Finite Group Databasebeta

← #29663493

Subgroups — #29663493

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#21456269C211yesmaximal, characteristic, complemented
3#21454563C211yesmaximal, characteristic, complemented
4#21445296C212yesmaximal, complemented
5#21445282C212yesmaximal, complemented
6#21445172C212yesmaximal, complemented
7#21445158C212yesmaximal, complemented
8#21444572C211yesmaximal, characteristic, complemented
9#21444466C211yesmaximal, characteristic, complemented
10C2×#16839506C211yesmaximal, characteristic, complemented
11C61×#3205C211yesmaximal, characteristic, complemented
12C2×#14519803—21no—
13C2×#14519803—21no—
14C2×#14519784—21no—
15#16840312C2211yescharacteristic
16#16840149—22no—
17#16840138C2212yes—
18#16840134C2212yes—
19#16840042C2212yescomplemented
20#16840039—21no—
21#16839996—22no—
22#16839957C2212yescomplemented
23#16839954—21no—
24#16839840C2212yescomplemented
25#16839834—22no—
26D4×#92418C2212yescomplemented
27C2×C4⋊#92418C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29#16839729—21no—
30C4×#11758194C2211yescharacteristic
31#16839514C2212yescomplemented
32#16839512C2212yescomplemented
33#16839507C2211yescharacteristic, complemented
34#16839507C2211yescharacteristic, complemented
35#16839506C2212yes—
36#16839506C2212yes—
37#16839505C2211yescharacteristic, complemented
38#16839505C2211yescharacteristic, complemented
39C61×#532C2211yescharacteristic
40C2×#127×C61C2211yescharacteristic
41C61×#493—21no—
42C61×#486—21no—
43C61×#459C2212yes—
44C61×#459C2212yes—
45C61×#447—22no—
46C61×#431C2211yescharacteristic
47C2×#11758222—21no—
48#14519803—44no—
49#14519803—44no—
50C2×#11758200—22no—
51C2×#11758200—22no—
52C2×#11758199—22no—
53C2×#11758199—22no—
54C2×#11758199—22no—
55#14519784—44no—
56C2×#11758194—22no—
57C2×#11758194D411yescharacteristic
58C2×#11758194D411yescharacteristic
59C22×#92418—21no—
60C22×#92418—21no—
61C22×#92418—22no—
62#14519777C2311yescharacteristic
63#14519774C2311yescharacteristic
64#14519765—21no—
65#14519751—21no—
66#14519751—21no—
67C4×D244—21no—
68C4×D244—21no—
69#14519710—22no—
70C4⋊#92418—21no—
71C4⋊#92418—21no—
72C4⋊#92418C2312yes—
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2311yescharacteristic
75#14519698—22no—
76#14519697—22no—
77C22⋊#92418—22no—
78C22⋊#92418—22no—
79#14519688D412yes—
80C4×#92418C2312yes—
81C2×#41×C61—21no—
82C22×D4×C61—21no—
83C22×D4×C61—21no—
84C22×D4×C61—21no—
85C2×D8×C61—22no—
86C2×D8×C61—22no—
87C4⋊Q8×C61D411yescharacteristic
88#127×C61—21no—
89#127×C61—21no—
90#127×C61—21no—
91#127×C61D411yescharacteristic
92#127×C61C2312yes—
93#124×C61—21no—
94#121×C61—21no—
95#121×C61D412yes—
96C4×D4×C61—21no—
97C4×D4×C61C2312yes—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61C2311yescharacteristic
100C4⋊C8×C61C2311yescharacteristic
101#102×C61—22no—
102#102×C61—22no—
103C22⋊C8×C61—21no—
104C4×C8×C61—21no—
105C24×C61—41no—
106#41×C61—42no—
107#41×C61—42no—
108C2×Q8×C61—21no—
109C2×D4×C61—22no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—41no—
116C2×D4×C61—41no—
117C2×D4×C61—41no—
118C2×D4×C61—42no—
119C2×D4×C61—42no—
120C2×D4×C61—81no—
121C2×D4×C61—81no—
122C2×D4×C61—82no—
123C2×D4×C61C2×D412yes—
124C22×C4×C61—22no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61#4111yescharacteristic
128C2×C22⋊D61—41no—
129#11758222—42no—
130#11758222—42no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—21no—
134C2×C4×D61—41no—
135C2×Dic122—21no—
136D8×C61—82no—
137D8×C61—82no—
138C2×C8×C61—21no—
139C2×C8×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61C2×D412yes—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—22no—
144C42×C61C2×D411yescharacteristic
145C42×C61C2×D411yescharacteristic
