Finite Group Databasebeta

← #29669631

Subgroups — #29669631

Order 15616.

712 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458044C211yesmaximal, characteristic, complemented
2#21457637C211yesmaximal, characteristic, complemented
3D8×D244C211yesmaximal, characteristic
4#21445461C211yesmaximal, characteristic, complemented
5#21445354C211yesmaximal, characteristic, complemented
6#21445267C211yesmaximal, characteristic, complemented
7#21445258C211yesmaximal, characteristic, complemented
8#21445179C211yesmaximal, characteristic, complemented
9#21444996C211yesmaximal, characteristic, complemented
10#21444934C211yesmaximal, characteristic, complemented
11#21444929C211yesmaximal, characteristic, complemented
12#21444891C211yesmaximal, characteristic, complemented
13#21444649C211yesmaximal, characteristic, complemented
14#21444629C211yesmaximal, characteristic, complemented
15C61×#3360C211yesmaximal, characteristic, complemented
16C2×D8×D61—21no—
17D4×D244C2211yescharacteristic
18D4×D244C2211yescharacteristic
19#16840139C2211yescharacteristic
20#16840136C2211yescharacteristic, complemented
21#16840110C2211yescharacteristic, complemented
22#16840110C2211yescharacteristic, complemented
23#16840110C2211yescharacteristic, complemented
24#16840086—21no—
25#16840080—21no—
26#16840049C2211yescharacteristic
27#16840039C2211yescharacteristic
28#16840010—21no—
29#16839999—21no—
30#16839993—21no—
31#16839962C2211yescharacteristic, complemented
32#16839962C2211yescharacteristic, complemented
33#16839957C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839912—21no—
36#16839912—21no—
37#16839897—21no—
38#16839897—21no—
39C4×#11758201C2211yescharacteristic
40#16839840C2211yescharacteristic, complemented
41C8×D244C2211yescharacteristic
42D4×#92418C2211yescharacteristic, complemented
43#16839766—21no—
44C2×#14519697—21no—
45#16839515C2211yescharacteristic, complemented
46#16839514C2211yescharacteristic, complemented
47#16839511C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839505C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic, complemented
51Q8⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic, complemented
56C61×#556C2211yescharacteristic, complemented
57C61×#554C2211yescharacteristic, complemented
58C61×#488—21no—
59C61×#487C2211yescharacteristic, complemented
60C61×#479—21no—
61C61×#476C2211yescharacteristic, complemented
62C61×#463—21no—
63C61×#461C2211yescharacteristic, complemented
64C8×Q8×C61C2211yescharacteristic, complemented
65C4×SD16×C61—21no—
66C4×D8×C61C2211yescharacteristic
67C22×D244—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70C2×D4×D61—21no—
71C2×#11758201—21no—
72C2×#11758201—21no—
73C2×#11758201—41no—
74C2×#11758201D411yescharacteristic
75C2×D488—21no—
76C2×D488—21no—
77C2×#11758199—21no—
78C2×#11758199—21no—
79D8×D61—44no—
80C2×#11758194—21no—
81C22×#92418—21no—
82C2×C8×D61—21no—
83#14519774C2311yescharacteristic
84#14519774C2311yescharacteristic
85#14519752—21no—
86#14519752—21no—
87#14519751—21no—
88#14519751—21no—
89#14519747—41no—
90#14519747—41no—
91C4×D244C2311yescharacteristic
92#14519710—21no—
93#14519708—21no—
94#14519708D411yescharacteristic
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519701D411yescharacteristic
99#14519697—21no—
100#14519697—21no—
101#14519697—21no—
102#14519697—21no—
103#14519697—21no—
104#14519697—21no—
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—41no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113#14519689—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×SD16×C61—21no—
117C2×D8×C61D411yescharacteristic
118C4⋊Q8×C61C2311yescharacteristic
119C4⋊Q8×C61C2311yescharacteristic
120#126×C61—41no—
121#125×C61—21no—
122#124×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126C4×Q8×C61—21no—
127C4×Q8×C61C2311yescharacteristic
128C4×D4×C61—21no—
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131#117×C61—21no—
132C2×C4⋊C4×C61—21no—
133#107×C61D411yescharacteristic
134#106×C61—21no—
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137#103×C61—21no—
138#103×C61—21no—
139#103×C61—21no—
140#102×C61—21no—
141#102×C61D411yescharacteristic
142#102×C61D411yescharacteristic
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—41no—
146C23×D61—41no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×D4×C61C2×D411yescharacteristic
