Finite Group Databasebeta

← #10877187

Subgroups — #10877187

Order 768.

704 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82139C211yesmaximal, characteristic
2C2×#4964C211yesmaximal, characteristic, complemented
3#78627C211yesmaximal, characteristic
4#78623C211yesmaximal, characteristic
5C3×#1821C211yesmaximal, characteristic, complemented
6#69983C211yesmaximal, characteristic, complemented
7#69974C211yesmaximal, characteristic, complemented
8#69011C211yesmaximal, characteristic, complemented
9#69010C211yesmaximal, characteristic, complemented
10#69001C211yesmaximal, characteristic, complemented
11#69000C211yesmaximal, characteristic, complemented
12#68985C211yesmaximal, characteristic, complemented
13#68984C211yesmaximal, characteristic, complemented
14#68981C211yesmaximal, characteristic, complemented
15#68980C211yesmaximal, characteristic, complemented
16#27676—31nomaximal, Sylow
17C22×C3×C4⋊C4C2211yescharacteristic
18#5791C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×#208C2211yescharacteristic
21C2×#907C2211yescharacteristic
22C3×#407C2211yescharacteristic
23C3×#407C2211yescharacteristic
24#5207C2211yescharacteristic
25#5133C2211yescharacteristic
26#5133C2211yescharacteristic
27#5129C2211yescharacteristic
28C2×#822C2211yescharacteristic
29C2×#822C2211yescharacteristic
30#5122C2211yescharacteristic
31#5121C2211yescharacteristic
32#5121C2211yescharacteristic
33#5119C2211yescharacteristic
34#4971—21no—
35#4970—21no—
36#4968—21no—
37#4967—21no—
38#4966C2211yescharacteristic
39#4966C2211yescharacteristic
40#4964—21no—
41#4964—21no—
42#4964—21no—
43#4964—21no—
44#4964C2211yescharacteristic
45#4964C2211yescharacteristic
46C2×#808C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#807C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4737—21no—
51#4737—21no—
52#4729—21no—
53#4729—21no—
54#4728—21no—
55#4728—21no—
56#4723—21no—
57#4723—21no—
58C3×#343C2211yescharacteristic
59C3×#343C2211yescharacteristic
60C3×#343C2211yescharacteristic
61C3×#343C2211yescharacteristic
62#4558C2211yescharacteristic
63#4558C2211yescharacteristic
64#4554C2211yescharacteristic
65#4554C2211yescharacteristic
66#4554C2211yescharacteristic
67#4554C2211yescharacteristic
68#3093—31no—
69#3092—31no—
70#2930—31no—
71C2×#428—31no—
72#2090—31no—
73#2089—31no—
74#2079—31no—
75#2078—31no—
76#1952—31no—
77#1951—31no—
78#1822—31no—
79#1821D311yescharacteristic, complemented
80#1816—31no—
81#1815—31no—
82#1811—31no—
83C23×C3×C4—21no—
84C23×C3×C4C2311yescharacteristic
85C2×#235C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C2×C3×M16C2311yescharacteristic
89C2×C3×M16C2311yescharacteristic
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97#953—21no—
98C2×#217—21no—
99#930D411yescharacteristic
100#930D411yescharacteristic
101#929—21no—
102C4×#67—21no—
103C4×#67—21no—
104#926—21no—
105#926—21no—
106#926—21no—
107#926—21no—
108#926C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111#924D411yescharacteristic
112#924D411yescharacteristic
113C2×C4×Dic3—21no—
114C2×#208—22no—
115C2×#208—22no—
116C2×#208D411yescharacteristic
117C2×#208D411yescharacteristic
