Finite Group Databasebeta

← #29665418

Subgroups — #29665418

Order 15616.

704 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456762C211yesmaximal, characteristic, complemented
2#21455609C211yesmaximal, characteristic, complemented
3#21455300C211yesmaximal, characteristic
4#21454927C211yesmaximal, characteristic, complemented
5#21445236C212yesmaximal, complemented
6#21445229C212yesmaximal, complemented
7#21445168C212yesmaximal, complemented
8#21445161C212yesmaximal, complemented
9#21444510C211yesmaximal, characteristic, complemented
10C2×#16839505C211yesmaximal, characteristic, complemented
11C61×#1570C211yesmaximal, characteristic, complemented
12C2×#14519805—21no—
13#16840319C2211yescharacteristic
14C2×#14519774C2211yescharacteristic
15C4×#11758222C2211yescharacteristic
16#16840135C2212yes—
17#16840109C2212yes—
18#16840070—21no—
19#16840063—21no—
20#16840058C2211yescharacteristic, complemented
21#16840053C2211yescharacteristic, complemented
22#16840048C2211yescharacteristic, complemented
23#16840043C2211yescharacteristic, complemented
24#16839963—21no—
25#16839963—21no—
26#16839963—21no—
27#16839963C2212yes—
28#16839958—21no—
29#16839958C2212yes—
30#16839915—22no—
31C8×Dic122C2212yescomplemented
32C8×D244C2212yescomplemented
33#16839825—22no—
34#16839814—21no—
35#16839800—21no—
36C2×C4⋊#92418C2211yescharacteristic
37C4⋊C8×D61—21no—
38#16839768—21no—
39#16839759—21no—
40#16839510C2212yescomplemented
41#16839507C2212yescomplemented
42#16839505C2212yes—
43#16839505C2212yes—
44C61×#431C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#334C2212yescomplemented
48C61×#334C2212yescomplemented
49C2×#11758222D411yescharacteristic
50C22×D244—21no—
51C22×D244—21no—
52C22×D244—21no—
53#14519805—44no—
54C2×#11758190—22no—
55C2×#11758190—22no—
56C2×#11758190—22no—
57C2×D488—22no—
58C22×#92418—21no—
59C2×C8×D61—22no—
60#14519777C2311yescharacteristic
61#14519777C2311yescharacteristic
62#14519774—21no—
63#14519774—21no—
64#14519774—21no—
65#14519774C2312yes—
66#14519769—21no—
67#14519756—21no—
68#14519756—21no—
69C4×Dic122—21no—
70C4×Dic122—21no—
71C4×Dic122D411yescharacteristic
72C4×Dic122C2311yescharacteristic
73C4×D244D411yescharacteristic
74C4×D244C2311yescharacteristic
75C4×C22⋊D61—21no—
76C4×C22⋊D61—21no—
77#14519736—21no—
78#14519736D411yescharacteristic
79#14519734—21no—
80C42×D61—21no—
81#14519711—21no—
82#14519711—21no—
83#14519709—21no—
84#14519709—21no—
85C4⋊#92418—21no—
86C4⋊#92418C2312yes—
87#14519706—22no—
88#14519702—21no—
89#14519702—21no—
90#14519699—21no—
91#14519699—21no—
92#14519699—22no—
93#14519699—22no—
94#14519699—22no—
95#14519699—22no—
96#14519689—22no—
97C8×Dic61—22no—
98C2×M16×C61—21no—
99C22×C8×C61—21no—
100C2×C42×C61C2311yescharacteristic
101C4⋊C8×C61—21no—
102C4⋊C8×C61—21no—
103C4⋊C8×C61D411yescharacteristic
104C4⋊C8×C61D411yescharacteristic
105C4⋊C8×C61C2312yes—
106C4⋊C8×C61C2311yescharacteristic
