Finite Group Databasebeta

← #29665415

Subgroups — #29665415

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#21456757C211yesmaximal, characteristic, complemented
2#21455601C211yesmaximal, characteristic, complemented
3#21455300C211yesmaximal, characteristic
4#21454928C211yesmaximal, characteristic, complemented
5#21445238C212yesmaximal, complemented
6#21445231C212yesmaximal, complemented
7#21445170C212yesmaximal, complemented
8#21445163C212yesmaximal, complemented
9#21444591C211yesmaximal, characteristic, complemented
10C2×#16839505C211yesmaximal, characteristic, complemented
11C61×#1550C211yesmaximal, characteristic, complemented
12C2×#14519805—21no—
13C2×#14519774C2211yescharacteristic
14C4×#11758222C2211yescharacteristic
15#16840136C2212yes—
16#16840110C2212yes—
17#16840066—21no—
18#16840059—21no—
19#16840054C2211yescharacteristic, complemented
20#16840049C2211yescharacteristic, complemented
21#16840044C2211yescharacteristic, complemented
22#16840039C2211yescharacteristic, complemented
23#16839963—21no—
24#16839963C2212yes—
25#16839962—21no—
26#16839962—21no—
27#16839958—21no—
28#16839958C2212yes—
29#16839912—22no—
30#16839886C2212yescomplemented
31#16839827C2212yescomplemented
32C8×C22⋊D61—22no—
33#16839815C2211yescharacteristic
34#16839814—21no—
35#16839798—21no—
36C2×C4⋊#92418C2211yescharacteristic
37C4⋊C8×D61—21no—
38#16839768—21no—
39#16839756—21no—
40#16839512C2212yescomplemented
41#16839512C2212yescomplemented
42#16839505C2212yes—
43#16839505C2212yes—
44C61×#533C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#333C2212yescomplemented
48C61×#332C2212yescomplemented
49C2×#11758222D411yescharacteristic
50C22×D244—21no—
51C22×D244—21no—
52C22×D244—21no—
53#14519805—44no—
54C2×#11758190—22no—
55C2×#11758201—22no—
56C2×#11758201—22no—
57C2×D488—22no—
58C22×#92418—21no—
59C2×C8×D61—22no—
60#14519774—21no—
61#14519774—21no—
62#14519774—21no—
63#14519774C2312yes—
64C4×Dic122D411yescharacteristic
65C4×Dic122C2311yescharacteristic
66C4×D244D411yescharacteristic
67C4×D244C2311yescharacteristic
68C4×C22⋊D61—21no—
69C4×C22⋊D61—21no—
70#14519736D411yescharacteristic
71#14519734—21no—
72C42×D61—21no—
73#14519710—21no—
74#14519710—21no—
75#14519708—21no—
76#14519708—21no—
77C4⋊#92418—21no—
78C4⋊#92418D411yescharacteristic
79C4⋊#92418D411yescharacteristic
80C4⋊#92418C2311yescharacteristic
81C4⋊#92418C2311yescharacteristic
82#14519706—22no—
83#14519700—21no—
84#14519700—21no—
85#14519699—22no—
86#14519699—22no—
87#14519697—21no—
88#14519697—21no—
89#14519697—22no—
90#14519697—22no—
91C22⋊#92418D412yes—
92#14519689—22no—
93C4×#92418C2312yes—
94C8×Dic61—22no—
95C2×M16×C61—21no—
96C22×C8×C61—21no—
97C4⋊Q8×C61C2311yescharacteristic
98C4⋊Q8×C61C2311yescharacteristic
99#125×C61—21no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—21no—
102C4×Q8×C61—21no—
103C4×Q8×C61—21no—
104#117×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—21no—
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109C4⋊C8×C61C2312yes—
110C23×D61—41no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C2×C22⋊D61—21no—
117#11758222—22no—
118#11758222SD1614yescomplemented
119C2×D244—22no—
120C2×D244—22no—
121C2×D244—22no—
122C2×D244—42no—
123C2×D244—42no—
124C2×D244—41no—
125C2×D244—41no—
126C2×D244—41no—
127C2×D244—81no—
128C2×D244—81no—
129C2×D244—82no—
130C2×D244C2×D411yescharacteristic
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×Dic122C2×D411yescharacteristic
135M16×C61—42no—
136C2×C8×C61—22no—
137C2×C8×C61—22no—
138C2×C8×C61—81no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C22⋊C4×C61—41no—
146C42×C61—41no—
147C42×C61C2×D411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C42×C61#4111yescharacteristic
150C42×C61C2411yescharacteristic
151C22⋊Dic61—21no—
152#11758201—82no—
153#11758201—82no—
154#11758198—21no—
155#11758198—21no—
156C4⋊Dic61C2×D411yescharacteristic
157C4⋊Dic61C2×D411yescharacteristic
158#11758196—21no—
159#11758196—21no—
160C4×Dic61—21no—
161C4×Dic61—21no—
162C4×Dic61—21no—
163C2×#92418—21no—
164C2×#92418—21no—
165C2×#92418—42no—
166C2×#92418C2×D412yes—
167C2×#92418C2×D412yes—
168D488—44no—
169#11758190—44no—
170C8×D61—82no—
171C23×C61#12111yescharacteristic
172C22×D61—21no—
173C22×D61—42no—
174C22×D61—82no—
175C22×D61—82no—
176C22×D61—82no—
177Q8×C61—41no—
178Q8×C61—41no—
179Q8×C61—81no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61#12111yescharacteristic
187C2×C4×C61#12111yescharacteristic
188C2×C4×C61C2×SD1612yes—
189C2×C4×C61#13512yes—
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C2×#4111yescharacteristic
193C22⋊D61—24no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—21no—
197D244—21no—
198D244—42no—
199D244—42no—
200D244—42no—
201D244—42no—
202D244—82no—
203D244—82no—
204D244—82no—
205D244—82no—
206D244C2×SD1612yes—
207C4×D61—24no—
208C4×D61—41no—
