Finite Group Databasebeta

← #29663500

Subgroups — #29663500

Order 15616.

790 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457126C211yesmaximal, characteristic, complemented
2#21456739C211yesmaximal, characteristic, complemented
3#21456272C211yesmaximal, characteristic, complemented
4#21454562C211yesmaximal, characteristic, complemented
5#21445297C212yesmaximal, complemented
6#21445287C212yesmaximal, complemented
7#21445173C212yesmaximal, complemented
8#21445163C212yesmaximal, complemented
9#21444572C211yesmaximal, characteristic, complemented
10C2×#16839505C211yesmaximal, characteristic, complemented
11C61×#1540C211yesmaximal, characteristic, complemented
12C2×#14519802—21no—
13C2×#14519802—21no—
14C2×#14519783—21no—
15#16840312C2211yescharacteristic
16C2×#14519774C2211yescharacteristic
17#16840152—22no—
18#16840139—21no—
19#16840138C2212yes—
20#16840136C2212yes—
21#16840109—21no—
22#16840042—21no—
23#16840039C2212yescomplemented
24#16839996—22no—
25#16839958C2212yes—
26#16839958C2212yes—
27#16839957—21no—
28#16839954C2212yescomplemented
29#16839898—22no—
30#16839839C2212yescomplemented
31#16839832—22no—
32#16839827C2212yescomplemented
33C2×C4⋊#92418C2211yescharacteristic
34#16839728—21no—
35C4×#11758194C2211yescharacteristic
36#16839514C2212yescomplemented
37#16839512C2212yescomplemented
38#16839505C2212yes—
39#16839505C2212yes—
40C61×#532C2211yescharacteristic
41C61×#431C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C61×#332C2211yescharacteristic, complemented
44C61×#332C2211yescharacteristic, complemented
45C61×#331C2211yescharacteristic, complemented
46C61×#331C2211yescharacteristic, complemented
47C2×#11758222—21no—
48C22×D244—21no—
49C22×D244—21no—
50C22×D244—21no—
51#14519802—44no—
52#14519802—44no—
53C2×#11758201—22no—
54C2×#11758201—22no—
55C2×D488—22no—
56C2×D488—22no—
57C2×#11758199—22no—
58C2×#11758199—22no—
59C2×#11758199—22no—
60#14519783—44no—
61C2×#11758194D411yescharacteristic
62C2×#11758194D411yescharacteristic
63C2×#11758189—22no—
64C22×#92418—21no—
65C2×C8×D61—22no—
66#14519777D411yescharacteristic
67#14519774—21no—
68#14519774—21no—
69#14519774—21no—
70#14519774D411yescharacteristic
71#14519774C2312yes—
72#14519765—21no—
73#14519751—21no—
74#14519751D412yes—
75C4×D244—21no—
76C4×D244C2312yes—
77#14519710—22no—
78C4⋊#92418—21no—
79C4⋊#92418C2312yes—
80#14519699—22no—
81#14519699—22no—
82#14519698—22no—
83#14519697—22no—
84#14519689—22no—
85#14519689—22no—
86#14519688D412yes—
87C4×#92418C2312yes—
88C2×#41×C61—21no—
89C22×C8×C61—21no—
90C4⋊Q8×C61C2311yescharacteristic
91#127×C61C2311yescharacteristic
92#124×C61—21no—
93#121×C61—21no—
94#121×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C2×C42×C61C2311yescharacteristic
98C4⋊C8×C61—21no—
99C4⋊C8×C61C2311yescharacteristic
100C4⋊C8×C61C2311yescharacteristic
101C4⋊C8×C61C2311yescharacteristic
102C4⋊C8×C61C2311yescharacteristic
103C22⋊C8×C61—21no—
104C4×C8×C61—21no—
105C23×D61—41no—
106#41×C61—42no—
107#41×C61—42no—
108C2×Q8×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—41no—
113C22×C4×C61—41no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61#4111yescharacteristic
117C2×C22⋊D61—22no—
118#11758222—42no—
119#11758222—42no—
120C2×D244—21no—
121C2×D244—22no—
122C2×D244—22no—
123C2×D244—21no—
124C2×D244—22no—
125C2×D244—42no—
126C2×D244—42no—
127C2×D244—41no—
128C2×D244—41no—
129C2×D244—41no—
130C2×D244—81no—
131C2×D244—81no—
132C2×D244—82no—
133C2×D244C2×D412yes—
134C2×C4×D61—22no—
135C2×Dic122—21no—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61—42no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C22⋊C4×C61—41no—
144C22⋊C4×C61—41no—
145C42×C61C2×D411yescharacteristic
146C42×C61C2×D411yescharacteristic
147C42×C61#4111yescharacteristic
148C42×C61C2411yescharacteristic
149#11758201—44no—
150#11758201—44no—
151#11758199—44no—
152#11758199—44no—
153#11758199—82no—
154#11758198—22no—
