Finite Group Databasebeta

← #29665431

Subgroups — #29665431

Order 15616.

758 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456737C211yesmaximal, characteristic, complemented
2#21455274C211yesmaximal, characteristic, complemented
3#21454930C211yesmaximal, characteristic, complemented
4#21445261C212yesmaximal, complemented
5#21445254C212yesmaximal, complemented
6#21445177C212yesmaximal, complemented
7#21445172C212yesmaximal, complemented
8#21444595C211yesmaximal, characteristic, complemented
9#21444497C211yesmaximal, characteristic, complemented
10C2×#16839506C211yesmaximal, characteristic, complemented
11C61×#3038C211yesmaximal, characteristic, complemented
12C2×#14519797—21no—
13#16840312C2211yescharacteristic
14#16840138C2212yes—
15#16840108C2212yes—
16#16839984—21no—
17#16839977—21no—
18#16839972C2211yescharacteristic, complemented
19#16839967C2211yescharacteristic, complemented
20#16839962C2211yescharacteristic, complemented
21#16839957C2211yescharacteristic, complemented
22#16839954—21no—
23#16839954—21no—
24#16839896—22no—
25Q8×#92418C2212yescomplemented
26D4×#92418—22no—
27D4×#92418C2212yescomplemented
28#16839815—21no—
29#16839815C2211yescharacteristic
30C2×C4⋊#92418—21no—
31C2×C4⋊#92418C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33#16839756—21no—
34#16839514C2212yescomplemented
35#16839514C2212yescomplemented
36#16839509C2212yescomplemented
37#16839506C2212yes—
38#16839506C2212yes—
39#16839505C2212yescomplemented
40C2×AGL(1,ℤ/8)×C61—21no—
41C2×#127×C61C2211yescharacteristic
42C4×#41×C61C2211yescharacteristic
43C61×#460—21no—
44C61×#460C2212yes—
45C61×#459—21no—
46C61×#459C2212yes—
47C61×#423C2211yescharacteristic
48C61×#418—21no—
49C2×#11758222—21no—
50#14519797—44no—
51C2×#11758202—21no—
52C2×#11758200—21no—
53C2×#11758201—21no—
54C2×#11758199—21no—
55C2×#11758199—22no—
56C2×#11758199—22no—
57C22×#92418—21no—
58C22×#92418—21no—
59C22×#92418—22no—
60#14519777C2311yescharacteristic
61#14519774C2311yescharacteristic
62#14519765—21no—
63#14519751—21no—
64#14519751—21no—
65C4×D244—21no—
66C4×D244—21no—
67C4⋊#92418—21no—
68C4⋊#92418—21no—
69C4⋊#92418—22no—
70C4⋊#92418D411yescharacteristic
71C4⋊#92418D411yescharacteristic
72C4⋊#92418C2311yescharacteristic
73C4⋊#92418C2311yescharacteristic
74C4⋊#92418C2312yes—
75#14519700—21no—
76#14519700—21no—
77#14519698—22no—
78#14519698—22no—
79#14519697—21no—
80#14519697—21no—
81C22⋊#92418—22no—
82C22⋊#92418D412yes—
83C4×#92418—22no—
84C4×#92418C2312yes—
85C2×#41×C61D411yescharacteristic
86C22×D4×C61—21no—
87C22×D4×C61—21no—
88C22×D4×C61—21no—
89AGL(1,ℤ/8)×C61—44no—
90C2×SD16×C61—22no—
91C2×D8×C61—22no—
92C2×M16×C61—21no—
93#127×C61—21no—
94#127×C61—21no—
95#127×C61—21no—
96#127×C61C2312yes—
97C4×Q8×C61D411yescharacteristic
98C4×Q8×C61C2311yescharacteristic
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C4×D4×C61D411yescharacteristic
102C4×D4×C61C2311yescharacteristic
103#117×C61—21no—
104#117×C61D411yescharacteristic
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109#102×C61—22no—
110#102×C61—22no—
111C24×C61—41no—
112#41×C61—22no—
113#41×C61D814yescomplemented
114C2×Q8×C61C2×D411yescharacteristic
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—41no—
120C2×D4×C61—41no—
121C2×D4×C61—41no—
122C2×D4×C61—42no—
123C2×D4×C61—42no—
124C2×D4×C61—81no—
125C2×D4×C61—81no—
126C2×D4×C61—82no—
127C2×D4×C61C2×D411yescharacteristic
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61#4111yescharacteristic
134C2×C22⋊D61—41no—
135#11758222—42no—
136#11758222—42no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×C4×D61—41no—
141C2×Dic122—21no—
142SD16×C61—44no—
143D8×C61—44no—
144M16×C61—42no—
145C2×C8×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61C2×D411yescharacteristic
149C4⋊C4×C61C2×D411yescharacteristic
150C22⋊C4×C61—21no—
151C22⋊C4×C61—21no—
152C22⋊C4×C61—21no—
153C42×C61—21no—
154C42×C61—21no—
155C42×C61—21no—
156C42×C61C2×D411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C42×C61#4111yescharacteristic