209C4×D61—41no—
210C4×D61—81no—
211Dic122—21no—
212Dic122C2×SD1612yes—
213C8×C61—42no—
214C8×C61—42no—
215#92418—22no—
216#92418—22no—
217#92418—41no—
218#92418—41no—
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—81no—
235C4×C61—81no—
236C4×C61#46512yes—
237C4×C61C22×SD1611yescharacteristic
238C4×C61C2×#13511yescharacteristic
239C4×C61#57611yescharacteristic
240C4×C61#57711yescharacteristic
241Dic61—22no—
242Dic61—41no—
243Dic61—41no—
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#1570—611no—
253#1550D6111yescharacteristic, complemented
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#533C2×D6111yescharacteristic
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—611no—
284#431—1221no—
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#334—612no—
293#334—612no—
294#333C2×D6112yescomplemented
295#332C2×D6112yescomplemented
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—1222no—
311C22×C8—1221no—
312C4⋊Q8C22×D6111yescharacteristic
313C4⋊Q8C22×D6111yescharacteristic
314#127—612no—
315#127—1221no—
316#127—1221no—
317#127—1221no—
318#125—21no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8—611no—
324C4×Q8—611no—
325C4×D4—611no—
326C4×D4—611no—
327C4×D4—1221no—
328C4×D4—1221no—
329#117—21no—
330#117—611no—
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—611no—
341C4⋊C8—611no—
342C4⋊C8—611no—
343C4⋊C8—611no—
344C4⋊C8—1222no—
345C4⋊C8—1221no—
346C4⋊C8C22×D6112yes—
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—612no—
357C22⋊C8—1222no—
358C4×C8—612no—
359C4×C8—1222no—
360C24—2441no—
361#41—614no—
362#41—1222no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—611no—
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×C4#1175822211yescharacteristic
383C22×C4D4×D6111yescharacteristic
384C22×C4D4×D6111yescharacteristic
385SD16—4882no—
386SD16—4882no—
387SD16—2444no—
388D8—2444no—
389M16—42no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—81no—
393C2×C8—612no—
394C2×C8—612no—
395C2×C8—1221no—
396C2×C8—1221no—
397C2×C8—4882no—
398C2×C8—2442no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—611no—
406C4⋊C4—611no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C22⋊C4—41no—
410C22⋊C4—1221no—
411C22⋊C4—1221no—
412C22⋊C4—1221no—
413C42—41no—
414C42—1221no—
415C42—1221no—
416C42—1221no—
417C42#1175822211yescharacteristic
418C42D4×D6111yescharacteristic
419C42D4×D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—1221no—
422C23—4882no—
423C23—4882no—
424C23—4882no—
425C23—2442no—
426C23#1451975011yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—81no—
430Q8—612no—
431Q8—1221no—
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—41no—
447C2×C4—41no—
448C2×C4—1221no—
449C2×C4—1224no—
450C2×C4—1221no—
451C2×C4—1221no—
452C2×C4—4881no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4#1451975011yescharacteristic
456C2×C4#1451975011yescharacteristic
457C2×C4#1451975011yescharacteristic
458C2×C4SD16×D6112yes—
459C2×C4#1451980012yes—
460C2×C4C2×D4×D6111yescharacteristic
461C2×C4C2×D4×D6111yescharacteristic
462C2×C4C2×#1175822211yescharacteristic
463C8—42no—
464C8—42no—
465C8—1222no—
466C8—1222no—
467C8—2441no—
468C8—2441no—
469C22—21no—
470C22—21no—
471C22—1222no—
472C22—9764no—
473C22—4882no—
474C22—4882no—
475C22—4882no—
476C22—2441no—
477C22—2441no—
478C22#1683999412yes—
479C22C2×#1451975011yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—81no—
484C4—81no—
485C4—1222no—
486C4—2441no—
487C4—2441no—
488C4#1683999412yes—
489C4C2×SD16×D6111yescharacteristic
490C4C2×#1451980011yescharacteristic
491C4#1684056611yescharacteristic
492C4#1684057111yescharacteristic
493C2—22no—
494C2—9762no—
495C2—2442no—
496C2#2145535711yescharacteristic
497C2C2×#1683999411yescharacteristic
498C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1550C2
socleC22×C61C2×#121
prime radical#21455300C2
hypercenter#1550D61
hypocenterC61#19409
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
1. #21456757C2semidirect
2. #21455601C2semidirect
4. #21454928C2semidirect
5. #21445238C2semidirect
6. #21445231C2semidirect
7. #21445170C2semidirect
8. #21445163C2semidirect
9. #21444591C2semidirect
10. C2×#16839505C2semidirect
11. C61×#1550C2semidirect
19. #16840054C22semidirect
20. #16840049C22semidirect
21. #16840044C22semidirect
22. #16840039C22semidirect
30. #16839886C22semidirect
31. #16839827C22semidirect
40. #16839512C22semidirect
41. #16839512C22semidirect
47. C61×#333C22semidirect
48. C61×#332C22semidirect
118. #11758222SD16semidirect
253. #1550D61semidirect
294. #333C2×D61semidirect
295. #332C2×D61semidirect
298. C61#19409semidirect

Maximal subgroups

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

subgroupindex
#214567572
#214556012
#214553002
#214549282
#214452382
#214452312
#214451702
#214451632
#214445912
C2×#168395052
C61×#15502
#1940961