Finite Group Databasebeta

← #29837046

Subgroups — #29837046

Order 15616.

610 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458188C211yesmaximal, characteristic, complemented
2D4×D488C211yesmaximal, characteristic
3#21448061C211yesmaximal, characteristic, complemented
4#21448011C211yesmaximal, characteristic, complemented
5#21448000C211yesmaximal, characteristic, complemented
6#21447977C211yesmaximal, characteristic, complemented
7#21447932C211yesmaximal, characteristic, complemented
8#21447915C211yesmaximal, characteristic, complemented
9#21447860C211yesmaximal, characteristic, complemented
10#21447750C211yesmaximal, characteristic, complemented
11#21447750C211yesmaximal, characteristic, complemented
12#21447685C211yesmaximal, characteristic, complemented
13#21447536C211yesmaximal, characteristic, complemented
14C2×#16839564C211yesmaximal, characteristic, complemented
15C61×#2168C211yesmaximal, characteristic, complemented
16C22×D488C2211yescharacteristic
17C22×D488C2211yescharacteristic
18#16840452—21no—
19D4×D244—21no—
20C2×#14519722C2211yescharacteristic
21C2×#14519722C2211yescharacteristic, complemented
22C2×#14519722C2211yescharacteristic, complemented
23#16840164C2211yescharacteristic, complemented
24#16840139C2211yescharacteristic
25#16840104—21no—
26#16840093—21no—
27#16840070—21no—
28#16840003C2211yescharacteristic, complemented
29#16840003C2211yescharacteristic, complemented
30#16839998C2211yescharacteristic
31#16839998C2211yescharacteristic
32#16839958—21no—
33#16839898—21no—
34#16839898—21no—
35C4×Dic244C2211yescharacteristic, complemented
36C4×D488C2211yescharacteristic
37#16839800C2211yescharacteristic, complemented
38C2×#14519711C2211yescharacteristic, complemented
39#16839590C2211yescharacteristic, complemented
40#16839588C2211yescharacteristic, complemented
41#16839588C2211yescharacteristic, complemented
42#16839567C2211yescharacteristic, complemented
43#16839564C2211yescharacteristic
44#16839564C2211yescharacteristic
45#16839564C2211yescharacteristic
46#16839564C2211yescharacteristic, complemented
47#16839564C2211yescharacteristic, complemented
48#16839564C2211yescharacteristic, complemented
49#16839564C2211yescharacteristic, complemented
50#16839564C2211yescharacteristic, complemented
51#16839564C2211yescharacteristic, complemented
52C2×M32×C61C2211yescharacteristic, complemented
53C22×C16×C61C2211yescharacteristic, complemented
54C8×D4×C61C2211yescharacteristic
55C61×C4⋊C16C2211yescharacteristic, complemented
56C61×C22⋊C16C2211yescharacteristic, complemented
57C61×C22⋊C16C2211yescharacteristic, complemented
58C61×#346C2211yescharacteristic, complemented
59C22×D244—21no—
60C22×D244—21no—
61C2×D4×D61—41no—
62C2×Dic244C2311yescharacteristic
63C2×D488—22no—
64C2×D488—22no—
65C2×D488—21no—
66C2×D488—21no—
67C2×D488—21no—
68C2×D488—41no—
69C2×D488—41no—
70C2×D488C2311yescharacteristic
71C2×D488C2311yescharacteristic
72C2×D488C2311yescharacteristic
73#14519774—21no—
74#14519770—41no—
75#14519769—21no—
76#14519767—41no—
77#14519758—41no—
78#14519756—21no—
79#14519756—21no—
80#14519752—41no—
81#14519751—21no—
82#14519751—21no—
83#14519747—41no—
84#14519747—41no—
85C4×Dic122—21no—
86C4×D244—21no—
87#14519736—21no—
88#14519733—41no—
89C2×C4⋊Dic61—21no—
90#14519722—22no—
91#14519722—41no—
92#14519722—41no—
93#14519711—21no—
94#14519711C2311yescharacteristic
95#14519711C2311yescharacteristic
96#14519711C2311yescharacteristic
97#14519711C2311yescharacteristic
98#14519709—21no—
99#14519709—21no—
100#14519702—21no—
101#14519702—21no—
102#14519702—21no—
103#14519699—21no—
104#14519699—21no—
105#14519699—21no—
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C4×D4×C61D411yescharacteristic
109M32×C61—41no—
110C2×C16×C61—41no—
111C2×C16×C61C2311yescharacteristic
112C2×C16×C61C2311yescharacteristic
113C2×C16×C61C2311yescharacteristic
114C2×C16×C61C2311yescharacteristic
115C4⋊C8×C61D411yescharacteristic
116C22⋊C8×C61D411yescharacteristic
117C22⋊C8×C61D411yescharacteristic
118C4×C8×C61C2311yescharacteristic
119C23×D61—41no—
120C23×D61—41no—
121C2×D4×C61D811yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C2×C22⋊D61—41no—
