Finite Group Databasebeta

← #29663793

Subgroups — #29663793

Order 15616.

744 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456378C211yesmaximal, characteristic, complemented
2#21455618C211yesmaximal, characteristic, complemented
3#21455285C211yesmaximal, characteristic
4C2×C8×D244C211yesmaximal, characteristic, complemented
5#21445243C212yesmaximal, complemented
6#21445229C212yesmaximal, complemented
7#21445195C212yesmaximal, complemented
8#21445161C212yesmaximal, complemented
9#21444510C211yesmaximal, characteristic, complemented
10#21444497C211yesmaximal, characteristic, complemented
11C61×#1566C211yesmaximal, characteristic, complemented
12C2×#14519805—21no—
13C22×C8×D61—21no—
14#16840319C2211yescharacteristic
15#16840312C2211yescharacteristic
16C2×C4×D244C2211yescharacteristic
17#16840160C2211yescharacteristic, complemented
18#16840135C2211yescharacteristic, complemented
19#16840113C2211yescharacteristic, complemented
20#16840109C2211yescharacteristic, complemented
21#16840063—22no—
22#16840053C2212yes—
23#16840043C2212yes—
24#16839991—21no—
25#16839991—21no—
26#16839978—22no—
27#16839968C2212yes—
28#16839958C2212yes—
29C4×#11758190—21no—
30C4×#11758190—21no—
31C8×D244—21no—
32C8×D244—21no—
33C8×D244C2212yes—
34C8×D244C2212yes—
35#16839804—21no—
36C2×C4⋊#92418C2211yescharacteristic
37#16839768—21no—
38#16839764—21no—
39C2×#14519689—21no—
40#16839510C2212yescomplemented
41#16839509C2212yescomplemented
42#16839507C2212yescomplemented
43#16839505C2212yescomplemented
44C61×#423C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4×C8×C61C2211yescharacteristic
47C61×#334C2212yescomplemented
48C61×#334C2212yescomplemented
49C2×#11758222—21no—
50C22×D244D411yescharacteristic
51C22×C4×D61—21no—
52#14519805—44no—
53C2×#11758190—22no—
54C2×#11758190—21no—
55C2×#11758190—21no—
56C2×D488—22no—
57C22×#92418—21no—
58C2×C8×D61—22no—
59C2×C8×D61—22no—
60C2×C8×D61—44no—
61#14519777C2311yescharacteristic
62#14519777C2311yescharacteristic
63#14519777C2311yescharacteristic
64#14519774C2311yescharacteristic
65#14519769—21no—
66#14519765—21no—
67#14519756—21no—
68#14519756—21no—
69#14519751—21no—
70#14519751—21no—
71C4×Dic122—21no—
72C4×Dic122—21no—
73C4×D244—21no—
74C4×D244—21no—
75C4×D244—21no—
76C4×D244—21no—
77C4×D244D412yes—
78C4×D244C2312yes—
79#14519736—21no—
80C2×C4⋊Dic61D411yescharacteristic
81C2×#11758198—21no—
82#14519711—22no—
83#14519709—21no—
84#14519709—21no—
85#14519709—22no—
86C4⋊#92418—21no—
87C4⋊#92418C2312yes—
88#14519702—21no—
89#14519702—21no—
90#14519702—22no—
91#14519699—22no—
92#14519699—22no—
93#14519699—21no—
94#14519699—21no—
95#14519699—22no—
96#14519689—22no—
97#14519689—22no—
98C2×M16×C61—21no—
99C2×M16×C61—21no—
100C22×C8×C61D411yescharacteristic
101C22×C8×C61D411yescharacteristic
102C2×C42×C61C2311yescharacteristic
103C4⋊C8×C61—21no—
104C4⋊C8×C61—21no—
105C4⋊C8×C61C2312yes—
106C4⋊C8×C61C2312yes—
107C4×C8×C61D411yescharacteristic
108C4×C8×C61D411yescharacteristic
