Finite Group Databasebeta

← #10858722

Subgroups — #10858722

Order 768.

740 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82539C211yesmaximal, characteristic, complemented
2#82537C211yesmaximal, characteristic
3#79244C211yesmaximal, characteristic, complemented
4#79209C211yesmaximal, characteristic, complemented
5C3×#1856C211yesmaximal, characteristic, complemented
6#70671C211yesmaximal, characteristic, complemented
7#70668C211yesmaximal, characteristic, complemented
8#70088C211yesmaximal, characteristic
9#70075C211yesmaximal, characteristic, complemented
10#69663C211yesmaximal, characteristic
11#69656C211yesmaximal, characteristic, complemented
12#68774C211yesmaximal, characteristic, complemented
13#68771C211yesmaximal, characteristic, complemented
14#68743C211yesmaximal, characteristic, complemented
15#68735C211yesmaximal, characteristic, complemented
16#26407—31nomaximal, Sylow
17C3×#518C2211yescharacteristic
18C2×#932C2211yescharacteristic
19C2×#907C2211yescharacteristic
20#5523C2211yescharacteristic
21#5519C2211yescharacteristic
22#5504C2211yescharacteristic
23C3×#428C2211yescharacteristic
24C3×#428C2211yescharacteristic
25C2×C3×#100C2211yescharacteristic
26C3×#390C2211yescharacteristic
27#5215C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30#5133C2211yescharacteristic
31C2×#822C2211yescharacteristic
32#5121C2211yescharacteristic
33#5119C2211yescharacteristic
34#5018—21no—
35C4×#214—21no—
36#5014—21no—
37C4×#213—21no—
38#5009—21no—
39#5009—21no—
40#5008—21no—
41#5007—21no—
42#5002C2211yescharacteristic
43#4999C2211yescharacteristic
44#4998C2211yescharacteristic
45#4997C2211yescharacteristic
46#4995C2211yescharacteristic
47#4995C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#808C2211yescharacteristic
50#4935C2211yescharacteristic
51C2×#805C2211yescharacteristic
52#4891—21no—
53#4881—21no—
54#4880—21no—
55#4877—21no—
56#4870—21no—
57#4861—21no—
58D3×#106—21no—
59#4854—21no—
60C3×#339C2211yescharacteristic
61C3×#339C2211yescharacteristic
62#4558C2211yescharacteristic
63#4554C2211yescharacteristic
64#4554C2211yescharacteristic
65#4530C2211yescharacteristic
66#4524C2211yescharacteristic
67#4523C2211yescharacteristic
68#3084—31no—
69#3081—31no—
70#2946—31no—
71#2912—31no—
72#2094—31no—
73#2090—31no—
74#2063—31no—
75#2060—31no—
76#1951—31no—
77#1922—31no—
78#1907—31no—
79#1883—31no—
80#1866—31no—
81#1859—31no—
82#1856D311yescharacteristic, complemented
83C22×C3×D4C2311yescharacteristic
84C2×#235C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C3×C4×D4—21no—
88C3×C4×D4—21no—
89C3×C4×D4—21no—
90C3×C4×D4—21no—
91C3×#117C2311yescharacteristic
92C3×#117C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—41no—
95C2×C3×C22⋊C4—41no—
96C2×C3×C42C2311yescharacteristic
97C2×#214—21no—
98#932—22no—
99#932Q814yescomplemented
100C2×#213—21no—
101C4×#67—41no—
102#926C2311yescharacteristic
103C2×C4⋊Dic3—41no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×#208C2311yescharacteristic
107C2×#208C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110#907—22no—
111#907—22no—
112C2×ΓL(1,𝔽25)—21no—
