Finite Group Databasebeta

← #10858699

Subgroups — #10858699

Order 768.

732 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83930C211yesmaximal, characteristic
2#83785C211yesmaximal, characteristic, complemented
3#78873C211yesmaximal, characteristic, complemented
4#78731C211yesmaximal, characteristic, complemented
5C3×#1907C211yesmaximal, characteristic
6#72088C211yesmaximal, characteristic
7#72017C211yesmaximal, characteristic, complemented
8#70298C211yesmaximal, characteristic, complemented
9#70181C211yesmaximal, characteristic, complemented
10#70075C211yesmaximal, characteristic, complemented
11#70068C211yesmaximal, characteristic, complemented
12#68984C211yesmaximal, characteristic, complemented
13#68979C211yesmaximal, characteristic, complemented
14#68925C211yesmaximal, characteristic, complemented
15#68923C211yesmaximal, characteristic, complemented
16#26405—31nomaximal, Sylow
17C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
18C2×C3×#131C2211yescharacteristic
19#5827C2211yescharacteristic
20#5827C2211yescharacteristic
21#5798C2211yescharacteristic
22#5500C2211yescharacteristic
23C3×#417C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#100C2211yescharacteristic
26C2×#837C2211yescharacteristic
27#5233C2211yescharacteristic
28#5221C2211yescharacteristic
29#5215C2211yescharacteristic
30C2×#832C2211yescharacteristic
31#5164—21no—
32Dic3×SD16—21no—
33#5152—21no—
34Dic3×D8—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×#818—21no—
47C2×#818C2211yescharacteristic
48#5102C2211yescharacteristic
49#4977C2211yescharacteristic
50#4964C2211yescharacteristic
51C2×#808C2211yescharacteristic
52#4796—21no—
53#4791—21no—
54#4766—21no—
55#4766—21no—
56#4763—21no—
57#4761—21no—
58#4761—21no—
59#4759—21no—
60C3×#343C2211yescharacteristic
61C3×#341C2211yescharacteristic
62#4555C2211yescharacteristic
63#4555C2211yescharacteristic
64#4555C2211yescharacteristic
65#4554C2211yescharacteristic
66#4554C2211yescharacteristic
67#4550C2211yescharacteristic
68#3084—31no—
69#3081—31no—
70#2946—31no—
71#2913—31no—
72#2092—31no—
73#2089—31no—
74#2063—31no—
75#2061—31no—
76#1951—31no—
77#1922—31no—
78#1907D311yescharacteristic
79#1883—31no—
80#1866—31no—
81#1860—31no—
82#1856—31no—
83C2×C3×#41C2311yescharacteristic
84C22×C3×D4C2311yescharacteristic
85C3×AGL(1,ℤ/8)—22no—
86C3×AGL(1,ℤ/8)Q814yescomplemented
87C2×C3×SD16—21no—
88C2×C3×D8—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
98C2×#217—41no—
99C2×#217—41no—
100Q8×Dic3—21no—
101Q8×Dic3—21no—
102D4×Dic3—21no—
103D4×Dic3—21no—
104D4×Dic3—21no—
105D4×Dic3—21no—
106D4×Dic3—21no—
107D4×Dic3—21no—
108D4×Dic3—41no—
109D4×Dic3—41no—
110#926—21no—
111#926—21no—
112#926C2311yescharacteristic
113#926C2311yescharacteristic
114#926C2311yescharacteristic
115C2×C4⋊Dic3—21no—
116C2×C4⋊Dic3—21no—
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4⋊Dic3C2311yescharacteristic
119#924—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×#102—21no—
129C3×#102—21no—
130C3×#102—21no—
131C3×#100—22no—
132#837—22no—
133#835D411yescharacteristic
134#835D411yescharacteristic
135#832—21no—
136#832—21no—
137#832—21no—
138#832—21no—
139#832D411yescharacteristic
140#832D411yescharacteristic
141C3⋊#95—41no—
142#822—22no—
143#818—21no—
144#818—21no—
145#818—21no—
146#818—21no—
147#818—21no—
148#818—21no—
149#818D411yescharacteristic
150#818D411yescharacteristic
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×AGL(1,ℤ/8)C2×D311yescharacteristic
168C2×#131C2×D311yescharacteristic
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—31no—
190#428—31no—
191#428—61no—
192#428—61no—
193#427—31no—
194#427—61no—
195C2×#107—31no—
196C2×#107—61no—
197C2×#106—31no—
198#424—31no—
199#424—31no—
200C2×#104—31no—
201#418—31no—