155#11758198—22no—
156C4⋊Dic61—21no—
157C4⋊Dic61C2×D412yes—
158#11758194—22no—
159#11758194D814yescomplemented
160C2×#92418—22no—
161C2×#92418—81no—
162C2×#92418C2×D412yes—
163C2×#92418C2×D412yes—
164D488—82no—
165D488—82no—
166#11758189—44no—
167C8×D61—44no—
168C23×C61—41no—
169C23×C61#12111yescharacteristic
170C22×D61—22no—
171C22×D61—42no—
172C22×D61—82no—
173C22×D61—82no—
174C22×D61—82no—
175Q8×C61—41no—
176Q8×C61—41no—
177D4×C61—41no—
178D4×C61—41no—
179D4×C61—41no—
180D4×C61—41no—
181D4×C61—81no—
182D4×C61—82no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—82no—
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61C2×D812yes—
192C2×C4×C61AGL(1,ℤ/8)12yes—
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C22⋊D61—44no—
197C2×Dic61—22no—
198D244—22no—
199D244—22no—
200D244—42no—
201D244—42no—
202D244—42no—
203D244—42no—
204D244—42no—
205D244—81no—
206D244—82no—
207D244—82no—
208D244—82no—
209D244—82no—
210C4×D61—42no—
211C4×D61—42no—
212Dic122—42no—
213C8×C61—41no—
214C8×C61—41no—
215C8×C61—41no—
216C8×C61—41no—
217#92418—22no—
218#92418—42no—
219#92418C2×D814yes—
220#19230—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—81no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61#45912yes—
227C22×C61C2×#12111yescharacteristic
228C2×D61—42no—
229C2×D61—42no—
230C2×D61—42no—
231C2×D61—82no—
232C2×D61—82no—
233C2×D61—82no—
234C2×D61—164no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—81no—
239C4×C61#45912yes—
240C4×C61C22×D811yescharacteristic
241C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
242C4×C61#57611yescharacteristic
243C4×C61#57711yescharacteristic
244Dic61—42no—
245#3245—611no—
246#3205—611no—
247#3151—611no—
248#2935—611no—
249#1687—612no—
250#1685—612no—
251#1667—612no—
252#1665—612no—
253#1565—611no—
254#1540D6111yescharacteristic, complemented
255C2×#331—611no—
256C2×C61—22no—
257C2×C61—161no—
258C2×C61C2×#45911yescharacteristic
259C2×C61#306211yescharacteristic
260C2×C61#306811yescharacteristic
261D61—82no—
262D61—162no—
263C2×AGL(1,ℤ/8)—1221no—
264C2×AGL(1,ℤ/8)—1221no—
265C2×#131—1221no—
266#532—611no—
267#532C2×D6111yescharacteristic
268C2×#127—611no—
269#496—1222no—
270#493—612no—
271#493—1221no—
272#492—612no—
273#486—1221no—
274#474—612no—
275#474—1221no—
276#466—1222no—
277#459—612no—
278#459—612no—
279#459—612no—
280#459—1221no—
281#447—1222no—
282#436—612no—
283#435—1222no—
284C8×D4—612no—
285#431C2×D6111yescharacteristic
286C2×C4⋊C8—611no—
287C2×C4⋊C8C2×D6111yescharacteristic
288#408—1221no—
289C4×M16—611no—
290#335—612no—
291#334—612no—
292#332C2×D6111yescharacteristic, complemented
293#332C2×D6111yescharacteristic, complemented
294#331—612no—
295#331—612no—
296#331C2×D6111yescharacteristic, complemented
297#331C2×D6111yescharacteristic, complemented
298C61#1923011yescharacteristic, complemented, Sylow
299C2×#41—21no—
300C2×#41—1221no—
301C22×D4—1221no—
302C22×D4—1221no—
303C22×D4—1221no—
304AGL(1,ℤ/8)—2444no—
305AGL(1,ℤ/8)—2444no—
306C2×SD16—1222no—
307C2×SD16—1222no—
308C2×D8—1222no—
309C2×D8—1222no—
310C2×D8—1222no—
311C2×D8—1222no—
312C2×D8—1222no—
313#131—2444no—
314C2×M16—611no—
315C2×M16—611no—
316C2×M16—1222no—
317C22×C8—21no—
318C22×C8—1222no—
319C22×C8—1221no—
320C4⋊Q8—611no—
321C4⋊Q8C22×D6111yescharacteristic
322#127—611no—
323#127—612no—
324#127—1221no—
325#127—1221no—
326#127—1221no—
327#127C22×D6111yescharacteristic
328#124—21no—
329#124—1221no—
330#121—21no—
331#121—21no—
332#121—612no—
333#121—1221no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—612no—
337C4×D4—1221no—
338C2×C42C22×D6111yescharacteristic
339#107—1222no—
340C4⋊C8—21no—
341C4⋊C8—612no—
342C4⋊C8—1221no—
343C4⋊C8C22×D6111yescharacteristic
344C4⋊C8C22×D6111yescharacteristic
345C4⋊C8C22×D6111yescharacteristic
346C4⋊C8C22×D6111yescharacteristic
347#102—1222no—
348#102—1222no—
349#102—1222no—
350#102—1222no—
351C22⋊C8—21no—
352C22⋊C8—1222no—
353C22⋊C8—1222no—
