Finite Group Databasebeta

← #10858715

Subgroups — #10858715

Order 768.

714 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79246C211yesmaximal, characteristic, complemented
2#79210C211yesmaximal, characteristic
3#79162C211yesmaximal, characteristic
4#78571C211yesmaximal, characteristic
5C3×#1952C211yesmaximal, characteristic, complemented
6#70722C211yesmaximal, characteristic
7#70717C211yesmaximal, characteristic, complemented
8#70680C211yesmaximal, characteristic
9#70677C211yesmaximal, characteristic, complemented
10#70644C211yesmaximal, characteristic, complemented
11#69899C211yesmaximal, characteristic, complemented
12#69007C211yesmaximal, characteristic
13#69004C211yesmaximal, characteristic
14#68980C211yesmaximal, characteristic, complemented
15#68978C211yesmaximal, characteristic, complemented
16#26406—31nomaximal, Sylow
17C2×#909C2211yescharacteristic
18C2×#907C2211yescharacteristic
19#5523C2211yescharacteristic
20#5516C2211yescharacteristic
21#5504C2211yescharacteristic
22#5500C2211yescharacteristic
23C2×C3×#108C2211yescharacteristic
24C3×#428C2211yescharacteristic
25C3×#428C2211yescharacteristic
26C2×C3×#107C2211yescharacteristic
27C3×#405C2211yescharacteristic
28C2×#825C2211yescharacteristic
29C2×#824C2211yescharacteristic
30#5121C2211yescharacteristic
31#5121C2211yescharacteristic
32#5120C2211yescharacteristic
33#5119C2211yescharacteristic
34#5113C2211yescharacteristic
35C2×#816C2211yescharacteristic
36#5102C2211yescharacteristic
37#4998C2211yescharacteristic
38#4975C2211yescharacteristic
39#4973C2211yescharacteristic
40C2×#810C2211yescharacteristic
41#4966C2211yescharacteristic
42#4964C2211yescharacteristic
43C2×#807C2211yescharacteristic
44#4891—21no—
45#4890—21no—
46#4881—21no—
47D3×#107—21no—
48#4877—21no—
49#4877—21no—
50#4875—21no—
51#4875—21no—
52#4861—21no—
53#4860—21no—
54#4857—21no—
55#4854—21no—
56#4716—21no—
57#4710—21no—
58C4×Dic12—21no—
59C4×#204—21no—
60C3×#343C2211yescharacteristic
61C3×#343C2211yescharacteristic
62#4558C2211yescharacteristic
63#4555C2211yescharacteristic
64#4555C2211yescharacteristic
65#4554C2211yescharacteristic
66#4550C2211yescharacteristic
67#4532C2211yescharacteristic
68#3083—31no—
69#3082—31no—
70#2947—31no—
71#2913—31no—
72#2095—31no—
73#2089—31no—
74#2064—31no—
75#2060—31no—
76#1952D311yescharacteristic, complemented
77#1922—31no—
78#1908—31no—
79#1882—31no—
80#1866—31no—
81#1859—31no—
82#1857—31no—
83C2×#235C2311yescharacteristic
84C22×Dic6C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×#117C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C4×#67—41no—
93#926C2311yescharacteristic
94#926C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96#924—41no—
97C2×Dic3⋊C4—41no—
98C2×C4×Dic3C2311yescharacteristic
99C2×#208C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101#909—22no—
102#909Q814yescomplemented
103#907—22no—
104#907—22no—
105C2×Dic12—21no—
106C2×#204—21no—
107C2×ΓL(1,𝔽25)—21no—
108C2×D3×C8—21no—
109#893—21no—
110#892—21no—
111#892—21no—
112D3×C4⋊C4—21no—
113D3×C4⋊C4—21no—
114#890—21no—
115#890—21no—
116#888—21no—
117#888—21no—
118#887—21no—