202#417C2×D311yescharacteristic
203#416—31no—
204C2×#102—31no—
205C2×#102C2×D311yescharacteristic
206C2×#100C2×D311yescharacteristic
207#405—31no—
208#390—31no—
209#382—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—
218C24×C3—21no—
219C3×#41—21no—
220C3×#41—21no—
221C3×#41C2×Q812yes—
222C3×#41#4112yes—
223C2×C3×Q8C2×D411yescharacteristic
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—21no—
227C2×C3×D4—21no—
228C2×C3×D4—21no—
229C2×C3×D4—21no—
230C2×C3×D4C2×D411yescharacteristic
231C2×C3×D4C2×Q812yes—
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—
240C22×Dic3—41no—
241C22×Dic3—41no—
242C2×Dic6—41no—
243C3×SD16—24no—
244C3×D8—24no—
245C3×M16—24no—
246C3×M16—21no—
247C3×M16—21no—
248C3×M16—21no—
249C3×M16—21no—
250C3×M16C2×Q812yes—
251C3×M16#4112yes—
252C2×C3×C8—24no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8C2×D411yescharacteristic
259C2×C3×C8C2×D411yescharacteristic
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262#217—21no—
263#217—21no—
264#217—21no—
265#217—41no—
266#217—41no—
267#217—41no—
268#217—41no—
269#217—82no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—41no—
275C4⋊Dic3—41no—
276C4⋊Dic3—41no—
277C4⋊Dic3C2×D411yescharacteristic
278C4⋊Dic3C2×D411yescharacteristic
279C4⋊Dic3C2×D411yescharacteristic
280C4⋊Dic3C2×D411yescharacteristic
281C4⋊Dic3C2×D411yescharacteristic
282C4⋊Dic3C2×D411yescharacteristic
283Dic3⋊C4—41no—
284Dic3⋊C4—41no—
285C4×Dic3—21no—
286C4×Dic3—21no—
287C4×Dic3—21no—
288C4×Dic3—21no—
289C4×Dic3—21no—
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×D4C22×D311yescharacteristic
301AGL(1,ℤ/8)—22no—
302AGL(1,ℤ/8)Dic614yescomplemented
303C2×SD16—21no—
304C2×D8—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×D4—61no—
323C4×D4—61no—
324C4×D4—61no—
325C4×D4—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—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4C22×D311yescharacteristic
340C2×C22⋊C4—121no—
341C2×C22⋊C4—121no—
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#107—61no—
352#107—61no—
353#106—31no—
354#106—31no—
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—31no—
369#103—31no—
370#102—21no—
371#102—21no—
372#102—21no—
373#102—31no—
374#102—31no—
375#102—61no—
376#102—61no—
377#102—61no—
378#102—61no—
379#100—22no—
380#97—61no—
381C4×C8—61no—
382#95—121no—
383C23×C3—21no—
384C23×C3—22no—
385C23×C3—41no—
386C23×C3—41no—
387C23×C3—81no—
388C23×C3—81no—
389C23×C3H(5,𝔽2)11yescharacteristic
390C3×Q8—21no—
391C3×Q8C22⋊Q812yes—
392C3×D4—22no—
393C3×D4—22no—
394C3×D4—22no—
395C3×D4—21no—
396C3×D4—22no—
397C3×D4—41no—
398C3×D4—41no—
399C3×D4C22⋊Q812yes—
400C2×C3×C4—21no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—41no—
404C2×C3×C4—81no—
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C22×D411yescharacteristic
407C2×C3×C4C22×Q811yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C42−1+411yescharacteristic
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—81no—
423C2×Dic3—81no—
424C2×Dic3—161no—
425C2×Dic3—161no—
426C2×Dic3—161no—
427C2×Dic3—161no—
428Dic6—161no—
429C3×C8—22no—
430C3×C8—21no—
431C3×C8—22no—
432C3×C8—21no—
433C3×C8—41no—
434C3⋊C8—21no—
435C3⋊C8—21no—
436C3⋊C8—41no—
437C24—21no—
438#41—21no—
439#41—21no—
440#41C2×Dic612yes—
441#41#23712yes—
442C2×Q8—121no—
443C2×Q8D3×D411yescharacteristic
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4C2×Dic612yes—
451C2×D4D3×D411yescharacteristic
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—121no—
458C22×C4—121no—
459C22×C4—121no—
460C22×C4—121no—
461C22×C4C23×D311yescharacteristic
462SD16—24no—
463D8—24no—
464M16—24no—
465M16—21no—
466M16—21no—
467M16—21no—
468M16—21no—
469M16—61no—
470M16—61no—
471M16C2×Dic612yes—
472M16#23712yes—
