Finite Group Databasebeta

← #10858718

Subgroups — #10858718

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#79244C211yesmaximal, characteristic, complemented
2#79201C211yesmaximal, characteristic
3#79116C211yesmaximal, characteristic, complemented
4#79090C211yesmaximal, characteristic, complemented
5C3×#2090C211yesmaximal, characteristic, complemented
6#70674C211yesmaximal, characteristic, complemented
7#70666C211yesmaximal, characteristic, complemented
8#70587C211yesmaximal, characteristic, complemented
9#70545C211yesmaximal, characteristic, complemented
10#69005C211yesmaximal, characteristic
11#69002C211yesmaximal, characteristic, complemented
12#68996C211yesmaximal, characteristic, complemented
13#68984C211yesmaximal, characteristic, complemented
14#68948C211yesmaximal, characteristic
15#68925C211yesmaximal, characteristic, complemented
16#26407—31nomaximal, Sylow
17C2×#908C2211yescharacteristic
18C2×#901C2211yescharacteristic
19#5521C2211yescharacteristic
20#5519C2211yescharacteristic
21#5504C2211yescharacteristic
22#5504C2211yescharacteristic
23C3×#428C2211yescharacteristic
24C3×#428C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C3×#424C2211yescharacteristic
27C2×#825C2211yescharacteristic
28#5133C2211yescharacteristic
29#5131C2211yescharacteristic
30C2×#823C2211yescharacteristic
31C2×#822C2211yescharacteristic
32#5122C2211yescharacteristic
33#5121C2211yescharacteristic
34#5119C2211yescharacteristic
35C2×#821C2211yescharacteristic
36#5106C2211yescharacteristic
37#5002C2211yescharacteristic
38#4988C2211yescharacteristic
39#4975C2211yescharacteristic
40C2×#809C2211yescharacteristic
41#4964C2211yescharacteristic
42#4964C2211yescharacteristic
43C2×#808C2211yescharacteristic
44#4891—21no—
45#4889—21no—
46#4877—21no—
47#4872—21no—
48#4870—21no—
49#4869—21no—
50#4861—21no—
51#4861—21no—
52#4860—21no—
53D3×#106—21no—
54#4854—21no—
55#4852—21no—
56#4847—21no—
57#4847—21no—
58#4846—21no—
59#4842—21no—
60C3×#343C2211yescharacteristic
61C3×#343C2211yescharacteristic
62C3×#341C2211yescharacteristic
63#4555C2211yescharacteristic
64#4555C2211yescharacteristic
65#4554C2211yescharacteristic
66#4550C2211yescharacteristic
67#4532C2211yescharacteristic
68#3084—31no—
69#3081—31no—
70#2946—31no—
71#2912—31no—
72#2094—31no—
73#2090D311yescharacteristic, complemented
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#1856—31no—
83C2×#235C2311yescharacteristic
84C22×D12C2311yescharacteristic
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—
93C4×#67—41no—
94C2×(C2×D3)⋊C4—41no—
95#926C2311yescharacteristic
96#926C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—41no—
99C2×C4×Dic3C2311yescharacteristic
100C2×#208C2311yescharacteristic
101C22×C3⋊C8C2311yescharacteristic
102#908—22no—
103#908Q814yescomplemented
104C2×D24—21no—
105C2×#204—21no—
106#901—22no—
107#901—22no—
108C2×ΓL(1,𝔽25)—21no—
109C2×D3×C8—21no—
110#893—21no—
111#893—21no—
112#893—21no—
113#893—21no—
114#892—21no—
115#892—21no—
116D3×C4⋊C4—21no—
117D3×C4⋊C4—21no—
118#890—21no—
119#888—21no—
120#887—21no—
121#887—21no—
122D3×C22⋊C4—41no—
123#878—41no—
124C4×D12—21no—
125C4×D12—21no—
126#870—21no—
127#869—21no—
128C3×#107—21no—
129C3×#107—21no—
130C3×#106—21no—
131C3×#106—21no—
132C3×#106—21no—
133C3×#106—21no—
134C3×C4⋊C8—21no—
135C3×C4⋊C8—21no—
136#825—22no—
137#823—22no—
138#822—22no—
139#821—21no—
140#821—21no—
141#821—21no—
142#821—21no—
143#821—21no—
144#818—21no—
145#818D411yescharacteristic
146#818D411yescharacteristic
147#817—21no—
148#817D411yescharacteristic
