Finite Group Databasebeta

← #29663491

Subgroups — #29663491

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#21456737C211yesmaximal, characteristic, complemented
3#21454565C211yesmaximal, characteristic, complemented
4#21445295C212yesmaximal, complemented
5#21445281C212yesmaximal, complemented
6#21445171C212yesmaximal, complemented
7#21445157C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444468C211yesmaximal, characteristic, complemented
10C2×#16839506C211yesmaximal, characteristic, complemented
11C61×#3151C211yesmaximal, characteristic, complemented
12C2×#14519785—21no—
13#16840312C2211yescharacteristic
14#16840138—21no—
15#16840108—21no—
16#16840042—21no—
17#16840039C2212yescomplemented
18#16839996—22no—
19#16839957—21no—
20#16839954C2212yes—
21#16839954C2212yes—
22#16839954C2212yescomplemented
23#16839896—22no—
24#16839841C2212yescomplemented
25#16839835—22no—
26C8×D244C2212yescomplemented
27#16839815C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29C2×C4⋊#92418C2211yescharacteristic
30#16839730—21no—
31#16839515C2212yescomplemented
32#16839513C2212yescomplemented
33#16839508C2211yescharacteristic, complemented
34#16839508C2211yescharacteristic, complemented
35#16839506C2212yes—
36#16839506C2212yes—
37#16839506C2211yescharacteristic, complemented
38#16839506C2211yescharacteristic, complemented
39C2×AGL(1,ℤ/8)×C61—21no—
40C2×AGL(1,ℤ/8)×C61—21no—
41C61×#532C2211yescharacteristic
42C2×#127×C61C2211yescharacteristic
43C61×#496—22no—
44C61×#493C2212yes—
45C61×#492C2212yes—
46C4×M16×C61C2211yescharacteristic
47C2×#11758222—21no—
48C2×#11758199—22no—
49C2×#11758199—22no—
50C2×#11758199—22no—
51#14519785—44no—
52C2×#11758189—22no—
53C22×#92418—21no—
54C22×#92418—21no—
55C2×C8×D61—22no—
56#14519777D411yescharacteristic
57#14519774D411yescharacteristic
58#14519765—21no—
59#14519751—21no—
60#14519751D412yes—
61C4×D244—21no—
62C4×D244C2312yes—
63#14519710—22no—
64C4⋊#92418—21no—
65C4⋊#92418—21no—
66C4⋊#92418C2311yescharacteristic
67C4⋊#92418C2311yescharacteristic
68C4⋊#92418C2312yes—
69C4⋊#92418C2311yescharacteristic
70C4⋊#92418C2311yescharacteristic
71#14519698—22no—
72#14519698—22no—
73#14519698—22no—
74#14519697—22no—
75C22⋊#92418—21no—
76#14519689—22no—
77#14519689—22no—
78C4×#92418—21no—
79C2×#41×C61—21no—
80C22×D4×C61—21no—
81C22×D4×C61—21no—
82C22×D4×C61—21no—
83AGL(1,ℤ/8)×C61—44no—
84AGL(1,ℤ/8)×C61—44no—
85C2×SD16×C61—22no—
86C2×SD16×C61—22no—
87C2×D8×C61—22no—
88C2×D8×C61—22no—
89C2×M16×C61D411yescharacteristic
90C2×M16×C61D411yescharacteristic
91C4⋊Q8×C61C2311yescharacteristic
92#127×C61—21no—
93#127×C61—21no—
94#127×C61—21no—
95#127×C61C2312yes—
96#127×C61C2311yescharacteristic
97#124×C61—21no—
98#121×C61—21no—
99#121×C61—21no—
100C4×D4×C61—21no—
101C4×D4×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#97×C61D412yes—
104C4×C8×C61C2312yes—
105C24×C61—41no—
106#41×C61—42no—
107#41×C61—42no—
108C2×Q8×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—21no—
114C2×D4×C61—22no—
115C2×D4×C61—41no—
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×C61—41no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61#4111yescharacteristic
128C2×C22⋊D61—22no—
129#11758222—42no—
130#11758222—42no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244C2×D412yes—
134C2×C4×D61—22no—
135C2×Dic122—21no—
136SD16×C61—44no—
137SD16×C61—44no—
138D8×C61—44no—
139D8×C61—44no—
140M16×C61—22no—
141M16×C61D814yescomplemented
142C2×C8×C61C2×D412yes—
