Finite Group Databasebeta

← #29647048

Subgroups — #29647048

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#21459434C211yesmaximal, characteristic
2#21454945C211yesmaximal, characteristic, complemented
3#21454464C211yesmaximal, characteristic, complemented
4#21444290C212yesmaximal, complemented
5#21444273C211yesmaximal, characteristic, complemented
6#21444250C212yesmaximal, complemented
7#21444241C211yesmaximal, characteristic, complemented
8#21444220C212yesmaximal, complemented
9#21444203C212yesmaximal, complemented
10#21444201C211yesmaximal, characteristic, complemented
11C61×#1489C211yesmaximal, characteristic, complemented
12#16840490C411yescharacteristic
13#16840485C411yescharacteristic
14#16840480C412yes—
15#16840452—21no—
16#16840445—22no—
17D4×Dic122C411yescharacteristic
18#16840421—21no—
19#16840416C411yescharacteristic
20#16840411C412yes—
21#16840319C2211yescharacteristic
22#16840316C2212yes—
23#16840296—21no—
24#16840292—22no—
25C4×#11758222C2211yescharacteristic
26C4×#11758224C2212yes—
27C2×C4×Dic122—21no—
28#16839889—21no—
29#16839831—22no—
30#16839827—21no—
31#16839824—22no—
32#16839818C2212yes—
33#16839815C2211yescharacteristic
34#16839813C2212yes—
35#16839813C2212yes—
36#16839790—21no—
37C4⋊C8×D61C2212yes—
38#16839730—21no—
39C2×C4×#92418C2211yescharacteristic
40#16839494C2212yes—
41#16839494C2212yes—
42Q8⋊#92418C2211yescharacteristic
43Q8⋊#92418C2211yescharacteristic
44#16839492C2212yes—
45#16839492C2212yes—
46D4⋊#92418C2211yescharacteristic
47D4⋊#92418C2211yescharacteristic
48C4×#41×C61C2211yescharacteristic
49C61×#423C2211yescharacteristic
50C4×M16×C61C2211yescharacteristic
51C61×Q8⋊C8C2212yes—
52C61×D4⋊C8C2212yes—
53C2×#11758222D411yescharacteristic
54C2×#11758224C2×C412yes—
55C22×Dic122—21no—
56C22×#92418—21no—
57C2×C8×D61—22no—
58#14519777C2×C411yescharacteristic
59#14519777C2×C411yescharacteristic
60#14519776C2×C412yes—
61#14519775C2×C412yes—
62#14519772—22no—
63#14519771—22no—
64#14519770—22no—
65#14519769—21no—
66#14519769—21no—
67#14519767—22no—
68#14519767—22no—
69#14519765—21no—
70#14519763—22no—
71#14519761—22no—
72#14519760—22no—
73#14519758—22no—
74#14519757C2×C412yes—
75#14519756—21no—
76#14519756—21no—
77#14519756C2×C412yes—
78#14519755—22no—
79#14519754—22no—
80#14519753C2×C412yes—
81#14519751—21no—
82#14519749—22no—
83C4×Dic122—21no—
84C4×Dic122—21no—
85C4×Dic122—21no—
86C4×Dic122—22no—
87C4×Dic122—22no—
88C4×Dic122C2×C411yescharacteristic
89C4×Dic122D411yescharacteristic
90C4×Dic122C2312yes—
91#14519742C2×C412yes—
92C4×D244C2×C411yescharacteristic
93C4×D244D411yescharacteristic
94D4×Dic61C2×C412yes—
95C4×C22⋊D61—22no—
96C4×C22⋊D61C2×C412yes—
97C4×C22⋊D61D412yes—
98#14519737—22no—
99#14519736—21no—
100#14519736D411yescharacteristic
101#14519735C2×C412yes—
102#14519734—22no—
103#14519734D412yes—
104#14519733—22no—
105C2×C4⋊Dic61—21no—
106C2×#11758196—22no—
107C2×C4×Dic61—22no—
108C42×D61C2312yes—
109C4⋊#92418C2311yescharacteristic
110C4⋊#92418C2311yescharacteristic
111#14519706—22no—
112#14519706—22no—
113#14519706—22no—
114C22⋊#92418—21no—
115#14519689—22no—
116#14519689—22no—
117#14519689—22no—
118#14519687—22no—
119C4×#92418—21no—
120C4×#92418—21no—
121C4×#92418C2311yescharacteristic
122C4×#92418C2311yescharacteristic
123C2×#41×C61C2×C411yescharacteristic
124C2×M16×C61—21no—
125C2×M16×C61—21no—
126C4×Q8×C61C2×C411yescharacteristic
127C4×Q8×C61C2311yescharacteristic
