Finite Group Databasebeta

← #29663497

Subgroups — #29663497

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#21456734C211yesmaximal, characteristic, complemented
3#21456271C211yesmaximal, characteristic, complemented
4#21445298C212yesmaximal, complemented
5#21445284C212yesmaximal, complemented
6#21445174C212yesmaximal, complemented
7#21445160C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444469C211yesmaximal, characteristic, complemented
10C2×#16839505C211yesmaximal, characteristic, complemented
11C61×#2935C211yesmaximal, characteristic, complemented
12C2×#14519805—21no—
13C2×#14519805—21no—
14#16840312C2211yescharacteristic
15C2×#14519774C2211yescharacteristic
16#16840151—22no—
17#16840139C2212yes—
18#16840138—21no—
19#16840135C2212yes—
20#16840110—21no—
21#16840042C2212yescomplemented
22#16840039—21no—
23#16839996—22no—
24#16839957C2212yes—
25#16839957C2212yes—
26#16839957C2212yescomplemented
27#16839954—21no—
28#16839895—22no—
29#16839815C2211yescharacteristic
30C2×C4⋊#92418C2211yescharacteristic
31#16839515C2212yescomplemented
32#16839513C2212yescomplemented
33#16839509C2211yescharacteristic, complemented
34#16839509C2211yescharacteristic, complemented
35#16839505C2212yes—
36#16839505C2212yes—
37#16839505C2211yescharacteristic, complemented
38#16839505C2211yescharacteristic, complemented
39C2×#131×C61—21no—
40C61×#532C2211yescharacteristic
41C61×#436C2212yescomplemented
42C61×#435—22no—
43C8×D4×C61C2212yescomplemented
44C2×C4⋊C8×C61C2211yescharacteristic
45C61×#408—21no—
46C4×M16×C61C2211yescharacteristic
47C2×#11758222—21no—
48C22×D244—21no—
49C22×D244—21no—
50C22×D244—21no—
51#14519805—44no—
52#14519805—44no—
53C2×#11758190—22no—
54C2×#11758190—22no—
55C2×D488—22no—
56C2×D488—22no—
57C2×#11758199—22no—
58C2×#11758199—22no—
59C2×#11758199—22no—
60C22×#92418—21no—
61#14519777C2311yescharacteristic
62#14519774—21no—
63#14519774—21no—
64#14519774—21no—
65#14519774C2312yes—
66#14519774C2311yescharacteristic
67#14519765—21no—
68#14519751—21no—
69#14519751—21no—
70C4×D244—21no—
71C4×D244—21no—
72#14519710—22no—
73C4⋊#92418—21no—
74C4⋊#92418C2311yescharacteristic
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2312yes—
77#14519698—22no—
78#14519697—22no—
79#14519697—22no—
80#14519697—22no—
81C22⋊#92418—21no—
82C4×#92418—21no—
83C2×#41×C61—21no—
84#131×C61—44no—
85C2×M16×C61—22no—
86C2×M16×C61D411yescharacteristic
87C2×M16×C61D411yescharacteristic
88C22×C8×C61—21no—
89C22×C8×C61—22no—
90C4⋊Q8×C61D411yescharacteristic
91#127×C61D411yescharacteristic
92#124×C61—21no—
93#121×C61—21no—
94#121×C61D412yes—
95C4×D4×C61—21no—
96C4×D4×C61C2312yes—
97C2×C42×C61C2311yescharacteristic
98C4⋊C8×C61—21no—
99C4⋊C8×C61C2311yescharacteristic
100C4⋊C8×C61C2311yescharacteristic
101C22⋊C8×C61—22no—
102C22⋊C8×C61—22no—
103#97×C61D412yes—
104C4×C8×C61C2312yes—
105C23×D61—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×C61C2×D412yes—
113C22×C4×C61—22no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61#4111yescharacteristic
117C2×C22⋊D61—41no—
118#11758222—42no—
119#11758222—42no—
120C2×D244—22no—
121C2×D244—22no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—22no—
125C2×D244—21no—
126C2×D244—42no—
127C2×D244—42no—
128C2×D244—41no—
129C2×D244—41no—
130C2×D244—41no—
131C2×D244—81no—
132C2×D244—81no—
133C2×D244—82no—
134C2×C4×D61—41no—
135C2×Dic122—21no—
136M16×C61—22no—
137M16×C61—44no—
138M16×C61D814yescomplemented
139C2×C8×C61—21no—
140C2×C8×C61—21no—
141C2×C8×C61—42no—
142C2×C8×C61—44no—
