Finite Group Databasebeta

← #29706591

Subgroups — #29706591

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#21456286C211yesmaximal, characteristic
2#21446313C211yesmaximal, characteristic, complemented
3#21446309C211yesmaximal, characteristic, complemented
4#21446307C211yesmaximal, characteristic, complemented
5#21446298C211yesmaximal, characteristic, complemented
6#21446297C211yesmaximal, characteristic, complemented
7#21446294C211yesmaximal, characteristic, complemented
8#21446289C211yesmaximal, characteristic, complemented
9#21446222C211yesmaximal, characteristic, complemented
10#21446204C211yesmaximal, characteristic, complemented
11#21446200C211yesmaximal, characteristic, complemented
12#21446199C211yesmaximal, characteristic, complemented
13#21445967C211yesmaximal, characteristic, complemented
14#21445966C211yesmaximal, characteristic, complemented
15C61×#1905C211yesmaximal, characteristic, complemented
16C2×#14519808C2211yescharacteristic
17C2×#14519805C2211yescharacteristic
18C2×#14519802C2211yescharacteristic
19#16840529C2211yescharacteristic
20#16840528C2211yescharacteristic
21#16840308C2211yescharacteristic
22C2×#14519766C2211yescharacteristic
23#16840236C2211yescharacteristic
24#16840155C412yescomplemented
25#16840148C412yescomplemented
26#16840147C412yescomplemented
27#16840143C412yescomplemented
28#16839780C2211yescharacteristic
29#16839780C2211yescharacteristic
30#16839779C2211yescharacteristic
31#16839771C2211yescharacteristic
32#16839766C2211yescharacteristic
33#16839766C2211yescharacteristic
34C2×#14519696C2211yescharacteristic
35C2×#14519693C2211yescharacteristic
36#16839735C2211yescharacteristic
37M16×Dic61C2211yescharacteristic
38#16839533C2211yescharacteristic
39#16839532C2211yescharacteristic
40#16839532C2211yescharacteristic
41#16839523C2211yescharacteristic
42#16839521C2211yescharacteristic
43#16839521C2211yescharacteristic
44#16839521C2211yescharacteristic
45#16839520C2211yescharacteristic
46#16839520C2211yescharacteristic
47#16839518C2211yescharacteristic
48C2×#137×C61C2211yescharacteristic
49C61×#519C2211yescharacteristic
50C61×#421C2211yescharacteristic
51C61×#419C2211yescharacteristic
52C61×#410C2211yescharacteristic
53C61×#344C2211yescharacteristic
54C61×H(3,ℤ/4)C2211yescharacteristic
55C2×#11758222C2311yescharacteristic
56C22×D244C2311yescharacteristic
57#14519808—41no—
58#14519808—41no—
59#14519805—41no—
60#14519805—41no—
61#14519802—41no—
62#14519802—41no—
63C2×#11758202C2×C412yescomplemented
64C2×#11758190C2×C412yes—
65C2×#11758201C2×C412yes—
66C2×#11758201C2×C412yescomplemented
67C2×D488C2×C412yes—
68C2×#11758199C2×C412yes—
69#14519785—41no—
70#14519784—41no—
71#14519784—41no—
72#14519783—41no—
73C2×#11758194—41no—
74C2×#11758194C2311yescharacteristic
75C2×#11758194C2311yescharacteristic
76C2×#11758194C2311yescharacteristic
77C2×#11758194C2311yescharacteristic
78M16×D61—41no—
79#14519775C2×C411yescharacteristic, complemented
80#14519773C2×C411yescharacteristic, complemented
81#14519766—41no—
82#14519766C2×C412yes—
83#14519766C2×C412yes—
84#14519766C2×C411yescharacteristic, complemented
85#14519764C2×C411yescharacteristic, complemented
86#14519749—41no—
87C4×D244—41no—
88#14519739—41no—
89C4×C22⋊D61—41no—
90#14519736C2311yescharacteristic
91C2×#11758198—41no—
