Finite Group Databasebeta

← #29665425

Subgroups — #29665425

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#21456734C211yesmaximal, characteristic, complemented
2#21455299C211yesmaximal, characteristic, complemented
3#21454930C211yesmaximal, characteristic, complemented
4#21445265C212yesmaximal, complemented
5#21445258C212yesmaximal, complemented
6#21445177C212yesmaximal, complemented
7#21445172C212yesmaximal, complemented
8#21444593C211yesmaximal, characteristic, complemented
9#21444496C211yesmaximal, characteristic, complemented
10C2×#16839505C211yesmaximal, characteristic, complemented
11C61×#3035C211yesmaximal, characteristic, complemented
12C2×#14519802—21no—
13C2×#14519774C2211yescharacteristic
14#16840138C2212yes—
15#16840110C2212yes—
16#16839962—21no—
17#16839962C2212yes—
18#16839957—21no—
19#16839957—21no—
20#16839957—21no—
21#16839957C2212yes—
22#16839895—22no—
23Q8×#92418C2212yescomplemented
24D4×#92418—22no—
25D4×#92418C2212yescomplemented
26#16839815—21no—
27#16839815C2211yescharacteristic
28#16839786C2211yescharacteristic
29C2×C4⋊#92418—21no—
30C2×C4⋊#92418C2211yescharacteristic
31#16839766—21no—
32#16839514C2212yescomplemented
33#16839514C2212yescomplemented
34#16839508C2212yescomplemented
35#16839506C2212yescomplemented
36#16839505C2212yes—
37#16839505C2212yes—
38C2×#135×C61—21no—
39C61×#532C2211yescharacteristic
40C4×#41×C61C2211yescharacteristic
41C61×#464—21no—
42C61×#463—21no—
43C61×#462C2211yescharacteristic, complemented
44C61×#461C2211yescharacteristic, complemented
45C61×#460C2211yescharacteristic, complemented
46C61×#459C2211yescharacteristic, complemented
47C2×C4⋊C8×C61C2211yescharacteristic
48C61×#416—21no—
49C22×D244—21no—
50C22×D244—21no—
51C22×D244—21no—
52#14519802—44no—
53C2×#11758201—22no—
54C2×#11758199—22no—
55C2×#11758199—22no—
56C2×#11758199—22no—
57C2×#11758194—21no—
58C22×#92418—21no—
59C22×#92418—22no—
60#14519774—21no—
61#14519774—21no—
62#14519774—21no—
63#14519774C2312yes—
64C4⋊#92418—21no—
65C4⋊#92418—21no—
66C4⋊#92418—22no—
67C4⋊#92418D411yescharacteristic
68C4⋊#92418D411yescharacteristic
69C4⋊#92418C2312yes—
70C4⋊#92418C2312yes—
71#14519697—22no—
72#14519697—22no—
73#14519697—22no—
74#14519697—22no—
75C22⋊#92418—22no—
76C22⋊#92418D412yes—
77C4×#92418—22no—
78C4×#92418C2312yes—
79C2×#41×C61—21no—
80C2×#41×C61D411yescharacteristic
81#135×C61—44no—
82C2×Q16×C61—21no—
83C2×SD16×C61—21no—
84C2×SD16×C61—21no—
85C2×D8×C61—21no—
86C22×C8×C61—21no—
87C4⋊Q8×C61C2311yescharacteristic
88#127×C61C2311yescharacteristic
89#124×C61—21no—
90#121×C61—21no—
91#121×C61—21no—
92C4×Q8×C61D411yescharacteristic
93C4×Q8×C61C2311yescharacteristic
94C4×D4×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61D411yescharacteristic
99C4×D4×C61C2311yescharacteristic
100#117×C61—21no—
101#117×C61D411yescharacteristic
102C2×C42×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—21no—
105C4⋊C8×C61C2311yescharacteristic
106C4⋊C8×C61C2311yescharacteristic
107#103×C61—21no—
108#103×C61—21no—
109#102×C61—21no—
110#102×C61—21no—
111C23×D61—41no—
112#41×C61—22no—
113#41×C61—42no—
114#41×C61—42no—
115#41×C61D814yescomplemented
116C2×Q8×C61—21no—
117C2×Q8×C61C2×D411yescharacteristic
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—41no—
123C2×D4×C61C2×D411yescharacteristic
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—41no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61#4111yescharacteristic
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—42no—
135C2×D244—42no—
136C2×D244—41no—
137C2×D244—41no—
138C2×D244—41no—
139C2×D244—81no—
140C2×D244—81no—
141C2×D244—82no—
142Q16×C61—42no—
143SD16×C61—42no—
144SD16×C61—42no—
145D8×C61—42no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—42no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61C2×D411yescharacteristic
