Finite Group Databasebeta

← #10858639

Subgroups — #10858639

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#83784C211yesmaximal, characteristic
3#78873C211yesmaximal, characteristic
4#78779C211yesmaximal, characteristic, complemented
5C3×#1908C211yesmaximal, characteristic
6#72100C211yesmaximal, characteristic
7#72017C211yesmaximal, characteristic
8#70297C211yesmaximal, characteristic
9#70250C211yesmaximal, characteristic, complemented
10#70109C211yesmaximal, characteristic, complemented
11#70102C211yesmaximal, characteristic, complemented
12#68984C211yesmaximal, characteristic, complemented
13#68978C211yesmaximal, characteristic, complemented
14#68925C211yesmaximal, characteristic, complemented
15#68924C211yesmaximal, characteristic, complemented
16#26401—31nomaximal, Sylow
17C2×C3×#137C2211yescharacteristic
18C2×C3×#131C2211yescharacteristic
19#5827C2211yescharacteristic
20#5827C2211yescharacteristic
21#5812C2211yescharacteristic
22#5500C2211yescharacteristic
23C3×#417C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#101C2211yescharacteristic
26C2×#837C2211yescharacteristic
27#5233C2211yescharacteristic
28#5229C2211yescharacteristic
29#5225C2211yescharacteristic
30C2×#835C2211yescharacteristic
31#5184—21no—
32Dic3×Q16—21no—
33#5164—21no—
34Dic3×SD16—21no—
35C2×#822C2211yescharacteristic
36#5122C2211yescharacteristic
37#5121—21no—
38#5121—21no—
39#5121C2211yescharacteristic
40#5121C2211yescharacteristic
41#5121C2211yescharacteristic
42#5121C2211yescharacteristic
43#5119C2211yescharacteristic
44#5106—21no—
45#5106C2211yescharacteristic
46C2×#817—21no—
47C2×#817C2211yescharacteristic
48#5102C2211yescharacteristic
49#4975C2211yescharacteristic
50#4964C2211yescharacteristic
51C2×#808C2211yescharacteristic
52#4799—21no—
53#4796—21no—
54#4796—21no—
55#4793—21no—
56#4791—21no—
57#4791—21no—
58#4766—21no—
59#4761—21no—
60C3×#343C2211yescharacteristic
61C3×#341C2211yescharacteristic
62#4555C2211yescharacteristic
63#4555C2211yescharacteristic
64#4555C2211yescharacteristic
65#4554C2211yescharacteristic
66#4554C2211yescharacteristic
67#4550C2211yescharacteristic
68#3084—31no—
69#3082—31no—
70#2947—31no—
71#2912—31no—
72#2096—31no—
73#2090—31no—
74#2064—31no—
75#2060—31no—
76#1951—31no—
77#1922—31no—
78#1908D311yescharacteristic
79#1882—31no—
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—
89C3×#131—22no—
90C3×#131—22no—
91C2×C3×M16—21no—
92C2×C3×M16C2311yescharacteristic
93C2×C3×M16C2311yescharacteristic
94C2×C3×M16C2311yescharacteristic
95C22×C3×C8—21no—
96C22×C3×C8C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98Q8×Dic3—21no—
99Q8×Dic3—21no—
100Q8×Dic3—21no—
101Q8×Dic3—21no—
102Q8×Dic3—21no—
103Q8×Dic3—21no—
104D4×Dic3—21no—
105D4×Dic3—21no—
106D4×Dic3—41no—
107D4×Dic3—41no—
108#926—21no—
109#926—21no—
110#926—41no—
111#926C2311yescharacteristic
112#926C2311yescharacteristic
113#926C2311yescharacteristic
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3—21no—
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118#924—41no—
119C2×Dic3⋊C4—41no—
120C2×C4×Dic3C2311yescharacteristic
121C2×#208C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#890—21no—
124#890—21no—
125#888—21no—
126#888—21no—
127C3×#103—21no—
128C3×#103—21no—
129C3×#103—21no—
130C3×#102—21no—
131C3×#101—22no—
132#837—22no—
133#835—21no—
134#835—21no—
135#835—21no—
136#835—21no—
137#835D411yescharacteristic
138#835D411yescharacteristic
139#832D411yescharacteristic
140#832D411yescharacteristic
141C3⋊#95—41no—
142#822—22no—
143#818—21no—
144#818—21no—
145#818—21no—
146#818—21no—
147#818D411yescharacteristic
