Finite Group Databasebeta

← #10764097

Subgroups — #10764097

Order 768.

988 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85848C211yesmaximal, characteristic
2C3×#2990C211yesmaximal, characteristic, complemented
3#82523C211yesmaximal, characteristic, complemented
4#70819C212yesmaximal, complemented
5#70817C212yesmaximal, complemented
6#70777C211yesmaximal, characteristic, complemented
7#70774C211yesmaximal, characteristic, complemented
8#70511C211yesmaximal, characteristic, complemented
9#70502C211yesmaximal, characteristic, complemented
10#70432C212yesmaximal, complemented
11#70431C212yesmaximal, complemented
12#17776—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14#5830C412yes—
15#5585—21no—
16#5585—21no—
17#5553—21no—
18#5553—21no—
19#5550—22no—
20#5550—22no—
21#5535C414yes—
22#5521C412yes—
23#5477C2212yes—
24#5477C2212yes—
25C2×#874C2211yescharacteristic
26#5470C2211yescharacteristic
27C3×#446—21no—
28C3×#435—22no—
29C3×C8×D4—21no—
30C3×#432C2212yescomplemented
31C3×#431C2211yescharacteristic
32C3×#423C2211yescharacteristic
33C2×C3×C4⋊C8C2212yescomplemented
34Q8×C3⋊C8—21no—
35#5012—21no—
36#5012—22no—
37#4925C2211yescharacteristic
38#4925C2212yescomplemented
39#4924C2211yescharacteristic
40#4924C2212yescomplemented
41#4486C2211yescharacteristic, complemented
42#4486C2212yescomplemented
43#4486C2211yescharacteristic, complemented
44#4485C2212yes—
45#4485C2212yes—
46#4485C2212yes—
47#4485C2212yes—
48#4482C2211yescharacteristic, complemented
49#4482C2212yescomplemented
50#4482C2211yescharacteristic, complemented
51#3508—31no—
52#2991—31no—
53#2990D311yescharacteristic, complemented
54#1561—31no—
55#1557—32no—
56#1553—32no—
57#1547—32no—
58#1545—31no—
59#1545—31no—
60#1544—31no—
61#1540—32no—
62C2×C3×#41C2×C411yescharacteristic
63C2×#235—22no—
64C2×#235—22no—
65C22×D12—21no—
66C22×D12—21no—
67C22×D12C2×C412yes—
68C2×C3×M16—21no—
69C22×C3×C8—22no—
70C3×C4×Q8D411yescharacteristic
71C3×C4×Q8D411yescharacteristic
72C3×C4×D4D412yes—
73C3×C4×D4D411yescharacteristic
74C3×C4×D4D412yes—
75C3×C4×D4D411yescharacteristic
76C3×#117C2×C411yescharacteristic
77C3×#117C2×C412yes—
78C2×C3×C42C2311yescharacteristic
79C2×C3×C42C2312yes—
80#947—22no—
81#940—22no—
82#937—24no—
83#930—22no—
84#930—22no—
85#930C2×C414yes—
86#930C2×C414yes—
87#926C2×C412yes—
88#924—21no—
89#924—21no—
90C2×#208—21no—
91C2×#208—22no—
92#894—24no—
93#894—24no—
94#893—22no—
95#893—22no—
96#884—24no—
97#884—24no—
98D3×C22⋊C4—24no—
99#875—22no—
100#875—22no—
101#874—21no—
102#874—21no—
103#874C2×C412yes—
104#874C2312yes—
105C4×D12—22no—
106C4×D12—22no—
107C4×D12C2×C414yes—
108C4×D12C2×C414yes—
109#870C2×C412yes—
110#869C2311yescharacteristic
111#869C2311yescharacteristic
112C4×Dic6—21no—
113C4×Dic6—21no—
114C3×C4⋊C8—22no—
115C3×C4⋊C8—21no—
116C3×C4⋊C8C2311yescharacteristic
117C3×C4⋊C8C2312yes—
118C3×C4⋊C8C2311yescharacteristic
119C3×C22⋊C8—21no—
120C3×C22⋊C8—22no—
121C3×#97—22no—
122C3×C4×C8—21no—
123#830—21no—
