Finite Group Databasebeta

← #29662870

Subgroups — #29662870

Order 15616.

766 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456271C211yesmaximal, characteristic
2#21455851C211yesmaximal, characteristic, complemented
3C2×#16839963C211yesmaximal, characteristic, complemented
4M16×Dic122C211yesmaximal, characteristic, complemented
5#21445304C212yesmaximal, complemented
6#21445292C212yesmaximal, complemented
7#21445180C212yesmaximal, complemented
8#21445168C212yesmaximal, complemented
9#21444494C211yesmaximal, characteristic, complemented
10#21444486C211yesmaximal, characteristic, complemented
11C61×#1565C211yesmaximal, characteristic, complemented
12C2×#14519805—21no—
13C2×#14519805—21no—
14C22×#11758190—21no—
15#16840312C2211yescharacteristic
16C2×#14519774C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18#16840151—22no—
19#16840139C2212yes—
20#16840135C2212yes—
21#16840081—21no—
22#16840048C2211yescharacteristic, complemented
23#16840048C2211yescharacteristic, complemented
24#16839963—21no—
25#16839963—21no—
26#16839963C2212yes—
27#16839963C2212yes—
28#16839963C2211yescharacteristic, complemented
29#16839963C2211yescharacteristic, complemented
30#16839903—21no—
31#16839892C2212yescomplemented
32#16839890—22no—
33C8×Dic122C2212yescomplemented
34C4×#11758190—21no—
35C4×#11758190—21no—
36#16839786C2211yescharacteristic
37#16839785—21no—
38C2×#14519699—21no—
39M16×Dic61—21no—
40#16839507C2212yescomplemented
41#16839505C2212yes—
42#16839505C2212yes—
43#16839505C2212yescomplemented
44C61×#431C2211yescharacteristic
45C2×C4⋊C8×C61C2211yescharacteristic
46C4×M16×C61C2211yescharacteristic
47C61×#335C2212yescomplemented
48C61×#334C2212yescomplemented
49C2×#11758222—21no—
50C22×Dic122D411yescharacteristic
51C22×D244—21no—
52C22×D244—21no—
53C22×D244—21no—
54#14519805—44no—
55#14519805—44no—
56C2×#11758190—22no—
57C2×#11758190—22no—
58C2×#11758190—21no—
59C2×#11758190—21no—
60C2×#11758190—22no—
61C2×#11758190—22no—
62C2×#11758190—44no—
63C2×D488—22no—
64C2×D488—22no—
65C2×#11758194—21no—
66#14519777C2311yescharacteristic
67#14519774—21no—
68#14519774—21no—
69#14519774—21no—
70#14519774C2312yes—
71#14519774C2311yescharacteristic
72#14519765—21no—
73#14519751—21no—
74#14519751—21no—
75C4×Dic122—21no—
76C4×Dic122—21no—
77C4×Dic122D411yescharacteristic
78C4×Dic122D411yescharacteristic
79C4×Dic122C2311yescharacteristic
80C4×Dic122C2311yescharacteristic
81C4×D244—21no—
82C4×D244—21no—
83C2×C4⋊Dic61D411yescharacteristic
84C2×#11758196—21no—
85C2×C4×Dic61—21no—
86#14519709—21no—
87#14519709—21no—
88C4⋊#92418—21no—
89C4⋊#92418C2312yes—
90#14519706—22no—
91#14519706—22no—
92#14519702—21no—
93#14519702—21no—
94#14519699—22no—
95#14519699—22no—
96#14519699—21no—
97#14519699—21no—
98#14519687—22no—
99C8×Dic61—22no—
100C2×M16×C61D411yescharacteristic
101C2×M16×C61D411yescharacteristic
102C22×C8×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—21no—
105C4⋊C8×C61C2312yes—
106C4⋊C8×C61C2311yescharacteristic
107C4⋊C8×C61C2311yescharacteristic
108C22⋊C8×C61—21no—
109#97×C61D412yes—
110C4×C8×C61—21no—
