Finite Group Databasebeta

← #29836864

Subgroups — #29836864

Order 15616.

568 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458190C211yesmaximal, characteristic, complemented
2#21457501C211yesmaximal, characteristic
3#21448052C211yesmaximal, characteristic, complemented
4#21448014C211yesmaximal, characteristic, complemented
5#21448013C211yesmaximal, characteristic, complemented
6#21447979C211yesmaximal, characteristic, complemented
7#21447918C211yesmaximal, characteristic, complemented
8#21447883C211yesmaximal, characteristic, complemented
9#21447758C211yesmaximal, characteristic, complemented
10#21447747C211yesmaximal, characteristic, complemented
11#21447668C211yesmaximal, characteristic, complemented
12C16×C22⋊D61C211yesmaximal, characteristic, complemented
13#21447525C211yesmaximal, characteristic, complemented
14C2×#16839564C211yesmaximal, characteristic, complemented
15C61×#2216C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C22×D488C2211yescharacteristic
18#16840454—21no—
19#16840403—21no—
20C2×#14519720C2211yescharacteristic
21C2×#14519720C2211yescharacteristic
22C2×C16×D61C2211yescharacteristic, complemented
23#16840163C2211yescharacteristic, complemented
24#16840147C2211yescharacteristic
25#16840102—21no—
26#16840090—21no—
27#16840071—21no—
28#16840000C2211yescharacteristic, complemented
29#16839999C2211yescharacteristic
30#16839980—21no—
31#16839912—21no—
32#16839910—21no—
33Q16×Dic61C2211yescharacteristic, complemented
34#16839844C2211yescharacteristic
35C8×C22⋊D61C2211yescharacteristic
36#16839799C2211yescharacteristic, complemented
37C2×#14519711C2211yescharacteristic
38#16839589C2211yescharacteristic, complemented
39#16839589C2211yescharacteristic, complemented
40#16839580C2211yescharacteristic, complemented
41#16839566C2211yescharacteristic, complemented
42#16839566C2211yescharacteristic, complemented
43#16839564C2211yescharacteristic
44#16839564C2211yescharacteristic
45#16839564C2211yescharacteristic
46#16839564C2211yescharacteristic
47#16839562C2211yescharacteristic
48#16839562C2211yescharacteristic
49C22⋊#11758185C2211yescharacteristic
50#16839541C2211yescharacteristic, complemented
51C16×Dic61C2211yescharacteristic, complemented
52C2×Q32×C61C2211yescharacteristic, complemented
53C22×C16×C61C2211yescharacteristic
54C61×#467C2211yescharacteristic
55C61×#467C2211yescharacteristic
56C61×#366C2211yescharacteristic, complemented
57C61×#358C2211yescharacteristic, complemented
58C61×#358C2211yescharacteristic, complemented
59C2×#11758226—21no—
60C22×D244—21no—
61#14519800—22no—
62#14519800—41no—
63C2×#11758201—21no—
64C2×#11758201—21no—
65C2×D488—22no—
66C2×D488—21no—
67C2×D488—41no—
68C2×D488C2311yescharacteristic
69C2×D488C2311yescharacteristic
70C2×C8×D61C2311yescharacteristic
71#14519772—41no—
72#14519768—21no—
73#14519768—41no—
74#14519766—21no—
75#14519762—41no—
76#14519761—41no—
77#14519753—21no—
78#14519752—21no—
79#14519752—41no—
80#14519747—41no—
81Q8×Dic61—21no—
82#14519739—21no—
83C4×C22⋊D61D411yescharacteristic
84#14519735—21no—
85#14519733—41no—
86C2×C4⋊Dic61—21no—
87C22⋊C4×D61—41no—
88#14519720—22no—
89#14519720—41no—
90C2×#11758185C2311yescharacteristic
91C2×#11758185C2311yescharacteristic
92C16×D61—41no—
93#14519711—21no—
94#14519711C2311yescharacteristic
95#14519711C2311yescharacteristic
96#14519708—21no—
97#14519708—21no—
98#14519706D411yescharacteristic
99#14519701—21no—
100#14519701—21no—
101#14519697—21no—
102#14519697—21no—
103C22⋊#92418D411yescharacteristic
104#14519689D411yescharacteristic
105C8×Dic61C2311yescharacteristic
106C2×Q16×C61C2311yescharacteristic
107C2×Q16×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22⋊Q8×C61—21no—
110C22⋊Q8×C61—21no—
111Q32×C61—41no—
112C2×C16×C61—41no—
113C2×C16×C61C2311yescharacteristic
114C2×C16×C61C2311yescharacteristic
115#107×C61C2311yescharacteristic
116#107×C61C2311yescharacteristic
117#103×C61—21no—
118#103×C61—21no—