119#887—21no—
120#887—21no—
121#67⋊C4—41no—
122#877—41no—
123C4×D12—21no—
124C4×Dic6—21no—
125C4×Dic6—21no—
126C4×Dic6—21no—
127C3×#108—22no—
128C3×#107—21no—
129C3×#107—21no—
130C3×#107—21no—
131C3×#107—21no—
132C3×#106—21no—
133C3×#106—21no—
134C3×#97—21no—
135C3×C4×C8—21no—
136C3⋊#95—41no—
137#825—22no—
138#824—22no—
139#821D411yescharacteristic
140#821D411yescharacteristic
141#818—21no—
142#818—21no—
143#817—21no—
144#817—21no—
145#816—21no—
146#816—21no—
147#816—21no—
148#816—21no—
149#816D411yescharacteristic
150#816D411yescharacteristic
151Dic3⋊C8—21no—
152Dic3⋊C8—21no—
153Dic3⋊C8—21no—
154Dic3⋊C8—21no—
155#810—21no—
156#810—21no—
157#810—21no—
158#809—21no—
159#808—21no—
160#808D411yescharacteristic
161#808D411yescharacteristic
162#807—21no—
163#807—21no—
164#807—21no—
165#807D411yescharacteristic
166#807D411yescharacteristic
167C2×#137—31no—
168C2×#131—31no—
169#544—31no—
170#520—31no—
171#520—31no—
172#519—31no—
173#479—61no—
174#479—61no—
175#479—61no—
176#478—61no—
177#477—61no—
178#477—61no—
179#475—61no—
180#474—61no—
181#441—61no—
182#440—61no—
183C4×Q16—61no—
184C4×SD16—61no—
185C2×#108C2×D311yescharacteristic
186#428—31no—
187#428—31no—
188#428—31no—
189#428—61no—
190#428—61no—
191#428C2×D311yescharacteristic
192#428C2×D311yescharacteristic
193#427—31no—
194#427—61no—
195C2×#107—31no—
196C2×#107—61no—
197C2×#107C2×D311yescharacteristic
198#424—31no—
199#424—31no—
200C2×#104—31no—
201#419—31no—
202#417—31no—
203#416—31no—
204C2×#103—31no—
205C2×#103—31no—
206C2×#101—31no—
207#405C2×D311yescharacteristic
208#391—31no—
209#384—31no—
210#343—31no—
211#343C2×D311yescharacteristic
212#343C2×D311yescharacteristic
213#341—31no—
214#339—31no—
215#339—31no—
216#339—31no—
217#336—31no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4C2411yescharacteristic
221C2×#67—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225#235—21no—
226#235—21no—
227#235C2×Q812yes—
228#235#4112yes—
229C2×D12C2×D411yescharacteristic
230C2×C4×D3—21no—
231C2×C4×D3—41no—
232C2×C4×D3—41no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—41no—
239C2×Dic6C2×D411yescharacteristic
240C2×Dic6C2×Q812yes—
241C3×M16—21no—
242C3×M16—21no—
243C3×M16—21no—
244C3×M16—21no—
245C3×M16C2×Q812yes—
246C3×M16#4112yes—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8C2×D411yescharacteristic
253C2×C3×C8C2×D411yescharacteristic
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4C2×D411yescharacteristic
259C3×C4⋊C4C2×D411yescharacteristic
260C3×C22⋊C4—21no—
261C3×C42—21no—
262#217—21no—
263#217—21no—
264(C2×D3)⋊C4—41no—
265(C2×D3)⋊C4—41no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—41no—
269C4⋊Dic3C2×D411yescharacteristic
270C4⋊Dic3C2×D411yescharacteristic
271C4⋊Dic3C2×D411yescharacteristic
272C4⋊Dic3C2×D411yescharacteristic
273Dic3⋊C4—41no—
274Dic3⋊C4—41no—
275Dic3⋊C4—41no—
276Dic3⋊C4—41no—
277Dic3⋊C4—41no—
278Dic3⋊C4—41no—
279Dic3⋊C4—82no—
280C4×Dic3—21no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—21no—
284C4×Dic3—41no—
285C4×Dic3—41no—
286C4×Dic3—41no—
287C4×Dic3—41no—
288#208—21no—
