Finite Group Databasebeta

← #29662867

Subgroups — #29662867

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#21456269C211yesmaximal, characteristic
2#21455851C211yesmaximal, characteristic, complemented
3C2×#16839959C211yesmaximal, characteristic, complemented
4#21454770C211yesmaximal, characteristic, complemented
5#21445302C212yesmaximal, complemented
6#21445290C212yesmaximal, complemented
7#21445178C212yesmaximal, complemented
8#21445166C212yesmaximal, complemented
9#21444572C211yesmaximal, characteristic, complemented
10#21444497C211yesmaximal, characteristic, complemented
11C61×#1545C211yesmaximal, characteristic, complemented
12C2×#14519803—21no—
13C2×#14519803—21no—
14C22×#11758200—21no—
15#16840312C2211yescharacteristic
16C2×C4×Dic122C2211yescharacteristic
17#16840149—22no—
18#16840138C2212yes—
19#16840134C2212yes—
20#16840081—21no—
21#16840048C2211yescharacteristic, complemented
22#16840048C2211yescharacteristic, complemented
23#16839963C2211yescharacteristic, complemented
24#16839963C2211yescharacteristic, complemented
25#16839959—21no—
26#16839959—21no—
27#16839959C2212yes—
28#16839959C2212yes—
29#16839903—21no—
30#16839891—22no—
31#16839889C2212yescomplemented
32#16839886C2212yescomplemented
33C4×#11758190—21no—
34C4×#11758190—21no—
35#16839785—21no—
36C2×C4⋊#92418C2211yescharacteristic
37C2×#14519698—21no—
38C4×#11758194C2211yescharacteristic
39M16×Dic61—21no—
40#16839514C2212yescomplemented
41#16839512C2212yescomplemented
42#16839509C2212yescomplemented
43#16839505C2212yescomplemented
44C2×#127×C61C2211yescharacteristic
45C61×#431C2211yescharacteristic
46C61×#423C2211yescharacteristic
47C61×#331C2212yes—
48C61×#331C2212yes—
49C2×#11758222—21no—
50C22×Dic122D411yescharacteristic
51#14519803—44no—
52#14519803—44no—
53C2×#11758190—21no—
54C2×#11758190—21no—
55C2×#11758200—22no—
56C2×#11758200—22no—
57C2×#11758200—22no—
58C2×#11758200—22no—
59C2×#11758200—44no—
60C2×#11758199—22no—
61C2×#11758199—22no—
62C2×#11758194D411yescharacteristic
63C2×#11758194D411yescharacteristic
64C22×#92418—21no—
65#14519777C2311yescharacteristic
66#14519774C2311yescharacteristic
67#14519765—21no—
68#14519751—21no—
69#14519751—21no—
70C4×Dic122—21no—
71C4×Dic122—21no—
72C4×Dic122D411yescharacteristic
73C4×Dic122D411yescharacteristic
74C4×Dic122C2311yescharacteristic
75C4×Dic122C2311yescharacteristic
76C4×D244—21no—
77C4×D244—21no—
78C2×C4⋊Dic61D411yescharacteristic
79C2×#11758196—21no—
80C2×C4×Dic61—21no—
81#14519709—21no—
82#14519709—21no—
83C4⋊#92418—21no—
84C4⋊#92418C2312yes—
85#14519706—22no—
86#14519706—22no—
87#14519702—21no—
88#14519702—21no—
89#14519699—21no—
90#14519699—21no—
91#14519698—22no—
92#14519698—22no—
93#14519688D412yes—
94#14519687—22no—
95C4×#92418C2312yes—
96C8×Dic61—22no—
97C22×D4×C61—21no—
98C22×D4×C61—21no—
99C22×D4×C61—21no—
100C2×M16×C61—21no—
101#127×C61—21no—
102#127×C61—21no—
103#127×C61—21no—
104#127×C61C2312yes—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—21no—
107C4⋊C8×C61C2311yescharacteristic
108C4⋊C8×C61C2311yescharacteristic
109C4⋊C8×C61C2312yes—
110C22⋊C8×C61—21no—
111C4×C8×C61—21no—
112C24×C61—41no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—42no—
117C2×D4×C61—42no—
118C2×D4×C61—41no—
119C2×D4×C61—41no—
120C2×D4×C61—41no—
121C2×D4×C61—42no—
122C2×D4×C61—42no—
123C2×D4×C61—82no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61#4111yescharacteristic
127C2×C22⋊D61—41no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130#11758222—42no—
131#11758222—42no—
132C2×D244—21no—
133C2×D244—21no—
134C2×D244—21no—
