Finite Group Databasebeta

← #29836527

Subgroups — #29836527

Order 15616.

702 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840198C211yesmaximal, characteristic, complemented
2C2×#16840192C211yesmaximal, characteristic, complemented
3#21457579C211yesmaximal, characteristic, complemented
4#21448086C211yesmaximal, characteristic, complemented
5#21447934C211yesmaximal, characteristic, complemented
6#21447903C211yesmaximal, characteristic, complemented
7#21447878C211yesmaximal, characteristic, complemented
8#21447854C211yesmaximal, characteristic, complemented
9#21447757C211yesmaximal, characteristic, complemented
10#21447742C211yesmaximal, characteristic, complemented
11#21447740C211yesmaximal, characteristic, complemented
12#21447731C211yesmaximal, characteristic, complemented
13#21447683C211yesmaximal, characteristic, complemented
14#21447645C211yesmaximal, characteristic, complemented
15C61×#3280C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic
17C2×#14519799C2211yescharacteristic, complemented
18#16840433—21no—
19#16840198—24no—
20#16840192—24no—
21C2×#14519723C2211yescharacteristic, complemented
22C2×#14519721C2211yescharacteristic
23C2×#14519721C2211yescharacteristic
24C2×#14519720C2211yescharacteristic, complemented
25C2×#14519720C2211yescharacteristic, complemented
26C2×#14519718C2211yescharacteristic
27C2×#14519718C2211yescharacteristic
28C2×#14519718C2211yescharacteristic
29C2×#14519718C2211yescharacteristic
30C2×#14519717C2211yescharacteristic
31C22×#11758185C2211yescharacteristic, complemented
32#16840139C2211yescharacteristic, complemented
33#16839998C2211yescharacteristic
34#16839998C2211yescharacteristic, complemented
35#16839963—21no—
36#16839909—21no—
37#16839909—21no—
38C4×Dic244C2211yescharacteristic, complemented
39#16839587C2211yescharacteristic, complemented
40C4⋊#11758185C2211yescharacteristic, complemented
41#16839566C2211yescharacteristic, complemented
42#16839565C2211yescharacteristic, complemented
43#16839563C2211yescharacteristic
44#16839563C2211yescharacteristic
45#16839562C2211yescharacteristic, complemented
46#16839562C2211yescharacteristic, complemented
47C22⋊#11758185C2211yescharacteristic
48C22⋊#11758185C2211yescharacteristic, complemented
49#16839538C2211yescharacteristic, complemented
50C2×#135×C61C2211yescharacteristic
51C22×D8×C61C2211yescharacteristic
52C61×#546—21no—
53C61×#495C2211yescharacteristic
54C61×#467C2211yescharacteristic
55C61×#466C2211yescharacteristic
56C61×#463—21no—
57C61×#450—21no—
58C61×#449—21no—
59C4×D8×C61C2211yescharacteristic
60C8×D4×C61C2211yescharacteristic
61C2×#11758222—21no—
62C2×#11758222—21no—
63#14519799—22no—
64#14519799—22no—
65#14519799—41no—
66#14519799—41no—
67C2×Dic244C2311yescharacteristic
68C2×#11758190—21no—
69C2×#11758190—21no—
70C2×D488—21no—
71C2×D488C2311yescharacteristic
72C2×D488C2311yescharacteristic
73#14519774—21no—
74#14519773—41no—
75#14519751—21no—
76#14519751—21no—
77#14519750—41no—
78#14519750—41no—
79C4×Dic122—21no—
80#14519739—41no—
81#14519723—22no—
82#14519721—22no—
83#14519721D414yescomplemented
84#14519720—22no—
85#14519720—41no—
86#14519718—22no—
87#14519718—22no—
88#14519718—22no—
89#14519718—41no—
90#14519717—22no—
91C2×#11758185—22no—
92C2×#11758185C2311yescharacteristic
93C2×#11758185C2311yescharacteristic
94C2×#11758185C2311yescharacteristic
95C2×#11758185C2311yescharacteristic
96#14519711C2311yescharacteristic
97#14519702—21no—
98#14519699—21no—
99#14519699—21no—
100C2×#41×C61—21no—
101C22×D4×C61—21no—
