Finite Group Databasebeta

← #29657723

Subgroups — #29657723

Order 15616.

1176 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457126C211yesmaximal, characteristic, complemented
2#21456394C211yesmaximal, characteristic, complemented
3#21455291C211yesmaximal, characteristic, complemented
4#21445068C212yesmaximal, complemented
5#21445012C212yesmaximal, complemented
6#21444922C212yesmaximal, complemented
7#21444884C212yesmaximal, complemented
8#21444496C211yesmaximal, characteristic, complemented
9#21444270C211yesmaximal, characteristic, complemented
10C2×Q8⋊#92418C211yesmaximal, characteristic, complemented
11C61×#3497C211yesmaximal, characteristic, complemented
12C2×#14519808—21no—
13#16840312C2211yescharacteristic
14#16840165C2211yescharacteristic
15#16840138C2211yescharacteristic
16#16840110C2211yescharacteristic
17#16840108C2211yescharacteristic
18#16840042C2212yes—
19#16840039—21no—
20#16839996—22no—
21#16839996—22no—
22#16839984—22no—
23#16839972C2212yes—
24#16839962C2212yes—
25#16839957C2212yes—
26#16839954—21no—
27C4×#11758199—22no—
28#16839786C2211yescharacteristic
29C2×C4⋊#92418C2211yescharacteristic
30#16839761—21no—
31C2×C4×#92418C2211yescharacteristic
32#16839508C2212yes—
33#16839506C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C2×2−1+4×C61—21no—
39C61×#550—21no—
40C61×#550C2212yes—
41D4×Q8×C61—21no—
42D4×Q8×C61C2212yes—
43C61×#548—22no—
44C61×#548—22no—
45C61×#537—22no—
46C61×#537—21no—
47C61×#532—22no—
48C61×#532C2211yescharacteristic
49C61×#532C2211yescharacteristic
50C2×C4×Q8×C61C2211yescharacteristic
51C2×#11758222—21no—
52#14519808—44no—
53C2×#11758202—22no—
54C2×#11758201—22no—
55C2×#11758199—22no—
56C2×#11758199—22no—
57C2×#11758194—21no—
58C22×#92418—21no—
59C22×#92418—21no—
60#14519777C2311yescharacteristic
61#14519774C2311yescharacteristic
62#14519765—21no—
63#14519751—21no—
64#14519751—21no—
65C4×D244—21no—
66C4×D244—21no—
67#14519710—22no—
68#14519710—22no—
69C4⋊#92418—21no—
70C4⋊#92418—21no—
71C4⋊#92418C2312yes—
72C4⋊#92418C2312yes—
73#14519700—22no—
74#14519698—22no—
75#14519698—22no—
76#14519697—22no—
77#14519697—22no—
78#14519697—22no—
79C4×#92418—21no—
80C4×#92418C2311yescharacteristic
81C4×#92418C2311yescharacteristic
822−1+4×C61—44no—
832−1+4×C61—44no—
84C2×#41×C61—24no—
85C2×#41×C61—22no—
86C2×#41×C61—22no—
87C2×#41×C61—21no—
88C2×#41×C61—21no—
89C22×Q8×C61—22no—
90C22×Q8×C61—22no—
91C22×Q8×C61D411yescharacteristic
92C4⋊Q8×C61—22no—
93C4⋊Q8×C61D411yescharacteristic
94C4⋊Q8×C61C2311yescharacteristic
95#127×C61—22no—
96#127×C61D411yescharacteristic
97#127×C61C2311yescharacteristic
98#124×C61—24no—
99#124×C61—21no—
100#124×C61—21no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61—24no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—24no—
105#121×C61—24no—
106#121×C61—22no—
107#121×C61—24no—
108#121×C61—24no—
109#121×C61—21no—
110#121×C61—21no—
111#121×C61—22no—
112#121×C61D412yes—
113#121×C61D412yes—
114C4×Q8×C61—21no—
115C4×Q8×C61—21no—
116C4×Q8×C61D412yes—
117C4×Q8×C61C2312yes—
118C4×D4×C61—24no—
119C4×D4×C61—22no—
120C4×D4×C61—22no—
121C4×D4×C61—21no—
122C4×D4×C61—21no—
123C4×D4×C61D412yes—
