Finite Group Databasebeta

← #29665412

Subgroups — #29665412

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#21455601C211yesmaximal, characteristic, complemented
2#21455295C211yesmaximal, characteristic
3#21454927C211yesmaximal, characteristic, complemented
4#21445232C212yesmaximal, complemented
5#21445225C212yesmaximal, complemented
6#21445164C212yesmaximal, complemented
7#21445157C212yesmaximal, complemented
8#21444590C211yesmaximal, characteristic, complemented
9#21444509C211yesmaximal, characteristic, complemented
10C2×#16839506C211yesmaximal, characteristic, complemented
11C61×#3208C211yesmaximal, characteristic, complemented
12C2×#14519803—21no—
13C4×#11758222C2211yescharacteristic
14#16840066—21no—
15#16840059—21no—
16#16840054C2211yescharacteristic, complemented
17#16840049C2211yescharacteristic, complemented
18#16840044C2211yescharacteristic, complemented
19#16840039C2211yescharacteristic, complemented
20#16839959—21no—
21#16839959C2212yes—
22#16839954—21no—
23#16839954C2212yes—
24C8×Dic122C2212yescomplemented
25C8×D244C2212yescomplemented
26#16839825—22no—
27#16839814—21no—
28#16839798—21no—
29#16839786C2211yescharacteristic
30C2×C4⋊#92418C2211yescharacteristic
31C2×C4⋊#92418C2211yescharacteristic
32C4⋊C8×D61—21no—
33#16839767—21no—
34#16839756—21no—
35#16839513C2212yescomplemented
36#16839513C2212yescomplemented
37#16839511C2212yescomplemented
38#16839508C2212yescomplemented
39#16839506C2212yes—
40#16839506C2212yes—
41C61×#533C2211yescharacteristic
42C2×#127×C61C2211yescharacteristic
43C61×#492C2212yes—
44C61×#486C2212yes—
45C61×#460—21no—
46C61×#460—21no—
47C61×#450—22no—
48C61×#431C2211yescharacteristic
49C2×#11758222D411yescharacteristic
50#14519803—44no—
51C2×#11758200—22no—
52C2×#11758199—22no—
53C2×#11758194—21no—
54C22×#92418—21no—
55C22×#92418—21no—
56C2×C8×D61—22no—
57C4×Dic122D411yescharacteristic
58C4×Dic122C2311yescharacteristic
59C4×D244D411yescharacteristic
60C4×D244C2311yescharacteristic
61C4×C22⋊D61—21no—
62C4×C22⋊D61—21no—
63#14519736D411yescharacteristic
64#14519734—21no—
65C42×D61—21no—
66#14519710—21no—
67#14519710—21no—
68#14519708—21no—
69#14519708—21no—
70C4⋊#92418—21no—
71C4⋊#92418—21no—
72C4⋊#92418—21no—
73C4⋊#92418C2312yes—
74C4⋊#92418C2312yes—
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2311yescharacteristic
77#14519706—22no—
78#14519700—21no—
79#14519700—21no—
80#14519698—22no—
81#14519698—22no—
82#14519697—21no—
83#14519697—21no—
84#14519689—22no—
85C8×Dic61—22no—
86C22×D4×C61—21no—
87C22×D4×C61—21no—
88C22×D4×C61—21no—
89C2×SD16×C61—22no—
90C2×SD16×C61—22no—
91C4⋊Q8×C61C2311yescharacteristic
92C4⋊Q8×C61C2311yescharacteristic
93#127×C61—21no—
94#127×C61—21no—
95#127×C61—21no—
96#127×C61C2312yes—
97#125×C61—21no—
98C22⋊Q8×C61—21no—
99C22⋊Q8×C61—21no—
100C4×Q8×C61—21no—
101C4×Q8×C61—21no—
102#117×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61D411yescharacteristic
105C4⋊C8×C61D411yescharacteristic
106#102×C61—22no—
107#102×C61—22no—
108C22⋊C8×C61D412yes—
109C4×C8×C61C2312yes—
110C24×C61—41no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—41no—
117C2×D4×C61—41no—
118C2×D4×C61—41no—
119C2×D4×C61—42no—
120C2×D4×C61—42no—
121C2×D4×C61—81no—
122C2×D4×C61—81no—
123C2×D4×C61—82no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61#4111yescharacteristic
127C2×C22⋊D61—21no—
