Finite Group Databasebeta

← #29647233

Subgroups — #29647233

Order 15616.

974 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#21459907C211yesmaximal, characteristic
2#21454906C211yesmaximal, characteristic, complemented
3#21448183C211yesmaximal, characteristic, complemented
4#21444343C212yesmaximal
5#21444342C211yesmaximal, characteristic
6#21444331C211yesmaximal, characteristic, complemented
7#21444329C212yesmaximal, complemented
8#21444316C212yesmaximal, complemented
9#21444306C211yesmaximal, characteristic
10#21444304C212yesmaximal
11C61×#1503C211yesmaximal, characteristic, complemented
12#16840488C412yescomplemented
13#16840487C411yescharacteristic, complemented
14#16840487C411yescharacteristic, complemented
15#16840486C411yescharacteristic, complemented
16#16840484C412yescomplemented
17#16840454—21no—
18#16840445—22no—
19#16840427C411yescharacteristic, complemented
20#16840425—21no—
21#16840371—22no—
22#16840370—21no—
23#16840314C2212yes—
24#16840298C2212yes—
25#16840296C2211yescharacteristic
26#16840238—21no—
27C2×C4×Dic122C2211yescharacteristic
28C2×#14519742—21no—
29#16839892—21no—
30#16839891—22no—
31C8×Dic122—21no—
32#16839819C2211yescharacteristic
33#16839816C2212yes—
34#16839812C2212yes—
35C4⋊C4×Dic61—21no—
36C4×#11758196—22no—
37#16839613C2212yes—
38#16839608C2211yescharacteristic
39#16839607C2212yes—
40#16839500C2211yescharacteristic
41#16839500C2211yescharacteristic
42#16839499C2212yes—
43#16839499C2212yes—
44#16839498C2212yes—
45#16839497C2211yescharacteristic
46#16839497C2211yescharacteristic
47#16839495C2212yes—
48C61×#533C2211yescharacteristic
49C61×#431C2211yescharacteristic
50C4×C4⋊C4×C61C2211yescharacteristic
51C61×#328C2212yes—
52C61×#328C2212yes—
53C22×Dic122D411yescharacteristic
54C2×Q8×D61—21no—
55#14519776—22no—
56#14519776C2×C412yes—
57#14519775—21no—
58#14519775C2×C412yes—
59#14519772—22no—
60#14519772—22no—
61#14519772—22no—
62#14519771—21no—
63#14519771—22no—
64#14519769—22no—
65#14519769C2×C411yescharacteristic
66#14519768—21no—
67#14519768—22no—
68#14519768C2×C412yes—
69#14519767—22no—
70#14519767—22no—
71#14519767—22no—
72#14519766C2×C412yes—
73#14519765C2×C411yescharacteristic
74#14519761—22no—
75#14519760—21no—
76#14519757—21no—
77#14519756C2311yescharacteristic
78#14519755—22no—
79#14519755—22no—
80#14519754—22no—
81#14519753—21no—
82#14519751C2311yescharacteristic
83Q8×Dic61—21no—
84Q8×Dic61C2×C412yes—
85Q8×Dic61C2×C412yes—
86C4×Dic122—22no—
87C4×Dic122C2×C411yescharacteristic
88C4×Dic122C2×C412yes—
89C4×Dic122C2×C411yescharacteristic
90C4×Dic122C2×C411yescharacteristic
91C4×Dic122D411yescharacteristic
92C4×Dic122D411yescharacteristic
93C4×Dic122D412yes—
94#14519742—28no—
95#14519742—21no—
96#14519742—22no—
97#14519742C2×C412yes—
98C4×D244C2×C411yescharacteristic
99#14519739C2×C412yes—
100C4×C22⋊D61—21no—
101#14519737—22no—
102#14519735—21no—
103#14519735—22no—
104#14519734—21no—
105#14519734—22no—
106#14519733—22no—
107C2×C4⋊Dic61D411yescharacteristic
108C2×#11758196—22no—
109C2×#11758196D412yes—
110C2×C4×Dic61—21no—
111C2×C4×Dic61—22no—
112C2×C4×Dic61—22no—
113C2×C4×Dic61C2312yes—
114C4⋊#92418—21no—