124#830—22no—
125C4⋊(C3⋊C8)—21no—
126C4⋊(C3⋊C8)—22no—
127C4⋊(C3⋊C8)C2312yes—
128C4⋊(C3⋊C8)C2311yescharacteristic
129C4⋊(C3⋊C8)C2311yescharacteristic
130C4×C3⋊C8—21no—
131C4×C3⋊C8—22no—
132#559—61no—
133#559—61no—
134#550—61no—
135#550—61no—
136#546—62no—
137#546—62no—
138#540—34no—
139#535—32no—
140#533—31no—
141#532—32no—
142#532—32no—
143C2×#127—31no—
144#518—32no—
145C4×#41C2×D311yescharacteristic
146#446—21no—
147C8×Q8—61no—
148#436—62no—
149#436—61no—
150#435—22no—
151C8×D4—21no—
152#432C2×D312yescomplemented
153#431—32no—
154#431—31no—
155#431C2×D311yescharacteristic
156#423—31no—
157#423—32no—
158#423C2×D311yescharacteristic
159C2×C4⋊C8C2×D312yescomplemented
160#332—31no—
161#332—32no—
162#332—31no—
163#332—31no—
164#332—31no—
165#332—32no—
166#331—32no—
167#331—32no—
168#331—32no—
169#331—32no—
170C23×D3—22no—
171C3×#41—42no—
172C2×C3×Q8C22⋊C411yescharacteristic
173C2×C3×D4C22⋊C411yescharacteristic
174C2×C3×D4C22⋊C412yes—
175C22×C3×C4C22×C411yescharacteristic
176C22×C3×C4C22×C412yes—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C2×#67—24no—
182C2×#67—24no—
183#235—82no—
184#235—82no—
185#235—82no—
186#235—82no—
187C2×D12—22no—
188C2×D12—22no—
189C2×D12—22no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—44no—
193C2×D12—81no—
194C2×D12—81no—
195C2×D12—81no—
196C2×D12—81no—
197C2×D12C22×C414yes—
198C2×C4×D3—24no—
199C2×C4×D3—24no—
200C2×Dic6—21no—
201C2×Dic6—21no—
202C3×M16—81no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—22no—
206C2×C3×C8—82no—
207C3×C4⋊C4C22⋊C411yescharacteristic
208C3×C4⋊C4C22⋊C411yescharacteristic
209C3×C4⋊C4C22⋊C412yes—
210C3×C4⋊C4C22⋊C412yes—
211C3×C22⋊C4C22⋊C412yes—
212C3×C22⋊C4C22⋊C412yes—
213C3×C22⋊C4C22⋊C412yes—
214C3×C42C22×C411yescharacteristic
215C3×C42C22×C412yes—
216C3×C42C2×D411yescharacteristic
217C3×C42C2×D412yes—
218C3×C42C2×D411yescharacteristic
219C3×C42C2×D412yes—
220C3×C42C2411yescharacteristic
221#217—22no—
222(C2×D3)⋊C4—24no—
223(C2×D3)⋊C4—24no—
224(C2×D3)⋊C4—24no—
225(C2×D3)⋊C4—24no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3C22×C412yes—
229C4⋊Dic3C22×C412yes—
230Dic3⋊C4—22no—
231Dic3⋊C4—22no—
232C4×Dic3—22no—
233#208—81no—
234#208—82no—
235C2×C3⋊C8—21no—
236C2×C3⋊C8—21no—
237C2×C3⋊C8—22no—
238C2×#41—62no—
239C2×#41—62no—
240C2×#41C4×D311yescharacteristic
241C22×D4—32no—
242C22×D4—61no—
243C22×D4—61no—
244C2×M16—21no—
245C2×M16—62no—
246C2×M16—61no—
247C22×C8—22no—
248C4⋊Q8—31no—
249C4⋊Q8—31no—
250#127—32no—
251#127—32no—
252#127—62no—
253#127—61no—
254#127—61no—
255#125—32no—
256#124—62no—
257#124—62no—
258#124—62no—
259#123—64no—
260#123—64no—
261C22⋊Q8—61no—
262C22⋊Q8—61no—
263#121—34no—
264#121—34no—
265#121—62no—
266#121—62no—
267#121—64no—
268#121—64no—
269#120—64no—
270C4×Q8—61no—
271C4×Q8—61no—
272C4×Q8D1211yescharacteristic
273C4×Q8#6711yescharacteristic
274C4×D4—34no—
275C4×D4—34no—
276C4×D4—62no—
277C4×D4—62no—
278C4×D4—62no—
279C4×D4—62no—
280C4×D4D1212yes—
281C4×D4D1211yescharacteristic