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—41no—
155C2×C22⋊D61—41no—
156C2×C22⋊D61—41no—
157D4×D61—42no—
158D4×D61—42no—
159D4×D61—81no—
160D4×D61—81no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—42no—
168C2×D244—42no—
169C2×D244—41no—
170C2×D244—41no—
171C2×D244—81no—
172C2×D244—81no—
173C2×D244C2×D411yescharacteristic
174C2×C4×D61—21no—
175SD16×C61—42no—
176D8×C61—22no—
177C2×C8×C61—21no—
178C2×C8×C61C2×D411yescharacteristic
179C2×C8×C61C2×D411yescharacteristic
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—41no—
186C4⋊C4×C61—41no—
187C4⋊C4×C61—41no—
188C4⋊C4×C61C2×D411yescharacteristic
189C4⋊C4×C61C2×D411yescharacteristic
190C4⋊C4×C61C2×D411yescharacteristic
191C22⋊C4×C61—21no—
192C22⋊C4×C61—21no—
193C22⋊C4×C61—41no—
194C22⋊C4×C61—41no—
195C22⋊C4×C61—41no—
196C42×C61—21no—
197C42×C61—41no—
198C42×C61C2411yescharacteristic
199#11758201—22no—
200#11758201—81no—
201#11758201—81no—
202#11758201—82no—
203#11758199—42no—
204#11758199—42no—
205#11758198—21no—
206#11758198—41no—
207#11758198—41no—
208C4⋊Dic61C2×D411yescharacteristic
209#11758194—81no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—81no—
215C2×#92418C2×D411yescharacteristic
216C2×#92418C2×D411yescharacteristic
217D488—42no—
218D488—81no—
219C8×D61—42no—
220C23×C61—21no—
221C23×C61—21no—
222C22×D61—21no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—82no—
226C22×D61—82no—
227C22×D61—82no—
228C22×D61—82no—
229C22×D61—161no—
230C22×D61—161no—
231Q8×C61—21no—
232Q8×C61—21no—
233Q8×C61—41no—
234Q8×C61—41no—
235Q8×C61—81no—
236D4×C61—21no—
237D4×C61—21no—
238D4×C61—21no—
239D4×C61AGL(1,ℤ/8)12yes—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—161no—
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61H(5,𝔽2)11yescharacteristic
254C22⋊D61—82no—
255C22⋊D61—82no—
256C2×Dic61—21no—
257D244—21no—
258D244—41no—
259D244—41no—
260D244—41no—
261D244—41no—
262D244—41no—
263D244—41no—
264D244—41no—
265D244—41no—
266D244—81no—
267D244—81no—
268D244—82no—
269D244—82no—
270D244#13512yes—
271C4×D61—41no—
272C4×D61—41no—
273C8×C61—21no—
274C8×C61—21no—
275C8×C61—41no—
276#92418—21no—
277#92418—21no—
278#92418—41no—
279#92418—41no—
280#92418—41no—
281#92418—41no—
282#19773—611nomaximal, Sylow
283C22×C61—22no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61D4211yescharacteristic
290C2×D61—22no—
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—81no—
299C2×D61—164no—
300C2×D61—164no—
301C4×C61—21no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61C2×#13511yescharacteristic
309C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
310C4×C61#57611yescharacteristic
311C4×C61#57711yescharacteristic
312Dic61—41no—
313#3360D6111yescharacteristic, complemented
314#3324—611no—
315#3295—611no—
316D4×D8—611no—
317#1717—611no—
318#1697—611no—
319#1683—611no—
320#1681—611no—
321#1668—611no—
322#1639—611no—
323#1629—611no—
324#1628—611no—
325#1622—611no—
326#1581—611no—
327SD16⋊C8—611no—
328C2×C61—42no—
329C2×C61—81no—
330C2×C61#326511yescharacteristic
331C2×C61#328011yescharacteristic
332C2×C61#328811yescharacteristic
333D61—42no—
334D61—161no—
335D61—161no—
336C22×D8—1221no—
337#556C2×D6111yescharacteristic, complemented
338#554C2×D6111yescharacteristic, complemented
339D42—611no—
340D42—611no—
341#493—611no—
342#492—611no—
343#488—21no—
344#487C2×D6111yescharacteristic, complemented
345#486—611no—
346#486—611no—
347#486—611no—
348#482—1221no—
349#481—1221no—
350#479—21no—
351#476—611no—
352#476C2×D6111yescharacteristic, complemented
353#474—611no—
354#468—1221no—
355#466—1221no—
356#465—1221no—
357#463—21no—
358#461C2×D6111yescharacteristic, complemented
359#460—611no—
360#460—611no—
361#459—611no—
362#459—611no—
363#450—1221no—
364#450—1221no—
365#447—1221no—
366#447—1221no—
367C8×Q8C2×D6111yescharacteristic, complemented
368C4×SD16—21no—
369C4×SD16—611no—
370C4×D8C2×D6111yescharacteristic
371#436—611no—
372C8×D4—611no—
373C8×D4—611no—
374#418—1221no—
375C2×#102—1221no—
376#335—611no—
377#335—611no—
378#333—611no—
379#332—611no—
380#331—611no—
381#331—611no—
382Q8⋊C8—611no—
383D4⋊C8—611no—
384D4⋊C8—611no—
385D4⋊C8—611no—
386#322—611no—
387C61#1977311yescharacteristic, complemented, Sylow
388C22×D4—1221no—
389C22×D4—1221no—
390C22×D4—1221no—
391C22×D4—1221no—
392C2×SD16—21no—
