Finite Group Databasebeta

← #10858716

Subgroups — #10858716

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#83476C211yesmaximal, characteristic, complemented
2#83474C211yesmaximal, characteristic, complemented
3#79209C211yesmaximal, characteristic, complemented
4#79201C211yesmaximal, characteristic
5C3×#2060C211yesmaximal, characteristic, complemented
6#70676C211yesmaximal, characteristic, complemented
7#70671C211yesmaximal, characteristic, complemented
8#69592C211yesmaximal, characteristic, complemented
9#69587C211yesmaximal, characteristic, complemented
10#69006C211yesmaximal, characteristic
11#68984C211yesmaximal, characteristic, complemented
12#68776C211yesmaximal, characteristic
13#68774C211yesmaximal, characteristic, complemented
14#68758C211yesmaximal, characteristic, complemented
15#68738C211yesmaximal, characteristic, complemented
16#26407—31nomaximal, Sylow
17C3×#544C2211yescharacteristic
18C2×#908C2211yescharacteristic
19C2×#907C2211yescharacteristic
20#5523C2211yescharacteristic
21#5521C2211yescharacteristic
22#5504C2211yescharacteristic
23C3×#428C2211yescharacteristic
24C3×#428C2211yescharacteristic
25C3×#384C2211yescharacteristic
26C2×#823C2211yescharacteristic
27C2×#822C2211yescharacteristic
28#5122C2211yescharacteristic
29#5121C2211yescharacteristic
30#5094—21no—
31#5087—21no—
32#5084—21no—
33#5084—21no—
34#5066—21no—
35#5066—21no—
36#5066—21no—
37#5063—21no—
38#5002C2211yescharacteristic
39#5001C2211yescharacteristic
40#4999C2211yescharacteristic
41#4999C2211yescharacteristic
42#4998C2211yescharacteristic
43#4995C2211yescharacteristic
44#4966C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#809C2211yescharacteristic
47#4964C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#808C2211yescharacteristic
50#4939C2211yescharacteristic
51C2×#805C2211yescharacteristic
52#4889—21no—
53#4881—21no—
54#4880—21no—
55#4872—21no—
56#4869—21no—
57#4861—21no—
58D3×#106—21no—
59#4852—21no—
60C3×#343C2211yescharacteristic
61C3×#339C2211yescharacteristic
62C3×#339C2211yescharacteristic
63#4558C2211yescharacteristic
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#2060D311yescharacteristic, complemented
76#1951—31no—
77#1922—31no—
78#1907—31no—
79#1883—31no—
80#1866—31no—
81#1859—31no—
82#1856—31no—
83C2×#235C2311yescharacteristic
84C22×D12C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C3×C4⋊Q8—21no—
88C3×C4⋊Q8—21no—
89C3×#125—21no—
90C3×#125—21no—
91C3×C22⋊Q8—41no—
92C3×#117C2311yescharacteristic
93C3×#117C2311yescharacteristic
94C2×C3×C4⋊C4—41no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C4×#67—41no—
99C2×(C2×D3)⋊C4—41no—
100#926C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×#208C2311yescharacteristic
103C2×#208C2311yescharacteristic
104C22×C3⋊C8C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106#908—22no—
107#908Q814yescomplemented
108#907—22no—
109#907—22no—
110C2×D24—21no—
111C2×#204—21no—
112C2×ΓL(1,𝔽25)—21no—
113C2×D3×C8—21no—
114#893—21no—
115#893—21no—
116#893—21no—
117#892—21no—
118#892—21no—
119D3×C4⋊C4—21no—
120D3×C4⋊C4—21no—
121#887—21no—
122#67⋊C4—41no—
123D3×C22⋊C4—41no—
124C4×D12—21no—
125C4×D12—21no—
126C4×D12—21no—
127C4×Dic6—21no—
128C3×#107—21no—
129C3×#107—21no—
130C3×#106—21no—
131C3×#106—21no—
132#823—22no—
133#822—22no—
134#818—21no—
135#817—21no—
136#815—21no—
137C8×Dic3—21no—
138#810—21no—
139#810D411yescharacteristic
140#810D411yescharacteristic
141#809—21no—
142#809—21no—
143#809—21no—
144#809—21no—
145#809—21no—
146#809—21no—
147#809—21no—
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—
165C4⋊(C3⋊C8)—21no—