124C4×D4×C61C2312yes—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61D411yescharacteristic
127C2×C42×C61—21no—
128C2×C42×C61C2311yescharacteristic
129#41×C61—416no—
130#41×C61—44no—
131#41×C61—48no—
132#41×C61—44no—
133#41×C61—42no—
134#41×C61—42no—
135#41×C61—42no—
136#41×C61—42no—
137C2×Q8×C61—24no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—22no—
141C2×Q8×C61—21no—
142C2×Q8×C61—44no—
143C2×Q8×C61—48no—
144C2×Q8×C61—44no—
145C2×Q8×C61D814yescomplemented
146C2×Q8×C61C2×D412yes—
147C2×D4×C61—24no—
148C2×D4×C61—22no—
149C2×D4×C61—22no—
150C2×D4×C61—24no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—21no—
155C2×D4×C61C2×D412yes—
156C2×D4×C61C2×D412yes—
157C22×C4×C61—21no—
158C22×C4×C61—24no—
159C22×C4×C61—24no—
160C22×C4×C61—21no—
161C22×C4×C61—22no—
162C22×C4×C61—22no—
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D411yescharacteristic
166C2×C22⋊D61—41no—
167#11758222—42no—
168#11758222—42no—
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—21no—
172C2×C4×D61—41no—
173C2×Dic122—21no—
174C4⋊C4×C61—24no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61C2×D412yes—
181C4⋊C4×C61C2×D412yes—
182C4⋊C4×C61C2×D412yes—
183C22⋊C4×C61—24no—
184C22⋊C4×C61—24no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—22no—
189C42×C61—22no—
190C42×C61—22no—
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2411yescharacteristic
195#11758202—44no—
196#11758201—44no—
197#11758199—44no—
198#11758199—82no—
199#11758198—41no—
200#11758198—41no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203#11758194—42no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—22no—
207C2×#92418—22no—
208C2×#92418—42no—
209C2×#92418—81no—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61#12011yescharacteristic
213C22×D61—41no—
214Q8×C61—22no—
215Q8×C61—24no—
216Q8×C61—48no—
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—42no—
220Q8×C61C2×D814yes—
221D4×C61—22no—
222D4×C61—22no—
223D4×C61—42no—
224D4×C61—48no—
225D4×C61—44no—
226D4×C61—44no—
227D4×C61—41no—
228D4×C61—41no—
229D4×C61—41no—
230D4×C61—41no—
231D4×C61—42no—
232D4×C61—81no—
233D4×C61—84no—
234C2×C4×C61—22no—
235C2×C4×C61—24no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—41no—
242C2×C4×C61—48no—
243C2×C4×C61—48no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—44no—
247C2×C4×C61—81no—
248C2×C4×C61#12011yescharacteristic
249C2×C4×C61#12011yescharacteristic
250C2×C4×C61#12011yescharacteristic
251C2×C4×C61C2×D812yes—
252C2×C4×C61AGL(1,ℤ/8)12yes—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C22⋊D61—82no—
257C2×Dic61—41no—
258D244—42no—
259D244—41no—
260D244—41no—
261D244—41no—
262D244—41no—
263C4×D61—82no—
264Dic122—41no—
265Dic122—41no—
266#92418—22no—
267#92418—41no—
268#92418—42no—
269#92418—42no—
270#18674—611nomaximal, Sylow
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61#44712yes—
280C22×C61C2×#12011yescharacteristic
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C4×C61—24no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—42no—
289C4×C61—42no—
290C4×C61—42no—
291C4×C61#44712yes—
292C4×C61C22×D811yescharacteristic
