Finite Group Databasebeta

← #29725153

Subgroups — #29725153

Order 15616.

750 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450739C211yesmaximal, characteristic
2#21448731C211yesmaximal, characteristic, complemented
3#21447380C211yesmaximal, characteristic, complemented
4#21447376C211yesmaximal, characteristic, complemented
5#21447225C211yesmaximal, characteristic
6#21446753C211yesmaximal, characteristic, complemented
7#21446576C211yesmaximal, characteristic, complemented
8#21446441C211yesmaximal, characteristic
9#21446436C211yesmaximal, characteristic
10#21446393C211yesmaximal, characteristic, complemented
11#21446391C211yesmaximal, characteristic, complemented
12#21445945C211yesmaximal, characteristic, complemented
13#21445817C211yesmaximal, characteristic
14#102×Dic61C211yesmaximal, characteristic
15C61×#2071C211yesmaximal, characteristic
16C2×#14519754—21no—
17C2×C4×Dic122C2211yescharacteristic
18C2×#14519742—21no—
19C2×D4×Dic61C2211yescharacteristic
20C2×#14519711C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×C8×Dic61C2211yescharacteristic
26#16839685—21no—
27#16839685—21no—
28#16839665C2211yescharacteristic
29#16839661C2211yescharacteristic
30#16839660C2211yescharacteristic
31#16839658—21no—
32#16839657—21no—
33#16839657—21no—
34#16839631C2211yescharacteristic
35#16839631C2211yescharacteristic
36#16839631C2211yescharacteristic
37#16839620—21no—
38C4⋊C4×Dic61C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4×C4⋊Dic61C2211yescharacteristic
41#16839525C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#102×C61C2211yescharacteristic
53C2×#102×C61C2211yescharacteristic
54C61×#390C2211yescharacteristic
55C61×#384C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122D411yescharacteristic
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519754—44no—
64C4×Dic122—22no—
65C4×Dic122—22no—
66#14519742—44no—
67D4×Dic61—22no—
68D4×Dic61—22no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196D411yescharacteristic
83C2×#11758196D411yescharacteristic
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92#14519711—22no—
93#14519710—22no—
94#14519706—22no—
95#14519698—22no—
96#14519698—22no—
97C8×Dic61—22no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103C22×D4×C61C2311yescharacteristic
104C22×C8×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C42×C61C2311yescharacteristic
114#102×C61—22no—
115#102×C61—22no—
116C24×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61#4112yes—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—42no—
134C22×Dic61—42no—
135C22×Dic61—42no—
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122D814yes—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—42no—
156C4⋊C4×C61—81no—
157C22⋊C4×C61—42no—
158C22⋊C4×C61—81no—
159C42×C61—21no—
160C42×C61—21no—
161C22⋊Dic61—42no—
162C22⋊Dic61—42no—
163C22⋊Dic61—81no—
164C4⋊Dic61—21no—
165C4⋊Dic61—22no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—42no—
173C4⋊Dic61#4112yes—
174#11758196—22no—
175#11758196—22no—
176#11758196—42no—
177#11758196—42no—
178#11758196—42no—
179#11758196—42no—
180#11758196—81no—
181#11758196—81no—
182#11758196—81no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—42no—
186C4×Dic61—42no—
187C4×Dic61—42no—
188C4×Dic61#4112yes—
189C4×Dic61#4112yes—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C23×C61—22no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61C22×D411yescharacteristic
204D4×C61—22no—
205D4×C61—22no—
206D4×C61—41no—
207D4×C61—41no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
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—41no—
222C2×C4×C61C2×D812yes—
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C612−1+411yescharacteristic
228C2×C4×C612−1+411yescharacteristic
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—22no—
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—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—84no—
249C2×Dic61#12312yes—
250C2×Dic61#12312yes—
251C2×Dic61C2×D812yes—
252C2×Dic61C2×D812yes—
253C2×Dic61#13712yes—
254Dic122—28no—
255C8×C61—41no—
256C8×C61—41no—
257#92418—41no—
258#92418—41no—
259#24990—611nomaximal, Sylow
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61C2×#12311yescharacteristic
268C22×C61#54811yescharacteristic
269C22×C61#54811yescharacteristic
270C22×C61C22×D811yescharacteristic
271C22×C61C2×#13711yescharacteristic
272C22×C61#57711yescharacteristic
273C22×C61#57711yescharacteristic
274C4×C61—22no—
275C4×C61—41no—
276C4×C61—81no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61#52911yescharacteristic
280C4×C61#54111yescharacteristic
281C4×C61#55211yescharacteristic
282C4×C61#55611yescharacteristic
283Dic61—22no—
284Dic61—22no—
285Dic61—41no—
286Dic61—42no—
287Dic61—81no—
288Dic61—81no—
289Dic61—81no—
290Dic61#48014yes—
291#2538—611no—
292Q8×C4⋊C4—611no—
293#2099—611no—
294#2099—611no—
295#2071—611no—
296#2071D6111yescharacteristic
297#1984—611no—
298#1947—611no—
299#1925—611no—
300#1925—611no—
301#1918—611no—
302#1918—611no—
303#1834—611no—
