Finite Group Databasebeta

← #10762188

Subgroups — #10762188

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86221C211yesmaximal, characteristic
2#79052C211yesmaximal, characteristic, complemented
3C4×#901C211yesmaximal, characteristic, complemented
4C3×#1492C211yesmaximal, characteristic, complemented
5#71801C211yesmaximal, characteristic, complemented
6#70497C211yesmaximal, characteristic, complemented
7#69734C211yesmaximal, characteristic, complemented
8#68176C212yesmaximal, complemented
9#68174C212yesmaximal, complemented
10#68171C212yesmaximal, complemented
11#68165C212yesmaximal, complemented
12#17520—31nomaximal, Sylow
13C3×#532C2211yescharacteristic
14#5832—22no—
15C2×#901—21no—
16#5734C411yescharacteristic
17#5722C411yescharacteristic
18#5718C411yescharacteristic
19#5718C411yescharacteristic
20#5670—22no—
21#5616—21no—
22#5616C412yes—
23#5599—21no—
24#5599C412yes—
25#5548—21no—
26#5548C2212yes—
27C4×#237—21no—
28C4×#237C2212yes—
29C4×#235C2211yescharacteristic
30C4×#235C2211yescharacteristic
31C3×#423C2211yescharacteristic
32C2×C3×C4×C8C2211yescharacteristic
33#5128—22no—
34#5101—21no—
35#4925C2211yescharacteristic
36C4×#208C2211yescharacteristic
37#4838—21no—
38#4837C2212yes—
39#4833C2212yes—
40#4830—21no—
41#4689—22no—
42C4×ΓL(1,𝔽25)C2212yes—
43C4×D3×C8C2212yes—
44C3×D4⋊C8C2212yes—
45C3×D4⋊C8C2212yes—
46D4⋊(C3⋊C8)C2212yes—
47D4⋊(C3⋊C8)C2212yes—
48#4484C2212yes—
49#4483C2212yes—
50#4459C2211yescharacteristic
51#4459C2211yescharacteristic
52#4456C2211yescharacteristic
53#4456C2211yescharacteristic
54#3483—31no—
55#2991—31no—
56C4×#131—31no—
57#1492—32no—
58#1492D311yescharacteristic, complemented
59#1489—31no—
60#1489—32no—
61#1484—32no—
62#1484—31no—
63#1476—31no—
64#1476—32no—
65C2×C3×#41—21no—
66C2×#237—21no—
67C2×#237C2×C412yes—
68C2×#235—22no—
69C2×#235C2×C411yescharacteristic
70C2×#235D411yescharacteristic
71C2×C3×M16—21no—
72C22×C3×C8—21no—
73C3×C4⋊Q8C2×C411yescharacteristic
74C3×#127C2×C411yescharacteristic
75C3×#124—21no—
76C3×#121—21no—
77C3×#121C2×C412yes—
78C3×C4×D4—21no—
79C3×C4×D4C2312yes—
80C2×C3×C42C2311yescharacteristic
81#947—22no—
82#944—22no—
83#940—22no—
84#940—21no—
85#940C2×C412yes—
86#939—22no—
87#939—22no—
88#939—22no—
89#939—22no—
90#938—21no—
91#938C2×C412yes—
92#936—22no—
93D4×Dic3—21no—
94D4×Dic3C2×C412yes—
95C4×#67—21no—
96C4×#67—21no—
97C4×#67—22no—
98C4×#67—22no—
99C4×#67C2×C412yes—
100C4×#67D412yes—
101#926C2×C411yescharacteristic
102#926D411yescharacteristic
103C2×C4×Dic3—22no—
104C2×#208—21no—
105#901—28no—
106C2×ΓL(1,𝔽25)—22no—
107C2×ΓL(1,𝔽25)—22no—
108C2×D3×C8—22no—
109#895—21no—
110#895C2×C412yes—
111#892—21no—
112#892C2×C412yes—
113#888—21no—
114#888C2×C412yes—
115#887—21no—
116#887C2×C412yes—
117#885—22no—
118#885—22no—
119#884—22no—
120#884—22no—
121#67⋊C4—22no—
122#67⋊C4—22no—
123#877—22no—
124#877—22no—
125C4×D12C2×C411yescharacteristic
126C4×D12C2×C411yescharacteristic
127C4×D12D411yescharacteristic
128C4×D12C2311yescharacteristic
129#872—21no—
130#872D412yes—
131C42×D3—21no—
132C42×D3C2312yes—
133C4×Dic6C2×C411yescharacteristic
134C4×Dic6C2×C411yescharacteristic
135C4×Dic6D411yescharacteristic
136C4×Dic6C2311yescharacteristic
137C3×C4⋊C8—21no—
138C3×C4⋊C8C2312yes—
139C3×C4×C8—21no—
140C3×C4×C8C2311yescharacteristic
