Finite Group Databasebeta

← #10763234

Subgroups — #10763234

Order 768.

974 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86257C211yesmaximal, characteristic
2#79331C211yesmaximal, characteristic, complemented
3D3×#432C211yesmaximal, characteristic, complemented
4C3×#1537C211yesmaximal, characteristic, complemented
5#71823C211yesmaximal, characteristic, complemented
6#70811C211yesmaximal, characteristic, complemented
7#70811C211yesmaximal, characteristic, complemented
8#68368C214yesmaximal, complemented
9#68361C214yesmaximal, complemented
10#17662—31nomaximal, Sylow
11C3×#533C2211yescharacteristic
12#5728C411yescharacteristic
13#5728C411yescharacteristic
14#5725C411yescharacteristic
15#5725C411yescharacteristic
16#5691C2212yes—
17#5691C2212yes—
18#5627—21no—
19#5627—21no—
20#5624—21no—
21#5624—21no—
22#5580—21no—
23#5580—21no—
24C4×#239—21no—
25C4×#239—21no—
26#5524C412yes—
27#5522C412yes—
28#5477C2211yescharacteristic
29C2×C42×D3C2211yescharacteristic
30C3×#432C2211yescharacteristic
31C3×#432C2211yescharacteristic
32#4926C2211yescharacteristic
33#4926C2211yescharacteristic
34#4837—21no—
35#4835—21no—
36#4833—21no—
37D3×C4⋊C8—21no—
38#4726—21no—
39D3×C22⋊C8—21no—
40#4719—21no—
41#4718—21no—
42#4707C2212yes—
43#4705C2212yes—
44#4705C2212yes—
45D3×#97C2212yes—
46C3×#330C2214yescomplemented
47#4543C2214yescomplemented
48#4464C2214yescomplemented
49#4464C2214yescomplemented
50#3444—31no—
51#2920—31no—
52C2×#432—31no—
53#1537—31no—
54#1537D311yescharacteristic, complemented
55#1536—31no—
56#1536—31no—
57#1535—34no—
58#1524—34no—
59C2×#239—21no—
60C2×#239—21no—
61C2×#235C2×C412yes—
62C22×C4×D3C2×C411yescharacteristic
63C22×C4×D3D411yescharacteristic
64C22×C4×D3D411yescharacteristic
65C3×C4⋊Q8C2×C411yescharacteristic
66C3×C4⋊Q8C2×C411yescharacteristic
67C3×#125C2312yes—
68C3×C22⋊Q8—21no—
69C3×C22⋊Q8—21no—
70C3×C4×Q8—21no—
71C3×C4×Q8—21no—
72C3×#117C2×C412yes—
73C2×C3×C42C2311yescharacteristic
74#947—21no—
75#947—21no—
76#946—21no—
77#946—21no—
78Q8×Dic3—21no—
79Q8×Dic3—21no—
80#930—21no—
81#930—21no—
82C2×C4×Dic3C2×C411yescharacteristic
83C2×C4×Dic3D411yescharacteristic
84C2×C4×Dic3D411yescharacteristic
85C2×ΓL(1,𝔽25)—22no—
86C2×D3×C8—22no—
87#898—22no—
88#898—22no—
89#897C2×C412yes—
90#897C2×C412yes—
91#895—21no—
92#895—21no—
93#895C2×C412yes—
94#895C2×C412yes—
95#894—22no—
96#894—22no—
97#893—22no—
98#893—22no—
99#893—21no—
100#893—21no—
101#893C2×C412yes—
102#893C2×C412yes—
103#892—21no—
104#892—21no—
105#892—22no—
106#892—22no—
107#890—21no—
108#890—21no—
109#887C2×C412yes—
110#887C2×C412yes—
111#886—22no—
112#882—22no—
113#67⋊C4—22no—
114#67⋊C4—22no—
115#875C2312yes—
116#874C2×C411yescharacteristic
117C4×D12—22no—
118C4×D12—22no—
119C4×D12—21no—
120C4×D12—21no—
121#872—22no—
122C42×D3—22no—
123C42×D3C2×C411yescharacteristic
124C42×D3C2×C411yescharacteristic
125C42×D3D411yescharacteristic
126C42×D3D411yescharacteristic
127C42×D3D411yescharacteristic
128C42×D3D411yescharacteristic
129C42×D3C2311yescharacteristic
130C42×D3C2311yescharacteristic
131#869C2×C411yescharacteristic
132C3×C4⋊C8—21no—
133C3×C4⋊C8—21no—
134C3×C22⋊C8—21no—
135C3×C22⋊C8—21no—
136C3×#97C2312yes—