111C4×C8×C61C2312yes—
112C23×D61—41no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C2×C22⋊D61—41no—
117C22×Dic61—21no—
118C22×Dic61—21no—
119#11758222—42no—
120#11758222—42no—
121C2×D244—22no—
122C2×D244—22no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244—22no—
126C2×D244—21no—
127C2×D244—42no—
128C2×D244—42no—
129C2×D244—42no—
130C2×D244—42no—
131C2×D244—41no—
132C2×D244—41no—
133C2×D244—41no—
134C2×D244—82no—
135C2×C4×D61—41no—
136C2×Dic122—22no—
137C2×Dic122—21no—
138C2×Dic122—21no—
139C2×Dic122—41no—
140C2×Dic122C2×D411yescharacteristic
141C2×Dic122C2×D411yescharacteristic
142M16×C61—22no—
143M16×C61SD1614yescomplemented
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—22no—
147C2×C8×C61—42no—
148C2×C8×C61C2×D412yes—
149C2×C8×C61C2×D412yes—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61#4111yescharacteristic
153C42×C61C2411yescharacteristic
154#11758198—41no—
155#11758198—41no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61C2×D411yescharacteristic
160C4⋊Dic61C2×D411yescharacteristic
161#11758196—21no—
162#11758196—21no—
163#11758196—21no—
164#11758196—21no—
165C4×Dic61—21no—
166C4×Dic61—21no—
167C4×Dic61—21no—
168C4×Dic61—21no—
169#11758194—42no—
170C2×#92418—22no—
171D488—44no—
172D488—44no—
173#11758190—44no—
174#11758190—42no—
175#11758190—44no—
176#11758190—42no—
177#11758190—44no—
178#11758190—44no—
179C23×C61#12111yescharacteristic
180C22×D61—41no—
181C22×D61—42no—
182C22×D61—82no—
183C22×D61—82no—
184C22×D61—82no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—41no—
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61C2×SD1612yes—
192C2×C4×C61AGL(1,ℤ/8)12yes—
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C22⋊D61—82no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—42no—
202C2×Dic61—41no—
203C2×Dic61—81no—
204D244—42no—
205D244—41no—
206D244—42no—
207D244—42no—
208D244—41no—
209D244—41no—
210D244—41no—
211D244—42no—
212D244—81no—
213D244—82no—
214D244—82no—
215D244—82no—
216D244—82no—
217C4×D61—82no—
218Dic122—21no—
219Dic122—21no—
220Dic122—21no—
221Dic122—42no—
222Dic122—41no—
223Dic122—41no—
224Dic122AGL(1,ℤ/8)12yescomplemented
225C8×C61—22no—
226C8×C61—42no—
227C8×C61—41no—
228C8×C61—41no—
229C8×C61C2×SD1614yescomplemented
230#92418—42no—
231#19170—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61#46012yes—
235C22×C61C2×#12111yescharacteristic
236C2×D61—81no—
237C2×D61—82no—
238C2×D61—82no—
239C2×D61—81no—
240C2×D61—82no—
241C2×D61—81no—
242C2×D61—164no—
243C4×C61—21no—
244C4×C61—21no—
245C4×C61—21no—
246C4×C61#46012yes—
247C4×C61C22×SD1611yescharacteristic
248C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
249C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
250C4×C61#57511yescharacteristic
251Dic61—22no—
252Dic61—41no—
253Dic61—41no—
254Dic61—41no—
255Dic61—81no—
256#3151—611no—
257#3103—611no—
258C2×#460—611no—
259Q8×M16—611no—
260#1688—612no—
261#1686—612no—
262#1668—612no—
263#1666—612no—
264#1565D6111yescharacteristic, complemented
265#1547—611no—
266#1545—611no—
267C2×C61—22no—