393C2×SD16—611no—
394C2×SD16—1221no—
395C2×SD16—1221no—
396C2×SD16—2441no—
397C2×D8—1221no—
398C2×D8—1221no—
399C2×D8—1221no—
400C2×D8—1221no—
401C2×D8—2444no—
402C2×D8C22⋊D6111yescharacteristic
403C2×M16—1221no—
404C22×C8—1221no—
405C22×C8—1221no—
406C4⋊Q8C22×D6111yescharacteristic
407C4⋊Q8C22×D6111yescharacteristic
408#127—611no—
409#127—611no—
410#126—41no—
411#125—21no—
412#124—21no—
413C22⋊Q8—21no—
414C22⋊Q8—21no—
415C22⋊Q8—41no—
416#121—1221no—
417#121—1221no—
418#121—1221no—
419#121—1221no—
420#120—2441no—
421#120—2441no—
422C4×Q8—21no—
423C4×Q8C22×D6111yescharacteristic
424C4×D4—21no—
425C4×D4—611no—
426C4×D4C22×D6111yescharacteristic
427C4×D4C22×D6111yescharacteristic
428#117—21no—
429C2×C4⋊C4—21no—
430#107—1221no—
431#107C22⋊D6111yescharacteristic
432#106—21no—
433#106—611no—
434#106—1221no—
435C4⋊C8—21no—
436C4⋊C8—611no—
437C4⋊C8—611no—
438C4⋊C8—611no—
439C4⋊C8C22×D6111yescharacteristic
440#103—21no—
441#103—21no—
442#103—21no—
443#103—611no—
444#102—21no—
445#102—611no—
446#102—1221no—
447#102—1221no—
448#102—1221no—
449#102—1221no—
450#102—1221no—
451#102—1221no—
452#102—1221no—
453#102—1221no—
454#102—1221no—
455#102—1221no—
456#102—2441no—
457#102C22⋊D6111yescharacteristic
458#102C22⋊D6111yescharacteristic
459C22⋊C8—1221no—
460C22⋊C8—1221no—
461C22⋊C8—1221no—
462C4×C8—21no—
463C4×C8—611no—
464C4×C8—611no—
465C4×C8C22×D6111yescharacteristic
466C24—2441no—
467C24—2441no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8D4×D6111yescharacteristic
471C2×D4—611no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—4881no—
479C2×D4—4881no—
480C2×D4—4881no—
481C2×D4—4881no—
482C2×D4—2441no—
483C2×D4—2441no—
484C2×D4—2442no—
485C2×D4—2442no—
486C2×D4—2441no—
487C2×D4—2441no—
488C2×D4—2442no—
489C2×D4—2442no—
490C2×D4C2×C22⋊D6111yescharacteristic
491C2×D4C2×C22⋊D6111yescharacteristic
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—41no—
495C22×C4—1221no—
496SD16—42no—
497SD16—1222no—
498SD16—4882no—
499SD16—4881no—
500SD16—4881no—
501D8—22no—
502D8—4881no—
503D8—2442no—
504D8—2442no—
505D8—2442no—
506M16—4881no—
507C2×C8—21no—
508C2×C8—611no—
509C2×C8—611no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—4881no—
515C2×C8—2442no—
516C2×C8C2×C22⋊D6111yescharacteristic
517C2×C8C2×C22⋊D6111yescharacteristic
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—611no—
527C4⋊C4D4×D6111yescharacteristic
528C4⋊C4C2×C22⋊D6111yescharacteristic
529C4⋊C4C2×C22⋊D6111yescharacteristic
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—1221no—
536C22⋊C4—2441no—
537C22⋊C4—2441no—
538C42—21no—
539C42—41no—
540C42C23×D6111yescharacteristic
541C23—21no—
542C23—21no—
543C23—1221no—
544C23—9761no—
545C23—9761no—
546C23—4882no—
547C23—4882no—
548C23—4882no—
549C23—4882no—
550C23—2441no—
551C23—2441no—
552Q8—21no—
553Q8—21no—
554Q8—41no—
555Q8—41no—
556Q8—81no—
557D4—21no—
558D4—21no—
559D4—21no—
560D4—612no—
561D4—1221no—
562D4—4881no—
563D4—4881no—
564D4—4882no—
565D4—4882no—
566D4—4882no—
567D4—4882no—
568D4—2441no—
569D4—2441no—
570D4—2441no—
571D4—2441no—
572D4—2441no—
573D4—2441no—
574D4—2441no—
575D4—2441no—
576D4#1451980212yes—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—161no—
588C2×C4—1221no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4C2×D4×D6111yescharacteristic
592C2×C4C22×C22⋊D6111yescharacteristic
593C2×C4#1451982211yescharacteristic
594C8—21no—
595C8—21no—
596C8—41no—
597C8—1221no—
598C8—1221no—
599C8—2441no—
600C8—2441no—
601C8—2441no—
602C8—2441no—
603C22—22no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—1222no—
610C22—9764no—
611C22—9764no—
612C22—4881no—
613C22—4881no—
614C22—4881no—
615C22—4881no—
616C22—4881no—
617C22—4881no—
618C22—2441no—
619C22—2441no—
620C22D4×C22⋊D6111yescharacteristic
621C4—21no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—81no—
627C4—81no—
628C4—2441no—
629C4#135×D6111yescharacteristic
630C4C2×#1451980211yescharacteristic
631C4#1684056411yescharacteristic
632C4#1684057011yescharacteristic
633C2—42no—
634C2—81no—
635C2—9761no—
636C2—9761no—
637C2—2442no—
638C2#2145731511yescharacteristic
639C2#2145749811yescharacteristic
640C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3360C2
socleC22×C61D42
prime radicalD8×D244C2
hypercenter#3360D61
hypocenterC61#19773
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3360

