C22×C4C2×C4×D311yescharacteristic
372C22×C4C2×C4×D312yes—
373C22×C4C2×D1211yescharacteristic
374C22×C4C2×D1212yes—
375C22×C4C2×#6711yescharacteristic
376C22×C4C2×#6712yes—
377M16—81no—
378M16—82no—
379M16—241no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—61no—
384C2×C8—61no—
385C2×C8—62no—
386C2×C8—242no—
387C4⋊C4—34no—
388C4⋊C4—62no—
389C4⋊C4—64no—
390C4⋊C4—62no—
391C4⋊C4—64no—
392C4⋊C4—62no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4(C2×D3)⋊C411yescharacteristic
396C4⋊C4(C2×D3)⋊C411yescharacteristic
397C4⋊C4(C2×D3)⋊C412yes—
398C4⋊C4(C2×D3)⋊C412yes—
399C22⋊C4—64no—
400C22⋊C4—64no—
401C22⋊C4—64no—
402C22⋊C4—62no—
403C22⋊C4—62no—
404C22⋊C4(C2×D3)⋊C412yes—
405C22⋊C4(C2×D3)⋊C412yes—
406C22⋊C4(C2×D3)⋊C412yes—
407C42—64no—
408C42C2×C4×D312yes—
409C42C2×C4×D311yescharacteristic
410C42C2×D1211yescharacteristic
411C42C2×D1212yes—
412C42C2×#6711yescharacteristic
413C42C2×#6712yes—
414C42C23×D311yescharacteristic
415C22×C3—21no—
416C22×C3—22no—
417C22×C3—21no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—21no—
421C22×C3C22×C22⋊C411yescharacteristic
422C2×D3—42no—
423C2×D3—42no—
424C2×D3—42no—
425C3×C4—21no—
426C3×C4—22no—
427C3×C4—21no—
428C3×C4—22no—
429C3×C4—22no—
430C3×C4—21no—
431C3×C4#41312yes—
432C3×C4#57611yescharacteristic
433C3×C4#57711yescharacteristic
434C3×C4#57711yescharacteristic
435C3×C4#57811yescharacteristic
436Dic3—42no—
437Dic3—44no—
438C23—62no—
439C23C2×(C2×D3)⋊C412yes—
440C23C2×(C2×D3)⋊C411yescharacteristic
441Q8—22no—
442Q8—122no—
443Q8—122no—
444Q8—122no—
445Q8—242no—
446D4—22no—
447D4—42no—
448D4—122no—
449D4—124no—
450D4—121no—
451D4—121no—
452D4—121no—
453D4—121no—
454D4—244no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—42no—
465C2×C4—62no—
466C2×C4—64no—
467C2×C4—124no—
468C2×C4—244no—
469C2×C4—242no—
470C2×C4C2×(C2×D3)⋊C412yes—
471C2×C4C2×(C2×D3)⋊C411yescharacteristic
472C2×C4C2×(C2×D3)⋊C412yes—
473C2×C4C2×(C2×D3)⋊C411yescharacteristic
474C2×C4C2×(C2×D3)⋊C412yes—
475C2×C4C2×(C2×D3)⋊C411yescharacteristic
476C2×C4C22×C4×D311yescharacteristic
477C2×C4C22×D1211yescharacteristic
478C2×C4C22×#6711yescharacteristic
479C8—42no—
480C8—41no—
481C8—41no—
482C8—121no—
483C8—122no—
484C8—121no—
485C2×C3—41no—
486C2×C3—42no—
487C2×C3C2×#41311yescharacteristic
488C2×C3#289711yescharacteristic
489C2×C3#289811yescharacteristic
490D3—82no—
491C22—21no—
492C22—22no—
493C22—22no—
494C22—21no—
495C22—21no—
496C22—22no—
497C22—122no—
498C22—122no—
499C22—122no—
500C22C22×(C2×D3)⋊C411yescharacteristic
501C4—21no—
502C4—22no—
503C4—22no—
504C4—21no—
505C4—22no—
506C4—21no—
507C4—122no—
508C4—124no—
509C4#513212yes—
510C4#575111yescharacteristic
511C4#575311yescharacteristic
512C4#583511yescharacteristic
513C4#583711yescharacteristic
514C3#1778111yescharacteristic, complemented, Sylow
515C2—41no—
516C2—42no—
517C2—242no—
518C2#8256311yescharacteristic
519C2C2×#513211yescharacteristic
520C2#8380611yescharacteristic
subgroupG/H
centerC22C22×(C2×D3)⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#2991C2
socleC22×C3C22×C22⋊C4
prime radical#85850C2
cosocleC3×C42C24
hypercenter#2991D3
hypocenterC3#17781
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2991

Complements

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

NH ≅ G/Ntype
2. C3×#2991C2semidirect
3. #82522C2semidirect
4. #70780C2semidirect
5. #70776C2semidirect
6. #70775C2semidirect
7. #70773C2semidirect
8. #70510C2semidirect
9. #70506C2semidirect
10. #70505C2semidirect
11. #70503C2semidirect
52. #2991D3semidirect
514. C3#17781semidirect

Maximal subgroups

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

subgroupindex
#858502
C3×#29912
#825222
#707802
#707762
#707752
#707732
#705102
#705062
#705052
#705032
#177813