Finite Group Databasebeta

← #29836672

Subgroups — #29836672

Order 15616.

718 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840192C211yesmaximal, characteristic, complemented
2C22×#14519718C211yesmaximal, characteristic, complemented
3#21457506C211yesmaximal, characteristic, complemented
4#21448078C211yesmaximal, characteristic, complemented
5#21447908C211yesmaximal, characteristic, complemented
6#21447903C211yesmaximal, characteristic, complemented
7#21447874C211yesmaximal, characteristic, complemented
8#21447871C211yesmaximal, characteristic, complemented
9#21447748C211yesmaximal, characteristic, complemented
10#21447747C211yesmaximal, characteristic, complemented
11#21447742C211yesmaximal, characteristic, complemented
12#21447740C211yesmaximal, characteristic, complemented
13C4×#14519718C211yesmaximal, characteristic, complemented
14D4×#11758185C211yesmaximal, characteristic, complemented
15C61×#3280C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic, complemented
17C22×D488C2211yescharacteristic
18#16840408—21no—
19#16840192—24no—
20C2×#14519723C2211yescharacteristic, complemented
21C2×#14519721C2211yescharacteristic, complemented
22C2×#14519721C2211yescharacteristic, complemented
23C2×#14519720C2211yescharacteristic, complemented
24C2×#14519720C2211yescharacteristic, complemented
25C2×#14519718—24no—
26C2×#14519718C2211yescharacteristic
27C2×#14519718C2211yescharacteristic
28C2×#14519718C2211yescharacteristic
29C2×#14519718C2211yescharacteristic
30C22×#11758185C2211yescharacteristic
31C22×#11758185C2211yescharacteristic, complemented
32#16840146C2211yescharacteristic, complemented
33#16840003C2211yescharacteristic
34#16839998C2211yescharacteristic, complemented
35#16839978—21no—
36#16839915—21no—
37#16839909—21no—
38C4×D488C2211yescharacteristic, complemented
39#16839587C2211yescharacteristic, complemented
40C4⋊#11758185C2211yescharacteristic, complemented
41#16839566C2211yescharacteristic, complemented
42#16839565C2211yescharacteristic, complemented
43#16839563C2211yescharacteristic, complemented
44#16839563C2211yescharacteristic, complemented
45#16839562C2211yescharacteristic, complemented
46#16839562C2211yescharacteristic, complemented
47C22⋊#11758185C2211yescharacteristic
48C22⋊#11758185C2211yescharacteristic, complemented
49C4×#11758185C2211yescharacteristic, complemented
50C2×#135×C61C2211yescharacteristic, complemented
51C22×D8×C61C2211yescharacteristic
52C61×#546—21no—
53C61×#495C2211yescharacteristic, complemented
54C61×#467C2211yescharacteristic
55C61×#466C2211yescharacteristic, complemented
56C61×#463—21no—
57C61×#450—21no—
58C61×#449—21no—
59C4×D8×C61C2211yescharacteristic, complemented
60C8×D4×C61C2211yescharacteristic
61C2×#11758222—21no—
62C22×D244—21no—
63#14519799—22no—
64#14519799—41no—
65C2×Dic244C2311yescharacteristic
66C2×#11758190—21no—
67C2×#11758190—21no—
68C2×D488—22no—
69C2×D488—21no—
70C2×D488—41no—
71C2×D488C2311yescharacteristic
72C2×D488C2311yescharacteristic
73#14519765—21no—
74#14519762—41no—
75#14519756—21no—
76#14519751—21no—
77#14519750—41no—
78#14519745—41no—
79C4×D244—21no—
80C22⋊C4×D61—41no—
81#14519723—22no—
82#14519721—22no—
83#14519721—41no—
84#14519720—22no—
85#14519720—41no—
86#14519718—22no—
87#14519718—22no—
88#14519718—22no—
89#14519718D414yescomplemented
