Finite Group Databasebeta

← #29665414

Subgroups — #29665414

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#21455299C211yesmaximal, characteristic
3#21454930C211yesmaximal, characteristic, complemented
4#21445237C212yesmaximal, complemented
5#21445230C212yesmaximal, complemented
6#21445169C212yesmaximal, complemented
7#21445162C212yesmaximal, complemented
8#21444589C211yesmaximal, characteristic, complemented
9#21444509C211yesmaximal, characteristic, complemented
10C2×#16839505C211yesmaximal, characteristic, complemented
11C61×#3076C211yesmaximal, characteristic, complemented
12C2×#14519802—21no—
13C2×#14519774C2211yescharacteristic
14#16840136C2212yes—
15#16840110C2212yes—
16#16839962—21no—
17#16839962—21no—
18#16839962—21no—
19#16839962C2212yes—
20#16839957—21no—
21#16839957C2212yes—
22#16839912—22no—
23Q8×#92418C2212yescomplemented
24D4×#92418—22no—
25D4×#92418C2212yescomplemented
26#16839815—21no—
27#16839815C2211yescharacteristic
28#16839786C2211yescharacteristic
29C2×C4⋊#92418—21no—
30C2×C4⋊#92418C2211yescharacteristic
31#16839766—21no—
32#16839512C2212yescomplemented
33#16839512C2212yescomplemented
34#16839511C2212yescomplemented
35#16839508C2212yescomplemented
36#16839505C2212yes—
37#16839505C2212yes—
38C61×#533C2211yescharacteristic
39C4×#41×C61C2211yescharacteristic
40C61×#479—21no—
41C61×#478—21no—
42C61×#477C2211yescharacteristic, complemented
43C61×#476C2211yescharacteristic, complemented
44C61×#475C2211yescharacteristic, complemented
45C61×#474C2211yescharacteristic, complemented
46C61×#427—21no—
47C2×C4⋊C8×C61C2211yescharacteristic
48C61×#416—21no—
49C22×D244—21no—
50C22×D244—21no—
51C22×D244—21no—
52#14519802—44no—
53C2×#11758201—22no—
54C2×#11758201—22no—
55C2×#11758201—22no—
56C2×#11758199—22no—
57C2×#11758194—21no—
58C22×#92418—21no—
59C22×#92418—22no—
60#14519774—21no—
61#14519774—21no—
62#14519774—21no—
63#14519774C2312yes—
64C4⋊#92418—21no—
65C4⋊#92418—21no—
66C4⋊#92418—22no—
67C4⋊#92418D411yescharacteristic
68C4⋊#92418D411yescharacteristic
69C4⋊#92418C2312yes—
70C4⋊#92418C2312yes—
71#14519697—22no—
72#14519697—22no—
73#14519697—22no—
74#14519697—22no—
75C22⋊#92418—22no—
76C22⋊#92418D412yes—
77C4×#92418—22no—
78C4×#92418C2312yes—
79C2×#41×C61D411yescharacteristic
80C22×C8×C61—21no—
81C4⋊Q8×C61C2311yescharacteristic
82C4⋊Q8×C61C2311yescharacteristic
83#125×C61—21no—
84C22⋊Q8×C61—21no—
85C22⋊Q8×C61—21no—
86C4×Q8×C61—21no—
87C4×Q8×C61—21no—
88C4×Q8×C61D411yescharacteristic
89C4×Q8×C61C2311yescharacteristic
90C4×D4×C61—21no—
91C4×D4×C61—21no—
92C4×D4×C61D411yescharacteristic
93C4×D4×C61C2311yescharacteristic
94#117×C61—21no—
95#117×C61—21no—
96#117×C61D411yescharacteristic
97C2×C42×C61—21no—
98C2×C42×C61C2311yescharacteristic
99#107×C61—21no—
100#107×C61—21no—
101#106×C61—21no—
102#106×C61—21no—
103C4⋊C8×C61—21no—
104C4⋊C8×C61C2311yescharacteristic
105C4⋊C8×C61C2311yescharacteristic
106#103×C61—21no—
107#103×C61—21no—
108#102×C61—21no—
109#102×C61—21no—
110C23×D61—41no—
111#41×C61—22no—
112#41×C61SD1614yescomplemented
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61C2×D411yescharacteristic
116C2×D4×C61—21no—
117C2×D4×C61C2×D411yescharacteristic
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C2×D244—22no—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—42no—
128C2×D244—42no—
129C2×D244—41no—
130C2×D244—41no—
131C2×D244—41no—
132C2×D244—81no—
133C2×D244—81no—
134C2×D244—82no—
135C2×C8×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61—42no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61—41no—
146C4⋊C4×C61C2×D411yescharacteristic
147C4⋊C4×C61C2×D411yescharacteristic
148C22⋊C4×C61—21no—