166C4⋊(C3⋊C8)—21no—
167C2×AGL(1,ℤ/8)—31no—
168C2×#131—31no—
169#544C2×D311yescharacteristic
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—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×#100—31no—
207#405—31no—
208#390—31no—
209#384C2×D311yescharacteristic
210#343—31no—
211#343—31no—
212#343C2×D311yescharacteristic
213#341—31no—
214#339—31no—
215#339C2×D311yescharacteristic
216#339C2×D311yescharacteristic
217#336—31no—
218C23×D3—21no—
219C2×C3×Q8—41no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4C2411yescharacteristic
224C2×#67—21no—
225C22×Dic3—21no—
226#235—21no—
227#235—21no—
228#235C2×Q812yes—
229#235#4112yes—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12C2×D411yescharacteristic
236C2×D12C2×Q812yes—
237C2×C4×D3—21no—
238C2×C4×D3—41no—
239C2×C4×D3—41no—
240C2×C4×D3—41no—
241C2×C4×D3—41no—
242C2×Dic6C2×D411yescharacteristic
243C3×M16—21no—
244C3×M16—21no—
245C3×M16C2×Q812yes—
246C3×M16#4112yes—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—41no—
254C3×C4⋊C4—41no—
255C3×C4⋊C4—41no—
256C3×C4⋊C4—82no—
257C3×C4⋊C4C2×D411yescharacteristic
258C3×C4⋊C4C2×D411yescharacteristic
259C3×C4⋊C4C2×D411yescharacteristic
260C3×C4⋊C4C2×D411yescharacteristic
261C3×C22⋊C4—21no—
262C3×C22⋊C4—21no—
263C3×C42—21no—
264C3×C42—21no—
265C3×C42—21no—
266C3×C42—21no—
267#217—21no—
268(C2×D3)⋊C4—41no—
269(C2×D3)⋊C4—41no—
270(C2×D3)⋊C4—41no—
271(C2×D3)⋊C4—41no—
272(C2×D3)⋊C4—82no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3C2×D411yescharacteristic
276C4⋊Dic3C2×D411yescharacteristic
277Dic3⋊C4—41no—
278Dic3⋊C4—41no—
279C4×Dic3—21no—
280C4×Dic3—41no—
281C4×Dic3—41no—
282#208—21no—
283#208—21no—
284#208—21no—
285#208—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8C2×D411yescharacteristic
293C2×C3⋊C8C2×D411yescharacteristic
294C2×C3⋊C8C2×D411yescharacteristic
295C2×C3⋊C8C2×D411yescharacteristic
296D24—24no—
297#204—24no—
298ΓL(1,𝔽25)—24no—
299D3×C8—24no—
300C2×#41—31no—
301C22×D4—31no—
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—21no—
317C4⋊Q8—21no—
318#125—21no—
319#125—21no—
320C22⋊Q8—41no—
321C4×Q8—61no—
322C4×Q8—61no—
323C4×D4—61no—
324C4×D4—61no—
325C4×D4—61no—
326C4×D4—61no—
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—41no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C22⋊C4—121no—
343C2×C22⋊C4—121no—
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—62no—
382#97—61no—
383C4×C8—61no—
384C23×C3H(5,𝔽2)11yescharacteristic
385C22×D3—21no—
386C22×D3—22no—
387C22×D3—41no—
388C22×D3—41no—
389C22×D3—81no—
390C22×D3—81no—
391C3×Q8—161no—
392C2×C3×C4—22no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
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—41no—
415C2×Dic3—81no—
416D12—22no—
417D12—21no—
418D12—22no—
419D12—41no—
420D12—41no—
421D12C22⋊Q812yes—
422C4×D3—22no—
423C4×D3—22no—
424C4×D3—161no—
425C4×D3—161no—
426C4×D3—161no—
427C4×D3—161no—
428Dic6—21no—
429Dic6C22⋊Q812yes—
430C3×C8—22no—
431C3×C8—22no—
432C3⋊C8—21no—
433C3⋊C8—21no—
434C3⋊C8—21no—
435C3⋊C8—21no—
436C3⋊C8—41no—
437C3⋊C8—41no—
438C24—61no—
439#41—32no—
440#41—32no—
441#41—61no—
442#41—61no—
443C2×Q8—31no—
444C2×Q8—41no—
445C2×D4—31no—
446C2×D4—32no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—61no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—121no—
459C22×C4—121no—
460C22×C4—121no—
461C22×C4—121no—
462C22×C4C23×D311yescharacteristic
463SD16—64no—
464D8—64no—
465M16—21no—
466M16—21no—
467M16—64no—
468M16—61no—
469M16—61no—
470M16—61no—
471M16—61no—
472M16D3×Q812yes—
473M16#23912yes—
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—21no—