282C4×D4#6712yes—
283C4×D4#6711yescharacteristic
284#117—32no—
285#117C4×D311yescharacteristic
286#117C4×D312yes—
287C2×C22⋊C4—64no—
288C2×C42C22×D311yescharacteristic
289C2×C42C22×D312yes—
290C4⋊C8—22no—
291C4⋊C8—21no—
292C4⋊C8—32no—
293C4⋊C8—31no—
294C4⋊C8—31no—
295C4⋊C8—62no—
296C4⋊C8—61no—
297C4⋊C8C22×D311yescharacteristic
298C4⋊C8C22×D312yes—
299C4⋊C8C22×D311yescharacteristic
300C22⋊C8—21no—
301C22⋊C8—22no—
302C22⋊C8—62no—
303C22⋊C8—61no—
304#97—22no—
305C4×C8—21no—
306C4×C8—62no—
307C4×C8—61no—
308C23×C3C2×C22⋊C411yescharacteristic
309C23×C3C2×C22⋊C412yes—
310C22×D3—24no—
311C22×D3—44no—
312C22×D3—82no—
313C22×D3—82no—
314C3×Q8—22no—
315C3×D4—22no—
316C3×D4—42no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—42no—
327C2×C3×C4C2×C22⋊C411yescharacteristic
328C2×C3×C4C2×C22⋊C411yescharacteristic
329C2×C3×C4C2×C22⋊C412yes—
330C2×C3×C4C2×C22⋊C412yes—
331C2×C3×C4C2×C22⋊C412yes—
332C2×C3×C4C2×C22⋊C411yescharacteristic
333C2×C3×C4C23×C411yescharacteristic
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336#67—84no—
337#67—84no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340D12—44no—
341D12—42no—
342D12—42no—
343D12—42no—
344D12—42no—
345D12—44no—
346D12—82no—
347D12—82no—
348C4×D3—84no—
349C4×D3—84no—
350Dic6—41no—
351Dic6—41no—
352Dic6—41no—
353Dic6—41no—
354C3×C8—41no—
355C3×C8—42no—
356C3×C8—41no—
357C3⋊C8—41no—
358C3⋊C8—42no—
359C3⋊C8—41no—
360C24—62no—
361#41—42no—
362#41—242no—
363#41—242no—
364#41—242no—
365#41—242no—
366C2×Q8—61no—
367C2×Q8—61no—
368C2×Q8(C2×D3)⋊C411yescharacteristic
369C2×D4—34no—
370C2×D4—64no—
371C2×D4—64no—
372C2×D4—62no—
373C2×D4—62no—
374C2×D4—62no—
375C2×D4—62no—
376C2×D4—62no—
377C2×D4—124no—
378C2×D4—241no—
379C2×D4—241no—
380C2×D4—241no—
381C2×D4—241no—
382C2×D4(C2×D3)⋊C411yescharacteristic
383C2×D4(C2×D3)⋊C412yes—
384C22×C4—64no—
385C22×C4—64no—
386C22×C4C2×C4×D311yescharacteristic
387C22×C4C2×C4×D312yes—
388C22×C4C2×D1212yes—
389C22×C4C2×D1211yescharacteristic
390C22×C4C2×#6712yes—
391C22×C4C2×#6711yescharacteristic
392M16—81no—
393M16—241no—
394M16—242no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—22no—
398C2×C8—61no—
399C2×C8—62no—
400C2×C8—61no—
401C2×C8—82no—
402C4⋊C4—32no—
403C4⋊C4—32no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—61no—
407C4⋊C4—61no—
408C4⋊C4(C2×D3)⋊C411yescharacteristic
409C4⋊C4(C2×D3)⋊C411yescharacteristic
410C4⋊C4(C2×D3)⋊C412yes—
411C4⋊C4(C2×D3)⋊C412yes—
412C22⋊C4—62no—
413C22⋊C4—64no—
414C22⋊C4—64no—
415C22⋊C4—64no—
416C22⋊C4—64no—
417C22⋊C4(C2×D3)⋊C412yes—
418C22⋊C4(C2×D3)⋊C412yes—
419C22⋊C4(C2×D3)⋊C412yes—
420C42—62no—
421C42C2×C4×D312yes—
422C42C2×C4×D311yescharacteristic
423C42C2×D1212yes—
424C42C2×D1211yescharacteristic
425C42C2×#6711yescharacteristic
426C42C2×#6712yes—
427C42C23×D311yescharacteristic
428C22×C3—21no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3C22×C22⋊C411yescharacteristic
435C2×D3—44no—
436C2×D3—44no—
437C2×D3—44no—
438C2×D3—164no—
