Finite Group Databasebeta

← #29836856

Subgroups — #29836856

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#21458183C211yesmaximal, characteristic, complemented
2#21457501C211yesmaximal, characteristic
3#21448052C211yesmaximal, characteristic, complemented
4#21448012C211yesmaximal, characteristic, complemented
5#21447975C211yesmaximal, characteristic, complemented
6#21447916C211yesmaximal, characteristic, complemented
7#21447908C211yesmaximal, characteristic, complemented
8#21447884C211yesmaximal, characteristic, complemented
9#21447762C211yesmaximal, characteristic, complemented
10#21447750C211yesmaximal, characteristic, complemented
11#21447670C211yesmaximal, characteristic, complemented
12#21447636C211yesmaximal, characteristic, complemented
13#21447525C211yesmaximal, characteristic, complemented
14C2×#16839562C211yesmaximal, characteristic, complemented
15C61×#2235C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C22×D488C2211yescharacteristic
18#16840447—21no—
19#16840403—21no—
20C2×#14519723C2211yescharacteristic, complemented
21C2×#14519722C2211yescharacteristic
22C2×#14519722C2211yescharacteristic
23C22×#11758185C2211yescharacteristic
24C2×C16×D61C2211yescharacteristic, complemented
25#16840163C2211yescharacteristic, complemented
26#16840147C2211yescharacteristic
27#16840105—21no—
28#16840092—21no—
29#16840068—21no—
30#16840004C2211yescharacteristic, complemented
31#16840003C2211yescharacteristic
32#16839999C2211yescharacteristic
33#16839980—21no—
34#16839912—21no—
35#16839910—21no—
36#16839856C2211yescharacteristic, complemented
37#16839844C2211yescharacteristic
38C8×C22⋊D61C2211yescharacteristic
39#16839799C2211yescharacteristic, complemented
40#16839587C2211yescharacteristic, complemented
41#16839580C2211yescharacteristic, complemented
42#16839566C2211yescharacteristic, complemented
43#16839565C2211yescharacteristic, complemented
44#16839564C2211yescharacteristic
45#16839564C2211yescharacteristic
46#16839562C2211yescharacteristic
47#16839562C2211yescharacteristic
48#16839562C2211yescharacteristic
49#16839562C2211yescharacteristic
50#16839541C2211yescharacteristic, complemented
51C16×Dic61C2211yescharacteristic, complemented
52C61×#467C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C61×#367C2211yescharacteristic, complemented
55C61×#367C2211yescharacteristic, complemented
56C61×#358C2211yescharacteristic, complemented
57C61×#358C2211yescharacteristic, complemented
58C61×C22⋊C16C2211yescharacteristic
59C2×#11758226—21no—
60C22×D244—21no—
61#14519800—22no—
62#14519800—41no—
63C2×Dic244C2311yescharacteristic
64C2×#11758201—21no—
65C2×#11758201—21no—
66C2×D488—22no—
67C2×D488—21no—
68C2×D488—41no—
69C2×D488C2311yescharacteristic
70C2×D488C2311yescharacteristic
71C2×C8×D61C2311yescharacteristic
72#14519772—41no—
73#14519768—21no—
74#14519767—41no—
75#14519766—21no—
76#14519762—41no—
77#14519760—41no—
78#14519757—21no—
79#14519756—21no—
80#14519752—21no—
81#14519752—41no—
82#14519747—41no—
83#14519742—21no—
84#14519739—21no—
85C4×C22⋊D61D411yescharacteristic
86#14519736—41no—
87#14519735—21no—
88C22⋊C4×D61—41no—
89#14519723—41no—
90#14519722—22no—
91#14519722—41no—
92C2×#11758185—41no—
93C2×#11758185C2311yescharacteristic
94C2×#11758185C2311yescharacteristic
95C16×D61—41no—
96#14519711C2311yescharacteristic
97#14519708—21no—
98#14519708—21no—
99#14519706D411yescharacteristic
100#14519702—21no—
101#14519700—21no—
102#14519700—21no—
103#14519697—21no—
104#14519697—21no—
105C22⋊#92418D411yescharacteristic
106#14519689D411yescharacteristic
107C8×Dic61C2311yescharacteristic
108C2×Q16×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22⋊Q8×C61—21no—
111C2×C4⋊C4×C61—21no—
112C2×C16×C61C2311yescharacteristic
113C2×C16×C61C2311yescharacteristic
114#107×C61—21no—
115#107×C61C2311yescharacteristic
116#107×C61C2311yescharacteristic
117#107×C61C2311yescharacteristic
118#103×C61—21no—
119C23×D61—41no—
120C2×Q8×C61—21no—
121C22×C4×C61—41no—
122C22×C4×C61C2×D411yescharacteristic
123C2×C22⋊D61D811yescharacteristic
124#11758226—42no—
125#11758226—81no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—42no—
129C2×D244—41no—
130C2×D244—41no—
131C2×D244—41no—
132C2×D244—81no—
133C2×C4×D61—41no—
134C2×C4×D61—41no—
135C2×C4×D61C2×D411yescharacteristic
136C2×Dic122—21no—
137Q16×C61—21no—
138Q16×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61—21no—
141C2×C8×C61C2×D411yescharacteristic
142C2×C8×C61C2411yescharacteristic
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—41no—
147C4⋊C4×C61—41no—
148C22⋊C4×C61—41no—
149C22⋊Dic61—41no—
150C22⋊Dic61D811yescharacteristic
151#11758201—42no—
152#11758201—81no—
153#11758198—41no—
154#11758198—41no—
155#11758198—41no—
156#11758198—41no—
157#11758198—41no—
158#11758198—41no—
159#11758198D811yescharacteristic
160C4⋊Dic61—21no—
161C4⋊Dic61—41no—
162C4⋊Dic61—41no—
163#11758196—41no—
164#11758196—41no—
165#11758196—41no—
166#11758196D811yescharacteristic
167C4×Dic61—41no—
168C4×Dic61—41no—
169C4×Dic61C2×D411yescharacteristic
170C2×#92418C2×D411yescharacteristic
171C2×#92418C2×D411yescharacteristic
172Dic244—21no—
173Dic244—21no—
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—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—82no—
201C2×C4×C61C2×D811yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C22⋊D61—22no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61C2×D811yescharacteristic
207C2×Dic61C2×D811yescharacteristic
208C2×Dic61C2×D811yescharacteristic
209D244—22no—
210D244—41no—
211D244—41no—
212D244—41no—
213D244—82no—
214D244—82no—
215D244—161no—
216C4×D61—21no—
217C4×D61—21no—
218C4×D61—82no—
219C4×D61—161no—
220Dic122—41no—
221Dic122—41no—
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#2235D6111yescharacteristic, complemented
258#2234—611no—
259#2228—611no—
260#2217—611no—
261#2216—611no—
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×Q32—611no—
281C2×SD32—611no—
282C2×SD32—611no—
283C22×C16—611no—
284C22×C16—611no—
285#499—611no—
286#495—611no—
287#485—1221no—
288#483—1221no—
289#479—1221no—
290#467—611no—
291#467—611no—
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×#107C2×D6111yescharacteristic
302#367C2×D6111yescharacteristic, complemented
303#367C2×D6111yescharacteristic, complemented
304#366—611no—
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⋊C16C2×D6111yescharacteristic
318C4×C16—611no—
319C61#3386311yescharacteristic, complemented, Sylow
320C2×#41—1221no—
321C22×D4—1221no—
322#135—1222no—
323#135—2441no—
324C2×Q16—611no—
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—1221no—
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—21no—
353C2×C22⋊C4—2441no—
354Q32—2441no—
355SD32—1222no—
356SD32—2441no—
357C2×C16—611no—
358C2×C16—611no—
359C2×C16—2441no—
360C2×C16—2441no—
361C2×C16C22×D6111yescharacteristic
362C2×C16C22×D6111yescharacteristic
363#107—21no—
364#107—611no—
365#107C22×D6111yescharacteristic
366#107C22×D6111yescharacteristic
367#107C22×D6111yescharacteristic
368#106—1221no—
369#106—1221no—
370C4⋊C8—611no—
371#103—21no—
372#103—1221no—
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—1221no—
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—41no—
394C22×C4—611no—
395C22×C4—2441no—
396C22×C4—2441no—
397C22×C4D4×D6111yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—1221no—
401Q16—1221no—
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—21no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—611no—
423C4⋊C4—1221no—
424C4⋊C4—2441no—
425C4⋊C4—2441no—
426C4⋊C4—2441no—
427C4⋊C4—2441no—
428C4⋊C4—2441no—
429C22⋊C4—41no—
430C22⋊C4—611no—
431C22⋊C4—611no—
432C22⋊C4—2441no—
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—2441no—
456Q8—2441no—
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—82no—
472C2×C4—611no—
473C2×C4—611no—
474C2×C4—611no—
475C2×C4—1221no—
476C2×C4—1221no—
477C2×C4—9761no—
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#1451982311yescharacteristic
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#1684019312yes—
501C22C2×D8×D6111yescharacteristic
502C4—21no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4—1221no—
508C4—1221no—
509C4—1221no—
510C4—4881no—
511C4—4881no—
512C4#1684040411yescharacteristic
513C4#1684057111yescharacteristic
514C2—22no—
515C2—9761no—
516C2—4882no—
517C2—2441no—
518C2#2145763011yescharacteristic
519C2C2×#1684019311yescharacteristic
520C2#2145896411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2235C2
socleC22×C61C22×D8
prime radical#21457501C2
hypercenter#2235D61
hypocenterC61#33863
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2235