149C22⋊C4×C61—21no—
150C22⋊C4×C61—21no—
151C22⋊C4×C61—41no—
152C42×C61—21no—
153C42×C61—21no—
154C42×C61—21no—
155C42×C61—41no—
156C42×C61C2×D411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C42×C61#4111yescharacteristic
159C42×C61C2411yescharacteristic
160#11758201—44no—
161#11758201—82no—
162#11758201—82no—
163#11758199—44no—
164#11758194—42no—
165C2×#92418—22no—
166C2×#92418—22no—
167C2×#92418—82no—
168C2×#92418—81no—
169C2×#92418C2×D412yes—
170C2×#92418C2×D412yes—
171C23×C61—21no—
172C23×C61#12111yescharacteristic
173C22×D61—42no—
174C22×D61—82no—
175C22×D61—82no—
176C22×D61—82no—
177Q8×C61—21no—
178Q8×C61—41no—
179Q8×C61—41no—
180Q8×C61—81no—
181Q8×C61C2×SD1612yes—
182D4×C61—24no—
183D4×C61—21no—
184D4×C61C2×SD1612yes—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—24no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—81no—
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61#12111yescharacteristic
199C2×C4×C61#12111yescharacteristic
200C2×C4×C61C2×SD1612yes—
201C2×C4×C61#13512yes—
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205D244—42no—
206D244—42no—
207D244—42no—
208D244—42no—
209D244—82no—
210D244—82no—
211D244—82no—
212D244—82no—
213C8×C61—41no—
214C8×C61—41no—
215#92418—22no—
216#92418—22no—
217#92418—42no—
218#92418—42no—
219#19409—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—22no—
223C22×C61—41no—
224C22×C61—41no—
225C22×C61#46512yes—
226C22×C61C2×#12111yescharacteristic
227C2×D61—82no—
228C2×D61—82no—
229C2×D61—82no—
230C2×D61—164no—
231C4×C61—21no—
232C4×C61—22no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—41no—
236C4×C61—41no—
237C4×C61—81no—
238C4×C61—81no—
239C4×C61#46512yes—
240C4×C61C22×SD1611yescharacteristic
241C4×C61C2×#13511yescharacteristic
242C4×C61#57611yescharacteristic
243C4×C61#57711yescharacteristic
244#3208—611no—
245#3076D6111yescharacteristic, complemented
246#3038—611no—
247#2988—611no—
248#1678—612no—
249#1677—612no—
250#1666—612no—
251#1665—612no—
252#1570—611no—
253#1550—611no—
254C2×#331—611no—
255C2×C61—22no—
256C2×C61—42no—
257C2×C61#304511yescharacteristic
258C2×C61C2×#46511yescharacteristic
259C2×C61#306811yescharacteristic
260D61—162no—
261C2×AGL(1,ℤ/8)—1221no—
262#533C2×D6111yescharacteristic
263C2×#127—611no—
264C4×#41C2×D6111yescharacteristic
265#492—612no—
266#486—612no—
267#479—21no—
268#478—21no—
269#477C2×D6111yescharacteristic, complemented
270#476C2×D6111yescharacteristic, complemented
271#475C2×D6111yescharacteristic, complemented
272#474C2×D6111yescharacteristic, complemented
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—21no—
286#423—611no—
287C2×C4⋊C8—611no—
288C2×C4⋊C8—1221no—
289C2×C4⋊C8C2×D6111yescharacteristic
290#418—1221no—
291#416—21no—
292#334—612no—
293#334—612no—
294#333—612no—
295#332—612no—
296#331—612no—
297#331—612no—
298C61#1940911yescharacteristic, complemented, Sylow
299C2×#41C22⋊D6111yescharacteristic
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—1221no—
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×Q8C22⋊D6111yescharacteristic
324C4×Q8C22×D6111yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4C22⋊D6111yescharacteristic
328C4×D4C22×D6111yescharacteristic
329#117—21no—
330#117—21no—
331#117C22⋊D6111yescharacteristic
332C2×C42—21no—
333C2×C42C22×D6111yescharacteristic
334#107—21no—
335#107—21no—
336#106—21no—
337#106—21no—
338C4⋊C8—21no—
339C4⋊C8—611no—
340C4⋊C8—611no—
341C4⋊C8—612no—
342C4⋊C8—612no—
343C4⋊C8—1222no—
344C4⋊C8—1221no—
345C4⋊C8—1221no—
346C4⋊C8C22×D6111yescharacteristic
347C4⋊C8C22×D6111yescharacteristic
348#103—21no—
349#103—21no—
350#102—21no—
351#102—21no—
352#102—1222no—
353#102—1222no—
