Finite Group Databasebeta

← #29665455

Subgroups — #29665455

Order 15616.

650 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456752C211yesmaximal, characteristic, complemented
2#21455879C211yesmaximal, characteristic, complemented
3#21454930C211yesmaximal, characteristic, complemented
4#21445275C211yesmaximal, characteristic, complemented
5#21445268C211yesmaximal, characteristic, complemented
6#21445233C211yesmaximal, characteristic, complemented
7#21445226C211yesmaximal, characteristic, complemented
8#21445177C211yesmaximal, characteristic, complemented
9#21445172C211yesmaximal, characteristic, complemented
10#21445169C211yesmaximal, characteristic, complemented
11#21445162C211yesmaximal, characteristic, complemented
12#21444600C211yesmaximal, characteristic, complemented
13#21444515C211yesmaximal, characteristic, complemented
14#21444468C211yesmaximal, characteristic, complemented
15C61×#3115C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17#16840138C2211yescharacteristic
18#16840136C2211yescharacteristic
19#16840112C2211yescharacteristic
20#16840108C2211yescharacteristic
21#16840086—21no—
22#16840057C2211yescharacteristic
23#16840052C2211yescharacteristic
24#16839974—21no—
25#16839974—21no—
26#16839962C2211yescharacteristic
27#16839957C2211yescharacteristic
28#16839907—21no—
29#16839907—21no—
30#16839907—21no—
31Q8×#92418C2211yescharacteristic
32Q8×#92418C2211yescharacteristic
33C4×#11758201—21no—
34C4×#11758199—21no—
35D4×#92418—21no—
36D4×#92418—21no—
37D4×#92418C2211yescharacteristic
38D4×#92418C2211yescharacteristic
39#16839815—21no—
40#16839815C2211yescharacteristic
41#16839815C2211yescharacteristic
42C2×C4⋊#92418—21no—
43C2×C4⋊#92418C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839508C2211yescharacteristic
51#16839508C2211yescharacteristic
52#16839506C2211yescharacteristic
53#16839506C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C61×#532C2211yescharacteristic
57C4×#41×C61C2211yescharacteristic
58C61×#482—21no—
59C61×#475C2211yescharacteristic
60C61×#474C2211yescharacteristic
61C61×#460C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#449—21no—
64C61×#442—21no—
65C61×#440—21no—
66C61×#432C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758201—21no—
69C2×#11758201—21no—
70C2×#11758201—21no—
71C2×#11758199—21no—
72C2×#11758199—21no—
73C2×#11758199—21no—
74C22×#92418—21no—
75C22×#92418—21no—
76C22×#92418—21no—
77#14519777C2311yescharacteristic
78#14519774C2311yescharacteristic
79#14519765—21no—
80#14519751—21no—
81#14519751—21no—
82C4×D244—21no—
83C4×D244—21no—
84#14519710—21no—
85#14519708—21no—
86C4⋊#92418—21no—
87C4⋊#92418—21no—
88C4⋊#92418—21no—
89C4⋊#92418D411yescharacteristic
90C4⋊#92418D411yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519701—21no—
96#14519701—21no—
97#14519701—21no—
98#14519698—21no—
99#14519698—21no—
100#14519698—21no—
101#14519697—21no—
102#14519697—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105C22⋊#92418—21no—
106C22⋊#92418D411yescharacteristic
107C22⋊#92418D411yescharacteristic
108C4×#92418—21no—
109C4×#92418—21no—
110C4×#92418—21no—
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C2×#41×C61—21no—
114C2×#41×C61D411yescharacteristic
115C2×SD16×C61—21no—
116C2×D8×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118#127×C61C2311yescharacteristic
119#124×C61—21no—
120#121×C61—21no—
121#121×C61—21no—
122C4×Q8×C61D411yescharacteristic
123C4×Q8×C61C2311yescharacteristic
124C4×D4×C61—21no—
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61—21no—
128C4×D4×C61D411yescharacteristic
129C4×D4×C61C2311yescharacteristic
130#117×C61—21no—
131#117×C61D411yescharacteristic
132C2×C42×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#107×C61—21no—
135#106×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143#97×C61—21no—
144#41×C61—22no—
145#41×C61—22no—
146#41×C61—81no—
147#41×C61—81no—
148C2×Q8×C61—21no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—41no—
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—41no—
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61#4111yescharacteristic
163C2×C22⋊D61—41no—
164#11758222—82no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×C4×D61—41no—
