Finite Group Databasebeta

← #10858712

Subgroups — #10858712

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
1#83965C211yesmaximal, characteristic
2#82562C211yesmaximal, characteristic
3#82559C211yesmaximal, characteristic
4#78950C211yesmaximal, characteristic, complemented
5C3×#1882C211yesmaximal, characteristic
6#72103C211yesmaximal, characteristic
7#70376C211yesmaximal, characteristic
8#70363C211yesmaximal, characteristic
9#70337C211yesmaximal, characteristic, complemented
10#69684C211yesmaximal, characteristic, complemented
11#69672C211yesmaximal, characteristic, complemented
12#68985C211yesmaximal, characteristic, complemented
13#68980C211yesmaximal, characteristic, complemented
14#68738C211yesmaximal, characteristic, complemented
15#68735C211yesmaximal, characteristic, complemented
16#26406—31nomaximal, Sylow
17C2×C3×#137C2211yescharacteristic
18C3×#520C2211yescharacteristic
19#5827C2211yescharacteristic
20C2×#948C2211yescharacteristic
21#5812C2211yescharacteristic
22#5519C2211yescharacteristic
23C2×C3×#104C2211yescharacteristic
24C3×#419C2211yescharacteristic
25C3×#391C2211yescharacteristic
26#5233C2211yescharacteristic
27#5232C2211yescharacteristic
28C2×#836C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31#5184—21no—
32Dic3×Q16—21no—
33#5164—21no—
34Dic3×SD16—21no—
35C2×#822C2211yescharacteristic
36#5122C2211yescharacteristic
37#5121C2211yescharacteristic
38#5121C2211yescharacteristic
39#5119C2211yescharacteristic
40#5022—21no—
41#5021—21no—
42#5009—21no—
43#5007—21no—
44#4999C2211yescharacteristic
45#4998—21no—
46#4998C2211yescharacteristic
47#4995—21no—
48#4995C2211yescharacteristic
49#4964—21no—
50#4964C2211yescharacteristic
51#4964C2211yescharacteristic
52C2×#807—21no—
53C2×#807C2211yescharacteristic
54C2×#807C2211yescharacteristic
55#4935C2211yescharacteristic
56#4799—21no—
57#4796—21no—
58#4793—21no—
59#4791—21no—
60C3×#343C2211yescharacteristic
61C3×#339C2211yescharacteristic
62#4558C2211yescharacteristic
63#4554C2211yescharacteristic
64#4554C2211yescharacteristic
65#4524C2211yescharacteristic
66#4523C2211yescharacteristic
67#4523C2211yescharacteristic
68#3083—31no—
69#3082—31no—
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#1882D311yescharacteristic
80#1866—31no—
81#1859—31no—
82#1857—31no—
83C2×C3×#41C2311yescharacteristic
84C22×C3×Q8C2311yescharacteristic
85C3×#137—22no—
86C3×#137Q814yescomplemented
87C2×C3×Q16—21no—
88C2×C3×SD16—21no—
89C2×C3×M16C2311yescharacteristic
90C2×C3×M16C2311yescharacteristic
91C3×C4×Q8—21no—
92C3×C4×D4—21no—
93C3×C4×D4—41no—
94C3×#117—21no—
95C3×#117C2311yescharacteristic
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99#948—22no—
100#948—22no—
101Q8×Dic3—21no—
102Q8×Dic3—21no—
103Q8×Dic3—21no—
104Q8×Dic3—21no—
105Q8×Dic3—21no—
106D4×Dic3—21no—
107D4×Dic3—41no—
108#926—21no—
109#926—41no—
110#926C2311yescharacteristic
111#926C2311yescharacteristic
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3—41no—
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×#208—21no—
117C2×#208C2311yescharacteristic
118C2×#208C2311yescharacteristic
