Finite Group Databasebeta

← #10849409

Subgroups — #10849409

Order 768.

1020 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85850C211yesmaximal, characteristic
2#82508C211yesmaximal, characteristic, complemented
3#78528C211yesmaximal, characteristic, complemented
4#73308C211yesmaximal, characteristic, complemented
5C3×C4×#104C211yesmaximal, characteristic, complemented
6#70362C214yesmaximal, complemented
7#69879C211yesmaximal, characteristic, complemented
8#69395C211yesmaximal, characteristic, complemented
9#68772C214yesmaximal, complemented
10#25667—31nomaximal, Sylow
11C3×C4×#41C2211yescharacteristic
12#5833C2212yes—
13#5584—21no—
14#5579—21no—
15#5565—22no—
16#5563—21no—
17#5552—22no—
18#5551—21no—
19#5534—22no—
20#5523C2212yes—
21#5477—21no—
22#5477C2211yescharacteristic
23#5470—21no—
24C2×#869C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C3×#104C2212yes—
27C3×C4×M16C2211yescharacteristic
28C3×C43C2211yescharacteristic
29#5232C2212yes—
30#5133C2214yes—
31#5021—21no—
32#5020—21no—
33#5008—21no—
34#5007—21no—
35#4995C2212yes—
36#4936C2214yes—
37#4929—22no—
38#4929—22no—
39C2×#805C2214yes—
40#4924C2211yescharacteristic
41#4714—22no—
42#4694—21no—
43#4681—21no—
44C3×#339C2212yes—
45#4554C2214yes—
46#3511—31no—
47#3141—31no—
48#3079—31no—
49#2862—31no—
50#2040—34no—
51#1997—31no—
52#1962—31no—
53#1860—34no—
54C4×#104D311yescharacteristic, complemented
55C2×C3×#41C2311yescharacteristic
56C2×#235—22no—
57C2×#235C2312yes—
58C22×Dic6—21no—
59C22×Dic6D412yes—
60C2×C3×M16C2312yes—
61C3×C4×Q8D411yescharacteristic
62C3×C4×Q8D411yescharacteristic
63C3×C4×D4—21no—
64C3×C4×D4—21no—
65C3×C4×D4D411yescharacteristic
66C3×C4×D4D411yescharacteristic
67C3×#117—21no—
68C3×#117C2311yescharacteristic
69C2×C3×C42—21no—
70C2×C3×C42—21no—
71C2×C3×C42—21no—
72C2×C3×C42C2312yes—
73C2×C3×C42C2311yescharacteristic
74#946—22no—
75#939—24no—
76#938—22no—
77#930—22no—
78#930—21no—
79#930—21no—
80#930—22no—
81#926—22no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3D412yes—
84C2×C4⋊Dic3C2312yes—
85C2×C4⋊Dic3C2312yes—
86#924—22no—
87#924—22no—
88C2×#208C2312yes—
89#898—24no—
90#896—24no—
91#892—22no—
92D3×C4⋊C4—22no—
93#885—24no—
94#883—24no—
95#67⋊C4—24no—
96#875—22no—
97#875D412yes—
98#874D411yescharacteristic
99C4×D12—22no—
100C4×D12—22no—
101C4×D12—21no—
102C4×D12—21no—
103#870—22no—
104#869—21no—
105#869—21no—
106#869—21no—
107#869—21no—
108#869—21no—
109#869—21no—
110#869—21no—
111#869—21no—
112#869D411yescharacteristic
113C4×Dic6—22no—
114C4×Dic6—22no—
115C3×#104—24no—
116C3×#97—21no—
117C3×C4×C8—21no—
118#835—22no—
119#832—22no—
120C3⋊#95—28no—
121#821—24no—
122#816—24no—
123#808—22no—
124#807—22no—
125#805—28no—
126C4⋊(C3⋊C8)—21no—
127C4⋊(C3⋊C8)—21no—
128#556—61no—
129#555—61no—
130#554—61no—
131#553—62no—
132#548—62no—
133#547—61no—
134#542—62no—
135#537—32no—
136#533—61no—
137#532—31no—
138#532—61no—
139C2×C4⋊Q8—31no—
140C2×C4⋊Q8—31no—
141#520—32no—
142C4×#41C2×D311yescharacteristic
143#489—62no—
144#487—61no—
145#486—61no—
146#477—61no—
147#476—61no—
148#475—61no—
149#474—61no—
150#428—32no—
151#423—31no—
152C2×#104—34no—
153C2×#104C2×D312yes—
154#417—32no—
155#417—34no—
156C4×M16C2×D311yescharacteristic
157#391—62no—
158#391—62no—
159#382—34no—
160C43C2×D311yescharacteristic
161#343—34no—
162#339C2×D312yes—
163C3×#41—22no—
164C3×#41—22no—
165C2×C3×Q8C2×D411yescharacteristic
166C2×C3×D4—21no—
167C2×C3×D4C2×D411yescharacteristic
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C2×#67—22no—
176C2×#67—24no—
177C22×Dic3—22no—
178C22×Dic3C2×D414yes—