268C2×C61C2×#46011yescharacteristic
269C2×C61#305511yescharacteristic
270C2×C61#306811yescharacteristic
271D61—161no—
272D61—162no—
273C2×AGL(1,ℤ/8)—1221no—
274C2×AGL(1,ℤ/8)—1221no—
275C22×SD16—1221no—
276#532—611no—
277C2×#127—611no—
278C2×C4×Q8—611no—
279#496—1222no—
280#493—612no—
281#492—612no—
282#481—1221no—
283#475—611no—
284#475—611no—
285#460—612no—
286#460—612no—
287#460—611no—
288#460—611no—
289#460—1221no—
290#460—1221no—
291#448—1221no—
292#446—612no—
293#446—1222no—
294C8×Q8—612no—
295C4×SD16—1221no—
296C4×SD16—1221no—
297#431C2×D6111yescharacteristic
298#423—611no—
299#423—1221no—
300C2×C4⋊C8C2×D6111yescharacteristic
301C2×#102—1221no—
302C4×M16—1221no—
303C4×M16C2×D6111yescharacteristic
304#335C2×D6112yescomplemented
305#334C2×D6112yescomplemented
306#332—612no—
307#331—612no—
308#331—612no—
309#331—612no—
310C61#1917011yescharacteristic, complemented, Sylow
311C2×#41—1221no—
312C22×Q8—611no—
313C22×D4—1221no—
314C22×D4—1221no—
315C22×D4—1221no—
316AGL(1,ℤ/8)—2444no—
317AGL(1,ℤ/8)—2444no—
318C2×SD16—1222no—
319C2×SD16—1222no—
320C2×SD16—1221no—
321C2×SD16—1221no—
322C2×SD16—1222no—
323C2×SD16—1222no—
324C2×SD16—2444no—
325C2×D8—1222no—
326C2×D8—1222no—
327C2×M16—1221no—
328C2×M16D24411yescharacteristic
329C2×M16D24411yescharacteristic
330C22×C8—21no—
331C4⋊Q8—611no—
332#127—612no—
333#127—611no—
334#127—1221no—
335#127—1221no—
336#127—1221no—
337#124—1221no—
338#121—1221no—
339#121—1221no—
340C4×Q8—611no—
341C4×Q8—611no—
342C4×Q8—611no—
343C4×Q8—611no—
344C4×Q8—1221no—
345C4×Q8—1221no—
346C4×D4—1221no—
347C4×D4—1221no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—1221no—
350C2×C42—1221no—
351C2×C42C22×D6111yescharacteristic
352#106—1221no—
353#106—1221no—
354C4⋊C8—21no—
355C4⋊C8—612no—
356C4⋊C8—1222no—
357C4⋊C8—1222no—
358C4⋊C8—1221no—
359C4⋊C8C22×D6111yescharacteristic
360C4⋊C8C22×D6111yescharacteristic
361C4⋊C8C22×D6112yes—
362#103—1221no—
363#103—1221no—
364#102—1222no—
365#102—1222no—
366#102—1221no—
367#102—1221no—
368C22⋊C8—21no—
369#97—1222no—
370#97D24412yes—
371C4×C8—21no—
372C4×C8—1222no—
373C4×C8C22×D6112yes—
374C24—2441no—
375#41—2442no—
376#41—2442no—
377C2×Q8—611no—
378C2×Q8—611no—
379C2×Q8—1222no—
380C2×Q8—1221no—
381C2×Q8—1221no—
382C2×Q8—2441no—
383C2×D4—1221no—
384C2×D4—1222no—
385C2×D4—1221no—
386C2×D4—1222no—
387C2×D4—1221no—
388C2×D4—1222no—
389C2×D4—4882no—
390C2×D4—2442no—
391C2×D4—2442no—
392C2×D4—2441no—
393C2×D4—2441no—
394C2×D4—2441no—
395C2×D4—2441no—
396C2×D4—2442no—
397C2×D4—2442no—
398C22×C4—1221no—
399C22×C4—1221no—
400C22×C4—2441no—
401C22×C4C2×D24411yescharacteristic
402C22×C4D4×D6111yescharacteristic
403C22×C4#1175822611yescharacteristic
404SD16—2444no—
405SD16—2442no—
406SD16—2444no—
407SD16—2442no—
408SD16—2444no—
409SD16—2444no—
410D8—2444no—
411D8—2444no—
412M16—22no—
413M16—2442no—
414M16#1175819014yescomplemented
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—22no—
418C2×C8—42no—
419C2×C8—1222no—
420C2×C8C2×D24412yes—
421C2×C8C2×D24412yes—
422C4⋊C4—611no—
423C4⋊C4—611no—
424C4⋊C4—1221no—
425C4⋊C4—1221no—
426C4⋊C4—1221no—