137C3×#97C2312yes—
138#830—21no—
139#830—21no—
140(C2×D3)⋊C8—22no—
141(C2×D3)⋊C8—22no—
142#815—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145C8×Dic3—22no—
146C4⋊(C3⋊C8)—21no—
147C4⋊(C3⋊C8)—21no—
148#803C2312yes—
149#803C2312yes—
150#533—31no—
151#533—31no—
152#533C2×D311yescharacteristic
153#532—31no—
154#532—31no—
155#532—31no—
156#529—32no—
157#529—32no—
158#529—61no—
159#529—61no—
160#529—61no—
161#529—61no—
162#525—32no—
163#525—61no—
164#525—61no—
165C4×#41—32no—
166C4×#41—61no—
167C4×#41—61no—
168C22×C42—31no—
169#432—31no—
170#432—31no—
171#432—32no—
172#432—32no—
173#432—61no—
174#432—61no—
175#432C2×D311yescharacteristic
176#432C2×D311yescharacteristic
177#431—61no—
178#431—61no—
179#423—61no—
180C2×C4⋊C8—61no—
181#407—61no—
182C2×C22⋊C8—61no—
183C4×M16—32no—
184C2×#97—32no—
185#330—34no—
186#330C2×D314yescomplemented
187#329—34no—
188#329—34no—
189C23×D3C22⋊C411yescharacteristic
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C22×C3×C4C22×C411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C2×#67—22no—
196C22×Dic3C22⋊C411yescharacteristic
197C22×Dic3C22⋊C411yescharacteristic
198C22×Dic3C22⋊C411yescharacteristic
199#239—44no—
200#239—44no—
201#235—44no—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12C22×C412yes—
207C2×C4×D3—22no—
208C2×C4×D3—22no—
209C2×C4×D3—22no—
210C2×C4×D3—42no—
211C2×C4×D3—42no—
212C2×C4×D3—42no—
213C2×C4×D3C22⋊C411yescharacteristic
214C2×C4×D3C22⋊C411yescharacteristic
215C2×C4×D3C22⋊C411yescharacteristic
216C2×C4×D3C22⋊C411yescharacteristic
217C2×C4×D3C22⋊C411yescharacteristic
218C2×C4×D3C22⋊C411yescharacteristic
219C2×C4×D3C22×C411yescharacteristic
220C2×C4×D3C22×C411yescharacteristic
221C2×C4×D3C2×D411yescharacteristic
222C2×C4×D3C2×D411yescharacteristic
223C2×C4×D3C2×D411yescharacteristic
224C2×C4×D3C2×D411yescharacteristic
225C2×Dic6C22×C412yes—
226C2×C3×C8—22no—
227C2×C3×C8—22no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4C22×C412yes—
233C3×C4⋊C4C22×C412yes—
234C3×C22⋊C4—22no—
235C3×C42—22no—
236C3×C42C22×C411yescharacteristic
237C3×C42C2×D411yescharacteristic
238C3×C42C2×D411yescharacteristic
239C3×C42C2411yescharacteristic
240(C2×D3)⋊C4—22no—
241(C2×D3)⋊C4—22no—
242(C2×D3)⋊C4—22no—
243(C2×D3)⋊C4—22no—
244(C2×D3)⋊C4—22no—
245(C2×D3)⋊C4—22no—
246(C2×D3)⋊C4—22no—
247(C2×D3)⋊C4—22no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—22no—
251C4⋊Dic3—22no—
252Dic3⋊C4—22no—
253Dic3⋊C4—22no—
254C4×Dic3—22no—
255C4×Dic3—22no—
256C4×Dic3C22⋊C411yescharacteristic
257C4×Dic3C22⋊C411yescharacteristic
258C4×Dic3C22⋊C411yescharacteristic
259C4×Dic3C22⋊C411yescharacteristic
260C4×Dic3C22⋊C411yescharacteristic
261C4×Dic3C22⋊C411yescharacteristic
262C4×Dic3C22×C411yescharacteristic
263C4×Dic3C22×C411yescharacteristic
264C4×Dic3C2×D411yescharacteristic
265C4×Dic3C2×D411yescharacteristic
266C4×Dic3C2×D411yescharacteristic
267C4×Dic3C2×D411yescharacteristic
268C2×C3⋊C8—22no—
269C2×C3⋊C8—22no—
270ΓL(1,𝔽25)—44no—
271D3×C8—44no—
272C2×#41—32no—
273C2×#41—61no—
274C2×#41—61no—
275C23×C4—31no—
276C23×C4—31no—
277C23×C4—31no—
278C2×M16—62no—
279C22×C8—62no—
280C4⋊Q8—31no—
281C4⋊Q8C4×D311yescharacteristic
282C4⋊Q8C4×D311yescharacteristic