148#818D411yescharacteristic
149#817—21no—
150#817—21no—
151#817—21no—
152#817—21no—
153#817—21no—
154#817—21no—
155#817D411yescharacteristic
156#817D411yescharacteristic
157#815—21no—
158Dic3⋊C8—21no—
159Dic3⋊C8—21no—
160Dic3⋊C8—21no—
161Dic3⋊C8—21no—
162C8×Dic3—21no—
163#808—21no—
164#808—21no—
165#808—21no—
166#807—21no—
167C2×#137C2×D311yescharacteristic
168C2×#131C2×D311yescharacteristic
169#544—31no—
170#520—31no—
171#520—31no—
172#519—31no—
173#479—61no—
174#479—61no—
175#479—61no—
176#478—61no—
177#477—61no—
178#476—61no—
179#475—61no—
180#475—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×#106—31no—
196C2×#106—31no—
197C2×#106—61no—
198#424—31no—
199#424—31no—
200C2×#104—31no—
201#419—31no—
202#417C2×D311yescharacteristic
203#416—31no—
204C2×#103—31no—
205C2×#103C2×D311yescharacteristic
206C2×#101C2×D311yescharacteristic
207#405—31no—
208#391—31no—
209#384—31no—
210#343—31no—
211#343—31no—
212#343C2×D311yescharacteristic
213#341C2×D311yescharacteristic
214#339—31no—
215#339—31no—
216#339—31no—
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×C4C2411yescharacteristic
235C22×Dic3—21no—
236C22×Dic3—21no—
237C22×Dic3—21no—
238C22×Dic3—41no—
239C22×Dic3—41no—
240C2×Dic6—41no—
241C3×Q16—24no—
242C3×SD16—24no—
243C3×M16—24no—
244C3×M16—21no—
245C3×M16—21no—
246C3×M16—21no—
247C3×M16—21no—
248C3×M16C2×Q812yes—
249C3×M16#4112yes—
250C2×C3×C8—24no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8C2×D411yescharacteristic
257C2×C3×C8C2×D411yescharacteristic
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260#217—21no—
261#217—21no—
262#217—21no—
263#217—41no—
264#217—41no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—41no—
270C4⋊Dic3—41no—
271C4⋊Dic3—41no—
272C4⋊Dic3—41no—
273C4⋊Dic3—41no—
274C4⋊Dic3C2×D411yescharacteristic
275C4⋊Dic3C2×D411yescharacteristic
276C4⋊Dic3C2×D411yescharacteristic
277C4⋊Dic3C2×D411yescharacteristic
278C4⋊Dic3C2×D411yescharacteristic
279C4⋊Dic3C2×D411yescharacteristic
280Dic3⋊C4—41no—
281Dic3⋊C4—41no—
282Dic3⋊C4—82no—
283C4×Dic3—21no—
284C4×Dic3—21no—
285C4×Dic3—21no—
286C4×Dic3—21no—
287C4×Dic3—21no—
288C4×Dic3—41no—
289C4×Dic3—41no—
290C4×Dic3—41no—
291C4×Dic3—41no—
292#208—21no—
293#208—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
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—22no—
306#131—22no—
307C2×M16—21no—
308C2×M16—31no—
309C2×M16C22×D311yescharacteristic
310C2×M16C22×D311yescharacteristic
311C2×M16C22×D311yescharacteristic
312C22×C8—21no—
313C22×C8—31no—
314C22×C8C22×D311yescharacteristic
315C4⋊Q8—61no—
316C4⋊Q8—61no—
317#125—61no—
318#125—61no—
319C22⋊Q8—121no—
320C4×Q8—61no—
321C4×Q8—61no—
322C4×Q8—61no—
323C4×Q8—61no—
324C4×Q8—61no—
325C4×Q8—61no—
326C4×D4—61no—
327C4×D4—61no—
328C4×D4—121no—
329C4×D4—121no—
330#117—31no—
331#117—31no—
332#117—31no—
333#117—61no—
334#117—61no—
335#117—121no—
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—31no—
345#107—31no—
346#107—61no—
347#107—61no—
348#107—61no—
349#107—61no—
350#107—61no—
351#106—31no—
352#106—31no—
353#106—61no—
354#106—61no—
355#106—61no—
356#106—61no—
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—62no—
367#103—21no—
368#103—21no—
369#103—21no—
370#103—31no—
371#103—31no—
372#103—61no—
373#103—61no—
374#103—61no—
375#103—61no—
376#102—21no—
377#102—31no—
378#102—31no—
379#101—22no—
380#97—61no—
381C4×C8—61no—
382#95—121no—
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—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
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
409C2×Dic3—22no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—81no—
