Finite Group Databasebeta

← #29651058

Subgroups — #29651058

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455879C211yesmaximal, characteristic
2#21454833C211yesmaximal, characteristic, complemented
3#21454562C211yesmaximal, characteristic, complemented
4#21444724C211yesmaximal, characteristic, complemented
5#21444706C211yesmaximal, characteristic, complemented
6#21444680C211yesmaximal, characteristic, complemented
7#21444679C211yesmaximal, characteristic, complemented
8#21444658C211yesmaximal, characteristic, complemented
9#21444645C211yesmaximal, characteristic, complemented
10#21444640C211yesmaximal, characteristic, complemented
11#21444611C211yesmaximal, characteristic, complemented
12#21444602C211yesmaximal, characteristic, complemented
13#21444289C211yesmaximal, characteristic, complemented
14#21444114C211yesmaximal, characteristic, complemented
15C61×#1477C211yesmaximal, characteristic, complemented
16C2×#14519783—21no—
17#16840312C2211yescharacteristic
18#16840086C411yescharacteristic
19#16840086C411yescharacteristic
20#16840057C411yescharacteristic
21#16840052C411yescharacteristic
22#16839962C411yescharacteristic
23#16839957C411yescharacteristic
24#16839907C411yescharacteristic
25#16839907C411yescharacteristic
26#16839893C2211yescharacteristic
27#16839893C2211yescharacteristic
28C4×#11758201C2211yescharacteristic
29C4×#11758201C2211yescharacteristic
30C4×#11758199C2211yescharacteristic
31C4×#11758199C2211yescharacteristic
32#16839839C2211yescharacteristic
33#16839839C2211yescharacteristic
34#16839834—21no—
35#16839834—21no—
36#16839832—21no—
37#16839832—21no—
38D4×#92418C2211yescharacteristic
39D4×#92418C2211yescharacteristic
40#16839827C2211yescharacteristic
41#16839827C2211yescharacteristic
42#16839819—21no—
43#16839819C2211yescharacteristic
44#16839815C2211yescharacteristic
45#16839728—21no—
46C4×#11758194C2211yescharacteristic
47C2×#14519688—21no—
48#16839514C2211yescharacteristic
49#16839514C2211yescharacteristic
50#16839512C2211yescharacteristic
51#16839512C2211yescharacteristic
52#16839491C2211yescharacteristic
53#16839491C2211yescharacteristic
54#16839491C2211yescharacteristic
55#16839491C2211yescharacteristic
56#16839484C2211yescharacteristic
57#16839484C2211yescharacteristic
58#16839483C2211yescharacteristic
59C8×#92418C2211yescharacteristic
60C4×#41×C61C2211yescharacteristic
61C61×#431C2211yescharacteristic
62C2×C4⋊C8×C61C2211yescharacteristic
63C61×Q8⋊C8C2211yescharacteristic
64C61×Q8⋊C8C2211yescharacteristic
65C61×D4⋊C8C2211yescharacteristic
66C61×D4⋊C8C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758201C2×C411yescharacteristic
69C2×#11758201C2×C411yescharacteristic
70C2×#11758199C2×C411yescharacteristic
71C2×#11758199C2×C411yescharacteristic
72#14519783—44no—
73C2×#11758194D411yescharacteristic
74C2×#11758194D411yescharacteristic
75C2×#11758189—21no—
76C2×#11758189—21no—
77C22×#92418—21no—
78C22×#92418—21no—
79C2×C8×D61—21no—
80C2×C8×D61—21no—
81#14519777C2×C411yescharacteristic
82#14519774C2×C411yescharacteristic
83#14519765—21no—
84#14519751—21no—
85#14519751C2×C411yescharacteristic
86#14519751C2×C411yescharacteristic
87C4×D244—21no—
88C4×D244C2311yescharacteristic
89C4×D244C2311yescharacteristic
90#14519710C2×C411yescharacteristic
91#14519710C2×C411yescharacteristic
92#14519708C2×C411yescharacteristic
93#14519708C2×C411yescharacteristic
94C4⋊#92418—21no—
95C4⋊#92418—21no—
96C4⋊#92418C2×C411yescharacteristic
97C4⋊#92418C2×C411yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100#14519701C2×C411yescharacteristic
101#14519701C2×C411yescharacteristic
