Finite Group Databasebeta

← #29748646

Subgroups — #29748646

Order 15616.

704 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450358C211yesmaximal, characteristic, complemented
2#21448892C211yesmaximal, characteristic
3#21447235C211yesmaximal, characteristic
4#21447191C211yesmaximal, characteristic
5#21447059C211yesmaximal, characteristic
6#21447058C211yesmaximal, characteristic
7#21446494C211yesmaximal, characteristic, complemented
8#21446454C211yesmaximal, characteristic, complemented
9#21446406C211yesmaximal, characteristic, complemented
10#21446396C211yesmaximal, characteristic, complemented
11#21446374C211yesmaximal, characteristic, complemented
12#21446084C211yesmaximal, characteristic, complemented
13#21445833C211yesmaximal, characteristic, complemented
14C4×#14519698C211yesmaximal, characteristic
15C61×#1864C211yesmaximal, characteristic
16C2×#14519764C2211yescharacteristic
17C2×#14519742C2211yescharacteristic
18C2×#14519709C2211yescharacteristic
19C2×#14519706C2211yescharacteristic
20C2×#14519702C2211yescharacteristic
21C2×#14519702C2211yescharacteristic
22C2×#14519700C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×C4×#92418C2211yescharacteristic
27#16839700—21no—
28#16839695—21no—
29#16839668C2211yescharacteristic
30#16839665C2211yescharacteristic
31#16839661C2211yescharacteristic
32#16839658C2211yescharacteristic
33#16839658C2211yescharacteristic
34#16839656—21no—
35#16839650—21no—
36#16839647—21no—
37#16839635—21no—
38#16839627C2211yescharacteristic
39#16839626—21no—
40C4⋊C4×Dic61C2211yescharacteristic
41C4×#11758196C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43C4×C22⋊Dic61—21no—
44#16839525C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839516C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519764—22no—
63#14519764—22no—
64#14519742—22no—
65#14519742—41no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196D411yescharacteristic
74C2×#11758196D411yescharacteristic
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61C2311yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84#14519709—22no—
85#14519706—41no—
86#14519702—22no—
87#14519702—22no—
88#14519700—22no—
89#14519700—41no—
90#14519698—22no—
91#14519698—22no—
92C4×#92418—41no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684D411yescharacteristic
103#14519684D411yescharacteristic
104C22×D4×C61C2311yescharacteristic
105C23×C4×C61—21no—
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C4×D4×C61—22no—
109C4×D4×C61—41no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61—21no—
115C2×C42×C61C2311yescharacteristic
116#106×C61—22no—
117#102×C61—22no—
118#102×C61—41no—
119#95×C61—21no—
120C24×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61#4112yes—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—42no—
131C22×C4×C61—42no—
132C22×C4×C61—81no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122#4112yes—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—81no—
162C4⋊C4×C61#4112yes—
163C22⋊C4×C61—42no—
164C42×C61—21no—
165C42×C61—21no—
166C22⋊Dic61—24no—
167C22⋊Dic61—24no—
168C22⋊Dic61—42no—
169C22⋊Dic61—81no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—42no—
176C4⋊Dic61#4112yes—
177#11758196—22no—
178#11758196—41no—
179#11758196—42no—
180#11758196—81no—
181#11758196—81no—
182C4×Dic61—21no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C4×Dic61—42no—
187C4×Dic61—42no—
188C4×Dic61—81no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C23×C61—22no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61C22×D411yescharacteristic
203D4×C61—22no—
204D4×C61—22no—
205D4×C61—41no—
206D4×C61—41no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—84no—
222C2×C4×C61#13512yes—
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61H(5,𝔽2)11yescharacteristic
228C2×C4×C612−1+411yescharacteristic
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—22no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—22no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253Dic122—22no—
254Dic122—22no—
255Dic122—41no—
256Dic122—41no—
257C8×C61—41no—
258C8×C61—41no—
259#92418—41no—
260#92418—41no—
261#26760—611nomaximal, Sylow
262C22×C61—42no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61#52511yescharacteristic
270C22×C61#54611yescharacteristic
271C22×C61#54811yescharacteristic
272C22×C61C2×#13511yescharacteristic
273C22×C61#57511yescharacteristic
274C22×C61#57711yescharacteristic
275C22×C61#57711yescharacteristic
276C4×C61—22no—
277C4×C61—42no—
278C4×C61—41no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61#52911yescharacteristic
282C4×C61#53911yescharacteristic
283C4×C61#55211yescharacteristic
284C4×C61#55311yescharacteristic
285Dic61—41no—
286Dic61—42no—
287Dic61—41no—
288Dic61—42no—
289Dic61—41no—
290Dic61—41no—
291Dic61—81no—
292Dic61—81no—
293#2508—611no—
294#2367—611no—
295#2073—611no—
296#2065—611no—
297#2042—611no—
298#2042—611no—
299#1933—611no—
300#1927—611no—
301#1920—611no—
302#1919—611no—
303#1916—611no—