154C4⋊C4×C61C2×D411yescharacteristic
155C22⋊C4×C61—21no—
156C22⋊C4×C61—21no—
157C22⋊C4×C61—21no—
158C22⋊C4×C61—41no—
159C22⋊C4×C61—41no—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61C2×D411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C42×C61#4111yescharacteristic
166C42×C61C2411yescharacteristic
167#11758201—44no—
168#11758199—44no—
169#11758199—82no—
170#11758199—82no—
171#11758194—42no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—82no—
175C2×#92418—81no—
176C2×#92418C2×D412yes—
177C2×#92418C2×D412yes—
178C23×C61—21no—
179C23×C61—41no—
180C23×C61#12111yescharacteristic
181C22×D61—42no—
182C22×D61—82no—
183C22×D61—82no—
184C22×D61—82no—
185Q8×C61—21no—
186Q8×C61—41no—
187Q8×C61—41no—
188Q8×C61C2×D812yes—
189D4×C61—24no—
190D4×C61—21no—
191D4×C61—41no—
192D4×C61—41no—
193D4×C61—41no—
194D4×C61—41no—
195D4×C61—81no—
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—41no—
208C2×C4×C61—81no—
209C2×C4×C61—82no—
210C2×C4×C61#12111yescharacteristic
211C2×C4×C61#12111yescharacteristic
212C2×C4×C61#12111yescharacteristic
213C2×C4×C61C2×D812yes—
214C2×C4×C61#13512yes—
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218D244—42no—
219D244—42no—
220D244—42no—
221D244—42no—
222D244—82no—
223D244—82no—
224D244—82no—
225D244—82no—
226C8×C61—41no—
227C8×C61—41no—
228#92418—22no—
229#92418—22no—
230#92418—42no—
231#92418—42no—
232#19410—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—22no—
236C22×C61—41no—
237C22×C61—41no—
238C22×C61—81no—
239C22×C61—81no—
240C22×C61—81no—
241C22×C61#46612yes—
242C22×C61C2×#12111yescharacteristic
243C2×D61—82no—
244C2×D61—82no—
245C2×D61—82no—
246C2×D61—164no—
247C4×C61—21no—
248C4×C61—22no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—41no—
252C4×C61—41no—
253C4×C61—81no—
254C4×C61#46612yes—
255C4×C61C22×D811yescharacteristic
256C4×C61C2×#13511yescharacteristic
257C4×C61#57611yescharacteristic
258C4×C61#57711yescharacteristic
259#3205—611no—
260#3038—611no—
261#3035D6111yescharacteristic, complemented
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—161no—
273C2×C61#304511yescharacteristic
274C2×C61C2×#46611yescharacteristic
275C2×C61#306811yescharacteristic
276D61—162no—
277C2×AGL(1,ℤ/8)—1221no—
278C2×#135—21no—
279#532C2×D6111yescharacteristic
280C2×#127—611no—
281C4×#41C2×D6111yescharacteristic
282#493—612no—
283#486—612no—
284#464—21no—
285#463—21no—
286#462C2×D6111yescharacteristic, complemented
287#461C2×D6111yescharacteristic, complemented
288#460—612no—
289#460—1221no—
290#460C2×D6111yescharacteristic, complemented
291#459—612no—
292#459—1221no—
293#459—1221no—
294#459—1221no—
295#459C2×D6111yescharacteristic, complemented
296#447—1222no—
297C8×Q8—612no—
298C8×D4—612no—
299C8×D4—1222no—
300#431—611no—
301#431—1221no—
302#423—611no—
303C2×C4⋊C8—611no—
304C2×C4⋊C8—1221no—
305C2×C4⋊C8C2×D6111yescharacteristic
306#418—1221no—
307#416—21no—
308#335—612no—
309#335—612no—
310#332—612no—
311#331—612no—
312#331—612no—
313#331—612no—
314C61#1941011yescharacteristic, complemented, Sylow
315C2×#41—21no—
316C2×#41C22⋊D6111yescharacteristic
317C22×D4—1221no—
318C22×D4—1221no—
319C22×D4—1221no—
320AGL(1,ℤ/8)—2444no—
321#135—44no—
322C2×Q16—21no—
323C2×SD16—21no—
324C2×SD16—21no—
325C2×SD16—1222no—
326C2×D8—21no—
327C2×D8—1222no—
328C2×D8—1222no—
329C2×D8—1222no—
330C2×M16—1221no—
331C22×C8—21no—
332C22×C8—1222no—
333C22×C8—1221no—
334C4⋊Q8C22×D6111yescharacteristic
335#127—612no—
336#127—1221no—
337#127—1221no—
338#127—1221no—
339#127C22×D6111yescharacteristic
340#124—21no—
341#121—21no—
342#121—21no—
343C4×Q8C22⋊D6111yescharacteristic
344C4×Q8C22×D6111yescharacteristic
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—21no—
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—612no—
360C4⋊C8—1222no—