Complements

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

NH ≅ G/Ntype
1. #21458044C2semidirect
2. #21457637C2semidirect
4. #21445461C2semidirect
5. #21445354C2semidirect
6. #21445267C2semidirect
7. #21445258C2semidirect
8. #21445179C2semidirect
9. #21444996C2semidirect
10. #21444934C2semidirect
11. #21444929C2semidirect
12. #21444891C2semidirect
13. #21444649C2semidirect
14. #21444629C2semidirect
15. C61×#3360C2semidirect
20. #16840136C22semidirect
21. #16840110C22semidirect
22. #16840110C22semidirect
23. #16840110C22semidirect
31. #16839962C22semidirect
32. #16839962C22semidirect
40. #16839840C22semidirect
42. D4×#92418C22semidirect
45. #16839515C22semidirect
46. #16839514C22semidirect
47. #16839511C22semidirect
48. #16839508C22semidirect
49. #16839505C22semidirect
50. #16839505C22semidirect
51. Q8⋊#92418C22semidirect
52. D4⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
55. #16839484C22semidirect
56. C61×#556C22semidirect
57. C61×#554C22semidirect
59. C61×#487C22semidirect
61. C61×#476C22semidirect
63. C61×#461C22semidirect
64. C8×Q8×C61C22semidirect
313. #3360D61semidirect
337. #556C2×D61semidirect
338. #554C2×D61semidirect
344. #487C2×D61semidirect
352. #476C2×D61semidirect
358. #461C2×D61semidirect
367. C8×Q8C2×D61semidirect
387. C61#19773semidirect

Maximal subgroups

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

subgroupindex
#214580442
#214576372
D8×D2442
#214454612
#214453542
#214452672
#214452582
#214451792
#214449962
#214449342
#214449292
#214448912
#214446492
#214446292
C61×#33602
#1977361