Finite Group Databasebeta

← #10804403

Subgroups — #10804403

Order 768.

1262 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86076C211yesmaximal, characteristic
2#82556C211yesmaximal, characteristic, complemented
3#78706C211yesmaximal, characteristic, complemented
4C3×#1888C211yesmaximal, characteristic, complemented
5#71724C211yesmaximal, characteristic, complemented
6#71663C211yesmaximal, characteristic, complemented
7#70330C212yesmaximal, complemented
8#70327C212yesmaximal, complemented
9#70105C211yesmaximal, characteristic, complemented
10#68777C212yesmaximal, complemented
11#68737C212yesmaximal, complemented
12#21785—31nomaximal, Sylow
13C2×#1007C2211yescharacteristic
14C2×#1004C2211yescharacteristic
15C2×C3×#124C2211yescharacteristic
16C3×#519C2211yescharacteristic, complemented
17#5816C2211yescharacteristic
18#5812C2211yescharacteristic, complemented
19#5805C2211yescharacteristic
20#5745C2212yescomplemented
21#5677—22no—
22#5676—22no—
23#5675C412yescomplemented
24#5674C412yescomplemented
25D3×#124C412yescomplemented
26#5668C412yescomplemented
27#5490—24no—
28#5477C2211yescharacteristic
29C4×#235C2211yescharacteristic
30C3×#418C2212yes—
31C2×C3×#101C2211yescharacteristic, complemented
32C2×#836C2211yescharacteristic, complemented
33#5215C2212yes—
34#5139C2212yes—
35#5132C2212yes—
36#5006—24no—
37#5006C2212yes—
38#5006C2212yescomplemented
39#5005C2212yes—
40#4751—22no—
41D3×#101—22no—
42C3×#339C2212yescomplemented
43#4557C2212yescomplemented
44#4531C2212yescomplemented
45#4523C2212yescomplemented
46#3500—31no—
47#2889—31no—
48#2884—31no—
49#1912—31no—
50#1912—31no—
51#1910—32no—
52#1888—31no—
53#1888D311yescharacteristic, complemented
54#1884—32no—
55#1827—32no—
56#1795—32no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C22×#67—22no—
60#1007—42no—
61#1007—42no—
62C2×#239C2×C412yes—
63C2×#239D412yes—
64C2×D3×Q8C2×C412yes—
65C2×D3×Q8D412yes—
66#1004—42no—
67#1004—42no—
68C2×#237—21no—
69C2×#237C2×C412yes—
70C2×D3×D4—21no—
71C2×D3×D4C2×C412yes—
72C2×#235D414yes—
73C2×#235C2311yescharacteristic
74C2×#235C2311yescharacteristic
75C2×#235C2311yescharacteristic
76C2×C3×M16C2312yes—
77C3×#124—22no—
78C3×#124C2×C412yes—
79C3×#124C2×C412yes—
80C3×C4×Q8—42no—
81C3×#117—22no—
82C3×#117C2312yes—
83C2×C3×C22⋊C4—22no—
84C2×C3×C42C2311yescharacteristic
85#953—22no—
86#947C2×C412yes—
87#946—22no—
88Q8×Dic3—42no—
89#944C2×C412yes—
90#940C2×C412yes—
91#938—22no—
92#936C2×C412yes—
93#930—22no—
94#929—22no—
95C4×#67—22no—
96#926—22no—
97#926C2312yes—
98#926C2311yescharacteristic
99C2×#208C2312yes—
100#907—42no—
101D3×M16—24no—
102#901—42no—
103#885—24no—
104#884—24no—
105#884—24no—
106#883—24no—
107#882—24no—
108#67⋊C4—24no—
109D3×C22⋊C4—24no—
110#878—24no—
111#875C2×C411yescharacteristic
112#875C2×C411yescharacteristic
113#874C2×C411yescharacteristic
114C4×D12—21no—
115C4×D12—22no—
116#872C2×C412yes—
117C42×D3C2×C412yes—
118#869C2×C411yescharacteristic
119C4×Dic6—21no—
120C3×#102—42no—
121C3×#101—22no—
122#836—22no—
123#832—42no—
124#825—42no—
125#824—22no—
126#823—22no—
127#806—24no—
128#806—24no—