Complements

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

NH ≅ G/Ntype
1. #21458183C2semidirect
3. #21448052C2semidirect
4. #21448012C2semidirect
5. #21447975C2semidirect
6. #21447916C2semidirect
7. #21447908C2semidirect
8. #21447884C2semidirect
9. #21447762C2semidirect
10. #21447750C2semidirect
11. #21447670C2semidirect
12. #21447636C2semidirect
13. #21447525C2semidirect
14. C2×#16839562C2semidirect
15. C61×#2235C2semidirect
20. C2×#14519723C22semidirect
24. C2×C16×D61C22semidirect
25. #16840163C22semidirect
30. #16840004C22semidirect
36. #16839856C22semidirect
39. #16839799C22semidirect
40. #16839587C22semidirect
41. #16839580C22semidirect
42. #16839566C22semidirect
43. #16839565C22semidirect
50. #16839541C22semidirect
51. C16×Dic61C22semidirect
54. C61×#367C22semidirect
55. C61×#367C22semidirect
56. C61×#358C22semidirect
57. C61×#358C22semidirect
257. #2235D61semidirect
302. #367C2×D61semidirect
303. #367C2×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
#214581832
#214575012
#214480522
#214480122
#214479752
#214479162
#214479082
#214478842
#214477622
#214477502
#214476702
#214476362
#214475252
C2×#168395622
C61×#22352
#3386361