92C22⋊C4×D61—41no—
93C2×C4×Dic61C2311yescharacteristic
94#14519705—41no—
95#14519705—41no—
96#14519704—41no—
97#14519701—41no—
98#14519697—41no—
99#14519697—41no—
100#14519696—41no—
101#14519693—41no—
102#14519691—41no—
103#14519687C2×C412yescomplemented
104C8×Dic61C2×C412yescomplemented
105C2×#41×C61C2311yescharacteristic
106C22×Q8×C61C2311yescharacteristic
107#137×C61—41no—
108#137×C61—41no—
109C2×Q16×C61C2×C412yescomplemented
110C2×SD16×C61C2×C412yescomplemented
111C2×M16×C61C2311yescharacteristic
112C2×M16×C61C2311yescharacteristic
113C4×Q8×C61—41no—
114C4×Q8×C61—41no—
115#117×C61—41no—
116#117×C61C2311yescharacteristic
117#117×C61C2311yescharacteristic
118#117×C61C2311yescharacteristic
119#104×C61—41no—
120#103×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—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61C22×C412yes—
133C2×Q8×C61C22×C411yescharacteristic
134C2×D4×C61—21no—
135C2×D4×C61C22×C411yescharacteristic
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C2411yescharacteristic
139C2×C22⋊D61—21no—
140C2×C22⋊D61—41no—
141C22×Dic61—21no—
142#11758222—21no—
143#11758222—21no—
144#11758222—21no—
145#11758222—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244C22×C412yes—
152C2×D244C22×C411yescharacteristic
153C2×C4×D61—21no—
154C2×C4×D61—42no—
155C2×Dic122C22×C411yescharacteristic
156Q16×C61—42no—
157SD16×C61—42no—
158M16×C61—21no—
159M16×C61—21no—
160M16×C61—21no—
161M16×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61C22×C412yes—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—42no—
168C22⋊C4×C61—21no—
169C22⋊C4×C61—21no—
170C22⋊C4×C61—21no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—42no—
175C22⋊Dic61—21no—
176C22⋊Dic61—41no—
177#11758202—42no—
178#11758201—42no—
179#11758201—42no—
180#11758199—42no—
181#11758198—42no—
182#11758198—42no—
183#11758198—41no—
184#11758198—42no—
185C4⋊Dic61—21no—
186#11758196—41no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61C22×C411yescharacteristic
190C4×Dic61C22×C411yescharacteristic
191#11758194—21no—
192#11758194—21no—
193#11758194—21no—
194#11758194—21no—
195#11758194—21no—
196#11758194—21no—
197#11758194—21no—
198#11758194—21no—
199#11758194—42no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—42no—
204C2×#92418C22×C412yes—
205D488—42no—
206#11758190—42no—
207#11758189—42no—
208C8×D61—42no—
209C23×C61—21no—
210C23×C61H(5,𝔽2)11yescharacteristic
211C22×D61—22no—
212C22×D61—21no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—82no—
216Q8×C61—21no—
217Q8×C61—22no—
218Q8×C61—22no—
219Q8×C61—21no—
220Q8×C61—41no—
221Q8×C61—41no—
222D4×C61—21no—
223D4×C61—21no—
224D4×C61—41no—
225D4×C61—41no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61C23×C411yescharacteristic
236C2×C4×C61H(5,𝔽2)11yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×C4×C61H(5,𝔽2)11yescharacteristic
239C2×C4×C61H(5,𝔽2)11yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C22⋊D61—41no—
242C22⋊D61—41no—
243C22⋊D61—82no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—41no—
248C2×Dic61—81no—
249D244—22no—
250D244—21no—
251D244—22no—
252D244—21no—
253D244—41no—
254D244—41no—
255C4×D61—41no—
256C4×D61—41no—
257C4×D61—162no—
258Dic122—21no—
259Dic122—21no—
260C8×C61—22no—
261C8×C61—41no—
262#92418—22no—
263#92418—41no—
