Finite Group Databasebeta

← #10858696

Subgroups — #10858696

Order 768.

732 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79146C211yesmaximal, characteristic, complemented
2#79130C211yesmaximal, characteristic, complemented
3#78731C211yesmaximal, characteristic, complemented
4#78726C211yesmaximal, characteristic
5C3×#2092C211yesmaximal, characteristic, complemented
6#70626C211yesmaximal, characteristic, complemented
7#70572C211yesmaximal, characteristic, complemented
8#70218C211yesmaximal, characteristic
9#70197C211yesmaximal, characteristic, complemented
10#69663C211yesmaximal, characteristic
11#69642C211yesmaximal, characteristic, complemented
12#69562C211yesmaximal, characteristic, complemented
13#69543C211yesmaximal, characteristic, complemented
14#68694C211yesmaximal, characteristic, complemented
15#68645C211yesmaximal, characteristic, complemented
16#26405—31nomaximal, Sylow
17#5798C2211yescharacteristic
18C2×#932C2211yescharacteristic
19C2×#901C2211yescharacteristic
20#5504C2211yescharacteristic
21#5504C2211yescharacteristic
22#5500C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#382C2211yescharacteristic
29C2×#833C2211yescharacteristic
30#5215C2211yescharacteristic
31#5113C2211yescharacteristic
32C2×#819C2211yescharacteristic
33#5106C2211yescharacteristic
34#5102C2211yescharacteristic
35#5102C2211yescharacteristic
36#5101C2211yescharacteristic
37#4966C2211yescharacteristic
38#4964C2211yescharacteristic
39#4964C2211yescharacteristic
40#4964C2211yescharacteristic
41#4964C2211yescharacteristic
42#4964C2211yescharacteristic
43C2×#805C2211yescharacteristic
44#4881—21no—
45#4881—21no—
46#4880—21no—
47D3×#107—21no—
48#4871—21no—
49#4870—21no—
50#4857—21no—
51#4854—21no—
52#4785—21no—
53#4766—21no—
54#4763—21no—
55#4761—21no—
56#4759—21no—
57#4758—21no—
58#4757—21no—
59#4756—21no—
60C3×#336C2211yescharacteristic
61#4552C2211yescharacteristic
62#4530C2211yescharacteristic
63#4530C2211yescharacteristic
64#4530C2211yescharacteristic
65#4523C2211yescharacteristic
66#4523C2211yescharacteristic
67#4523C2211yescharacteristic
68#3084—31no—
69#3081—31no—
70#2946—31no—
71#2913—31no—
72#2092D311yescharacteristic, complemented
73#2089—31no—
74#2063—31no—
75#2061—31no—
76#1951—31no—
77#1922—31no—
78#1907—31no—
79#1883—31no—
80#1866—31no—
81#1860—31no—
82#1856—31no—
83C22×C3×D4C2311yescharacteristic
84C2×#235C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—41no—
91C2×C3×C42C2311yescharacteristic
92C2×#217—41no—
93D4×Dic3—21no—
94D4×Dic3—21no—
95D4×Dic3—21no—
96D4×Dic3—21no—
97C2×#214—21no—
98#932—22no—
99#932Q814yescomplemented
100C2×#213—21no—
101C4×#67—41no—
102C4×#67—41no—
103#926C2311yescharacteristic
104#926C2311yescharacteristic
105#926C2311yescharacteristic
106#924—41no—
107C2×#208C2311yescharacteristic
108C2×#208C2311yescharacteristic
109C2×#208C2311yescharacteristic
110C2×#208C2311yescharacteristic
111#901—22no—
112#901—22no—
113C2×ΓL(1,𝔽25)—21no—
114C2×D3×C8—21no—
115#893—21no—
116#893—21no—
117#892—21no—
118#892—21no—
119D3×C4⋊C4—21no—
120D3×C4⋊C4—21no—
121#890—21no—
122#890—21no—
123#888—21no—