115#14519706—22no—
116#14519706—22no—
117C22⋊#92418C2312yes—
118#14519688—21no—
119#14519687—22no—
120C8×Dic61—22no—
121#14519684—21no—
122#14519684—22no—
123#14519684C2311yescharacteristic
124#14519684C2311yescharacteristic
125C4⋊Q8×C61C2×C411yescharacteristic
126C4⋊Q8×C61C2×C411yescharacteristic
127#125×C61—21no—
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61C2312yes—
130C4×Q8×C61—21no—
131C4×Q8×C61C2×C412yes—
132#117×C61—21no—
133C2×C4⋊C4×C61—21no—
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C22⋊C8×C61C2311yescharacteristic
138C22⋊C8×C61C2311yescharacteristic
139C4×C8×C61—21no—
140#95×C61C2312yes—
141C2×Q8×C61—21no—
142C2×Q8×C61C22×C412yes—
143C22×C4×C61—22no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C2×C22⋊D61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61C2×D412yes—
151C22×Dic61C2×D412yes—
152Q8×D61—44no—
153C2×D244C22×C411yescharacteristic
154C2×C4×D61—21no—
155C2×C4×D61—22no—
156C2×Dic122—24no—
157C2×Dic122—22no—
158C2×Dic122C22⋊C412yes—
159C2×Dic122C22⋊C411yescharacteristic
160C2×Dic122C22⋊C411yescharacteristic
161C2×Dic122C22×C411yescharacteristic
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C4⋊C4×C61—24no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61C22×C412yes—
171C22⋊C4×C61—22no—
172C42×C61—22no—
173C42×C61—81no—
174C42×C61C22×C411yescharacteristic
175C42×C61C22×C411yescharacteristic
176C42×C61C2×D411yescharacteristic
177C42×C61C2×D411yescharacteristic
178C22⋊Dic61—21no—
179#11758198—21no—
180#11758198—21no—
181#11758198—21no—
182#11758198—21no—
183#11758198—22no—
184#11758198—22no—
185#11758198—22no—
186C4⋊Dic61—22no—
187C4⋊Dic61—22no—
188C4⋊Dic61—22no—
189C4⋊Dic61C22⋊C411yescharacteristic
190C4⋊Dic61C22⋊C411yescharacteristic
191C4⋊Dic61C22⋊C412yes—
192C4⋊Dic61C22×C411yescharacteristic
193C4⋊Dic61C22×C411yescharacteristic
194#11758196—28no—
195#11758196—21no—
196#11758196—21no—
197#11758196—21no—
198#11758196—21no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196C22⋊C412yes—
205#11758196C22⋊C412yes—
206#11758196C22⋊C412yes—
207#11758196C22⋊C412yes—
208C4×Dic61—28no—
209C4×Dic61—24no—
210C4×Dic61—21no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214C4×Dic61—82no—
215C4×Dic61C22×C412yes—
216C4×Dic61C22×C412yes—
217C4×Dic61C2×D412yes—
218C4×Dic61C2×D412yes—
219C2×#92418—22no—
220C23×C61C22×D411yescharacteristic
221C22×D61—21no—
222Q8×C61—42no—
223Q8×C61—42no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—24no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—41no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C22⋊D61—41no—
240C22⋊D61—41no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—24no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61C2×C22⋊C412yes—
254C2×Dic61C2×C22⋊C412yes—
255C2×Dic61C2×C22⋊C412yes—
256C2×Dic61C2×C22⋊C412yes—
257D244—41no—
258C4×D61—42no—
259C4×D61—44no—
260Dic122—22no—
261Dic122—24no—
262Dic122—44no—
263Dic122—41no—
264Dic122#10414yes—
265C8×C61—41no—
266C8×C61—41no—
267#92418—42no—
268#17549—611nomaximal, Sylow
269C22×C61—21no—
270C22×C61—21no—
271C22×C61C22×C22⋊C411yescharacteristic
272C22×C61#57711yescharacteristic
273C22×C61#57811yescharacteristic