125C2×C22⋊D61—41no—
126C22×Dic61—41no—
127D4×D61—84no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—42no—
132C2×D244—42no—
133C2×D244—41no—
134C2×D244—41no—
135C2×D244—41no—
136C2×D244—41no—
137C2×D244—81no—
138C2×D244—81no—
139C2×C4×D61—41no—
140C2×Dic122—21no—
141C2×C8×C61—21no—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61C2×D411yescharacteristic
146C2×C8×C61C2×D411yescharacteristic
147C2×C8×C61C2×D411yescharacteristic
148C2×C8×C61C2411yescharacteristic
149C4⋊C4×C61D811yescharacteristic
150C22⋊C4×C61D811yescharacteristic
151C22⋊C4×C61D811yescharacteristic
152C42×C61C2×D411yescharacteristic
153C22⋊Dic61—41no—
154C22⋊Dic61—41no—
155C22⋊Dic61—41no—
156#11758198—41no—
157#11758198—41no—
158#11758198—41no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—41no—
164#11758196—41no—
165#11758196—41no—
166#11758196—41no—
167#11758196—41no—
168#11758196—41no—
169C4×Dic61—41no—
170C4×Dic61—41no—
171Dic244—21no—
172Dic244—21no—
173D488—21no—
174D488—21no—
175D488—21no—
176D488—21no—
177D488—21no—
178D488—42no—
179D488—42no—
180D488—81no—
181D488#4112yes—
182C16×C61—21no—
183C16×C61—21no—
184C16×C61—21no—
185C16×C61—21no—
186C23×C61C2×D811yescharacteristic
187C23×C61C2×D811yescharacteristic
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—82no—
192C22×D61—82no—
193C22×D61—82no—
194C22×D61—82no—
195C22×D61—161no—
196C22×D61—161no—
197D4×C61—22no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61C2×D811yescharacteristic
203C2×C4×C61C2×D811yescharacteristic
204C2×C4×C61C2×D811yescharacteristic
205C2×C4×C61C2×D811yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C22⋊D61—82no—
208C22⋊D61—82no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—82no—
215D244—22no—
216D244—41no—
217D244—41no—
218D244—41no—
219D244—41no—
220D244—41no—
221D244—82no—
222D244—82no—
223D244—82no—
224D244—161no—
225C4×D61—82no—
226Dic122—41no—
227Dic122—41no—
228C8×C61—21no—
229C8×C61—21no—
230C8×C61—21no—
231C8×C61C2×#4111yescharacteristic
232C8×C61H(5,𝔽2)11yescharacteristic
233#33877—611nomaximal, Sylow
234C22×C61—21no—
235C22×C61—21no—
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—21no—
239C22×C61#50912yes—
240C22×C61C22×D811yescharacteristic
241C2×D61—42no—
242C2×D61—81no—
243C2×D61—81no—
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—81no—
247C2×D61—81no—
248C2×D61—81no—
249C2×D61—81no—
250C2×D61—164no—
251C2×D61—164no—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—21no—
255C4×C61—21no—
256C4×C61#54611yescharacteristic
257C4×C61#57711yescharacteristic
258Dic61—81no—
259Dic61—81no—
260Dic61—81no—
261Dic61—81no—
262Dic61—81no—
263#3336—611no—
264D4×D8—611no—
265#2244—611no—
266#2234—611no—
267#2232—611no—
268#2228—611no—
269#2220—611no—
270#2217—611no—
271#2207—611no—
272#2187—611no—
273#2187—611no—
274#2175—611no—
275#2168D6111yescharacteristic, complemented
276#2140—611no—
277C2×#357—611no—
278C2×C61—22no—
279C2×C61—41no—
280C2×C61#329411yescharacteristic
281C2×C61C2×#50911yescharacteristic
282C2×C61#341611yescharacteristic
283D61—82no—
284D61—161no—
285D61—161no—
286C22×D8—611no—
287C22×D8—611no—
288#552—1221no—
289D42—1221no—
290C2×SD32—611no—
291C2×SD32—611no—
292C2×SD32—611no—
293C2×M32C2×D6111yescharacteristic, complemented
294C22×C16C2×D6111yescharacteristic, complemented
295#499—611no—
296#493—611no—
297#485—1221no—
298#483—1221no—
299#479—1221no—
300#467—611no—
301#467—611no—
302#466—611no—
303#466—611no—
304#459—1221no—
305#447—1221no—
306#447—1221no—
307C4×Q16—611no—
308C4×D8—611no—
309C8×D4C2×D6111yescharacteristic
310#427—611no—
311C2×#107—611no—
312#367—611no—
313#366—611no—
314#366—611no—
315C4⋊C16C2×D6111yescharacteristic, complemented
316#358—611no—
317#357—611no—
318#357—611no—
319#357—611no—
320#357—611no—
321#357—611no—
322#357—611no—
323#357—611no—
324#357—611no—
325#357—611no—
326C22⋊C16C2×D6111yescharacteristic, complemented
327C22⋊C16C2×D6111yescharacteristic, complemented
328#346C2×D6111yescharacteristic, complemented
329C61#3387711yescharacteristic, complemented, Sylow
330C22×D4—1221no—
331C22×D4—1221no—
332C22×D4—2441no—
333C2×Q16—611no—