124#888—21no—
125#887—21no—
126#887—21no—
127C3×#107—21no—
128C3×#107—21no—
129C3×#106—21no—
130C3×#106—21no—
131C3×#102—21no—
132C3×#102—21no—
133C3×#102—21no—
134C3×#102—21no—
135C3×#95—41no—
136#833—22no—
137#832—21no—
138#832—21no—
139#821D411yescharacteristic
140#821D411yescharacteristic
141#819—22no—
142#818D411yescharacteristic
143#818D411yescharacteristic
144#817D411yescharacteristic
145#817D411yescharacteristic
146#816D411yescharacteristic
147#816D411yescharacteristic
148#815—21no—
149Dic3⋊C8—21no—
150Dic3⋊C8—21no—
151Dic3⋊C8—21no—
152Dic3⋊C8—21no—
153C8×Dic3—21no—
154#810—21no—
155#809—21no—
156#808—21no—
157#808—21no—
158#808—21no—
159#808—21no—
160#808—21no—
161#807—21no—
162#807—21no—
163#807—21no—
164#807—21no—
165#807—21no—
166#805—22no—
167C2×AGL(1,ℤ/8)—31no—
168C2×#131—31no—
169#544—31no—
170#520—31no—
171#520—31no—
172#518—31no—
173#479—61no—
174#478—61no—
175#478—61no—
176#478—61no—
177#476—61no—
178#476—61no—
179#475—61no—
180#474—61no—
181#442—61no—
182#440—61no—
183C4×SD16—61no—
184C4×D8—61no—
185C2×#108—31no—
186#428—31no—
187#428—31no—
188#428—31no—
189#428—31no—
190#428—31no—
191#428—61no—
192#428—61no—
193#427—31no—
194#427—61no—
195C2×#107—61no—
196C2×#107C2×D311yescharacteristic
197C2×#106C2×D311yescharacteristic
198#424—31no—
199#424—31no—
200C2×#104—31no—
201#418—31no—
202#417—31no—
203#416—31no—
204C2×#102C2×D311yescharacteristic
205C2×#102C2×D311yescharacteristic
206C2×#100—31no—
207#405—31no—
208#390C2×D311yescharacteristic
209#382C2×D311yescharacteristic
210#343—31no—
211#343—31no—
212#343—31no—
213#341—31no—
214#339—31no—
215#339—31no—
216#339—31no—
217#336C2×D311yescharacteristic
218C24×C3—21no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4C2×Q812yes—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4C2411yescharacteristic
230C2×#67—21no—
231C22×Dic3—41no—
232C22×Dic3—41no—
233#235—21no—
234#235—21no—
235#235C2×Q812yes—
236#235#4112yes—
237C2×D12C2×D411yescharacteristic
238C2×C4×D3—21no—
239C2×C4×D3—41no—
240C2×C4×D3—41no—
241C2×Dic6—41no—
242C2×Dic6C2×D411yescharacteristic
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8C2×D411yescharacteristic
248C2×C3×C8C2×D411yescharacteristic
249C2×C3×C8C2×D411yescharacteristic
250C2×C3×C8C2×D411yescharacteristic
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C22⋊C4—82no—
258C3×C42—21no—
259C3×C42—21no—
260#217—21no—
261#217—21no—
262#217—21no—
263#217—41no—
264#217—41no—
265#214—24no—
266#213—24no—
267(C2×D3)⋊C4—41no—
268(C2×D3)⋊C4—41no—
269C4⋊Dic3—41no—
270C4⋊Dic3C2×D411yescharacteristic
271C4⋊Dic3C2×D411yescharacteristic
272C4⋊Dic3C2×D411yescharacteristic
273C4⋊Dic3C2×D411yescharacteristic
274C4⋊Dic3C2×D411yescharacteristic
275C4⋊Dic3C2×D411yescharacteristic
276Dic3⋊C4—41no—
277Dic3⋊C4—41no—
278Dic3⋊C4—41no—
279Dic3⋊C4—41no—
280C4×Dic3—21no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—41no—
284C4×Dic3—41no—
285#208—21no—
286#208—21no—
287#208—21no—
288#208—21no—
289#208—21no—
290#208—21no—
291#208C2×Q812yes—
292#208#4112yes—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297ΓL(1,𝔽25)—24no—
298D3×C8—24no—
