Finite Group Databasebeta

← #29663774

Subgroups — #29663774

Order 15616.

590 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457145C211yesmaximal, characteristic
2#21456538C211yesmaximal, characteristic, complemented
3#21454824C211yesmaximal, characteristic, complemented
4#21445399C211yesmaximal, characteristic, complemented
5#21445361C211yesmaximal, characteristic, complemented
6#21445340C211yesmaximal, characteristic, complemented
7#21445286C211yesmaximal, characteristic, complemented
8#21445268C211yesmaximal, characteristic, complemented
9#21445230C211yesmaximal, characteristic, complemented
10#21445221C211yesmaximal, characteristic, complemented
11#21445169C211yesmaximal, characteristic, complemented
12#21444572C211yesmaximal, characteristic, complemented
13#21444479C211yesmaximal, characteristic, complemented
14#21444476C211yesmaximal, characteristic, complemented
15C61×#3233C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17#16840165C2211yescharacteristic
18#16840136C2211yescharacteristic
19#16840112C2211yescharacteristic
20#16840110C2211yescharacteristic
21#16840059—21no—
22#16840057C2211yescharacteristic
23#16840052C2211yescharacteristic
24#16840010—21no—
25#16840010—21no—
26#16839996—21no—
27#16839993—21no—
28#16839974—21no—
29#16839962C2211yescharacteristic
30#16839957C2211yescharacteristic
31#16839893C2211yescharacteristic
32Q8×#92418C2211yescharacteristic
33#16839860—21no—
34#16839860—21no—
35#16839840—21no—
36#16839840C2211yescharacteristic
37#16839834—21no—
38D4×#92418C2211yescharacteristic
39#16839823—21no—
40#16839815C2211yescharacteristic
41C2×C4⋊#92418C2211yescharacteristic
42#16839718—21no—
43C4×#11758194C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839510C2211yescharacteristic
51#16839510C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C61×#529C2211yescharacteristic
58C61×#499—21no—
59C61×#488—21no—
60C61×#485—21no—
61C61×#482—21no—
62C61×#477C2211yescharacteristic
63C61×#476C2211yescharacteristic
64C61×#475C2211yescharacteristic
65C61×#474C2211yescharacteristic
66C61×#431C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758201—21no—
69C2×#11758201—21no—
70C2×#11758201—21no—
71C2×#11758199—21no—
72C2×#11758194—21no—
73C2×#11758194D411yescharacteristic
74C2×#11758194D411yescharacteristic
75C22×#92418—21no—
76C22×#92418—21no—
77#14519777C2311yescharacteristic
78#14519774C2311yescharacteristic
79#14519765—21no—
80#14519751—21no—
81#14519751—21no—
82C4×D244—21no—
83C4×D244—21no—
84#14519710—21no—
85#14519710—21no—
86#14519710—21no—
87#14519708—21no—
88C4⋊#92418—21no—
89C4⋊#92418—21no—
90C4⋊#92418—21no—
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519701—21no—
96#14519701—21no—
97#14519701—21no—
98#14519698—21no—
99#14519697—21no—
100#14519697—21no—
101#14519697—21no—
102#14519697—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105C22⋊#92418—21no—
106#14519688—21no—
107#14519688D411yescharacteristic
108#14519688D411yescharacteristic
109C4×#92418—21no—
110C4×#92418—21no—
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C4⋊Q8×C61C2311yescharacteristic
114C4⋊Q8×C61C2311yescharacteristic
115#126×C61—21no—
116#125×C61—21no—
117#125×C61D411yescharacteristic
118#124×C61D411yescharacteristic
119#123×C61—21no—
120C22⋊Q8×C61—21no—
121C22⋊Q8×C61—21no—
122C22⋊Q8×C61D411yescharacteristic
123#121×C61D411yescharacteristic
124C4×Q8×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61C2311yescharacteristic
127C4×D4×C61—21no—
128C4×D4×C61C2311yescharacteristic
129#117×C61—21no—
130#117×C61—21no—
131C2×C42×C61C2311yescharacteristic
132#107×C61—21no—
133#107×C61—21no—
134#106×C61—21no—
135#106×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143C4×C8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61#4111yescharacteristic
153C2×C22⋊D61—41no—
154#11758222—82no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×C4×D61—41no—