119C22×C3⋊C8—21no—
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122#890—21no—
123#888—21no—
124#878—41no—
125#870—21no—
126#869—21no—
127C3×#104—22no—
128C3×#103—21no—
129C3×#103—21no—
130C3×#95—41no—
131#836—22no—
132#835—21no—
133#835—21no—
134#835—21no—
135#835—21no—
136#835—21no—
137#835D411yescharacteristic
138#835D411yescharacteristic
139#832—21no—
140#832D411yescharacteristic
141#832D411yescharacteristic
142#822—22no—
143#818—21no—
144#818—21no—
145#817—21no—
146#817—21no—
147#815—21no—
148Dic3⋊C8—21no—
149Dic3⋊C8—21no—
150C8×Dic3—21no—
151#808—21no—
152#808—21no—
153#808—21no—
154#808D411yescharacteristic
155#808D411yescharacteristic
156#807—21no—
157#807—21no—
158#807—21no—
159#807—21no—
160#807—21no—
161#807—21no—
162#807—21no—
163#807D411yescharacteristic
164#807D411yescharacteristic
165C4⋊(C3⋊C8)—21no—
166C4⋊(C3⋊C8)—21no—
167C2×#137C2×D311yescharacteristic
168C2×#131—31no—
169#544—31no—
170#520—31no—
171#520C2×D311yescharacteristic
172#519—31no—
173#479—61no—
174#479—61no—
175#479—61no—
176#478—61no—
177#477—61no—
178#477—61no—
179#475—61no—
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—31no—
191#428—61no—
192#428—61no—
193#427—31no—
194#427—61no—
195C2×#107—31no—
196C2×#107—31no—
197C2×#107—61no—
198#424—31no—
199#424—31no—
200C2×#104C2×D311yescharacteristic
201#419C2×D311yescharacteristic
202#417—31no—
203#416—31no—
204C2×#103—31no—
205C2×#103—31no—
206C2×#101—31no—
207#405—31no—
208#391C2×D311yescharacteristic
209#384—31no—
210#343—31no—
211#343—31no—
212#343C2×D311yescharacteristic
213#341—31no—
214#339—31no—
215#339—31no—
216#339C2×D311yescharacteristic
217#336—31no—
218C3×#41—21no—
219C3×#41—21no—
220C3×#41C2×Q812yes—
221C3×#41#4112yes—
222C2×C3×Q8—21no—
223C2×C3×Q8—21no—
224C2×C3×Q8—21no—
225C2×C3×Q8—21no—
226C2×C3×Q8—21no—
227C2×C3×Q8C2×D411yescharacteristic
228C2×C3×Q8C2×Q812yes—
229C2×C3×D4—21no—
230C2×C3×D4C2×D411yescharacteristic
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4—21no—
234C22×C3×C4—21no—
235C22×C3×C4—41no—
236C22×C3×C4C2411yescharacteristic
237C22×Dic3—21no—
238C22×Dic3—21no—
239C22×Dic3—41no—
240C2×Dic6—41no—
241C3×Q16—24no—
242C3×SD16—24no—
243C3×M16—21no—
244C3×M16—21no—
245C3×M16—21no—
246C3×M16C2×Q812yes—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—41no—
252C3×C4⋊C4C2×D411yescharacteristic
253C3×C4⋊C4C2×D411yescharacteristic
254C3×C22⋊C4—21no—
255C3×C22⋊C4—41no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42—41no—
260#217—21no—
261#217—21no—
262#217—41no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—41no—
268C4⋊Dic3—41no—
269C4⋊Dic3—41no—
270C4⋊Dic3—41no—
271C4⋊Dic3—82no—
272C4⋊Dic3C2×D411yescharacteristic
273C4⋊Dic3C2×D411yescharacteristic
274C4⋊Dic3C2×D411yescharacteristic
275C4⋊Dic3C2×D411yescharacteristic
276Dic3⋊C4—41no—
277Dic3⋊C4—41no—
278C4×Dic3—21no—
279C4×Dic3—21no—
280C4×Dic3—41no—
281C4×Dic3—41no—
282C4×Dic3—41no—
283#208—24no—
284#208—21no—
285#208—21no—
286#208—21no—
287#208#4112yes—
288C2×C3⋊C8—24no—
289C2×C3⋊C8—21no—
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
