Finite Group Databasebeta

← #29836865

Subgroups — #29836865

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#21458189C211yesmaximal, characteristic, complemented
2#21457503C211yesmaximal, characteristic
3#21448050C211yesmaximal, characteristic, complemented
4#21448015C211yesmaximal, characteristic, complemented
5#21448008C211yesmaximal, characteristic, complemented
6#21447977C211yesmaximal, characteristic, complemented
7#21447912C211yesmaximal, characteristic, complemented
8#21447907C211yesmaximal, characteristic, complemented
9#21447758C211yesmaximal, characteristic, complemented
10#21447746C211yesmaximal, characteristic, complemented
11C4×#14519720C211yesmaximal, characteristic, complemented
12#21447638C211yesmaximal, characteristic, complemented
13#21447525C211yesmaximal, characteristic, complemented
14#357×D61C211yesmaximal, characteristic, complemented
15C61×#2211C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×D8×D61C2211yescharacteristic
18#16840453—21no—
19#16840405—21no—
20C2×#14519720C2211yescharacteristic
21C2×C16×D61C2211yescharacteristic
22C2×C16×D61C2211yescharacteristic, complemented
23#16840164C2211yescharacteristic, complemented
24#16840101—21no—
25#16840091—21no—
26#16840069—21no—
27#16840000C2211yescharacteristic
28#16840000C2211yescharacteristic
29#16840000C2211yescharacteristic, complemented
30#16839995C2211yescharacteristic
31#16839974—21no—
32#16839911—21no—
33#16839906—21no—
34C4×D488C2211yescharacteristic
35C8×D244C2211yescharacteristic
36#16839799C2211yescharacteristic, complemented
37#107×D61C2211yescharacteristic
38#16839590C2211yescharacteristic, complemented
39#16839589C2211yescharacteristic, complemented
40#16839588C2211yescharacteristic, complemented
41#16839566C2211yescharacteristic, complemented
42#16839566C2211yescharacteristic, complemented
43#16839566C2211yescharacteristic, complemented
44#16839564C2211yescharacteristic
45#16839564C2211yescharacteristic
46#16839563C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839562C2211yescharacteristic
49#16839541C2211yescharacteristic
50#16839541C2211yescharacteristic, complemented
51C4×#11758185C2211yescharacteristic, complemented
52C2×Q32×C61C2211yescharacteristic, complemented
53C2×SD32×C61C2211yescharacteristic
54C61×#495C2211yescharacteristic
55C4×Q16×C61C2211yescharacteristic, complemented
56C61×C4⋊C16C2211yescharacteristic, complemented
57C61×#358C2211yescharacteristic, complemented
58C61×#357C2211yescharacteristic
59C2×#11758226—21no—
60C2×D4×D61—21no—
61#14519800—22no—
62#14519800—41no—
63C2×#11758201—21no—
64C2×D488C2311yescharacteristic
65C2×#11758199—21no—
66D8×D61—22no—
67D8×D61—41no—
68C2×C8×D61C2311yescharacteristic
69C2×C8×D61C2311yescharacteristic
70#14519771—41no—
71#14519769—21no—
72#14519767—41no—
73#14519762—41no—
74#14519759—41no—
75#14519753—21no—
76#14519753—21no—
77#14519753—21no—
78#14519750—41no—
79#14519748—21no—
80#14519747—41no—
81C4×D244—21no—
82C4×D244D411yescharacteristic
83#14519735—21no—
84#14519734—41no—
85C4⋊C4×D61—21no—