264#92418—41no—
265#92418—41no—
266#23362—611nomaximal, Sylow
267C22×C61—41no—
268C22×C61—41no—
269C22×C61—81no—
270C22×C61#51811yescharacteristic
271C22×C61#56011yescharacteristic
272C22×C61#56011yescharacteristic
273C2×D61—41no—
274C2×D61—42no—
275C2×D61—82no—
276C2×D61—81no—
277C2×D61—82no—
278C2×D61—161no—
279C4×C61—41no—
280C4×C61—42no—
281C4×C61—81no—
282C4×C61—81no—
283C4×C61—81no—
284C4×C61#51811yescharacteristic
285C4×C61#51811yescharacteristic
286C4×C61#56011yescharacteristic
287C4×C61#56011yescharacteristic
288Dic61—41no—
289Dic61—41no—
290Dic61—41no—
291Dic61—81no—
292#3153—611no—
293#1905—611no—
294#1905D6111yescharacteristic, complemented
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#1884—611no—
305#1841—611no—
306#1841—611no—
307C2×C61—21no—
308C2×C61—21no—
309C2×C61—21no—
310C2×C61—81no—
311C2×C61#238011yescharacteristic
312D61—81no—
313D61—82no—
314C2×#137—611no—
315C2×#137C2×D6111yescharacteristic
316C2×AGL(1,ℤ/8)—611no—
317C2×AGL(1,ℤ/8)—611no—
318#568—611no—
319#568—611no—
320#532—611no—
321C2×#124—611no—
322#519C2×D6111yescharacteristic
323#518—611no—
324#497—612no—
325#496—612no—
326#495—612no—
327#495—612no—
328#421—611no—
329#421—611no—
330#421—611no—
331#421C2×D6111yescharacteristic
332#419—611no—
333#419C2×D6111yescharacteristic
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#344C2×D6111yescharacteristic
345#339—611no—
346#339—611no—
347#339—611no—
348#339—611no—
349#338—611no—
350#338—611no—
351H(3,ℤ/4)—611no—
352H(3,ℤ/4)C2×D6111yescharacteristic
353C61#2336211yescharacteristic, complemented, Sylow
354C2×#41—611no—
355C2×#41C22×D6111yescharacteristic
356C22×Q8C22×D6111yescharacteristic
357C22×D4—611no—
358#137—41no—
359#137—41no—
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×Q16C4×D6112yescomplemented
368C2×SD16—612no—
369C2×SD16—612no—
370C2×SD16—612no—
371C2×SD16C4×D6112yescomplemented
372C2×D8—612no—
373C2×D8—612no—
374#131—2441no—
375#131—2441no—
376#131—2441no—
377#131—2441no—
378C2×M16—611no—
379C2×M16—611no—
380C2×M16—611no—
381C2×M16—611no—
382C2×M16—2441no—
383C2×M16—2441no—
384C2×M16C22×D6111yescharacteristic
385C2×M16C22×D6111yescharacteristic
386C4⋊Q8—611no—
387#127—611no—
388#124—612no—
389#124—612no—
390#124—611no—
391#124—611no—
392#124—2441no—
393#121—2441no—
394C4×Q8—41no—
395C4×Q8—41no—
396C4×D4—2441no—
397C4×D4—2441no—
398C4×D4—2441no—
399#117—41no—
400#117—611no—
401#117C22×D6111yescharacteristic
402#117C22×D6111yescharacteristic
403#117C22×D6111yescharacteristic
404C2×C22⋊C4—2441no—
405C2×C22⋊C4—2441no—
406C2×C42—611no—
407#104—41no—
408#104—2441no—
409#104—2441no—
410#104—2441no—
411#103—41no—
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—1221no—
427#41—1221no—
428#41—1221no—
429#41—1221no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—611no—
436C2×Q8C2×C4×D6112yes—
437C2×Q8C2×C4×D6111yescharacteristic
438C2×D4—21no—
439C2×D4—612no—
440C2×D4—611no—
441C2×D4—1221no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—2441no—
448C2×D4C2×C4×D6111yescharacteristic
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—2442no—
454C22×C4C23×D6111yescharacteristic
455Q16—42no—
456Q16—2442no—
457SD16—42no—
458SD16—2442no—
459SD16—2442no—
460SD16—2442no—
461D8—2442no—
462D8—2442no—
463M16—21no—
464M16—21no—
465M16—21no—
