Finite Group Databasebeta

← #29836674

Subgroups — #29836674

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
2C2×D16×D61C211yesmaximal, characteristic, complemented
3#21457507C211yesmaximal, characteristic, complemented
4#21457498C211yesmaximal, characteristic, complemented
5#21448079C211yesmaximal, characteristic, complemented
6#21447909C211yesmaximal, characteristic, complemented
7#21447903C211yesmaximal, characteristic, complemented
8#21447877C211yesmaximal, characteristic, complemented
9#21447872C211yesmaximal, characteristic, complemented
10#21447749C211yesmaximal, characteristic, complemented
11#21447746C211yesmaximal, characteristic, complemented
12#21447744C211yesmaximal, characteristic, complemented
13#21447663C211yesmaximal, characteristic, complemented
14#21447636C211yesmaximal, characteristic, complemented
15C61×#2186C211yesmaximal, characteristic, complemented
16C2×#14519799C2211yescharacteristic, complemented
17C2×D8×D61C2211yescharacteristic
18C2×D8×D61C2211yescharacteristic
19#16840409—21no—
20#16840400—21no—
21#16840192—24no—
22C2×#14519723C2211yescharacteristic, complemented
23C2×#14519722C2211yescharacteristic, complemented
24C2×#14519721C2211yescharacteristic, complemented
25C2×#14519720C2211yescharacteristic, complemented
26C2×D976C2211yescharacteristic
27C2×#14519718C2211yescharacteristic
28C2×#14519718C2211yescharacteristic
29D16×D61—24no—
30C22×#11758185C2211yescharacteristic, complemented
31C2×C16×D61C2211yescharacteristic
32#16840147C2211yescharacteristic, complemented
33#16840145C2211yescharacteristic, complemented
34#16840004C2211yescharacteristic
35#16840000C2211yescharacteristic
36#16839998C2211yescharacteristic, complemented
37#16839979—21no—
38#16839976—21no—
39#16839914—21no—
40#16839911—21no—
41#16839907—21no—
42#16839905—21no—
43D8×Dic61C2211yescharacteristic, complemented
44#16839844C2211yescharacteristic, complemented
45C8×C22⋊D61C2211yescharacteristic
46#16839587C2211yescharacteristic, complemented
47#16839580C2211yescharacteristic, complemented
48#16839567C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic, complemented
50#16839563C2211yescharacteristic, complemented
51#16839562C2211yescharacteristic, complemented
52#16839541C2211yescharacteristic
53C16×Dic61C2211yescharacteristic, complemented
54C2×#135×C61C2211yescharacteristic, complemented
55C2×SD32×C61C2211yescharacteristic, complemented
56C2×D16×C61C2211yescharacteristic
57C61×#466C2211yescharacteristic, complemented
58C61×#358C2211yescharacteristic, complemented
59C61×#357C2211yescharacteristic, complemented
60C61×C22⋊C16C2211yescharacteristic, complemented
61C2×#11758222—21no—
62C2×D4×D61—21no—
63C2×D4×D61—21no—
64#14519799—22no—
65#14519799—41no—
66C2×Dic244C2311yescharacteristic
67C2×#11758190—21no—
68C2×#11758200—21no—
69C2×#11758201—21no—
70C2×D488C2311yescharacteristic
71C2×D488C2311yescharacteristic
72C2×#11758199—21no—
73C2×#11758199—21no—
74D8×D61—22no—
75D8×D61—22no—
76D8×D61—41no—
77D8×D61—41no—
78C2×C8×D61C2311yescharacteristic
79#14519766—21no—
80#14519764—21no—
81#14519760—41no—
82#14519759—41no—
83#14519757—21no—
84#14519753—21no—
85#14519751—21no—
86#14519749—41no—
87#14519748—41no—
88#14519745—41no—
89#14519745—41no—
90D4×Dic61—21no—
91#14519739—21no—
92C4×C22⋊D61D411yescharacteristic
93C2×#11758198—41no—
94C22⋊C4×D61—41no—
95#14519723—22no—
96#14519722—41no—
97#14519721—22no—
98#14519720—22no—
99D976—22no—
100#14519718—22no—
101#14519718D414yescomplemented
102C2×#11758185—22no—
103C2×#11758185C2311yescharacteristic
104C2×#11758185C2311yescharacteristic
105C16×D61—22no—
106#14519711C2311yescharacteristic
107#14519706D411yescharacteristic
108#14519701—21no—
109#14519700—21no—
110#14519699—21no—
111#14519698—21no—
112#14519697—21no—
113C22⋊#92418D411yescharacteristic
114#14519689D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C2×#41×C61—21no—
117#135×C61—22no—
118#135×C61—41no—
119C2×Q16×C61C2311yescharacteristic
120C2×SD16×C61—21no—
121C2×D8×C61C2311yescharacteristic
122C2×D8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124#121×C61—21no—
125SD32×C61—41no—
126D16×C61—22no—
127C2×C16×C61C2311yescharacteristic
128C2×C16×C61C2311yescharacteristic
129#107×C61C2311yescharacteristic
130#102×C61—21no—
131C23×D61—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—41no—