179#235—24no—
180#235—24no—
181#235—84no—
182C2×D12—21no—
183C2×D12—21no—
184C2×D12C2×D412yes—
185C2×C4×D3—22no—
186C2×C4×D3—24no—
187C2×Dic6—22no—
188C2×Dic6—22no—
189C2×Dic6—44no—
190C2×Dic6—44no—
191C2×Dic6—41no—
192C2×Dic6—41no—
193C2×Dic6C2×D412yes—
194C3×M16—22no—
195C3×M16—22no—
196C2×C3×C8—22no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4C2×D411yescharacteristic
200C3×C4⋊C4C2×D411yescharacteristic
201C3×C22⋊C4—21no—
202C3×C22⋊C4—21no—
203C3×C22⋊C4—21no—
204C3×C42—21no—
205C3×C42—21no—
206C3×C42—21no—
207C3×C42—22no—
208C3×C42—42no—
209C3×C42—42no—
210C3×C42C2×D411yescharacteristic
211C3×C42C2×D411yescharacteristic
212C3×C42C2×D411yescharacteristic
213C3×C42C2×D411yescharacteristic
214C3×C42#4114yescomplemented
215#217—24no—
216(C2×D3)⋊C4—24no—
217(C2×D3)⋊C4—24no—
218(C2×D3)⋊C4—22no—
219(C2×D3)⋊C4—22no—
220C4⋊Dic3—21no—
221C4⋊Dic3—22no—
222C4⋊Dic3—22no—
223C4⋊Dic3—21no—
224C4⋊Dic3—22no—
225C4⋊Dic3—22no—
226C4⋊Dic3—22no—
227C4⋊Dic3—22no—
228C4⋊Dic3—41no—
229C4⋊Dic3—41no—
230Dic3⋊C4—24no—
231Dic3⋊C4—24no—
232C4×Dic3—24no—
233#208—24no—
234C2×C3⋊C8—22no—
235C2×#41—32no—
236C2×#41—62no—
237C2×#41C22×D311yescharacteristic
238C22×Q8—32no—
239C22×Q8—61no—
240C2×M16—32no—
241C2×M16C22×D312yes—
242C4⋊Q8—31no—
243C4⋊Q8—61no—
244C4⋊Q8—61no—
245C4⋊Q8—61no—
246C4⋊Q8—61no—
247C4⋊Q8—61no—
248C4⋊Q8—61no—
249C4⋊Q8—61no—
250C4⋊Q8—61no—
251#127—31no—
252#126—64no—
253#125—62no—
254#124—32no—
255#124—62no—
256#124—64no—
257#123—64no—
258C22⋊Q8—64no—
259C22⋊Q8—62no—
260C22⋊Q8—62no—
261C22⋊Q8—62no—
262#121—62no—
263#121—62no—
264#121—64no—
265#121—62no—
266#121—61no—
267#121—61no—
268C4×Q8—62no—
269C4×Q8—62no—
270C4×Q8D1211yescharacteristic
271C4×Q8D1211yescharacteristic
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—62no—
275C4×D4—62no—
276C4×D4—64no—
277C4×D4—61no—
278C4×D4—61no—
279C4×D4D1211yescharacteristic
280C4×D4D1211yescharacteristic
281#117—21no—
282#117—62no—
283#117—62no—
284#117C22×D311yescharacteristic
285C2×C4⋊C4—32no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4—32no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4—61no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42C22×D312yes—
294C2×C42C22×D311yescharacteristic
295#107—62no—
296#106—62no—
297C4⋊C8—61no—
298C4⋊C8—61no—
299#104—24no—
300#104—68no—
301#103—64no—
302#103—62no—
303#102—64no—
304#102—62no—
305#97—21no—
306C4×C8—21no—
307#95—68no—
308C23×C3—21no—
309C23×C3C22×D411yescharacteristic
310C22×D3—22no—
311C3×Q8—21no—
312C3×Q8—21no—
313C3×D4—21no—
314C3×D4—21no—
315C3×D4—42no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—24no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4#12011yescharacteristic
328C2×C3×C4#12011yescharacteristic
329C2×C3×C4#12011yescharacteristic
330C2×C3×C4#12011yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C2×#4112yes—
334C2×C3×C42−1+412yes—
335#67—44no—
336#67—84no—
337C2×Dic3—22no—
338C2×Dic3—24no—
339C2×Dic3—44no—
340C2×Dic3—44no—
341C2×Dic3—42no—
342D12—24no—
343D12—42no—
344D12—42no—
345C4×D3—42no—
346C4×D3—42no—
347C4×D3—84no—
348Dic6—24no—
349Dic6—44no—
350Dic6—82no—
351Dic6—82no—
352Dic6—82no—
353C3×C8—42no—
354C3⋊C8—42no—
355#41—22no—
356#41—22no—
357#41—64no—
358#41—64no—
359#41—244no—
360C2×Q8—32no—
361C2×Q8—62no—
362C2×Q8—62no—
363C2×Q8—124no—
364C2×Q8—124no—
365C2×Q8—121no—
366C2×Q8—121no—
367C2×Q8C2×D1211yescharacteristic
368C2×D4—21no—
369C2×D4—32no—
370C2×D4—62no—
371C2×D4—64no—
372C2×D4—61no—
373C2×D4—61no—
374C2×D4C2×D1211yescharacteristic
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—34no—