107C4⋊C8×C61C2311yescharacteristic
108C22⋊C8×C61D412yes—
109C4×C8×C61C2312yes—
110C23×D61—41no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C2×C22⋊D61—21no—
115#11758222—22no—
116#11758222SD1614yescomplemented
117C2×D244—22no—
118C2×D244—22no—
119C2×D244—22no—
120C2×D244—42no—
121C2×D244—42no—
122C2×D244—41no—
123C2×D244—41no—
124C2×D244—41no—
125C2×D244—81no—
126C2×D244—81no—
127C2×D244—82no—
128C2×D244C2×D411yescharacteristic
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×Dic122—21no—
133C2×Dic122—21no—
134C2×Dic122C2×D411yescharacteristic
135M16×C61—42no—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61—22no—
139C2×C8×C61—42no—
140C2×C8×C61C2×D412yes—
141C2×C8×C61C2×D412yes—
142C42×C61C2×D411yescharacteristic
143C42×C61C2×D411yescharacteristic
144C42×C61#4111yescharacteristic
145C42×C61C2411yescharacteristic
146C22⋊Dic61—21no—
147C22⋊Dic61—41no—
148#11758198—21no—
149#11758198—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61C2×D411yescharacteristic
155C4⋊Dic61C2×D411yescharacteristic
156#11758196—21no—
157#11758196—21no—
158#11758196—41no—
159#11758196—41no—
160C4×Dic61—21no—
161C4×Dic61—21no—
162C4×Dic61—21no—
163C4×Dic61—41no—
164C2×#92418—22no—
165C2×#92418—81no—
166D488—44no—
167#11758190—44no—
168#11758190—82no—
169#11758190—82no—
170C8×D61—82no—
171C23×C61#12111yescharacteristic
172C22×D61—21no—
173C22×D61—42no—
174C22×D61—82no—
175C22×D61—82no—
176C22×D61—82no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—41no—
180C2×C4×C61#12111yescharacteristic
181C2×C4×C61#12111yescharacteristic
182C2×C4×C61#12111yescharacteristic
183C2×C4×C61C2×SD1612yes—
184C2×C4×C61#13512yes—
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C61C22×D411yescharacteristic
187C2×C4×C61C2×#4111yescharacteristic
188C22⋊D61—24no—
189C2×Dic61—21no—
190C2×Dic61—21no—
191C2×Dic61—21no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194D244—21no—
195D244—42no—
196D244—42no—
197D244—42no—
198D244—42no—
199D244—82no—
200D244—82no—
201D244—82no—
202D244—82no—
203D244C2×SD1612yes—
204C4×D61—24no—
205C4×D61—41no—
206C4×D61—41no—
207C4×D61—81no—
208Dic122—21no—
209Dic122—41no—
210Dic122—41no—
211Dic122—81no—
212Dic122C2×SD1612yes—
213C8×C61—22no—
214C8×C61—22no—
215C8×C61—42no—
216C8×C61—41no—
217C8×C61—41no—
218#92418—42no—
219#19409—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61#46512yes—
223C22×C61C2×#12111yescharacteristic
224C2×D61—22no—
225C2×D61—41no—
226C2×D61—41no—
227C2×D61—82no—
228C2×D61—82no—
229C2×D61—82no—
230C2×D61—164no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61#46512yes—
235C4×C61C22×SD1611yescharacteristic
236C4×C61C2×#13511yescharacteristic
237C4×C61#57611yescharacteristic
238C4×C61#57711yescharacteristic
239Dic61—22no—
240Dic61—41no—
241Dic61—41no—
242Dic61—81no—
243Dic61—81no—
244#3208—611no—
245#3076—611no—
246#3038—611no—
247#2988—611no—
248#1678—612no—
249#1677—612no—
250#1666—612no—