149#817D411yescharacteristic
150#816—21no—
151#815—21no—
152Dic3⋊C8—21no—
153Dic3⋊C8—21no—
154C8×Dic3—21no—
155#810D411yescharacteristic
156#810D411yescharacteristic
157#809—21no—
158#809—21no—
159#809D411yescharacteristic
160#809D411yescharacteristic
161#808—21no—
162#808—21no—
163#808—21no—
164#808—21no—
165#807—21no—
166#807—21no—
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—61no—
190#428—61no—
191#428C2×D311yescharacteristic
192#428C2×D311yescharacteristic
193#427—31no—
194#427—61no—
195C2×#106—31no—
196C2×#106—61no—
197C2×#106C2×D311yescharacteristic
198#424—31no—
199#424C2×D311yescharacteristic
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#384—31no—
210#343—31no—
211#343C2×D311yescharacteristic
212#343C2×D311yescharacteristic
213#341C2×D311yescharacteristic
214#339—31no—
215#339—31no—
216#339—31no—
217#336—31no—
218C23×D3—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4C2411yescharacteristic
222C2×#67—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225#235—21no—
226#235—21no—
227#235C2×Q812yes—
228#235#4112yes—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12C2×D411yescharacteristic
235C2×D12C2×Q812yes—
236C2×C4×D3—21no—
237C2×C4×D3—41no—
238C2×C4×D3—41no—
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—21no—
247C3×M16—21no—
248C3×M16C2×Q812yes—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8C2×D411yescharacteristic
255C2×C3×C8C2×D411yescharacteristic
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4C2×D411yescharacteristic
261C3×C4⋊C4C2×D411yescharacteristic
262C3×C22⋊C4—21no—
263C3×C42—21no—
264#217—21no—
265#217—21no—
266(C2×D3)⋊C4—41no—
267(C2×D3)⋊C4—41no—
268(C2×D3)⋊C4—41no—
269(C2×D3)⋊C4—41no—
270(C2×D3)⋊C4—82no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—41no—
274C4⋊Dic3C2×D411yescharacteristic
275C4⋊Dic3C2×D411yescharacteristic
276C4⋊Dic3C2×D411yescharacteristic
277C4⋊Dic3C2×D411yescharacteristic
278Dic3⋊C4—41no—
279Dic3⋊C4—41no—
280Dic3⋊C4—41no—
281Dic3⋊C4—41no—
282Dic3⋊C4—82no—
283C4×Dic3—21no—
284C4×Dic3—21no—
285C4×Dic3—21no—
286C4×Dic3—21no—
287C4×Dic3—41no—
288C4×Dic3—41no—
289#208—21no—
290#208#4112yes—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
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—61no—
310C2×M16C22×D311yescharacteristic
311C2×M16C22×D311yescharacteristic
312C2×M16C22×D311yescharacteristic
313C22×C8—31no—
314C22×C8—61no—
315C22×C8C22×D311yescharacteristic
316C4⋊Q8—61no—
317C4⋊Q8—61no—
318#125—61no—
319#125—61no—
320C22⋊Q8—121no—
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—31no—
333#117—61no—
334#117—61no—
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×C22⋊C4—121no—
343C2×C22⋊C4—121no—
344C2×C42—31no—
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—21no—
356#106—21no—
357#106—31no—
358#106—31no—
359#106—61no—
360#106—61no—
361#106—61no—
362#106—61no—
363#106—61no—
364C4⋊C8—21no—
365C4⋊C8—21no—
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—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—81no—
396C2×C3×C4—81no—
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C4C22×Q811yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C42−1+411yescharacteristic
403#67—22no—
404#67—22no—
405C2×Dic3—21no—
406C2×Dic3—22no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—81no—
415D12—22no—
416D12—21no—
417D12—22no—
418D12—41no—
419D12—41no—
420D12C22⋊Q812yes—
421C4×D3—22no—
422C4×D3—22no—
423C4×D3—161no—
424C4×D3—161no—
425C4×D3—161no—
426C4×D3—161no—
427Dic6—21no—
428Dic6—161no—
429Dic6C22⋊Q812yes—
430C3×C8—22no—
431C3×C8—21no—