159C2×Dic122—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—41no—
171C4⋊C4×C61—41no—
172C4⋊C4×C61C2×D411yescharacteristic
173C4⋊C4×C61C2×D411yescharacteristic
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—41no—
180C42×C61—21no—
181C42×C61—41no—
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61#4111yescharacteristic
185C42×C61C2411yescharacteristic
186#11758201—42no—
187#11758201—42no—
188#11758201—81no—
189#11758199—81no—
190#11758198—41no—
191#11758198—41no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194#11758194—22no—
195#11758194—41no—
196#11758194—42no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—42no—
202C2×#92418—81no—
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C23×C61—21no—
208C23×C61#12111yescharacteristic
209C22×D61—41no—
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—81no—
215D4×C61—21no—
216D4×C61—21no—
217D4×C61—81no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61#12111yescharacteristic
231C2×C4×C61#12111yescharacteristic
232C2×C4×C61#12111yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C22⋊D61—82no—
237C2×Dic61—41no—
238D244—41no—
239D244—41no—
240D244—41no—
241D244—41no—
242D244—81no—
243C4×D61—82no—
244Dic122—81no—
245C8×C61—41no—
246C8×C61—41no—
247#92418—21no—
248#92418—21no—
249#92418—21no—
250#92418—41no—
251#92418—41no—
252#92418—41no—
253#92418—41no—
254#92418#13512yes—
255#19253—611nomaximal, Sylow
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61C2×#12111yescharacteristic
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C4×C61—21no—
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—21no—
270C4×C61—41no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61—81no—
274C4×C61—81no—
275C4×C61C2×#13511yescharacteristic
276C4×C61#57511yescharacteristic
277C4×C61#57711yescharacteristic
278C4×C61#57811yescharacteristic
279Dic61—81no—
280#3247—611no—
281#3233D6111yescharacteristic, complemented
282#3182—611no—
283#2969—611no—
284#1704—611no—
285#1698—611no—
286#1695—611no—
287#1685—611no—
288#1683—611no—
289#1677—611no—
290#1676—611no—
291#1666—611no—
292#1565—611no—
293#1543—611no—
294#1542—611no—
295C2×C61—41no—
296C2×C61—81no—
297C2×C61#303511yescharacteristic
298C2×C61#306311yescharacteristic
299C2×C61#306811yescharacteristic
300D61—161no—
301#533C2×D6111yescharacteristic
302#532—611no—
303#529C2×D6111yescharacteristic
304#500—611no—
305#499—21no—
306#492—611no—
307#488—21no—
308#487—611no—
309#486—611no—
310#485—21no—
311#482—21no—
312#478—1221no—
313#477—611no—
314#477C2×D6111yescharacteristic
315#476—611no—
316#476C2×D6111yescharacteristic
317#475C2×D6111yescharacteristic
318#474C2×D6111yescharacteristic
319#468—1221no—
320#468—1221no—
321#466—1221no—
322#465—1221no—
323#463—1221no—
324#460—611no—
325#459—611no—
326#446—611no—
327C8×Q8—611no—
328#440—1221no—
329#440—1221no—
330#436—611no—
331#436—1221no—
332#435—1221no—
333C8×D4—611no—
334#433—1221no—
335#431—611no—
336#431C2×D6111yescharacteristic
337C2×C4⋊C8—611no—
338#405—1221no—
339C4×M16—611no—
340#335—611no—
341#335—611no—
342#334—611no—
343#334—611no—
344#333—611no—
345#333—611no—
346#333—611no—
347#333—611no—
348#332—611no—
349#332—611no—
350#331—611no—
351#331—611no—
352C61#1925311yescharacteristic, complemented, Sylow
353C2×#41—1221no—
354C2×SD16—1221no—
355C2×SD16—1221no—
356C2×SD16—1221no—
357C2×D8—1221no—
358C2×M16—611no—
359C2×M16—611no—
360C2×M16—1221no—
361C22×C8—1221no—
362C22×C8—1221no—
363C4⋊Q8—611no—
364C4⋊Q8C22×D6111yescharacteristic
365C4⋊Q8C22×D6111yescharacteristic
366#127—611no—
367#126—21no—
368#125—21no—
369#125C22⋊D6111yescharacteristic
370#124—1221no—
371#124C22⋊D6111yescharacteristic
372#123—21no—
373C22⋊Q8—21no—
374C22⋊Q8—21no—
375C22⋊Q8C22⋊D6111yescharacteristic
376#121—1221no—
377#121—1221no—
378#121C22⋊D6111yescharacteristic
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8C22×D6111yescharacteristic
382C4×D4—21no—
383C4×D4—1221no—