293C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
294C4×C61#57711yescharacteristic
295C4×C61#57811yescharacteristic
296Dic61—81no—
297#3497D6111yescharacteristic, complemented
298#3245—611no—
299#3169—611no—
300#3037—611no—
301#1649—612no—
302#1641—612no—
303#1627—612no—
304#1621—612no—
305#1547—611no—
306#1492—611no—
307C2×Q8⋊C8—611no—
308C2×C61—22no—
309C2×C61—82no—
310C2×C61—82no—
311C2×C61C2×#44711yescharacteristic
312C2×C61#301311yescharacteristic
313C2×C61#301811yescharacteristic
314D61—161no—
315C2×2−1+4—21no—
316C2×#137—1221no—
317#550—21no—
318#550C2×D6112yes—
319D4×Q8—21no—
320D4×Q8C2×D6112yes—
321#548—22no—
322#548—22no—
323#537—22no—
324#537—21no—
325#532—22no—
326#532—611no—
327#532C2×D6111yescharacteristic
328#532C2×D6111yescharacteristic
329C2×C4×Q8C2×D6111yescharacteristic
330#500—611no—
331#493—611no—
332#486—611no—
333#486—611no—
334#474—612no—
335#474—1221no—
336#466—1222no—
337#466—1222no—
338#464—1222no—
339#462—612no—
340#460—612no—
341#459—612no—
342#459—1221no—
343C4×D8—1222no—
344#423—611no—
345C2×C4⋊C8—611no—
346#417—1221no—
347C2×C4×C8—611no—
348#332—612no—
349#331—612no—
350Q8⋊C8—612no—
351Q8⋊C8—612no—
352D4⋊C8—612no—
353D4⋊C8—612no—
354C61#1867411yescharacteristic, complemented, Sylow
3552−1+4—44no—
3562−1+4—44no—
357C2×#41—24no—
358C2×#41—22no—
359C2×#41—22no—
360C2×#41—21no—
361C2×#41—21no—
362C2×#41—1221no—
363C22×Q8—22no—
364C22×Q8—22no—
365C22×Q8C22⋊D6111yescharacteristic
366#137—2444no—
367C2×Q16—1222no—
368C2×SD16—1222no—
369C2×D8—1222no—
370C2×D8—1222no—
371C2×M16—1221no—
372C22×C8—1221no—
373C22×C8—1221no—
374C4⋊Q8—22no—
375C4⋊Q8—611no—
376C4⋊Q8C22⋊D6111yescharacteristic
377C4⋊Q8C22×D6111yescharacteristic
378#127—22no—
379#127—611no—
380#127C22⋊D6111yescharacteristic
381#127C22×D6111yescharacteristic
382#124—24no—
383#124—21no—
384#124—21no—
385#124—1221no—
386C22⋊Q8—22no—
387C22⋊Q8—24no—
388C22⋊Q8—22no—
389C22⋊Q8—24no—
390#121—24no—
391#121—22no—
392#121—24no—
393#121—24no—
394#121—21no—
395#121—21no—
396#121—22no—
397#121—1221no—
398#121—1221no—
399#121C22⋊D6112yes—
400#121C22⋊D6112yes—
401C4×Q8—21no—
402C4×Q8—21no—
403C4×Q8C22⋊D6112yes—
404C4×Q8C22×D6112yes—
405C4×D4—24no—
406C4×D4—22no—
407C4×D4—22no—
408C4×D4—21no—
409C4×D4—21no—
410C4×D4—1221no—
411C4×D4—1221no—
412C4×D4C22⋊D6112yes—
413C4×D4C22×D6112yes—
414C2×C4⋊C4—21no—
415C2×C4⋊C4C22⋊D6111yescharacteristic
416C2×C42—21no—
417C2×C42C22×D6111yescharacteristic
418#107—1222no—
419#107—1222no—
420C4⋊C8—612no—
421C4⋊C8—612no—
422C4⋊C8—1221no—
423C4⋊C8—1221no—
424#103—1222no—
425#102—1222no—
426#102—1222no—
427#102—1222no—
428#102—1222no—
429#102—1222no—
430C4×C8—611no—
431C4×C8—611no—
432C4×C8—1221no—
433#41—416no—
434#41—44no—
435#41—48no—
436#41—44no—
437#41—42no—
438#41—42no—
439#41—42no—
440#41—42no—
441#41—2442no—
442#41—2442no—
443C2×Q8—24no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—22no—
448C2×Q8—44no—
449C2×Q8—48no—
450C2×Q8—44no—
451C2×Q8—1221no—
452C2×Q8#1175819914yescomplemented
453C2×Q8C2×C22⋊D6112yes—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—1221no—
465C2×D4—2441no—
466C2×D4C2×C22⋊D6112yes—
467C2×D4C2×C22⋊D6112yes—