439C3×C4—21no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—22no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4#41312yes—
446C3×C4#57611yescharacteristic
447C3×C4#57611yescharacteristic
448C3×C4#57711yescharacteristic
449C3×C4#57711yescharacteristic
450Dic3—42no—
451Dic3—42no—
452C23—64no—
453C23—124no—
454C23—242no—
455C23—242no—
456C23C2×(C2×D3)⋊C411yescharacteristic
457C23C2×(C2×D3)⋊C412yes—
458Q8—22no—
459Q8—121no—
460Q8—121no—
461Q8—121no—
462Q8—121no—
463D4—22no—
464D4—42no—
465D4—124no—
466D4—124no—
467D4—122no—
468D4—122no—
469D4—122no—
470D4—122no—
471D4—242no—
472D4—242no—
473D4—244no—
474D4—244no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—42no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—244no—
488C2×C4—244no—
489C2×C4C2×(C2×D3)⋊C412yes—
490C2×C4C2×(C2×D3)⋊C411yescharacteristic
491C2×C4C2×(C2×D3)⋊C412yes—
492C2×C4C2×(C2×D3)⋊C411yescharacteristic
493C2×C4C2×(C2×D3)⋊C412yes—
494C2×C4C2×(C2×D3)⋊C411yescharacteristic
495C2×C4C22×C4×D311yescharacteristic
496C2×C4C22×D1211yescharacteristic
497C2×C4C22×#6711yescharacteristic
498C8—41no—
499C8—42no—
500C8—41no—
501C8—121no—
502C8—122no—
503C8—121no—
504C2×C3—41no—
505C2×C3—42no—
506C2×C3C2×#41311yescharacteristic
507C2×C3#289711yescharacteristic
508C2×C3#289711yescharacteristic
509D3—84no—
510C22—21no—
511C22—22no—
512C22—22no—
513C22—21no—
514C22—22no—
515C22—21no—
516C22—124no—
517C22—124no—
518C22—124no—
519C22—484no—
520C22C22×(C2×D3)⋊C411yescharacteristic
521C4—21no—
522C4—22no—
523C4—22no—
524C4—21no—
525C4—22no—
526C4—21no—
527C4—122no—
528C4—122no—
529C4#513212yes—
530C4#575111yescharacteristic
531C4#575211yescharacteristic
532C4#583511yescharacteristic
533C4#583711yescharacteristic
534C3#1777611yescharacteristic, complemented, Sylow
535C2—41no—
536C2—42no—
537C2—244no—
538C2#8256311yescharacteristic
539C2C2×#513211yescharacteristic
540C2#8379711yescharacteristic
subgroupG/H
centerC22C22×(C2×D3)⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#2990C2
socleC22×C3C22×C22⋊C4
prime radical#85848C2
cosocleC3×C42C24
hypercenter#2990D3
hypocenterC3#17776
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2990

Complements

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

NH ≅ G/Ntype
2. C3×#2990C2semidirect
3. #82523C2semidirect
4. #70819C2semidirect
5. #70817C2semidirect
6. #70777C2semidirect
7. #70774C2semidirect
8. #70511C2semidirect
9. #70502C2semidirect
10. #70432C2semidirect
11. #70431C2semidirect
30. C3×#432C22semidirect
33. C2×C3×C4⋊C8C22semidirect
38. #4925C22semidirect
40. #4924C22semidirect
41. #4486C22semidirect
42. #4486C22semidirect
43. #4486C22semidirect
48. #4482C22semidirect
49. #4482C22semidirect
50. #4482C22semidirect
53. #2990D3semidirect
152. #432C2×D3semidirect
159. C2×C4⋊C8C2×D3semidirect
534. C3#17776semidirect

Maximal subgroups

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

subgroupindex
#858482
C3×#29902
#825232
#708192
#708172
#707772
#707742
#705112
#705022
#704322
#704312
#177763