143C2×C8×C61C2×D412yes—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C22⋊C4×C61—41no—
147C22⋊C4×C61—41no—
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C42×C61#4111yescharacteristic
151C42×C61C2411yescharacteristic
152#11758199—82no—
153#11758199—82no—
154#11758199—82no—
155#11758198—22no—
156#11758198—22no—
157C4⋊Dic61—21no—
158C4⋊Dic61C2×D412yes—
159C2×#92418—21no—
160C2×#92418—21no—
161C2×#92418—21no—
162C2×#92418—21no—
163C2×#92418—22no—
164C2×#92418—42no—
165C2×#92418—81no—
166#11758189—44no—
167C8×D61—44no—
168C23×C61—41no—
169C23×C61—42no—
170C23×C61—82no—
171C23×C61—82no—
172C23×C61—82no—
173C23×C61#12111yescharacteristic
174C22×D61—22no—
175Q8×C61—41no—
176Q8×C61—41no—
177D4×C61—41no—
178D4×C61—42no—
179D4×C61—41no—
180D4×C61—42no—
181D4×C61—41no—
182D4×C61—42no—
183D4×C61—42no—
184D4×C61—41no—
185D4×C61—81no—
186D4×C61—82no—
187D4×C61—82no—
188D4×C61—82no—
189D4×C61—82no—
190D4×C61—82no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—82no—
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61#12111yescharacteristic
199C2×C4×C61C2×D812yes—
200C2×C4×C61AGL(1,ℤ/8)12yes—
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C22⋊D61—44no—
205C2×Dic61—22no—
206D244—22no—
207D244—22no—
208D244—42no—
209D244—81no—
210C4×D61—42no—
211C4×D61—42no—
212Dic122—42no—
213C8×C61—22no—
214C8×C61C2×D814yes—
215#92418—41no—
216#92418—41no—
217#92418—42no—
218#92418—41no—
219#92418—41no—
220#19230—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—81no—
224C22×C61—82no—
225C22×C61—82no—
226C22×C61—81no—
227C22×C61—82no—
228C22×C61—81no—
229C22×C61—164no—
230C22×C61#45912yes—
231C22×C61C2×#12111yescharacteristic
232C2×D61—42no—
233C2×D61—42no—
234C2×D61—42no—
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#3151D6111yescharacteristic, complemented
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—161no—
258C2×C61—162no—
259C2×C61C2×#45911yescharacteristic
260C2×C61#306211yescharacteristic
261C2×C61#306811yescharacteristic
262D61—82no—
263C2×AGL(1,ℤ/8)—21no—
264C2×AGL(1,ℤ/8)—21no—
265C2×#131—1221no—
266#532—611no—
267#532C2×D6111yescharacteristic
268C2×#127C2×D6111yescharacteristic
269#496—22no—
270#493—1221no—
271#493C2×D6112yes—
272#492C2×D6112yes—
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#431—611no—
286C2×C4⋊C8—611no—
287C2×C4⋊C8—611no—
288#408—1221no—
289C4×M16C2×D6111yescharacteristic
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)—44no—
305AGL(1,ℤ/8)—44no—
306C2×SD16—22no—
307C2×SD16—22no—
308C2×D8—22no—
309C2×D8—22no—
310C2×D8—1222no—
311C2×D8—1222no—
312C2×D8—1222no—
313#131—2444no—
314C2×M16—1222no—
315C2×M16C22⋊D6111yescharacteristic
316C2×M16C22⋊D6111yescharacteristic
317C22×C8—1221no—
318C22×C8—1221no—
319C22×C8—1222no—
320C4⋊Q8—611no—
321C4⋊Q8C22×D6111yescharacteristic
322#127—21no—
323#127—21no—
324#127—21no—
325#127—611no—
326#127C22×D6112yes—
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—611no—
341C4⋊C8—611no—
342C4⋊C8—612no—
343C4⋊C8—611no—
344C4⋊C8—611no—
345C4⋊C8—1221no—
346C4⋊C8—1221no—
347#102—1222no—
348#102—1222no—
349#102—1222no—
350#102—1222no—
351C22⋊C8—1221no—
352C22⋊C8—1222no—
353C22⋊C8—1222no—
354#97C22⋊D6112yes—
355C4×C8—1221no—
356C4×C8C22×D6112yes—
357C24—41no—
358#41—42no—
359#41—42no—
360#41—2442no—
361#41—2442no—
362C2×Q8—21no—
363C2×Q8—1221no—
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—21no—
369C2×D4—22no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—81no—