143C2×C8×C61C2×D412yes—
144C2×C8×C61C2×D412yes—
145C4⋊C4×C61—21no—
146C4⋊C4×C61C2×D412yes—
147C22⋊C4×C61—22no—
148C22⋊C4×C61—22no—
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61#4111yescharacteristic
152C42×C61C2411yescharacteristic
153#11758199—82no—
154#11758199—82no—
155#11758199—82no—
156#11758198—41no—
157#11758198—41no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C2×#92418—21no—
161C2×#92418—21no—
162C2×#92418—22no—
163C2×#92418—81no—
164D488—44no—
165D488—44no—
166#11758190—44no—
167#11758190—44no—
168C23×C61—22no—
169C23×C61#12111yescharacteristic
170C22×D61—41no—
171C22×D61—42no—
172C22×D61—82no—
173C22×D61—82no—
174C22×D61—82no—
175Q8×C61—42no—
176D4×C61—22no—
177D4×C61—22no—
178D4×C61—44no—
179D4×C61—42no—
180D4×C61—81no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—22no—
184C2×C4×C61—41no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61#12111yescharacteristic
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61C2×D812yes—
191C2×C4×C61AGL(1,ℤ/8)12yes—
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C22⋊D61—82no—
196C2×Dic61—41no—
197D244—42no—
198D244—41no—
199D244—42no—
200D244—41no—
201D244—41no—
202D244—42no—
203D244—42no—
204D244—41no—
205D244—81no—
206D244—82no—
207D244—82no—
208D244—82no—
209D244—82no—
210C4×D61—82no—
211Dic122—41no—
212Dic122—41no—
213C8×C61—22no—
214C8×C61—41no—
215C8×C61—41no—
216C8×C61C2×D814yes—
217#92418—42no—
218#92418—41no—
219#92418—41no—
220#19230—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—42no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61#45912yes—
227C22×C61C2×#12111yescharacteristic
228C2×D61—81no—
229C2×D61—82no—
230C2×D61—82no—
231C2×D61—81no—
232C2×D61—82no—
233C2×D61—81no—
234C2×D61—164no—
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#3205—611no—
247#3151—611no—
248#2935D6111yescharacteristic, complemented
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×C61C2×#45911yescharacteristic
259C2×C61#306211yescharacteristic
260C2×C61#306811yescharacteristic
261D61—161no—
262D61—162no—
263C2×AGL(1,ℤ/8)—1221no—
264C2×AGL(1,ℤ/8)—1221no—
265C2×#131—21no—
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#436C2×D6112yescomplemented
283#435—22no—
284C8×D4C2×D6112yescomplemented
285#431—611no—
286C2×C4⋊C8—611no—
287C2×C4⋊C8C2×D6111yescharacteristic
288#408—21no—
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—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—44no—
314C2×M16—22no—
315C2×M16C22⋊D6111yescharacteristic
316C2×M16C22⋊D6111yescharacteristic
317C22×C8—21no—
318C22×C8—22no—
319C22×C8—1221no—
320C4⋊Q8—611no—
321C4⋊Q8D24411yescharacteristic
322#127—612no—
323#127—611no—
324#127—1221no—
325#127—1221no—
326#127—1221no—
327#127D24411yescharacteristic
328#124—21no—
329#124—1221no—
330#121—21no—
331#121—1221no—
332#121—1221no—
333#121D24412yes—
334C4×D4—21no—
335C4×D4—1221no—
336C4×D4—1221no—
337C4×D4C22×D6112yes—
338C2×C42C22×D6111yescharacteristic
339#107—1222no—
340C4⋊C8—21no—
341C4⋊C8—612no—
342C4⋊C8—611no—
343C4⋊C8—611no—
344C4⋊C8—1221no—
345C4⋊C8C22×D6111yescharacteristic
346C4⋊C8C22×D6111yescharacteristic
347#102—1222no—
348#102—1222no—
349#102—1222no—
350#102—1222no—
351C22⋊C8—22no—
352C22⋊C8—22no—
353C22⋊C8—1221no—
354#97C22⋊D6112yes—
355C4×C8—1221no—
356C4×C8C22×D6112yes—
357C24—2441no—
358#41—42no—
359#41—42no—
360#41—2442no—
361#41—2442no—
362C2×Q8—21no—
363C2×Q8—1221no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—1221no—
368C2×D4—1222no—
369C2×D4—1221no—
370C2×D4—1222no—