159C42×C61C2411yescharacteristic
160#11758202—42no—
161#11758201—42no—
162#11758200—42no—
163#11758199—42no—
164#11758199—82no—
165#11758199—82no—
166#11758198—41no—
167#11758198—41no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C2×#92418—22no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—42no—
174C2×#92418—82no—
175C2×#92418—81no—
176C2×#92418C2×D412yes—
177C2×#92418C2×D412yes—
178C23×C61—21no—
179C23×C61—42no—
180C23×C61—82no—
181C23×C61—82no—
182C23×C61—82no—
183C23×C61#12111yescharacteristic
184C22×D61—41no—
185Q8×C61—21no—
186Q8×C61C2×D812yes—
187D4×C61—24no—
188D4×C61—21no—
189D4×C61—42no—
190D4×C61—42no—
191D4×C61—42no—
192D4×C61—42no—
193D4×C61—82no—
194D4×C61—82no—
195D4×C61—82no—
196D4×C61—82no—
197D4×C61C2×D812yes—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—24no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—81no—
208C2×C4×C61#12111yescharacteristic
209C2×C4×C61#12111yescharacteristic
210C2×C4×C61#12111yescharacteristic
211C2×C4×C61C2×D812yes—
212C2×C4×C61#13512yes—
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C22⋊D61—82no—
217C2×Dic61—41no—
218D244—41no—
219D244—41no—
220D244—41no—
221D244—41no—
222D244—81no—
223C4×D61—82no—
224Dic122—41no—
225Dic122—41no—
226C8×C61—42no—
227#92418—22no—
228#92418—22no—
229#92418—41no—
230#92418—41no—
231#92418—42no—
232#19410—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—22no—
236C22×C61—41no—
237C22×C61—41no—
238C22×C61—82no—
239C22×C61—82no—
240C22×C61—82no—
241C22×C61—164no—
242C22×C61#46612yes—
243C22×C61C2×#12111yescharacteristic
244C2×D61—81no—
245C2×D61—81no—
246C2×D61—81no—
247C4×C61—21no—
248C4×C61—22no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—41no—
252C4×C61—41no—
253C4×C61#46612yes—
254C4×C61C22×D811yescharacteristic
255C4×C61C2×#13511yescharacteristic
256C4×C61#57611yescharacteristic
257C4×C61#57711yescharacteristic
258Dic61—81no—
259#3205—611no—
260#3038D6111yescharacteristic, complemented
261#3035—611no—
262#2988—611no—
263#1682—612no—
264#1681—612no—
265#1668—612no—
266#1667—612no—
267#1571—611no—
268#1547—611no—
269C2×#331—611no—
270C2×C61—22no—
271C2×C61—42no—
272C2×C61—162no—
273C2×C61#304511yescharacteristic
274C2×C61C2×#46611yescharacteristic
275C2×C61#306811yescharacteristic
276D61—161no—
277C2×AGL(1,ℤ/8)—21no—
278C2×#135—1221no—
279#532—611no—
280C2×#127C2×D6111yescharacteristic
281C4×#41C2×D6111yescharacteristic
282#493—612no—
283#486—612no—
284#464—1221no—
285#463—1221no—
286#462—611no—
287#461—611no—
288#460—21no—
289#460—611no—
290#460C2×D6112yes—
291#459—21no—
292#459—611no—
293#459—1221no—
294#459—1221no—
295#459C2×D6112yes—
296#447—1222no—
297C8×Q8—612no—
298C8×D4—612no—
299C8×D4—1222no—
300#431—611no—
301#431—1221no—
302#423C2×D6111yescharacteristic
303C2×C4⋊C8—611no—
304C2×C4⋊C8—611no—
305C2×C4⋊C8—1221no—
306#418—21no—
307#416—1221no—
308#335—612no—
309#335—612no—
310#332—612no—
311#331—612no—
312#331—612no—
313#331—612no—
314C61#1941011yescharacteristic, complemented, Sylow
315C2×#41—1221no—
316C2×#41C22⋊D6111yescharacteristic
317C22×D4—21no—
318C22×D4—21no—
319C22×D4—21no—
320AGL(1,ℤ/8)—44no—
321#135—2444no—
322C2×Q16—1221no—
323C2×SD16—22no—
324C2×SD16—1221no—
325C2×SD16—1221no—
326C2×D8—22no—
327C2×D8—1222no—
328C2×D8—1222no—
329C2×D8—1221no—
330C2×M16—21no—
331C22×C8—1222no—
332C22×C8—1221no—
333C22×C8—1221no—
334C4⋊Q8—611no—
335#127—21no—
336#127—21no—
337#127—21no—
338#127—611no—
339#127C22×D6112yes—
340#124—1221no—
341#121—1221no—
342#121—1221no—
343C4×Q8C22⋊D6111yescharacteristic
344C4×Q8C22×D6111yescharacteristic
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—1221no—
348C4×D4—1221no—
349C4×D4C22⋊D6111yescharacteristic
350C4×D4C22×D6111yescharacteristic
351#117—21no—
352#117C22⋊D6111yescharacteristic
353C2×C42—21no—
354C2×C42C22×D6111yescharacteristic
355C4⋊C8—21no—
356C4⋊C8—611no—
357C4⋊C8—611no—
358C4⋊C8—612no—
359C4⋊C8—611no—
360C4⋊C8—611no—
361C4⋊C8—1222no—
362C4⋊C8—1221no—
