Finite Group Databasebeta

← #29667636

Subgroups — #29667636

Order 15616.

590 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457037C211yesmaximal, characteristic, complemented
2#21456937C211yesmaximal, characteristic, complemented
3#21454979C211yesmaximal, characteristic, complemented
4#21445357C211yesmaximal, characteristic, complemented
5#21445350C211yesmaximal, characteristic, complemented
6#21445345C211yesmaximal, characteristic, complemented
7#21445340C211yesmaximal, characteristic, complemented
8#21445233C211yesmaximal, characteristic, complemented
9#21445226C211yesmaximal, characteristic, complemented
10#21445221C211yesmaximal, characteristic, complemented
11#21445216C211yesmaximal, characteristic, complemented
12#21444602C211yesmaximal, characteristic, complemented
13#21444514C211yesmaximal, characteristic, complemented
14C2×#16839510C211yesmaximal, characteristic, complemented
15C61×#3247C211yesmaximal, characteristic
16C2×#14519777C2211yescharacteristic
17#16840172—21no—
18#16840165C2211yescharacteristic
19#16840165C2211yescharacteristic
20#16840121—21no—
21#16840108C2211yescharacteristic
22#16840108C2211yescharacteristic
23#16840090—21no—
24#16840073—21no—
25#16840073—21no—
26#16840073—21no—
27#16840057C2211yescharacteristic
28#16840057C2211yescharacteristic
29#16840052C2211yescharacteristic
30#16840052C2211yescharacteristic
31#16840042—21no—
32#16840042—21no—
33#16839893C2211yescharacteristic
34Q8×#92418C2211yescharacteristic
35#16839840C2211yescharacteristic
36#16839834—21no—
37D4×#92418—21no—
38D4×#92418C2211yescharacteristic
39#16839823—21no—
40#16839819C2211yescharacteristic
41#16839815C2211yescharacteristic
42#16839789—21no—
43C2×C4⋊#92418C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839510C2211yescharacteristic
51#16839510C2211yescharacteristic
52#16839510C2211yescharacteristic
53#16839510C2211yescharacteristic
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
67C22×Dic122—21no—
68C2×#11758194—21no—
69C22×#92418—21no—
70C22×#92418—21no—
71#14519777—21no—
72#14519777—21no—
73#14519777—21no—
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78#14519710—21no—
79#14519710—21no—
80#14519710—21no—
81#14519710—21no—
82#14519710—21no—
83#14519710—21no—
84#14519708—21no—
85#14519708—21no—
86C4⋊#92418—21no—
87C4⋊#92418—21no—
88C4⋊#92418—21no—
89C4⋊#92418D411yescharacteristic
90C4⋊#92418D411yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519701—21no—
96#14519701—21no—
97#14519698—21no—
98#14519698—21no—
99#14519698—21no—
100#14519698—21no—
101#14519698—21no—
102#14519698—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105C22⋊#92418—21no—
106C22⋊#92418D411yescharacteristic
107C22⋊#92418D411yescharacteristic
108#14519688—21no—
109#14519688—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
158C22×Dic61—41no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—41no—
162C2×Dic122—82no—
163SD16×C61—81no—
164D8×C61—81no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—81no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—41no—
182C22⋊C4×C61—41no—
183C42×C61—21no—
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D411yescharacteristic
186C42×C61#4111yescharacteristic
187C42×C61C2411yescharacteristic
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—41no—
193C4⋊Dic61—41no—
194#11758194—81no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—81no—
200C2×#92418—81no—
201C2×#92418C2×D411yescharacteristic
202C2×#92418C2×D411yescharacteristic
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205C23×C61—21no—