251#1665—612no—
252#1570D6111yescharacteristic, complemented
253#1550—611no—
254C2×#331—611no—
255C2×C61—22no—
256C2×C61#304511yescharacteristic
257C2×C61C2×#46511yescharacteristic
258C2×C61#306811yescharacteristic
259D61—42no—
260D61—162no—
261C2×AGL(1,ℤ/8)—1221no—
262#533—611no—
263C2×#127—611no—
264C4×#41—611no—
265#492—612no—
266#486—612no—
267#479—1221no—
268#478—1221no—
269#477—611no—
270#476—611no—
271#475—611no—
272#474—611no—
273#460—612no—
274#460—1221no—
275#460—1221no—
276#460—1221no—
277#459—612no—
278#459—1221no—
279#450—1222no—
280C8×Q8—612no—
281C8×D4—612no—
282C8×D4—1222no—
283#431—1221no—
284#431C2×D6111yescharacteristic
285#427—1221no—
286#423C2×D6111yescharacteristic
287C2×C4⋊C8—611no—
288C2×C4⋊C8—1221no—
289C2×C4⋊C8C2×D6111yescharacteristic
290#418—1221no—
291#416—1221no—
292#334C2×D6112yescomplemented
293#334C2×D6112yescomplemented
294#333—612no—
295#332—612no—
296#331—612no—
297#331—612no—
298C61#1940911yescharacteristic, complemented, Sylow
299C2×#41—611no—
300C22×D4—1221no—
301C22×D4—1221no—
302C22×D4—1221no—
303AGL(1,ℤ/8)—2444no—
304C2×SD16—1222no—
305C2×SD16—1222no—
306C2×SD16—1222no—
307C2×D8—1222no—
308C2×M16—21no—
309C22×C8—21no—
310C22×C8—1221no—
311C22×C8—1222no—
312C4⋊Q8—611no—
313C4⋊Q8—611no—
314#127—612no—
315#127—1221no—
316#127—1221no—
317#127—1221no—
318#125—1221no—
319C22⋊Q8—1221no—
320C22⋊Q8—1221no—
321C4×Q8—611no—
322C4×Q8—611no—
323C4×Q8—1221no—
324C4×Q8—1221no—
325C4×D4—611no—
326C4×D4—611no—
327C4×D4—1221no—
328C4×D4—1221no—
329#117—611no—
330#117—1221no—
331#117—1221no—
332C2×C42—1221no—
333C2×C42C22×D6111yescharacteristic
334#107—1221no—
335#107—1221no—
336#106—1221no—
337#106—1221no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—612no—
341C4⋊C8—1222no—
342C4⋊C8—1221no—
343C4⋊C8D24411yescharacteristic
344C4⋊C8D24411yescharacteristic
345C4⋊C8C22×D6111yescharacteristic
346C4⋊C8C22×D6111yescharacteristic
347C4⋊C8C22×D6112yes—
348#103—1221no—
349#103—1221no—
350#102—1222no—
351#102—1222no—
352#102—1221no—
353#102—1221no—
354#102—1222no—
355#102—1222no—
356C22⋊C8—1222no—
357C22⋊C8D24412yes—
358C4×C8—1222no—
359C4×C8C22×D6112yes—
360C24—2441no—
361#41—614no—
362#41—1222no—
363C2×Q8—611no—
364C2×Q8—1221no—
365C2×Q8—1221no—
366C2×D4—611no—
367C2×D4—1221no—
368C2×D4—1222no—
369C2×D4—1222no—
370C2×D4—1222no—
371C2×D4—4881no—
372C2×D4—4881no—
373C2×D4—4882no—
374C2×D4—2441no—
375C2×D4—2441no—
376C2×D4—2441no—
377C2×D4—2442no—
378C2×D4—2442no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4C2×D24411yescharacteristic
383C22×C4D4×D6111yescharacteristic
384C22×C4#1175822611yescharacteristic
385SD16—4882no—
386SD16—4882no—
387SD16—2444no—
388D8—2444no—
389M16—42no—
390C2×C8—22no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—42no—
394C2×C8—1222no—
395C2×C8—4882no—
396C2×C8—4881no—
397C2×C8C2×D24412yes—
398C2×C8C2×D24412yes—
399C4⋊C4—611no—
400C4⋊C4—611no—
401C4⋊C4—1221no—
402C4⋊C4—1221no—