102#14519698C2×C411yescharacteristic
103#14519698C2×C411yescharacteristic
104#14519697C2×C411yescharacteristic
105#14519697C2×C411yescharacteristic
106#14519697C2×C411yescharacteristic
107#14519697C2×C411yescharacteristic
108C22⋊#92418—21no—
109C22⋊#92418—21no—
110C22⋊#92418—21no—
111C22⋊#92418C2×C411yescharacteristic
112C22⋊#92418C2×C411yescharacteristic
113#14519689—21no—
114#14519689—21no—
115#14519689—21no—
116#14519689—21no—
117#14519688—21no—
118#14519688—21no—
119#14519688—21no—
120#14519688D411yescharacteristic
121#14519688D411yescharacteristic
122C4×#92418C2311yescharacteristic
123C4×#92418C2311yescharacteristic
124C4×#92418C2311yescharacteristic
125C4×#92418C2311yescharacteristic
126C2×#41×C61C2×C411yescharacteristic
127C22×C8×C61—21no—
128C4×Q8×C61C2×C411yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61C2×C411yescharacteristic
133C4×D4×C61C2311yescharacteristic
134#117×C61—21no—
135#117×C61C2×C411yescharacteristic
136C2×C42×C61—21no—
137C2×C42×C61C2311yescharacteristic
138C4⋊C8×C61—21no—
139C4⋊C8×C61—21no—
140C4⋊C8×C61C2311yescharacteristic
141C4⋊C8×C61C2311yescharacteristic
142C22⋊C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145#41×C61—22no—
146#41×C61—41no—
147C2×Q8×C61C22×C411yescharacteristic
148C2×D4×C61—21no—
149C2×D4×C61C22×C411yescharacteristic
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61C22×C411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61#4111yescharacteristic
156C2×C22⋊D61—21no—
157C2×C22⋊D61—21no—
158#11758222—42no—
159#11758222—42no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244C22×C411yescharacteristic
163C2×D244C22×C411yescharacteristic
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×Dic122—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—42no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C22×C411yescharacteristic
175C4⋊C4×C61C22×C411yescharacteristic
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61C22×C411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61#4111yescharacteristic
185C42×C61C2411yescharacteristic
186#11758201—41no—
187#11758201—41no—
188#11758199—22no—
189#11758199—22no—
190#11758198—21no—
191#11758198—21no—
192#11758198—21no—
193#11758198—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61C22×C411yescharacteristic
196C4⋊Dic61C22×C411yescharacteristic
197#11758194—22no—
198#11758194—41no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—42no—
202C2×#92418—42no—
203C2×#92418C22×C411yescharacteristic
204C2×#92418C22×C411yescharacteristic
205C2×#92418C22×C411yescharacteristic
206C2×#92418C22×C411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211#11758189—42no—
212#11758189—42no—
213C8×D61—42no—
214C8×D61—42no—
215C23×C61—21no—
216C23×C61C4×D411yescharacteristic
217C22×D61—21no—
218C22×D61—21no—
219Q8×C61—21no—
220Q8×C61—21no—
221D4×C61—21no—
222D4×C61—42no—
223D4×C61#13112yescomplemented
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—42no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—81no—
236C2×C4×C61C4×D411yescharacteristic
237C2×C4×C61C4×D411yescharacteristic
238C2×C4×C61C4×D411yescharacteristic
239C2×C4×C61C23×C411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C22⋊D61—42no—
243C22⋊D61—42no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246D244—21no—
247D244—21no—
248D244—21no—
249D244—21no—
250D244—42no—
251D244—81no—
252C4×D61—41no—
253C4×D61—41no—
254C4×D61—41no—
255C4×D61—41no—
256Dic122—42no—
257C8×C61—41no—
258C8×C61—41no—
259C8×C61—41no—
260C8×C61—41no—
261#92418—21no—
262#92418—21no—
263#92418—21no—
264#92418—21no—