334C2×D8—611no—
335C2×D8—611no—
336C2×D8—611no—
337C2×D8—1222no—
338C2×D8—1222no—
339C2×D8—1221no—
340C2×D8—1221no—
341C2×D8—1221no—
342C2×D8—2441no—
343C2×D8—2441no—
344C22×C8C22×D6111yescharacteristic
345C22×C8C22×D6111yescharacteristic
346#127—1221no—
347#126—2441no—
348#125—1221no—
349#125—2441no—
350#123—2441no—
351C22⋊Q8—1221no—
352C22⋊Q8—1221no—
353#121—1221no—
354#121—1221no—
355#121—2441no—
356#120—2441no—
357#120—2441no—
358C4×Q8—1221no—
359C4×D4—1221no—
360C4×D4D24411yescharacteristic
361#117—1221no—
362#117—2441no—
363C2×C4⋊C4—1221no—
364SD32—1222no—
365SD32—2441no—
366SD32—2441no—
367M32—41no—
368C2×C16—41no—
369C2×C16C22×D6111yescharacteristic
370C2×C16C22×D6111yescharacteristic
371C2×C16C22×D6111yescharacteristic
372C2×C16C22×D6111yescharacteristic
373#107—611no—
374#107—611no—
375#107—611no—
376#107—611no—
377#107—1221no—
378#106—1221no—
379#106—1221no—
380C4⋊C8D24411yescharacteristic
381#103—1221no—
382#103—1221no—
383#103—1221no—
384#102—1221no—
385#102—1221no—
386#102—1221no—
387C22⋊C8D24411yescharacteristic
388C22⋊C8D24411yescharacteristic
389C4×C8C22×D6111yescharacteristic
390C24—2441no—
391C24—2441no—
392C2×Q8—1221no—
393C2×D4—1221no—
394C2×D4—1221no—
395C2×D4—1221no—
396C2×D4—4881no—
397C2×D4—4881no—
398C2×D4—4884no—
399C2×D4—2442no—
400C2×D4—2442no—
401C2×D4—2441no—
402C2×D4—2441no—
403C2×D4—2441no—
404C2×D4—2441no—
405C2×D4—2441no—
406C2×D4—2441no—
407C2×D4D48811yescharacteristic
408C22×C4—2441no—
409C22×C4—2441no—
410C22×C4C2×D24411yescharacteristic
411C22×C4C2×D24411yescharacteristic
412Q16—1221no—
413Q16—1221no—
414D8—612no—
415D8—1221no—
416D8—1221no—
417D8—1221no—
418D8—1221no—
419D8—1221no—
420D8—4881no—
421D8—2442no—
422D8—2442no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8C2×D24411yescharacteristic
428C2×C8C2×D24411yescharacteristic
429C2×C8C2×D24411yescharacteristic
430C2×C8C23×D6111yescharacteristic
431C4⋊C4—1221no—
432C4⋊C4—1221no—
433C4⋊C4—1221no—
434C4⋊C4—1221no—
435C4⋊C4—2441no—
436C4⋊C4—2441no—
437C4⋊C4—2441no—
438C4⋊C4—2441no—
439C4⋊C4—2441no—
440C4⋊C4—2441no—
441C4⋊C4D48811yescharacteristic
442C22⋊C4—2441no—
443C22⋊C4—2441no—
444C22⋊C4—2441no—
445C22⋊C4—2441no—
446C22⋊C4—2441no—
447C22⋊C4—2441no—
448C22⋊C4D48811yescharacteristic
449C22⋊C4D48811yescharacteristic
450C42—2441no—
451C42—2441no—
452C42C2×D24411yescharacteristic
453C16—21no—
454C16—21no—
455C16—21no—
456C16—21no—
457C23—9761no—
458C23—9761no—
459C23—4882no—
460C23—4882no—
461C23—4882no—
462C23—4882no—
463C23—2441no—
464C23—2441no—
465C23—2441no—
466C23C2×D48811yescharacteristic
467C23C2×D48811yescharacteristic
468Q8—2441no—
469Q8—2441no—
470D4—22no—
471D4—1222no—
472D4—9761no—
473D4—4882no—
474D4—4882no—
475D4—4882no—
476D4—4882no—
477D4—4882no—
478D4—2441no—
479D4—2441no—
480D4—2441no—
481D4—2441no—
482D4—2441no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—4882no—
488C2×C4—4882no—
489C2×C4—2441no—
490C2×C4—2441no—
491C2×C4—2441no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4C2×D48811yescharacteristic
495C2×C4C2×D48811yescharacteristic
496C2×C4C2×D48811yescharacteristic
497C2×C4C2×D48811yescharacteristic
498C2×C4C22×D24411yescharacteristic
499C8—21no—
500C8—21no—
501C8—21no—
502C8#41×D6111yescharacteristic
503C8#1451982211yescharacteristic
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—9764no—
510C22—9764no—
511C22—4881no—
512C22—4881no—
513C22—4881no—
514C22—4881no—
515C22—4881no—
516C22—4881no—
517C22—4881no—
518C22—4881no—
519C22—2442no—
520C22#1684020012yes—
521C22C22×D48811yescharacteristic
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—4881no—
527C4—4881no—
528C4—4881no—
529C4—4881no—
530C4—4881no—
531C4#1684040711yescharacteristic
532C4#1684057411yescharacteristic
533C2—22no—
534C2—41no—
535C2—9761no—
536C2—9761no—
537C2—4882no—
538C2#2145763311yescharacteristic
539C2C2×#1684020011yescharacteristic
540C2#2145896711yescharacteristic
subgroupG/H
centerC22C22×D488
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2168C2
socleC22×C61C22×D8
prime radicalD4×D488C2
hypercenter#2168D61
hypocenterC61#33877
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #2168