380C22×C4—64no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4D3×D411yescharacteristic
384C22×C4D3×D411yescharacteristic
385C22×C4C23×D311yescharacteristic
386M16—22no—
387M16—22no—
388M16—64no—
389C2×C8—22no—
390C2×C8—62no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—64no—
394C4⋊C4—64no—
395C4⋊C4—62no—
396C4⋊C4—62no—
397C4⋊C4—61no—
398C4⋊C4—61no—
399C4⋊C4—62no—
400C4⋊C4—62no—
401C4⋊C4—62no—
402C4⋊C4—62no—
403C4⋊C4—121no—
404C4⋊C4—121no—
405C4⋊C4C2×D1211yescharacteristic
406C4⋊C4C2×D1211yescharacteristic
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4—64no—
411C22⋊C4—64no—
412C22⋊C4—64no—
413C22⋊C4—62no—
414C22⋊C4—62no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—22no—
419C42—42no—
420C42—42no—
421C42—64no—
422C42C2×D1211yescharacteristic
423C42C2×D1211yescharacteristic
424C42#23514yescomplemented
425C42D3×D411yescharacteristic
426C42D3×D411yescharacteristic
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3C2×#12011yescharacteristic
433C22×C3#54812yes—
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—42no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—44no—
440C3×C4—41no—
441C3×C4—41no—
442C3×C4—41no—
443C3×C4#45412yes—
444C3×C4#45512yes—
445C3×C4#53211yescharacteristic
446C3×C4#53611yescharacteristic
447C3×C4#54712yes—
448Dic3—42no—
449Dic3—44no—
450Dic3—82no—
451C23—21no—
452C23—62no—
453C23C2×D3×D411yescharacteristic
454Q8—21no—
455Q8—21no—
456Q8—64no—
457Q8—124no—
458Q8—242no—
459Q8—242no—
460Q8—242no—
461D4—21no—
462D4—21no—
463D4—42no—
464D4—64no—
465D4—124no—
466D4—122no—
467D4—122no—
468D4—244no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—24no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—62no—
481C2×C4—64no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—124no—
485C2×C4—124no—
486C2×C4—122no—
487C2×C4—244no—
488C2×C4#88211yescharacteristic
489C2×C4#88211yescharacteristic
490C2×C4#88211yescharacteristic
491C2×C4#88211yescharacteristic
492C2×C4C22×D1211yescharacteristic
493C2×C4C2×#23512yes—
494C2×C4C2×D3×D411yescharacteristic
495C2×C4#101012yes—
496C8—42no—
497C8—122no—
498C2×C3—22no—
499C2×C3—81no—
500C2×C3#243211yescharacteristic
501C2×C3C2×#45511yescharacteristic
502C2×C3C2×#45411yescharacteristic
503D3—82no—
504C22—21no—
505C22—21no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—122no—
510C22—122no—
511C22—122no—
512C22C2×#88211yescharacteristic
513C22#555212yes—
514C4—21no—
515C4—21no—
516C4—44no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—122no—
521C4—124no—
522C4—242no—
523C4#482512yes—
524C4#482712yes—
525C4#547711yescharacteristic
526C4#552611yescharacteristic
527C4#555112yes—
528C3#2566711yescharacteristic, complemented, Sylow
529C2—22no—
530C2—81no—
531C2—242no—
532C2#7348911yescharacteristic
533C2C2×#482511yescharacteristic
534C2C2×#482711yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#104C2
socleC22×C3C2×#120
prime radical#85850C2
hypercenterC4×#104D3
hypocenterC3#25667
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#104

Complements

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

NH ≅ G/Ntype
2. #82508C2semidirect
3. #78528C2semidirect
4. #73308C2semidirect
5. C3×C4×#104C2semidirect
6. #70362C2semidirect
7. #69879C2semidirect
8. #69395C2semidirect
9. #68772C2semidirect
54. C4×#104D3semidirect
214. C3×C42#41semidirect
424. C42#235semidirect
528. C3#25667semidirect

Maximal subgroups

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

subgroupindex
#858502
#825082
#785282
#733082
C3×C4×#1042
#703622
#698792
#693952
#687722
#256673