206C23×C61—41no—
207C23×C61#12111yescharacteristic
208Q8×C61—21no—
209Q8×C61—21no—
210Q8×C61—81no—
211D4×C61—21no—
212D4×C61—21no—
213D4×C61—42no—
214D4×C61—41no—
215D4×C61—41no—
216D4×C61—41no—
217D4×C61—41no—
218D4×C61—81no—
219D4×C61—82no—
220C2×C4×C61—22no—
221C2×C4×C61—21no—
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—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—82no—
232C2×C4×C61#12111yescharacteristic
233C2×C4×C61#12111yescharacteristic
234C2×C4×C61#12111yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—82no—
241Dic122—82no—
242Dic122—81no—
243Dic122—81no—
244C8×C61—41no—
245C8×C61—41no—
246#92418—21no—
247#92418—21no—
248#92418—21no—
249#92418—21no—
250#92418—41no—
251#92418—41no—
252#92418—41no—
253#92418—41no—
254#19591—611nomaximal, Sylow
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61—41no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61—81no—
263C22×C61#47212yes—
264C22×C61C2×#12111yescharacteristic
265C4×C61—21no—
266C4×C61—21no—
267C4×C61—21no—
268C4×C61—21no—
269C4×C61—41no—
270C4×C61—41no—
271C4×C61—41no—
272C4×C61—81no—
273C4×C61#57711yescharacteristic
274C4×C61#57711yescharacteristic
275C4×C61#57811yescharacteristic
276C4×C61#57811yescharacteristic
277Dic61—81no—
278Dic61—81no—
279#3247D6111yescharacteristic
280#3238—611no—
281#3226—611no—
282#2994—611no—
283#1698—611no—
284#1697—611no—
285#1696—611no—
286#1695—611no—
287#1678—611no—
288#1677—611no—
289#1676—611no—
290#1675—611no—
291#1573—611no—
292#1551—611no—
293C2×#333—611no—
294C2×C61—22no—
295C2×C61—81no—
296C2×C61—161no—
297C2×C61#304611yescharacteristic
298C2×C61#306311yescharacteristic
299C2×C61C2×#47211yescharacteristic
300#532C2×D6111yescharacteristic
301C2×C4⋊Q8—611no—
302#529C2×D6111yescharacteristic
303#501—1221no—
304#500—611no—
305#500—611no—
306#489—1221no—
307#486—611no—
308#486—611no—
309#483—1221no—
310#480—1221no—
311#480—1221no—
312#480—1221no—
313#478—21no—
314#477—611no—
315#477—611no—
316#477C2×D6111yescharacteristic
317#476—611no—
318#476—611no—
319#476C2×D6111yescharacteristic
320#474—1221no—
321#474—1221no—
322#466—21no—
323#465—21no—
324#463—21no—
325#460C2×D6111yescharacteristic
326#459C2×D6111yescharacteristic
327#446—611no—
328C8×Q8—611no—
329#436—611no—
330#435—1221no—
331C8×D4—611no—
332C8×D4—1221no—
333#433—1221no—
334#432—611no—
335#431—611no—
336#424—1221no—
337C2×C4⋊C8—611no—
338C2×C4⋊C8C2×D6111yescharacteristic
339#335—611no—
340#335—611no—
341#334—611no—
342#334—611no—
343#333—611no—
344#333—611no—
345#333—611no—
346#333—611no—
347#333—611no—
348#333—611no—
349#331—611no—
350#331—611no—
351C61#1959111yescharacteristic, complemented, Sylow
352C2×#41—21no—
353C22×Q8—1221no—
354C2×SD16—21no—
355C2×D8—21no—
356C2×M16—1221no—
357C22×C8—21no—
358C22×C8—1221no—
359C22×C8—1221no—
360C4⋊Q8—611no—
361C4⋊Q8—611no—
362C4⋊Q8—1221no—
363C4⋊Q8—1221no—
364C4⋊Q8—1221no—
365C4⋊Q8C22×D6111yescharacteristic
366#127C22×D6111yescharacteristic
367#126—21no—
368#125C22⋊D6111yescharacteristic
369#124—21no—
370#124C22⋊D6111yescharacteristic
371#123—21no—
372C22⋊Q8C22⋊D6111yescharacteristic
373#121—21no—
374#121—21no—
375#121C22⋊D6111yescharacteristic
376C4×Q8C22×D6111yescharacteristic
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4C22×D6111yescharacteristic
381#117—21no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C42C22×D6111yescharacteristic
385#107—21no—
386#107—1221no—
387#107—1221no—
388#107—1221no—
389#107—1221no—
390#107—1221no—
391#107—1221no—
392#106—21no—
393#106—1221no—
394#106—1221no—
395C4⋊C8—21no—