109C4×C8×C61C2311yescharacteristic
110C4×C8×C61C2311yescharacteristic
111C23×D61—21no—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C2×C22⋊D61—41no—
116C22×Dic61—21no—
117#11758222—42no—
118#11758222—42no—
119C2×D244—21no—
120C2×D244—22no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244SD1614yescomplemented
125C2×D244C2×D412yes—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×C4×D61—41no—
131C2×Dic122—21no—
132C2×Dic122—21no—
133C2×Dic122—21no—
134M16×C61—42no—
135M16×C61—42no—
136C2×C8×C61—22no—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C2×C8×C61—41no—
140C2×C8×C61C2×D411yescharacteristic
141C2×C8×C61C2×D411yescharacteristic
142C2×C8×C61C2×D411yescharacteristic
143C2×C8×C61C2×D411yescharacteristic
144C42×C61C2×D411yescharacteristic
145C42×C61C2×D411yescharacteristic
146C42×C61#4111yescharacteristic
147C42×C61C2411yescharacteristic
148C22⋊Dic61—41no—
149#11758198—22no—
150#11758198—22no—
151#11758198—41no—
152#11758198—41no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61C2×D412yes—
161#11758196—41no—
162#11758196—41no—
163C4×Dic61—41no—
164C2×#92418—22no—
165C2×#92418—42no—
166D488—44no—
167#11758190—42no—
168#11758190—44no—
169#11758190—42no—
170C8×D61—44no—
171C8×D61—44no—
172C23×C61#12111yescharacteristic
173C22×D61—22no—
174C22×D61—24no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—41no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—41no—
181C2×C4×C61#12111yescharacteristic
182C2×C4×C61#12111yescharacteristic
183C2×C4×C61#12111yescharacteristic
184C2×C4×C61C2×SD1612yes—
185C2×C4×C61#13512yes—
186C2×C4×C61C22×D411yescharacteristic
187C2×C4×C61C22×D411yescharacteristic
188C2×C4×C61C2×#4111yescharacteristic
189C22⋊D61—82no—
190C2×Dic61—22no—
191C2×Dic61—42no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195D244—22no—
196D244—24no—
197D244—41no—
198D244—41no—
199D244—41no—
200D244—41no—
201D244—81no—
202D244C2×SD1614yes—
203C4×D61—44no—
204C4×D61—42no—
205C4×D61—42no—
206C4×D61—82no—
207Dic122—41no—
208Dic122—41no—
209Dic122—41no—
210Dic122—41no—
211Dic122—81no—
212C8×C61—21no—
213C8×C61—21no—
214C8×C61—21no—
215C8×C61—42no—
216C8×C61—42no—
217C8×C61AGL(1,ℤ/8)12yescomplemented
218#92418—42no—
219#19254—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61#46512yes—
223C22×C61C2×#12111yescharacteristic
224C2×D61—24no—
225C2×D61—48no—
226C2×D61—42no—
227C2×D61—42no—
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—81no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61#46512yes—
235C4×C61C22×SD1611yescharacteristic
236C4×C61C2×#13511yescharacteristic
237C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
238C4×C61#57511yescharacteristic
239Dic61—42no—
240Dic61—81no—
241Dic61—81no—
242Dic61—81no—
243#3167—611no—
244#3077—611no—
245#3036—611no—
246C2×C8×D4—611no—
247#1679—612no—
248#1677—612no—
249#1671—612no—
250#1665—612no—
251#1566D6111yescharacteristic, complemented
252#1550—611no—
253#1547—611no—
254C2×C61—22no—