141C3×C4×C8C2311yescharacteristic
142#830—21no—
143(C2×D3)⋊C8—22no—
144#815—22no—
145Dic3⋊C8—22no—
146C8×Dic3—22no—
147C8×Dic3—22no—
148C4⋊(C3⋊C8)C2311yescharacteristic
149C4⋊(C3⋊C8)C2311yescharacteristic
150#803—21no—
151C4×C3⋊C8—21no—
152C4×C3⋊C8C2312yes—
153C2×#131—61no—
154Q82—31no—
155#555—31no—
156#555—32no—
157#555—61no—
158#555—62no—
159#550—31no—
160#550—31no—
161#550—32no—
162#550—61no—
163#532—32no—
164#532—61no—
165#532—62no—
166#532C2×D311yescharacteristic
167C4×#41—31no—
168C4×#41—32no—
169C4×#41—31no—
170C4×#41—61no—
171C8×Q8—61no—
172#436—61no—
173#436—62no—
174#431—31no—
175#431—32no—
176#423—32no—
177#423C2×D311yescharacteristic
178#405—62no—
179#405—61no—
180C4×M16—31no—
181C4×M16—32no—
182C2×C4×C8—32no—
183C2×C4×C8C2×D311yescharacteristic
184Q8⋊C8—32no—
185Q8⋊C8—31no—
186Q8⋊C8—31no—
187D4⋊C8—32no—
188D4⋊C8—32no—
189D4⋊C8—32no—
190D4⋊C8—31no—
191D4⋊C8—31no—
192D4⋊C8C2×D312yes—
193D4⋊C8C2×D312yes—
194C3×#41—81no—
195C3×#41—81no—
196C2×C3×Q8—21no—
197C2×C3×D4—22no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4C22×C412yes—
201C22×C3×C4—22no—
202C22×C3×C4C22×C411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C2×#67—21no—
206C2×#67—22no—
207C2×#67—22no—
208C2×#67—22no—
209C2×#67—22no—
210C2×#67C22⋊C412yes—
211C22×Dic3—22no—
212C22×Dic3—22no—
213#237—24no—
214#237—42no—
215#237—44no—
216#235—24no—
217#235—22no—
218#235—41no—
219#235—82no—
220#235—82no—
221C2×D12—22no—
222C2×D12C22⋊C411yescharacteristic
223C2×D12C22×C411yescharacteristic
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3C22×C412yes—
228C2×C4×D3C2×D412yes—
229C2×C4×D3C2×D412yes—
230C2×Dic6—22no—
231C2×Dic6C22⋊C411yescharacteristic
232C2×Dic6C22×C411yescharacteristic
233C3×M16—81no—
234C2×C3×C8—24no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—22no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4C22×C412yes—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C42C22×C411yescharacteristic
243C3×C42C2×D411yescharacteristic
244C3×C42C2×D411yescharacteristic
245C3×C42C2411yescharacteristic
246#217—21no—
247#217—22no—
248#217—22no—
249#217C22⋊C412yes—
250(C2×D3)⋊C4—22no—
251(C2×D3)⋊C4—21no—
252(C2×D3)⋊C4—21no—
253(C2×D3)⋊C4—22no—
254(C2×D3)⋊C4C22⋊C412yes—
255(C2×D3)⋊C4C22⋊C412yes—
256C4⋊Dic3C22⋊C411yescharacteristic
257C4⋊Dic3C22⋊C411yescharacteristic
258C4⋊Dic3C22×C411yescharacteristic
259C4⋊Dic3C22×C411yescharacteristic
260Dic3⋊C4—21no—
261Dic3⋊C4—21no—
262Dic3⋊C4—22no—
263Dic3⋊C4—22no—
264Dic3⋊C4C22⋊C412yes—
265Dic3⋊C4C22⋊C412yes—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—22no—
270C4×Dic3—22no—
271C4×Dic3C22×C412yes—
272C4×Dic3C2×D412yes—
273C4×Dic3C2×D412yes—
274#208—24no—
275C2×C3⋊C8—22no—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278ΓL(1,𝔽25)—28no—
279ΓL(1,𝔽25)—82no—
280D3×C8—28no—
281C2×#41—21no—
282C2×#41—31no—
283C2×#41—32no—
284C2×#41—31no—
285C2×#41—62no—
286C2×#41—61no—
287#131—68no—
288C2×M16—21no—
289C2×M16—62no—
290C2×M16—62no—
291C2×M16—61no—
292C22×C8—21no—
293C22×C8—62no—
294C4⋊Q8—32no—
295C4⋊Q8—61no—
296C4⋊Q8—62no—
297C4⋊Q8C4×D311yescharacteristic
298#127—32no—
299#127—62no—
300#127—61no—
301#127C4×D311yescharacteristic
302#124—21no—
303#124—62no—
304#124—62no—
305#124—62no—
306#124—62no—
307#121—21no—
308#121—32no—
309#121—32no—