371C2×D4—1221no—
372C2×D4—1222no—
373C2×D4—4881no—
374C2×D4—4881no—
375C2×D4—4882no—
376C2×D4—2441no—
377C2×D4—2441no—
378C2×D4—2441no—
379C2×D4—2441no—
380C2×D4—2442no—
381C2×D4—2442no—
382C2×D4C2×D24412yes—
383C22×C4—22no—
384C22×C4—2441no—
385C22×C4C2×D24411yescharacteristic
386C22×C4#1175822211yescharacteristic
387C22×C4C2×C22⋊D6111yescharacteristic
388SD16—2444no—
389SD16—2444no—
390D8—4882no—
391D8—4882no—
392D8—4882no—
393D8—2444no—
394D8—2444no—
395M16—22no—
396M16—44no—
397M16#1175819914yescomplemented
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—42no—
401C2×C8—44no—
402C2×C8—1221no—
403C2×C8—1221no—
404C2×C8—1222no—
405C2×C8—4881no—
406C2×C8C2×C22⋊D6112yes—
407C2×C8C2×C22⋊D6112yes—
408C4⋊C4—21no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4C2×D24412yes—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C42C2×D24411yescharacteristic
417C42#1175822211yescharacteristic
418C42C2×C22⋊D6111yescharacteristic
419C42C23×D6111yescharacteristic
420C23—22no—
421C23—4882no—
422C23—4882no—
423C23—4882no—
424C23—2441no—
425C23—2442no—
426C23#1451975111yescharacteristic
427Q8—42no—
428Q8—2441no—
429Q8—2441no—
430D4—22no—
431D4—22no—
432D4—44no—
433D4—42no—
434D4—81no—
435D4—4881no—
436D4—4882no—
437D4—4882no—
438D4—4882no—
439D4—4882no—
440D4—4882no—
441D4—2441no—
442D4—2442no—
443D4—2441no—
444D4—2442no—
445D4—2441no—
446D4—2442no—
447D4—2442no—
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#1451975111yescharacteristic
458C2×C4#1451975111yescharacteristic
459C2×C4#1451975111yescharacteristic
460C2×C4C2×#1175819912yes—
461C2×C4#1451980212yes—
462C2×C4C22×C22⋊D6111yescharacteristic
463C2×C4C22×D24411yescharacteristic
464C2×C4C2×#1175822211yescharacteristic
465C8—22no—
466C8—41no—
467C8—41no—
468C8—2442no—
469C8—2441no—
470C8—2441no—
471C8C2×#1175819914yes—
472C22—21no—
473C22—21no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—9764no—
478C22—4881no—
479C22—4882no—
480C22—4882no—
481C22—4881no—
482C22—4882no—
483C22—4881no—
484C22#1683995712yes—
485C22C2×#1451975111yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—42no—
490C4—4881no—
491C4#1683995712yes—
492C4C22×#1175819911yescharacteristic
493C4C2×#1451980211yescharacteristic
494C4#1684056711yescharacteristic
495C4#1684057411yescharacteristic
496C2—22no—
497C2—82no—
498C2—9762no—
499C2—9761no—
500C2C2×#1683995711yescharacteristic
501C2#2145549911yescharacteristic
502C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2935C2
socleC22×C61C2×#121
hypercenter#2935D61
hypocenterC61#19230
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2935

Complements

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

NH ≅ G/Ntype
1. #21457126C2semidirect
2. #21456734C2semidirect
3. #21456271C2semidirect
4. #21445298C2semidirect
5. #21445284C2semidirect
6. #21445174C2semidirect
7. #21445160C2semidirect
8. #21444571C2semidirect
9. #21444469C2semidirect
10. C2×#16839505C2semidirect
11. C61×#2935C2semidirect
21. #16840042C22semidirect
26. #16839957C22semidirect
31. #16839515C22semidirect
32. #16839513C22semidirect
33. #16839509C22semidirect
34. #16839509C22semidirect
37. #16839505C22semidirect
38. #16839505C22semidirect
41. C61×#436C22semidirect
43. C8×D4×C61C22semidirect
138. M16×C61D8semidirect
248. #2935D61semidirect
282. #436C2×D61semidirect
284. C8×D4C2×D61semidirect
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
#214567342
#214562712
#214452982
#214452842
#214451742
#214451602
#214445712
#214444692
C2×#168395052
C61×#29352
#1923061