118C2×#208D411yescharacteristic
119C2×#208D411yescharacteristic
120C2×#208C2311yescharacteristic
121C2×#208C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#907—22no—
125#907—22no—
126C2×ΓL(1,𝔽25)—21no—
127C2×D3×C8—21no—
128C3×C22⋊C8—21no—
129C3×C22⋊C8—21no—
130C3×C22⋊C8—21no—
131C3×C22⋊C8—21no—
132#830—21no—
133#830—21no—
134#822—22no—
135#822—22no—
136#821—21no—
137#821—21no—
138(C2×D3)⋊C8—21no—
139(C2×D3)⋊C8—21no—
140#818—21no—
141#818—21no—
142#817—21no—
143#817—21no—
144#816—21no—
145#816—21no—
146#815—21no—
147Dic3⋊C8—21no—
148Dic3⋊C8—21no—
149C8×Dic3—21no—
150#810—21no—
151#810—21no—
152#809—21no—
153#809—21no—
154#808—21no—
155#808—21no—
156#808—21no—
157#808—21no—
158#808—21no—
159#808—21no—
160#807—21no—
161#807—21no—
162#807—21no—
163#807—21no—
164#807—21no—
165#807—21no—
166C2×#131—31no—
167C22×M16—31no—
168#525—31no—
169C2×#117—31no—
170C22×C4⋊C4C2×D311yescharacteristic
171#485—61no—
172#485—61no—
173#484—61no—
174#483—61no—
175#482—61no—
176#482—61no—
177#481—61no—
178#480—61no—
179#436—61no—
180#436—61no—
181#435—61no—
182#435—61no—
183C2×#108—31no—
184C2×#108—31no—
185#428—31no—
186#428—31no—
187#428—31no—
188#428—31no—
189#428—61no—
190#428—61no—
191#428—61no—
192#428—61no—
193C2×#107—31no—
194C2×#107—31no—
195C2×#106—31no—
196C2×#106—31no—
197#424—31no—
198#417—31no—
199#417—31no—
200#417—31no—
201#417—31no—
202#408—31no—
203#407—31no—
204#407C2×D311yescharacteristic
205#407C2×D311yescharacteristic
206#405—31no—
207#343—31no—
208#343—31no—
209#343—31no—
210#343—31no—
211#343C2×D311yescharacteristic
212#343C2×D311yescharacteristic
213#343C2×D311yescharacteristic
214#343C2×D311yescharacteristic
215#341—31no—
216#341—31no—
217C24×C3#4111yescharacteristic
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—41no—
225C22×C3×C4—41no—
226C22×C3×C4—82no—
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4C2×D411yescharacteristic
231C22×C3×C4#4111yescharacteristic
232C22×C3×C4C2411yescharacteristic
233C2×#67—21no—
234C2×#67—21no—
235C2×#67—21no—
236C22×Dic3—21no—
237C22×Dic3—21no—
238#235—21no—
239#235—21no—
240#235—21no—
241#235—21no—
242C2×D12C2×D411yescharacteristic
243C2×C4×D3—21no—
244C2×Dic6C2×D411yescharacteristic
245C3×M16—21no—
246C3×M16—21no—
247C3×M16—21no—
248C3×M16—21no—
249C3×M16—21no—
250C3×M16—21no—
251C3×M16—21no—
252C3×M16—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—41no—
262C3×C4⋊C4—41no—
263#217—21no—
264#217—21no—
265#217—21no—
266#217—41no—
267#217—41no—
268(C2×D3)⋊C4—21no—
269(C2×D3)⋊C4—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3C2×D411yescharacteristic
275C4⋊Dic3C2×D411yescharacteristic
276Dic3⋊C4—21no—
277Dic3⋊C4—21no—
278C4×Dic3—21no—
279C4×Dic3—41no—
280C4×Dic3—41no—
281#208—22no—
282#208—22no—
283#208—22no—
284#208—22no—
285#208—21no—
286#208—21no—
287#208—21no—
288#208—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8C2×D411yescharacteristic
294C2×C3⋊C8C2×D411yescharacteristic
295C2×C3⋊C8C2×D411yescharacteristic
296C2×C3⋊C8C2×D411yescharacteristic
297C2×C3⋊C8#4112yes—
298C2×C3⋊C8#4112yes—
299ΓL(1,𝔽25)—42no—
300D3×C8—42no—
301C2×#41—31no—
302C23×C4—21no—