90C2×#11758185—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—
132C23×D61—41no—
133#41×C61—42no—
134#41×C61—81no—
135C2×Q8×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—42no—
139C2×D4×C61—41no—
140C2×D4×C61—41no—
141C2×D4×C61—41no—
142C2×D4×C61—81no—
143C2×D4×C61D811yescharacteristic
144C22×C4×C61—41no—
145C22×C4×C61—41no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C2×C22⋊D61—41no—
149C2×C22⋊D61—41no—
150#11758222—42no—
151#11758222—81no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—42no—
155C2×D244—41no—
156C2×D244—81no—
157C2×C4×D61—41no—
158C2×C4×D61—41no—
159C2×Dic122—21no—
160Q16×C61—21no—
161Q16×C61—21no—
162SD16×C61—42no—
163SD16×C61—81no—
164D8×C61—21no—
165D8×C61—21no—
166D8×C61—21no—
167D8×C61—42no—
168D8×C61C2×D412yes—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61C2×D411yescharacteristic
174C2×C8×C61C2×D411yescharacteristic
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2411yescharacteristic
177C4⋊C4×C61—21no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61D811yescharacteristic
180C22⋊C4×C61—41no—
181C22⋊C4×C61—41no—
182C22⋊C4×C61—41no—
183C22⋊C4×C61—41no—
184C22⋊C4×C61—41no—
185C22⋊C4×C61D811yescharacteristic
186C22⋊C4×C61D811yescharacteristic
187C42×C61C2×D411yescharacteristic
188C22⋊Dic61—41no—
189#11758198—41no—
190#11758198—41no—
191#11758198—41no—
192#11758198—41no—
193C4⋊Dic61—21no—
194#11758196—41no—
195Dic244—21no—
196Dic244—21no—
197D488—21no—
198D488—21no—
199D488—21no—
200D488—42no—
201D488C2×D412yes—
202#11758190—42no—
203#11758190—81no—
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—
217C22×D61—82no—
218C22×D61—82no—
219C22×D61—161no—
220Q8×C61—41no—
221Q8×C61—41no—
222D4×C61—22no—
223D4×C61—22no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—82no—
228D4×C61—82no—
229D4×C61—161no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—82no—
237C2×C4×C61—161no—
238C2×C4×C61C2×D811yescharacteristic
239C2×C4×C61C2×D811yescharacteristic
240C2×C4×C61C2×D811yescharacteristic
241C2×C4×C61C2×D811yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C22⋊D61—82no—
244C22⋊D61—161no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247D244—22no—
248D244—41no—
249D244—41no—
250D244—41no—
251D244—82no—
252C4×D61—82no—
253C4×D61—161no—
254Dic122—41no—
255Dic122—41no—
256C8×C61—21no—
257C8×C61—21no—
258C8×C61—21no—
259C8×C61C22×D411yescharacteristic
260C8×C61H(5,𝔽2)11yescharacteristic
261#33846—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—42no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—164no—
274C22×C61#50812yescomplemented
275C22×C61C22×D811yescharacteristic
276C2×D61—42no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—164no—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61D4211yescharacteristic
290C4×C61#57611yescharacteristic
291Dic61—81no—
292Dic61—81no—
293C2×#508—611no—
294C22×D16—611no—
295#3280—611no—
296#3280D6111yescharacteristic, complemented
297#2249—611no—
298#2216—611no—
299#2215—611no—
300#2210—611no—
301#2210—611no—
302#2187—611no—
303#2187—611no—
304#2186—611no—
305#2186—611no—
306C4×D16—611no—
307D4×C16—611no—
308C2×C61—22no—
309C2×C61—41no—
310C2×C61—82no—
311C2×C61—161no—
312C2×C61D4×D811yescharacteristic
313C2×C61C2×#50811yescharacteristic
314C2×C61#341511yescharacteristic
315D61—82no—
316D61—161no—
317C2×#135—611no—
318C2×#135C2×D6111yescharacteristic, complemented