86C22⋊C4×D61—41no—
87#14519720—22no—
88C2×#11758185C2311yescharacteristic
89C2×#11758185C2311yescharacteristic
90C16×D61—41no—
91C16×D61—41no—
92#14519711C2311yescharacteristic
93#14519711C2311yescharacteristic
94#14519711C2311yescharacteristic
95#14519710—21no—
96#14519709—21no—
97#14519708—21no—
98C4⋊#92418D411yescharacteristic
99#14519701—21no—
100#14519701—21no—
101#14519701—21no—
102#14519699—21no—
103#14519698—21no—
104#14519689D411yescharacteristic
105#14519689D411yescharacteristic
106C2×Q16×C61C2311yescharacteristic
107C2×Q16×C61C2311yescharacteristic
108C2×SD16×C61—21no—
109C2×D8×C61C2311yescharacteristic
110#124×C61—21no—
111C4×Q8×C61—21no—
112Q32×C61—41no—
113SD32×C61—41no—
114C2×C16×C61C2311yescharacteristic
115C2×C16×C61C2311yescharacteristic
116#107×C61C2311yescharacteristic
117#103×C61—21no—
118C4×C8×C61C2311yescharacteristic
119C23×D61—41no—
120C2×Q8×C61—21no—
121C2×Q8×C61—21no—
122C2×D4×C61—21no—
123C2×C22⋊D61—41no—
124C2×C22⋊D61—41no—
125#11758226—42no—
126#11758226—81no—
127D4×D61—42no—
128D4×D61—81no—
129C2×D244—21no—
130C2×D244—41no—
131C2×D244D811yescharacteristic
132C2×C4×D61—41no—
133C2×C4×D61—41no—
134C2×C4×D61—41no—
135C2×C4×D61C2×D411yescharacteristic
136C2×C4×D61C2×D411yescharacteristic
137Q16×C61—21no—
138Q16×C61—21no—
139Q16×C61—21no—
140Q16×C61—21no—
141SD16×C61—81no—
142D8×C61—21no—
143D8×C61—21no—
144C2×C8×C61C2×D411yescharacteristic
145C2×C8×C61C2411yescharacteristic
146C4⋊C4×C61—21no—
147C4⋊C4×C61—41no—
148C22⋊C4×C61—41no—
149C22⋊C4×C61—41no—
150C42×C61—41no—
151C42×C61C2×D411yescharacteristic
152C22⋊Dic61—41no—
153#11758201—42no—
154#11758199—42no—
155#11758198—41no—
156#11758198—41no—
157#11758198—41no—
158#11758198—41no—
159#11758198—41no—
160#11758198D811yescharacteristic
161#11758198D811yescharacteristic
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61D811yescharacteristic
166#11758196—41no—
167#11758196—41no—
168#11758196—41no—
169#11758196—41no—
170#11758196—41no—
171#11758196—41no—
172C4×Dic61—41no—
173C2×#92418C2×D411yescharacteristic
174C2×#92418C2×D411yescharacteristic
175D488—21no—
176D488#4112yes—
177C8×D61—21no—
178C8×D61—21no—
179C8×D61—21no—
180C8×D61—21no—
181C16×C61—21no—
182C16×C61—21no—
183#11758185—21no—
184#11758185—21no—
185C23×C61—41no—
186C22×D61—41no—
187C22×D61—82no—
188C22×D61—82no—
189C22×D61—161no—
190C22×D61C2×D811yescharacteristic
191C22×D61C2×D811yescharacteristic
192Q8×C61—41no—
193Q8×C61—41no—
194Q8×C61—41no—
195Q8×C61—41no—
196D4×C61—41no—
197D4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61C2×D811yescharacteristic
202C2×C4×C61C2×D811yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C22⋊D61—82no—
205C22⋊D61—161no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61C2×D811yescharacteristic
210C2×Dic61C2×D811yescharacteristic
211D244—22no—
212D244—22no—
213D244—41no—
214D244—82no—
215C4×D61—21no—
216C4×D61—21no—
217C4×D61—21no—
218C4×D61—21no—
219C4×D61—82no—
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—81no—
229C22×C61—81no—
230C22×C61—81no—
231C22×C61C22×D811yescharacteristic
