Finite Group Databasebeta

← #29706592

Subgroups — #29706592

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456287C211yesmaximal, characteristic
2#21446316C211yesmaximal, characteristic, complemented
3#21446310C211yesmaximal, characteristic, complemented
4#21446309C211yesmaximal, characteristic, complemented
5#21446303C211yesmaximal, characteristic, complemented
6#21446299C211yesmaximal, characteristic, complemented
7#21446297C211yesmaximal, characteristic, complemented
8#21446292C211yesmaximal, characteristic, complemented
9#21446290C211yesmaximal, characteristic, complemented
10#21446221C211yesmaximal, characteristic, complemented
11#21446206C211yesmaximal, characteristic, complemented
12#21446203C211yesmaximal, characteristic, complemented
13#21445965C211yesmaximal, characteristic, complemented
14#21445965C211yesmaximal, characteristic, complemented
15C61×#1884C211yesmaximal, characteristic, complemented
16C2×#14519807C2211yescharacteristic
17C2×#14519807C2211yescharacteristic
18C2×#14519802C2211yescharacteristic
19C2×#14519802C2211yescharacteristic
20#16840529C2211yescharacteristic
21C2×#14519765C2211yescharacteristic
22#16840239C2211yescharacteristic
23#16840236C2211yescharacteristic
24#16840154C412yescomplemented
25#16840149C412yescomplemented
26#16840144C412yescomplemented
27#16840144C412yescomplemented
28#16839779C2211yescharacteristic
29#16839779C2211yescharacteristic
30#16839772C2211yescharacteristic
31#16839770C2211yescharacteristic
32#16839768C2211yescharacteristic
33#16839766C2211yescharacteristic
34C2×#14519695C2211yescharacteristic
35C2×#14519693C2211yescharacteristic
36#16839736C2211yescharacteristic
37C4×#11758194C2211yescharacteristic
38#16839534C2211yescharacteristic
39#16839534C2211yescharacteristic
40#16839532C2211yescharacteristic
41#16839532C2211yescharacteristic
42#16839522C2211yescharacteristic
43#16839522C2211yescharacteristic
44#16839520C2211yescharacteristic
45#16839520C2211yescharacteristic
46#16839518C2211yescharacteristic
47#16839518C2211yescharacteristic
48C61×#568C2211yescharacteristic
49C61×#532C2211yescharacteristic
50C61×#421C2211yescharacteristic
51C61×#421C2211yescharacteristic
52C61×#410C2211yescharacteristic
53C61×#339C2211yescharacteristic
54C61×#339C2211yescharacteristic
55C22×Dic122C2311yescharacteristic
56C22×D244C2311yescharacteristic
57#14519807—41no—
58#14519807—41no—
59#14519807—41no—
60#14519807—41no—
61#14519802—41no—
62#14519802—41no—
63#14519802—41no—
64#14519802—41no—
65C2×#11758202C2×C412yescomplemented
66C2×#11758202C2×C412yescomplemented
67C2×#11758200C2×C412yescomplemented
68C2×#11758200C2×C412yescomplemented
69C2×#11758201C2×C412yes—
70C2×#11758201C2×C412yes—
71C2×#11758199C2×C412yes—
72C2×#11758199C2×C412yes—
73#14519784—41no—
74#14519784—41no—
75C2×#11758194—41no—
76C2×#11758194C2311yescharacteristic
77C2×#11758194C2311yescharacteristic
78C2×#11758194C2311yescharacteristic
79C2×#11758194C2311yescharacteristic
80#14519765—41no—
81#14519765C2×C412yes—
82#14519765C2×C412yes—
83C4×Dic122—41no—
84#14519742—41no—
85C4×D244—41no—
86#14519739—41no—
87#14519736C2311yescharacteristic
88#14519736C2311yescharacteristic
89#14519733—41no—
90C2×#11758198—41no—
91C22⋊C4×D61—41no—
92#14519704—41no—
93#14519704—41no—
94#14519702—41no—
95#14519700—41no—
96#14519699—41no—
97#14519697—41no—
98#14519695—41no—
99#14519693—41no—
100#14519692—41no—
101#14519688C2×C412yescomplemented
102C4×#92418C2×C412yescomplemented
103C2×#41×C61C2311yescharacteristic
