Finite Group Databasebeta

← #29836596

Subgroups — #29836596

Order 15616.

712 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×#16840196C211yesmaximal, characteristic, complemented
2#21457504C211yesmaximal, characteristic, complemented
3#21457498C211yesmaximal, characteristic, complemented
4#21448079C211yesmaximal, characteristic, complemented
5#21447932C211yesmaximal, characteristic, complemented
6#21447929C211yesmaximal, characteristic, complemented
7#21447876C211yesmaximal, characteristic, complemented
8#21447872C211yesmaximal, characteristic, complemented
9#21447759C211yesmaximal, characteristic, complemented
10#21447756C211yesmaximal, characteristic, complemented
11#21447731C211yesmaximal, characteristic, complemented
12#21447729C211yesmaximal, characteristic, complemented
13D16×Dic61C211yesmaximal, characteristic, complemented
14#21447649C211yesmaximal, characteristic, complemented
15C2×C61×#509C211yesmaximal, characteristic, complemented
16C2×#14519796C2211yescharacteristic
17C2×D8×D61C2211yescharacteristic, complemented
18#16840406—21no—
19#16840400—21no—
20#16840196—24no—
21C2×#14519723C2211yescharacteristic, complemented
22C2×#14519722C2211yescharacteristic, complemented
23C2×#14519721C2211yescharacteristic, complemented
24C2×#14519718C2211yescharacteristic
25C2×#14519718C2211yescharacteristic
26C2×#14519716C2211yescharacteristic, complemented
27#16840147C2211yescharacteristic, complemented
28#16840145C2211yescharacteristic, complemented
29#16840003C2211yescharacteristic, complemented
30#16840000C2211yescharacteristic, complemented
31#16839998C2211yescharacteristic
32#16839995C2211yescharacteristic
33#16839979—21no—
34#16839975—21no—
35#16839913—21no—
36#16839911—21no—
37#16839907—21no—
38#16839904—21no—
39D8×Dic61C2211yescharacteristic, complemented
40D8×Dic61C2211yescharacteristic, complemented
41C8×C22⋊D61C2211yescharacteristic
42#16839588C2211yescharacteristic, complemented
43#16839580C2211yescharacteristic, complemented
44#16839567C2211yescharacteristic, complemented
45#16839566C2211yescharacteristic, complemented
46#16839564C2211yescharacteristic, complemented
47#16839563C2211yescharacteristic, complemented
48#16839563C2211yescharacteristic, complemented
49#16839563C2211yescharacteristic, complemented
50C22⋊#11758185C2211yescharacteristic, complemented
51#16839541C2211yescharacteristic, complemented
52C16×Dic61C2211yescharacteristic, complemented
53C2×#135×C61C2211yescharacteristic
54C22×D8×C61C2211yescharacteristic, complemented
55C61×#509—24no—
56C2×SD32×C61C2211yescharacteristic, complemented
57C2×SD32×C61C2211yescharacteristic, complemented
58C2×D16×C61C2211yescharacteristic
59C2×D16×C61C2211yescharacteristic
60C2×M32×C61C2211yescharacteristic, complemented
61C2×#11758224—21no—
62C2×D4×D61—21no—
63#14519796—22no—
64#14519796—41no—
65C2×Dic244C2311yescharacteristic
66C2×#11758200—21no—
67C2×#11758200—21no—
68C2×#11758201—21no—
69C2×D488C2311yescharacteristic
70C2×#11758199—21no—
71D8×D61—22no—
72D8×D61—41no—
73C2×C8×D61C2311yescharacteristic
74#14519766—21no—
75#14519764—21no—
76#14519760—41no—
77#14519758—41no—
78#14519756—21no—
79#14519753—21no—
80#14519751—21no—
81#14519749—41no—
82#14519749—41no—
83#14519748—21no—
84#14519746—41no—
85#14519745—41no—
86D4×Dic61—21no—