310#121—62no—
311#121—62no—
312#121—62no—
313#121—62no—
314#121—61no—
315#121—61no—
316#121—62no—
317#121—62no—
318#121C4×D312yes—
319C4×Q8—31no—
320C4×Q8—31no—
321C4×Q8—32no—
322C4×Q8—31no—
323C4×Q8—31no—
324C4×Q8—61no—
325C4×D4—21no—
326C4×D4—32no—
327C4×D4—32no—
328C4×D4—32no—
329C4×D4—31no—
330C4×D4—31no—
331C4×D4—31no—
332C4×D4—31no—
333C4×D4—62no—
334C4×D4—62no—
335C4×D4—62no—
336C4×D4—61no—
337C4×D4—62no—
338C4×D4—61no—
339C4×D4—61no—
340C4×D4C22×D312yes—
341#117—31no—
342#117—32no—
343#117—31no—
344#117—32no—
345#117—62no—
346#117—62no—
347#117—61no—
348#117—61no—
349C2×C42—32no—
350C2×C42—62no—
351C2×C42—61no—
352C2×C42C22×D311yescharacteristic
353C4⋊C8—21no—
354C4⋊C8—31no—
355C4⋊C8—31no—
356C4⋊C8—62no—
357C4⋊C8C22×D312yes—
358C22⋊C8—62no—
359C22⋊C8—61no—
360#97—62no—
361#97—61no—
362C4×C8—21no—
363C4×C8—32no—
364C4×C8—62no—
365C4×C8—62no—
366C4×C8—61no—
367C4×C8C22×D311yescharacteristic
368C4×C8C22×D311yescharacteristic
369C23×C3—22no—
370C23×C3C2×C22⋊C411yescharacteristic
371C22×D3—21no—
372C22×D3C2×C22⋊C412yes—
373C3×Q8—81no—
374C3×D4—22no—
375C3×D4—22no—
376C3×D4—42no—
377C3×D4—81no—
378C3×D4—82no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—42no—
386C2×C3×C4—42no—
387C2×C3×C4C2×C22⋊C411yescharacteristic
388C2×C3×C4C2×C22⋊C411yescharacteristic
389C2×C3×C4C2×C22⋊C411yescharacteristic
390C2×C3×C4C23×C411yescharacteristic
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C22×D411yescharacteristic
393#67—24no—
394#67—44no—
395#67—42no—
396#67—44no—
397#67—82no—
398#67—82no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—44no—
403C2×Dic3—44no—
404C2×Dic3C2×C22⋊C412yes—
405C2×Dic3C2×C22⋊C412yes—
406C2×Dic3C2×C22⋊C412yes—
407D12—21no—
408D12—21no—
409D12—22no—
410D12—82no—
411C4×D3—22no—
412C4×D3—22no—
413C4×D3—22no—
414C4×D3—22no—
415C4×D3—24no—
416C4×D3—42no—
417C4×D3—41no—
418C4×D3—41no—
419C4×D3—42no—
420Dic6—21no—
421Dic6—82no—
422Dic6#10414yes—
423Dic6#13112yescomplemented
424C3×C8—22no—
425C3×C8—22no—
426C3×C8—42no—
427C3⋊C8—24no—
428C3⋊C8—41no—
429C3⋊C8—41no—
430#41—64no—
431#41—62no—
432#41—64no—
433#41—81no—
434#41—81no—
435#41—121no—
436#41—122no—
437#41—124no—
438#41—242no—
439#41—242no—
440C2×Q8—21no—
441C2×Q8—31no—
442C2×Q8—31no—
443C2×Q8—62no—
444C2×D4—22no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—32no—
448C2×D4—31no—
449C2×D4—31no—
450C2×D4—62no—
451C2×D4—62no—
452C2×D4—62no—
453C2×D4—62no—
454C2×D4—61no—
455C2×D4—62no—
456C2×D4C2×C4×D312yes—
457C22×C4—22no—
458C22×C4—32no—
459C22×C4—32no—
460C22×C4—32no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4C2×C4×D311yescharacteristic
467C22×C4D3×D411yescharacteristic
468C22×C4D3×D411yescharacteristic
469M16—68no—
470M16—64no—
471M16—81no—
472M16—242no—
473C2×C8—24no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—22no—
477C2×C8—68no—
478C2×C8—62no—
479C2×C8—61no—
480C2×C8—61no—
481C4⋊C4—21no—
482C4⋊C4—32no—
483C4⋊C4—31no—
484C4⋊C4—32no—
485C4⋊C4—31no—
486C4⋊C4—31no—
487C4⋊C4—31no—
488C4⋊C4—62no—
489C4⋊C4—62no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4C2×C4×D312yes—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—32no—
496C22⋊C4—32no—
497C22⋊C4—32no—
498C22⋊C4—62no—
499C22⋊C4—62no—
500C22⋊C4—62no—
501C22⋊C4—62no—
502C22⋊C4—61no—
503C22⋊C4—61no—
504C22⋊C4—61no—