432C3×C8—21no—
433C3×C8—41no—
434C3×C8—41no—
435C3⋊C8—22no—
436C3⋊C8—21no—
437C3⋊C8—21no—
438C24—61no—
439#41—32no—
440#41—32no—
441#41—61no—
442#41—61no—
443C2×Q8—31no—
444C2×Q8—121no—
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—61no—
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—21no—
468M16—21no—
469M16—21no—
470M16—32no—
471M16—64no—
472M16—61no—
473M16D3×Q812yes—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—31no—
480C2×C8—31no—
481C2×C8—64no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8D3×D411yescharacteristic
486C2×C8D3×D411yescharacteristic
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—31no—
492C4⋊C4—31no—
493C4⋊C4—31no—
494C4⋊C4—31no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—121no—
498C4⋊C4—121no—
499C4⋊C4—121no—
500C4⋊C4—121no—
501C4⋊C4—121no—
502C4⋊C4—242no—
503C4⋊C4D3×D411yescharacteristic
504C4⋊C4D3×D411yescharacteristic
505C22⋊C4—21no—
506C22⋊C4—61no—
507C22⋊C4—61no—
508C22⋊C4—121no—
509C22⋊C4—121no—
510C22⋊C4—121no—
511C22⋊C4—121no—
512C22⋊C4—242no—
513C42—21no—
514C42—61no—
515C42—61no—
516C42—61no—
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—41no—
534C3×C4—41no—
535C3×C4—81no—
536C3×C4—81no—
537C3×C4C2×C22⋊Q811yescharacteristic
538C3×C4#52511yescharacteristic
539C3×C4#54711yescharacteristic
540C3×C4D4×Q811yescharacteristic
541Dic3—22no—
542Dic3—42no—
543Dic3—41no—
544Dic3—41no—
545Dic3—41no—
546Dic3—41no—
547Dic3—81no—
548C23—61no—
549C23—62no—
550C23—121no—
551C23—121no—
552C23—241no—
553C23—241no—
554C23#100911yescharacteristic
555Q8—32no—
556Q8—61no—
557Q8—481no—
558D4—32no—
559D4—62no—
560D4—62no—
561D4—62no—
562D4—62no—
563D4—61no—
564D4—121no—
565D4—121no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—61no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—121no—
581C2×C4—121no—
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×C4#100711yescharacteristic
592C2×C4D3×#4111yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C8—21no—
595C8—21no—
596C8—22no—
597C8—41no—
598C8—41no—
599C8—62no—
600C8—61no—
601C8—61no—
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#560911yescharacteristic
620C22#569011yescharacteristic
621C22#569211yescharacteristic
622C4—41no—
623C4—41no—
624C4—62no—
625C4—81no—
626C4—81no—
627C4—122no—
628C4—121no—
629C4—121no—
630C4—121no—
631C4—121no—
632C4—241no—
633C4D3×C22⋊Q811yescharacteristic
634C4#563111yescharacteristic
635C4#565611yescharacteristic
636C4#572711yescharacteristic
637C3#2640711yescharacteristic, complemented, Sylow
638C2—21no—
639C2—21no—
640C2—122no—
641C2—242no—
642C2#7795211yescharacteristic
643C2#8093211yescharacteristic
644C2#8093511yescharacteristic
subgroupG/H
centerC22#5609
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2090C2
socleC22×C3D42
prime radicalC2×#908C22
hypercenter#2090D3
hypocenterC3#26407
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2090

Complements

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

NH ≅ G/Ntype
1. #79244C2semidirect
3. #79116C2semidirect
4. #79090C2semidirect
5. C3×#2090C2semidirect
6. #70674C2semidirect
7. #70666C2semidirect
8. #70587C2semidirect
9. #70545C2semidirect
11. #69002C2semidirect
12. #68996C2semidirect
13. #68984C2semidirect
15. #68925C2semidirect
73. #2090D3semidirect
103. #908Q8semidirect
637. C3#26407semidirect

Maximal subgroups

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

subgroupindex
#792442
#792012
#791162
#790902
C3×#20902
#706742
#706662
#705872
#705452
#690052
#690022
#689962
#689842
#689482
#689252
#264073