Finite Group Databasebeta

← #10849400

Subgroups — #10849400

Order 768.

1092 conjugacy classes of subgroups.

The subgroup lattice has 540 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#73311C211yesmaximal, characteristic, complemented
5C3×C4×#104C211yesmaximal, characteristic, complemented
6#70361C214yesmaximal, complemented
7#69879C211yesmaximal, characteristic, complemented
8#69395C211yesmaximal, characteristic, complemented
9#68771C214yesmaximal, complemented
10#25666—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×#869—22no—
25C2×#869C2211yescharacteristic
26C2×#869C2211yescharacteristic
27C2×C3×#104C2212yes—
28C3×C4×M16C2211yescharacteristic
29C3×C43C2211yescharacteristic
30#5232C2212yes—
31#5133C2214yes—
32#5021—21no—
33#5020—21no—
34#5008—21no—
35#5007—21no—
36#4995C2212yes—
37#4935—22no—
38#4935C2214yes—
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#2868—31no—
50#2039—34no—
51#1997—31no—
52#1962—31no—
53#1859—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⋊Dic3—28no—
84C2×C4⋊Dic3D412yes—
85C2×C4⋊Dic3C2312yes—
86C2×C4⋊Dic3C2312yes—
87#924—22no—
88#924—22no—
89C2×#208C2312yes—
90#898—24no—
91#896—24no—
92#892—22no—
93D3×C4⋊C4—22no—
94#885—24no—
95#883—24no—
96#67⋊C4—24no—
97#875—22no—
98#875D412yes—
99#874D411yescharacteristic
100C4×D12—22no—
101C4×D12—22no—
102C4×D12—21no—
103C4×D12—21no—
104#870—22no—
105#869—21no—
106#869—21no—
107#869—21no—
108#869—21no—
109#869—21no—
110#869—21no—
111#869—21no—
112#869—21no—
113#869—48no—
114#869D411yescharacteristic
115C4×Dic6—22no—
116C4×Dic6—22no—
117C3×#104—24no—
118C3×#97—21no—
119C3×C4×C8—21no—
120#835—22no—
121#832—22no—
122#821—24no—
123#816—24no—
124#808—22no—
125#807—22no—
126#805—28no—
127C4⋊(C3⋊C8)—21no—
128C4⋊(C3⋊C8)—21no—
129#556—61no—
130#555—61no—
131#554—61no—
132#553—62no—
133#548—62no—
134#547—61no—
135#542—62no—
136#537—32no—
137#533—61no—
138#532—31no—
139#532—61no—
140C2×C4⋊Q8—31no—
141C2×C4⋊Q8—31no—
142C2×C4⋊Q8—62no—
143#520—32no—
144C4×#41C2×D311yescharacteristic
145#489—62no—
146#487—61no—
147#486—61no—
148#477—61no—
149#476—61no—
150#475—61no—
151#474—61no—
152#428—32no—
153#423—31no—
154C2×#104—34no—
155C2×#104C2×D312yes—
156#417—32no—
157#417—34no—
158C4×M16C2×D311yescharacteristic
159#384—34no—
160#384—62no—
161C43C2×D311yescharacteristic
162#343—34no—
163#339C2×D312yes—
164C3×#41—22no—
165C3×#41—22no—
166C2×C3×Q8C2×D411yescharacteristic
167C2×C3×D4—21no—
168C2×C3×D4C2×D411yescharacteristic
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C2×#67—22no—
177C2×#67—24no—
178C22×Dic3—22no—
179C22×Dic3C2×D414yes—
180#235—24no—
181#235—24no—
182#235—84no—
183C2×D12—21no—
184C2×D12—21no—
185C2×D12C2×D412yes—
186C2×C4×D3—22no—
187C2×C4×D3—24no—
188C2×Dic6—22no—
189C2×Dic6—22no—
190C2×Dic6—44no—
191C2×Dic6—44no—
192C2×Dic6—41no—
193C2×Dic6—41no—
194C2×Dic6C2×D412yes—
195C3×M16—22no—
196C3×M16—22no—
197C2×C3×C8—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4C2×D411yescharacteristic
201C3×C4⋊C4C2×D411yescharacteristic
202C3×C22⋊C4—21no—
203C3×C22⋊C4—21no—
204C3×C22⋊C4—21no—
205C3×C42—24no—
206C3×C42—24no—
207C3×C42—21no—
208C3×C42—21no—
209C3×C42—21no—
