Finite Group Databasebeta

← #29663782

Subgroups — #29663782

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#21456994C211yesmaximal, characteristic, complemented
2#21456533C211yesmaximal, characteristic, complemented
3#21454824C211yesmaximal, characteristic, complemented
4#21445403C211yesmaximal, characteristic, complemented
5#21445357C211yesmaximal, characteristic, complemented
6#21445340C211yesmaximal, characteristic, complemented
7#21445282C211yesmaximal, characteristic, complemented
8#21445272C211yesmaximal, characteristic, complemented
9#21445226C211yesmaximal, characteristic, complemented
10#21445221C211yesmaximal, characteristic, complemented
11#21445165C211yesmaximal, characteristic, complemented
12#21444569C211yesmaximal, characteristic, complemented
13#21444481C211yesmaximal, characteristic, complemented
14#21444477C211yesmaximal, characteristic, complemented
15C61×#3247C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17#16840165C2211yescharacteristic
18#16840161—21no—
19#16840134C2211yescharacteristic
20#16840116—21no—
21#16840114C2211yescharacteristic
22#16840108C2211yescharacteristic
23#16840100—21no—
24#16840083—21no—
25#16840057C2211yescharacteristic
26#16840052C2211yescharacteristic
27#16840047C2211yescharacteristic
28#16840042C2211yescharacteristic
29#16840006—21no—
30#16840006—21no—
31#16839893C2211yescharacteristic
32Q8×#92418C2211yescharacteristic
33#16839862—21no—
34#16839862—21no—
35#16839840—21no—
36#16839840C2211yescharacteristic
37#16839834—21no—
38D4×#92418C2211yescharacteristic
39#16839823—21no—
40#16839815C2211yescharacteristic
41#16839815C2211yescharacteristic
42#16839718—21no—
43C4×#11758194C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839510C2211yescharacteristic
51#16839510C2211yescharacteristic
52#16839507C2211yescharacteristic
53#16839507C2211yescharacteristic
54#16839506C2211yescharacteristic
55#16839506C2211yescharacteristic
56C61×#532C2211yescharacteristic
57C61×#529C2211yescharacteristic
58C61×#478—21no—
59C61×#477C2211yescharacteristic
60C61×#476C2211yescharacteristic
61C61×#466—21no—
62C61×#465—21no—
63C61×#463—21no—
64C61×#460C2211yescharacteristic
65C61×#459C2211yescharacteristic
66C2×C4⋊C8×C61C2211yescharacteristic
67C2×#11758200—21no—
68C2×#11758200—21no—
69C2×#11758194—21no—
70C2×#11758194D411yescharacteristic
71C2×#11758194D411yescharacteristic
72C22×#92418—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519769—21no—
76#14519756—21no—
77#14519756—21no—
78C4×Dic122—21no—
79C4×Dic122—21no—
80#14519736—21no—
81#14519710—21no—
82#14519710—21no—
83#14519710—21no—
84#14519710—21no—
85#14519708—21no—
86#14519708—21no—
87C4⋊#92418—21no—
88C4⋊#92418—21no—
89C4⋊#92418C2311yescharacteristic
90C4⋊#92418C2311yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93#14519701—21no—
94#14519701—21no—
95#14519700—21no—
96#14519700—21no—
97#14519698—21no—
98#14519698—21no—
99#14519698—21no—
100#14519698—21no—
101C22⋊#92418—21no—
102C22⋊#92418—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105#14519688—21no—
106#14519688D411yescharacteristic
107#14519688D411yescharacteristic
108C4×#92418—21no—
109C4×#92418—21no—
110C4×#92418—21no—
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C2×#41×C61—21no—
114C2×SD16×C61—21no—
115C2×D8×C61—21no—
116C22×C8×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118#127×C61C2311yescharacteristic
119#126×C61—21no—
120#125×C61D411yescharacteristic
121#124×C61—21no—
122#124×C61D411yescharacteristic
123#123×C61—21no—
124C22⋊Q8×C61D411yescharacteristic
125#121×C61—21no—
