Finite Group Databasebeta

← #10858705

Subgroups — #10858705

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
1C3×#3082C211yesmaximal, characteristic, complemented
2#83475C211yesmaximal, characteristic
3#82547C211yesmaximal, characteristic
4#79179C211yesmaximal, characteristic
5#71900C211yesmaximal, characteristic, complemented
6#71658C211yesmaximal, characteristic, complemented
7#70680C211yesmaximal, characteristic
8#70669C211yesmaximal, characteristic, complemented
9#70661C211yesmaximal, characteristic, complemented
10#70545C211yesmaximal, characteristic
11#70532C211yesmaximal, characteristic, complemented
12#70348C211yesmaximal, characteristic
13#70337C211yesmaximal, characteristic, complemented
14#70286C211yesmaximal, characteristic
15#70249C211yesmaximal, characteristic, complemented
16#26406—31nomaximal, Sylow
17C3×#544C2211yescharacteristic
18C3×#519C2211yescharacteristic
19C2×#942C2211yescharacteristic
20C2×#901C2211yescharacteristic
21#5523C2211yescharacteristic
22#5523C2211yescharacteristic
23C3×#479—21no—
24C3×#479—21no—
25C3×#477—21no—
26C3×#475—21no—
27C3×#428C2211yescharacteristic
28C3×#427C2211yescharacteristic
29C3×#424C2211yescharacteristic
30C3×#419C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32#5229C2211yescharacteristic
33C2×#836C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#825C2211yescharacteristic
36C2×#822C2211yescharacteristic
37C2×#818C2211yescharacteristic
38#5091—21no—
39#5087—21no—
40#5066—21no—
41#5063—21no—
42#5030—21no—
43C4×#216—21no—
44#5026—21no—
45C4×#215—21no—
46#5002C2211yescharacteristic
47#4999C2211yescharacteristic
48#4997C2211yescharacteristic
49#4995C2211yescharacteristic
50#4995C2211yescharacteristic
51#4995C2211yescharacteristic
52#4995C2211yescharacteristic
53#4975C2211yescharacteristic
54#4966C2211yescharacteristic
55C2×#807C2211yescharacteristic
56#4880—21no—
57D3×#107—21no—
58#4861—21no—
59#4861—21no—
60#4554C2211yescharacteristic
61#4554C2211yescharacteristic
62#4552C2211yescharacteristic
63#4550C2211yescharacteristic
64#4532C2211yescharacteristic
65#4532C2211yescharacteristic
66#4532C2211yescharacteristic
67#4530C2211yescharacteristic
68#3083—31no—
69#3082D311yescharacteristic, complemented
70#2947—31no—
71#2913—31no—
72#2095—31no—
73#2089—31no—
74#2064—31no—
75#2060—31no—
76#1952—31no—
77#1922—31no—
78#1908—31no—
79#1882—31no—
80#1866—31no—
81#1859—31no—
82#1857—31no—
83C22×C3×Q8C2311yescharacteristic
84C2×#235C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4⋊Q8—21no—
88C3×C4⋊Q8—21no—
89C3×#125—21no—
90C3×#125—21no—
91C3×C22⋊Q8—41no—
92C3×C4×Q8—21no—
93C3×C4×Q8—21no—
94C3×C4×Q8—21no—
95C3×C4×Q8—21no—
96C3×#117—41no—
97C3×#117C2311yescharacteristic
98C3×#117C2311yescharacteristic
99C3×#117C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×#216—21no—
102#942—22no—
103#942Q814yescomplemented
104C2×#215—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—41no—
108C2×C4×Dic3C2311yescharacteristic
109C2×#208C2311yescharacteristic
110C2×#208C2311yescharacteristic
111C2×#208C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113#901—22no—
114#901—22no—
115C2×ΓL(1,𝔽25)—21no—
116C2×D3×C8—21no—
117#892—21no—
118#892—21no—
119D3×C4⋊C4—21no—
120D3×C4⋊C4—21no—
121#67⋊C4—41no—
122#67⋊C4—41no—
123C4×D12—21no—
124C4×D12—21no—
125C4×Dic6—21no—
126C4×Dic6—21no—
127C3×#107—21no—
128C3×#107D411yescharacteristic
129C3×#107D411yescharacteristic
130C3×#106—21no—
131C3×#106D411yescharacteristic
132C3×#106D411yescharacteristic
133C3×C4⋊C8—21no—
134C3×C4⋊C8—21no—
135C3×#103—21no—
136C3×#103—21no—
137C3×#103—21no—
138C3×#103—21no—
139#836—22no—
140#835—21no—
141#835—21no—
142C3⋊#95—41no—
143#825—22no—
144#822—22no—
145#818—21no—
146#818—21no—
147#810—21no—
148#810D411yescharacteristic
149#810D411yescharacteristic
150#809—21no—
151#809D411yescharacteristic
152#809D411yescharacteristic
153#808—21no—
154#808—21no—
155#808—21no—
156#808—21no—
157#807—21no—
158#807—21no—