427C4⋊C4—1221no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—1221no—
431C22⋊C4—2441no—
432C22⋊C4—2441no—
433C42—1221no—
434C42—1221no—
435C42—1221no—
436C42—1221no—
437C42C2×D24411yescharacteristic
438C42D4×D6111yescharacteristic
439C42#1175822611yescharacteristic
440C42C23×D6111yescharacteristic
441C23—4882no—
442C23—4882no—
443C23—4882no—
444C23—2441no—
445C23—2442no—
446C23#1451975211yescharacteristic
447Q8—612no—
448Q8—1221no—
449Q8—1221no—
450Q8—1221no—
451Q8—2442no—
452Q8—2441no—
453Q8—2441no—
454D4—4881no—
455D4—4882no—
456D4—4882no—
457D4—4882no—
458D4—4882no—
459D4—4882no—
460D4—2441no—
461D4—2442no—
462D4—2442no—
463D4—2441no—
464D4—2441no—
465D4—2442no—
466D4—2441no—
467D4—2442no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—1221no—
472C2×C4—1221no—
473C2×C4—1221no—
474C2×C4—1221no—
475C2×C4—4882no—
476C2×C4—4881no—
477C2×C4—2442no—
478C2×C4—2441no—
479C2×C4#1451975211yescharacteristic
480C2×C4#1451975211yescharacteristic
481C2×C4#1451975211yescharacteristic
482C2×C4C2×#1175819012yes—
483C2×C4#1451980512yes—
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4C22×D24411yescharacteristic
486C2×C4C2×#1175822611yescharacteristic
487C8—22no—
488C8—42no—
489C8—41no—
490C8—41no—
491C8—2442no—
492C8C2×#1175819014yescomplemented
493C22—21no—
494C22—21no—
495C22—9764no—
496C22—4882no—
497C22—4881no—
498C22—4881no—
499C22—4882no—
500C22—4882no—
501C22—4881no—
502C22#1683996312yes—
503C22C2×#1451975211yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—1222no—
508C4—4881no—
509C4—2441no—
510C4—2441no—
511C4—2441no—
512C4#1683996312yes—
513C4C22×#1175819011yescharacteristic
514C4AGL(1,ℤ/8)×D6111yescharacteristic
515C4C2×#1451980511yescharacteristic
516C4#1684056111yescharacteristic
517C2—22no—
518C2—9762no—
519C2—9761no—
520C2C2×#1683996311yescharacteristic
521C2#2145543911yescharacteristic
522C2#2145554811yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1565C2
socleC22×C61C2×#121
prime radical#21456271C2
hypercenter#1565D61
hypocenterC61#19170
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
2. #21455851C2semidirect
3. C2×#16839963C2semidirect
4. M16×Dic122C2semidirect
5. #21445304C2semidirect
6. #21445292C2semidirect
7. #21445180C2semidirect
8. #21445168C2semidirect
9. #21444494C2semidirect
10. #21444486C2semidirect
11. C61×#1565C2semidirect
22. #16840048C22semidirect
23. #16840048C22semidirect
28. #16839963C22semidirect
29. #16839963C22semidirect
31. #16839892C22semidirect
33. C8×Dic122C22semidirect
40. #16839507C22semidirect
43. #16839505C22semidirect
47. C61×#335C22semidirect
48. C61×#334C22semidirect
143. M16×C61SD16semidirect
224. Dic122AGL(1,ℤ/8)semidirect
229. C8×C61C2×SD16semidirect
264. #1565D61semidirect
304. #335C2×D61semidirect
305. #334C2×D61semidirect
310. C61#19170semidirect
414. M16#11758190semidirect
492. C8C2×#11758190semidirect

Maximal subgroups

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

subgroupindex
#214562712
#214558512
C2×#168399632
M16×Dic1222
#214453042
#214452922
#214451802
#214451682
#214444942
#214444862
C61×#15652
#1917061