87D4×Dic61—21no—
88C4×C22⋊D61D411yescharacteristic
89C2×C22⋊Dic61—41no—
90C2×#11758198—41no—
91#14519723—41no—
92#14519722—22no—
93#14519721—22no—
94#14519718—22no—
95#14519718—41no—
96#14519716—22no—
97C2×#11758185C2311yescharacteristic
98C2×#11758185C2311yescharacteristic
99#14519711C2311yescharacteristic
100#14519711C2311yescharacteristic
101#14519706D411yescharacteristic
102#14519702—21no—
103#14519701—21no—
104#14519699—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698—21no—
108C22⋊#92418D411yescharacteristic
109#14519689D411yescharacteristic
110C8×Dic61C2311yescharacteristic
111C2×#41×C61—21no—
112C22×D4×C61—21no—
113#135×C61—22no—
114#135×C61—41no—
115C2×Q16×C61C2311yescharacteristic
116C2×SD16×C61—21no—
117C2×D8×C61—22no—
118C2×D8×C61—21no—
119C2×D8×C61—41no—
120C2×D8×C61C2311yescharacteristic
121C2×D8×C61C2311yescharacteristic
122C2×D8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124SD32×C61—22no—
125SD32×C61—22no—
126D16×C61—22no—
127D16×C61D414yescomplemented
128M32×C61—22no—
129C2×C16×C61C2311yescharacteristic
130C2×C16×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—21no—
139C2×D4×C61—42no—
140C2×D4×C61—41no—
141C2×D4×C61—41no—
142C2×D4×C61—81no—
143C22×C4×C61—41no—
144C22×C4×C61C2×D411yescharacteristic
145C2×C22⋊D61—41no—
146C2×C22⋊D61—41no—
147C2×C22⋊D61—41no—
148C2×C22⋊D61—41no—
149C2×C22⋊D61D811yescharacteristic
150C22×Dic61—41no—
151C22×Dic61—41no—
152C22×Dic61—41no—
153#11758224—42no—
154#11758224—81no—
155D4×D61—42no—
156D4×D61—81no—
157C2×D244—21no—
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×C61—21no—
167D8×C61—42no—
168D8×C61C2×D412yes—
169D8×C61C2×D412yes—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2411yescharacteristic
174C22⋊Dic61—41no—
175C22⋊Dic61—41no—
176C22⋊Dic61—41no—
177C22⋊Dic61—41no—
178C22⋊Dic61—41no—
179C22⋊Dic61—41no—
180C22⋊Dic61D811yescharacteristic
181#11758201—81no—
182#11758200—42no—
183#11758200—81no—
184#11758199—42no—
185#11758198—41no—
186#11758198—41no—
187#11758198—41no—
188#11758198—41no—
189#11758198D811yescharacteristic
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192#11758196—41no—
193#11758196—41no—
194#11758196D811yescharacteristic
195C4×Dic61C2×D411yescharacteristic
196C2×#92418C2×D411yescharacteristic
197C2×#92418C2×D411yescharacteristic
198Dic244—21no—
199Dic244—21no—
200D488—21no—
201D488—21no—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61C2×D412yes—
206#11758185—21no—
207#11758185—21no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—82no—
212C23×C61—82no—
213C23×C61—161no—
214C23×C61C2×D811yescharacteristic
215C22×D61—41no—
216C22×D61—82no—
217C22×D61—82no—
218C22×D61—161no—
219C22×D61C2×D811yescharacteristic
220Q8×C61—41no—
221Q8×C61—41no—
222D4×C61—22no—
223D4×C61—22no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—82no—
229D4×C61—82no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—82no—
234C2×C4×C61C2×D811yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C22⋊D61—41no—
237C22⋊D61—82no—
238C22⋊D61—82no—
239C22⋊D61—161no—
240C22⋊D61—161no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—82no—
245C2×Dic61—161no—
246C2×Dic61—161no—
247C2×Dic61C2×D811yescharacteristic
248C2×Dic61C2×D811yescharacteristic
249C2×Dic61C2×D811yescharacteristic
250D244—41no—
