Finite Group Databasebeta

← #29665423

Subgroups — #29665423

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#21455295C211yesmaximal, characteristic, complemented
2#21455268C211yesmaximal, characteristic, complemented
3#21454927C211yesmaximal, characteristic, complemented
4#21445260C212yesmaximal, complemented
5#21445253C212yesmaximal, complemented
6#21445176C212yesmaximal, complemented
7#21445171C212yesmaximal, complemented
8#21444594C211yesmaximal, characteristic, complemented
9#21444496C211yesmaximal, characteristic, complemented
10C2×#16839506C211yesmaximal, characteristic, complemented
11C61×#3205C211yesmaximal, characteristic, complemented
12C2×#14519803—21no—
13C2×#14519797—21no—
14C4×#11758222C2211yescharacteristic
15#16839981—21no—
16#16839974—21no—
17#16839969C2211yescharacteristic, complemented
18#16839964C2211yescharacteristic, complemented
19#16839959—21no—
20#16839959C2212yes—
21#16839959C2211yescharacteristic, complemented
22#16839954—21no—
23#16839954C2212yes—
24#16839954C2211yescharacteristic, complemented
25C8×Dic122C2212yescomplemented
26C8×D244C2212yescomplemented
27#16839825—22no—
28#16839814—21no—
29#16839786C2211yescharacteristic
30C2×C4⋊#92418C2211yescharacteristic
31C2×C4⋊#92418C2211yescharacteristic
32C4⋊C8×D61—21no—
33#16839767—21no—
34#16839758—21no—
35#16839515C2212yescomplemented
36#16839515C2212yescomplemented
37#16839508C2212yescomplemented
38#16839506C2212yes—
39#16839506C2212yes—
40#16839506C2212yescomplemented
41C61×#532C2211yescharacteristic
42C2×#127×C61C2211yescharacteristic
43C61×#493C2212yes—
44C61×#486C2212yes—
45C61×#459—21no—
46C61×#459—21no—
47C61×#447—22no—
48C61×#431C2211yescharacteristic
49C2×#11758222D411yescharacteristic
50#14519803—44no—
51#14519797—44no—
52C2×#11758202—21no—
53C2×#11758200—22no—
54C2×#11758200—21no—
55C2×#11758201—21no—
56C2×#11758199—22no—
57C2×#11758199—21no—
58C2×#11758194—21no—
59C22×#92418—21no—
60C22×#92418—21no—
61C2×C8×D61—22no—
62C4×Dic122D411yescharacteristic
63C4×Dic122C2311yescharacteristic
64C4×D244D411yescharacteristic
65C4×D244C2311yescharacteristic
66C4×C22⋊D61—21no—
67C4×C22⋊D61—21no—
68#14519736D411yescharacteristic
69#14519734—21no—
70C42×D61—21no—
71C4⋊#92418—21no—
72C4⋊#92418—21no—
73C4⋊#92418—21no—
74C4⋊#92418C2312yes—
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2311yescharacteristic
77C4⋊#92418C2312yes—
78#14519706—22no—
79#14519701—21no—
80#14519701—21no—
81#14519698—22no—
82#14519698—22no—
83#14519698—21no—
84#14519698—21no—
85#14519689—22no—
86C8×Dic61—22no—
87C2×#41×C61—21no—
88C22×D4×C61—21no—
89C22×D4×C61—21no—
90C22×D4×C61—21no—
91C2×D8×C61—22no—
92C2×D8×C61—22no—
93C4⋊Q8×C61C2311yescharacteristic
94#127×C61—21no—
95#127×C61—21no—
96#127×C61—21no—
97#127×C61C2312yes—
98#127×C61C2311yescharacteristic
99#124×C61—21no—
100#121×C61—21no—
101#121×C61—21no—
102C4×D4×C61—21no—
103C4×D4×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61D411yescharacteristic
106C4⋊C8×C61D411yescharacteristic
107#102×C61—22no—
108#102×C61—22no—
109C22⋊C8×C61D412yes—
110C4×C8×C61C2312yes—
111C24×C61—41no—
112#41×C61—42no—
113#41×C61—42no—
114C2×Q8×C61—21no—
115C2×D4×C61—22no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—22no—
119C2×D4×C61—21no—
120C2×D4×C61—22no—
121C2×D4×C61—41no—
122C2×D4×C61—41no—
123C2×D4×C61—41no—
124C2×D4×C61—41no—
125C2×D4×C61—42no—
126C2×D4×C61—42no—
127C2×D4×C61—81no—
128C2×D4×C61—81no—
129C2×D4×C61—82no—
130C22×C4×C61—41no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61#4111yescharacteristic
134C2×C22⋊D61—21no—
135#11758222—22no—
136#11758222D814yescomplemented
137C2×D244C2×D411yescharacteristic
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×Dic122C2×D411yescharacteristic
142D8×C61—82no—
143D8×C61—82no—
144C2×C8×C61C2×D412yes—
145C2×C8×C61C2×D412yes—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C22⋊C4×C61—41no—