361C4⋊C8—1221no—
362C4⋊C8—1221no—
363C4⋊C8C22×D6111yescharacteristic
364C4⋊C8C22×D6111yescharacteristic
365#103—21no—
366#103—21no—
367#102—21no—
368#102—21no—
369#102—1222no—
370#102—1222no—
371#102—1222no—
372#102—1222no—
373C22⋊C8—612no—
374C22⋊C8—1222no—
375C4×C8—612no—
376C4×C8—1222no—
377C24—2441no—
378#41—22no—
379#41—42no—
380#41—42no—
381#41#1175819914yescomplemented
382C2×Q8—21no—
383C2×Q8C2×C22⋊D6111yescharacteristic
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—41no—
389C2×D4—1222no—
390C2×D4—1222no—
391C2×D4—1222no—
392C2×D4—4881no—
393C2×D4—4881no—
394C2×D4—4882no—
395C2×D4—2441no—
396C2×D4—2441no—
397C2×D4—2441no—
398C2×D4—2442no—
399C2×D4—2442no—
400C2×D4C2×C22⋊D6111yescharacteristic
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—41no—
405C22×C4D4×D6111yescharacteristic
406C22×C4#1175822411yescharacteristic
407C22×C4C2×C22⋊D6111yescharacteristic
408Q16—42no—
409SD16—42no—
410SD16—42no—
411SD16—2444no—
412D8—42no—
413D8—4882no—
414D8—4882no—
415D8—2444no—
416M16—2442no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—42no—
420C2×C8—612no—
421C2×C8—612no—
422C2×C8—1222no—
423C2×C8—1222no—
424C2×C8—4882no—
425C2×C8—4881no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4C2×C22⋊D6111yescharacteristic
431C4⋊C4C2×C22⋊D6111yescharacteristic
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42D4×D6111yescharacteristic
441C42#1175822411yescharacteristic
442C42C2×C22⋊D6111yescharacteristic
443C42C23×D6111yescharacteristic
444C23—21no—
445C23—41no—
446C23—4882no—
447C23—4882no—
448C23—4882no—
449C23—2442no—
450C23#1451974911yescharacteristic
451Q8—21no—
452Q8—41no—
453Q8—41no—
454Q8C2×#1175819912yes—
455D4—24no—
456D4—21no—
457D4—41no—
458D4—41no—
459D4—41no—
460D4—41no—
461D4—81no—
462D4—82no—
463D4—4882no—
464D4—4882no—
465D4—4882no—
466D4—4882no—
467D4—2442no—
468D4—2442no—
469D4—2442no—
470D4—2442no—
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—41no—
482C2×C4—81no—
483C2×C4—82no—
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—41no—
493C8—41no—
494C8—1222no—
495C8—1222no—
496C8—2442no—
497C8—2442no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—41no—
502C22—41no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—9764no—
507C22—4882no—
508C22—4882no—
509C22—4882no—
510C22#1683999612yes—
511C22C2×#1451974911yescharacteristic
512C4—22no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—41no—
517C4—41no—
518C4—81no—
519C4#1683999612yes—
520C4C22×#1175819911yescharacteristic
521C4C2×#1451979711yescharacteristic
522C4#1684056411yescharacteristic
523C4#1684056911yescharacteristic
524C2—22no—
525C2—42no—
526C2—161no—
527C2—9762no—
528C2#2145535611yescharacteristic
529C2C2×#1683999611yescharacteristic
530C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3035C2
socleC22×C61C2×#121
hypercenter#3035D61
hypocenterC61#19410
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #21456734C2semidirect
2. #21455299C2semidirect
3. #21454930C2semidirect
4. #21445265C2semidirect
5. #21445258C2semidirect
6. #21445177C2semidirect
7. #21445172C2semidirect
8. #21444593C2semidirect
9. #21444496C2semidirect
10. C2×#16839505C2semidirect
11. C61×#3035C2semidirect
23. Q8×#92418C22semidirect
25. D4×#92418C22semidirect
32. #16839514C22semidirect
33. #16839514C22semidirect
34. #16839508C22semidirect
35. #16839506C22semidirect
43. C61×#462C22semidirect
44. C61×#461C22semidirect
45. C61×#460C22semidirect
46. C61×#459C22semidirect
115. #41×C61D8semidirect
261. #3035D61semidirect
286. #462C2×D61semidirect
287. #461C2×D61semidirect
290. #460C2×D61semidirect
295. #459C2×D61semidirect
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
#214567342
#214552992
#214549302
#214452652
#214452582
#214451772
#214451722
#214445932
#214444962
C2×#168395052
C61×#30352
#1941061