319C22×D8—611no—
320C22×D8C2×D6111yescharacteristic
321#546—21no—
322#546—1221no—
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×D16—1224no—
334C22×C16—611no—
335C22×C16—611no—
336#495—611no—
337#495C2×D6111yescharacteristic, complemented
338#467—611no—
339#467C2×D6111yescharacteristic
340#466—611no—
341#466C2×D6111yescharacteristic, complemented
342#463—21no—
343#463—1221no—
344#450—21no—
345#450—1221no—
346#449—21no—
347#449—1221no—
348C4×D8—611no—
349C4×D8C2×D6111yescharacteristic, complemented
350C8×D4C2×D6111yescharacteristic
351#366—611no—
352C4⋊C16—611no—
353#358—611no—
354#358—611no—
355#357—611no—
356#357—611no—
357#357—611no—
358#357—611no—
359C22⋊C16—611no—
360C22⋊C16—611no—
361C4×C16—611no—
362C61#3384611yescharacteristic, complemented, Sylow
363C2×#41—21no—
364C2×#41—1221no—
365C22×D4—21no—
366C22×D4—1221no—
367#135—22no—
368#135—41no—
369#135—1222no—
370#135—2441no—
371C2×Q16—611no—
372C2×Q16C22×D6111yescharacteristic
373C2×SD16—21no—
374C2×SD16—21no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×D8—22no—
378C2×D8—21no—
379C2×D8—41no—
380C2×D8—611no—
381C2×D8—611no—
382C2×D8—1222no—
383C2×D8—1221no—
384C2×D8—2441no—
385C2×D8C22×D6111yescharacteristic
386C2×D8C22×D6111yescharacteristic
387C22×C8C22×D6111yescharacteristic
388C22×C8C22×D6111yescharacteristic
389#124—21no—
390#124—1221no—
391#123—41no—
392#123—2441no—
393C22⋊Q8—21no—
394C22⋊Q8—1221no—
395#121—21no—
396#121—41no—
397#121—1221no—
398#121—2441no—
399#120—41no—
400#120—2441no—
401C4×D4—21no—
402C4×D4—1221no—
403C4×D4C22⋊D6111yescharacteristic
404C2×C22⋊C4—41no—
405C2×C22⋊C4—2441no—
406Q32—1222no—
407SD32—1222no—
408SD32—1222no—
409SD32—2441no—
410SD32—2441no—
411D16—614no—
412D16—1222no—
413D16—1222no—
414D16—1222no—
415C2×C16—611no—
416C2×C16—611no—
417C2×C16—611no—
418C2×C16—611no—
419C2×C16—1222no—
420C2×C16—1222no—
421#107—611no—
422#107C22×D6111yescharacteristic
423C4⋊C8C22⋊D6111yescharacteristic
424#103—21no—
425#103—1221no—
426#102—21no—
427#102—21no—
428#102—1221no—
429#102—1221no—
430C22⋊C8C22⋊D6111yescharacteristic
431C22⋊C8C22⋊D6111yescharacteristic
432C4×C8C22×D6111yescharacteristic
433C24—41no—
434C24—2441no—
435#41—42no—
436#41—81no—
437#41—4881no—
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—4881no—
451C2×D4—2442no—
452C2×D4—2441no—
453C2×D4—2441no—
454C2×D4—2441no—
455C2×D4#1175819911yescharacteristic
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—2441no—
459C22×C4—2441no—
460C22×C4C2×C22⋊D6111yescharacteristic
461C22×C4C2×C22⋊D6111yescharacteristic
462Q16—21no—
463Q16—21no—
464Q16—1221no—
465Q16—1221no—
466SD16—42no—
467SD16—81no—
468SD16—4881no—
469SD16—2442no—
470D8—21no—
471D8—21no—
472D8—21no—
473D8—42no—
474D8—612no—
475D8—1221no—
476D8—1221no—
477D8—1221no—
478D8—2442no—
479D8D4×D6112yes—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8C2×C22⋊D6111yescharacteristic
485C2×C8C2×C22⋊D6111yescharacteristic
486C2×C8C2×C22⋊D6111yescharacteristic
487C2×C8C23×D6111yescharacteristic
488C4⋊C4—21no—
489C4⋊C4—41no—
490C4⋊C4—1221no—
491C4⋊C4—2441no—
492C4⋊C4#1175819911yescharacteristic
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—2441no—
499C22⋊C4—2441no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C22⋊C4—2441no—
503C22⋊C4#1175819911yescharacteristic
504C22⋊C4#1175819911yescharacteristic