232C2×D61—21no—
233C2×D61—21no—
234C2×D61—21no—
235C2×D61—21no—
236C2×D61—21no—
237C2×D61—42no—
238C2×D61—81no—
239C2×D61—81no—
240C2×D61—164no—
241C2×D61#50812yes—
242C4×C61—21no—
243C4×C61—21no—
244C4×C61—81no—
245C4×C61—81no—
246C4×C61—81no—
247C4×C61#54611yescharacteristic
248C4×C61#57711yescharacteristic
249Dic61—21no—
250Dic61—21no—
251Dic61—81no—
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#2216—611no—
262#2211D6111yescharacteristic, complemented
263#2189—611no—
264#2187—611no—
265C4×SD32—611no—
266D4×C16—611no—
267#2138—611no—
268C2×#357—611no—
269C2×C61—161no—
270C2×C61#329411yescharacteristic
271C2×C61C2×#50811yescharacteristic
272C2×C61#341611yescharacteristic
273D61—22no—
274D61—41no—
275D61—82no—
276C2×#135—611no—
277C22×D8—611no—
278#552—1221no—
279#546—1221no—
280C2×Q32C2×D6111yescharacteristic, complemented
281C2×SD32—611no—
282C2×SD32C2×D6111yescharacteristic
283C22×C16—611no—
284C22×C16—611no—
285#499—611no—
286#495C2×D6111yescharacteristic
287#485—1221no—
288#483—1221no—
289#479—1221no—
290#467—611no—
291#467—611no—
292#467—611no—
293#466—611no—
294#463—1221no—
295#450—1221no—
296#449—1221no—
297C4×Q16C2×D6111yescharacteristic, complemented
298C4×D8—611no—
299C8×D4—611no—
300#427—611no—
301C2×#107—611no—
302#367—611no—
303#367—611no—
304#366—611no—
305C4⋊C16C2×D6111yescharacteristic, complemented
306#358—611no—
307#358—611no—
308#358—611no—
309#358C2×D6111yescharacteristic, complemented
310#357—611no—
311#357—611no—
312#357—611no—
313#357—611no—
314#357—611no—
315#357C2×D6111yescharacteristic
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—21no—
327C2×SD16—1221no—
328C2×D8—611no—
329C2×D8—1221no—
330C2×D8—1222no—
331C2×D8—2441no—
332C2×D8C22×D6111yescharacteristic
333C22×C8—611no—
334C22×C8—611no—
335#126—2441no—
336#125—1221no—
337#125—2441no—
338#124—21no—
339#123—2441no—
340#123—2441no—
341C22⋊Q8—1221no—
342C22⋊Q8—1221no—
343C22⋊Q8—1221no—
344#121—1221no—
345#121—2441no—
346#120—2441no—
347C4×Q8—21no—
348C4×D4—611no—
349C4×D4—1221no—
350#117—1221no—
351#117—2441no—
352C2×C4⋊C4—1221no—
353C2×C22⋊C4—2441no—
354Q32—41no—
355SD32—41no—
356SD32—1222no—
357C2×C16—611no—
358C2×C16—611no—
359C2×C16—2441no—
360C2×C16—2441no—
361C2×C16C22×D6111yescharacteristic
362C2×C16C22×D6111yescharacteristic
363#107—611no—
364#107—611no—
365#107—611no—
366#107—1221no—
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×C8C22×D6111yescharacteristic
380C24—2441no—
381#41—4881no—
382#41—2442no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×D4—21no—
386C2×D4—611no—
387C2×D4—1221no—
388C2×D4—4881no—
389C2×D4—2441no—
390C2×D4—2441no—
391C2×D4—2442no—
392C2×D4—2441no—
393C22×C4—611no—
394C22×C4—611no—
395C22×C4—2441no—
396C22×C4—2441no—
397C22×C4—2441no—
398Q16—21no—
399Q16—21no—
400Q16—21no—
401Q16—21no—
402SD16—81no—
403SD16—2442no—
404D8—21no—
405D8—21no—
406D8—612no—
407D8—1221no—
408D8—2442no—
409C2×C8—611no—
410C2×C8—611no—
411C2×C8—1221no—
412C2×C8—1221no—
413C2×C8—1221no—
414C2×C8—1221no—
415C2×C8D4×D6111yescharacteristic