403C4⋊C4—1221no—
404C4⋊C4—1221no—
405C4⋊C4—1221no—
406C4⋊C4—1221no—
407C4⋊C4—2441no—
408C4⋊C4—2441no—
409C22⋊C4—1221no—
410C22⋊C4—1221no—
411C22⋊C4—1221no—
412C22⋊C4—2441no—
413C42—1221no—
414C42—1221no—
415C42—1221no—
416C42—2441no—
417C42C2×D24411yescharacteristic
418C42D4×D6111yescharacteristic
419C42#1175822611yescharacteristic
420C42C23×D6111yescharacteristic
421C23—1221no—
422C23—4882no—
423C23—4882no—
424C23—4882no—
425C23—2442no—
426C23#1451975211yescharacteristic
427Q8—612no—
428Q8—1221no—
429Q8—4881no—
430Q8—2441no—
431Q8—2441no—
432D4—612no—
433D4—1224no—
434D4—1221no—
435D4—4882no—
436D4—4882no—
437D4—4882no—
438D4—4882no—
439D4—2442no—
440D4—2442no—
441D4—2442no—
442D4—2442no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—41no—
446C2×C4—1221no—
447C2×C4—1224no—
448C2×C4—1221no—
449C2×C4—1221no—
450C2×C4—4881no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4#1451975211yescharacteristic
456C2×C4#1451975211yescharacteristic
457C2×C4#1451975211yescharacteristic
458C2×C4SD16×D6112yes—
459C2×C4#1451979812yes—
460C2×C4C2×D4×D6111yescharacteristic
461C2×C4C22×D24411yescharacteristic
462C2×C4C2×#1175822611yescharacteristic
463C8—22no—
464C8—22no—
465C8—42no—
466C8—41no—
467C8—41no—
468C8—2442no—
469C22—21no—
470C22—21no—
471C22—1222no—
472C22—9764no—
473C22—4882no—
474C22—4882no—
475C22—4882no—
476C22—2441no—
477C22—2441no—
478C22#1683999212yes—
479C22C2×#1451975211yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—1222no—
484C4—4881no—
485C4—4881no—
486C4—2441no—
487C4—2441no—
488C4#1683999212yes—
489C4C2×SD16×D6111yescharacteristic
490C4C2×#1451979811yescharacteristic
491C4#1684056711yescharacteristic
492C4#1684057411yescharacteristic
493C2—22no—
494C2—9762no—
495C2—2442no—
496C2#2145536011yescharacteristic
497C2C2×#1683999211yescharacteristic
498C2#2145554811yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1570C2
socleC22×C61C2×#121
prime radical#21455300C2
hypercenter#1570D61
hypocenterC61#19409
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
1. #21456762C2semidirect
2. #21455609C2semidirect
4. #21454927C2semidirect
5. #21445236C2semidirect
6. #21445229C2semidirect
7. #21445168C2semidirect
8. #21445161C2semidirect
9. #21444510C2semidirect
10. C2×#16839505C2semidirect
11. C61×#1570C2semidirect
20. #16840058C22semidirect
21. #16840053C22semidirect
22. #16840048C22semidirect
23. #16840043C22semidirect
31. C8×Dic122C22semidirect
32. C8×D244C22semidirect
40. #16839510C22semidirect
41. #16839507C22semidirect
47. C61×#334C22semidirect
48. C61×#334C22semidirect
116. #11758222SD16semidirect
252. #1570D61semidirect
292. #334C2×D61semidirect
293. #334C2×D61semidirect
298. C61#19409semidirect

Maximal subgroups

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

subgroupindex
#214567622
#214556092
#214553002
#214549272
#214452362
#214452292
#214451682
#214451612
#214445102
C2×#168395052
C61×#15702
#1940961