119C23×D61—41no—
120C2×Q8×C61—21no—
121C2×Q8×C61—21no—
122C22×C4×C61C2×D411yescharacteristic
123C2×C22⋊D61D811yescharacteristic
124C22×Dic61—41no—
125#11758226—42no—
126#11758226—81no—
127C2×D244—21no—
128C2×D244—21no—
129C2×D244—42no—
130C2×D244—41no—
131C2×D244—41no—
132C2×D244—41no—
133C2×D244—81no—
134C2×C4×D61—41no—
135C2×C4×D61—41no—
136C2×C4×D61C2×D411yescharacteristic
137Q16×C61—21no—
138Q16×C61—21no—
139Q16×C61—21no—
140Q16×C61—21no—
141C2×C8×C61—21no—
142C2×C8×C61—21no—
143C2×C8×C61C2×D411yescharacteristic
144C2×C8×C61C2411yescharacteristic
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—41no—
148C4⋊C4×C61—41no—
149C22⋊C4×C61—41no—
150C22⋊C4×C61—41no—
151C22⋊Dic61D811yescharacteristic
152#11758201—42no—
153#11758201—81no—
154#11758198—41no—
155#11758198—41no—
156#11758198—41no—
157#11758198—41no—
158#11758198—41no—
159#11758198—41no—
160#11758198D811yescharacteristic
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—41no—
164C4⋊Dic61—41no—
165C4⋊Dic61—41no—
166#11758196—41no—
167#11758196—41no—
168#11758196D811yescharacteristic
169C4×Dic61—41no—
170C4×Dic61—41no—
171C4×Dic61C2×D411yescharacteristic
172C2×#92418C2×D411yescharacteristic
173C2×#92418C2×D411yescharacteristic
174D488—21no—
175D488—21no—
176D488—21no—
177D488—42no—
178D488#4112yes—
179C8×D61—21no—
180C8×D61—21no—
181C16×C61—21no—
182C16×C61—21no—
183#11758185—21no—
184#11758185—21no—
185C23×C61C2×D811yescharacteristic
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—82no—
189C22×D61—82no—
190C22×D61—161no—
191C22×D61C2×D811yescharacteristic
192Q8×C61—41no—
193Q8×C61—41no—
194Q8×C61—41no—
195Q8×C61—41no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61C2×D811yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C22⋊D61—22no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—82no—
208C2×Dic61C2×D811yescharacteristic
209C2×Dic61C2×D811yescharacteristic
210C2×Dic61C2×D811yescharacteristic
211D244—22no—
212D244—41no—
213D244—41no—
214D244—41no—
215D244—82no—
216D244—82no—
217D244—161no—
218C4×D61—21no—
219C4×D61—21no—
220C4×D61—82no—
221C4×D61—161no—
222C8×C61—21no—
223C8×C61C2×#4111yescharacteristic
224C8×C61H(5,𝔽2)11yescharacteristic
225#92418—21no—
226#92418—21no—
227#33863—611nomaximal, Sylow
228C22×C61—21no—
229C22×C61—21no—
230C22×C61#50812yes—
231C22×C61C22×D811yescharacteristic
232C2×D61—21no—
233C2×D61—21no—
234C2×D61—21no—
235C2×D61—42no—
236C2×D61—81no—
237C2×D61—81no—
238C2×D61—81no—
239C2×D61—81no—
240C2×D61—81no—
241C2×D61—164no—
242C4×C61—21no—
243C4×C61—81no—
244C4×C61—81no—
245C4×C61—81no—
246C4×C61—81no—
247C4×C61#54611yescharacteristic
248C4×C61#57711yescharacteristic
249Dic61—21no—
250Dic61—21no—
251Dic61—21no—
252Dic61—81no—
253Dic61—81no—
254#3336—611no—
255#3280—611no—
256#2242—611no—
257#2235—611no—
258#2234—611no—
259#2228—611no—
260#2217—611no—
261#2216D6111yescharacteristic, complemented
262#2211—611no—
263#2189—611no—
264#2187—611no—
265C4×SD32—611no—
266D4×C16—611no—
267#2138—611no—
268C2×#357—611no—
269C2×C61—22no—
270C2×C61#329411yescharacteristic
271C2×C61C2×#50811yescharacteristic
272C2×C61#341611yescharacteristic
273D61—41no—
274D61—82no—
275D61—161no—
276C2×#135—611no—
277C22×D8—611no—
278#552—1221no—
279#546—1221no—
280C2×Q32C2×D6111yescharacteristic, complemented
281C2×SD32—611no—
282C2×SD32—611no—
283C22×C16—611no—
284C22×C16C2×D6111yescharacteristic
285#499—611no—
286#495—611no—
287#485—1221no—
288#483—1221no—
289#479—1221no—
290#467—611no—
291#467C2×D6111yescharacteristic
292#467C2×D6111yescharacteristic
293#466—611no—
294#463—1221no—
295#450—1221no—
296#449—1221no—
297C4×Q16—611no—
298C4×D8—611no—
299C8×D4—611no—
300#427—611no—
301C2×#107—611no—
302#367—611no—
303#367—611no—
304#366C2×D6111yescharacteristic, complemented
305C4⋊C16—611no—
306#358—611no—
307#358—611no—
308#358C2×D6111yescharacteristic, complemented
309#358C2×D6111yescharacteristic, complemented
310#357—611no—