297C2×C3⋊C8C2×D411yescharacteristic
298C2×C3⋊C8C2×D411yescharacteristic
299C2×#41C22×D311yescharacteristic
300C22×Q8C22×D311yescharacteristic
301#137—22no—
302#137Dic614yescomplemented
303C2×Q16—21no—
304C2×SD16—21no—
305#131—62no—
306#131—62no—
307C2×M16—31no—
308C2×M16—31no—
309C2×M16—61no—
310C2×M16C22×D311yescharacteristic
311C2×M16C22×D311yescharacteristic
312C22×C8—31no—
313C22×C8—31no—
314C22×C8—61no—
315C4⋊Q8—61no—
316C4⋊Q8—61no—
317#125—61no—
318#125—61no—
319C22⋊Q8—121no—
320C4×Q8—21no—
321C4×Q8—61no—
322C4×Q8—61no—
323C4×Q8—61no—
324C4×Q8—61no—
325C4×Q8—61no—
326C4×D4—21no—
327C4×D4—41no—
328C4×D4—61no—
329C4×D4—121no—
330#117—21no—
331#117—31no—
332#117—31no—
333#117—61no—
334#117—121no—
335#117C22×D311yescharacteristic
336C2×C4⋊C4—21no—
337C2×C4⋊C4—31no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—121no—
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C42C22×D311yescharacteristic
343#108—62no—
344#107—31no—
345#107—31no—
346#107—61no—
347#107—61no—
348#107—61no—
349#107—61no—
350#107—61no—
351#107—61no—
352#107—61no—
353#107—61no—
354#107—61no—
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—22no—
367#103—21no—
368#103—21no—
369#103—31no—
370#103—31no—
371#103—61no—
372#103—61no—
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—41no—
383C23×C3—21no—
384C23×C3H(5,𝔽2)11yescharacteristic
385C3×Q8—21no—
386C3×Q8—22no—
387C3×Q8—22no—
388C3×Q8—41no—
389C3×Q8—41no—
390C3×Q8C22⋊Q812yes—
391C3×D4—22no—
392C3×D4—22no—
393C3×D4—21no—
394C3×D4C22⋊Q812yes—
395C2×C3×C4—21no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—22no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—81no—
408C2×C3×C4—161no—
409C2×C3×C4C22×D411yescharacteristic
410C2×C3×C4C22×D411yescharacteristic
411C2×C3×C4C22×Q811yescharacteristic
412C2×C3×C4C2×#4111yescharacteristic
413C2×C3×C4C2×#4111yescharacteristic
414C2×C3×C42−1+411yescharacteristic
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—81no—
422C2×Dic3—81no—
423C2×Dic3—161no—
424Dic6—161no—
425C3×C8—22no—
426C3×C8—41no—
427C3⋊C8—21no—
428C3⋊C8—22no—
429C3⋊C8—21no—
430C3⋊C8—21no—
431C3⋊C8—21no—
432C3⋊C8—41no—
433#41—21no—
434#41—21no—
435#41C2×Dic612yes—
436#41#23712yes—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—121no—
443C2×Q8C2×Dic612yes—
444C2×Q8D3×D411yescharacteristic
445C2×D4—21no—
446C2×D4D3×D411yescharacteristic
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—41no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—121no—
455C22×C4C23×D311yescharacteristic
456Q16—24no—
457SD16—24no—
458M16—21no—
459M16—21no—
460M16—21no—
461M16—32no—
462M16—64no—
463M16—61no—
464M16—61no—
465M16—61no—
466M16C2×Dic612yes—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—31no—
470C2×C8—31no—
471C2×C8—31no—
472C2×C8—31no—
473C2×C8—64no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8—61no—
479C2×C8—61no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—31no—
483C4⋊C4—31no—
484C4⋊C4—31no—
485C4⋊C4—31no—
486C4⋊C4—41no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—121no—
492C4⋊C4—121no—
493C4⋊C4—121no—
494C4⋊C4—121no—
495C4⋊C4—121no—