129#805—42no—
130C2×2−1+4—31no—
131C2×H(5,𝔽2)—31no—
132#568—32no—
133#548—62no—
134#546—64no—
135#546—62no—
136#536—31no—
137#536—32no—
138#535—31no—
139#535—32no—
140#532—32no—
141#532—31no—
142C2×#124—32no—
143C2×#124C2×D311yescharacteristic
144#519—31no—
145#519C2×D311yescharacteristic, complemented
146C4×#41—31no—
147#421—32no—
148#421—32no—
149#418—32no—
150#418C2×D312yes—
151#413—32no—
152#413—62no—
153C2×#101—31no—
154C2×#101—62no—
155C2×#101C2×D311yescharacteristic, complemented
156#410—32no—
157#410—32no—
158#410—64no—
159#339—32no—
160#339C2×D312yescomplemented
161H(3,ℤ/4)—32no—
162H(3,ℤ/4)—32no—
163C24×C3—21no—
164C23×D3—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8C22×C412yes—
168C2×C3×Q8C2×D412yes—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4C22×C412yes—
173C22×C3×C4—21no—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2411yescharacteristic
176C2×#67—21no—
177C2×#67—22no—
178C2×#67—24no—
179C2×#67—22no—
180C2×#67—44no—
181C2×#67—44no—
182C2×#67C2×D412yes—
183C2×#67C2×D412yes—
184C22×Dic3—22no—
185#239—44no—
186#239—44no—
187D3×Q8—44no—
188D3×Q8—44no—
189#237—44no—
190#237—44no—
191D3×D4—44no—
192D3×D4—44no—
193#235—22no—
194#235—22no—
195#235—22no—
196#235—22no—
197#235—22no—
198#235—22no—
199#235—44no—
200#235—44no—
201C2×D12—22no—
202C2×D12C22⋊C414yes—
203C2×D12C22×C411yescharacteristic
204C2×D12C22×C411yescharacteristic
205C2×D12C2×D411yescharacteristic
206C2×C4×D3—21no—
207C2×C4×D3—22no—
208C2×C4×D3C22⋊C414yes—
209C2×C4×D3C22⋊C414yes—
210C2×C4×D3C22×C412yes—
211C2×C4×D3C2×D412yes—
212C2×Dic6C22⋊C414yes—
213C2×Dic6C22×C411yescharacteristic
214C2×Dic6C22×C411yescharacteristic
215C2×Dic6C2×D411yescharacteristic
216C3×M16—22no—
217C3×M16—22no—
218C2×C3×C8—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—42no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—22no—
224C3×C42—21no—
225C3×C42—22no—
226C3×C42—42no—
227C3×C42C22×C411yescharacteristic
228C3×C42C22×C411yescharacteristic
229#217—21no—
230#217—24no—
231#217—22no—
232(C2×D3)⋊C4—22no—
233(C2×D3)⋊C4—22no—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—22no—
236C4⋊Dic3—21no—
237C4⋊Dic3—22no—
238C4⋊Dic3—22no—
239C4⋊Dic3—42no—
240Dic3⋊C4—24no—
241Dic3⋊C4—22no—
242C4×Dic3—22no—
243C4×Dic3—22no—
244C4×Dic3—42no—
245C4×Dic3C22×C412yes—
246#208—22no—
247#208—22no—
248C2×C3⋊C8—22no—
249ΓL(1,𝔽25)—44no—
250D3×C8—44no—
2512−1+4—122no—
2522−1+4—122no—
253H(5,𝔽2)—122no—
254H(5,𝔽2)—122no—
255C2×#41—32no—
256C2×#41—32no—
257C2×#41—34no—
258C2×#41—31no—
259C2×#41—31no—
260C2×#41—32no—
261C2×#41—31no—
262C2×#41—61no—
263C22×Q8—32no—
264C22×Q8—32no—
265C22×Q8C22×D311yescharacteristic
266C22×D4—32no—
267C22×D4—61no—
268C22×D4—62no—
269C22×D4C22×D311yescharacteristic
270#131—122no—
271#131—122no—
272C2×M16—32no—
273C2×M16—64no—
274C2×M16C22×D312yes—
275C4⋊Q8—32no—
276C4⋊Q8—31no—
277#127—32no—
278#127—31no—
279#124—22no—
280#124—32no—
281#124—32no—
282#124—31no—
283#124—31no—
284#124—64no—
285#124C4×D312yes—
286#124C4×D312yes—
287#123—64no—
288#123—62no—
289C22⋊Q8—64no—
290C22⋊Q8—62no—
291#121—64no—
292#121—64no—