159#807—21no—
160#807—21no—
161#807—21no—
162#807—21no—
163C4⋊(C3⋊C8)—21no—
164C4⋊(C3⋊C8)—21no—
165#803—21no—
166C4×C3⋊C8—21no—
167C2×#137—31no—
168C2×#131—31no—
169#544C2×D311yescharacteristic
170#520—31no—
171#520—31no—
172#519C2×D311yescharacteristic
173#479—21no—
174#479—21no—
175#479—61no—
176#478—61no—
177#477—21no—
178#477—61no—
179#475—21no—
180#474—61no—
181#441—61no—
182#440—61no—
183C4×Q16—61no—
184C4×SD16—61no—
185C2×#108—31no—
186#428—31no—
187#428—31no—
188#428—31no—
189#428—31no—
190#428—61no—
191#428—61no—
192#428C2×D311yescharacteristic
193#427—61no—
194#427C2×D311yescharacteristic
195C2×#107—31no—
196C2×#107—31no—
197C2×#107—61no—
198#424—31no—
199#424C2×D311yescharacteristic
200C2×#104—31no—
201#419C2×D311yescharacteristic
202#417—31no—
203#416—31no—
204C2×#103—31no—
205C2×#103C2×D311yescharacteristic
206C2×#101—31no—
207#405—31no—
208#391—31no—
209#384—31no—
210#343—31no—
211#343—31no—
212#343—31no—
213#341—31no—
214#339—31no—
215#339—31no—
216#339—31no—
217#336—31no—
218C2×C3×Q8—21no—
219C2×C3×Q8—21no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8—21no—
223C2×C3×Q8—41no—
224C2×C3×Q8C2×Q812yes—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4C2411yescharacteristic
228C2×#67—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232#235—21no—
233#235—21no—
234#235C2×Q812yes—
235#235#4112yes—
236C2×D12C2×D411yescharacteristic
237C2×C4×D3—21no—
238C2×C4×D3—41no—
239C2×C4×D3—41no—
240C2×Dic6C2×D411yescharacteristic
241C3×M16—21no—
242C3×M16—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8C2×D411yescharacteristic
247C2×C3×C8C2×D411yescharacteristic
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—41no—
251C3×C4⋊C4—41no—
252C3×C4⋊C4—41no—
253C3×C4⋊C4—41no—
254C3×C4⋊C4—41no—
255C3×C4⋊C4C2×D411yescharacteristic
256C3×C4⋊C4C2×D411yescharacteristic
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×C22⋊C4—21no—
264C3×C42—21no—
265C3×C42—21no—
266C3×C42—21no—
267C3×C42—41no—
268C3×C42—41no—
269#216—24no—
270#215—24no—
271(C2×D3)⋊C4—41no—
272(C2×D3)⋊C4—41no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277Dic3⋊C4—41no—
278Dic3⋊C4—41no—
279Dic3⋊C4—82no—
280C4×Dic3—21no—
281C4×Dic3—21no—
282C4×Dic3—41no—
283C4×Dic3—41no—
284#208—21no—
285#208—21no—
286#208—21no—
287#208—21no—
288#208C2×Q812yes—
289#208#4112yes—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8C2×D411yescharacteristic
296C2×C3⋊C8C2×D411yescharacteristic
297ΓL(1,𝔽25)—24no—
298D3×C8—24no—
299C2×#41—31no—
300C22×Q8C22×D311yescharacteristic
301#137—34no—
302#137—62no—
303C2×Q16—61no—
304C2×SD16—61no—
305#131—62no—
306#131—62no—
307C2×M16—31no—
308C2×M16—31no—
309C2×M16—31no—
310C2×M16—61no—
311C2×M16C22×D311yescharacteristic
312C22×C8—31no—
313C22×C8—61no—
314C22×C8C22×D311yescharacteristic
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317#125—21no—
318#125—21no—
319C22⋊Q8—41no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8—61no—
325C4×Q8—61no—
326C4×D4—61no—
327C4×D4—61no—
328C4×D4—121no—
329C4×D4—121no—
330#117—41no—
331#117—61no—
332#117—61no—
333#117C22×D311yescharacteristic
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×C42—31no—
343#108—62no—
344#107—21no—
345#107—61no—
346#107—61no—
347#107—61no—
348#107—61no—
349#107—61no—
350#107—61no—
351#107—61no—
352#107—61no—
353#107#6711yescharacteristic
354#107#6711yescharacteristic
355#106—21no—
356#106—61no—
357#106—61no—
358#106—61no—
359#106—61no—
360#106#6711yescharacteristic
361#106#6711yescharacteristic
362C4⋊C8—21no—
363C4⋊C8—21no—
364C4⋊C8—61no—
365C4⋊C8—61no—
366#104—62no—
367#103—21no—
368#103—21no—
369#103—21no—
370#103—21no—
371#103—31no—
372#103—31no—
373#103—61no—
374#103—61no—
375#103—61no—
376#102—31no—
377#102—31no—
378#102—61no—
379#101—62no—
380#97—61no—
381C4×C8—61no—
382#95—121no—
383C23×C3H(5,𝔽2)11yescharacteristic
384C22×D3—21no—
385C3×Q8—22no—
386C3×Q8—22no—
387C3×Q8—41no—
388C3×Q8—41no—