251D244—41no—
252C4×D61—21no—
253C4×D61—21no—
254Dic122—41no—
255Dic122—41no—
256C8×C61—21no—
257C8×C61C22×D411yescharacteristic
258C8×C61H(5,𝔽2)11yescharacteristic
259#92418—21no—
260#92418—21no—
261#33839—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—164no—
275C22×C61C22×D811yescharacteristic
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—81no—
280C2×D61—81no—
281C2×D61—81no—
282C2×D61—164no—
283C4×C61—21no—
284C4×C61—81no—
285C4×C61D4211yescharacteristic
286C4×C61#57611yescharacteristic
287Dic61—21no—
288Dic61—21no—
289Dic61—21no—
290Dic61—81no—
291Dic61—81no—
292Dic61—81no—
293C2×#509—611no—
294C2×#509D6111yescharacteristic, complemented
295#3280—611no—
296#3280—611no—
297#2249—611no—
298#2220—611no—
299#2220—611no—
300#2210—611no—
301#2210—611no—
302#2189—611no—
303#2189—611no—
304#2184—611no—
305#2184—611no—
306C4×D16—611no—
307#2169—611no—
308C2×C61—41no—
309C2×C61—82no—
310C2×C61—82no—
311C2×C61—161no—
312C2×C61D4×D811yescharacteristic
313C2×C61#341411yescharacteristic
314C2×C61#341511yescharacteristic
315D61—41no—
316D61—161no—
317C2×#135—611no—
318C2×#135C2×D6111yescharacteristic
319C22×D8—611no—
320C22×D8C2×D6111yescharacteristic, complemented
321#546—1221no—
322#546—1221no—
323#509—24no—
324#509—1224no—
325C2×Q32—611no—
326C2×SD32—611no—
327C2×SD32—611no—
328C2×SD32C2×D6111yescharacteristic, complemented
329C2×SD32C2×D6111yescharacteristic, complemented
330C2×D16—611no—
331C2×D16—611no—
332C2×D16C2×D6111yescharacteristic
333C2×D16C2×D6111yescharacteristic
334C2×M32—611no—
335C2×M32C2×D6111yescharacteristic, complemented
336#495—611no—
337#495—611no—
338#467—611no—
339#467—611no—
340#466—611no—
341#466—611no—
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#358—611no—
355#357—611no—
356#357—611no—
357#357—611no—
358#357—611no—
359C22⋊C16—611no—
360C22⋊C16—611no—
361C4×C16—611no—
362C61#3383911yescharacteristic, 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—1221no—
375C2×SD16—1221no—
376C2×SD16—1221no—
377C2×D8—22no—
378C2×D8—21no—
379C2×D8—41no—
380C2×D8—611no—
381C2×D8—1221no—
382C2×D8—1222no—
383C2×D8—2441no—
384C2×D8C22×D6111yescharacteristic
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—1221no—
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—2441no—
407SD32—22no—
408SD32—22no—
409SD32—1222no—
410SD32—1222no—
411D16—22no—
412D16—1222no—
413D16—2441no—
414D16C22⋊D6114yescomplemented
415M32—22no—
416M32—1222no—
417C2×C16—611no—
418C2×C16—611no—
419C2×C16C22×D6111yescharacteristic
420C2×C16C22×D6111yescharacteristic
421#107—611no—
422#107—611no—
423C4⋊C8—611no—
424#103—1221no—
425#103—1221no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430C22⋊C8—611no—
431C22⋊C8—611no—
432C4×C8—611no—
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—21no—
444C2×D4—42no—
445C2×D4—41no—
446C2×D4—41no—
447C2×D4—81no—
448C2×D4—611no—
449C2×D4—1221no—
450C2×D4—4881no—
451C2×D4—2441no—
452C2×D4—2442no—
453C2×D4—2441no—
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—21no—
474D8—42no—
475D8—1221no—
476D8—1221no—
477D8—2442no—
478D8C2×C22⋊D6112yes—
479D8C2×C22⋊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—611no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—2441no—
492C4⋊C4—2441no—
493C22⋊C4—611no—
494C22⋊C4—611no—
495C22⋊C4—2441no—