139C2×D4×C61—41no—
140C22×C4×C61—41no—
141C22×C4×C61C2×D411yescharacteristic
142C2×C22⋊D61—41no—
143C2×C22⋊D61—41no—
144C2×C22⋊D61—41no—
145C2×C22⋊D61—41no—
146C2×C22⋊D61D811yescharacteristic
147C22×Dic61—41no—
148D4×D61—42no—
149D4×D61—42no—
150D4×D61—81no—
151D4×D61—81no—
152#11758222—42no—
153#11758222—81no—
154C2×D244—21no—
155C2×D244—21no—
156C2×C4×D61—41no—
157C2×C4×D61—41no—
158C2×C4×D61C2×D411yescharacteristic
159C2×Dic122—21no—
160Q16×C61—21no—
161Q16×C61—21no—
162SD16×C61—42no—
163D8×C61—21no—
164D8×C61—21no—
165D8×C61—21no—
166D8×C61C2×D412yes—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2411yescharacteristic
171C4⋊C4×C61—21no—
172C22⋊C4×C61—41no—
173C22⋊Dic61—41no—
174C22⋊Dic61—41no—
175C22⋊Dic61—41no—
176C22⋊Dic61D811yescharacteristic
177#11758201—81no—
178#11758200—81no—
179#11758199—42no—
180#11758199—42no—
181#11758198—41no—
182#11758198—41no—
183#11758198—41no—
184#11758198—41no—
185#11758198—41no—
186#11758198—41no—
187#11758198D811yescharacteristic
188C4⋊Dic61—21no—
189C4⋊Dic61—41no—
190#11758196—41no—
191#11758196D811yescharacteristic
192C4×Dic61C2×D411yescharacteristic
193C2×#92418C2×D411yescharacteristic
194C2×#92418C2×D411yescharacteristic
195Dic244—21no—
196Dic244—21no—
197D488—21no—
198D488—21no—
199D488—21no—
200D488C2×D412yes—
201#11758190—42no—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61—21no—
206#11758185—21no—
207#11758185C2×D412yes—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61C2×D811yescharacteristic
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—82no—
214C22×D61—82no—
215C22×D61—82no—
216C22×D61—82no—
217C22×D61—161no—
218C22×D61—161no—
219C22×D61C2×D811yescharacteristic
220Q8×C61—41no—
221Q8×C61—41no—
222D4×C61—22no—
223D4×C61—41no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—82no—
227D4×C61—161no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—82no—
233C2×C4×C61C2×D811yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C22⋊D61—22no—
236C22⋊D61—82no—
237C22⋊D61—82no—
238C22⋊D61—82no—
239C22⋊D61—161no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—161no—
243C2×Dic61C2×D811yescharacteristic
244C2×Dic61C2×D811yescharacteristic
245C2×Dic61C2×D811yescharacteristic
246D244—22no—
247D244—41no—
248D244—41no—
249D244—41no—
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—82no—
253C4×D61—161no—
254Dic122—41no—
255Dic122—41no—
256C8×C61—21no—
257C8×C61C22×D411yescharacteristic
258C8×C61H(5,𝔽2)11yescharacteristic
259#92418—21no—
260#92418—21no—
261#33846—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—42no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61C22×D811yescharacteristic
272C2×D61—21no—
273C2×D61—21no—
274C2×D61—42no—
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C2×D61—81no—
280C2×D61—164no—
281C2×D61—164no—
282C2×D61#50812yescomplemented
283C4×C61—21no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61D4211yescharacteristic
287C4×C61#57611yescharacteristic
288Dic61—21no—
289Dic61—21no—
290Dic61—21no—
291Dic61—81no—
292Dic61—81no—
293C2×#508—611no—
294C22×D16—611no—
295#3280—611no—
296#3280—611no—
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#2186D6111yescharacteristic, complemented
306C4×D16—611no—
307D4×C16—611no—
308C2×C61—41no—
309C2×C61—82no—
310C2×C61—161no—
311C2×C61D4×D811yescharacteristic
312C2×C61C2×#50811yescharacteristic
313C2×C61#341511yescharacteristic
314D61—22no—
315D61—82no—
316D61—161no—
317C2×#135—611no—
318C2×#135C2×D6111yescharacteristic, complemented
319C22×D8—611no—
320C22×D8—611no—
321#546—1221no—
322#546—1221no—
323#508—1224no—
324C2×Q32—611no—
325C2×SD32—611no—
326C2×SD32—611no—
327C2×SD32—611no—
328C2×SD32C2×D6111yescharacteristic, complemented
329C2×D16—611no—
330C2×D16—611no—
331C2×D16—611no—
332C2×D16—1224no—
333C2×D16C2×D6111yescharacteristic
334C22×C16—611no—
335C22×C16—611no—
336#495—611no—
337#495—611no—
338#467—611no—
339#467—611no—
340#466—611no—
341#466C2×D6111yescharacteristic, complemented
342#463—1221no—
343#463—1221no—
344#450—1221no—
345#450—1221no—
346#449—1221no—
347#449—1221no—
348C4×D8—611no—
349C4×D8—611no—
350C8×D4—611no—
351#366—611no—
352C4⋊C16—611no—
353#358—611no—
354#358C2×D6111yescharacteristic, complemented
355#357—611no—
356#357—611no—
357#357—611no—