299C2×#41—31no—
300C22×D4C22×D311yescharacteristic
301AGL(1,ℤ/8)—34no—
302AGL(1,ℤ/8)—62no—
303C2×SD16—61no—
304C2×D8—61no—
305#131—62no—
306#131—62no—
307C2×M16—31no—
308C2×M16—31no—
309C2×M16—31no—
310C2×M16—31no—
311C2×M16—61no—
312C22×C8—61no—
313C22×C8C22×D311yescharacteristic
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×D4—61no—
323C4×D4—61no—
324C4×D4—61no—
325C4×D4—61no—
326C4×D4—61no—
327C4×D4—61no—
328C4×D4—121no—
329C4×D4—121no—
330#117—31no—
331#117—31no—
332#117—31no—
333#117—61no—
334#117—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4C22×D311yescharacteristic
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C4⋊C4C22×D311yescharacteristic
340C2×C22⋊C4—41no—
341C2×C22⋊C4—121no—
342C2×C42C22×D311yescharacteristic
343#108—62no—
344#107—21no—
345#107—21no—
346#107—31no—
347#107—31no—
348#107—61no—
349#107—61no—
350#107—61no—
351#107—61no—
352#107—61no—
353#106—21no—
354#106—21no—
355#106—31no—
356#106—31no—
357#106—61no—
358#106—61no—
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#102—21no—
371#102—21no—
372#102—21no—
373#102—21no—
374#102—31no—
375#102—31no—
376#102—61no—
377#102—61no—
378#102—61no—
379#100—62no—
380#97—61no—
381C4×C8—61no—
382#95—41no—
383C23×C3—22no—
384C23×C3—41no—
385C23×C3—41no—
386C23×C3—81no—
387C23×C3—81no—
388C23×C3H(5,𝔽2)11yescharacteristic
389C22×D3—21no—
390C3×D4—22no—
391C3×D4—22no—
392C3×D4—41no—
393C3×D4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—81no—
401C2×C3×C4—81no—
402C2×C3×C4—81no—
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C22×Q811yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C42−1+411yescharacteristic
409#67—22no—
410#67—22no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—161no—
419C2×Dic3—161no—
420D12—21no—
421D12C22⋊Q812yes—
422C4×D3—22no—
423C4×D3—22no—
424C4×D3—161no—
425C4×D3—161no—
426Dic6—21no—
427Dic6—161no—
428Dic6C22⋊Q812yes—
429C3×C8—21no—
430C3×C8—21no—
431C3×C8—21no—
432C3×C8—21no—
433C3⋊C8—22no—
434C3⋊C8—22no—
435C3⋊C8—41no—
436C3⋊C8—41no—
437C24—21no—
438#41—32no—
439#41—32no—
440#41—61no—
441#41—61no—
442C2×Q8—31no—
443C2×Q8—121no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—31no—
450C2×D4—61no—
451C2×D4D3×Q812yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—61no—
457C22×C4—121no—
458C22×C4—121no—
459C22×C4—121no—
460C22×C4—121no—
461C22×C4C23×D311yescharacteristic
462SD16—64no—
463D8—64no—
464M16—32no—
465M16—32no—
466M16—64no—
467M16—61no—
468M16—61no—
469M16—61no—
470M16—61no—
471M16—61no—
472M16—61no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—64no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8D3×D411yescharacteristic
483C2×C8D3×D411yescharacteristic
484C2×C8D3×D411yescharacteristic
485C2×C8D3×D411yescharacteristic
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—31no—
493C4⋊C4—31no—
494C4⋊C4—31no—
495C4⋊C4—31no—
496C4⋊C4—31no—
497C4⋊C4—31no—
498C4⋊C4—121no—
499C4⋊C4—121no—
500C4⋊C4—121no—
501C4⋊C4—121no—
502C4⋊C4—121no—
503C22⋊C4—61no—
504C22⋊C4—61no—
505C22⋊C4—61no—
506C22⋊C4—82no—
507C22⋊C4—121no—
508C22⋊C4—121no—
509C22⋊C4—121no—
510C22⋊C4—121no—