293#121—62no—
294#121—62no—
295#120—64no—
296#120—62no—
297C4×Q8—42no—
298C4×Q8—61no—
299C4×Q8—122no—
300C4×D4—62no—
301C4×D4—61no—
302C4×D4—64no—
303C4×D4—62no—
304#117—22no—
305#117—31no—
306#117—32no—
307#117—32no—
308#117—62no—
309#117C22×D312yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—64no—
312C2×C42—32no—
313C2×C42C22×D311yescharacteristic
314#104—122no—
315#104—122no—
316#102—42no—
317#102—122no—
318#101—22no—
319#101—62no—
320#101—62no—
321#100—62no—
322#99—64no—
323#99—64no—
324C23×C3—22no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—82no—
328C23×C3C22×D411yescharacteristic
329C22×D3—21no—
330C22×D3—44no—
331C22×D3—84no—
332C22×D3C2×C22⋊C412yes—
333C22×D3C2×C22⋊C412yes—
334C3×Q8—24no—
335C3×Q8—24no—
336C3×D4—24no—
337C3×D4—42no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—42no—
343C2×C3×C4—81no—
344C2×C3×C4C2×C22⋊C414yes—
345C2×C3×C4C23×C411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4H(5,𝔽2)12yes—
348C2×C3×C4H(5,𝔽2)12yes—
349#67—22no—
350#67—22no—
351#67—22no—
352#67—22no—
353#67—44no—
354#67—44no—
355#67—41no—
356#67—41no—
357#67—84no—
358C2×Dic3—22no—
359C2×Dic3—21no—
360C2×Dic3—44no—
361C2×Dic3—42no—
362C2×Dic3C2×C22⋊C412yes—
363C2×Dic3C2×C22⋊C412yes—
364D12—22no—
365D12—22no—
366D12—22no—
367D12—44no—
368D12—82no—
369C4×D3—24no—
370C4×D3—24no—
371C4×D3—44no—
372C4×D3—44no—
373C4×D3—42no—
374C4×D3—82no—
375Dic6—22no—
376Dic6—22no—
377Dic6—22no—
378Dic6—44no—
379C3×C8—42no—
380C3⋊C8—42no—
381C24—21no—
382C24—62no—
383#41—62no—
384#41—62no—
385#41—62no—
386#41—62no—
387#41—62no—
388#41—62no—
389#41—124no—
390#41—124no—
391#41—124no—
392#41—124no—
393#41—124no—
394#41—124no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—34no—
398C2×Q8—31no—
399C2×Q8—31no—
400C2×Q8—31no—
401C2×Q8—124no—
402C2×Q8—124no—
403C2×Q8C2×C4×D312yes—
404C2×Q8D3×D412yes—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—32no—
409C2×D4—32no—
410C2×D4—34no—
411C2×D4—31no—
412C2×D4—31no—
413C2×D4—31no—
414C2×D4—62no—
415C2×D4—62no—
416C2×D4—62no—
417C2×D4—61no—
418C2×D4—64no—
419C2×D4—124no—
420C2×D4—124no—
421C2×D4—124no—
422C2×D4—124no—
423C2×D4C2×C4×D312yes—
424C22×C4—21no—
425C22×C4—32no—
426C22×C4—32no—
427C22×C4—34no—
428C22×C4—34no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—61no—
432C22×C4D3×D412yes—
433C22×C4C23×D311yescharacteristic
434M16—22no—
435M16—22no—
436M16—62no—
437M16—62no—
438M16—124no—
439C2×C8—22no—
440C2×C8—62no—
441C2×C8—124no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—62no—
445C4⋊C4—62no—
446C4⋊C4—64no—
447C4⋊C4—61no—
448C4⋊C4—62no—
449C4⋊C4—122no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—22no—
453C22⋊C4—62no—
454C22⋊C4—62no—
455C22⋊C4—62no—
456C22⋊C4—64no—
457C22⋊C4—61no—
458C22⋊C4—64no—
459C22⋊C4—62no—
460C42—21no—
461C42—22no—
462C42—32no—
463C42—42no—
464C42—62no—
465C42—62no—
466C42—122no—
467C42C2×C4×D311yescharacteristic
468C42C2×C4×D311yescharacteristic
469C22×C3—41no—
470C22×C3—44no—
471C22×C3—42no—
472C22×C3—161no—
473C22×C3—161no—
474C22×C3C22×C22⋊C411yescharacteristic
475C22×C3#53411yescharacteristic