128C4×D4×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61C2×C411yescharacteristic
131C4×D4×C61C2311yescharacteristic
132#117×C61—21no—
133#117×C61C2×C411yescharacteristic
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2312yes—
138#97×C61—21no—
139C4×C8×C61C2312yes—
140#41×C61—22no—
141#41×C61—41no—
142C2×Q8×C61C22×C411yescharacteristic
143C2×D4×C61—21no—
144C2×D4×C61C22×C411yescharacteristic
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61—22no—
152C2×C22⋊D61C22⋊C412yes—
153C22×Dic61—22no—
154C22×Dic61—22no—
155#11758224—24no—
156#11758224—42no—
157#11758222—42no—
158C2×D244C22⋊C411yescharacteristic
159C2×C4×D61C22×C412yes—
160C2×C4×D61C2×D412yes—
161C2×C4×D61C2×D412yes—
162C2×Dic122—21no—
163C2×Dic122—22no—
164C2×Dic122—44no—
165C2×Dic122C22⋊C411yescharacteristic
166C2×Dic122C22×C412yes—
167M16×C61—42no—
168M16×C61—81no—
169C2×C8×C61—22no—
170C2×C8×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C22×C411yescharacteristic
174C4⋊C4×C61C22×C411yescharacteristic
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61C22×C411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61C2411yescharacteristic
185C22⋊Dic61—22no—
186C22⋊Dic61—22no—
187C22⋊Dic61C22⋊C412yes—
188#11758198—22no—
189#11758198—22no—
190#11758198—22no—
191#11758198C22⋊C412yes—
192#11758198C22⋊C412yes—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—22no—
197C4⋊Dic61C22⋊C411yescharacteristic
198C4⋊Dic61C22⋊C411yescharacteristic
199C4⋊Dic61C22×C412yes—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196C22⋊C412yes—
208#11758196C22⋊C412yes—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61C22×C412yes—
213C4×Dic61C2×D412yes—
214C4×Dic61C2×D412yes—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—42no—
220C8×D61—82no—
221C23×C61—21no—
222C23×C61C2×C22⋊C411yescharacteristic
223C22×D61C2×C22⋊C412yes—
224Q8×C61—21no—
225Q8×C61—21no—
226D4×C61—21no—
227D4×C61—42no—
228D4×C61#13112yes—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—42no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—42no—
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C22⋊D61—42no—
248C22⋊D61—44no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—44no—
252C2×Dic61—44no—
253C2×Dic61C2×C22⋊C412yes—
254C2×Dic61C2×C22⋊C412yes—
255C2×Dic61C2×C22⋊C412yes—
256D244—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—42no—
262Dic122—22no—
263Dic122—22no—
264Dic122—22no—
265Dic122—44no—
266Dic122—42no—
267C8×C61—42no—
268C8×C61—42no—
269#92418—41no—
270#92418—41no—
271#92418—41no—
272#92418—41no—
273#17523—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—21no—
276C22×C61—21no—
277C22×C61—21no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61C22×C22⋊C411yescharacteristic
281C2×D61—22no—
282C2×D61—22no—
283C2×D61—22no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—21no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61—41no—
290C4×C61#42112yes—
291C4×C61C2×#13111yescharacteristic
292C4×C61#56811yescharacteristic
293C4×C61#57711yescharacteristic
294C4×C61#57811yescharacteristic
295Dic61—22no—
296Dic61—22no—
297Dic61—22no—
298Dic61—42no—
299Dic61—42no—
300#3482—611no—
301#2990—611no—
302#2922—611no—
303#1497—612no—
304#1493—611no—
305#1489D6111yescharacteristic, complemented
306#1488—612no—
307#1485—611no—
308#1481—612no—
309#1477—612no—
310#1476—611no—
311C2×C61—41no—
312C2×C61—42no—
313C2×C61#288011yescharacteristic