490C4⋊C4—21no—
491C4⋊C4—31no—
492C4⋊C4—31no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—82no—
499C4⋊C4—121no—
500C4⋊C4—121no—
501C4⋊C4D3×D411yescharacteristic
502C4⋊C4D3×D411yescharacteristic
503C4⋊C4D3×D411yescharacteristic
504C4⋊C4D3×D411yescharacteristic
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—61no—
508C22⋊C4—121no—
509C22⋊C4—121no—
510C22⋊C4—121no—
511C22⋊C4—121no—
512C22⋊C4—242no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—61no—
518C42—121no—
519C42—121no—
520C22×C3—21no—
521C22×C3—21no—
522C22×C3#53911yescharacteristic
523C22×C3D4211yescharacteristic
524C22×C3#55411yescharacteristic
525C2×D3—22no—
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—42no—
529C2×D3—81no—
530C2×D3—81no—
531C2×D3—82no—
532C2×D3—161no—
533C3×C4—42no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4—81no—
539C3×C4—81no—
540C3×C4C2×C22⋊Q811yescharacteristic
541C3×C4#52511yescharacteristic
542C3×C4#54711yescharacteristic
543C3×C4D4×Q811yescharacteristic
544Dic3—22no—
545Dic3—41no—
546Dic3—41no—
547Dic3—81no—
548C23—61no—
549C23—62no—
550C23—121no—
551C23—121no—
552C23—241no—
553C23—241no—
554C23#100411yescharacteristic
555Q8—32no—
556Q8—61no—
557Q8—161no—
558D4—32no—
559D4—62no—
560D4—62no—
561D4—61no—
562D4—62no—
563D4—62no—
564D4—121no—
565D4—121no—
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—41no—
575C2×C4—61no—
576C2×C4—62no—
577C2×C4—62no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—121no—
583C2×C4—241no—
584C2×C4—481no—
585C2×C4—481no—
586C2×C4—481no—
587C2×C4—481no—
588C2×C4C2×D3×D411yescharacteristic
589C2×C4C2×D3×D411yescharacteristic
590C2×C4C2×D3×Q811yescharacteristic
591C2×C4C2×#23911yescharacteristic
592C2×C4#100711yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C8—22no—
595C8—22no—
596C8—61no—
597C8—61no—
598C8—61no—
599C8—61no—
600C8—121no—
601C8—121no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3#244911yescharacteristic
605C2×C3#328811yescharacteristic
606C2×C3#329111yescharacteristic
607D3—42no—
608D3—82no—
609C22—21no—
610C22—21no—
611C22—62no—
612C22—121no—
613C22—121no—
614C22—122no—
615C22—242no—
616C22—241no—
617C22—241no—
618C22—481no—
619C22#559011yescharacteristic
620C22#563811yescharacteristic
621C22#573011yescharacteristic
622C4—42no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—62no—
628C4—81no—
629C4—81no—
630C4—121no—
631C4—121no—
632C4—241no—
633C4D3×C22⋊Q811yescharacteristic
634C4#562711yescharacteristic
635C4#572611yescharacteristic
636C4#572711yescharacteristic
637C3#2640711yescharacteristic, complemented, Sylow
638C2—21no—
639C2—21no—
640C2—122no—
641C2—242no—
642C2#7725211yescharacteristic
643C2#8028011yescharacteristic
644C2#8030211yescharacteristic
subgroupG/H
centerC22#5590
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2060C2
socleC22×C3D42
prime radicalC2×#908C22
hypercenter#2060D3
hypocenterC3#26407
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2060

Complements

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

NH ≅ G/Ntype
1. #83476C2semidirect
2. #83474C2semidirect
3. #79209C2semidirect
5. C3×#2060C2semidirect
6. #70676C2semidirect
7. #70671C2semidirect
8. #69592C2semidirect
9. #69587C2semidirect
11. #68984C2semidirect
13. #68774C2semidirect
14. #68758C2semidirect
15. #68738C2semidirect
75. #2060D3semidirect
107. #908Q8semidirect
637. C3#26407semidirect

Maximal subgroups

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

subgroupindex
#834762
#834742
#792092
#792012
C3×#20602
#706762
#706712
#695922
#695872
#690062
#689842
#687762
#687742
#687582
#687382
#264073