303C23×C4C22×D311yescharacteristic
304#131—62no—
305#131—62no—
306C2×M16—31no—
307C2×M16—31no—
308C2×M16—31no—
309C2×M16—31no—
310C2×M16—31no—
311C2×M16—31no—
312C2×M16—61no—
313C2×M16—62no—
314C2×M16—62no—
315C2×M16C22×D311yescharacteristic
316C2×M16C22×D311yescharacteristic
317C2×M16C22×D311yescharacteristic
318C2×M16C22×D311yescharacteristic
319C22×C8—31no—
320C22×C8—31no—
321C22×C8—61no—
322#123—61no—
323C22⋊Q8—31no—
324C22⋊Q8—31no—
325#121—31no—
326#121—31no—
327#120—61no—
328C4×D4—61no—
329C4×D4—61no—
330#117—31no—
331#117—61no—
332#117—61no—
333#117—61no—
334#117—61no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4C22×D311yescharacteristic
343C2×C4⋊C4C22×D311yescharacteristic
344C2×C22⋊C4—61no—
345C2×C42—61no—
346#108—62no—
347#108—62no—
348#107—61no—
349#107—61no—
350#107—61no—
351#107—61no—
352#107—61no—
353#107—61no—
354#107—61no—
355#107—61no—
356#106—61no—
357#106—61no—
358#106—61no—
359#106—61no—
360#106—61no—
361#106—61no—
362#106—61no—
363#106—61no—
364C4⋊C8—61no—
365C4⋊C8—61no—
366#103—61no—
367#103—61no—
368#103—61no—
369#103—61no—
370#102—61no—
371#102—61no—
372#102—61no—
373#102—61no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376C22⋊C8—21no—
377C22⋊C8—21no—
378C22⋊C8—61no—
379C22⋊C8—61no—
380C22⋊C8—61no—
381C22⋊C8—61no—
382#97—61no—
383C4×C8—61no—
384C23×C3—21no—
385C23×C3—21no—
386C23×C3—41no—
387C23×C3—41no—
388C23×C3#12111yescharacteristic
389C23×C3#12111yescharacteristic
390C23×C3C2×#4111yescharacteristic
391C22×D3—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—82no—
405C2×C3×C4—82no—
406C2×C3×C4—81no—
407C2×C3×C4—81no—
408C2×C3×C4#12111yescharacteristic
409C2×C3×C4#12111yescharacteristic
410C2×C3×C4C22×D411yescharacteristic
411C2×C3×C4C22×D411yescharacteristic
412C2×C3×C4C2×#4111yescharacteristic
413C2×C3×C4C2×#4111yescharacteristic
414C2×C3×C42−1+411yescharacteristic
415C2×C3×C42−1+411yescharacteristic
416#67—41no—
417#67—41no—
418#67—81no—
419#67—81no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—81no—
426C2×Dic3—81no—
427D12—21no—
428D12—21no—
429C4×D3—41no—
430C4×D3—41no—
431Dic6—21no—
432Dic6—21no—
433C3×C8—41no—
434C3×C8—41no—
435C3×C8—41no—
436C3×C8—41no—
437C3⋊C8—21no—
438C3⋊C8—21no—
439C3⋊C8—41no—
440C3⋊C8—41no—
441C3⋊C8#12312yes—
442C3⋊C8#12312yes—
443C24#23511yescharacteristic
444#41—61no—
445#41—61no—
446#41—61no—
447#41—61no—
448C2×Q8—31no—
449C2×D4—31no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—82no—
465C22×C4#23511yescharacteristic
466C22×C4D3×D411yescharacteristic
467C22×C4D3×D411yescharacteristic
468C22×C4D3×D411yescharacteristic
469C22×C4D3×D411yescharacteristic
470C22×C4C23×D311yescharacteristic
471M16—21no—
472M16—21no—
473M16—21no—
474M16—21no—
475M16—21no—
476M16—21no—
477M16—21no—
478M16—21no—
479M16—62no—
480M16—62no—
481M16—62no—
482M16—62no—
483M16—61no—
484M16—61no—
485M16—61no—
486M16—61no—
487M16—122no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—31no—
493C2×C8—31no—
494C2×C8—32no—
495C2×C8—32no—
496C2×C8—31no—
497C2×C8—31no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—61no—
502C2×C8—122no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—31no—