104C2×#41×C61C2311yescharacteristic
105#131×C61—41no—
106#131×C61—41no—
107C2×M16×C61—41no—
108C2×M16×C61C2311yescharacteristic
109C2×M16×C61C2311yescharacteristic
110C4⋊Q8×C61C2×C411yescharacteristic, complemented
111#127×C61C2×C411yescharacteristic, complemented
112#124×C61C2×C411yescharacteristic, complemented
113#124×C61C2×C411yescharacteristic, complemented
114#121×C61—41no—
115C4×D4×C61—41no—
116#117×C61C2311yescharacteristic
117#117×C61C2311yescharacteristic
118C2×C42×C61C2311yescharacteristic
119#104×C61—41no—
120#104×C61—41no—
121#99×C61—41no—
122C23×D61—21no—
123#41×C61—21no—
124#41×C61—21no—
125#41×C61—21no—
126#41×C61—21no—
127#41×C61—21no—
128#41×C61—21no—
129#41×C61—21no—
130#41×C61—21no—
131C2×Q8×C61C22×C411yescharacteristic
132C2×Q8×C61C22×C411yescharacteristic
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—41no—
136C2×D4×C61C22×C411yescharacteristic
137C2×D4×C61C22×C411yescharacteristic
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244C22×C412yes—
149C2×C4×D61—42no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—21no—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122C22×C412yes—
156M16×C61—21no—
157M16×C61—21no—
158M16×C61—21no—
159M16×C61—21no—
160M16×C61—42no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—42no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—41no—
167C22⋊C4×C61—21no—
168C22⋊C4×C61—21no—
169C22⋊C4×C61—41no—
170C22⋊C4×C61—41no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61C22×C411yescharacteristic
175C42×C61C22×C411yescharacteristic
176C22⋊Dic61—21no—
177C22⋊Dic61—21no—
178#11758202—42no—
179#11758202—42no—
180#11758201—42no—
181#11758201—42no—
182#11758200—42no—
183#11758200—42no—
184#11758199—42no—
185#11758199—42no—
186#11758198—42no—
187#11758198—42no—
188#11758198—42no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191#11758196—42no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—42no—
195#11758194—21no—
196#11758194—21no—
197#11758194—21no—
198#11758194—21no—
199#11758194—21no—
200#11758194—21no—
201#11758194—21no—
202#11758194—21no—
203#11758194—42no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—42no—
207C2×#92418C22×C412yes—
208C2×#92418C22×C412yes—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61H(5,𝔽2)11yescharacteristic
212C22×D61—22no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—82no—
216Q8×C61—21no—
217Q8×C61—21no—
218Q8×C61—21no—
219Q8×C61—21no—
220D4×C61—21no—
221D4×C61—21no—
222D4×C61—21no—
223D4×C61—21no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—82no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—81no—
240C2×C4×C61C23×C411yescharacteristic
241C2×C4×C61H(5,𝔽2)11yescharacteristic
242C2×C4×C61H(5,𝔽2)11yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×Dic61—22no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251D244—22no—
252D244—22no—
253D244—41no—
254D244—41no—
255C4×D61—162no—
256Dic122—22no—
257Dic122—22no—
258Dic122—41no—
259Dic122—41no—
260C8×C61—41no—
261C8×C61—41no—
262#92418—22no—
263#92418—22no—
264#92418—41no—
265#92418—41no—
266#23362—611nomaximal, Sylow
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—81no—
271C22×C61—81no—
272C22×C61#51811yescharacteristic
273C22×C61#56011yescharacteristic
274C22×C61#56011yescharacteristic
275C2×D61—42no—
276C2×D61—82no—
277C2×D61—82no—
278C2×D61—161no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61#51811yescharacteristic
286C4×C61#51811yescharacteristic
287C4×C61#56011yescharacteristic
288C4×C61#56011yescharacteristic
