Finite Group Databasebeta

← #29752632

Subgroups — #29752632

Order 15616.

690 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21449687C211yesmaximal, characteristic
2#21448917C211yesmaximal, characteristic, complemented
3#21447206C211yesmaximal, characteristic
4#21447160C211yesmaximal, characteristic, complemented
5#21447154C211yesmaximal, characteristic, complemented
6#21446964C211yesmaximal, characteristic
7#21446561C211yesmaximal, characteristic, complemented
8#21446446C211yesmaximal, characteristic
9#21446437C211yesmaximal, characteristic
10#21446400C211yesmaximal, characteristic, complemented
11#21446389C211yesmaximal, characteristic, complemented
12#21446380C211yesmaximal, characteristic, complemented
13#21446081C211yesmaximal, characteristic
14#21445643C211yesmaximal, characteristic
15C61×#1989C211yesmaximal, characteristic, complemented
16C2×#14519768C2211yescharacteristic
17C2×#14519739C2211yescharacteristic
18C2×#14519711C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519699C2211yescharacteristic
23C2×#14519699C2211yescharacteristic
24C2×#14519697C2211yescharacteristic
25C2×#14519697C2211yescharacteristic
26C2×#14519687C2211yescharacteristic
27#16839697—21no—
28#16839684—21no—
29#16839664C2211yescharacteristic
30#16839663C2211yescharacteristic
31#16839661C2211yescharacteristic
32#16839661C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839659—21no—
35#16839658C2211yescharacteristic
36#16839646—21no—
37#16839645—21no—
38#16839633—21no—
39#16839619—21no—
40C4⋊C4×Dic61C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×#11758196C2211yescharacteristic
43C4×C4⋊Dic61—21no—
44#16839525C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839516C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#125×C61C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C4×C4⋊C4×C61C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×D244C2311yescharacteristic
60C22×C4×D61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519768—22no—
64#14519768—22no—
65#14519739—22no—
66#14519739—41no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196D411yescharacteristic
77C2×#11758196D411yescharacteristic
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519711—22no—
88#14519710—41no—
89#14519708—22no—
90#14519708—22no—
91#14519699—22no—
92#14519699—41no—
93#14519697—22no—
94#14519697Q814yes—
95#14519687—41no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684D411yescharacteristic
103#14519684D411yescharacteristic
104C22×C8×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106#125×C61—22no—
107#125×C61—41no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C42×C61—21no—
115C2×C42×C61C2311yescharacteristic
116C4⋊C8×C61—41no—
117#95×C61—21no—
118C23×D61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61C2×D411yescharacteristic
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244C2×Q812yes—
139C2×C4×D61—42no—
140C2×C4×D61—42no—
141C2×C4×D61—81no—
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—42no—
154C4⋊C4×C61—42no—
155C4⋊C4×C61C2×Q812yes—
156C4⋊C4×C61#4112yes—
157C4⋊C4×C61#4112yes—
158C42×C61—21no—
159C42×C61—21no—
160C42×C61—42no—
161#11758198—42no—
162#11758198—42no—
163#11758198—81no—
164#11758198—81no—
165#11758198—81no—
166C4⋊Dic61—24no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—42no—
173C4⋊Dic61—81no—
174C4⋊Dic61#4112yes—
175#11758196—24no—
176#11758196—41no—
177#11758196—41no—
178#11758196—81no—
179#11758196—81no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—21no—
183C4×Dic61—21no—
184C4×Dic61—81no—
185C4×Dic61—81no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418C2×Q812yes—
192C23×C61C22×D411yescharacteristic
193C22×D61—22no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—22no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61C22×Q811yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61H(5,𝔽2)11yescharacteristic
223C2×C4×C612−1+411yescharacteristic
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—22no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245D244—22no—
246D244—22no—
247D244—41no—
248D244—41no—
249C4×D61—84no—
250C8×C61—41no—
251C8×C61—41no—
252#92418—22no—
253#92418—41no—
254#27027—611nomaximal, Sylow
255C22×C61#52511yescharacteristic
256C22×C61#54611yescharacteristic
257C22×C61D4×Q811yescharacteristic
258C22×C61#57511yescharacteristic
259C22×C61#57511yescharacteristic
260C22×C61#57611yescharacteristic
261C22×C61#57711yescharacteristic
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—81no—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C4×C61—42no—
270C4×C61—42no—
271C4×C61—41no—
272C4×C61—42no—
273C4×C61—41no—
274C4×C61—81no—
275C4×C61#53311yescharacteristic
276C4×C61#54211yescharacteristic
277C4×C61#55411yescharacteristic
278C4×C61#55611yescharacteristic
279Dic61—41no—
280Dic61—41no—
281Dic61—42no—
282Dic61—41no—
283Dic61—41no—
284Dic61—81no—
285Dic61—81no—
286#2454—611no—
287#2370—611no—
288#2067—611no—
289#2059—611no—
290#2059—611no—
291#2025—611no—
292#1989D6111yescharacteristic, complemented
293#1944—611no—
294#1926—611no—
295#1925—611no—
