Finite Group Databasebeta

← #29647045

Subgroups — #29647045

Order 15616.

936 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459437C211yesmaximal, characteristic
2#21454940C211yesmaximal, characteristic, complemented
3#21454464C211yesmaximal, characteristic, complemented
4#21444290C212yesmaximal, complemented
5#21444274C211yesmaximal, characteristic, complemented
6#21444259C211yesmaximal, characteristic, complemented
7#21444252C212yesmaximal, complemented
8#21444220C212yesmaximal, complemented
9#21444205C212yesmaximal, complemented
10#21444199C211yesmaximal, characteristic, complemented
11C61×#1485C211yesmaximal, characteristic, complemented
12#16840487C411yescharacteristic
13#16840486C411yescharacteristic
14#16840483C412yes—
15#16840446—21no—
16#16840444—22no—
17#16840427C411yescharacteristic
18#16840422—21no—
19#16840415C411yescharacteristic
20#16840414C412yes—
21#16840316C2212yes—
22#16840299—21no—
23#16840290—22no—
24C4×#11758226C2212yes—
25C4×#11758222C2211yescharacteristic
26C4×#11758222C2211yescharacteristic
27C4×Q8×D61—21no—
28#16839892—21no—
29#16839836—22no—
30C8×D244—21no—
31#16839824—22no—
32#16839818C2212yes—
33#16839813C2212yes—
34#16839813C2212yes—
35#16839790—21no—
36#16839786C2211yescharacteristic
37C4⋊C8×D61C2212yes—
38#16839730—21no—
39C2×C4×#92418C2211yescharacteristic
40#16839494C2212yes—
41Q8⋊#92418C2212yes—
42Q8⋊#92418C2212yes—
43#16839492C2211yescharacteristic
44#16839492C2211yescharacteristic
45#16839491C2212yes—
46#16839489C2211yescharacteristic
47#16839489C2211yescharacteristic
48C61×#533C2211yescharacteristic
49C61×#431C2211yescharacteristic
50C4×M16×C61C2211yescharacteristic
51C61×Q8⋊C8C2212yes—
52C61×Q8⋊C8C2212yes—
53C2×#11758226C2×C412yes—
54C2×#11758222C2×C411yescharacteristic
55C2×#11758222D411yescharacteristic
56C2×Q8×D61—21no—
57C2×#11758194—21no—
58C22×#92418—21no—
59C2×C8×D61—22no—
60#14519776C2×C412yes—
61#14519775C2×C412yes—
62#14519772—22no—
63#14519772—22no—
64#14519770—22no—
65#14519768—21no—
66#14519767—22no—
67#14519767—22no—
68#14519766—21no—
69#14519763—22no—
70#14519762—22no—
71#14519762—22no—
72#14519759—22no—
73#14519757C2×C412yes—
74#14519755—22no—
75#14519755—22no—
76#14519753—21no—
77#14519753C2×C412yes—
78#14519752—21no—
79#14519750—22no—
80Q8×Dic61—21no—
81Q8×Dic61C2×C412yes—
82C4×Dic122—22no—
83C4×Dic122C2×C411yescharacteristic
84C4×Dic122C2×C411yescharacteristic
85C4×Dic122D411yescharacteristic
86C4×Dic122C2311yescharacteristic
87#14519742—21no—
88#14519742—22no—
89C4×D244—22no—
90C4×D244C2×C411yescharacteristic
91C4×D244C2×C411yescharacteristic
92C4×D244D411yescharacteristic
93C4×D244C2311yescharacteristic
94#14519739—22no—
95#14519739C2×C412yes—
96C4×C22⋊D61—21no—
97C4×C22⋊D61—21no—
98C4×C22⋊D61C2×C412yes—
99C4×C22⋊D61D412yes—
100#14519736C2×C411yescharacteristic
101#14519736D411yescharacteristic
102#14519735—22no—
103#14519735C2×C412yes—
104#14519734—22no—
105#14519734—21no—
106#14519734D412yes—
107C4⋊C4×D61—21no—
108C4⋊C4×D61—22no—
109C42×D61—21no—
110C42×D61—22no—
111C42×D61C2312yes—
112C4⋊#92418—21no—
113C4⋊#92418C2312yes—
114#14519706—22no—
115#14519706—22no—
116#14519706—22no—
117#14519689—22no—
118#14519689—22no—
119#14519689—22no—