255C2×C61#303611yescharacteristic
256C2×C61C2×#46511yescharacteristic
257C2×C61#306711yescharacteristic
258D61—44no—
259D61—161no—
260C2×AGL(1,ℤ/8)—1221no—
261C23×C8—1221no—
262#533—611no—
263#532—611no—
264C2×C4×D4—611no—
265#498—611no—
266#492—611no—
267#487—611no—
268#486—611no—
269#478—1222no—
270#476—612no—
271#474—612no—
272#465—1221no—
273#465—1221no—
274#463—1222no—
275#461—612no—
276#459—612no—
277C4×SD16—1221no—
278C4×SD16—1221no—
279C8×D4—612no—
280C8×D4—612no—
281C8×D4—1221no—
282C8×D4—1221no—
283#428—1221no—
284#423C2×D6111yescharacteristic
285#423C2×D6111yescharacteristic
286C2×C4⋊C8—611no—
287#418—1221no—
288#417—1221no—
289C2×C22⋊C8—1221no—
290C2×C4×C8C2×D6111yescharacteristic
291#334C2×D6112yescomplemented
292#334C2×D6112yescomplemented
293#333—612no—
294#332—612no—
295#332—612no—
296#331—612no—
297C61#1925411yescharacteristic, complemented, Sylow
298C2×#41—1221no—
299C22×D4—611no—
300C23×C4—1221no—
301AGL(1,ℤ/8)—2444no—
302C2×SD16—1222no—
303C2×SD16—1221no—
304C2×SD16—1221no—
305C2×D8—1222no—
306C2×M16—21no—
307C2×M16—21no—
308C22×C8—1221no—
309C22×C8—1222no—
310C22×C8—1222no—
311C22×C8—2444no—
312C22×C8D24411yescharacteristic
313C22×C8D24411yescharacteristic
314C4⋊Q8—611no—
315C4⋊Q8—611no—
316C4⋊Q8—611no—
317#127—611no—
318#125—1221no—
319#124—1221no—
320C22⋊Q8—1221no—
321C22⋊Q8—1221no—
322#121—1221no—
323#121—1221no—
324C4×Q8—1221no—
325C4×Q8—1221no—
326C4×D4—612no—
327C4×D4—612no—
328C4×D4—1221no—
329C4×D4—1221no—
330C4×D4—1221no—
331C4×D4—1221no—
332#117—1221no—
333C2×C4⋊C4—611no—
334C2×C22⋊C4—1221no—
335C2×C42C22×D6111yescharacteristic
336#107—1222no—
337#106—1222no—
338#106—1221no—
339#106—1221no—
340C4⋊C8—21no—
341C4⋊C8—21no—
342C4⋊C8—612no—
343C4⋊C8—1221no—
344C4⋊C8C22×D6112yes—
345C4⋊C8C22×D6112yes—
346#103—1222no—
347#103—1221no—
348#103—1221no—
349#102—1222no—
350#102—1222no—
351#102—1222no—
352#102—1221no—
353#102—1221no—
354C22⋊C8—1222no—
355C22⋊C8—1222no—
356C4×C8D24411yescharacteristic
357C4×C8D24411yescharacteristic
358C4×C8C22×D6111yescharacteristic
359C4×C8C22×D6111yescharacteristic
360C24—1221no—
361#41—2442no—
362#41—2442no—
363C2×Q8—1221no—
364C2×Q8—1221no—
365C2×Q8—1221no—
366C2×D4—612no—
367C2×D4—614no—
368C2×D4—1221no—
369C2×D4—1221no—
370C2×D4—1221no—
371C2×D4—1221no—
372C2×D4—1222no—
373C2×D4—2441no—
374C22×C4—1221no—
375C22×C4—1222no—
376C22×C4—1222no—
377C22×C4—2442no—
378C22×C4—2442no—
379C22×C4—2441no—
380C22×C4C2×D24411yescharacteristic
381C22×C4D4×D6111yescharacteristic
382C22×C4#1175822611yescharacteristic
383SD16—2442no—
384SD16—2444no—
385SD16—2442no—
386D8—2444no—
387M16—42no—
388M16—42no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—41no—
393C2×C8—1222no—
394C2×C8—2444no—
395C2×C8—2442no—
396C2×C8—2444no—
397C2×C8C2×D24411yescharacteristic
398C2×C8C2×D24411yescharacteristic
399C2×C8C2×D24411yescharacteristic
400C2×C8C2×D24411yescharacteristic
401C4⋊C4—612no—
402C4⋊C4—1221no—
403C4⋊C4—1221no—
404C4⋊C4—1221no—
405C4⋊C4—1221no—