210C3×C42—22no—
211C3×C42C2×D411yescharacteristic
212C3×C42C2×D411yescharacteristic
213C3×C42C2×D411yescharacteristic
214C3×C42C2×D411yescharacteristic
215C3×C42#4114yescomplemented
216#217—24no—
217(C2×D3)⋊C4—24no—
218(C2×D3)⋊C4—24no—
219(C2×D3)⋊C4—22no—
220(C2×D3)⋊C4—22no—
221C4⋊Dic3—21no—
222C4⋊Dic3—22no—
223C4⋊Dic3—22no—
224C4⋊Dic3—21no—
225C4⋊Dic3—22no—
226C4⋊Dic3—22no—
227C4⋊Dic3—22no—
228C4⋊Dic3—22no—
229C4⋊Dic3—416no—
230C4⋊Dic3—41no—
231C4⋊Dic3—41no—
232Dic3⋊C4—24no—
233Dic3⋊C4—24no—
234C4×Dic3—24no—
235#208—24no—
236C2×C3⋊C8—22no—
237C2×#41—32no—
238C2×#41—62no—
239C2×#41C22×D311yescharacteristic
240C22×Q8—32no—
241C22×Q8—61no—
242C2×M16—32no—
243C2×M16C22×D312yes—
244C4⋊Q8—31no—
245C4⋊Q8—61no—
246C4⋊Q8—61no—
247C4⋊Q8—61no—
248C4⋊Q8—61no—
249C4⋊Q8—61no—
250C4⋊Q8—61no—
251C4⋊Q8—61no—
252C4⋊Q8—61no—
253C4⋊Q8—128no—
254#127—31no—
255#126—64no—
256#125—62no—
257#124—32no—
258#124—62no—
259#124—64no—
260#123—64no—
261C22⋊Q8—64no—
262C22⋊Q8—62no—
263C22⋊Q8—62no—
264C22⋊Q8—62no—
265#121—62no—
266#121—62no—
267#121—64no—
268#121—62no—
269#121—61no—
270#121—61no—
271C4×Q8—62no—
272C4×Q8—62no—
273C4×Q8D1211yescharacteristic
274C4×Q8D1211yescharacteristic
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—62no—
278C4×D4—62no—
279C4×D4—64no—
280C4×D4—61no—
281C4×D4—61no—
282C4×D4D1211yescharacteristic
283C4×D4D1211yescharacteristic
284#117—21no—
285#117—62no—
286#117—62no—
287#117C22×D311yescharacteristic
288C2×C4⋊C4—32no—
289C2×C4⋊C4—32no—
290C2×C4⋊C4—32no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—68no—
293C2×C4⋊C4—61no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42C22×D312yes—
298C2×C42C22×D311yescharacteristic
299#107—62no—
300#106—62no—
301C4⋊C8—61no—
302C4⋊C8—61no—
303#104—24no—
304#104—68no—
305#103—64no—
306#103—62no—
307#102—64no—
308#102—62no—
309#97—21no—
310C4×C8—21no—
311C23×C3—21no—
312C23×C3C22×D411yescharacteristic
313C22×D3—22no—
314C3×Q8—21no—
315C3×Q8—21no—
316C3×D4—21no—
317C3×D4—21no—
318C3×D4—42no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4#12011yescharacteristic
331C2×C3×C4#12011yescharacteristic
332C2×C3×C4#12011yescharacteristic
333C2×C3×C4#12011yescharacteristic
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C2×#4112yes—
337C2×C3×C42−1+412yes—
338#67—44no—
339#67—84no—
340C2×Dic3—22no—
341C2×Dic3—24no—
342C2×Dic3—44no—
343C2×Dic3—44no—
344C2×Dic3—42no—
345D12—24no—
346D12—42no—
347D12—42no—
348C4×D3—42no—
349C4×D3—42no—
350C4×D3—84no—
351Dic6—24no—
352Dic6—44no—
353Dic6—82no—
354Dic6—82no—
355Dic6—82no—
356C3×C8—42no—
357C3⋊C8—42no—
358#41—22no—
359#41—22no—
360#41—64no—
361#41—64no—
362#41—244no—
363C2×Q8—32no—
364C2×Q8—62no—
365C2×Q8—62no—
366C2×Q8—124no—
367C2×Q8—124no—
368C2×Q8—121no—
369C2×Q8—121no—
370C2×Q8C2×D1211yescharacteristic
371C2×D4—21no—
372C2×D4—32no—
373C2×D4—62no—
374C2×D4—64no—
375C2×D4—61no—
376C2×D4—61no—
377C2×D4C2×D1211yescharacteristic
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—34no—
383C22×C4—64no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4D3×D411yescharacteristic
387C22×C4D3×D411yescharacteristic
388C22×C4C23×D311yescharacteristic
389M16—22no—
390M16—22no—