508C4⋊C4—31no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C22⋊C4—61no—
518C22⋊C4—61no—
519C22⋊C4—61no—
520C22⋊C4—61no—
521C22⋊C4—61no—
522C22⋊C4—121no—
523C22⋊C4—121no—
524C42—61no—
525C42—121no—
526C42—121no—
527C22×C3—21no—
528C22×C3—21no—
529C22×C3—21no—
530C22×C3—21no—
531C22×C3—21no—
532C22×C3—21no—
533C22×C3—21no—
534C22×C3—21no—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3C2×#12111yescharacteristic
538C22×C3#55211yescharacteristic
539C22×C3#55211yescharacteristic
540C2×D3—41no—
541C2×D3—41no—
542C2×D3—41no—
543C3×C4—21no—
544C3×C4—21no—
545C3×C4—81no—
546C3×C4—81no—
547C3×C4C2×#12311yescharacteristic
548C3×C4#53611yescharacteristic
549C3×C4#54711yescharacteristic
550C3×C4#54711yescharacteristic
551Dic3—41no—
552Dic3—41no—
553Dic3—41no—
554Dic3—81no—
555Dic3—81no—
556C23—21no—
557C23—21no—
558C23—41no—
559C23—41no—
560C23—61no—
561C23#88411yescharacteristic
562C23#88411yescharacteristic
563C23C2×#23511yescharacteristic
564Q8—61no—
565Q8—61no—
566D4—61no—
567D4—61no—
568D4—121no—
569D4—121no—
570D4—241no—
571D4—241no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—241no—
596C2×C4—241no—
597C2×C4#88411yescharacteristic
598C2×C4#88411yescharacteristic
599C2×C4C2×D3×D411yescharacteristic
600C2×C4C2×D3×D411yescharacteristic
601C2×C4D3×#4111yescharacteristic
602C2×C4D3×#4111yescharacteristic
603C2×C4#101011yescharacteristic
604C2×C4#101011yescharacteristic
605C8—32no—
606C8—32no—
607C8—41no—
608C8—41no—
609C8—41no—
610C8—41no—
611C8—61no—
612C8—61no—
613C8—121no—
614C8—121no—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3—41no—
618C2×C3—41no—
619C2×C3#248611yescharacteristic
620C2×C3#306811yescharacteristic
621C2×C3#306911yescharacteristic
622D3—81no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—42no—
632C22—42no—
633C22—121no—
634C22—121no—
635C22—121no—
636C22C2×#88411yescharacteristic
637C22#553111yescharacteristic
638C22#553111yescharacteristic
639C4—21no—
640C4—21no—
641C4—81no—
642C4—81no—
643C4—121no—
644C4—121no—
645C4—121no—
646C4—241no—
647C4—241no—
648C4#564811yescharacteristic
649C4D3×#12311yescharacteristic
650C4#565511yescharacteristic
651C4#565511yescharacteristic
652C3#2767611yescharacteristic, complemented, Sylow
653C2—21no—
654C2—21no—
655C2—41no—
656C2—41no—
657C2—241no—
658C2#7407511yescharacteristic
659C2#7870311yescharacteristic
660C2#7871211yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1821C2
socleC22×C3C2×#121
prime radical#5791C22
hypercenter#1821D3
hypocenterC3#27676
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
2. C2×#4964C2semidirect
5. C3×#1821C2semidirect
6. #69983C2semidirect
7. #69974C2semidirect
8. #69011C2semidirect
9. #69010C2semidirect
10. #69001C2semidirect
11. #69000C2semidirect
12. #68985C2semidirect
13. #68984C2semidirect
14. #68981C2semidirect
15. #68980C2semidirect
79. #1821D3semidirect
652. C3#27676semidirect

Maximal subgroups

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

subgroupindex
#821392
C2×#49642
#786272
#786232
C3×#18212
#699832
#699742
#690112
#690102
#690012
#690002
#689852
#689842
#689812
#689802
#276763