Complements

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

NH ≅ G/Ntype
1. #21458188C2semidirect
3. #21448061C2semidirect
4. #21448011C2semidirect
5. #21448000C2semidirect
6. #21447977C2semidirect
7. #21447932C2semidirect
8. #21447915C2semidirect
9. #21447860C2semidirect
10. #21447750C2semidirect
11. #21447750C2semidirect
12. #21447685C2semidirect
13. #21447536C2semidirect
14. C2×#16839564C2semidirect
15. C61×#2168C2semidirect
21. C2×#14519722C22semidirect
22. C2×#14519722C22semidirect
23. #16840164C22semidirect
28. #16840003C22semidirect
29. #16840003C22semidirect
35. C4×Dic244C22semidirect
37. #16839800C22semidirect
38. C2×#14519711C22semidirect
39. #16839590C22semidirect
40. #16839588C22semidirect
41. #16839588C22semidirect
42. #16839567C22semidirect
46. #16839564C22semidirect
47. #16839564C22semidirect
48. #16839564C22semidirect
49. #16839564C22semidirect
50. #16839564C22semidirect
51. #16839564C22semidirect
52. C2×M32×C61C22semidirect
53. C22×C16×C61C22semidirect
55. C61×C4⋊C16C22semidirect
56. C61×C22⋊C16C22semidirect
57. C61×C22⋊C16C22semidirect
58. C61×#346C22semidirect
275. #2168D61semidirect
293. C2×M32C2×D61semidirect
294. C22×C16C2×D61semidirect
315. C4⋊C16C2×D61semidirect
326. C22⋊C16C2×D61semidirect
327. C22⋊C16C2×D61semidirect
328. #346C2×D61semidirect
329. C61#33877semidirect

Maximal subgroups

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

subgroupindex
#214581882
D4×D4882
#214480612
#214480112
#214480002
#214479772
#214479322
#214479152
#214478602
#214477502
#214476852
#214475362
C2×#168395642
C61×#21682
#3387761