120#14519687—22no—
121C4×#92418—21no—
122C4×#92418C2311yescharacteristic
123C4×#92418C2311yescharacteristic
124C2×M16×C61—21no—
125C4⋊Q8×C61C2×C411yescharacteristic
126C4⋊Q8×C61C2×C411yescharacteristic
127#125×C61—21no—
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61C2×C412yes—
130C4×Q8×C61—21no—
131C4×Q8×C61C2312yes—
132#117×C61—21no—
133C2×C42×C61C2311yescharacteristic
134C4⋊C8×C61C2311yescharacteristic
135C4⋊C8×C61C2311yescharacteristic
136C22⋊C8×C61—21no—
137#97×C61—21no—
138C4×C8×C61—21no—
139C4×C8×C61C2312yes—
140C2×Q8×C61—21no—
141C2×Q8×C61C22×C412yes—
142C22×C4×C61C22×C411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C2×C22⋊D61—21no—
146C2×C22⋊D61C22⋊C412yes—
147#11758226—24no—
148#11758226—42no—
149Q8×D61—44no—
150#11758222—22no—
151#11758222—41no—
152#11758222—42no—
153C2×D244—22no—
154C2×D244C22⋊C411yescharacteristic
155C2×D244C22×C411yescharacteristic
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61C22×C412yes—
162C2×C4×D61C2×D412yes—
163C2×C4×D61C2×D412yes—
164C2×Dic122—22no—
165C2×Dic122C22⋊C411yescharacteristic
166C2×Dic122C22×C411yescharacteristic
167M16×C61—42no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61C22×C412yes—
177C22⋊C4×C61—22no—
178C42×C61—22no—
179C42×C61C22×C411yescharacteristic
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2411yescharacteristic
183C22⋊Dic61—21no—
184C22⋊Dic61C22⋊C412yes—
185#11758198—22no—
186#11758198—22no—
187#11758198—22no—
188#11758198—21no—
189#11758198—21no—
190#11758198—22no—
191#11758198C22⋊C412yes—
192#11758198C22⋊C412yes—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61C22⋊C411yescharacteristic
196C4⋊Dic61C22⋊C411yescharacteristic
197C4⋊Dic61C22×C411yescharacteristic
198C4⋊Dic61C22×C411yescharacteristic
199#11758196—21no—
200#11758196—21no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196C22⋊C412yes—
206#11758196C22⋊C412yes—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61C22×C412yes—
213C4×Dic61C2×D412yes—
214C4×Dic61C2×D412yes—
215#11758194—81no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—22no—
219C2×#92418—42no—
220C8×D61—82no—
221C23×C61C2×C22⋊C411yescharacteristic
222C22×D61—21no—
223C22×D61C2×C22⋊C412yes—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—42no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C22⋊D61—42no—
241C22⋊D61—42no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61C2×C22⋊C412yes—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C2×C22⋊C412yes—
248D244—21no—
249D244—22no—
250D244—44no—
251D244#13112yes—
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—42no—
257C4×D61—44no—
258C4×D61—44no—
259C4×D61—42no—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61—42no—
263Dic122—22no—
264Dic122—21no—
265Dic122—21no—
266Dic122—44no—
267C8×C61—42no—
268C8×C61—41no—
269C8×C61—41no—
270#92418—41no—
271#92418—41no—
272#92418—42no—
273#17523—611nomaximal, Sylow
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—21no—
277C22×C61C22×C22⋊C411yescharacteristic
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—22no—
281C2×D61—22no—
282C2×D61—41no—
283C2×D61—41no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—42no—
288C4×C61—42no—
289C4×C61#42112yes—