421C2×Dic3—81no—
422C2×Dic3—161no—
423C2×Dic3—161no—
424Dic6—161no—
425C3×C8—22no—
426C3×C8—21no—
427C3×C8—22no—
428C3×C8—21no—
429C3×C8—41no—
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—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—121no—
454C22×C4—121no—
455C22×C4C23×D311yescharacteristic
456Q16—24no—
457SD16—24no—
458M16—24no—
459M16—21no—
460M16—21no—
461M16—21no—
462M16—21no—
463M16—61no—
464M16—61no—
465M16C2×Dic612yes—
466M16#23712yes—
467C2×C8—24no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—31no—
474C2×C8—31no—
475C2×C8—61no—
476C2×C8—61no—
477C2×C8—61no—
478C2×C8D3×D411yescharacteristic
479C2×C8D3×D411yescharacteristic
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—31no—
483C4⋊C4—31no—
484C4⋊C4—31no—
485C4⋊C4—31no—
486C4⋊C4—31no—
487C4⋊C4—31no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—121no—
493C4⋊C4—121no—
494C4⋊C4—121no—
495C4⋊C4—121no—
496C4⋊C4—121no—
497C4⋊C4—121no—
498C4⋊C4—121no—
499C4⋊C4—242no—
500C22⋊C4—61no—
501C22⋊C4—61no—
502C22⋊C4—61no—
503C22⋊C4—121no—
504C22⋊C4—121no—
505C42—61no—
506C42—61no—
507C42—61no—
508C42—61no—
509C42—61no—
510C42—121no—
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—81no—
525C3×C4C2×C22⋊Q811yescharacteristic
526C3×C4#52511yescharacteristic
527C3×C4#54711yescharacteristic
528C3×C4D4×Q811yescharacteristic
529Dic3—42no—
530Dic3—41no—
531Dic3—41no—
532Dic3—41no—
533Dic3—41no—
534Dic3—41no—
535Dic3—41no—
536Dic3—81no—
537Dic3—81no—
538C23—21no—
539C23#100911yescharacteristic
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—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—62no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
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×C4C22×Dic611yescharacteristic
575C2×C4C2×D3×D411yescharacteristic
576C2×C4C2×D3×D411yescharacteristic
577C2×C4C2×#23711yescharacteristic
578C2×C4C2×#23711yescharacteristic
579C2×C4#101011yescharacteristic
580C8—22no—
581C8—21no—
582C8—22no—
583C8—21no—
584C8—41no—
585C8—61no—
586C8—61no—
587C8—121no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—42no—
591C2×C3#244911yescharacteristic
592C2×C3#328911yescharacteristic
593C2×C3#329111yescharacteristic
594C22—21no—
595C22—21no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22#554311yescharacteristic
600C22#560911yescharacteristic
601C22#566511yescharacteristic
602C4—22no—
603C4—42no—
604C4—81no—
605C4—122no—
606C4—121no—
607C4—121no—
608C4—121no—
609C4—121no—
610C4—121no—
611C4—121no—
612C4—241no—
613C4—241no—
614C4C2×#87811yescharacteristic
615C4D4×Dic611yescharacteristic
616C4#558611yescharacteristic
617C4#561111yescharacteristic
618C3#2640111yescharacteristic, complemented, Sylow
619C2—21no—
620C2—21no—
621C2—42no—
622C2#7471011yescharacteristic
623C2#8093511yescharacteristic
624C2#8094511yescharacteristic
subgroupG/H
centerC22#5609
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1908C2
socleC22×C3D42
prime radicalC2×C3×#41C23
hypercenter#1908D3
hypocenterC3#26401
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1908

Complements

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

NH ≅ G/Ntype
4. #78779C2semidirect
9. #70250C2semidirect
10. #70109C2semidirect
11. #70102C2semidirect
12. #68984C2semidirect
13. #68978C2semidirect
14. #68925C2semidirect
15. #68924C2semidirect
86. C3×#137Q8semidirect
302. #137Dic6semidirect
618. C3#26401semidirect

Maximal subgroups

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

subgroupindex
#839652
#837842
#788732
#787792
C3×#19082
#721002
#720172
#702972
#702502
#701092
#701022
#689842
#689782
#689252
#689242
#264013