265#92418—21no—
266#92418—21no—
267#92418—21no—
268#92418—41no—
269#92418—41no—
270#92418AGL(1,ℤ/8)12yes—
271#18007—611nomaximal, Sylow
272C22×C61—21no—
273C22×C61—22no—
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61C2×C4×D411yescharacteristic
279C2×D61—41no—
280C2×D61—41no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—41no—
290C4×C61—41no—
291C4×C61—41no—
292C4×C61C2×#13111yescharacteristic
293C4×C61#56811yescharacteristic
294C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
295C4×C61#57511yescharacteristic
296Dic61—41no—
297Dic61—41no—
298#3106—611no—
299#2970—611no—
300#2935—611no—
301#1592—611no—
302#1590—611no—
303#1586—611no—
304#1585—611no—
305#1582—611no—
306#1581—611no—
307#1580—611no—
308C8×D8—611no—
309#1573—611no—
310#1496—611no—
311#1477D6111yescharacteristic, complemented
312#1453—611no—
313C2×C61—41no—
314C2×C61—42no—
315C2×C61#293311yescharacteristic
316C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
317C2×C61#295711yescharacteristic
318D61—81no—
319D61—81no—
320C2×#131—1221no—
321#532—611no—
322C4×#41C2×D6111yescharacteristic
323#482—611no—
324#482—611no—
325#477—611no—
326#476—611no—
327#460—611no—
328#459—611no—
329#449—611no—
330#449—611no—
331#446—611no—
332#446—611no—
333C4×SD16—611no—
334C4×SD16—611no—
335C4×D8—611no—
336C4×D8—611no—
337#436—611no—
338#436—611no—
339#435—1221no—
340#435—1221no—
341#435—1221no—
342#435—1221no—
343C8×D4—611no—
344C8×D4—611no—
345C8×D4—611no—
346C8×D4—611no—
347#432—611no—
348#432—1221no—
349#431—611no—
350#431C2×D6111yescharacteristic
351C2×C4⋊C8C2×D6111yescharacteristic
352#408—1221no—
353C4×M16—611no—
354C2×#97—1221no—
355#335—611no—
356#335—611no—
357#334—611no—
358#334—611no—
359Q8⋊C8C2×D6111yescharacteristic
360Q8⋊C8C2×D6111yescharacteristic
361D4⋊C8—611no—
362D4⋊C8—611no—
363D4⋊C8—611no—
364D4⋊C8—611no—
365D4⋊C8C2×D6111yescharacteristic
366D4⋊C8C2×D6111yescharacteristic
367#322—611no—
368#322—611no—
369#322—611no—
370C82—611no—
371C61#1800711yescharacteristic, complemented, Sylow
372C2×#41—1221no—
373C2×#41C4×D6111yescharacteristic
374C2×SD16—611no—
375C2×SD16—611no—
376C2×D8—611no—
377C2×D8—611no—
378#131—2444no—
379C2×M16—611no—
380C2×M16—611no—
381C2×M16—1221no—
382C2×M16—1221no—
383C22×C8—21no—
384C22×C8—1221no—
385C22×C8—1221no—
386C22×C8—1221no—
387C22×C8—1221no—
388C4⋊Q8—611no—
389#127—611no—
390#124—1221no—
391#121—611no—
392#121—611no—
393#121—1221no—
394C4×Q8C4×D6111yescharacteristic
395C4×Q8C22×D6111yescharacteristic
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—611no—
399C4×D4—611no—
400C4×D4—1221no—
401C4×D4C4×D6111yescharacteristic
402C4×D4C22×D6111yescharacteristic
403#117—21no—
404#117C4×D6111yescharacteristic
405C2×C42—21no—
406C2×C42C22×D6111yescharacteristic
407#107—611no—
408#107—611no—
409#106—611no—
410#106—611no—
411C4⋊C8—21no—
412C4⋊C8—21no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—611no—
417C4⋊C8—1221no—
418C4⋊C8—1221no—
419C4⋊C8C22×D6111yescharacteristic
420C4⋊C8C22×D6111yescharacteristic
421#103—611no—
422#103—611no—
423#102—611no—
424#102—611no—
425#102—611no—
426#102—611no—
427#102—611no—
428#102—611no—
429C22⋊C8—21no—
430C22⋊C8—611no—
431C22⋊C8—611no—
432C22⋊C8—1221no—
433C22⋊C8—1221no—
434C22⋊C8—1221no—
435C22⋊C8—1221no—
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439#97—611no—
440#97—611no—
441#97—1221no—
442#97—1221no—
443#97—1221no—
444C4×C8—611no—
445C4×C8—611no—
446C4×C8—611no—
447C4×C8—611no—
448C4×C8C22×D6111yescharacteristic
449C4×C8C22×D6111yescharacteristic
450#41—22no—
451#41—41no—
452#41—2442no—
453#41—2442no—
454C2×Q8—1221no—