396C4⋊C8—611no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8—1221no—
405C4⋊C8C22×D6111yescharacteristic
406C4⋊C8C22×D6111yescharacteristic
407#103—21no—
408#103—1221no—
409#103—1221no—
410#102—21no—
411#102—21no—
412#102—21no—
413#102—1221no—
414#102—1221no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419C22⋊C8—611no—
420C22⋊C8—611no—
421C22⋊C8—1221no—
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424#97—1221no—
425#97—1221no—
426C4×C8—611no—
427C4×C8—611no—
428C4×C8—1221no—
429#41—82no—
430C2×Q8—21no—
431C2×Q8—1221no—
432C2×Q8—1221no—
433C2×Q8—4882no—
434C2×Q8—2441no—
435C2×Q8C2×C22⋊D6111yescharacteristic
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—41no—
441C2×D4C2×C22⋊D6111yescharacteristic
442C22×C4—21no—
443C22×C4—41no—
444C22×C4—2441no—
445C22×C4D4×D6111yescharacteristic
446C22×C4#1175822411yescharacteristic
447C22×C4C2×C22⋊D6111yescharacteristic
448SD16—81no—
449D8—81no—
450M16—4881no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—81no—
454C2×C8—611no—
455C2×C8—611no—
456C2×C8—611no—
457C2×C8—611no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—4881no—
463C2×C8—4881no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—2441no—
475C4⋊C4—2441no—
476C4⋊C4C2×C22⋊D6111yescharacteristic
477C4⋊C4C2×C22⋊D6111yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C42—21no—
486C42D4×D6111yescharacteristic
487C42#1175822411yescharacteristic
488C42C2×C22⋊D6111yescharacteristic
489C42C23×D6111yescharacteristic
490C23—21no—
491C23—41no—
492C23#1451974911yescharacteristic
493Q8—21no—
494Q8—21no—
495Q8—81no—
496Q8—4881no—
497Q8—4881no—
498Q8—4882no—
499D4—21no—
500D4—21no—
501D4—42no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—81no—
507D4—82no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—82no—
520C2×C4—4882no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4#1451974911yescharacteristic
524C2×C4#1451974911yescharacteristic
525C2×C4#1451974911yescharacteristic
526C2×C4C2×D4×D6111yescharacteristic
527C2×C4C22×C22⋊D6111yescharacteristic
528C2×C4C2×#1175822411yescharacteristic
529C8—41no—
530C8—41no—
531C8—1221no—
532C8—1221no—
533C8—1221no—
534C8—1221no—
535C8—2441no—
536C8—2441no—
537C8—2441no—
538C8—2441no—
539C22—21no—
540C22—21no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22#1684003012yes—
548C22C2×#1451974911yescharacteristic
549C4—21no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—81no—
557C4—4881no—
558C4—4881no—
559C4#1684056911yescharacteristic
560C4#1684057311yescharacteristic
561C4#1684057611yescharacteristic
562C4#1684057711yescharacteristic
563C2—22no—
564C2—81no—
565C2—161no—
566C2#2145536711yescharacteristic
567C2#2145551211yescharacteristic
568C2C2×#1684003011yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3247C2
socleC22×C61C2×#121
hypercenter#3247D61
hypocenterC61#19591
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. #21457037C2semidirect
2. #21456937C2semidirect
3. #21454979C2semidirect
4. #21445357C2semidirect
5. #21445350C2semidirect
6. #21445345C2semidirect
7. #21445340C2semidirect
8. #21445233C2semidirect
9. #21445226C2semidirect
10. #21445221C2semidirect
11. #21445216C2semidirect
12. #21444602C2semidirect
13. #21444514C2semidirect
14. C2×#16839510C2semidirect
351. C61#19591semidirect

Maximal subgroups

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

subgroupindex
#214570372
#214569372
#214549792
#214453572
#214453502
#214453452
#214453402
#214452332
#214452262
#214452212
#214452162
#214446022
#214445142
C2×#168395102
C61×#32472
#1959161