169C2×Dic122—21no—
170SD16×C61—42no—
171D8×C61—42no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C42×C61#4111yescharacteristic
191C42×C61C2411yescharacteristic
192#11758201—42no—
193#11758201—81no—
194#11758201—81no—
195#11758199—42no—
196#11758199—81no—
197#11758199—81no—
198#11758198—41no—
199#11758198—41no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—81no—
207C2×#92418—81no—
208C2×#92418—81no—
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213C23×C61—21no—
214C23×C61—41no—
215C23×C61#12111yescharacteristic
216C22×D61—41no—
217Q8×C61—21no—
218Q8×C61—41no—
219Q8×C61—41no—
220Q8×C61#13512yes—
221D4×C61—24no—
222D4×C61—21no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—81no—
228D4×C61—82no—
229D4×C61#13512yes—
230C2×C4×C61—24no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—81no—
243C2×C4×C61—82no—
244C2×C4×C61#12111yescharacteristic
245C2×C4×C61#12111yescharacteristic
246C2×C4×C61#12111yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C22⋊D61—82no—
251C2×Dic61—41no—
252D244—41no—
253D244—41no—
254D244—41no—
255D244—41no—
256D244—81no—
257C4×D61—82no—
258Dic122—81no—
259C8×C61—41no—
260C8×C61—41no—
261#92418—21no—
262#92418—21no—
263#92418—21no—
264#92418—21no—
265#92418—41no—
266#92418—41no—
267#92418—41no—
268#92418—41no—
269#19412—611nomaximal, Sylow
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—22no—
273C22×C61—21no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—81no—
277C22×C61—81no—
278C22×C61—81no—
279C22×C61C2×#12111yescharacteristic
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—81no—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—22no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61—81no—
291C4×C61C2×#13511yescharacteristic
292C4×C61C2×#13511yescharacteristic
293C4×C61#57611yescharacteristic
294C4×C61#57711yescharacteristic
295Dic61—81no—
296#3207—611no—
297#3115D6111yescharacteristic, complemented
298#3106—611no—
299#2988—611no—
300#1684—611no—
301#1683—611no—
302#1678—611no—
303#1677—611no—
304#1668—611no—
305#1667—611no—
306#1666—611no—
307#1665—611no—
308#1572—611no—
309#1551—611no—
310#1540—611no—
311C2×C61—41no—
312C2×C61—42no—
313C2×C61—161no—
314C2×C61#304511yescharacteristic
315C2×C61#305011yescharacteristic
316C2×C61#306811yescharacteristic
317D61—161no—
318#532—611no—
319#532C2×D6111yescharacteristic
320C4×#41C2×D6111yescharacteristic
321#493—611no—
322#492—611no—
323#487—611no—
324#486—611no—
325#482—21no—
326#482—1221no—
327#477—611no—
328#476—611no—
329#475C2×D6111yescharacteristic
330#474C2×D6111yescharacteristic
331#463—1221no—
332#463—1221no—
333#460—611no—
334#460C2×D6111yescharacteristic
335#459—611no—
336#459C2×D6111yescharacteristic
337#449—21no—
338#449—1221no—
339#449—1221no—
340#449—1221no—
341C8×Q8—611no—
342C8×Q8—611no—
343#442—21no—
344#440—21no—
345C4×SD16—1221no—
346C4×D8—1221no—
347C8×D4—611no—
348C8×D4—611no—
349C8×D4—1221no—
350C8×D4—1221no—
351#432C2×D6111yescharacteristic
352#431—611no—
353#431—611no—
354#431—1221no—
355C2×C4⋊C8—611no—
356C2×C4⋊C8—1221no—
357#335—611no—
358#335—611no—
359#334—611no—
360#334—611no—
361#333—611no—
362#333—611no—
363#332—611no—
364#332—611no—
365#331—611no—
366#331—611no—
367#331—611no—
368#331—611no—
369C61#1941211yescharacteristic, complemented, Sylow
370C2×#41—21no—
371C2×#41—1221no—
372C2×#41C22⋊D6111yescharacteristic
373C2×SD16—21no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×D8—21no—
378C2×D8—1221no—
379C2×D8—1221no—
380C2×D8—1221no—
381C22×C8—1221no—
382C22×C8—1221no—
383C22×C8—1221no—
384C4⋊Q8—611no—
385C4⋊Q8C22×D6111yescharacteristic
386#127—611no—
387#127C22×D6111yescharacteristic
388#124—21no—
389#124—1221no—
390#121—21no—
391#121—21no—
392#121—1221no—
393#121—1221no—
394C4×Q8C22⋊D6111yescharacteristic
395C4×Q8C22×D6111yescharacteristic
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—21no—
399C4×D4—21no—
400C4×D4—1221no—
401C4×D4—1221no—
402C4×D4C22⋊D6111yescharacteristic
403C4×D4C22×D6111yescharacteristic
404#117—21no—
405#117C22⋊D6111yescharacteristic
406C2×C42—21no—
407C2×C42C22×D6111yescharacteristic
408#107—21no—
409#107—1221no—
410#106—21no—