128#11758222—22no—
129#11758222SD1614yescomplemented
130C2×D244C2×D411yescharacteristic
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×Dic122C2×D411yescharacteristic
135SD16×C61—82no—
136SD16×C61—82no—
137C2×C8×C61C2×D412yes—
138C2×C8×C61C2×D412yes—
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#11758200—44no—
153#11758199—44no—
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—
163#11758194—42no—
164C2×#92418—22no—
165C2×#92418—21no—
166C2×#92418—21no—
167C2×#92418—22no—
168C2×#92418—42no—
169C2×#92418—81no—
170C8×D61—82no—
171C23×C61—42no—
172C23×C61—82no—
173C23×C61—82no—
174C23×C61—82no—
175C23×C61#12111yescharacteristic
176C22×D61—21no—
177Q8×C61—41no—
178Q8×C61—41no—
179Q8×C61—81no—
180D4×C61—42no—
181D4×C61—42no—
182D4×C61—42no—
183D4×C61—42no—
184D4×C61—82no—
185D4×C61—82no—
186D4×C61—82no—
187D4×C61—82no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61#12111yescharacteristic
194C2×C4×C61#12111yescharacteristic
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61C2×SD1612yes—
197C2×C4×C61#13512yes—
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C2×#4111yescharacteristic
201C22⋊D61—24no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205D244—21no—
206D244C2×SD1612yes—
207C4×D61—24no—
208C4×D61—41no—
209C4×D61—41no—
210C4×D61—81no—
211Dic122—21no—
212Dic122C2×SD1612yes—
213C8×C61—22no—
214C8×C61—22no—
215#92418—42no—
216#92418—41no—
217#92418—41no—
218#92418—42no—
219#19409—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—82no—
223C22×C61—82no—
224C22×C61—82no—
225C22×C61—164no—
226C22×C61#46512yes—
227C22×C61C2×#12111yescharacteristic
228C2×D61—22no—
229C2×D61—41no—
230C2×D61—41no—
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#3208D6111yescharacteristic, complemented
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#1550—611no—
254C2×#331—611no—
255C2×C61—22no—
256C2×C61—162no—
257C2×C61#304511yescharacteristic
258C2×C61C2×#46511yescharacteristic
259C2×C61#306811yescharacteristic
260D61—42no—
261C2×AGL(1,ℤ/8)—1221no—
262#533C2×D6111yescharacteristic
263C2×#127C2×D6111yescharacteristic
264C4×#41—611no—
265#492C2×D6112yes—
266#486C2×D6112yes—
267#479—1221no—
268#478—1221no—
269#477—611no—
270#476—611no—
271#475—611no—
272#474—611no—
273#460—21no—
274#460—21no—
275#460—612no—
276#460—1221no—
277#459—612no—
278#459—1221no—
279#450—22no—
280C8×Q8—612no—
281C8×D4—612no—
282C8×D4—1222no—
283#431—1221no—
284#431C2×D6111yescharacteristic
285#427—1221no—
286#423—611no—
287C2×C4⋊C8—611no—
288C2×C4⋊C8—611no—
289C2×C4⋊C8—1221no—
290#418—1221no—
291#416—1221no—
292#334—612no—
293#334—612no—
294#333—612no—
295#332—612no—
296#331—612no—
297#331—612no—
298C61#1940911yescharacteristic, complemented, Sylow
299C2×#41—611no—
300C22×D4—21no—
301C22×D4—21no—
302C22×D4—21no—
303AGL(1,ℤ/8)—2444no—
304C2×SD16—22no—
305C2×SD16—22no—
306C2×SD16—1222no—
307C2×D8—1222no—
308C2×M16—1221no—
309C22×C8—1221no—
310C22×C8—1221no—
311C22×C8—1222no—
312C4⋊Q8C22×D6111yescharacteristic
313C4⋊Q8C22×D6111yescharacteristic
314#127—21no—
315#127—21no—
316#127—21no—
317#127C22×D6112yes—
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—612no—
339C4⋊C8—611no—
340C4⋊C8—611no—
341C4⋊C8—612no—
342C4⋊C8—1222no—
343C4⋊C8—1221no—
344C4⋊C8—1221no—
345C4⋊C8—1221no—
346C4⋊C8C22⋊D6111yescharacteristic
347C4⋊C8C22⋊D6111yescharacteristic