354#102—1222no—
355#102—1222no—
356C22⋊C8—612no—
357C22⋊C8—1222no—
358C4×C8—612no—
359C4×C8—1222no—
360C24—2441no—
361#41—22no—
362#41#1175820114yescomplemented
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8C2×C22⋊D6111yescharacteristic
366C2×D4—21no—
367C2×D4—1222no—
368C2×D4—1222no—
369C2×D4—1222no—
370C2×D4—4881no—
371C2×D4—4881no—
372C2×D4—4882no—
373C2×D4—2441no—
374C2×D4—2441no—
375C2×D4—2441no—
376C2×D4—2442no—
377C2×D4—2442no—
378C2×D4C2×C22⋊D6111yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4D4×D6111yescharacteristic
383C22×C4#1175822411yescharacteristic
384C22×C4C2×C22⋊D6111yescharacteristic
385SD16—4882no—
386SD16—4882no—
387SD16—2444no—
388D8—2444no—
389M16—2442no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—42no—
393C2×C8—612no—
394C2×C8—612no—
395C2×C8—1222no—
396C2×C8—1222no—
397C2×C8—4882no—
398C2×C8—4881no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4C2×C22⋊D6111yescharacteristic
408C4⋊C4C2×C22⋊D6111yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—41no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—41no—
417C42D4×D6111yescharacteristic
418C42#1175822411yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—21no—
422C23—4882no—
423C23—4882no—
424C23—4882no—
425C23—2442no—
426C23#1451974911yescharacteristic
427Q8—21no—
428Q8—41no—
429Q8—41no—
430Q8—81no—
431Q8C2×#1175820112yes—
432D4—24no—
433D4—21no—
434D4—4882no—
435D4—4882no—
436D4—4882no—
437D4—4882no—
438D4—2442no—
439D4—2442no—
440D4—2442no—
441D4—2442no—
442D4C2×#1175820112yes—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—81no—
455C2×C4#1451974911yescharacteristic
456C2×C4#1451974911yescharacteristic
457C2×C4#1451974911yescharacteristic
458C2×C4C2×#1175820112yes—
459C2×C4#1451979712yes—
460C2×C4C2×D4×D6111yescharacteristic
461C2×C4C22×C22⋊D6111yescharacteristic
462C2×C4C2×#1175822411yescharacteristic
463C8—41no—
464C8—41no—
465C8—1222no—
466C8—1222no—
467C8—2442no—
468C8—2442no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—9764no—
475C22—4882no—
476C22—4882no—
477C22—4882no—
478C22#1683999312yes—
479C22C2×#1451974911yescharacteristic
480C4—22no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—81no—
487C4—81no—
488C4#1683999312yes—
489C4C22×#1175820111yescharacteristic
490C4C2×#1451979711yescharacteristic
491C4#1684056411yescharacteristic
492C4#1684056911yescharacteristic
493C2—22no—
494C2—42no—
495C2—9762no—
496C2#2145535611yescharacteristic
497C2C2×#1683999311yescharacteristic
498C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3076C2
socleC22×C61C2×#121
prime radical#21455299C2
hypercenter#3076D61
hypocenterC61#19409
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #21456757C2semidirect
3. #21454930C2semidirect
4. #21445237C2semidirect
5. #21445230C2semidirect
6. #21445169C2semidirect
7. #21445162C2semidirect
8. #21444589C2semidirect
9. #21444509C2semidirect
10. C2×#16839505C2semidirect
11. C61×#3076C2semidirect
23. Q8×#92418C22semidirect
25. D4×#92418C22semidirect
32. #16839512C22semidirect
33. #16839512C22semidirect
34. #16839511C22semidirect
35. #16839508C22semidirect
42. C61×#477C22semidirect
43. C61×#476C22semidirect
44. C61×#475C22semidirect
45. C61×#474C22semidirect
112. #41×C61SD16semidirect
245. #3076D61semidirect
269. #477C2×D61semidirect
270. #476C2×D61semidirect
271. #475C2×D61semidirect
272. #474C2×D61semidirect
298. C61#19409semidirect
362. #41#11758201semidirect

Maximal subgroups

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

subgroupindex
#214567572
#214552992
#214549302
#214452372
#214452302
#214451692
#214451622
#214445892
#214445092
C2×#168395052
C61×#30762
#1940961