126#121×C61—21no—
127#121×C61D411yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61C2311yescharacteristic
133#117×C61—21no—
134C2×C42×C61C2311yescharacteristic
135#107×C61—21no—
136#106×C61—21no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#102×C61—21no—
142#102×C61—21no—
143#102×C61—21no—
144#41×C61—82no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—41no—
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—41no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61#4111yescharacteristic
158C2×Dic122—21no—
159C2×Dic122—21no—
160SD16×C61—81no—
161D8×C61—81no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—81no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61C2×D411yescharacteristic
172C4⋊C4×C61C2×D411yescharacteristic
173C22⋊C4×C61—21no—
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—41no—
179C22⋊C4×C61—41no—
180C42×C61—21no—
181C42×C61C2×D411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61#4111yescharacteristic
184C42×C61C2411yescharacteristic
185C22⋊Dic61—41no—
186#11758200—42no—
187#11758200—42no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192#11758196—41no—
193#11758196—41no—
194C4×Dic61—41no—
195#11758194—22no—
196#11758194—41no—
197#11758194—42no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—42no—
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C23×C61—21no—
208C23×C61—41no—
209C23×C61#12111yescharacteristic
210Q8×C61—21no—
211Q8×C61—21no—
212Q8×C61—81no—
213D4×C61—21no—
214D4×C61—21no—
215D4×C61—41no—
216D4×C61—41no—
217D4×C61—41no—
218D4×C61—41no—
219D4×C61—81no—
220D4×C61—81no—
221D4×C61—82no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—82no—
234C2×C4×C61#12111yescharacteristic
235C2×C4×C61#12111yescharacteristic
236C2×C4×C61#12111yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×Dic61—41no—
241C2×Dic61—41no—
242Dic122—41no—
243Dic122—41no—
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×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61C2×#12111yescharacteristic
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×C61C2×#13511yescharacteristic
275C4×C61#57511yescharacteristic
276C4×C61#57711yescharacteristic
277C4×C61#57811yescharacteristic
278Dic61—81no—
279Dic61—81no—
280#3247D6111yescharacteristic
281#3233—611no—
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—161no—
298C2×C61#303511yescharacteristic
299C2×C61#306311yescharacteristic
300C2×C61#306811yescharacteristic
301#533—611no—
302#532C2×D6111yescharacteristic
303#529C2×D6111yescharacteristic
304#500—611no—
305#499—1221no—
306#492—611no—
307#488—1221no—
308#487—611no—
309#486—611no—
310#485—1221no—
311#482—1221no—
312#478—21no—
313#477—611no—
314#477C2×D6111yescharacteristic
315#476—611no—
316#476C2×D6111yescharacteristic
317#475—611no—
318#474—611no—
319#468—1221no—
320#468—1221no—
321#466—21no—
322#465—21no—
323#463—21no—
324#460C2×D6111yescharacteristic
325#459C2×D6111yescharacteristic
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#431—611no—
337C2×C4⋊C8C2×D6111yescharacteristic
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—21no—
354C2×SD16—21no—
355C2×SD16—1221no—
356C2×SD16—1221no—
357C2×D8—21no—
358C2×M16—611no—
359C2×M16—611no—
360C2×M16—1221no—
361C22×C8—21no—
362C22×C8—1221no—
363C4⋊Q8—611no—
364C4⋊Q8—611no—
365C4⋊Q8C22×D6111yescharacteristic
366#127C22×D6111yescharacteristic
367#126—21no—
368#125—1221no—
369#125C22⋊D6111yescharacteristic
370#124—21no—
371#124C22⋊D6111yescharacteristic
372#123—21no—
373C22⋊Q8—1221no—
374C22⋊Q8—1221no—
375C22⋊Q8C22⋊D6111yescharacteristic
376#121—21no—