311#357—611no—
312#357—611no—
313#357—611no—
314#357—611no—
315#357—611no—
316C22⋊C16—611no—
317C22⋊C16—611no—
318C4×C16—611no—
319C61#3386311yescharacteristic, complemented, Sylow
320C2×#41—1221no—
321C22×D4—1221no—
322#135—1222no—
323#135—2441no—
324C2×Q16C22×D6111yescharacteristic
325C2×Q16C22×D6111yescharacteristic
326C2×SD16—1221no—
327C2×SD16—1221no—
328C2×D8—611no—
329C2×D8—611no—
330C2×D8—1222no—
331C2×D8—1221no—
332C2×D8—2441no—
333C22×C8—611no—
334C22×C8C22×D6111yescharacteristic
335#126—2441no—
336#125—1221no—
337#125—2441no—
338#124—1221no—
339#123—2441no—
340#123—2441no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—1221no—
344#121—1221no—
345#121—2441no—
346#120—2441no—
347C4×Q8—1221no—
348C4×D4—611no—
349C4×D4—1221no—
350#117—1221no—
351#117—2441no—
352C2×C4⋊C4—1221no—
353C2×C22⋊C4—2441no—
354Q32—41no—
355SD32—1222no—
356SD32—2441no—
357C2×C16—41no—
358C2×C16—611no—
359C2×C16—611no—
360C2×C16—2441no—
361C2×C16C22×D6111yescharacteristic
362C2×C16C22×D6111yescharacteristic
363#107—611no—
364#107—611no—
365#107—1221no—
366#107C22×D6111yescharacteristic
367#107C22×D6111yescharacteristic
368#106—1221no—
369#106—1221no—
370C4⋊C8—611no—
371#103—21no—
372#103—21no—
373#103—1221no—
374#103—1221no—
375#102—1221no—
376#102—1221no—
377C22⋊C8—611no—
378C22⋊C8—611no—
379C4×C8—611no—
380C24—2441no—
381#41—4881no—
382#41—2442no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×D4—611no—
386C2×D4—1221no—
387C2×D4—1221no—
388C2×D4—4881no—
389C2×D4—2442no—
390C2×D4—2441no—
391C2×D4—2441no—
392C2×D4—2441no—
393C22×C4—611no—
394C22×C4—2441no—
395C22×C4—2441no—
396C22×C4—2441no—
397C22×C4D4×D6111yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—21no—
401Q16—21no—
402SD16—4881no—
403SD16—2442no—
404D8—612no—
405D8—1221no—
406D8—1221no—
407D8—1221no—
408D8—2442no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—611no—
412C2×C8—611no—
413C2×C8—1221no—
414C2×C8—1221no—
415C2×C8D4×D6111yescharacteristic
416C2×C8C23×D6111yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—611no—
422C4⋊C4—1221no—
423C4⋊C4—1221no—
424C4⋊C4—2441no—
425C4⋊C4—2441no—
426C4⋊C4—2441no—
427C4⋊C4—2441no—
428C4⋊C4—2441no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—611no—
432C22⋊C4—611no—
433C22⋊C4—2441no—
434C22⋊C4—2441no—
435C22⋊C4—2441no—
436C22⋊C4—2441no—
437C22⋊C4—2441no—
438C22⋊C4—2441no—
439C42—611no—
440C42—2441no—
441C42—2441no—
442C16—21no—
443C16—21no—
444C16—1221no—
445C16—1221no—
446C23—611no—
447C23—9761no—
448C23—4882no—
449C23—4882no—
450C23—2441no—
451C23—2441no—
452C23D8×D6111yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—41no—
456Q8—41no—
457D4—1222no—
458D4—1222no—
459D4—9761no—
460D4—4882no—
461D4—4882no—
462D4—2441no—
463D4—2441no—
464D4—2441no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—611no—
472C2×C4—611no—
473C2×C4—611no—
474C2×C4—1221no—
475C2×C4—1221no—
476C2×C4—9761no—
477C2×C4—4882no—
478C2×C4—4882no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4D8×D6111yescharacteristic
482C2×C4C2×D4×D6111yescharacteristic
483C8—21no—
484C8—1221no—
485C8—1221no—
486C8#41×D6111yescharacteristic
487C8#1451982211yescharacteristic
488C22—21no—
489C22—21no—
490C22—1221no—
491C22—1221no—
492C22—1221no—
493C22—9764no—
494C22—4881no—
495C22—4881no—
496C22—4881no—
497C22—4881no—
498C22—4881no—
499C22—2442no—
500C22#1684019512yes—
501C22C2×D8×D6111yescharacteristic
502C4—21no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4—1221no—
508C4—1221no—
509C4—1221no—
510C4—4881no—
511C4—4881no—
512C4#1684040111yescharacteristic
513C4#1684056911yescharacteristic
514C2—22no—
515C2—9761no—
516C2—4882no—
517C2—2441no—
518C2#2145762711yescharacteristic
519C2C2×#1684019511yescharacteristic
520C2#2145896211yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2216C2
socleC22×C61C22×D8
prime radical#21457501C2
hypercenter#2216D61
hypocenterC61#33863
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2216