289Dic61—42no—
290Dic61—81no—
291Dic61—81no—
292#3153—611no—
293#1905—611no—
294#1905—611no—
295#1904—611no—
296#1904—611no—
297#1903—611no—
298#1903—611no—
299#1902—611no—
300#1902—611no—
301#1888—611no—
302#1885—611no—
303#1884—611no—
304#1884D6111yescharacteristic, complemented
305#1841—611no—
306#1841—611no—
307C2×C61—21no—
308C2×C61—21no—
309C2×C61—21no—
310C2×C61—81no—
311C2×C61—81no—
312C2×C61#238011yescharacteristic
313D61—82no—
314C2×#137—611no—
315C2×#137—611no—
316C2×AGL(1,ℤ/8)—611no—
317C2×AGL(1,ℤ/8)—611no—
318#568—611no—
319#568C2×D6111yescharacteristic
320#532C2×D6111yescharacteristic
321C2×#124—611no—
322#519—611no—
323#518—611no—
324#497—612no—
325#496—612no—
326#495—612no—
327#495—612no—
328#421—611no—
329#421—611no—
330#421C2×D6111yescharacteristic
331#421C2×D6111yescharacteristic
332#419—611no—
333#419—611no—
334#418—611no—
335#418—611no—
336C2×#101—611no—
337C2×#100—611no—
338#410—611no—
339#410C2×D6111yescharacteristic
340C4×M16—611no—
341#344—611no—
342#344—611no—
343#344—611no—
344#344—611no—
345#339—611no—
346#339—611no—
347#339C2×D6111yescharacteristic
348#339C2×D6111yescharacteristic
349#338—611no—
350#338—611no—
351H(3,ℤ/4)—611no—
352H(3,ℤ/4)—611no—
353C61#2336211yescharacteristic, complemented, Sylow
354C2×#41C22×D6111yescharacteristic
355C2×#41C22×D6111yescharacteristic
356C22×Q8—611no—
357C22×D4—611no—
358#137—2441no—
359#137—2441no—
360#137—2441no—
361#137—2441no—
362AGL(1,ℤ/8)—2441no—
363AGL(1,ℤ/8)—2441no—
364AGL(1,ℤ/8)—2441no—
365AGL(1,ℤ/8)—2441no—
366C2×Q16—612no—
367C2×Q16—612no—
368C2×SD16—612no—
369C2×SD16—612no—
370C2×SD16—612no—
371C2×SD16—612no—
372C2×D8—612no—
373C2×D8—612no—
374#131—41no—
375#131—41no—
376#131—2441no—
377#131—2441no—
378C2×M16—41no—
379C2×M16—611no—
380C2×M16—611no—
381C2×M16—611no—
382C2×M16—611no—
383C2×M16—2441no—
384C2×M16C22×D6111yescharacteristic
385C2×M16C22×D6111yescharacteristic
386C4⋊Q8C4×D6111yescharacteristic, complemented
387#127C4×D6111yescharacteristic, complemented
388#124—612no—
389#124—612no—
390#124—2441no—
391#124C4×D6111yescharacteristic, complemented
392#124C4×D6111yescharacteristic, complemented
393#121—41no—
394C4×Q8—2441no—
395C4×Q8—2441no—
396C4×D4—41no—
397C4×D4—2441no—
398C4×D4—2441no—
399#117—611no—
400#117—611no—
401#117—2441no—
402#117C22×D6111yescharacteristic
403#117C22×D6111yescharacteristic
404C2×C22⋊C4—2441no—
405C2×C22⋊C4—2441no—
406C2×C42C22×D6111yescharacteristic
407#104—41no—
408#104—41no—
409#104—2441no—
410#104—2441no—
411#103—2441no—
412#103—2441no—
413#102—2441no—
414#102—2441no—
415#101—2441no—
416#100—2441no—
417#99—41no—
418#99—2441no—
419#97—612no—
420C4×C8—612no—
421C24—1221no—
422#41—21no—
423#41—21no—
424#41—21no—
425#41—21no—
426#41—21no—
427#41—21no—
428#41—21no—
429#41—21no—
430C2×Q8—612no—
431C2×Q8—1221no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8C2×C4×D6111yescharacteristic
437C2×Q8C2×C4×D6111yescharacteristic
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—41no—
441C2×D4—612no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4C2×C4×D6111yescharacteristic
448C2×D4C2×C4×D6111yescharacteristic
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—1221no—
453C22×C4—2442no—
454C22×C4C23×D6111yescharacteristic
455Q16—2442no—
456Q16—2442no—
457SD16—2442no—
458SD16—2442no—
459SD16—2442no—
460SD16—2442no—
461D8—2442no—
462D8—2442no—
463M16—21no—
464M16—21no—
465M16—21no—
466M16—21no—
467M16—42no—
468M16—1221no—
469M16—1221no—
470M16—1221no—
471M16—1221no—
472M16—1221no—
473M16—1221no—