363C4⋊C8—1221no—
364C4⋊C8C22×D6112yes—
365#103—1221no—
366#103—1221no—
367#102—22no—
368#102—22no—
369#102—1222no—
370#102—1222no—
371#102—1221no—
372#102—1221no—
373C22⋊C8—612no—
374C22⋊C8—1222no—
375C4×C8—612no—
376C4×C8—1222no—
377C24—41no—
378#41—22no—
379#41—2442no—
380#41—2442no—
381#41#1175819914yescomplemented
382C2×Q8—1221no—
383C2×Q8C2×C22⋊D6111yescharacteristic
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—81no—
394C2×D4—81no—
395C2×D4—82no—
396C2×D4—1221no—
397C2×D4—1221no—
398C2×D4—1221no—
399C2×D4—2441no—
400C2×D4C2×C22⋊D6111yescharacteristic
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—2441no—
405C22×C4D4×D6111yescharacteristic
406C22×C4#1175822411yescharacteristic
407C22×C4C2×C22⋊D6111yescharacteristic
408Q16—2442no—
409SD16—44no—
410SD16—2442no—
411SD16—2442no—
412D8—44no—
413D8—4882no—
414D8—4882no—
415D8—2442no—
416M16—42no—
417C2×C8—22no—
418C2×C8—612no—
419C2×C8—612no—
420C2×C8—1222no—
421C2×C8—1221no—
422C2×C8—1221no—
423C2×C8—4882no—
424C2×C8—4881no—
425C2×C8—2442no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4C2×C22⋊D6111yescharacteristic
431C4⋊C4C2×C22⋊D6111yescharacteristic
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—2441no—
436C22⋊C4—2441no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42D4×D6111yescharacteristic
441C42#1175822411yescharacteristic
442C42C2×C22⋊D6111yescharacteristic
443C42C23×D6111yescharacteristic
444C23—21no—
445C23—42no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—2441no—
450C23#1451974911yescharacteristic
451Q8—21no—
452Q8—2441no—
453Q8—2441no—
454Q8C2×#1175819912yes—
455D4—24no—
456D4—21no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—82no—
462D4—82no—
463D4—82no—
464D4—82no—
465D4—4881no—
466D4—4882no—
467D4—2441no—
468D4—2441no—
469D4—2441no—
470D4—2441no—
471D4C2×#1175819912yes—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—81no—
482C2×C4—4882no—
483C2×C4—2441no—
484C2×C4#1451974911yescharacteristic
485C2×C4#1451974911yescharacteristic
486C2×C4#1451974911yescharacteristic
487C2×C4C2×#1175819912yes—
488C2×C4#1451979712yes—
489C2×C4C2×D4×D6111yescharacteristic
490C2×C4C22×C22⋊D6111yescharacteristic
491C2×C4C2×#1175822411yescharacteristic
492C8—42no—
493C8—1222no—
494C8—1222no—
495C8—2441no—
496C8—2441no—
497C8—2442no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—41no—
502C22—41no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—164no—
507C22—4881no—
508C22—4881no—
509C22—4881no—
510C22#1683999612yes—
511C22C2×#1451974911yescharacteristic
512C4—21no—
513C4—22no—
514C4—21no—
515C4—21no—
516C4—41no—
517C4—41no—
518C4—4881no—
519C4#1683999612yes—
520C4C22×#1175819911yescharacteristic
521C4C2×#1451979711yescharacteristic
522C4#1684056611yescharacteristic
523C4#1684057211yescharacteristic
524C2—22no—
525C2—42no—
526C2—162no—
527C2—9761no—
528C2#2145536211yescharacteristic
529C2C2×#1683999611yescharacteristic
530C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3038C2
socleC22×C61C2×#121
hypercenter#3038D61
hypocenterC61#19410
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #21456737C2semidirect
2. #21455274C2semidirect
3. #21454930C2semidirect
4. #21445261C2semidirect
5. #21445254C2semidirect
6. #21445177C2semidirect
7. #21445172C2semidirect
8. #21444595C2semidirect
9. #21444497C2semidirect
10. C2×#16839506C2semidirect
11. C61×#3038C2semidirect
18. #16839972C22semidirect
19. #16839967C22semidirect
20. #16839962C22semidirect
21. #16839957C22semidirect
25. Q8×#92418C22semidirect
27. D4×#92418C22semidirect
34. #16839514C22semidirect
35. #16839514C22semidirect
36. #16839509C22semidirect
39. #16839505C22semidirect
113. #41×C61D8semidirect
260. #3038D61semidirect
314. C61#19410semidirect
381. #41#11758199semidirect

Maximal subgroups

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

subgroupindex
#214567372
#214552742
#214549302
#214452612
#214452542
#214451772
#214451722
#214445952
#214444972
C2×#168395062
C61×#30382
#1941061