473C2×C8—24no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—31no—
480C2×C8—31no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8D3×D411yescharacteristic
485C2×C8D3×D411yescharacteristic
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—31no—
489C4⋊C4—31no—
490C4⋊C4—31no—
491C4⋊C4—31no—
492C4⋊C4—31no—
493C4⋊C4—31no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—121no—
499C4⋊C4—121no—
500C4⋊C4—121no—
501C4⋊C4—121no—
502C4⋊C4—121no—
503C22⋊C4—61no—
504C22⋊C4—61no—
505C22⋊C4—61no—
506C22⋊C4—121no—
507C22⋊C4—121no—
508C22⋊C4—121no—
509C22⋊C4—121no—
510C22⋊C4—242no—
511C42—61no—
512C42—61no—
513C42—61no—
514C42—61no—
515C42—61no—
516C42—121no—
517C42—121no—
518C22×C3—21no—
519C22×C3—21no—
520C22×C3—22no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—42no—
524C22×C3—82no—
525C22×C3—81no—
526C22×C3—81no—
527C22×C3—161no—
528C22×C3#53911yescharacteristic
529C22×C3D4211yescharacteristic
530C22×C3#55411yescharacteristic
531C3×C4—22no—
532C3×C4—81no—
533C3×C4C2×C22⋊Q811yescharacteristic
534C3×C4#52511yescharacteristic
535C3×C4#54711yescharacteristic
536C3×C4D4×Q811yescharacteristic
537Dic3—41no—
538Dic3—41no—
539Dic3—41no—
540Dic3—41no—
541Dic3—41no—
542Dic3—41no—
543Dic3—81no—
544Dic3—81no—
545Dic3—81no—
546C23—21no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—81no—
551C23—81no—
552C23#100911yescharacteristic
553Q8—21no—
554Q8—481no—
555Q8#87812yes—
556D4—22no—
557D4—22no—
558D4—22no—
559D4—22no—
560D4—21no—
561D4—41no—
562D4—41no—
563D4#87812yes—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—81no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—241no—
581C2×C4—241no—
582C2×C4—481no—
583C2×C4—481no—
584C2×C4—481no—
585C2×C4—481no—
586C2×C4C22×Dic611yescharacteristic
587C2×C4C2×D3×D411yescharacteristic
588C2×C4C2×D3×D411yescharacteristic
589C2×C4C2×#23711yescharacteristic
590C2×C4C2×#23711yescharacteristic
591C2×C4#101011yescharacteristic
592C8—22no—
593C8—21no—
594C8—22no—
595C8—21no—
596C8—41no—
597C8—61no—
598C8—61no—
599C8—121no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—42no—
603C2×C3—82no—
604C2×C3#244911yescharacteristic
605C2×C3#328811yescharacteristic
606C2×C3#329011yescharacteristic
607C22—21no—
608C22—21no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—42no—
613C22—82no—
614C22—81no—
615C22—81no—
616C22—161no—
617C22#554311yescharacteristic
618C22#560911yescharacteristic
619C22#566511yescharacteristic
620C4—22no—
621C4—81no—
622C4—121no—
623C4—121no—
624C4—121no—
625C4—121no—
626C4—121no—
627C4—121no—
628C4—241no—
629C4—241no—
630C4—241no—
631C4C2×#87811yescharacteristic
632C4D4×Dic611yescharacteristic
633C4#558611yescharacteristic
634C4#561111yescharacteristic
635C3#2640511yescharacteristic, complemented, Sylow
636C2—21no—
637C2—21no—
638C2—42no—
639C2—82no—
640C2#7471011yescharacteristic
641C2#8093211yescharacteristic
642C2#8094011yescharacteristic
subgroupG/H
centerC22#5609
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1907C2
socleC22×C3D42
prime radicalC2×C3×AGL(1,ℤ/8)C22
hypercenter#1907D3
hypocenterC3#26405
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1907

Complements

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

NH ≅ G/Ntype
2. #83785C2semidirect
3. #78873C2semidirect
4. #78731C2semidirect
7. #72017C2semidirect
8. #70298C2semidirect
9. #70181C2semidirect
10. #70075C2semidirect
11. #70068C2semidirect
12. #68984C2semidirect
13. #68979C2semidirect
14. #68925C2semidirect
15. #68923C2semidirect
86. C3×AGL(1,ℤ/8)Q8semidirect
302. AGL(1,ℤ/8)Dic6semidirect
635. C3#26405semidirect

Maximal subgroups

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

subgroupindex
#839302
#837852
#788732
#787312
C3×#19072
#720882
#720172
#702982
#701812
#700752
#700682
#689842
#689792
#689252
#689232
#264053