384C4×D4—1221no—
385C4×D4C22×D6111yescharacteristic
386#117—21no—
387#117—21no—
388C2×C42C22×D6111yescharacteristic
389#107—21no—
390#107—21no—
391#107—1221no—
392#107—1221no—
393#107—1221no—
394#106—21no—
395#106—21no—
396#106—1221no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8C22×D6111yescharacteristic
405C4⋊C8C22×D6111yescharacteristic
406#103—21no—
407#103—21no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#102—21no—
412#102—21no—
413#102—1221no—
414#102—1221no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418C22⋊C8—21no—
419C22⋊C8—1221no—
420C22⋊C8—1221no—
421C22⋊C8—1221no—
422#97—611no—
423#97—611no—
424#97—1221no—
425C4×C8—21no—
426C4×C8—611no—
427C4×C8—611no—
428C4×C8—1221no—
429C4×C8—1221no—
430#41—4882no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—1221no—
434C2×Q8C2×C22⋊D6111yescharacteristic
435C2×D4—21no—
436C2×D4—1221no—
437C2×D4—1221no—
438C2×D4—1221no—
439C2×D4—2441no—
440C2×D4C2×C22⋊D6111yescharacteristic
441C22×C4—21no—
442C22×C4—2441no—
443C22×C4D4×D6111yescharacteristic
444C22×C4#1175822411yescharacteristic
445C22×C4C2×C22⋊D6111yescharacteristic
446SD16—4881no—
447SD16—2442no—
448SD16—2442no—
449D8—4881no—
450M16—1222no—
451M16—2442no—
452M16—2441no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—611no—
456C2×C8—611no—
457C2×C8—611no—
458C2×C8—611no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—4881no—
464C2×C8—2442no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4C2×C22⋊D6111yescharacteristic
478C4⋊C4C2×C22⋊D6111yescharacteristic
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—41no—
485C22⋊C4—2441no—
486C22⋊C4—2441no—
487C42—21no—
488C42—41no—
489C42D4×D6111yescharacteristic
490C42#1175822411yescharacteristic
491C42C2×C22⋊D6111yescharacteristic
492C42C23×D6111yescharacteristic
493C23—21no—
494C23—2441no—
495C23#1451974911yescharacteristic
496Q8—21no—
497Q8—21no—
498Q8—41no—
499Q8—41no—
500Q8—81no—
501Q8—4881no—
502D4—21no—
503D4—21no—
504D4—81no—
505D4—4881no—
506D4—4882no—
507D4—2441no—
508D4—2441no—
509D4—2441no—
510D4—2441no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—4882no—
524C2×C4—2441no—
525C2×C4#1451974911yescharacteristic
526C2×C4#1451974911yescharacteristic
527C2×C4#1451974911yescharacteristic
528C2×C4C2×D4×D6111yescharacteristic
529C2×C4C22×C22⋊D6111yescharacteristic
530C2×C4C2×#1175822411yescharacteristic
531C8—41no—
532C8—41no—
533C8—612no—
534C8—1221no—
535C8—1221no—
536C8—1221no—
537C8—2441no—
538C8—2441no—
539C8—2441no—
540C8—2441no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—4881no—
548C22—4881no—
549C22—4881no—
550C22C2×#1451974911yescharacteristic
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—4881no—
561C4#135×D6111yescharacteristic
562C4#1684055911yescharacteristic
563C4#1684057011yescharacteristic
564C4#1684057711yescharacteristic
565C2—41no—
566C2—81no—
567C2—9761no—
568C2#2145527011yescharacteristic
569C2#2145550611yescharacteristic
570C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3233C2
socleC22×C61C2×#121
prime radical#21457145C2
hypercenter#3233D61
hypocenterC61#19253
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3233

Complements

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

NH ≅ G/Ntype
2. #21456538C2semidirect
3. #21454824C2semidirect
4. #21445399C2semidirect
5. #21445361C2semidirect
6. #21445340C2semidirect
7. #21445286C2semidirect
8. #21445268C2semidirect
9. #21445230C2semidirect
10. #21445221C2semidirect
11. #21445169C2semidirect
12. #21444572C2semidirect
13. #21444479C2semidirect
14. #21444476C2semidirect
15. C61×#3233C2semidirect
281. #3233D61semidirect
352. C61#19253semidirect

Maximal subgroups

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

subgroupindex
#214571452
#214565382
#214548242
#214453992
#214453612
#214453402
#214452862
#214452682
#214452302
#214452212
#214451692
#214445722
#214444792
#214444762
C61×#32332
#1925361