274C2×D61—41no—
275C2×D61—41no—
276C2×D61—41no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—44no—
283C4×C61—42no—
284C4×C61—42no—
285C4×C61C2×#10412yes—
286Dic61—22no—
287Dic61—22no—
288Dic61—22no—
289Dic61—42no—
290Dic61—42no—
291Dic61—41no—
292Dic61—41no—
293Dic61—41no—
294Dic61C2×#10414yes—
295#3523—611no—
296#2984—611no—
297#2276—611no—
298#1509—611no—
299#1509—612no—
300#1506—612no—
301#1506—611no—
302#1503—612no—
303#1503D6111yescharacteristic, complemented
304#1500—611no—
305#1500—612no—
306C2×C61—21no—
307C2×C61—21no—
308C2×C61#288411yescharacteristic
309C2×C61#289811yescharacteristic
310C2×C61C22×#10411yescharacteristic
311D61—81no—
312#557—611no—
313#557—611no—
314#557—612no—
315#556—611no—
316#556—612no—
317#552—1222no—
318#552—1221no—
319D4×Q8—611no—
320D4×Q8—1221no—
321#541—1221no—
322#541—1222no—
323#533—612no—
324#533C2×D6111yescharacteristic
325#529—612no—
326#529—611no—
327#519—1221no—
328C2×C4×Q8—611no—
329C2×C4×Q8—1221no—
330#446—1221no—
331#446—1222no—
332C8×Q8—1221no—
333#432—611no—
334#432—612no—
335#431—612no—
336#431C2×D6111yescharacteristic
337C4×C4⋊C4—612no—
338C4×C4⋊C4—1222no—
339C4×C4⋊C4—1221no—
340C4×C4⋊C4C2×D6111yescharacteristic
341#376—611no—
342#376—612no—
343#328—612no—
344#328—612no—
345#328—611no—
346#328—611no—
347#328—612no—
348#328C2×D6112yes—
349#328C2×D6112yes—
350#327—611no—
351#327—611no—
352#327—612no—
353C61#1754911yescharacteristic, complemented, Sylow
354C22×Q8—611no—
355C22×Q8—1221no—
356C4⋊Q8—612no—
357C4⋊Q8—612no—
358C4⋊Q8—1222no—
359C4⋊Q8—1221no—
360C4⋊Q8C4×D6111yescharacteristic
361C4⋊Q8C4×D6111yescharacteristic
362#126—1222no—
363#126—1222no—
364#126—1222no—
365#126—1222no—
366#126—1221no—
367#125—21no—
368#125—612no—
369#125—611no—
370#125—1222no—
371#125—1222no—
372#125—1222no—
373#125—1222no—
374#125—1222no—
375#125—1221no—
376#124—612no—
377#124—611no—
378#123—1222no—
379#123—1221no—
380C22⋊Q8—21no—
381C22⋊Q8—611no—
382C22⋊Q8—1222no—
383C22⋊Q8—1222no—
384C22⋊Q8—1222no—
385C22⋊Q8—1221no—
386C22⋊Q8—1221no—
387C22⋊Q8C22×D6112yes—
388#121—611no—
389C4×Q8—21no—
390C4×Q8—611no—
391C4×Q8—612no—
392C4×Q8—611no—
393C4×Q8—611no—
394C4×Q8—612no—
395C4×Q8—611no—
396C4×Q8—612no—
397C4×Q8—612no—
398C4×Q8—612no—
399C4×Q8—611no—
400C4×Q8—1222no—
401C4×Q8—1222no—
402C4×Q8—1221no—
403C4×Q8—1228no—
404C4×Q8—1221no—
405C4×Q8C4×D6112yes—
406C4×D4—612no—
407C4×D4—611no—
408C4×D4—1222no—
409C4×D4—1221no—
410#117—21no—
411#117—1222no—
412#117—1222no—
413#117—1222no—
414#117—1221no—
415#117—1221no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—612no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—1222no—
420C2×C42—21no—
421C2×C42—612no—
422C2×C42—1222no—
423C2×C42—1221no—
424C2×C42—1222no—
425C2×C42C22×D6111yescharacteristic
426C4⋊C8—21no—
427C4⋊C8—1222no—
428C4⋊C8—1222no—
429C4⋊C8—1221no—
430C22⋊C8—612no—
431C22⋊C8C22×D6111yescharacteristic
432C22⋊C8C22×D6111yescharacteristic
433#97—1222no—
434#97—1221no—
435C4×C8—21no—
436C4×C8—1222no—
437#95—611no—
438#95—611no—
439#95—1222no—
440#95—1221no—
441#95C22×D6112yes—
442C2×Q8—21no—
443C2×Q8—611no—
444C2×Q8—612no—
445C2×Q8—611no—
446C2×Q8—611no—