113C2×D3×C8—21no—
114#893—21no—
115#892—21no—
116#892—21no—
117D3×C4⋊C4—21no—
118D3×C4⋊C4—21no—
119#890—21no—
120#888—21no—
121#887—21no—
122#67⋊C4—41no—
123#878—41no—
124C4×D12—21no—
125#870—21no—
126#869—21no—
127C4×Dic6—21no—
128C3×#107—21no—
129C3×#107—21no—
130C3×#106—21no—
131C3×#106—21no—
132C3×#100—22no—
133#832—21no—
134#832—21no—
135#832—21no—
136#832—21no—
137#832—21no—
138#832—21no—
139#822—22no—
140#821—21no—
141#818—21no—
142#817—21no—
143#816—21no—
144Dic3⋊C8—21no—
145Dic3⋊C8—21no—
146#810D411yescharacteristic
147#810D411yescharacteristic
148#809D411yescharacteristic
149#809D411yescharacteristic
150#808—21no—
151#808—21no—
152#808—21no—
153#808—21no—
154#808—21no—
155#808—21no—
156#808D411yescharacteristic
157#808D411yescharacteristic
158#807—21no—
159#807—21no—
160#807D411yescharacteristic
161#807D411yescharacteristic
162#805—22no—
163C4⋊(C3⋊C8)—21no—
164C4⋊(C3⋊C8)—21no—
165#803—21no—
166C4×C3⋊C8—21no—
167C2×AGL(1,ℤ/8)—31no—
168C2×#131—31no—
169#544—31no—
170#520—31no—
171#520—31no—
172#518C2×D311yescharacteristic
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—61no—
190#428—61no—
191#428C2×D311yescharacteristic
192#428C2×D311yescharacteristic
193#427—31no—
194#427—61no—
195C2×#106—31no—
196C2×#106—31no—
197C2×#106—61no—
198#424—31no—
199#424—31no—
200C2×#104—31no—
201#418—31no—
202#417—31no—
203#416—31no—
204C2×#102—31no—
205C2×#102—31no—
206C2×#100C2×D311yescharacteristic
207#405—31no—
208#390C2×D311yescharacteristic
209#384—31no—
210#343—31no—
211#343—31no—
212#343—31no—
213#341—31no—
214#339—31no—
215#339C2×D311yescharacteristic
216#339C2×D311yescharacteristic
217#336—31no—
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—41no—
228C22×C3×C4—41no—
229C22×C3×C4C2411yescharacteristic
230C2×#67—21no—
231C22×Dic3—21no—
232C22×Dic3—21no—
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
243C3×M16—21no—
244C3×M16—21no—
245C3×M16—21no—
246C3×M16#4112yes—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4C2×D411yescharacteristic
252C3×C4⋊C4C2×D411yescharacteristic
253C3×C4⋊C4C2×D411yescharacteristic
254C3×C4⋊C4C2×D411yescharacteristic
255C3×C22⋊C4—21no—
256C3×C22⋊C4—21no—
257C3×C22⋊C4—41no—
258C3×C22⋊C4—41no—
259C3×C22⋊C4—82no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—21no—
264#217—21no—
265#214—24no—
266#213—24no—
267(C2×D3)⋊C4—41no—
268(C2×D3)⋊C4—41no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—41no—
274C4⋊Dic3—82no—
275C4⋊Dic3C2×D411yescharacteristic
276C4⋊Dic3C2×D411yescharacteristic
277Dic3⋊C4—41no—
278Dic3⋊C4—41no—
279Dic3⋊C4—41no—
280Dic3⋊C4—41no—
281C4×Dic3—21no—
282C4×Dic3—41no—
283C4×Dic3—41no—
284#208—21no—
285#208—21no—
286#208—21no—
287#208C2×Q812yes—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8C2×D411yescharacteristic
295C2×C3⋊C8C2×D411yescharacteristic
296C2×C3⋊C8C2×D411yescharacteristic
297C2×C3⋊C8C2×D411yescharacteristic
298ΓL(1,𝔽25)—24no—
299D3×C8—24no—
300C2×#41—31no—
301C22×D4C22×D311yescharacteristic
302AGL(1,ℤ/8)—34no—
303AGL(1,ℤ/8)—62no—
304C2×SD16—61no—
305C2×D8—61no—
306#131—62no—
307#131—62no—
308C2×M16—31no—