102#135×C61—22no—
103#135×C61—41no—
104C2×Q16×C61C2311yescharacteristic
105C2×SD16×C61—21no—
106C2×SD16×C61—21no—
107C2×D8×C61—22no—
108C2×D8×C61—21no—
109C2×D8×C61—41no—
110C2×D8×C61C2311yescharacteristic
111C2×D8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114#124×C61—21no—
115#123×C61—41no—
116C22⋊Q8×C61—21no—
117#121×C61—21no—
118#121×C61—41no—
119#120×C61—41no—
120C4×D4×C61—21no—
121C4×D4×C61D411yescharacteristic
122C2×C22⋊C4×C61—41no—
123#107×C61C2311yescharacteristic
124C4⋊C8×C61D411yescharacteristic
125#103×C61—21no—
126#102×C61—21no—
127#102×C61—21no—
128C22⋊C8×C61D411yescharacteristic
129C22⋊C8×C61D411yescharacteristic
130C4×C8×C61C2311yescharacteristic
131C24×C61—41no—
132#41×C61—42no—
133#41×C61—81no—
134C2×Q8×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—42no—
138C2×D4×C61—41no—
139C2×D4×C61—41no—
140C2×D4×C61—41no—
141C2×D4×C61—81no—
142C2×D4×C61D811yescharacteristic
143C22×C4×C61—41no—
144C22×C4×C61—41no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C2×C22⋊D61—41no—
148C2×C22⋊D61—41no—
149C2×C22⋊D61—41no—
150C2×C22⋊D61—41no—
151#11758222—42no—
152#11758222—42no—
153#11758222—81no—
154#11758222—81no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—41no—
158C2×D244—41no—
159C2×C4×D61—41no—
160C2×C4×D61—41no—
161C2×Dic122—21no—
162Q16×C61—21no—
163Q16×C61—21no—
164SD16×C61—42no—
165SD16×C61—81no—
166D8×C61—21no—
167D8×C61—21no—
168D8×C61—21no—
169D8×C61—42no—
170D8×C61C2×D412yes—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
177C2×C8×C61C2×D411yescharacteristic
178C2×C8×C61C2411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61D811yescharacteristic
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61—41no—
186C22⋊C4×C61—41no—
187C22⋊C4×C61D811yescharacteristic
188C22⋊C4×C61D811yescharacteristic
189C42×C61C2×D411yescharacteristic
190#11758198—41no—
191#11758198—41no—
192C4⋊Dic61—21no—
193#11758196—41no—
194C4×Dic61—41no—
195Dic244—21no—
196Dic244C2×D412yes—
197D488—21no—
198D488—21no—
199D488—21no—
200D488—21no—
201D488—81no—
202#11758190—42no—
203#11758190—42no—
204#11758185—21no—
205#11758185—21no—
206#11758185—21no—
207#11758185C2×D412yes—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—82no—
211C23×C61—82no—
212C23×C61—161no—
213C23×C61C2×D811yescharacteristic
214C23×C61C2×D811yescharacteristic
215C22×D61—41no—
216C22×D61—41no—
217Q8×C61—41no—
218Q8×C61—41no—
219D4×C61—22no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—41no—
223D4×C61—41no—
224D4×C61—82no—
225D4×C61—82no—
226D4×C61—161no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—82no—
234C2×C4×C61—161no—
235C2×C4×C61C2×D811yescharacteristic
236C2×C4×C61C2×D811yescharacteristic
237C2×C4×C61C2×D811yescharacteristic
238C2×C4×C61C2×D811yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C22⋊D61—82no—
241C22⋊D61—82no—
242C22⋊D61—161no—
243C22⋊D61—161no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246D244—41no—
247D244—41no—
248D244—41no—
249D244—41no—
250D244—82no—
251D244—161no—
252C4×D61—82no—
253C4×D61—82no—
254Dic122—22no—
255Dic122—41no—
256C8×C61—21no—
257C8×C61—21no—
258C8×C61—21no—
259C8×C61C22×D411yescharacteristic
260C8×C61H(5,𝔽2)11yescharacteristic
261#33834—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—42no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—164no—
275C22×C61C22×D811yescharacteristic
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C4×C61—21no—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—81no—
287C4×C61—81no—
288C4×C61D4211yescharacteristic