511C42—21no—
512C42—21no—
513C42—61no—
514C42—61no—
515C42—61no—
516C42—121no—
517C42—121no—
518C22×C3—21no—
519C22×C3—21no—
520C22×C3—42no—
521C22×C3—82no—
522C22×C3—81no—
523C22×C3—81no—
524C22×C3—161no—
525C22×C3#53911yescharacteristic
526C22×C3D4211yescharacteristic
527C22×C3#55411yescharacteristic
528C2×D3—22no—
529C2×D3—41no—
530C2×D3—41no—
531C3×C4—81no—
532C3×C4—81no—
533C3×C4—81no—
534C3×C4—81no—
535C3×C4C2×C22⋊Q811yescharacteristic
536C3×C4#52511yescharacteristic
537C3×C4#54711yescharacteristic
538C3×C4D4×Q811yescharacteristic
539Dic3—22no—
540Dic3—41no—
541Dic3—41no—
542Dic3—41no—
543Dic3—41no—
544Dic3—41no—
545Dic3—41no—
546C23—22no—
547C23—41no—
548C23—41no—
549C23—61no—
550C23—81no—
551C23—81no—
552C23#100411yescharacteristic
553Q8—32no—
554Q8—61no—
555Q8—481no—
556D4—22no—
557D4—22no—
558D4—32no—
559D4—41no—
560D4—41no—
561D4—62no—
562D4—62no—
563D4—61no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—61no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—481no—
583C2×C4—481no—
584C2×C4—481no—
585C2×C4—481no—
586C2×C4C2×D3×D411yescharacteristic
587C2×C4C2×D3×D411yescharacteristic
588C2×C4C2×D3×Q811yescharacteristic
589C2×C4D3×#4111yescharacteristic
590C2×C4D3×#4111yescharacteristic
591C2×C4#101011yescharacteristic
592C8—21no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—62no—
597C8—62no—
598C8—121no—
599C8—121no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—82no—
603C2×C3#244911yescharacteristic
604C2×C3#328811yescharacteristic
605C2×C3#329011yescharacteristic
606D3—42no—
607C22—21no—
608C22—21no—
609C22—42no—
610C22—62no—
611C22—82no—
612C22—81no—
613C22—81no—
614C22—121no—
615C22—121no—
616C22—161no—
617C22#563611yescharacteristic
618C22#566011yescharacteristic
619C22#571711yescharacteristic
620C4—62no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—121no—
626C4—121no—
627C4—121no—
628C4—121no—
629C4—121no—
630C4—121no—
631C4D3×C22⋊Q811yescharacteristic
632C4#563211yescharacteristic
633C4#565511yescharacteristic
634C4#565611yescharacteristic
635C3#2640511yescharacteristic, complemented, Sylow
636C2—21no—
637C2—21no—
638C2—82no—
639C2—122no—
640C2#7810411yescharacteristic
641C2#8183511yescharacteristic
642C2#8186011yescharacteristic
subgroupG/H
centerC22#5717
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2092C2
socleC22×C3D42
prime radicalC2×#932C22
hypercenter#2092D3
hypocenterC3#26405
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2092

Complements

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

NH ≅ G/Ntype
1. #79146C2semidirect
2. #79130C2semidirect
3. #78731C2semidirect
5. C3×#2092C2semidirect
6. #70626C2semidirect
7. #70572C2semidirect
9. #70197C2semidirect
11. #69642C2semidirect
12. #69562C2semidirect
13. #69543C2semidirect
14. #68694C2semidirect
15. #68645C2semidirect
72. #2092D3semidirect
99. #932Q8semidirect
635. C3#26405semidirect

Maximal subgroups

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

subgroupindex
#791462
#791302
#787312
#787262
C3×#20922
#706262
#705722
#702182
#701972
#696632
#696422
#695622
#695432
#686942
#686452
#264053