358#357C2×D6111yescharacteristic, complemented
359C22⋊C16—611no—
360C22⋊C16C2×D6111yescharacteristic, complemented
361C4×C16—611no—
362C61#3384611yescharacteristic, complemented, Sylow
363C2×#41—21no—
364C2×#41—1221no—
365C22×D4—1221no—
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—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×D8—611no—
378C2×D8—611no—
379C2×D8—1221no—
380C2×D8—1221no—
381C2×D8—1222no—
382C2×D8—1222no—
383C2×D8—2441no—
384C2×D8—2441no—
385C2×D8C22×D6111yescharacteristic
386C2×D8C22×D6111yescharacteristic
387C22×C8—611no—
388C22×C8C22×D6111yescharacteristic
389#124—1221no—
390#124—1221no—
391#123—2441no—
392#123—2441no—
393C22⋊Q8—1221no—
394C22⋊Q8—1221no—
395#121—21no—
396#121—1221no—
397#121—2441no—
398#121—2441no—
399#120—2441no—
400#120—2441no—
401C4×D4—611no—
402C4×D4—1221no—
403C4×D4—1221no—
404C2×C22⋊C4—2441no—
405C2×C22⋊C4—2441no—
406Q32—1222no—
407SD32—41no—
408SD32—1222no—
409SD32—1222no—
410SD32—2441no—
411D16—22no—
412D16—614no—
413D16—1222no—
414D16—1222no—
415C2×C16—611no—
416C2×C16—611no—
417C2×C16—1222no—
418C2×C16—1222no—
419C2×C16C22×D6111yescharacteristic
420C2×C16C22×D6111yescharacteristic
421#107—611no—
422#107C22×D6111yescharacteristic
423C4⋊C8—611no—
424#103—1221no—
425#103—1221no—
426#102—21no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430C22⋊C8—611no—
431C22⋊C8—611no—
432C4×C8—611no—
433C24—2441no—
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—41no—
444C2×D4—41no—
445C2×D4—611no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—4881no—
449C2×D4—4881no—
450C2×D4—2441no—
451C2×D4—2441no—
452C2×D4—2442no—
453C2×D4—2442no—
454C2×D4—2441no—
455C2×D4—2441no—
456C22×C4—41no—
457C22×C4—611no—
458C22×C4—2441no—
459C22×C4—2441no—
460C22×C4—2441no—
461C22×C4D4×D6111yescharacteristic
462Q16—21no—
463Q16—21no—
464Q16—1221no—
465Q16—1221no—
466SD16—42no—
467SD16—4881no—
468SD16—4881no—
469SD16—2442no—
470D8—21no—
471D8—21no—
472D8—21no—
473D8—612no—
474D8—1221no—
475D8—1221no—
476D8—1221no—
477D8—2442no—
478D8—2442no—
479D8D4×D6112yes—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—611no—
483C2×C8—611no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8D4×D6111yescharacteristic
487C2×C8C23×D6111yescharacteristic
488C4⋊C4—21no—
489C4⋊C4—611no—
490C4⋊C4—1221no—
491C4⋊C4—2441no—
492C4⋊C4—2441no—
493C22⋊C4—41no—
494C22⋊C4—611no—
495C22⋊C4—611no—
496C22⋊C4—2441no—
497C22⋊C4—2441no—
498C22⋊C4—2441no—
499C22⋊C4—2441no—
500C22⋊C4—2441no—
501C22⋊C4—2441no—
502C22⋊C4—2441no—
503C22⋊C4—2441no—
504C22⋊C4—2441no—
505C42—611no—
506C16—21no—
507C16—21no—
508C16—612no—
509C16—1221no—
510C23—41no—
511C23—41no—
512C23—611no—
513C23—9761no—
514C23—9761no—
515C23—4882no—
516C23—4882no—
517C23—4882no—
518C23—4882no—
519C23—2441no—
520C23—2441no—
521C23D8×D6111yescharacteristic
522Q8—41no—
523Q8—41no—
524Q8—2441no—
525Q8—2441no—
526D4—22no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—82no—
531D4—161no—
532D4—1222no—
533D4—1222no—
534D4—9761no—
535D4—4882no—
536D4—4882no—
537D4—4882no—
538D4—2441no—
539D4—2441no—
540D4—2441no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—82no—
546C2×C4—611no—
547C2×C4—611no—
548C2×C4—611no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—9761no—
552C2×C4—9761no—
553C2×C4—4882no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4D8×D6111yescharacteristic
557C2×C4C2×D4×D6111yescharacteristic
558C8—21no—
559C8—1221no—
560C8—1221no—
561C8C2×D4×D6111yescharacteristic
562C8#1451982211yescharacteristic
563C22—21no—
564C22—21no—
565C22—21no—
566C22—42no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—612no—
573C22—1221no—
574C22—1221no—
575C22—9764no—
576C22—9764no—
577C22—4881no—
578C22—4881no—
579C22—4881no—
580C22—4881no—
581C22—4881no—
582C22—2442no—
583C22C2×D8×D6111yescharacteristic
584C4—21no—
585C4—81no—
586C4—81no—
587C4—1221no—
588C4—1221no—
589C4—1221no—
590C4—4881no—
591C4—4881no—
592C4#1684039411yescharacteristic
593C4#1684056611yescharacteristic
594C2—41no—
595C2—82no—
596C2—161no—
597C2—1222no—
598C2—9761no—
599C2—4882no—
600C2#2145748911yescharacteristic
601C2#508×D6111yescharacteristic
602C2#2145895911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2186C2
socleC22×C61C22×D8
hypercenter#2186D61
hypocenterC61#33846
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2186