505C42—32no—
506C42—32no—
507C42—32no—
508C42—62no—
509C42—62no—
510C42—61no—
511C42—61no—
512C42—61no—
513C42C2×C4×D311yescharacteristic
514C42D3×D411yescharacteristic
515C42D3×D411yescharacteristic
516C42C23×D311yescharacteristic
517C22×C3—21no—
518C22×C3—21no—
519C22×C3—21no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—42no—
523C22×C3C22×C22⋊C411yescharacteristic
524C2×D3—22no—
525C2×D3—22no—
526C2×D3—22no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—41no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—42no—
534C3×C4C2×#10412yes—
535C3×C4C2×#13111yescharacteristic
536C3×C4#56811yescharacteristic
537C3×C4#57611yescharacteristic
538C3×C4#57711yescharacteristic
539Dic3—22no—
540Dic3—22no—
541Dic3—22no—
542Dic3—41no—
543Dic3—41no—
544Dic3C2×#10414yes—
545C23—22no—
546C23—32no—
547C23—61no—
548C23D3×C22⋊C411yescharacteristic
549Q8—34no—
550Q8—32no—
551Q8—61no—
552Q8—81no—
553Q8—242no—
554D4—22no—
555D4—22no—
556D4—42no—
557D4—64no—
558D4—61no—
559D4—61no—
560D4—62no—
561D4—81no—
562D4—82no—
563D4—124no—
564D4—122no—
565D4—124no—
566D4—242no—
567D4—242no—
568D4—242no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—32no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—64no—
581C2×C4—61no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—122no—
589C2×C4—124no—
590C2×C4—124no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—122no—
594C2×C4D3×C22⋊C411yescharacteristic
595C2×C4D3×C22⋊C411yescharacteristic
596C2×C4D3×C22⋊C411yescharacteristic
597C2×C4C22×C4×D311yescharacteristic
598C2×C4C2×D3×D411yescharacteristic
599C2×C4C2×D3×D411yescharacteristic
600C8—22no—
601C8—22no—
602C8—42no—
603C8—64no—
604C8—121no—
605C8—121no—
606C2×C3—41no—
607C2×C3—82no—
608C2×C3#288011yescharacteristic
609C2×C3#289711yescharacteristic
610C2×C3C22×#10411yescharacteristic
611D3—42no—
612D3—81no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—62no—
620C22—62no—
621C22—62no—
622C22—121no—
623C22—121no—
624C22—121no—
625C22C2×D3×C22⋊C411yescharacteristic
626C4—21no—
627C4—21no—
628C4—21no—
629C4—34no—
630C4—42no—
631C4—62no—
632C4—62no—
633C4—62no—
634C4—121no—
635C4—121no—
636C4D3×#10412yes—
637C4D3×#13111yescharacteristic
638C4#575011yescharacteristic
639C4#575711yescharacteristic
640C4#576211yescharacteristic
641C3#1752011yescharacteristic, complemented, Sylow
642C2—41no—
643C2—82no—
644C2—122no—
645C2—241no—
646C2#7866311yescharacteristic
647C2#7874211yescharacteristic
648C2C2×D3×#10411yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1492C2
socleC22×C3C22×C22⋊C4
prime radical#86221C2
cosocleC3×C42C24
hypercenter#1492D3
hypocenterC3#17520
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1492

Complements

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

NH ≅ G/Ntype
2. #79052C2semidirect
3. C4×#901C2semidirect
4. C3×#1492C2semidirect
5. #71801C2semidirect
6. #70497C2semidirect
7. #69734C2semidirect
8. #68176C2semidirect
9. #68174C2semidirect
10. #68171C2semidirect
11. #68165C2semidirect
58. #1492D3semidirect
423. Dic6#131semidirect
641. C3#17520semidirect

Maximal subgroups

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

subgroupindex
#862212
#790522
C4×#9012
C3×#14922
#718012
#704972
#697342
#681762
#681742
#681712
#681652
#175203