135C2×C4×D61—41no—
136C2×Dic122—22no—
137C2×Dic122—21no—
138C2×Dic122—21no—
139C2×Dic122—41no—
140C2×Dic122C2×D411yescharacteristic
141C2×Dic122C2×D411yescharacteristic
142M16×C61—42no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61—22no—
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C42×C61#4111yescharacteristic
149C42×C61C2411yescharacteristic
150#11758200—44no—
151#11758200—44no—
152#11758200—44no—
153#11758200—44no—
154#11758199—44no—
155#11758199—44no—
156#11758198—41no—
157#11758198—41no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61C2×D411yescharacteristic
162C4⋊Dic61C2×D411yescharacteristic
163#11758196—21no—
164#11758196—21no—
165#11758196—21no—
166#11758196—21no—
167C4×Dic61—21no—
168C4×Dic61—21no—
169C4×Dic61—21no—
170C4×Dic61—21no—
171#11758194—22no—
172#11758194SD1614yescomplemented
173C2×#92418—22no—
174C2×#92418—42no—
175C2×#92418C2×D412yes—
176C2×#92418C2×D412yes—
177#11758190—42no—
178#11758190—42no—
179C23×C61—42no—
180C23×C61—82no—
181C23×C61—82no—
182C23×C61—82no—
183C23×C61#12111yescharacteristic
184C22×D61—41no—
185D4×C61—42no—
186D4×C61—42no—
187D4×C61—42no—
188D4×C61—42no—
189D4×C61—82no—
190D4×C61—82no—
191D4×C61—82no—
192D4×C61—82no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—41no—
196C2×C4×C61#12111yescharacteristic
197C2×C4×C61#12111yescharacteristic
198C2×C4×C61#12111yescharacteristic
199C2×C4×C61C2×SD1612yes—
200C2×C4×C61AGL(1,ℤ/8)12yes—
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C22⋊D61—82no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—21no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—81no—
212D244—41no—
213D244—41no—
214D244—41no—
215D244—41no—
216D244—81no—
217C4×D61—82no—
218Dic122—21no—
219Dic122—21no—
220Dic122—21no—
221Dic122—42no—
222Dic122—41no—
223Dic122—41no—
224Dic122AGL(1,ℤ/8)12yescomplemented
225C8×C61—41no—
226C8×C61—42no—
227C8×C61—41no—
228#92418—22no—
229#92418—42no—
230#92418C2×SD1614yescomplemented
231#19170—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—82no—
235C22×C61—82no—
236C22×C61—82no—
237C22×C61—164no—
238C22×C61#46012yes—
239C22×C61C2×#12111yescharacteristic
240C2×D61—81no—
241C2×D61—81no—
242C2×D61—81no—
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#1565—611no—
265#1547—611no—
266#1545D6111yescharacteristic, complemented
267C2×C61—22no—
268C2×C61—162no—
269C2×C61C2×#46011yescharacteristic
270C2×C61#305511yescharacteristic
271C2×C61#306811yescharacteristic
272D61—161no—
273C2×AGL(1,ℤ/8)—1221no—
274C2×AGL(1,ℤ/8)—1221no—
275C22×SD16—1221no—
276#532—611no—
277C2×#127C2×D6111yescharacteristic
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—1221no—
299#423C2×D6111yescharacteristic
300C2×C4⋊C8—611no—
301C2×#102—1221no—
302C4×M16—611no—
303C4×M16—1221no—
304#335—612no—
305#334—612no—
306#332—612no—
307#331—612no—
308#331C2×D6112yes—
309#331C2×D6112yes—
310C61#1917011yescharacteristic, complemented, Sylow
311C2×#41—1221no—
312C22×Q8—611no—
313C22×D4—21no—
314C22×D4—21no—
315C22×D4—21no—
316AGL(1,ℤ/8)—2444no—
317AGL(1,ℤ/8)—2444no—
318C2×SD16—1221no—
319C2×SD16—1221no—
320C2×SD16—1222no—
321C2×SD16—1222no—
322C2×SD16—1222no—
323C2×SD16—1222no—
324C2×SD16—2444no—
325C2×D8—1222no—
326C2×D8—1222no—
327C2×M16—21no—
328C2×M16—611no—
329C2×M16—611no—
330C22×C8—1221no—
331C4⋊Q8—611no—
332#127—21no—
333#127—21no—
334#127—21no—
335#127—611no—
336#127C22×D6112yes—
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×D6112yes—
360C4⋊C8C22×D6111yescharacteristic
361C4⋊C8C22×D6111yescharacteristic
362#103—1221no—
363#103—1221no—
364#102—1221no—