Complements

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

NH ≅ G/Ntype
1. #21458190C2semidirect
3. #21448052C2semidirect
4. #21448014C2semidirect
5. #21448013C2semidirect
6. #21447979C2semidirect
7. #21447918C2semidirect
8. #21447883C2semidirect
9. #21447758C2semidirect
10. #21447747C2semidirect
11. #21447668C2semidirect
12. C16×C22⋊D61C2semidirect
13. #21447525C2semidirect
14. C2×#16839564C2semidirect
15. C61×#2216C2semidirect
22. C2×C16×D61C22semidirect
23. #16840163C22semidirect
28. #16840000C22semidirect
33. Q16×Dic61C22semidirect
36. #16839799C22semidirect
38. #16839589C22semidirect
39. #16839589C22semidirect
40. #16839580C22semidirect
41. #16839566C22semidirect
42. #16839566C22semidirect
50. #16839541C22semidirect
51. C16×Dic61C22semidirect
52. C2×Q32×C61C22semidirect
56. C61×#366C22semidirect
57. C61×#358C22semidirect
58. C61×#358C22semidirect
261. #2216D61semidirect
280. C2×Q32C2×D61semidirect
304. #366C2×D61semidirect
308. #358C2×D61semidirect
309. #358C2×D61semidirect
319. C61#33863semidirect

Maximal subgroups

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

subgroupindex
#214581902
#214575012
#214480522
#214480142
#214480132
#214479792
#214479182
#214478832
#214477582
#214477472
#214476682
C16×C22⋊D612
#214475252
C2×#168395642
C61×#22162
#3386361