Complements

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

NH ≅ G/Ntype
1. C2×#16840192C2semidirect
2. C2×D16×D61C2semidirect
3. #21457507C2semidirect
4. #21457498C2semidirect
5. #21448079C2semidirect
6. #21447909C2semidirect
7. #21447903C2semidirect
8. #21447877C2semidirect
9. #21447872C2semidirect
10. #21447749C2semidirect
11. #21447746C2semidirect
12. #21447744C2semidirect
13. #21447663C2semidirect
14. #21447636C2semidirect
15. C61×#2186C2semidirect
16. C2×#14519799C22semidirect
22. C2×#14519723C22semidirect
23. C2×#14519722C22semidirect
24. C2×#14519721C22semidirect
25. C2×#14519720C22semidirect
30. C22×#11758185C22semidirect
32. #16840147C22semidirect
33. #16840145C22semidirect
36. #16839998C22semidirect
43. D8×Dic61C22semidirect
44. #16839844C22semidirect
46. #16839587C22semidirect
47. #16839580C22semidirect
48. #16839567C22semidirect
49. #16839564C22semidirect
50. #16839563C22semidirect
51. #16839562C22semidirect
53. C16×Dic61C22semidirect
54. C2×#135×C61C22semidirect
55. C2×SD32×C61C22semidirect
57. C61×#466C22semidirect
58. C61×#358C22semidirect
59. C61×#357C22semidirect
60. C61×C22⋊C16C22semidirect
101. #14519718D4semidirect
282. C2×D61#508semidirect
305. #2186D61semidirect
318. C2×#135C2×D61semidirect
328. C2×SD32C2×D61semidirect
341. #466C2×D61semidirect
354. #358C2×D61semidirect
358. #357C2×D61semidirect
360. C22⋊C16C2×D61semidirect
362. C61#33846semidirect

Maximal subgroups

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

subgroupindex
C2×#168401922
C2×D16×D612
#214575072
#214574982
#214480792
#214479092
#214479032
#214478772
#214478722
#214477492
#214477462
#214477442
#214476632
#214476362
C61×#21862
#3384661