365#102—1221no—
366#102—1222no—
367#102—1222no—
368C22⋊C8—21no—
369#97—612no—
370#97—1222no—
371C4×C8—21no—
372C4×C8—612no—
373C4×C8—1222no—
374C24—41no—
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—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—82no—
394C2×D4—1221no—
395C2×D4—1221no—
396C2×D4—1221no—
397C2×D4—2441no—
398C22×C4—1221no—
399C22×C4—1221no—
400C22×C4—2441no—
401C22×C4#1175822211yescharacteristic
402C22×C4D4×D6111yescharacteristic
403C22×C4D4×D6111yescharacteristic
404SD16—2442no—
405SD16—2442no—
406SD16—2444no—
407SD16—2444no—
408SD16—2444no—
409SD16—2444no—
410D8—2444no—
411D8—2444no—
412M16—42no—
413M16—614no—
414M16—1222no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—22no—
418C2×C8—612no—
419C2×C8—612no—
420C2×C8—1222no—
421C2×C8—2442no—
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—
437C42#1175822211yescharacteristic
438C42D4×D6111yescharacteristic
439C42D4×D6111yescharacteristic
440C42C23×D6111yescharacteristic
441C23—42no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—2441no—
446C23#1451975011yescharacteristic
447Q8—612no—
448Q8—1221no—
449Q8—1221no—
450Q8—1221no—
451Q8—2442no—
452Q8—2441no—
453Q8—2441no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—82no—
459D4—82no—
460D4—82no—
461D4—82no—
462D4—4881no—
463D4—4882no—
464D4—2441no—
465D4—2441no—
466D4—2441no—
467D4—2441no—
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#1451975011yescharacteristic
480C2×C4#1451975011yescharacteristic
481C2×C4#1451975011yescharacteristic
482C2×C4SD16×D6112yes—
483C2×C4#1451980112yes—
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4C2×D4×D6111yescharacteristic
486C2×C4C2×#1175822211yescharacteristic
487C8—41no—
488C8—42no—
489C8—41no—
490C8—614no—
491C8—1222no—
492C8—2442no—
493C22—21no—
494C22—21no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—164no—
499C22—4881no—
500C22—4881no—
501C22—4881no—
502C22#1683996112yes—
503C22C2×#1451975011yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—1222no—
508C4—4881no—
509C4—2441no—
510C4—2441no—
511C4—2441no—
512C4#1683996112yes—
513C4C2×SD16×D6111yescharacteristic
514C4AGL(1,ℤ/8)×D6111yescharacteristic
515C4C2×#1451980111yescharacteristic
516C4#1684055811yescharacteristic
517C2—22no—
518C2—162no—
519C2—9761no—
520C2C2×#1683996111yescharacteristic
521C2#2145543611yescharacteristic
522C2#2145554611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1545C2
socleC22×C61C2×#121
prime radical#21456269C2
hypercenter#1545D61
hypocenterC61#19170
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1545

Complements

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

NH ≅ G/Ntype
2. #21455851C2semidirect
3. C2×#16839959C2semidirect
4. #21454770C2semidirect
5. #21445302C2semidirect
6. #21445290C2semidirect
7. #21445178C2semidirect
8. #21445166C2semidirect
9. #21444572C2semidirect
10. #21444497C2semidirect
11. C61×#1545C2semidirect
21. #16840048C22semidirect
22. #16840048C22semidirect
23. #16839963C22semidirect
24. #16839963C22semidirect
31. #16839889C22semidirect
32. #16839886C22semidirect
40. #16839514C22semidirect
41. #16839512C22semidirect
42. #16839509C22semidirect
43. #16839505C22semidirect
172. #11758194SD16semidirect
224. Dic122AGL(1,ℤ/8)semidirect
230. #92418C2×SD16semidirect
266. #1545D61semidirect
310. C61#19170semidirect

Maximal subgroups

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

subgroupindex
#214562692
#214558512
C2×#168399592
#214547702
#214453022
#214452902
#214451782
#214451662
#214445722
#214444972
C61×#15452
#1917061