290C4×C61C2×#13111yescharacteristic
291C4×C61#56811yescharacteristic
292C4×C61#57711yescharacteristic
293C4×C61#57811yescharacteristic
294Dic61—22no—
295Dic61—22no—
296Dic61—22no—
297Dic61—41no—
298Dic61—41no—
299Dic61—41no—
300#3482—611no—
301#2990—611no—
302#2922—611no—
303#1497—612no—
304#1493—611no—
305#1489—611no—
306#1488—612no—
307#1485D6111yescharacteristic, complemented
308#1481—612no—
309#1477—612no—
310#1476—611no—
311C2×C61—41no—
312C2×C61#288011yescharacteristic
313C2×C61#289811yescharacteristic
314C2×C61C2×#42111yescharacteristic
315D61—42no—
316D61—42no—
317#557—611no—
318#556—611no—
319#556—612no—
320#552—1221no—
321#552—1222no—
322D4×Q8—611no—
323#548—1221no—
324#547—611no—
325#547—612no—
326#533—612no—
327#533C2×D6111yescharacteristic
328#529—1222no—
329#529—1221no—
330C4×#41—611no—
331C4×#41—612no—
332C4×#41—611no—
333C2×C4×Q8—1221no—
334#446—1221no—
335#435—1222no—
336C8×D4—1221no—
337#433—1222no—
338#432—612no—
339#431—612no—
340#431—612no—
341#431C2×D6111yescharacteristic
342#424—1221no—
343#423—611no—
344C2×C4⋊C8—612no—
345#408—1221no—
346C4×M16C2×D6111yescharacteristic
347C2×C4×C8—611no—
348Q8⋊C8—612no—
349Q8⋊C8—612no—
350Q8⋊C8—611no—
351Q8⋊C8—611no—
352Q8⋊C8—612no—
353Q8⋊C8C2×D6112yes—
354Q8⋊C8C2×D6112yes—
355D4⋊C8—611no—
356D4⋊C8—611no—
357D4⋊C8—612no—
358C61#1752311yescharacteristic, complemented, Sylow
359C2×#41—611no—
360C2×#41—612no—
361C2×#41—611no—
362C22×Q8—1221no—
363C2×M16—21no—
364C2×M16—1221no—
365C22×C8—1222no—
366C22×C8—1221no—
367C4⋊Q8—612no—
368C4⋊Q8—612no—
369C4⋊Q8C4×D6111yescharacteristic
370C4⋊Q8C4×D6111yescharacteristic
371#126—1222no—
372#126—1222no—
373#126—1222no—
374#125—21no—
375#125—1222no—
376#125—1222no—
377#125—1221no—
378#124—1222no—
379#124—1221no—
380#123—1222no—
381#123—1222no—
382#123—1222no—
383C22⋊Q8—21no—
384C22⋊Q8—612no—
385C22⋊Q8—612no—
386C22⋊Q8—1222no—
387C22⋊Q8—1222no—
388C22⋊Q8—1221no—
389C22⋊Q8C4×D6112yes—
390#121—1222no—
391#121—1221no—
392C4×Q8—21no—
393C4×Q8—611no—
394C4×Q8—611no—
395C4×Q8—612no—
396C4×Q8—611no—
397C4×Q8—611no—
398C4×Q8—1222no—
399C4×Q8—1222no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×Q8C22×D6112yes—
403C4×D4—612no—
404C4×D4—612no—
405C4×D4—612no—
406C4×D4—611no—
407C4×D4—611no—
408C4×D4—611no—
409C4×D4—611no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—1221no—
413C4×D4—1221no—
414#117—21no—
415#117—611no—
416#117—612no—
417#117—611no—
418#117—612no—
419#117—1222no—
420#117—1222no—
421#117—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1222no—
424C2×C42—612no—
425C2×C42—1222no—
426C2×C42—1221no—
427C2×C42C22×D6111yescharacteristic
428C4⋊C8—612no—
429C4⋊C8—1222no—
430C4⋊C8—1222no—
431C4⋊C8—1222no—
432C4⋊C8—1221no—
433C4⋊C8C22×D6111yescharacteristic
434C4⋊C8C22×D6111yescharacteristic
435C22⋊C8—21no—
436C22⋊C8—1222no—
437C22⋊C8—1222no—
438C22⋊C8—1222no—
439#97—21no—
440#97—1222no—
441C4×C8—21no—
442C4×C8—611no—
443C4×C8—611no—
444C4×C8—1221no—
445C4×C8C22×D6112yes—
446#41—1222no—
447#41—1224no—
448#41—2441no—
449#41—2442no—
450#41—2442no—
451C2×Q8—21no—
452C2×Q8—611no—
453C2×Q8—611no—
454C2×Q8—1222no—
455C2×Q8—2444no—
456C2×Q8C2×C4×D6112yes—
457C2×D4—612no—
458C2×D4—611no—
459C2×D4—611no—