391M16—64no—
392C2×C8—22no—
393C2×C8—62no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—64no—
397C4⋊C4—64no—
398C4⋊C4—62no—
399C4⋊C4—62no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4—62no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—1216no—
407C4⋊C4—121no—
408C4⋊C4—121no—
409C4⋊C4C2×D1211yescharacteristic
410C4⋊C4C2×D1211yescharacteristic
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—64no—
415C22⋊C4—64no—
416C22⋊C4—64no—
417C22⋊C4—62no—
418C22⋊C4—62no—
419C42—24no—
420C42—24no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—22no—
425C42—64no—
426C42C2×D1211yescharacteristic
427C42C2×D1211yescharacteristic
428C42#23514yescomplemented
429C42D3×D411yescharacteristic
430C42D3×D411yescharacteristic
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3C2×#12011yescharacteristic
437C22×C3#54812yes—
438C2×D3—42no—
439C2×D3—42no—
440C2×D3—42no—
441C3×C4—28no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4#45312yes—
448C3×C4#45612yes—
449C3×C4#53211yescharacteristic
450C3×C4#53611yescharacteristic
451C3×C4#54712yes—
452Dic3—42no—
453Dic3—44no—
454Dic3—82no—
455C23—21no—
456C23—62no—
457C23C2×D3×D411yescharacteristic
458Q8—21no—
459Q8—21no—
460Q8—64no—
461Q8—124no—
462Q8—242no—
463Q8—242no—
464Q8—242no—
465D4—21no—
466D4—21no—
467D4—42no—
468D4—64no—
469D4—124no—
470D4—122no—
471D4—122no—
472D4—244no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—62no—
485C2×C4—64no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—124no—
489C2×C4—124no—
490C2×C4—122no—
491C2×C4—244no—
492C2×C4#88211yescharacteristic
493C2×C4#88211yescharacteristic
494C2×C4#88211yescharacteristic
495C2×C4#88211yescharacteristic
496C2×C4C22×D1211yescharacteristic
497C2×C4C2×#23512yes—
498C2×C4C2×D3×D411yescharacteristic
499C2×C4#101012yes—
500C8—42no—
501C8—122no—
502C2×C3—22no—
503C2×C3—81no—
504C2×C3#243211yescharacteristic
505C2×C3C2×#45311yescharacteristic
506C2×C3C2×#45611yescharacteristic
507D3—82no—
508C22—21no—
509C22—21no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—122no—
514C22—122no—
515C22—122no—
516C22C2×#88211yescharacteristic
517C22#555212yes—
518C4—28no—
519C4—21no—
520C4—21no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—122no—
525C4—124no—
526C4—242no—
527C4#482212yes—
528C4#482612yes—
529C4#547711yescharacteristic
530C4#552611yescharacteristic
531C4#555112yes—
532C3#2566611yescharacteristic, complemented, Sylow
533C2—22no—
534C2—81no—
535C2—242no—
536C2#7348911yescharacteristic
537C2C2×#482211yescharacteristic
538C2C2×#482611yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#104C2
socleC22×C3C2×#120
prime radical#85850C2
hypercenterC4×#104D3
hypocenterC3#25666
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. #73311C2semidirect
5. C3×C4×#104C2semidirect
6. #70361C2semidirect
7. #69879C2semidirect
8. #69395C2semidirect
9. #68771C2semidirect
54. C4×#104D3semidirect
215. C3×C42#41semidirect
428. C42#235semidirect
532. C3#25666semidirect

Maximal subgroups

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

subgroupindex
#858502
#825082
#785282
#733112
C3×C4×#1042
#703612
#698792
#693952
#687712
#256663