377#121—21no—
378#121C22⋊D6111yescharacteristic
379C4×Q8—1221no—
380C4×Q8—1221no—
381C4×Q8C22×D6111yescharacteristic
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4C22×D6111yescharacteristic
386#117—21no—
387#117—1221no—
388C2×C42C22×D6111yescharacteristic
389#107—21no—
390#107—1221no—
391#107—1221no—
392#107—1221no—
393#107—1221no—
394#106—21no—
395#106—1221no—
396#106—1221no—
397C4⋊C8—21no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8C22×D6111yescharacteristic
405C4⋊C8C22×D6111yescharacteristic
406#103—21no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#102—21no—
412#102—21no—
413#102—21no—
414#102—1221no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418C22⋊C8—1221no—
419C22⋊C8—1221no—
420C22⋊C8—1221no—
421C22⋊C8—1221no—
422#97—611no—
423#97—611no—
424#97—1221no—
425C4×C8—611no—
426C4×C8—611no—
427C4×C8—1221no—
428C4×C8—1221no—
429C4×C8—1221no—
430#41—82no—
431C2×Q8—21no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8C2×C22⋊D6111yescharacteristic
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—41no—
440C2×D4C2×C22⋊D6111yescharacteristic
441C22×C4—21no—
442C22×C4—41no—
443C22×C4D4×D6111yescharacteristic
444C22×C4#1175822411yescharacteristic
445C22×C4C2×C22⋊D6111yescharacteristic
446SD16—81no—
447SD16—2442no—
448SD16—2442no—
449D8—81no—
450M16—1222no—
451M16—2442no—
452M16—2441no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—81no—
456C2×C8—611no—
457C2×C8—611no—
458C2×C8—611no—
459C2×C8—611no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—2442no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—2441no—
476C4⋊C4—2441no—
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—41no—
486C22⋊C4—2441no—
487C42—21no—
488C42—2441no—
489C42D4×D6111yescharacteristic
490C42#1175822411yescharacteristic
491C42C2×C22⋊D6111yescharacteristic
492C42C23×D6111yescharacteristic
493C23—21no—
494C23—41no—
495C23#1451974911yescharacteristic
496Q8—21no—
497Q8—21no—
498Q8—81no—
499Q8—4881no—
500Q8—2441no—
501Q8—2441no—
502D4—21no—
503D4—21no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—81no—
509D4—81no—
510D4—82no—
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—82no—
523C2×C4—2441no—
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—81no—
548C22—81no—
549C22—81no—
550C22C2×#1451974911yescharacteristic
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—81no—
559C4—4881no—
560C4—4881no—
561C4#135×D6111yescharacteristic
562C4#1684055911yescharacteristic
563C4#1684057311yescharacteristic
564C4#1684057711yescharacteristic
565C2—41no—
566C2—81no—
567C2—161no—
568C2#2145527011yescharacteristic
569C2#2145551211yescharacteristic
570C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3247C2
socleC22×C61C2×#121
hypercenter#3247D61
hypocenterC61#19253
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3247

Complements

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

NH ≅ G/Ntype
1. #21456994C2semidirect
2. #21456533C2semidirect
3. #21454824C2semidirect
4. #21445403C2semidirect
5. #21445357C2semidirect
6. #21445340C2semidirect
7. #21445282C2semidirect
8. #21445272C2semidirect
9. #21445226C2semidirect
10. #21445221C2semidirect
11. #21445165C2semidirect
12. #21444569C2semidirect
13. #21444481C2semidirect
14. #21444477C2semidirect
352. C61#19253semidirect

Maximal subgroups

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

subgroupindex
#214569942
#214565332
#214548242
#214454032
#214453572
#214453402
#214452822
#214452722
#214452262
#214452212
#214451652
#214445692
#214444812
#214444772
C61×#32472
#1925361