289C4×C61#57611yescharacteristic
290Dic61—42no—
291Dic61—81no—
292C2×#509—611no—
293C2×#508—611no—
294#3287—611no—
295#3280D6111yescharacteristic, complemented
296#2250—611no—
297#2220—611no—
298#2215—611no—
299#2211—611no—
300#2206—611no—
301#2189—611no—
302#2186—611no—
303#2186—611no—
304#2184—611no—
305#2175—611no—
306#2168—611no—
307C2×C61—41no—
308C2×C61—41no—
309C2×C61—82no—
310C2×C61—161no—
311C2×C61D4×D811yescharacteristic
312C2×C61#341411yescharacteristic
313C2×C61#341511yescharacteristic
314D61—161no—
315D61—161no—
316C2×#135—611no—
317C2×#135—611no—
318C2×#135C2×D6111yescharacteristic
319C22×D8C2×D6111yescharacteristic
320#550—1221no—
321#546—21no—
322#509—1224no—
323#508—1224no—
324C2×Q32—611no—
325C2×SD32—611no—
326C2×SD32—611no—
327C2×SD32—611no—
328C2×SD32—611no—
329C2×D16—611no—
330C2×D16—611no—
331C2×D16—611no—
332C2×D16—611no—
333C2×M32—611no—
334C22×C16—611no—
335#495C2×D6111yescharacteristic
336#493—611no—
337#467C2×D6111yescharacteristic
338#466—611no—
339#466—611no—
340#466C2×D6111yescharacteristic
341#463—21no—
342#460—1221no—
343#450—21no—
344#449—21no—
345#449—1221no—
346#449—1221no—
347C4×Q16—611no—
348C4×D8C2×D6111yescharacteristic
349C8×D4C2×D6111yescharacteristic
350#366—611no—
351C4⋊C16—611no—
352#358—611no—
353#358—611no—
354#357—611no—
355#357—611no—
356#357—611no—
357#357—611no—
358C22⋊C16—611no—
359C22⋊C16—611no—
360#346—611no—
361C61#3383411yescharacteristic, complemented, Sylow
362C2×#41—21no—
363C2×#41—1221no—
364C2×#41—1221no—
365C22×D4—21no—
366#135—22no—
367#135—41no—
368#135—1222no—
369#135—1222no—
370#135—2441no—
371#135—2441no—
372C2×Q16—611no—
373C2×Q16C22×D6111yescharacteristic
374C2×SD16—21no—
375C2×SD16—21no—
376C2×SD16—1221no—
377C2×SD16—1221no—
378C2×D8—22no—
379C2×D8—21no—
380C2×D8—41no—
381C2×D8—611no—
382C2×D8—611no—
383C2×D8—1221no—
384C2×D8C22×D6111yescharacteristic
385C2×D8C22×D6111yescharacteristic
386C22×C8C22×D6111yescharacteristic
387C22×C8C22×D6111yescharacteristic
388#127—1221no—
389#127—2441no—
390#124—21no—
391#123—41no—
392C22⋊Q8—21no—
393#121—21no—
394#121—41no—
395#121—1221no—
396#121—1221no—
397#121—2441no—
398#121—2441no—
399#120—41no—
400C4×Q8—1221no—
401C4×D4—21no—
402C4×D4—2441no—
403C4×D4C22⋊D6111yescharacteristic
404C2×C22⋊C4—41no—
405Q32—1222no—
406SD32—614no—
407SD32—1222no—
408SD32—1222no—
409SD32—2441no—
410D16—1222no—
411D16—1222no—
412D16—1222no—
413D16—2441no—
414M32—1222no—
415C2×C16—611no—
416C2×C16—611no—
417C2×C16—611no—
418C2×C16—611no—
419C2×C16—1222no—
420#107—611no—
421#107C22×D6111yescharacteristic
422C4⋊C8C22⋊D6111yescharacteristic
423#103—21no—
424#103—1221no—
425#102—21no—
426#102—21no—
427#102—1221no—
428#102—1221no—
429C22⋊C8C22⋊D6111yescharacteristic
430C22⋊C8C22⋊D6111yescharacteristic
431C4×C8C22×D6111yescharacteristic
432C24—41no—
433#41—42no—
434#41—81no—
435#41—4881no—
436#41—4881no—
437#41—2442no—
438#41—2442no—
439C2×Q8—21no—
440C2×Q8—1221no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—42no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—41no—
447C2×D4—81no—
448C2×D4—1221no—
449C2×D4—1221no—
450C2×D4—2441no—
451C2×D4—2441no—
452C2×D4—2441no—
453C2×D4—2441no—
454C2×D4—2441no—
455C2×D4—2441no—
456C2×D4#1175819911yescharacteristic
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—2441no—
460C22×C4—2441no—
461C22×C4C2×C22⋊D6111yescharacteristic
462C22×C4C2×C22⋊D6111yescharacteristic
463Q16—21no—
464Q16—21no—
465Q16—612no—
466Q16—1221no—
467SD16—42no—
468SD16—81no—
469SD16—2442no—
470SD16—2442no—
471D8—21no—