304#1802—611no—
305C4×#102—611no—
306C2×C61—82no—
307C2×C61#242111yescharacteristic
308C2×C61C2×#48011yescharacteristic
309C2×C61#309911yescharacteristic
310C2×C61#335811yescharacteristic
311C2×C61#335811yescharacteristic
312C2×C61#336911yescharacteristic
313C2×C61#337211yescharacteristic
314C2×C22⋊Q8—1221no—
315C2×C4×Q8—611no—
316C2×C4×Q8—1221no—
317C2×C4×D4—611no—
318C2×#107—611no—
319C2×#107—611no—
320C2×C4⋊C8—611no—
321C2×#102—611no—
322C2×#102—611no—
323C2×#102C2×D6111yescharacteristic
324C2×#102C2×D6111yescharacteristic
325C2×C4×C8—611no—
326#396—1221no—
327#396—1221no—
328#390—611no—
329#390C2×D6111yescharacteristic
330#389—611no—
331#389—611no—
332#389—1221no—
333#389—1221no—
334#389—1221no—
335#384—611no—
336#384—611no—
337#384—611no—
338#384C2×D6111yescharacteristic
339#380—1221no—
340C4×C4⋊C4—611no—
341C4×C4⋊C4—611no—
342C4×C4⋊C4—611no—
343#340—611no—
344#340—611no—
345#340—611no—
346#340—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340C2×D6111yescharacteristic
352#340C2×D6111yescharacteristic
353#336—611no—
354#336—611no—
355#336—611no—
356#336C2×D6111yescharacteristic
357C61#2499011yescharacteristic, complemented, Sylow
358C22×Q8—611no—
359C22×D4C22×D6111yescharacteristic
360C23×C4—1221no—
361C22×C8—611no—
362C22×C8—611no—
363C22×C8C22×D6111yescharacteristic
364C22×C8C22×D6111yescharacteristic
365C22⋊Q8—2444no—
366C4×Q8—1222no—
367C4×Q8—1222no—
368C4×Q8—2444no—
369C4×D4—1222no—
370C4×D4—1222no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6111yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—1221no—
394C2×C22⋊C4—1221no—
395C2×C22⋊C4—1221no—
396C2×C42—611no—
397C2×C42—611no—
398C2×C42—1221no—
399C2×C42—1221no—
400C2×C42—1221no—
401C2×C42C22×D6111yescharacteristic
402#107—1222no—
403#107—1222no—
404C4⋊C8—1222no—
405#102—22no—
406#102—22no—
407#102—1222no—
408#102—1222no—
409C4×C8—1222no—
410#95—1221no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415C24—21no—
416C2×Q8—614no—
417C2×Q8—1222no—
418C2×Q8—1222no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4#1175822412yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—2442no—
437C22×C4—2442no—
438C22×C4—2442no—
439C22×C4D4×D6111yescharacteristic
440C22×C4D4×D6111yescharacteristic
441C22×C4C23×D6111yescharacteristic
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—81no—
462C4⋊C4—612no—
463C4⋊C4—1222no—
464C4⋊C4—1222no—
465C4⋊C4—1222no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—4881no—
474C4⋊C4—4881no—
475C4⋊C4—4881no—
476C4⋊C4—2442no—
477C4⋊C4—2442no—
478C4⋊C4—2442no—
479C4⋊C4—2442no—
480C4⋊C4—2442no—
481C22⋊C4—42no—
482C22⋊C4—81no—
483C22⋊C4—4881no—
484C22⋊C4—2442no—
485C22⋊C4—2442no—
486C42—21no—
487C42—21no—
488C42—612no—
489C42—612no—
490C42—1221no—
491C42—1221no—
492C42—2442no—
493C42—2442no—
494C42—2442no—
495C23—22no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23C2×D4×D6111yescharacteristic
503Q8—1228no—
504D4—22no—
505D4—22no—
506D4—41no—
507D4—41no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—612no—
523C2×C4—612no—
524C2×C4—612no—
525C2×C4—612no—
526C2×C4—612no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1222no—
535C2×C4—4884no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4D8×D6112yes—
548C2×C4#41×D6111yescharacteristic
549C2×C4#41×D6111yescharacteristic
550C2×C4#41×D6111yescharacteristic
551C2×C4C2×#1175822411yescharacteristic
552C2×C4#1451982411yescharacteristic
553C2×C4#1451982411yescharacteristic
554C8—41no—
555C8—41no—
556C8—2441no—
557C8—2441no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22#123×D6111yescharacteristic
566C22#1684041811yescharacteristic
567C22#1684042011yescharacteristic
568C22C2×D8×D6111yescharacteristic
569C22#137×D6111yescharacteristic
570C22#1684056811yescharacteristic
571C22#1684057211yescharacteristic
572C4—22no—
573C4—41no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—614no—
578C4—1222no—
579C4—1222no—
580C4—4881no—
581C4—4881no—
582C4—4881no—
583C4—2442no—
584C4—2441no—
585C4#1684028911yescharacteristic
586C4#1684036911yescharacteristic
587C4#1684044911yescharacteristic
588C4#1684047911yescharacteristic
589C2—82no—
590C2#2144931511yescharacteristic
591C2#480×D6111yescharacteristic
592C2#2145580911yescharacteristic
593C2#2145844911yescharacteristic
594C2#2145845111yescharacteristic
595C2#2145855611yescharacteristic
596C2#2145859511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2071C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2071D61
hypocenterC61#24990
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
2. #21448731C2semidirect
3. #21447380C2semidirect
4. #21447376C2semidirect
6. #21446753C2semidirect
7. #21446576C2semidirect
10. #21446393C2semidirect
11. #21446391C2semidirect
12. #21445945C2semidirect
357. C61#24990semidirect

Maximal subgroups

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

subgroupindex
#214507392
#214487312
#214473802
#214473762
#214472252
#214467532
#214465762
#214464412
#214464362
#214463932
#214463912
#214459452
#214458172
#102×Dic612
C61×#20712
#2499061