389C3×Q8—161no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—22no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—81no—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C22×Q811yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C42−1+411yescharacteristic
407#67—22no—
408#67—22no—
409C2×Dic3—21no—
410C2×Dic3—22no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—81no—
416C2×Dic3—81no—
417D12—21no—
418D12C22⋊Q812yes—
419C4×D3—22no—
420C4×D3—22no—
421C4×D3—161no—
422C4×D3—161no—
423Dic6—21no—
424Dic6C22⋊Q812yes—
425C3×C8—21no—
426C3×C8—21no—
427C3×C8—41no—
428C3⋊C8—22no—
429C3⋊C8—22no—
430C3⋊C8—21no—
431C3⋊C8—21no—
432C3⋊C8—41no—
433#41—32no—
434#41—32no—
435#41—61no—
436#41—61no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—31no—
443C2×Q8—41no—
444C2×Q8D3×Q812yes—
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—32no—
461M16—32no—
462M16—64no—
463M16—61no—
464M16—61no—
465M16—61no—
466M16—61no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—31no—
471C2×C8—31no—
472C2×C8—64no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8C2×#6711yescharacteristic
479C2×C8C2×#6711yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—121no—
492C4⋊C4—121no—
493C4⋊C4—242no—
494C4⋊C4D3×D411yescharacteristic
495C4⋊C4D3×D411yescharacteristic
496C4⋊C4C2×#6711yescharacteristic
497C4⋊C4C2×#6711yescharacteristic
498C4⋊C4C2×#6711yescharacteristic
499C4⋊C4C2×#6711yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—121no—
504C22⋊C4—121no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—41no—
509C42—41no—
510C42—61no—
511C42—61no—
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—42no—
523C3×C4—41no—
524C3×C4—41no—
525C3×C4—41no—
526C3×C4—41no—
527C3×C4—41no—
528C3×C4—41no—
529C3×C4—81no—
530C3×C4C2×C22⋊Q811yescharacteristic
531C3×C4#52511yescharacteristic
532C3×C4#54711yescharacteristic
533C3×C4D4×Q811yescharacteristic
534Dic3—22no—
535Dic3—42no—
536Dic3—81no—
537Dic3—81no—
538C23—61no—
539C23#100411yescharacteristic
540Q8—22no—
541Q8—22no—
542Q8—32no—
543Q8—41no—
544Q8—41no—
545Q8—61no—
546Q8—161no—
547D4—32no—
548D4—62no—
549D4—62no—
550D4—61no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—61no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—81no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—241no—
571C2×C4—241no—
572C2×C4—481no—
573C2×C4—481no—
574C2×C4C2×D3×D411yescharacteristic
575C2×C4C2×D3×Q811yescharacteristic
576C2×C4#100711yescharacteristic
577C2×C4D3×#4111yescharacteristic
578C2×C4D3×#4111yescharacteristic
579C2×C4C22×#6711yescharacteristic
580C8—21no—
581C8—21no—
582C8—41no—
583C8—62no—
584C8—62no—
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#563611yescharacteristic
600C22#563811yescharacteristic
601C22D4×#6711yescharacteristic
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—42no—
609C4—62no—
610C4—81no—
611C4—122no—
612C4—241no—
613C4—241no—
614C4D3×C22⋊Q811yescharacteristic
615C4#563111yescharacteristic
616C4Q8×#6711yescharacteristic
617C4#582811yescharacteristic
618C3#2640611yescharacteristic, complemented, Sylow
619C2—21no—
620C2—21no—
621C2—122no—
622C2#8281711yescharacteristic
623C2#8401511yescharacteristic
624C2#8401811yescharacteristic
subgroupG/H
centerC22D4×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3082C2
socleC22×C3D42
prime radicalC2×#235C23
hypercenter#3082D3
hypocenterC3#26406
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3082

Complements

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

NH ≅ G/Ntype
1. C3×#3082C2semidirect
5. #71900C2semidirect
6. #71658C2semidirect
8. #70669C2semidirect
9. #70661C2semidirect
11. #70532C2semidirect
13. #70337C2semidirect
15. #70249C2semidirect
69. #3082D3semidirect
103. #942Q8semidirect
618. C3#26406semidirect

Maximal subgroups

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

subgroupindex
C3×#30822
#834752
#825472
#791792
#719002
#716582
#706802
#706692
#706612
#705452
#705322
#703482
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