149C22⋊C4×C61—41no—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61#4111yescharacteristic
153C42×C61C2411yescharacteristic
154C22⋊Dic61—21no—
155#11758202—42no—
156#11758201—42no—
157#11758200—42no—
158#11758200—44no—
159#11758199—42no—
160#11758199—44no—
161#11758198—21no—
162#11758198—21no—
163C4⋊Dic61C2×D411yescharacteristic
164C4⋊Dic61C2×D411yescharacteristic
165#11758196—21no—
166#11758196—21no—
167C4×Dic61—21no—
168C4×Dic61—21no—
169C4×Dic61—21no—
170#11758194—42no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—21no—
174C2×#92418—21no—
175C2×#92418—42no—
176C2×#92418—81no—
177C8×D61—82no—
178C23×C61—41no—
179C23×C61—42no—
180C23×C61—82no—
181C23×C61—82no—
182C23×C61—82no—
183C23×C61#12111yescharacteristic
184C22×D61—21no—
185Q8×C61—41no—
186Q8×C61—41no—
187D4×C61—42no—
188D4×C61—41no—
189D4×C61—42no—
190D4×C61—41no—
191D4×C61—41no—
192D4×C61—42no—
193D4×C61—41no—
194D4×C61—42no—
195D4×C61—81no—
196D4×C61—82no—
197D4×C61—82no—
198D4×C61—82no—
199D4×C61—82no—
200D4×C61—82no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—82no—
206C2×C4×C61#12111yescharacteristic
207C2×C4×C61#12111yescharacteristic
208C2×C4×C61#12111yescharacteristic
209C2×C4×C61C2×D812yes—
210C2×C4×C61#13512yes—
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C22⋊D61—24no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218D244—21no—
219D244C2×D812yes—
220C4×D61—24no—
221C4×D61—41no—
222C4×D61—41no—
223C4×D61—81no—
224Dic122—21no—
225Dic122C2×D812yes—
226C8×C61—22no—
227C8×C61—22no—
228#92418—41no—
229#92418—41no—
230#92418—42no—
231#92418—42no—
232#19410—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—82no—
236C22×C61—81no—
237C22×C61—81no—
238C22×C61—82no—
239C22×C61—82no—
240C22×C61—81no—
241C22×C61—164no—
242C22×C61#46612yes—
243C22×C61C2×#12111yescharacteristic
244C2×D61—22no—
245C2×D61—41no—
246C2×D61—41no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—81no—
251C4×C61#46612yes—
252C4×C61C22×D811yescharacteristic
253C4×C61C2×#13511yescharacteristic
254C4×C61#57611yescharacteristic
255C4×C61#57711yescharacteristic
256Dic61—22no—
257Dic61—41no—
258Dic61—41no—
259#3205D6111yescharacteristic, complemented
260#3038—611no—
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—161no—
272C2×C61—162no—
273C2×C61#304511yescharacteristic
274C2×C61C2×#46611yescharacteristic
275C2×C61#306811yescharacteristic
276D61—42no—
277C2×AGL(1,ℤ/8)—1221no—
278C2×#135—1221no—
279#532C2×D6111yescharacteristic
280C2×#127C2×D6111yescharacteristic
281C4×#41—611no—
282#493C2×D6112yes—
283#486C2×D6112yes—
284#464—1221no—
285#463—1221no—
286#462—611no—
287#461—611no—
288#460—612no—
289#460—611no—
290#460—1221no—
291#459—21no—
292#459—21no—
293#459—612no—
294#459—611no—
295#459—1221no—
296#447—22no—
297C8×Q8—612no—
298C8×D4—612no—
299C8×D4—1222no—
300#431—1221no—
301#431C2×D6111yescharacteristic
302#423—611no—
303C2×C4⋊C8—611no—
304C2×C4⋊C8—611no—
305C2×C4⋊C8—1221no—
306#418—1221no—
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—21no—
316C2×#41—611no—
317C22×D4—21no—
318C22×D4—21no—
319C22×D4—21no—
320AGL(1,ℤ/8)—2444no—
321#135—2444no—
322C2×Q16—1221no—
323C2×SD16—1221no—
324C2×SD16—1222no—
325C2×SD16—1221no—
326C2×D8—22no—
327C2×D8—22no—
328C2×D8—1221no—
329C2×D8—1222no—
330C2×M16—1221no—
331C22×C8—1221no—
332C22×C8—1221no—
333C22×C8—1222no—
334C4⋊Q8C22×D6111yescharacteristic
335#127—21no—
336#127—21no—
337#127—21no—
338#127C22×D6112yes—
339#127C22×D6111yescharacteristic
340#124—21no—
341#121—21no—
342#121—21no—
343C4×Q8—611no—
344C4×Q8—611no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—611no—
348C4×D4—611no—
349C4×D4—1221no—
350C4×D4—1221no—
351#117—611no—
352#117—1221no—
353C2×C42—1221no—
354C2×C42C22×D6111yescharacteristic
355C4⋊C8—612no—
356C4⋊C8—611no—
357C4⋊C8—611no—
358C4⋊C8—612no—
359C4⋊C8—1222no—
360C4⋊C8—1221no—
361C4⋊C8—1221no—
362C4⋊C8—1221no—
363C4⋊C8C22⋊D6111yescharacteristic