289#208—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8C2×D411yescharacteristic
294C2×C3⋊C8C2×D411yescharacteristic
295Dic12—24no—
296#204—24no—
297ΓL(1,𝔽25)—24no—
298D3×C8—24no—
299C2×#41—31no—
300C22×Q8—31no—
301#137—34no—
302#137—62no—
303C2×Q16—61no—
304C2×SD16—61no—
305#131—62no—
306#131—62no—
307C2×M16—31no—
308C2×M16—61no—
309C2×M16C22×D311yescharacteristic
310C2×M16C22×D311yescharacteristic
311C2×M16C22×D311yescharacteristic
312C22×C8—31no—
313C22×C8—61no—
314C22×C8C22×D311yescharacteristic
315C4⋊Q8—61no—
316C4⋊Q8—61no—
317#125—61no—
318#125—61no—
319C22⋊Q8—121no—
320C4×Q8—61no—
321C4×Q8—61no—
322C4×Q8—61no—
323C4×Q8—61no—
324C4×Q8—61no—
325C4×Q8—61no—
326C4×D4—61no—
327C4×D4—61no—
328C4×D4—121no—
329C4×D4—121no—
330#117—31no—
331#117—31no—
332#117—61no—
333#117—61no—
334#117—121no—
335#117C22×D311yescharacteristic
336C2×C4⋊C4—31no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—121no—
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C42—31no—
343#108—22no—
344#107—21no—
345#107—21no—
346#107—21no—
347#107—21no—
348#107—31no—
349#107—31no—
350#107—61no—
351#107—61no—
352#107—61no—
353#107—61no—
354#107—61no—
355#106—21no—
356#106—21no—
357#106—31no—
358#106—31no—
359#106—61no—
360#106—61no—
361#106—61no—
362C4⋊C8—61no—
363C4⋊C8—61no—
364C4⋊C8—61no—
365C4⋊C8—61no—
366#104—62no—
367#103—31no—
368#103—31no—
369#103—61no—
370#103—61no—
371#103—61no—
372#103—61no—
373#103—61no—
374#103—61no—
375#103—61no—
376#102—31no—
377#102—31no—
378#102—61no—
379#101—62no—
380#97—21no—
381C4×C8—21no—
382#95—121no—
383C23×C3H(5,𝔽2)11yescharacteristic
384C22×D3—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—81no—
390C2×C3×C4—81no—
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C22×Q811yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C42−1+411yescharacteristic
397#67—22no—
398#67—22no—
399C2×Dic3—21no—
400C2×Dic3—22no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—22no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—81no—
412D12—21no—
413D12C22⋊Q812yes—
414C4×D3—22no—
415C4×D3—22no—
416C4×D3—161no—
417C4×D3—161no—
418Dic6—21no—
419Dic6—22no—
420Dic6—22no—
421Dic6—41no—
422Dic6—41no—
423Dic6—161no—
424Dic6C22⋊Q812yes—
425C3×C8—22no—
426C3×C8—22no—
427C3×C8—21no—
428C3×C8—21no—
429C3×C8—41no—
430C3⋊C8—21no—
431C3⋊C8—21no—
432C3⋊C8—41no—
433#41—32no—
434#41—32no—
435#41—61no—
436#41—61no—
437C2×Q8—31no—
438C2×Q8—32no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×Q8—61no—
442C2×Q8—61no—
443C2×Q8—61no—
444C2×Q8—121no—
445C2×D4—31no—
446C2×D4—61no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—121no—
454C22×C4—121no—
455C22×C4C23×D311yescharacteristic
456Q16—64no—
457SD16—64no—
458M16—21no—
459M16—21no—
460M16—21no—
461M16—21no—
462M16—64no—
463M16—61no—
464M16—61no—
465M16#23512yes—
466M16D3×Q812yes—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—31no—
473C2×C8—31no—
474C2×C8—64no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8D3×D411yescharacteristic
479C2×C8D3×D411yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—31no—
485C4⋊C4—31no—
486C4⋊C4—31no—
487C4⋊C4—31no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—121no—