146C42×C61#4111yescharacteristic
147C42×C61C2411yescharacteristic
148#11758200—44no—
149#11758200—44no—
150#11758199—44no—
151#11758199—44no—
152#11758199—82no—
153#11758198—41no—
154#11758198—41no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157#11758194—22no—
158#11758194—44no—
159#11758194D814yescomplemented
160C2×#92418—21no—
161C2×#92418—21no—
162C2×#92418—22no—
163C2×#92418—42no—
164C2×#92418—44no—
165C2×#92418—81no—
166C2×#92418C2×D412yes—
167C2×#92418C2×D412yes—
168C23×C61—22no—
169C23×C61—42no—
170C23×C61—82no—
171C23×C61—82no—
172C23×C61—82no—
173C23×C61#12111yescharacteristic
174C22×D61—41no—
175Q8×C61—42no—
176D4×C61—22no—
177D4×C61—22no—
178D4×C61—44no—
179D4×C61—42no—
180D4×C61—42no—
181D4×C61—42no—
182D4×C61—42no—
183D4×C61—42no—
184D4×C61—81no—
185D4×C61—82no—
186D4×C61—82no—
187D4×C61—82no—
188D4×C61—82no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—22no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61C2×D812yes—
199C2×C4×C61AGL(1,ℤ/8)12yes—
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C2×#4111yescharacteristic
203C22⋊D61—82no—
204C2×Dic61—41no—
205D244—41no—
206D244—41no—
207D244—41no—
208D244—41no—
209D244—81no—
210C4×D61—82no—
211Dic122—41no—
212Dic122—41no—
213C8×C61—41no—
214C8×C61—41no—
215#92418—22no—
216#92418—41no—
217#92418—41no—
218#92418—42no—
219#92418C2×D814yes—
220#19230—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—42no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—82no—
227C22×C61—82no—
228C22×C61—82no—
229C22×C61—164no—
230C22×C61#45912yes—
231C22×C61C2×#12111yescharacteristic
232C2×D61—81no—
233C2×D61—81no—
234C2×D61—81no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—42no—
239C4×C61#45912yes—
240C4×C61C22×D811yescharacteristic
241C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
242C4×C61#57611yescharacteristic
243C4×C61#57711yescharacteristic
244Dic61—81no—
245#3245—611no—
246#3205D6111yescharacteristic, complemented
247#3151—611no—
248#2935—611no—
249#1687—612no—
250#1685—612no—
251#1667—612no—
252#1665—612no—
253#1565—611no—
254#1540—611no—
255C2×#331—611no—
256C2×C61—22no—
257C2×C61—82no—
258C2×C61—162no—
259C2×C61C2×#45911yescharacteristic
260C2×C61#306211yescharacteristic
261C2×C61#306811yescharacteristic
262D61—161no—
263C2×AGL(1,ℤ/8)—1221no—
264C2×AGL(1,ℤ/8)—1221no—
265C2×#131—1221no—
266#532—611no—
267#532C2×D6111yescharacteristic
268C2×#127C2×D6111yescharacteristic
269#496—1222no—
270#493—21no—
271#493—612no—
272#492—612no—
273#486—21no—
274#474—612no—
275#474—1221no—
276#466—1222no—
277#459—612no—
278#459—1221no—
279#459C2×D6112yes—
280#459C2×D6112yes—
281#447—22no—
282#436—612no—
283#435—1222no—
284C8×D4—612no—
285#431C2×D6111yescharacteristic
286C2×C4⋊C8—611no—
287C2×C4⋊C8—611no—
288#408—1221no—
289C4×M16—611no—
290#335—612no—
291#334—612no—
292#332—611no—
293#332—611no—
294#331—612no—
295#331—612no—
296#331—611no—
297#331—611no—
298C61#1923011yescharacteristic, complemented, Sylow
299C2×#41—21no—
300C2×#41—1221no—
301C22×D4—21no—
302C22×D4—21no—
303C22×D4—21no—
304AGL(1,ℤ/8)—2444no—
305AGL(1,ℤ/8)—2444no—
306C2×SD16—1222no—
307C2×SD16—1222no—
308C2×D8—22no—
309C2×D8—22no—
310C2×D8—1222no—
311C2×D8—1222no—
312C2×D8—1222no—
313#131—2444no—
314C2×M16—611no—
315C2×M16—611no—
316C2×M16—1222no—
317C22×C8—1222no—
318C22×C8—1221no—
319C22×C8—1221no—
320C4⋊Q8—611no—
321C4⋊Q8C22⋊D6111yescharacteristic
322#127—21no—
323#127—21no—
324#127—21no—
325#127—611no—
326#127C22⋊D6111yescharacteristic
327#127C22×D6112yes—
328#124—21no—
329#124—1221no—
330#121—21no—
331#121—1221no—
332#121—1221no—
333#121C22⋊D6112yes—
334C4×D4—21no—
335C4×D4—1221no—
336C4×D4—1221no—
337C4×D4C22×D6112yes—
338C2×C42C22×D6111yescharacteristic