377C2×D4—81no—
378C2×D4—82no—
379C2×D4—612no—
380C2×D4—1222no—
381C2×D4—1221no—
382C2×D4—1221no—
383C22×C4—41no—
384C22×C4—1222no—
385C22×C4D4×D6111yescharacteristic
386C22×C4#1175822411yescharacteristic
387C22×C4C2×C22⋊D6111yescharacteristic
388SD16—44no—
389SD16—44no—
390D8—44no—
391D8—44no—
392D8—4882no—
393D8—4882no—
394D8—4882no—
395M16—22no—
396M16—2444no—
397M16#1175819914yescomplemented
398C2×C8—1221no—
399C2×C8—1221no—
400C2×C8—1222no—
401C2×C8—1221no—
402C2×C8—1221no—
403C2×C8—4881no—
404C2×C8—2442no—
405C2×C8—2444no—
406C2×C8C2×C22⋊D6112yes—
407C2×C8C2×C22⋊D6112yes—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—612no—
411C4⋊C4—1221no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—1222no—
415C22⋊C4—1222no—
416C42D4×D6111yescharacteristic
417C42#1175822411yescharacteristic
418C42C2×C22⋊D6111yescharacteristic
419C42C23×D6111yescharacteristic
420C23—41no—
421C23—42no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23—1222no—
426C23#1451974811yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—2442no—
430D4—41no—
431D4—42no—
432D4—41no—
433D4—42no—
434D4—41no—
435D4—42no—
436D4—42no—
437D4—41no—
438D4—81no—
439D4—82no—
440D4—82no—
441D4—82no—
442D4—82no—
443D4—82no—
444D4—1222no—
445D4—1222no—
446D4—4881no—
447D4—2444no—
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#1451974811yescharacteristic
458C2×C4#1451974811yescharacteristic
459C2×C4#1451974811yescharacteristic
460C2×C4C2×#1175819912yes—
461C2×C4#1451980312yes—
462C2×C4C2×D4×D6111yescharacteristic
463C2×C4C22×C22⋊D6111yescharacteristic
464C2×C4C2×#1175822411yescharacteristic
465C8—22no—
466C8—2441no—
467C8—2441no—
468C8—2442no—
469C8—2441no—
470C8—2441no—
471C8C2×#1175819914yes—
472C22—21no—
473C22—21no—
474C22—81no—
475C22—82no—
476C22—82no—
477C22—81no—
478C22—82no—
479C22—81no—
480C22—164no—
481C22—2442no—
482C22—2442no—
483C22—2442no—
484C22#1683995412yes—
485C22C2×#1451974811yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—81no—
490C4—2442no—
491C4#1683995412yes—
492C4C22×#1175819911yescharacteristic
493C4C2×#1451980311yescharacteristic
494C4#1684056611yescharacteristic
495C4#1684057211yescharacteristic
496C2—22no—
497C2—162no—
498C2—161no—
499C2—4882no—
500C2C2×#1683995411yescharacteristic
501C2#2145549311yescharacteristic
502C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3151C2
socleC22×C61C2×#121
hypercenter#3151D61
hypocenterC61#19230
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3151

Complements

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

NH ≅ G/Ntype
1. #21457126C2semidirect
2. #21456737C2semidirect
3. #21454565C2semidirect
4. #21445295C2semidirect
5. #21445281C2semidirect
6. #21445171C2semidirect
7. #21445157C2semidirect
8. #21444571C2semidirect
9. #21444468C2semidirect
10. C2×#16839506C2semidirect
11. C61×#3151C2semidirect
17. #16840039C22semidirect
22. #16839954C22semidirect
24. #16839841C22semidirect
26. C8×D244C22semidirect
31. #16839515C22semidirect
32. #16839513C22semidirect
33. #16839508C22semidirect
34. #16839508C22semidirect
37. #16839506C22semidirect
38. #16839506C22semidirect
141. M16×C61D8semidirect
247. #3151D61semidirect
298. C61#19230semidirect
397. M16#11758199semidirect

Maximal subgroups

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

subgroupindex
#214571262
#214567372
#214545652
#214452952
#214452812
#214451712
#214451572
#214445712
#214444682
C2×#168395062
C61×#31512
#1923061