491C4⋊C4—121no—
492C4⋊C4—121no—
493C4⋊C4—121no—
494C4⋊C4—121no—
495C4⋊C4—121no—
496C4⋊C4—121no—
497C4⋊C4—242no—
498C4⋊C4D3×D411yescharacteristic
499C4⋊C4D3×D411yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—61no—
502C22⋊C4—61no—
503C22⋊C4—121no—
504C22⋊C4—121no—
505C42—21no—
506C42—61no—
507C42—61no—
508C42—61no—
509C42—61no—
510C42—121no—
511C42—121no—
512C42—121no—
513C42—121no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3#53911yescharacteristic
517C22×C3D4211yescharacteristic
518C22×C3#55411yescharacteristic
519C2×D3—22no—
520C2×D3—41no—
521C2×D3—41no—
522C3×C4—41no—
523C3×C4—41no—
524C3×C4—81no—
525C3×C4—81no—
526C3×C4C2×C22⋊Q811yescharacteristic
527C3×C4#52511yescharacteristic
528C3×C4#54711yescharacteristic
529C3×C4D4×Q811yescharacteristic
530Dic3—22no—
531Dic3—42no—
532Dic3—42no—
533Dic3—41no—
534Dic3—41no—
535Dic3—41no—
536Dic3—41no—
537Dic3—81no—
538C23—61no—
539C23#100911yescharacteristic
540Q8—32no—
541Q8—61no—
542Q8—62no—
543Q8—62no—
544Q8—121no—
545Q8—121no—
546Q8—481no—
547D4—32no—
548D4—62no—
549D4—62no—
550D4—61no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—61no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—241no—
572C2×C4—481no—
573C2×C4—481no—
574C2×C4C2×#23511yescharacteristic
575C2×C4C2×D3×D411yescharacteristic
576C2×C4C2×D3×D411yescharacteristic
577C2×C4C2×D3×Q811yescharacteristic
578C2×C4#100711yescharacteristic
579C2×C4D3×#4111yescharacteristic
580C8—21no—
581C8—21no—
582C8—22no—
583C8—22no—
584C8—41no—
585C8—61no—
586C8—61no—
587C8—121no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3#244911yescharacteristic
591C2×C3#328811yescharacteristic
592C2×C3#329311yescharacteristic
593D3—42no—
594C22—21no—
595C22—21no—
596C22—62no—
597C22—121no—
598C22—121no—
599C22#557911yescharacteristic
600C22#560911yescharacteristic
601C22#569211yescharacteristic
602C4—41no—
603C4—41no—
604C4—62no—
605C4—81no—
606C4—81no—
607C4—122no—
608C4—121no—
609C4—121no—
610C4—121no—
611C4—121no—
612C4—122no—
613C4—241no—
614C4#550711yescharacteristic
615C4#554711yescharacteristic
616C4D3×C22⋊Q811yescharacteristic
617C4#572711yescharacteristic
618C3#2640611yescharacteristic, complemented, Sylow
619C2—21no—
620C2—21no—
621C2—122no—
622C2#7622511yescharacteristic
623C2#8093911yescharacteristic
624C2#8094711yescharacteristic
subgroupG/H
centerC22#5609
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1952C2
socleC22×C3D42
prime radicalC2×#235C23
hypercenter#1952D3
hypocenterC3#26406
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1952

Complements

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

NH ≅ G/Ntype
1. #79246C2semidirect
5. C3×#1952C2semidirect
7. #70717C2semidirect
9. #70677C2semidirect
10. #70644C2semidirect
11. #69899C2semidirect
14. #68980C2semidirect
15. #68978C2semidirect
76. #1952D3semidirect
102. #909Q8semidirect
618. C3#26406semidirect

Maximal subgroups

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

subgroupindex
#792462
#792102
#791622
#785712
C3×#19522
#707222
#707172
#706802
#706772
#706442
#698992
#690072
#690042
#689802
#689782
#264063