309C2×M16—31no—
310C2×M16—61no—
311C2×M16C22×D311yescharacteristic
312C2×M16C22×D311yescharacteristic
313C22×C8—31no—
314C22×C8—31no—
315C22×C8—61no—
316C4⋊Q8—61no—
317C4⋊Q8—61no—
318#125—61no—
319#125—61no—
320C22⋊Q8—121no—
321C4×Q8—61no—
322C4×Q8—61no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—61no—
328C4×D4—61no—
329C4×D4—121no—
330C4×D4—121no—
331#117—31no—
332#117—61no—
333#117—61no—
334#117C22×D311yescharacteristic
335#117C22×D311yescharacteristic
336C2×C4⋊C4—31no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—121no—
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C22⋊C4—41no—
343C2×C22⋊C4—41no—
344C2×C42C22×D311yescharacteristic
345#108—62no—
346#107—21no—
347#107—21no—
348#107—31no—
349#107—31no—
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—
362#106—61no—
363#106—61no—
364C4⋊C8—61no—
365C4⋊C8—61no—
366C4⋊C8—61no—
367C4⋊C8—61no—
368#104—62no—
369#103—31no—
370#103—31no—
371#103—61no—
372#102—31no—
373#102—31no—
374#102—61no—
375#102—61no—
376#102—61no—
377#102—61no—
378#102—61no—
379#102—61no—
380#102—61no—
381#100—22no—
382#97—61no—
383C4×C8—61no—
384C23×C3—22no—
385C23×C3—41no—
386C23×C3—41no—
387C23×C3—81no—
388C23×C3—81no—
389C23×C3H(5,𝔽2)11yescharacteristic
390C22×D3—21no—
391C3×D4—22no—
392C3×D4—22no—
393C3×D4—41no—
394C3×D4—41no—
395C2×C3×C4—22no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—81no—
404C2×C3×C4—161no—
405C2×C3×C4—161no—
406C2×C3×C4C22×D411yescharacteristic
407C2×C3×C4C22×D411yescharacteristic
408C2×C3×C4C22×Q811yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×C3×C42−1+411yescharacteristic
412#67—22no—
413#67—22no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—81no—
420C2×Dic3—81no—
421D12—21no—
422D12C22⋊Q812yes—
423C4×D3—22no—
424C4×D3—22no—
425C4×D3—161no—
426C4×D3—161no—
427Dic6—21no—
428Dic6—161no—
429Dic6C22⋊Q812yes—
430C3×C8—22no—
431C3×C8—41no—
432C3⋊C8—22no—
433C3⋊C8—21no—
434C3⋊C8—21no—
435C3⋊C8—21no—
436C3⋊C8—21no—
437C3⋊C8—41no—
438C24—21no—
439#41—32no—
440#41—32no—
441#41—61no—
442#41—61no—
443C2×Q8—31no—
444C2×Q8—121no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—31no—
451C2×D4—61no—
452C2×D4D3×Q812yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—121no—
461C22×C4—121no—
462C22×C4C23×D311yescharacteristic
463SD16—64no—
464D8—64no—
465M16—21no—
466M16—21no—
467M16—21no—
468M16—32no—
469M16—64no—
470M16—61no—
471M16—61no—
472M16—61no—
473M16#23512yes—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—31no—
477C2×C8—31no—
478C2×C8—31no—
479C2×C8—31no—
480C2×C8—64no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—31no—
490C4⋊C4—31no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—121no—
496C4⋊C4—121no—
497C4⋊C4—121no—
498C4⋊C4—121no—
499C4⋊C4—121no—
500C4⋊C4—242no—
501C4⋊C4D3×D411yescharacteristic
502C4⋊C4D3×D411yescharacteristic
503C4⋊C4D3×D411yescharacteristic
504C4⋊C4D3×D411yescharacteristic
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—61no—
510C22⋊C4—82no—
511C22⋊C4—121no—
512C22⋊C4—121no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—61no—
518C42—121no—