468C22×C4—21no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—2441no—
475C22×C4C2×C22⋊D6111yescharacteristic
476C22×C4C2×C22⋊D6111yescharacteristic
477C22×C4C2×C22⋊D6111yescharacteristic
478Q16—2444no—
479SD16—2444no—
480D8—4882no—
481D8—2444no—
482M16—2442no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1222no—
486C2×C8—1222no—
487C2×C8—4881no—
488C2×C8—2442no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4C2×C22⋊D6112yes—
498C4⋊C4C2×C22⋊D6112yes—
499C4⋊C4C2×C22⋊D6112yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—2441no—
507C22⋊C4—2441no—
508C42—22no—
509C42—22no—
510C42C2×C22⋊D6111yescharacteristic
511C42C2×C22⋊D6111yescharacteristic
512C42C2×C22⋊D6111yescharacteristic
513C42C23×D6111yescharacteristic
514C23—22no—
515C23—22no—
516C23—2441no—
517C23#1451974611yescharacteristic
518Q8—22no—
519Q8—24no—
520Q8—48no—
521Q8—41no—
522Q8—41no—
523Q8—42no—
524Q8—2441no—
525Q8—2441no—
526Q8C2×#1175819914yes—
527D4—22no—
528D4—22no—
529D4—42no—
530D4—48no—
531D4—44no—
532D4—44no—
533D4—41no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—42no—
538D4—81no—
539D4—84no—
540D4—4882no—
541D4—2442no—
542D4—2441no—
543D4—2441no—
544D4—2441no—
545D4—2441no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—41no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—81no—
560C2×C4—4882no—
561C2×C4—2441no—
562C2×C4#1451974611yescharacteristic
563C2×C4#1451974611yescharacteristic
564C2×C4#1451974611yescharacteristic
565C2×C4C2×#1175819912yes—
566C2×C4#1451980312yes—
567C2×C4C22×C22⋊D6111yescharacteristic
568C2×C4C22×C22⋊D6111yescharacteristic
569C2×C4C22×C22⋊D6111yescharacteristic
570C8—1222no—
571C8—2441no—
572C8—2442no—
573C8—2442no—
574C22—21no—
575C22—21no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—4881no—
583C22—4881no—
584C22—4881no—
585C22#1683989612yes—
586C22C2×#1451974611yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—24no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—4881no—
595C4#1683989612yes—
596C4C22×#1175819911yescharacteristic
597C4C2×#1451980311yescharacteristic
598C4#1684057011yescharacteristic
599C4#1684057711yescharacteristic
600C2—22no—
601C2—82no—
602C2—82no—
603C2—9761no—
604C2C2×#1683989611yescharacteristic
605C2#2145513411yescharacteristic
606C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3497C2
socleC22×C61C2×#120
hypercenter#3497D61
hypocenterC61#18674
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3497

Complements

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

NH ≅ G/Ntype
1. #21457126C2semidirect
2. #21456394C2semidirect
3. #21455291C2semidirect
4. #21445068C2semidirect
5. #21445012C2semidirect
6. #21444922C2semidirect
7. #21444884C2semidirect
8. #21444496C2semidirect
9. #21444270C2semidirect
10. C2×Q8⋊#92418C2semidirect
11. C61×#3497C2semidirect
145. C2×Q8×C61D8semidirect
297. #3497D61semidirect
354. C61#18674semidirect
452. C2×Q8#11758199semidirect

Maximal subgroups

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

subgroupindex
#214571262
#214563942
#214552912
#214450682
#214450122
#214449222
#214448842
#214444962
#214442702
C2×Q8⋊#924182
C61×#34972
#1867461