455C2×Q8C2×C4×D6111yescharacteristic
456C2×D4—21no—
457C2×D4—611no—
458C2×D4—611no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4C2×C4×D6111yescharacteristic
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4C2×C4×D6111yescharacteristic
470C22×C4D4×D6111yescharacteristic
471C22×C4#1175822411yescharacteristic
472SD16—2441no—
473SD16—2441no—
474D8—1222no—
475D8—1222no—
476M16—1222no—
477M16—2442no—
478M16—2442no—
479M16—2441no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—42no—
485C2×C8—611no—
486C2×C8—611no—
487C2×C8—611no—
488C2×C8—611no—
489C2×C8—611no—
490C2×C8—611no—
491C2×C8—611no—
492C2×C8—611no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—2442no—
496C2×C8—2442no—
497C2×C8—2442no—
498C2×C8—2442no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—611no—
502C4⋊C4—611no—
503C4⋊C4—1221no—
504C4⋊C4C2×C4×D6111yescharacteristic
505C4⋊C4C2×C4×D6111yescharacteristic
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—1221no—
510C22⋊C4—1221no—
511C22⋊C4—1221no—
512C22⋊C4—1221no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42C2×C4×D6111yescharacteristic
517C42D4×D6111yescharacteristic
518C42#1175822411yescharacteristic
519C42C23×D6111yescharacteristic
520C23—21no—
521C23—1221no—
522C23—1221no—
523C23#1451973711yescharacteristic
524Q8—21no—
525Q8—21no—
526Q8—2442no—
527D4—21no—
528D4—42no—
529D4—1221no—
530D4—1221no—
531D4—1221no—
532D4—1221no—
533D4—4881no—
534D4—2442no—
535D4—2442no—
536D4—2442no—
537D4#1451978312yescomplemented
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—81no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4#1451973711yescharacteristic
557C2×C4#1451973711yescharacteristic
558C2×C4#1451973711yescharacteristic
559C2×C4C22×C4×D6111yescharacteristic
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4C2×#1175822411yescharacteristic
562C8—41no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—612no—
567C8—1221no—
568C8—1221no—
569C8—1221no—
570C8—1221no—
571C8—1221no—
572C8—1221no—
573C8—1221no—
574C8—2441no—
575C8—2441no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—2441no—
583C22—2441no—
584C22—2441no—
585C22—2441no—
586C22—2441no—
587C22—2441no—
588C22C2×#1451973711yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—2441no—
597C4—2441no—
598C4C2×#1451978311yescharacteristic
599C4#1684052811yescharacteristic
600C4AGL(1,ℤ/8)×D6111yescharacteristic
601C4#1684056111yescharacteristic
602C2—41no—
603C2—42no—
604C2—4881no—
605C2—4881no—
606C2#2145454011yescharacteristic
607C2#2145463211yescharacteristic
608C2#2145473211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1477C2
socleC22×C61C2×C4×D4
prime radical#21455879C2
cosocleC42×C61C24
hypercenter#1477D61
hypocenterC61#18007
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1477

Complements

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

NH ≅ G/Ntype
2. #21454833C2semidirect
3. #21454562C2semidirect
4. #21444724C2semidirect
5. #21444706C2semidirect
6. #21444680C2semidirect
7. #21444679C2semidirect
8. #21444658C2semidirect
9. #21444645C2semidirect
10. #21444640C2semidirect
11. #21444611C2semidirect
12. #21444602C2semidirect
13. #21444289C2semidirect
14. #21444114C2semidirect
15. C61×#1477C2semidirect
223. D4×C61#131semidirect
311. #1477D61semidirect
371. C61#18007semidirect
537. D4#14519783semidirect

Maximal subgroups

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

subgroupindex
#214558792
#214548332
#214545622
#214447242
#214447062
#214446802
#214446792
#214446582
#214446452
#214446402
#214446112
#214446022
#214442892
#214441142
C61×#14772
#1800761