460C2×D4—1222no—
461C2×D4—1221no—
462C22×C4—612no—
463C22×C4—612no—
464C22×C4—612no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4C2×C4×D6111yescharacteristic
471C22×C4D4×D6111yescharacteristic
472C22×C4D4×D6111yescharacteristic
473M16—42no—
474M16—4881no—
475C2×C8—22no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1222no—
481C2×C8—4882no—
482C2×C8—2442no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—612no—
489C4⋊C4—611no—
490C4⋊C4—612no—
491C4⋊C4—611no—
492C4⋊C4—611no—
493C4⋊C4—611no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4C2×C4×D6112yes—
503C22⋊C4—22no—
504C22⋊C4—612no—
505C22⋊C4—612no—
506C22⋊C4—612no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1221no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C42—22no—
515C42—612no—
516C42—612no—
517C42—612no—
518C42—1222no—
519C42—1222no—
520C42—1221no—
521C42—1221no—
522C42—1221no—
523C42C2×C4×D6111yescharacteristic
524C42D4×D6111yescharacteristic
525C42D4×D6111yescharacteristic
526C42C23×D6111yescharacteristic
527C23—612no—
528C23—1221no—
529C23C22⋊C4×D6111yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—42no—
533Q8—1222no—
534Q8—1221no—
535Q8—1221no—
536Q8—2444no—
537D4—612no—
538D4—1221no—
539D4—1222no—
540D4—2444no—
541D4—2442no—
542D4—2442no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—612no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2444no—
563C2×C4—2444no—
564C2×C4—2442no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4C22⋊C4×D6111yescharacteristic
568C2×C4C22⋊C4×D6111yescharacteristic
569C2×C4C22⋊C4×D6111yescharacteristic
570C2×C4C22×C4×D6111yescharacteristic
571C2×C4C2×D4×D6111yescharacteristic
572C2×C4C2×D4×D6111yescharacteristic
573C8—42no—
574C8—41no—
575C8—41no—
576C8—2441no—
577C8—2441no—
578C8—2442no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—1222no—
583C22—1222no—
584C22—1222no—
585C22—1222no—
586C22—2441no—
587C22—2441no—
588C22C2×C22⋊C4×D6111yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—42no—
593C4—42no—
594C4—1222no—
595C4—1222no—
596C4—1222no—
597C4—2441no—
598C4—2441no—
599C4—2441no—
600C4#1683977712yes—
601C4#131×D6111yescharacteristic
602C4#1684052711yescharacteristic
603C4#1684057111yescharacteristic
604C4#1684057811yescharacteristic
605C2—41no—
606C2—2442no—
607C2—2442no—
608C2#2145426711yescharacteristic
609C2#2145435111yescharacteristic
610C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1485C2
socleC22×C61C22×C22⋊C4
prime radical#21459437C2
cosocleC42×C61C24
hypercenter#1485D61
hypocenterC61#17523
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1485

Complements

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

NH ≅ G/Ntype
2. #21454940C2semidirect
3. #21454464C2semidirect
4. #21444290C2semidirect
5. #21444274C2semidirect
6. #21444259C2semidirect
7. #21444252C2semidirect
8. #21444220C2semidirect
9. #21444205C2semidirect
10. #21444199C2semidirect
11. C61×#1485C2semidirect
307. #1485D61semidirect
358. C61#17523semidirect

Maximal subgroups

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

subgroupindex
#214594372
#214549402
#214544642
#214442902
#214442742
#214442592
#214442522
#214442202
#214442052
#214441992
C61×#14852
#1752361