476C22×C3#53411yescharacteristic
477C2×D3—22no—
478C2×D3—22no—
479C2×D3—42no—
480C2×D3—42no—
481C2×D3—42no—
482C2×D3—41no—
483C2×D3—164no—
484C3×C4—24no—
485C3×C4—41no—
486C3×C4—41no—
487C3×C4—82no—
488C3×C4#51812yes—
489C3×C4#53412yes—
490Dic3—22no—
491Dic3—22no—
492Dic3—42no—
493Dic3—41no—
494Dic3—82no—
495C23—22no—
496C23—32no—
497C23—32no—
498C23—41no—
499C23—41no—
500C23—61no—
501C23—82no—
502C23—124no—
503C23—244no—
504C23C2×D3×D411yescharacteristic
505Q8—24no—
506Q8—24no—
507Q8—62no—
508Q8—62no—
509Q8—62no—
510Q8—124no—
511D4—24no—
512D4—42no—
513D4—62no—
514D4—62no—
515D4—62no—
516D4—62no—
517D4—62no—
518D4—62no—
519D4—62no—
520D4—124no—
521D4—124no—
522D4—124no—
523D4—121no—
524D4—121no—
525D4—242no—
526D4—244no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—32no—
532C2×C4—32no—
533C2×C4—42no—
534C2×C4—64no—
535C2×C4—64no—
536C2×C4—62no—
537C2×C4—61no—
538C2×C4—81no—
539C2×C4—124no—
540C2×C4—124no—
541C2×C4—124no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—242no—
545C2×C4D3×C22⋊C414yes—
546C2×C4C22×C4×D311yescharacteristic
547C2×C4C2×D3×D411yescharacteristic
548C2×C4#100412yes—
549C2×C4#100912yes—
550C8—42no—
551C8—122no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—82no—
556C2×C3#230811yescharacteristic
557D3—42no—
558D3—42no—
559D3—81no—
560C22—41no—
561C22—44no—
562C22—42no—
563C22—62no—
564C22—62no—
565C22—122no—
566C22—122no—
567C22—122no—
568C22—121no—
569C22—161no—
570C22—161no—
571C22—484no—
572C22C2×D3×C22⋊C411yescharacteristic
573C22#558911yescharacteristic
574C22#565311yescharacteristic
575C4—24no—
576C4—41no—
577C4—41no—
578C4—62no—
579C4—62no—
580C4—82no—
581C4—122no—
582C4—121no—
583C4—242no—
584C4#554512yes—
585C4#560512yes—
586C3#2178511yescharacteristic, complemented, Sylow
587C2—21no—
588C2—21no—
589C2—21no—
590C2—82no—
591C2—122no—
592C2—122no—
593C2—241no—
594C2#7487411yescharacteristic
subgroupG/H
centerC2#74874
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1888C2
socleC2×C3#2308
prime radical#86076C2
cosocleC22×C3×C4C24
hypercenter#1888D3
hypocenterC3#21785
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1888

Complements

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

NH ≅ G/Ntype
2. #82556C2semidirect
3. #78706C2semidirect
4. C3×#1888C2semidirect
5. #71724C2semidirect
6. #71663C2semidirect
7. #70330C2semidirect
8. #70327C2semidirect
9. #70105C2semidirect
10. #68777C2semidirect
11. #68737C2semidirect
16. C3×#519C22semidirect
18. #5812C22semidirect
20. #5745C22semidirect
23. #5675C4semidirect
24. #5674C4semidirect
25. D3×#124C4semidirect
26. #5668C4semidirect
31. C2×C3×#101C22semidirect
32. C2×#836C22semidirect
38. #5006C22semidirect
42. C3×#339C22semidirect
43. #4557C22semidirect
44. #4531C22semidirect
45. #4523C22semidirect
53. #1888D3semidirect
145. #519C2×D3semidirect
155. C2×#101C2×D3semidirect
160. #339C2×D3semidirect
586. C3#21785semidirect

Maximal subgroups

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

subgroupindex
#860762
#825562
#787062
C3×#18882
#717242
#716632
#703302
#703272
#701052
#687772
#687372
#217853