411#106—1221no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—611no—
417C4⋊C8—611no—
418C4⋊C8—1221no—
419C4⋊C8—1221no—
420C4⋊C8—1221no—
421C4⋊C8C22×D6111yescharacteristic
422C4⋊C8C22×D6111yescharacteristic
423#103—21no—
424#103—1221no—
425#103—1221no—
426#103—1221no—
427#102—21no—
428#102—21no—
429#102—21no—
430#102—1221no—
431#102—1221no—
432#102—1221no—
433#102—1221no—
434#102—1221no—
435C22⋊C8—21no—
436C22⋊C8—611no—
437C22⋊C8—611no—
438C22⋊C8—1221no—
439C22⋊C8—1221no—
440C22⋊C8—1221no—
441#97—21no—
442C4×C8—611no—
443C4×C8—611no—
444C4×C8—1221no—
445C4×C8—1221no—
446C4×C8—1221no—
447#41—22no—
448#41—22no—
449#41—81no—
450#41—81no—
451#41—4882no—
452C2×Q8—21no—
453C2×Q8—1221no—
454C2×Q8C2×C22⋊D6111yescharacteristic
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—41no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—2441no—
464C2×D4C2×C22⋊D6111yescharacteristic
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—41no—
469C22×C4—2441no—
470C22×C4D4×D6111yescharacteristic
471C22×C4#1175822411yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473SD16—42no—
474SD16—4881no—
475SD16—4881no—
476SD16—2442no—
477D8—42no—
478D8—4881no—
479D8—4881no—
480D8—2442no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—611no—
484C2×C8—611no—
485C2×C8—611no—
486C2×C8—611no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—1221no—
490C2×C8—1221no—
491C2×C8—4881no—
492C2×C8—4881no—
493C2×C8—4881no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4C2×C22⋊D6111yescharacteristic
501C4⋊C4C2×C22⋊D6111yescharacteristic
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C22⋊C4—2441no—
508C22⋊C4—2441no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42D4×D6111yescharacteristic
513C42#1175822411yescharacteristic
514C42C2×C22⋊D6111yescharacteristic
515C42C23×D6111yescharacteristic
516C23—21no—
517C23—41no—
518C23—2441no—
519C23#1451974911yescharacteristic
520Q8—21no—
521Q8—41no—
522Q8—41no—
523Q8—4881no—
524Q8#1451979712yes—
525D4—24no—
526D4—21no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—81no—
532D4—82no—
533D4—4881no—
534D4—4882no—
535D4—2441no—
536D4—2441no—
537D4—2441no—
538D4—2441no—
539D4#1451979712yes—
540C2×C4—24no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4—4882no—
555C2×C4—2441no—
556C2×C4#1451974911yescharacteristic
557C2×C4#1451974911yescharacteristic
558C2×C4#1451974911yescharacteristic
559C2×C4C2×D4×D6111yescharacteristic
560C2×C4C22×C22⋊D6111yescharacteristic
561C2×C4C2×#1175822411yescharacteristic
562C8—41no—
563C8—41no—
564C8—1221no—
565C8—1221no—
566C8—1221no—
567C8—1221no—
568C8—2441no—
569C8—2441no—
570C8—2441no—
571C8—2441no—
572C22—21no—
573C22—21no—
574C22—22no—
575C22—21no—
576C22—41no—
577C22—41no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—4881no—
582C22—4881no—
583C22—4881no—
584C22C2×#1451974911yescharacteristic
585C4—21no—
586C4—21no—
587C4—22no—
588C4—21no—
589C4—21no—
590C4—41no—
591C4—41no—
592C4—81no—
593C4—4881no—
594C4C2×#1451979711yescharacteristic
595C4C2×#1451979711yescharacteristic
596C4#1684056611yescharacteristic
597C4#1684057211yescharacteristic
598C2—41no—
599C2—42no—
600C2—161no—
601C2—9761no—
602C2#2145536211yescharacteristic
603C2#2145539711yescharacteristic
604C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3115C2
socleC22×C61C2×#121
hypercenter#3115D61
hypocenterC61#19412
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3115

Complements

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

NH ≅ G/Ntype
1. #21456752C2semidirect
2. #21455879C2semidirect
3. #21454930C2semidirect
4. #21445275C2semidirect
5. #21445268C2semidirect
6. #21445233C2semidirect
7. #21445226C2semidirect
8. #21445177C2semidirect
9. #21445172C2semidirect
10. #21445169C2semidirect
11. #21445162C2semidirect
12. #21444600C2semidirect
13. #21444515C2semidirect
14. #21444468C2semidirect
15. C61×#3115C2semidirect
297. #3115D61semidirect
369. C61#19412semidirect

Maximal subgroups

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

subgroupindex
#214567522
#214558792
#214549302
#214452752
#214452682
#214452332
#214452262
#214451772
#214451722
#214451692
#214451622
#214446002
#214445152
#214444682
C61×#31152
#1941261