348#103—1221no—
349#103—1221no—
350#102—22no—
351#102—22no—
352#102—1221no—
353#102—1221no—
354#102—1222no—
355#102—1222no—
356C22⋊C8—1222no—
357C22⋊C8C22⋊D6112yes—
358C4×C8—1222no—
359C4×C8C22×D6112yes—
360C24—41no—
361#41—614no—
362#41—1222no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—611no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—81no—
375C2×D4—81no—
376C2×D4—82no—
377C2×D4—611no—
378C2×D4—1221no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4D4×D6111yescharacteristic
383C22×C4#1175822411yescharacteristic
384C22×C4C2×C22⋊D6111yescharacteristic
385SD16—82no—
386SD16—82no—
387SD16—2444no—
388D8—2444no—
389M16—2442no—
390C2×C8—1222no—
391C2×C8—1221no—
392C2×C8—1221no—
393C2×C8—1222no—
394C2×C8—4882no—
395C2×C8—4881no—
396C2×C8—2442no—
397C2×C8C2×C22⋊D6112yes—
398C2×C8C2×C22⋊D6112yes—
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—
417C42D4×D6111yescharacteristic
418C42#1175822411yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—42no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—1221no—
426C23#1451974811yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—81no—
430Q8—612no—
431Q8—1221no—
432D4—42no—
433D4—42no—
434D4—42no—
435D4—42no—
436D4—82no—
437D4—82no—
438D4—82no—
439D4—82no—
440D4—612no—
441D4—1224no—
442D4—1221no—
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#1451974811yescharacteristic
456C2×C4#1451974811yescharacteristic
457C2×C4#1451974811yescharacteristic
458C2×C4SD16×D6112yes—
459C2×C4#1451979812yes—
460C2×C4C2×D4×D6111yescharacteristic
461C2×C4C22×C22⋊D6111yescharacteristic
462C2×C4C2×#1175822411yescharacteristic
463C8—22no—
464C8—22no—
465C8—2442no—
466C8—2441no—
467C8—2441no—
468C8—2442no—
469C22—21no—
470C22—21no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22—164no—
475C22—1222no—
476C22—2441no—
477C22—2441no—
478C22#1683998812yes—
479C22C2×#1451974811yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—81no—
484C4—81no—
485C4—1222no—
486C4—2441no—
487C4—2441no—
488C4#1683998812yes—
489C4C2×SD16×D6111yescharacteristic
490C4C2×#1451979811yescharacteristic
491C4#1684056511yescharacteristic
492C4#1684057311yescharacteristic
493C2—22no—
494C2—162no—
495C2—2442no—
496C2#2145535411yescharacteristic
497C2C2×#1683998811yescharacteristic
498C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3208C2
socleC22×C61C2×#121
prime radical#21455295C2
hypercenter#3208D61
hypocenterC61#19409
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
1. #21455601C2semidirect
3. #21454927C2semidirect
4. #21445232C2semidirect
5. #21445225C2semidirect
6. #21445164C2semidirect
7. #21445157C2semidirect
8. #21444590C2semidirect
9. #21444509C2semidirect
10. C2×#16839506C2semidirect
11. C61×#3208C2semidirect
16. #16840054C22semidirect
17. #16840049C22semidirect
18. #16840044C22semidirect
19. #16840039C22semidirect
24. C8×Dic122C22semidirect
25. C8×D244C22semidirect
35. #16839513C22semidirect
36. #16839513C22semidirect
37. #16839511C22semidirect
38. #16839508C22semidirect
129. #11758222SD16semidirect
244. #3208D61semidirect
298. C61#19409semidirect

Maximal subgroups

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

subgroupindex
#214556012
#214552952
#214549272
#214452322
#214452252
#214451642
#214451572
#214445902
#214445092
C2×#168395062
C61×#32082
#1940961