505C42C2×C22⋊D6111yescharacteristic
506C16—612no—
507C16—1221no—
508C16—1221no—
509C16—1221no—
510C23—41no—
511C23—41no—
512C23—82no—
513C23—82no—
514C23—161no—
515C23—9761no—
516C23—4882no—
517C23—4882no—
518C23—2441no—
519C23—2441no—
520C23C2×#1175819911yescharacteristic
521C23C2×#1175819911yescharacteristic
522Q8—41no—
523Q8—41no—
524Q8—2441no—
525Q8—2441no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—82no—
532D4—82no—
533D4—161no—
534D4—1222no—
535D4—9761no—
536D4—4882no—
537D4—4882no—
538D4—2441no—
539D4—2441no—
540D4—2441no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—82no—
548C2×C4—161no—
549C2×C4—9761no—
550C2×C4—4882no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4C2×#1175819911yescharacteristic
554C2×C4C2×#1175819911yescharacteristic
555C2×C4C2×#1175819911yescharacteristic
556C2×C4C2×#1175819911yescharacteristic
557C2×C4C22×C22⋊D6111yescharacteristic
558C8—21no—
559C8—21no—
560C8—21no—
561C8C2×D4×D6111yescharacteristic
562C8#1451982211yescharacteristic
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—9764no—
576C22—4881no—
577C22—4881no—
578C22—4881no—
579C22—4881no—
580C22—4881no—
581C22—2442no—
582C22#1684019212yescomplemented
583C22C22×#1175819911yescharacteristic
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—81no—
589C4—81no—
590C4—4881no—
591C4—4881no—
592C4D4×C22⋊D6111yescharacteristic
593C4#1684056511yescharacteristic
594C2—22no—
595C2—41no—
596C2—82no—
597C2—161no—
598C2—9761no—
599C2—4882no—
600C2D4×#1175819911yescharacteristic
601C2C2×#1684019211yescharacteristic
602C2#2145895811yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3280C2
socleC22×C61C22×D8
hypercenter#3280D61
hypocenterC61#33846
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×#16840192C2semidirect
2. C22×#14519718C2semidirect
3. #21457506C2semidirect
4. #21448078C2semidirect
5. #21447908C2semidirect
6. #21447903C2semidirect
7. #21447874C2semidirect
8. #21447871C2semidirect
9. #21447748C2semidirect
10. #21447747C2semidirect
11. #21447742C2semidirect
12. #21447740C2semidirect
13. C4×#14519718C2semidirect
14. D4×#11758185C2semidirect
15. C61×#3280C2semidirect
16. C2×#14519799C22semidirect
20. C2×#14519723C22semidirect
21. C2×#14519721C22semidirect
22. C2×#14519721C22semidirect
23. C2×#14519720C22semidirect
24. C2×#14519720C22semidirect
31. C22×#11758185C22semidirect
32. #16840146C22semidirect
34. #16839998C22semidirect
38. C4×D488C22semidirect
39. #16839587C22semidirect
40. C4⋊#11758185C22semidirect
41. #16839566C22semidirect
42. #16839565C22semidirect
43. #16839563C22semidirect
44. #16839563C22semidirect
45. #16839562C22semidirect
46. #16839562C22semidirect
48. C22⋊#11758185C22semidirect
49. C4×#11758185C22semidirect
50. C2×#135×C61C22semidirect
53. C61×#495C22semidirect
55. C61×#466C22semidirect
59. C4×D8×C61C22semidirect
89. #14519718D4semidirect
274. C22×C61#508semidirect
296. #3280D61semidirect
318. C2×#135C2×D61semidirect
337. #495C2×D61semidirect
341. #466C2×D61semidirect
349. C4×D8C2×D61semidirect
362. C61#33846semidirect
582. C22#16840192semidirect

Maximal subgroups

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

subgroupindex
C2×#168401922
C22×#145197182
#214575062
#214480782
#214479082
#214479032
#214478742
#214478712
#214477482
#214477472
#214477422
#214477402
C4×#145197182
D4×#117581852
C61×#32802
#3384661