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—1221no—
508C16—1221no—
509C16C2×C22⋊D6112yes—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—82no—
514C23—82no—
515C23—161no—
516C23—611no—
517C23—9761no—
518C23—4882no—
519C23—4882no—
520C23—2441no—
521C23D8×D6111yescharacteristic
522Q8—41no—
523Q8—41no—
524Q8—2441no—
525Q8—2441no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—82no—
533D4—82no—
534D4—9761no—
535D4—9761no—
536D4—4882no—
537D4—4882no—
538D4—2441no—
539D4—2441no—
540D4—2441no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—82no—
545C2×C4—611no—
546C2×C4—611no—
547C2×C4—611no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—9761no—
551C2×C4—9761no—
552C2×C4—4882no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4D8×D6111yescharacteristic
557C2×C4C2×D4×D6111yescharacteristic
558C8—21no—
559C8—1221no—
560C8—1221no—
561C8C22×C22⋊D6111yescharacteristic
562C8#1451982211yescharacteristic
563C22—21no—
564C22—21no—
565C22—21no—
566C22—42no—
567C22—42no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—164no—
576C22—1221no—
577C22—1221no—
578C22—1221no—
579C22—9764no—
580C22—4881no—
581C22—4881no—
582C22—4881no—
583C22C2×D8×D6111yescharacteristic
584C4—21no—
585C4—81no—
586C4—1221no—
587C4—1221no—
588C4—1221no—
589C4—4881no—
590C4—4881no—
591C4—4881no—
592C4D4×C22⋊D6111yescharacteristic
593C4#1684056311yescharacteristic
594C2—41no—
595C2—82no—
596C2—82no—
597C2—161no—
598C2—9761no—
599C2—2441no—
600C2D8×C22⋊D6111yescharacteristic
601C2#2145895111yescharacteristic
602C2#2145895911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#509C2
socleC22×C61C22×D8
hypercenterC2×#509D61
hypocenterC61#33839
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. C2×#16840196C2semidirect
2. #21457504C2semidirect
3. #21457498C2semidirect
4. #21448079C2semidirect
5. #21447932C2semidirect
6. #21447929C2semidirect
7. #21447876C2semidirect
8. #21447872C2semidirect
9. #21447759C2semidirect
10. #21447756C2semidirect
11. #21447731C2semidirect
12. #21447729C2semidirect
13. D16×Dic61C2semidirect
14. #21447649C2semidirect
15. C2×C61×#509C2semidirect
17. C2×D8×D61C22semidirect
21. C2×#14519723C22semidirect
22. C2×#14519722C22semidirect
23. C2×#14519721C22semidirect
26. C2×#14519716C22semidirect
27. #16840147C22semidirect
28. #16840145C22semidirect
29. #16840003C22semidirect
30. #16840000C22semidirect
39. D8×Dic61C22semidirect
40. D8×Dic61C22semidirect
42. #16839588C22semidirect
43. #16839580C22semidirect
44. #16839567C22semidirect
45. #16839566C22semidirect
46. #16839564C22semidirect
47. #16839563C22semidirect
48. #16839563C22semidirect
49. #16839563C22semidirect
50. C22⋊#11758185C22semidirect
51. #16839541C22semidirect
52. C16×Dic61C22semidirect
54. C22×D8×C61C22semidirect
56. C2×SD32×C61C22semidirect
57. C2×SD32×C61C22semidirect
60. C2×M32×C61C22semidirect
127. D16×C61D4semidirect
294. C2×#509D61semidirect
320. C22×D8C2×D61semidirect
328. C2×SD32C2×D61semidirect
329. C2×SD32C2×D61semidirect
335. C2×M32C2×D61semidirect
362. C61#33839semidirect
414. D16C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401962
#214575042
#214574982
#214480792
#214479322
#214479292
#214478762
#214478722
#214477592
#214477562
#214477312
#214477292
D16×Dic612
#214476492
C2×C61×#5092
#3383961