283#127—31no—
284#126—62no—
285#126—62no—
286#125—61no—
287#125—61no—
288#125C22×D312yes—
289#124—32no—
290#124—61no—
291#124—61no—
292#123—62no—
293#123—62no—
294#123—62no—
295C22⋊Q8—21no—
296C22⋊Q8—21no—
297C22⋊Q8—32no—
298C22⋊Q8—32no—
299C22⋊Q8—61no—
300C22⋊Q8—61no—
301#121—32no—
302#121—32no—
303#121—61no—
304#121—61no—
305#121—61no—
306#121—61no—
307#120—62no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8—32no—
311C4×Q8—32no—
312C4×Q8—61no—
313C4×Q8—61no—
314C4×D4—32no—
315C4×D4—32no—
316C4×D4—62no—
317C4×D4—62no—
318C4×D4—61no—
319C4×D4—61no—
320C4×D4—62no—
321C4×D4—62no—
322C4×D4—62no—
323C4×D4—62no—
324C4×D4—61no—
325C4×D4—61no—
326#117—62no—
327#117—62no—
328#117—62no—
329#117—61no—
330#117—61no—
331#117C4×D312yes—
332C2×C42—31no—
333C2×C42—31no—
334C2×C42—31no—
335C2×C42—31no—
336C2×C42—31no—
337C2×C42—31no—
338C2×C42—31no—
339C2×C42—31no—
340C2×C42—31no—
341C2×C42—31no—
342C2×C42—31no—
343C2×C42—62no—
344C2×C42C22×D311yescharacteristic
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8—62no—
348C4⋊C8—62no—
349C4⋊C8—61no—
350C4⋊C8—61no—
351C22⋊C8—21no—
352C22⋊C8—21no—
353C22⋊C8—62no—
354C22⋊C8—62no—
355C22⋊C8—61no—
356C22⋊C8—61no—
357#97—32no—
358#97—32no—
359#97—62no—
360#97C22×D312yes—
361#97C22×D312yes—
362C4×C8—62no—
363C23×C3C2×C22⋊C411yescharacteristic
364C22×D3—22no—
365C22×D3—22no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3C2×C22⋊C411yescharacteristic
369C22×D3C2×C22⋊C411yescharacteristic
370C3×Q8—42no—
371C3×Q8—42no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4C2×C22⋊C411yescharacteristic
378C2×C3×C4C2×C22⋊C411yescharacteristic
379C2×C3×C4C2×C22⋊C411yescharacteristic
380C2×C3×C4C23×C411yescharacteristic
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383#67—44no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3C2×C22⋊C411yescharacteristic
389C2×Dic3C2×C22⋊C411yescharacteristic
390C2×Dic3C2×C22⋊C411yescharacteristic
391C2×Dic3C2×C22⋊C411yescharacteristic
392C2×Dic3C2×C22⋊C411yescharacteristic
393C2×Dic3C2×C22⋊C411yescharacteristic
394D12—42no—
395D12—42no—
396D12—42no—
397D12—44no—
398D12—44no—
399C4×D3—22no—
400C4×D3—22no—
401C4×D3—22no—
402C4×D3—22no—
403C4×D3—22no—
404C4×D3—22no—
405C4×D3—41no—
406C4×D3—41no—
407C4×D3—41no—
408C4×D3—41no—
409C4×D3—41no—
410C4×D3—41no—
411C4×D3—44no—
412C4×D3—44no—
413C4×D3—44no—
414Dic6—42no—
415C3×C8—42no—
416C3×C8—42no—
417C3⋊C8—42no—
418C3⋊C8—42no—
419C24—31no—
420#41—124no—
421#41—124no—
422#41—124no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—32no—
426C2×D4—32no—
427C2×D4—62no—
428C2×D4—62no—
429C2×D4—62no—
430C2×D4—61no—
431C2×D4—61no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—122no—
451C22×C4—122no—
452C22×C4—122no—
453C22×C4C2×C4×D311yescharacteristic
454C22×C4D3×D411yescharacteristic
455C22×C4D3×D411yescharacteristic
456M16—124no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—62no—
460C2×C8—62no—
461C2×C8—124no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—62no—
467C4⋊C4—62no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4C2×C4×D312yes—
473C4⋊C4C2×C4×D312yes—
474C22⋊C4—22no—
475C22⋊C4—62no—
476C22⋊C4—62no—
477C22⋊C4—62no—
478C22⋊C4—62no—