416C2×C8C23×D6111yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—41no—
419C4⋊C4—611no—
420C4⋊C4—1221no—
421C4⋊C4—1221no—
422C4⋊C4—1221no—
423C4⋊C4—2441no—
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—41no—
440C42—2441no—
441C42D4×D6111yescharacteristic
442C16—21no—
443C16—21no—
444C16—1221no—
445C16—1221no—
446C23—41no—
447C23—611no—
448C23—611no—
449C23—9761no—
450C23—4882no—
451C23—4882no—
452C23—2441no—
453Q8—41no—
454Q8—41no—
455Q8—41no—
456Q8—41no—
457D4—41no—
458D4—41no—
459D4—1222no—
460D4—1222no—
461D4—9761no—
462D4—4882no—
463D4—4882no—
464D4—2441no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—611no—
469C2×C4—611no—
470C2×C4—1221no—
471C2×C4—1221no—
472C2×C4—1221no—
473C2×C4—1221no—
474C2×C4—9761no—
475C2×C4—4882no—
476C2×C4—4882no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4D8×D6111yescharacteristic
481C2×C4D8×D6111yescharacteristic
482C2×C4C2×D4×D6111yescharacteristic
483C8—21no—
484C8—1221no—
485C8—1221no—
486C8#41×D6111yescharacteristic
487C8#1451982211yescharacteristic
488C22—81no—
489C22—81no—
490C22—81no—
491C22—612no—
492C22—1221no—
493C22—1221no—
494C22—1221no—
495C22—1221no—
496C22—1221no—
497C22—9764no—
498C22—4881no—
499C22—4881no—
500C22—2442no—
501C22C2×D8×D6111yescharacteristic
502C4—21no—
503C4—21no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4—1221no—
508C4—1221no—
509C4—4881no—
510C4—4881no—
511C4—4881no—
512C4#1684040911yescharacteristic
513C4#1684057211yescharacteristic
514C2—161no—
515C2—1222no—
516C2—4882no—
517C2—2441no—
518C2#2145763511yescharacteristic
519C2#508×D6111yescharacteristic
520C2#2145896411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2211C2
socleC22×C61C22×D8
prime radical#21457503C2
hypercenter#2211D61
hypocenterC61#33863
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2211

Complements

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

NH ≅ G/Ntype
1. #21458189C2semidirect
3. #21448050C2semidirect
4. #21448015C2semidirect
5. #21448008C2semidirect
6. #21447977C2semidirect
7. #21447912C2semidirect
8. #21447907C2semidirect
9. #21447758C2semidirect
10. #21447746C2semidirect
11. C4×#14519720C2semidirect
12. #21447638C2semidirect
13. #21447525C2semidirect
14. #357×D61C2semidirect
15. C61×#2211C2semidirect
22. C2×C16×D61C22semidirect
23. #16840164C22semidirect
29. #16840000C22semidirect
36. #16839799C22semidirect
38. #16839590C22semidirect
39. #16839589C22semidirect
40. #16839588C22semidirect
41. #16839566C22semidirect
42. #16839566C22semidirect
43. #16839566C22semidirect
50. #16839541C22semidirect
51. C4×#11758185C22semidirect
52. C2×Q32×C61C22semidirect
55. C4×Q16×C61C22semidirect
56. C61×C4⋊C16C22semidirect
57. C61×#358C22semidirect
262. #2211D61semidirect
280. C2×Q32C2×D61semidirect
297. C4×Q16C2×D61semidirect
305. C4⋊C16C2×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
#214581892
#214575032
#214480502
#214480152
#214480082
#214479772
#214479122
#214479072
#214477582
#214477462
C4×#145197202
#214476382
#214475252
#357×D612
C61×#22112
#3386361