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C4⋊C4—2441no—
410C4⋊C4—2441no—
411C22⋊C4—1222no—
412C22⋊C4—1222no—
413C22⋊C4—2441no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C42—2441no—
417C42C2×D24411yescharacteristic
418C42D4×D6111yescharacteristic
419C42#1175822611yescharacteristic
420C42C23×D6111yescharacteristic
421C23—1224no—
422C23—1222no—
423C23—2442no—
424C23—2442no—
425C23—2441no—
426C23#1451975211yescharacteristic
427Q8—4881no—
428Q8—2441no—
429Q8—2441no—
430Q8—2441no—
431Q8—2441no—
432D4—614no—
433D4—1224no—
434D4—1222no—
435D4—4881no—
436D4—4882no—
437D4—2441no—
438D4—2441no—
439D4—2441no—
440D4—2441no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—1222no—
445C2×C4—4882no—
446C2×C4—2444no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4#1451975211yescharacteristic
454C2×C4#1451975211yescharacteristic
455C2×C4#1451975211yescharacteristic
456C2×C4SD16×D6112yes—
457C2×C4#1451979812yes—
458C2×C4C2×D4×D6111yescharacteristic
459C2×C4C22×D24411yescharacteristic
460C2×C4C2×#1175822611yescharacteristic
461C8—21no—
462C8—21no—
463C8—21no—
464C8—42no—
465C8—42no—
466C8—2442no—
467C8#1451980512yescomplemented
468C22—21no—
469C22—21no—
470C22—1224no—
471C22—4881no—
472C22—4881no—
473C22—4881no—
474C22—2448no—
475C22—2442no—
476C22—2442no—
477C22#1683999212yes—
478C22C2×#1451975211yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—4881no—
483C4—4881no—
484C4—4881no—
485C4—2442no—
486C4#1683999212yes—
487C4C2×SD16×D6111yescharacteristic
488C4C2×#1451979811yescharacteristic
489C4C2×#1451980511yescharacteristic
490C4#1684056211yescharacteristic
491C2—22no—
492C2—9761no—
493C2—2444no—
494C2#2145528511yescharacteristic
495C2C2×#1683999211yescharacteristic
496C2#2145554011yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1566C2
socleC22×C61C2×#121
prime radical#21455285C2
hypercenter#1566D61
hypocenterC61#19254
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1566

Complements

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

NH ≅ G/Ntype
1. #21456378C2semidirect
2. #21455618C2semidirect
4. C2×C8×D244C2semidirect
5. #21445243C2semidirect
6. #21445229C2semidirect
7. #21445195C2semidirect
8. #21445161C2semidirect
9. #21444510C2semidirect
10. #21444497C2semidirect
11. C61×#1566C2semidirect
17. #16840160C22semidirect
18. #16840135C22semidirect
19. #16840113C22semidirect
20. #16840109C22semidirect
40. #16839510C22semidirect
41. #16839509C22semidirect
42. #16839507C22semidirect
43. #16839505C22semidirect
47. C61×#334C22semidirect
48. C61×#334C22semidirect
124. C2×D244SD16semidirect
217. C8×C61AGL(1,ℤ/8)semidirect
251. #1566D61semidirect
291. #334C2×D61semidirect
292. #334C2×D61semidirect
297. C61#19254semidirect
467. C8#14519805semidirect

Maximal subgroups

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

subgroupindex
#214563782
#214556182
#214552852
C2×C8×D2442
#214452432
#214452292
#214451952
#214451612
#214445102
#214444972
C61×#15662
#1925461