339#107—1222no—
340C4⋊C8—611no—
341C4⋊C8—611no—
342C4⋊C8—612no—
343C4⋊C8—1221no—
344C4⋊C8—1221no—
345C4⋊C8C22×D6111yescharacteristic
346C4⋊C8C22×D6111yescharacteristic
347#102—22no—
348#102—22no—
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—41no—
358#41—42no—
359#41—42no—
360#41—2442no—
361#41—2442no—
362C2×Q8—21no—
363C2×Q8—1221no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—81no—
376C2×D4—81no—
377C2×D4—82no—
378C2×D4—1221no—
379C2×D4—1221no—
380C2×D4—1221no—
381C2×D4—2441no—
382C2×D4C2×C22⋊D6112yes—
383C22×C4—22no—
384C22×C4—2441no—
385C22×C4D4×D6111yescharacteristic
386C22×C4#1175822411yescharacteristic
387C22×C4C2×C22⋊D6111yescharacteristic
388SD16—2444no—
389SD16—2444no—
390D8—82no—
391D8—82no—
392D8—4882no—
393D8—2444no—
394D8—2444no—
395M16—614no—
396M16—1222no—
397M16—2444no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—612no—
401C2×C8—612no—
402C2×C8—1221no—
403C2×C8—1221no—
404C2×C8—1222no—
405C2×C8—4881no—
406C2×C8—2444no—
407C2×C8—2442no—
408C4⋊C4—21no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4C2×C22⋊D6112yes—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C42D4×D6111yescharacteristic
417C42#1175822411yescharacteristic
418C42C2×C22⋊D6111yescharacteristic
419C42C23×D6111yescharacteristic
420C23—22no—
421C23—42no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—2441no—
426C23#1451974911yescharacteristic
427Q8—42no—
428Q8—2441no—
429Q8—2441no—
430D4—22no—
431D4—22no—
432D4—44no—
433D4—42no—
434D4—42no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—81no—
439D4—82no—
440D4—82no—
441D4—82no—
442D4—82no—
443D4—4881no—
444D4—4882no—
445D4—2441no—
446D4—2441no—
447D4—2441no—
448D4—2441no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—4882no—
456C2×C4—2441no—
457C2×C4#1451974911yescharacteristic
458C2×C4#1451974911yescharacteristic
459C2×C4#1451974911yescharacteristic
460C2×C4D8×D6112yes—
461C2×C4#1451980112yes—
462C2×C4C2×D4×D6111yescharacteristic
463C2×C4C22×C22⋊D6111yescharacteristic
464C2×C4C2×#1175822411yescharacteristic
465C8—41no—
466C8—41no—
467C8—614no—
468C8—1222no—
469C8—2441no—
470C8—2441no—
471C8—2442no—
472C22—21no—
473C22—21no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—164no—
481C22—4881no—
482C22—4881no—
483C22—4881no—
484C22#1683995512yes—
485C22C2×#1451974911yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—42no—
490C4—4881no—
491C4#1683995512yes—
492C4C2×D8×D6111yescharacteristic
493C4C2×#1451980111yescharacteristic
494C4#1684056511yescharacteristic
495C4#1684057311yescharacteristic
496C2—22no—
497C2—82no—
498C2—162no—
499C2—9761no—
500C2C2×#1683995511yescharacteristic
501C2#2145549511yescharacteristic
502C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3205C2
socleC22×C61C2×#121
hypercenter#3205D61
hypocenterC61#19230
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #21457126C2semidirect
2. #21456269C2semidirect
3. #21454563C2semidirect
4. #21445296C2semidirect
5. #21445282C2semidirect
6. #21445172C2semidirect
7. #21445158C2semidirect
8. #21444572C2semidirect
9. #21444466C2semidirect
10. C2×#16839506C2semidirect
11. C61×#3205C2semidirect
19. #16840042C22semidirect
22. #16839957C22semidirect
24. #16839840C22semidirect
26. D4×#92418C22semidirect
31. #16839514C22semidirect
32. #16839512C22semidirect
33. #16839507C22semidirect
34. #16839507C22semidirect
37. #16839505C22semidirect
38. #16839505C22semidirect
159. #11758194D8semidirect
246. #3205D61semidirect
298. C61#19230semidirect

Maximal subgroups

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

subgroupindex
#214571262
#214562692
#214545632
#214452962
#214452822
#214451722
#214451582
#214445722
#214444662
C2×#168395062
C61×#32052
#1923061