466M16—21no—
467M16—1221no—
468M16—1221no—
469M16—1221no—
470M16—1221no—
471M16—1221no—
472M16—1221no—
473M16—1221no—
474M16—1221no—
475M16—2442no—
476M16—2442no—
477C2×C8—21no—
478C2×C8—612no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—2442no—
483C2×C8—2442no—
484C2×C8C2×C4×D6112yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—1221no—
490C4⋊C4—2441no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—1221no—
495C22⋊C4—2442no—
496C22⋊C4—2442no—
497C22⋊C4—2441no—
498C22⋊C4—2442no—
499C22⋊C4—2441no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—42no—
504C42—611no—
505C42—611no—
506C42—1221no—
507C42—1221no—
508C23—21no—
509C23—1222no—
510C23—1221no—
511C23—4882no—
512C23—2441no—
513C23—2441no—
514C23#1451982211yescharacteristic
515Q8—21no—
516Q8—22no—
517Q8—22no—
518Q8—21no—
519Q8—41no—
520Q8—41no—
521Q8—1221no—
522Q8—1221no—
523D4—21no—
524D4—21no—
525D4—41no—
526D4—41no—
527D4—1222no—
528D4—1221no—
529D4—1221no—
530D4—1222no—
531D4—4882no—
532D4—2441no—
533D4—2441no—
534D4—2441no—
535D4—2441no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—9762no—
549C2×C4—4881no—
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—22no—
560C8—41no—
561C8—1222no—
562C8—2441no—
563C8—2441no—
564C8—2441no—
565C22—41no—
566C22—41no—
567C22—81no—
568C22—9761no—
569C22—4881no—
570C22—4882no—
571C22—4882no—
572C22—2441no—
573C22—2442no—
574C22#1684023411yescharacteristic
575C22#1684050011yescharacteristic
576C22#1684050111yescharacteristic
577C4—41no—
578C4—42no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—4881no—
583C4—2441no—
584C4—2441no—
585C4—2441no—
586C4#1684023411yescharacteristic
587C4#1684023511yescharacteristic
588C4#1684050011yescharacteristic
589C4#1684050211yescharacteristic
590C2—21no—
591C2—21no—
592C2—21no—
593C2—81no—
594C2—4882no—
595C2—4881no—
596C2#2144899411yescharacteristic
subgroupG/H
centerC2#21448994
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1905C2
socleC2×C61#2380
prime radical#21456286C2
cosocleC22×C4×C61C24
hypercenter#1905D61
hypocenterC61#23362
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1905

Complements

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

NH ≅ G/Ntype
2. #21446313C2semidirect
3. #21446309C2semidirect
4. #21446307C2semidirect
5. #21446298C2semidirect
6. #21446297C2semidirect
7. #21446294C2semidirect
8. #21446289C2semidirect
9. #21446222C2semidirect
10. #21446204C2semidirect
11. #21446200C2semidirect
12. #21446199C2semidirect
13. #21445967C2semidirect
14. #21445966C2semidirect
15. C61×#1905C2semidirect
24. #16840155C4semidirect
25. #16840148C4semidirect
26. #16840147C4semidirect
27. #16840143C4semidirect
63. C2×#11758202C2×C4semidirect
66. C2×#11758201C2×C4semidirect
79. #14519775C2×C4semidirect
80. #14519773C2×C4semidirect
84. #14519766C2×C4semidirect
85. #14519764C2×C4semidirect
103. #14519687C2×C4semidirect
104. C8×Dic61C2×C4semidirect
109. C2×Q16×C61C2×C4semidirect
110. C2×SD16×C61C2×C4semidirect
294. #1905D61semidirect
353. C61#23362semidirect
367. C2×Q16C4×D61semidirect
371. C2×SD16C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214562862
#214463132
#214463092
#214463072
#214462982
#214462972
#214462942
#214462892
#214462222
#214462042
#214462002
#214461992
#214459672
#214459662
C61×#19052
#2336261