479C22⋊C4—62no—
480C22⋊C4—62no—
481C22⋊C4—62no—
482C22⋊C4—62no—
483C42—22no—
484C42—31no—
485C42—31no—
486C42—31no—
487C42—31no—
488C42—31no—
489C42—31no—
490C42—31no—
491C42—31no—
492C42—31no—
493C42—31no—
494C42—31no—
495C42—31no—
496C42—62no—
497C42—62no—
498C42C2×C4×D311yescharacteristic
499C42D3×D411yescharacteristic
500C42D3×D411yescharacteristic
501C42C23×D311yescharacteristic
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3C22×C22⋊C411yescharacteristic
506C2×D3—21no—
507C2×D3—21no—
508C2×D3—21no—
509C2×D3—21no—
510C2×D3—21no—
511C2×D3—44no—
512C2×D3—42no—
513C2×D3—42no—
514C2×D3—42no—
515C2×D3#42112yes—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—42no—
522C3×C4—42no—
523C3×C4#41312yes—
524Dic3—21no—
525Dic3—21no—
526Dic3—21no—
527Dic3—21no—
528Dic3—21no—
529Dic3—42no—
530Dic3#42112yes—
531C23—31no—
532C23—31no—
533C23—62no—
534C23—62no—
535C23—121no—
536C23—121no—
537C23D3×C22⋊C411yescharacteristic
538Q8—42no—
539Q8—42no—
540Q8—122no—
541D4—122no—
542D4—122no—
543D4—122no—
544D4—124no—
545D4—124no—
546D4—124no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—31no—
551C2×C4—31no—
552C2×C4—31no—
553C2×C4—31no—
554C2×C4—31no—
555C2×C4—31no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—124no—
573C2×C4—124no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—124no—
577C2×C4D3×C22⋊C411yescharacteristic
578C2×C4D3×C22⋊C411yescharacteristic
579C2×C4D3×C22⋊C411yescharacteristic
580C2×C4C22×C4×D311yescharacteristic
581C2×C4C2×D3×D411yescharacteristic
582C2×C4C2×D3×D411yescharacteristic
583C8—42no—
584C8—42no—
585C8—122no—
586C8—122no—
587C2×C3—41no—
588C2×C3C2×#41311yescharacteristic
589C2×C3C2×#42111yescharacteristic
590C2×C3C2×#42111yescharacteristic
591D3—22no—
592D3—41no—
593D3—82no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—32no—
598C22—61no—
599C22—61no—
600C22—61no—
601C22—61no—
602C22—61no—
603C22—124no—
604C22—122no—
605C22—122no—
606C22—122no—
607C22C2×D3×C22⋊C411yescharacteristic
608C4—21no—
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—32no—
614C4—42no—
615C4—42no—
616C4—61no—
617C4—61no—
618C4—61no—
619C4—61no—
620C4—61no—
621C4—122no—
622C4#475112yes—
623C3#1766211yescharacteristic, complemented, Sylow
624C2—41no—
625C2—62no—
626C2—121no—
627C2—242no—
628C2C2×#475111yescharacteristic
629C2D3×#42111yescharacteristic
630C2D3×#42111yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1537C2
socleC22×C3C22×C22⋊C4
prime radical#86257C2
cosocleC3×C42C24
hypercenter#1537D3
hypocenterC3#17662
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1537

Complements

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

NH ≅ G/Ntype
2. #79331C2semidirect
3. D3×#432C2semidirect
4. C3×#1537C2semidirect
5. #71823C2semidirect
6. #70811C2semidirect
7. #70811C2semidirect
8. #68368C2semidirect
9. #68361C2semidirect
46. C3×#330C22semidirect
47. #4543C22semidirect
48. #4464C22semidirect
49. #4464C22semidirect
54. #1537D3semidirect
186. #330C2×D3semidirect
623. C3#17662semidirect

Maximal subgroups

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

subgroupindex
#862572
#793312
D3×#4322
C3×#15372
#718232
#708112
#683682
#683612
#176623