472D8—21no—
473D8—21no—
474D8—42no—
475D8—1221no—
476D8—1221no—
477D8—1221no—
478D8—1221no—
479D8—4881no—
480D8D4×D6112yes—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8C2×C22⋊D6111yescharacteristic
486C2×C8C2×C22⋊D6111yescharacteristic
487C2×C8C2×C22⋊D6111yescharacteristic
488C2×C8C23×D6111yescharacteristic
489C4⋊C4—21no—
490C4⋊C4—41no—
491C4⋊C4—1221no—
492C4⋊C4—2441no—
493C4⋊C4#1175819911yescharacteristic
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—2441no—
500C22⋊C4—2441no—
501C22⋊C4#1175819911yescharacteristic
502C22⋊C4#1175819911yescharacteristic
503C42—2441no—
504C42C2×C22⋊D6111yescharacteristic
505C16—612no—
506C16—1221no—
507C16—1221no—
508C16—1221no—
509C23—41no—
510C23—41no—
511C23—82no—
512C23—82no—
513C23—161no—
514C23—2441no—
515C23—2441no—
516C23C2×#1175819911yescharacteristic
517C23C2×#1175819911yescharacteristic
518Q8—41no—
519Q8—41no—
520Q8—1222no—
521Q8—2441no—
522D4—22no—
523D4—41no—
524D4—41no—
525D4—41no—
526D4—41no—
527D4—82no—
528D4—82no—
529D4—161no—
530D4—9761no—
531D4—9761no—
532D4—9761no—
533D4—4882no—
534D4—4882no—
535D4—4882no—
536D4—2441no—
537D4—2441no—
538D4—2441no—
539D4—2441no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—82no—
547C2×C4—161no—
548C2×C4—4882no—
549C2×C4—4882no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4C2×#1175819911yescharacteristic
553C2×C4C2×#1175819911yescharacteristic
554C2×C4C2×#1175819911yescharacteristic
555C2×C4C2×#1175819911yescharacteristic
556C2×C4C22×C22⋊D6111yescharacteristic
557C8—21no—
558C8—21no—
559C8—21no—
560C8C2×D4×D6111yescharacteristic
561C8#1451982211yescharacteristic
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—164no—
575C22—4881no—
576C22—4881no—
577C22—4881no—
578C22—4881no—
579C22—4881no—
580C22—4881no—
581C22C22×#1175819911yescharacteristic
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—81no—
587C4—81no—
588C4—4881no—
589C4—2442no—
590C4D4×C22⋊D6111yescharacteristic
591C4#1684056511yescharacteristic
592C2—41no—
593C2—41no—
594C2—82no—
595C2—161no—
596C2—9761no—
597C2—9761no—
598C2D4×#1175819911yescharacteristic
599C2#2145895311yescharacteristic
600C2#2145895811yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3280C2
socleC22×C61C22×D8
hypercenter#3280D61
hypocenterC61#33834
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
1. C2×#16840198C2semidirect
2. C2×#16840192C2semidirect
3. #21457579C2semidirect
4. #21448086C2semidirect
5. #21447934C2semidirect
6. #21447903C2semidirect
7. #21447878C2semidirect
8. #21447854C2semidirect
9. #21447757C2semidirect
10. #21447742C2semidirect
11. #21447740C2semidirect
12. #21447731C2semidirect
13. #21447683C2semidirect
14. #21447645C2semidirect
15. C61×#3280C2semidirect
17. C2×#14519799C22semidirect
21. C2×#14519723C22semidirect
24. C2×#14519720C22semidirect
25. C2×#14519720C22semidirect
31. C22×#11758185C22semidirect
32. #16840139C22semidirect
34. #16839998C22semidirect
38. C4×Dic244C22semidirect
39. #16839587C22semidirect
40. C4⋊#11758185C22semidirect
41. #16839566C22semidirect
42. #16839565C22semidirect
45. #16839562C22semidirect
46. #16839562C22semidirect
48. C22⋊#11758185C22semidirect
49. #16839538C22semidirect
83. #14519721D4semidirect
295. #3280D61semidirect
361. C61#33834semidirect

Maximal subgroups

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

subgroupindex
C2×#168401982
C2×#168401922
#214575792
#214480862
#214479342
#214479032
#214478782
#214478542
#214477572
#214477422
#214477402
#214477312
#214476832
#214476452
C61×#32802
#3383461