304#1865—611no—
305#1864D6111yescharacteristic
306#1805—611no—
307C4×#102—611no—
308C2×C61—82no—
309C2×C61#246011yescharacteristic
310C2×C61#310211yescharacteristic
311C2×C61#312511yescharacteristic
312C2×C61#329611yescharacteristic
313C2×C61#332111yescharacteristic
314C2×C61#336011yescharacteristic
315C2×C61#337111yescharacteristic
316C2×#124—611no—
317C2×C4×Q8—611no—
318C2×C4×D4C2×D6111yescharacteristic
319C2×#106—611no—
320C2×#106C2×D6111yescharacteristic
321C2×C4⋊C8—611no—
322C2×#103—611no—
323C2×#103—611no—
324C2×#103—611no—
325C2×#103—611no—
326C2×#102—611no—
327C2×#102—611no—
328C2×#102C2×D6111yescharacteristic
329C2×#102C2×D6111yescharacteristic
330C2×C4×C8—611no—
331#400—1221no—
332#398—1221no—
333#391—611no—
334#390—611no—
335#389—611no—
336#389—611no—
337#389—611no—
338#389C2×D6111yescharacteristic
339#388—1221no—
340#388—1221no—
341#387—1221no—
342#385—1221no—
343#382—611no—
344#382—1221no—
345C4×C4⋊C4—611no—
346C4×C4⋊C4—611no—
347C4×C4⋊C4—611no—
348C4×C22⋊C4—1221no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340C2×D6111yescharacteristic
355#336—611no—
356#336—611no—
357#336—611no—
358#336C2×D6111yescharacteristic
359C61#2676011yescharacteristic, complemented, Sylow
360C22×Q8—611no—
361C22×D4C22×D6111yescharacteristic
362C23×C4—21no—
363C22×C8—611no—
364C22×C8—611no—
365C22×C8C22×D6111yescharacteristic
366C22×C8C22×D6111yescharacteristic
367#124—1222no—
368#124—1222no—
369C4×Q8—1222no—
370C4×Q8—2441no—
371C4×D4—22no—
372C4×D4—41no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—1221no—
394C2×C42—1221no—
395C2×C42—1221no—
396C2×C42C22×D6111yescharacteristic
397#106—22no—
398#106—1222no—
399C4⋊C8—2441no—
400#103—1222no—
401#103—1222no—
402#103—1222no—
403#103—2441no—
404#102—22no—
405#102—41no—
406#102—1222no—
407#102—1222no—
408C4×C8—2441no—
409#95—21no—
410#95—611no—
411#95—611no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420#95—1221no—
421C24—21no—
422C2×Q8—612no—
423C2×Q8—1221no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—1221no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4#1175822212yes—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—81no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4D4×D6111yescharacteristic
449C22×C4D4×D6111yescharacteristic
450C22×C4C23×D6111yescharacteristic
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—81no—
469C4⋊C4—612no—
470C4⋊C4—1221no—
471C4⋊C4—1222no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—4881no—
477C4⋊C4—4881no—
478C4⋊C4—2441no—
479C4⋊C4—2442no—
480C4⋊C4—2442no—
481C4⋊C4#1175822412yes—
482C22⋊C4—42no—
483C22⋊C4—1224no—
484C22⋊C4—1224no—
485C22⋊C4—4881no—
486C22⋊C4—2442no—
487C42—21no—
488C42—21no—
489C42—1221no—
490C42—1221no—
491C42—1221no—
492C42—1221no—
493C42—4881no—
494C42—2442no—
495C42—2442no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23C2×D4×D6111yescharacteristic
504Q8—1222no—
505Q8—1222no—
506Q8—2441no—
507Q8—2441no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—84no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1222no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1222no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4#1451979812yes—
552C2×C4#41×D6111yescharacteristic
553C2×C4#41×D6111yescharacteristic
554C2×C4C2×#1175822411yescharacteristic
555C2×C4C2×#1175822211yescharacteristic
556C2×C4#1451982211yescharacteristic
557C2×C4#1451982511yescharacteristic
558C8—41no—
559C8—41no—
560C8—2441no—
561C8—2441no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22#1684027111yescharacteristic
570C22#1684039911yescharacteristic
571C22#1684042211yescharacteristic
572C22C2×#1451979811yescharacteristic
573C22#1684055911yescharacteristic
574C22#1684057211yescharacteristic
575C22#1684057211yescharacteristic
576C4—22no—
577C4—41no—
578C4—42no—
579C4—81no—
580C4—81no—
581C4—4881no—
582C4—4881no—
583C4—2442no—
584C4—2442no—
585C4—2441no—
586C4—2441no—
587C4—2441no—
588C4—2441no—
589C4#1684029211yescharacteristic
590C4#1684035911yescharacteristic
591C4#1684044611yescharacteristic
592C4#1684045711yescharacteristic
593C2—82no—
594C2#2144977311yescharacteristic
595C2#2145584111yescharacteristic
596C2#2145610011yescharacteristic
597C2#2145765811yescharacteristic
598C2#2145800911yescharacteristic
599C2#2145846711yescharacteristic
600C2#2145858511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1864C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1864D61
hypocenterC61#26760
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
1. #21450358C2semidirect
7. #21446494C2semidirect
8. #21446454C2semidirect
9. #21446406C2semidirect
10. #21446396C2semidirect
11. #21446374C2semidirect
12. #21446084C2semidirect
13. #21445833C2semidirect
359. C61#26760semidirect

Maximal subgroups

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

subgroupindex
#214503582
#214488922
#214472352
#214471912
#214470592
#214470582
#214464942
#214464542
#214464062
#214463962
#214463742
#214460842
#214458332
C4×#145196982
C61×#18642
#2676061