364C4⋊C8C22⋊D6111yescharacteristic
365#103—1221no—
366#103—1221no—
367#102—22no—
368#102—22no—
369#102—1221no—
370#102—1221no—
371#102—1222no—
372#102—1222no—
373C22⋊C8—1222no—
374C22⋊C8C22⋊D6112yes—
375C4×C8—1222no—
376C4×C8C22×D6112yes—
377C24—41no—
378#41—42no—
379#41—42no—
380#41—614no—
381#41—1222no—
382C2×Q8—21no—
383C2×Q8—611no—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—81no—
397C2×D4—81no—
398C2×D4—82no—
399C2×D4—611no—
400C2×D4—1221no—
401C22×C4—41no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4—1221no—
405C22×C4D4×D6111yescharacteristic
406C22×C4#1175822411yescharacteristic
407C22×C4C2×C22⋊D6111yescharacteristic
408Q16—2442no—
409SD16—2442no—
410SD16—2444no—
411SD16—2442no—
412D8—82no—
413D8—82no—
414D8—2442no—
415D8—2444no—
416M16—2442no—
417C2×C8—1222no—
418C2×C8—1221no—
419C2×C8—1221no—
420C2×C8—1222no—
421C2×C8—4882no—
422C2×C8—4881no—
423C2×C8—2442no—
424C2×C8C2×C22⋊D6112yes—
425C2×C8C2×C22⋊D6112yes—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—611no—
429C4⋊C4—611no—
430C4⋊C4—1221no—
431C4⋊C4—1221no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—1221no—
435C22⋊C4—1221no—
436C22⋊C4—1221no—
437C42—1221no—
438C42—1221no—
439C42—1221no—
440C42D4×D6111yescharacteristic
441C42#1175822411yescharacteristic
442C42C2×C22⋊D6111yescharacteristic
443C42C23×D6111yescharacteristic
444C23—41no—
445C23—42no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—1221no—
450C23#1451974811yescharacteristic
451Q8—41no—
452Q8—41no—
453Q8—612no—
454Q8—1221no—
455D4—42no—
456D4—41no—
457D4—42no—
458D4—41no—
459D4—41no—
460D4—42no—
461D4—41no—
462D4—42no—
463D4—81no—
464D4—82no—
465D4—82no—
466D4—82no—
467D4—82no—
468D4—82no—
469D4—612no—
470D4—1224no—
471D4—1221no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—82no—
477C2×C4—1221no—
478C2×C4—1224no—
479C2×C4—1221no—
480C2×C4—1221no—
481C2×C4—4881no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4#1451974811yescharacteristic
485C2×C4#1451974811yescharacteristic
486C2×C4#1451974811yescharacteristic
487C2×C4D8×D6112yes—
488C2×C4#1451979612yes—
489C2×C4C2×D4×D6111yescharacteristic
490C2×C4C22×C22⋊D6111yescharacteristic
491C2×C4C2×#1175822411yescharacteristic
492C8—22no—
493C8—22no—
494C8—2441no—
495C8—2441no—
496C8—2442no—
497C8—2442no—
498C22—21no—
499C22—21no—
500C22—81no—
501C22—82no—
502C22—82no—
503C22—81no—
504C22—82no—
505C22—81no—
506C22—164no—
507C22—1222no—
508C22—2441no—
509C22—2441no—
510C22#1683999512yes—
511C22C2×#1451974811yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—81no—
516C4—1222no—
517C4—2441no—
518C4—2441no—
519C4#1683999512yes—
520C4C2×D8×D6111yescharacteristic
521C4C2×#1451979611yescharacteristic
522C4#1684056511yescharacteristic
523C4#1684057311yescharacteristic
524C2—22no—
525C2—162no—
526C2—161no—
527C2—2442no—
528C2#2145535411yescharacteristic
529C2C2×#1683999511yescharacteristic
530C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3205C2
socleC22×C61C2×#121
hypercenter#3205D61
hypocenterC61#19410
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #21455295C2semidirect
2. #21455268C2semidirect
3. #21454927C2semidirect
4. #21445260C2semidirect
5. #21445253C2semidirect
6. #21445176C2semidirect
7. #21445171C2semidirect
8. #21444594C2semidirect
9. #21444496C2semidirect
10. C2×#16839506C2semidirect
11. C61×#3205C2semidirect
17. #16839969C22semidirect
18. #16839964C22semidirect
21. #16839959C22semidirect
24. #16839954C22semidirect
25. C8×Dic122C22semidirect
26. C8×D244C22semidirect
35. #16839515C22semidirect
36. #16839515C22semidirect
37. #16839508C22semidirect
40. #16839506C22semidirect
136. #11758222D8semidirect
259. #3205D61semidirect
314. C61#19410semidirect

Maximal subgroups

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

subgroupindex
#214552952
#214552682
#214549272
#214452602
#214452532
#214451762
#214451712
#214445942
#214444962
C2×#168395062
C61×#32052
#1941061