474M16—1221no—
475M16—1221no—
476M16—2442no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—42no—
480C2×C8—612no—
481C2×C8—612no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—2442no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—41no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—2442no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—1221no—
496C22⋊C4—1221no—
497C22⋊C4—2442no—
498C22⋊C4—2442no—
499C22⋊C4—2442no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—1221no—
504C42—1221no—
505C42—2442no—
506C42C2×C4×D6111yescharacteristic
507C42C2×C4×D6111yescharacteristic
508C23—21no—
509C23—21no—
510C23—1222no—
511C23—4882no—
512C23—2441no—
513C23—2441no—
514C23#1451982211yescharacteristic
515Q8—21no—
516Q8—21no—
517Q8—21no—
518Q8—21no—
519Q8—1222no—
520Q8—1222no—
521Q8—2441no—
522Q8—2441no—
523D4—21no—
524D4—21no—
525D4—21no—
526D4—21no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—82no—
532D4—1222no—
533D4—1222no—
534D4—2441no—
535D4—2441no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—81no—
547C2×C4—1222no—
548C2×C4—9762no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4C22×C4×D6111yescharacteristic
554C2×C4#1451982211yescharacteristic
555C2×C4#1451982211yescharacteristic
556C2×C4#1451982211yescharacteristic
557C2×C4#1451982311yescharacteristic
558C2×C4#1451982311yescharacteristic
559C8—41no—
560C8—41no—
561C8—1222no—
562C8—1222no—
563C8—2441no—
564C8—2441no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—81no—
569C22—81no—
570C22—9761no—
571C22—4882no—
572C22—4882no—
573C22—2442no—
574C22#1684023511yescharacteristic
575C22#1684050011yescharacteristic
576C22#1684050011yescharacteristic
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—81no—
582C4—81no—
583C4—4881no—
584C4—4881no—
585C4—2442no—
586C4#1684023411yescharacteristic
587C4#1684023411yescharacteristic
588C4#1684050111yescharacteristic
589C4#1684050211yescharacteristic
590C2—21no—
591C2—21no—
592C2—21no—
593C2—81no—
594C2—81no—
595C2—4882no—
596C2#2144899411yescharacteristic
subgroupG/H
centerC2#21448994
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1884C2
socleC2×C61#2380
prime radical#21456287C2
cosocleC22×C4×C61C24
hypercenter#1884D61
hypocenterC61#23362
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1884

Complements

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

NH ≅ G/Ntype
2. #21446316C2semidirect
3. #21446310C2semidirect
4. #21446309C2semidirect
5. #21446303C2semidirect
6. #21446299C2semidirect
7. #21446297C2semidirect
8. #21446292C2semidirect
9. #21446290C2semidirect
10. #21446221C2semidirect
11. #21446206C2semidirect
12. #21446203C2semidirect
13. #21445965C2semidirect
14. #21445965C2semidirect
15. C61×#1884C2semidirect
24. #16840154C4semidirect
25. #16840149C4semidirect
26. #16840144C4semidirect
27. #16840144C4semidirect
65. C2×#11758202C2×C4semidirect
66. C2×#11758202C2×C4semidirect
67. C2×#11758200C2×C4semidirect
68. C2×#11758200C2×C4semidirect
101. #14519688C2×C4semidirect
102. C4×#92418C2×C4semidirect
110. C4⋊Q8×C61C2×C4semidirect
111. #127×C61C2×C4semidirect
112. #124×C61C2×C4semidirect
113. #124×C61C2×C4semidirect
304. #1884D61semidirect
353. C61#23362semidirect
386. C4⋊Q8C4×D61semidirect
387. #127C4×D61semidirect
391. #124C4×D61semidirect
392. #124C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214562872
#214463162
#214463102
#214463092
#214463032
#214462992
#214462972
#214462922
#214462902
#214462212
#214462062
#214462032
#214459652
C61×#18842
#2336261