496C4⋊C4—121no—
497C4⋊C4—242no—
498C4⋊C4D3×D411yescharacteristic
499C4⋊C4D3×D411yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—41no—
502C22⋊C4—61no—
503C22⋊C4—61no—
504C22⋊C4—121no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—41no—
509C42—61no—
510C42—61no—
511C42—121no—
512C42—121no—
513C42—121no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3—22no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3#53911yescharacteristic
520C22×C3D4211yescharacteristic
521C22×C3#55411yescharacteristic
522C3×C4—22no—
523C3×C4—42no—
524C3×C4—42no—
525C3×C4—41no—
526C3×C4—41no—
527C3×C4—81no—
528C3×C4C2×C22⋊Q811yescharacteristic
529C3×C4#52511yescharacteristic
530C3×C4#54711yescharacteristic
531C3×C4D4×Q811yescharacteristic
532Dic3—41no—
533Dic3—41no—
534Dic3—41no—
535Dic3—41no—
536Dic3—81no—
537Dic3—81no—
538C23—21no—
539C23#100411yescharacteristic
540Q8—21no—
541Q8—22no—
542Q8—22no—
543Q8—41no—
544Q8—41no—
545Q8—481no—
546Q8#87812yes—
547D4—22no—
548D4—22no—
549D4—21no—
550D4#87812yes—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—81no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—161no—
571C2×C4—241no—
572C2×C4—241no—
573C2×C4—481no—
574C2×C4C22×Dic611yescharacteristic
575C2×C4C2×D3×D411yescharacteristic
576C2×C4C2×D3×D411yescharacteristic
577C2×C4C2×#23711yescharacteristic
578C2×C4D3×#4111yescharacteristic
579C2×C4#101011yescharacteristic
580C8—22no—
581C8—41no—
582C8—61no—
583C8—61no—
584C8—61no—
585C8—61no—
586C8—62no—
587C8—121no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—42no—
591C2×C3#244911yescharacteristic
592C2×C3#328811yescharacteristic
593C2×C3#329311yescharacteristic
594C22—21no—
595C22—21no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22#553911yescharacteristic
600C22#559011yescharacteristic
601C22#560211yescharacteristic
602C4—22no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—42no—
607C4—81no—
608C4—121no—
609C4—121no—
610C4—121no—
611C4—121no—
612C4—241no—
613C4—241no—
614C4C2×#87811yescharacteristic
615C4D4×Dic611yescharacteristic
616C4#560611yescharacteristic
617C4#565511yescharacteristic
618C3#2640611yescharacteristic, complemented, Sylow
619C2—21no—
620C2—21no—
621C2—42no—
622C2#7470911yescharacteristic
623C2#8029411yescharacteristic
624C2#8030311yescharacteristic
subgroupG/H
centerC22#5590
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1882C2
socleC22×C3D42
prime radicalC2×C3×#41C23
hypercenter#1882D3
hypocenterC3#26406
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1882

Complements

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

NH ≅ G/Ntype
4. #78950C2semidirect
9. #70337C2semidirect
10. #69684C2semidirect
11. #69672C2semidirect
12. #68985C2semidirect
13. #68980C2semidirect
14. #68738C2semidirect
15. #68735C2semidirect
86. C3×#137Q8semidirect
302. #137Dic6semidirect
618. C3#26406semidirect

Maximal subgroups

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

subgroupindex
#839652
#825622
#825592
#789502
C3×#18822
#721032
#703762
#703632
#703372
#696842
#696722
#689852
#689802
#687382
#687352
#264063