519C42—121no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—42no—
523C22×C3—82no—
524C22×C3—81no—
525C22×C3—81no—
526C22×C3—161no—
527C22×C3#53911yescharacteristic
528C22×C3D4211yescharacteristic
529C22×C3#55411yescharacteristic
530C2×D3—22no—
531C2×D3—41no—
532C2×D3—41no—
533C3×C4—42no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4—81no—
539C3×C4C2×C22⋊Q811yescharacteristic
540C3×C4#52511yescharacteristic
541C3×C4#54711yescharacteristic
542C3×C4D4×Q811yescharacteristic
543Dic3—22no—
544Dic3—41no—
545Dic3—41no—
546Dic3—81no—
547Dic3—81no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—61no—
552C23—81no—
553C23—81no—
554C23#100411yescharacteristic
555Q8—32no—
556Q8—61no—
557Q8—481no—
558D4—22no—
559D4—22no—
560D4—32no—
561D4—41no—
562D4—41no—
563D4—62no—
564D4—62no—
565D4—61no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—61no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—81no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—161no—
583C2×C4—161no—
584C2×C4—241no—
585C2×C4—241no—
586C2×C4—481no—
587C2×C4—481no—
588C2×C4C2×#23511yescharacteristic
589C2×C4C2×D3×D411yescharacteristic
590C2×C4C2×D3×D411yescharacteristic
591C2×C4C2×D3×Q811yescharacteristic
592C2×C4D3×#4111yescharacteristic
593C2×C4#101011yescharacteristic
594C8—22no—
595C8—41no—
596C8—61no—
597C8—61no—
598C8—61no—
599C8—61no—
600C8—62no—
601C8—121no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—82no—
605C2×C3#244911yescharacteristic
606C2×C3#328811yescharacteristic
607C2×C3#329111yescharacteristic
608D3—42no—
609C22—21no—
610C22—21no—
611C22—42no—
612C22—62no—
613C22—82no—
614C22—81no—
615C22—81no—
616C22—121no—
617C22—121no—
618C22—161no—
619C22#551111yescharacteristic
620C22#559011yescharacteristic
621C22#566011yescharacteristic
622C4—42no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—62no—
628C4—81no—
629C4—121no—
630C4—121no—
631C4—241no—
632C4—241no—
633C4#554711yescharacteristic
634C4#555111yescharacteristic
635C4D3×C22⋊Q811yescharacteristic
636C4#563211yescharacteristic
637C3#2640711yescharacteristic, complemented, Sylow
638C2—21no—
639C2—21no—
640C2—82no—
641C2—122no—
642C2#7460911yescharacteristic
643C2#8028111yescharacteristic
644C2#8029411yescharacteristic
subgroupG/H
centerC22#5590
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1856C2
socleC22×C3D42
prime radicalC2×#932C22
hypercenter#1856D3
hypocenterC3#26407
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1856

Complements

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

NH ≅ G/Ntype
1. #82539C2semidirect
3. #79244C2semidirect
4. #79209C2semidirect
5. C3×#1856C2semidirect
6. #70671C2semidirect
7. #70668C2semidirect
9. #70075C2semidirect
11. #69656C2semidirect
12. #68774C2semidirect
13. #68771C2semidirect
14. #68743C2semidirect
15. #68735C2semidirect
82. #1856D3semidirect
99. #932Q8semidirect
637. C3#26407semidirect

Maximal subgroups

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

subgroupindex
#825392
#825372
#792442
#792092
C3×#18562
#706712
#706682
#700882
#700752
#696632
#696562
#687742
#687712
#687432
#687352
#264073