296#1919—611no—
297#1918—611no—
298#1917—611no—
299#1864—611no—
300#1762—611no—
301C2×C61#246511yescharacteristic
302C2×C61#311111yescharacteristic
303C2×C61#311911yescharacteristic
304C2×C61#330911yescharacteristic
305C2×C61#332311yescharacteristic
306C2×C61#338811yescharacteristic
307C2×C61#338911yescharacteristic
308D61—82no—
309C2×#125—611no—
310C2×#125C2×D6111yescharacteristic
311C2×C4×D4—611no—
312C2×#107—611no—
313C2×#107—611no—
314C2×#106—611no—
315C2×#106—611no—
316C2×C4⋊C8C2×D6111yescharacteristic
317C2×#102—611no—
318C2×#102—611no—
319C2×#102—611no—
320C2×#102—611no—
321C2×#97—611no—
322#399—1221no—
323#396—1221no—
324#390—611no—
325#390—611no—
326#389—611no—
327#389—611no—
328#389—611no—
329#389—611no—
330#389—1221no—
331#387—1221no—
332#387—1221no—
333#385—1221no—
334#380—1221no—
335C4×C4⋊C4—611no—
336C4×C4⋊C4—611no—
337C4×C4⋊C4—611no—
338C4×C4⋊C4—1221no—
339C4×C4⋊C4C2×D6111yescharacteristic
340#340—611no—
341#340—611no—
342#340—611no—
343#340—611no—
344#340C2×D6111yescharacteristic
345#340C2×D6111yescharacteristic
346#340C2×D6111yescharacteristic
347#340C2×D6111yescharacteristic
348#336—611no—
349#336—611no—
350#336—611no—
351#336—611no—
352C61#2702711yescharacteristic, complemented, Sylow
353C22×D4—611no—
354C23×C4—1221no—
355C22×C8—611no—
356C22×C8—611no—
357C22×C8C22×D6111yescharacteristic
358C22×C8C22×D6111yescharacteristic
359#125—22no—
360#125—41no—
361#125—1222no—
362#125—1222no—
363C4×D4—1222no—
364C4×D4—2441no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C42—21no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—1221no—
391C2×C42—1221no—
392C2×C42C22×D6111yescharacteristic
393#107—1222no—
394#107—2441no—
395#106—1222no—
396#106—1222no—
397C4⋊C8—41no—
398#102—614no—
399#102—1222no—
400#102—1222no—
401#102—2441no—
402#97—2441no—
403#95—21no—
404#95—611no—
405#95—611no—
406#95—1221no—
407#95—1221no—
408#95—1221no—
409#95—1221no—
410#95—1221no—
411#95—1221no—
412C24—1221no—
413C2×D4—612no—
414C2×D4—1221no—
415C2×D4—1221no—
416C2×D4—1221no—
417C2×D4—1221no—
418C2×D4—1221no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—4881no—
431C22×C4—2442no—
432C22×C4—2442no—
433C22×C4D4×D6111yescharacteristic
434C22×C4D4×D6111yescharacteristic
435C22×C4C23×D6111yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—612no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—612no—
456C4⋊C4—1224no—
457C4⋊C4—1224no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—4881no—
464C4⋊C4—4881no—
465C4⋊C4—4881no—
466C4⋊C4—2441no—
467C4⋊C4—2441no—
468C4⋊C4—2442no—
469C4⋊C4#1175822212yes—
470C4⋊C4Q8×D6112yes—
471C4⋊C4#1175822612yes—
472C22⋊C4—4881no—
473C22⋊C4—4881no—
474C22⋊C4—4881no—
475C22⋊C4—2442no—
476C22⋊C4—2442no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42—1221no—
481C42—1221no—
482C42—1221no—
483C42—1221no—
484C42—4881no—
485C42—4881no—
486C23—1222no—
487C23—2441no—
488C23—2441no—
489C23—2441no—
490C23—2441no—
491C23—2441no—
492C23—2441no—
493C23C2×D4×D6111yescharacteristic
494D4—1222no—
495D4—1222no—
496D4—2441no—
497D4—2441no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1222no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—4884no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4C2×Q8×D6111yescharacteristic
539C2×C4#41×D6111yescharacteristic
540C2×C4C2×#1175822211yescharacteristic
541C2×C4C2×#1175822611yescharacteristic
542C2×C4#1451982211yescharacteristic
543C2×C4#1451982411yescharacteristic
544C8—41no—
545C8—41no—
546C8—1222no—
547C8—2441no—
548C22—4881no—
549C22—4881no—
550C22—4881no—
551C22—4881no—
552C22—2442no—
553C22—2442no—
554C22—2442no—
555C22#1684027511yescharacteristic
556C22#1684040211yescharacteristic
557C22#1684042511yescharacteristic
558C22#1684055811yescharacteristic
559C22#1684055911yescharacteristic
560C22#1684056411yescharacteristic
561C22#1684057111yescharacteristic
562C4—42no—
563C4—42no—
564C4—41no—
565C4—42no—
566C4—41no—
567C4—81no—
568C4—4881no—
569C4—4881no—
570C4—2441no—
571C4—2441no—
572C4—2441no—
573C4—2442no—
574C4—2441no—
575C4#1684031611yescharacteristic
576C4#1684037611yescharacteristic
577C4#1684047311yescharacteristic
578C4#1684048511yescharacteristic
579C2—4882no—
580C2#2144984711yescharacteristic
581C2#2145593311yescharacteristic
582C2#2145602511yescharacteristic
583C2#2145782111yescharacteristic
584C2#2145803611yescharacteristic
585C2#2145876211yescharacteristic
586C2#2145877611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1989C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1989D61
hypocenterC61#27027
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
2. #21448917C2semidirect
4. #21447160C2semidirect
5. #21447154C2semidirect
7. #21446561C2semidirect
10. #21446400C2semidirect
11. #21446389C2semidirect
12. #21446380C2semidirect
15. C61×#1989C2semidirect
292. #1989D61semidirect
352. C61#27027semidirect

Maximal subgroups

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

subgroupindex
#214496872
#214489172
#214472062
#214471602
#214471542
#214469642
#214465612
#214464462
#214464372
#214464002
#214463892
#214463802
#214460812
#214456432
C61×#19892
#2702761