354#97—612no—
355C4×C8—21no—
356C4×C8—612no—
357C24—2441no—
358#41—42no—
359#41—42no—
360#41—2442no—
361#41—2442no—
362C2×Q8—21no—
363C2×Q8—1221no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—41no—
368C2×D4—612no—
369C2×D4—1222no—
370C2×D4—1222no—
371C2×D4—1222no—
372C2×D4—1222no—
373C2×D4—1221no—
374C2×D4—1221no—
375C2×D4—4881no—
376C2×D4—4881no—
377C2×D4—4882no—
378C2×D4—2441no—
379C2×D4—2441no—
380C2×D4—2441no—
381C2×D4—2442no—
382C2×D4—2442no—
383C22×C4—41no—
384C22×C4—1222no—
385C22×C4#1175822211yescharacteristic
386C22×C4D4×D6111yescharacteristic
387C22×C4D4×D6111yescharacteristic
388SD16—2444no—
389SD16—2444no—
390D8—4882no—
391D8—4882no—
392D8—4882no—
393D8—2444no—
394D8—2444no—
395M16—614no—
396M16—1222no—
397M16—2444no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—42no—
403C2×C8—612no—
404C2×C8—612no—
405C2×C8—1222no—
406C2×C8—4881no—
407C2×C8—2444no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—612no—
411C4⋊C4—1221no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—1222no—
415C22⋊C4—1222no—
416C42#1175822211yescharacteristic
417C42D4×D6111yescharacteristic
418C42D4×D6111yescharacteristic
419C42C23×D6111yescharacteristic
420C23—41no—
421C23—1222no—
422C23—4882no—
423C23—4882no—
424C23—4882no—
425C23—2442no—
426C23#1451975011yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—2442no—
430D4—41no—
431D4—41no—
432D4—41no—
433D4—41no—
434D4—81no—
435D4—82no—
436D4—1222no—
437D4—1222no—
438D4—4882no—
439D4—4882no—
440D4—4882no—
441D4—4881no—
442D4—4882no—
443D4—2444no—
444D4—2442no—
445D4—2442no—
446D4—2442no—
447D4—2442no—
448D4—2442no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—82no—
454C2×C4—1222no—
455C2×C4—2442no—
456C2×C4—2442no—
457C2×C4#1451975011yescharacteristic
458C2×C4#1451975011yescharacteristic
459C2×C4#1451975011yescharacteristic
460C2×C4D8×D6112yes—
461C2×C4#1451980412yes—
462C2×C4C2×D4×D6111yescharacteristic
463C2×C4C2×D4×D6111yescharacteristic
464C2×C4C2×#1175822211yescharacteristic
465C8—41no—
466C8—41no—
467C8—41no—
468C8—41no—
469C8—614no—
470C8—1222no—
471C8—2442no—
472C22—21no—
473C22—21no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—9764no—
478C22—4882no—
479C22—4882no—
480C22—4882no—
481C22—2442no—
482C22—2442no—
483C22—2442no—
484C22#1683995612yes—
485C22C2×#1451975011yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—81no—
490C4—2442no—
491C4#1683995612yes—
492C4C2×D8×D6111yescharacteristic
493C4C2×#1451980411yescharacteristic
494C4#1684056411yescharacteristic
495C4#1684056911yescharacteristic
496C2—22no—
497C2—161no—
498C2—9762no—
499C2—4882no—
500C2C2×#1683995611yescharacteristic
501C2#2145550211yescharacteristic
502C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1540C2
socleC22×C61C2×#121
hypercenter#1540D61
hypocenterC61#19230
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1540

Complements

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

NH ≅ G/Ntype
1. #21457126C2semidirect
2. #21456739C2semidirect
3. #21456272C2semidirect
4. #21454562C2semidirect
5. #21445297C2semidirect
6. #21445287C2semidirect
7. #21445173C2semidirect
8. #21445163C2semidirect
9. #21444572C2semidirect
10. C2×#16839505C2semidirect
11. C61×#1540C2semidirect
23. #16840039C22semidirect
28. #16839954C22semidirect
30. #16839839C22semidirect
32. #16839827C22semidirect
36. #16839514C22semidirect
37. #16839512C22semidirect
43. C61×#332C22semidirect
44. C61×#332C22semidirect
45. C61×#331C22semidirect
46. C61×#331C22semidirect
159. #11758194D8semidirect
254. #1540D61semidirect
292. #332C2×D61semidirect
293. #332C2×D61semidirect
296. #331C2×D61semidirect
297. #331C2×D61semidirect
298. C61#19230semidirect

Maximal subgroups

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

subgroupindex
#214571262
#214567392
#214562722
#214545622
#214452972
#214452872
#214451732
#214451632
#214445722
C2×#168395052
C61×#15402
#1923061