447C2×Q8—1222no—
448C2×Q8—1224no—
449C2×Q8—2444no—
450C2×Q8C2×C4×D6112yes—
451C2×D4—611no—
452C2×D4—1221no—
453C22×C4—22no—
454C22×C4—612no—
455C22×C4—612no—
456C22×C4—1222no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4D4×D6111yescharacteristic
461C22×C4D4×D6111yescharacteristic
462C22×C4C23×D6111yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—1222no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—612no—
473C4⋊C4—612no—
474C4⋊C4—612no—
475C4⋊C4—612no—
476C4⋊C4—611no—
477C4⋊C4—611no—
478C4⋊C4—612no—
479C4⋊C4—611no—
480C4⋊C4—611no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1228no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4C2×C4×D6112yes—
495C22⋊C4—22no—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C22⋊C4—1221no—
500C22⋊C4—1221no—
501C22⋊C4—1221no—
502C22⋊C4—1221no—
503C22⋊C4—1221no—
504C42—22no—
505C42—81no—
506C42—612no—
507C42—612no—
508C42—612no—
509C42—612no—
510C42—1222no—
511C42—1222no—
512C42—1222no—
513C42—1224no—
514C42—1228no—
515C42—1221no—
516C42—4882no—
517C42C2×C4×D6111yescharacteristic
518C42C2×C4×D6111yescharacteristic
519C42D4×D6111yescharacteristic
520C42D4×D6111yescharacteristic
521C23—1221no—
522C23C2×D4×D6111yescharacteristic
523Q8—42no—
524Q8—42no—
525Q8—614no—
526Q8—1224no—
527Q8—1222no—
528Q8—2441no—
529Q8—2444no—
530D4—2441no—
531D4—2441no—
532D4—2441no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—1224no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—2444no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2442no—
560C2×C4C22⋊C4×D6111yescharacteristic
561C2×C4C22⋊C4×D6111yescharacteristic
562C2×C4C22⋊C4×D6111yescharacteristic
563C2×C4C22⋊C4×D6111yescharacteristic
564C2×C4C22×C4×D6111yescharacteristic
565C2×C4C2×D4×D6111yescharacteristic
566C8—41no—
567C8—41no—
568C8—2442no—
569C22—21no—
570C22—21no—
571C22—2441no—
572C22—2441no—
573C22—2441no—
574C22C2×C22⋊C4×D6111yescharacteristic
575C22#1684056811yescharacteristic
576C22#1684057511yescharacteristic
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—44no—
583C4—42no—
584C4—42no—
585C4—614no—
586C4—1222no—
587C4—1222no—
588C4—1222no—
589C4—2442no—
590C4—2442no—
591C4—2441no—
592C4—2441no—
593C4—2441no—
594C4#104×D6112yes—
595C2—21no—
596C2—21no—
597C2—4881no—
598C2#2145428111yescharacteristic
599C2#2145434711yescharacteristic
600C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1503C2
socleC22×C61C22×C22⋊C4
prime radical#14519751C23
cosocleC22×C4×C61C24
hypercenter#1503D61
hypocenterC61#17549
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
2. #21454906C2semidirect
3. #21448183C2semidirect
6. #21444331C2semidirect
7. #21444329C2semidirect
8. #21444316C2semidirect
11. C61×#1503C2semidirect
12. #16840488C4semidirect
13. #16840487C4semidirect
14. #16840487C4semidirect
15. #16840486C4semidirect
16. #16840484C4semidirect
19. #16840427C4semidirect
303. #1503D61semidirect
353. C61#17549semidirect

Maximal subgroups

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

subgroupindex
#214599072
#214549062
#214481832
#214443432
#214443422
#214443312
#214443292
#214443162
#214443062
#214443042
C61×#15032
#1754961