314C2×C61#289811yescharacteristic
315C2×C61C2×#42111yescharacteristic
316D61—42no—
317#557—611no—
318#556—611no—
319#556—612no—
320#552—1222no—
321#552—1221no—
322D4×Q8—611no—
323#548—1221no—
324#547—612no—
325#547—611no—
326#533—612no—
327#533—611no—
328#529—1221no—
329#529—1222no—
330C4×#41—611no—
331C4×#41—612no—
332C4×#41C2×D6111yescharacteristic
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#431—611no—
342#424—1221no—
343#423C2×D6111yescharacteristic
344C2×C4⋊C8—612no—
345#408—1221no—
346C4×M16C2×D6111yescharacteristic
347C2×C4×C8—611no—
348Q8⋊C8—611no—
349Q8⋊C8—611no—
350Q8⋊C8—612no—
351Q8⋊C8—612no—
352Q8⋊C8—612no—
353Q8⋊C8—612no—
354Q8⋊C8C2×D6112yes—
355D4⋊C8—611no—
356D4⋊C8—611no—
357D4⋊C8C2×D6112yes—
358C61#1752311yescharacteristic, complemented, Sylow
359C2×#41—611no—
360C2×#41—612no—
361C2×#41C4×D6111yescharacteristic
362C22×Q8—1221no—
363C2×M16—21no—
364C2×M16—21no—
365C22×C8—1222no—
366C22×C8—1221no—
367C4⋊Q8—612no—
368C4⋊Q8—612no—
369C4⋊Q8—611no—
370C4⋊Q8—611no—
371#126—1222no—
372#126—1222no—
373#126—1222no—
374#125—1221no—
375#125—1222no—
376#125—1222no—
377#125—1221no—
378#124—1221no—
379#124—1222no—
380#123—1222no—
381#123—1222no—
382#123—1222no—
383C22⋊Q8—612no—
384C22⋊Q8—612no—
385C22⋊Q8—612no—
386C22⋊Q8—1222no—
387C22⋊Q8—1222no—
388C22⋊Q8—1221no—
389C22⋊Q8—1221no—
390#121—1221no—
391#121—1222no—
392C4×Q8—611no—
393C4×Q8—611no—
394C4×Q8—612no—
395C4×Q8—612no—
396C4×Q8—1222no—
397C4×Q8—1222no—
398C4×Q8—1221no—
399C4×Q8—1221no—
400C4×Q8—1221no—
401C4×Q8C4×D6111yescharacteristic
402C4×Q8C22×D6111yescharacteristic
403C4×D4—21no—
404C4×D4—21no—
405C4×D4—612no—
406C4×D4—612no—
407C4×D4—612no—
408C4×D4—611no—
409C4×D4—611no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4C4×D6111yescharacteristic
413C4×D4C22×D6111yescharacteristic
414#117—21no—
415#117—611no—
416#117—612no—
417#117—612no—
418#117—1222no—
419#117—1222no—
420#117—1221no—
421#117C4×D6111yescharacteristic
422C2×C4⋊C4—1222no—
423C2×C4⋊C4—1221no—
424C2×C42—21no—
425C2×C42—612no—
426C2×C42—1222no—
427C2×C42C22×D6111yescharacteristic
428C4⋊C8—21no—
429C4⋊C8—611no—
430C4⋊C8—611no—
431C4⋊C8—1222no—
432C4⋊C8—1222no—
433C4⋊C8—1222no—
434C4⋊C8C22×D6112yes—
435C22⋊C8—1222no—
436C22⋊C8—1222no—
437C22⋊C8—1222no—
438C22⋊C8—1221no—
439#97—21no—
440#97—1222no—
441C4×C8—611no—
442C4×C8—611no—
443C4×C8—1221no—
444C4×C8—1221no—
445C4×C8C22×D6112yes—
446#41—22no—
447#41—41no—
448#41—1224no—
449#41—2442no—
450#41—2442no—
451C2×Q8—611no—
452C2×Q8—612no—
453C2×Q8—1222no—
454C2×Q8—1221no—
455C2×Q8—2444no—
456C2×Q8C2×C4×D6111yescharacteristic
457C2×D4—21no—
458C2×D4—612no—
459C2×D4—611no—
460C2×D4—1222no—
461C2×D4C2×C4×D6111yescharacteristic
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—612no—
466C22×C4—612no—
467C22×C4—612no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4C2×C4×D6111yescharacteristic
471C22×C4D4×D6111yescharacteristic
472C22×C4D4×D6111yescharacteristic
473M16—42no—
474M16—81no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—4882no—
482C2×C8—2442no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—612no—
486C4⋊C4—611no—
487C4⋊C4—612no—
488C4⋊C4—611no—
489C4⋊C4—612no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4C2×C4×D6111yescharacteristic
502C4⋊C4C2×C4×D6111yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—612no—
507C22⋊C4—612no—
508C22⋊C4—612no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—612no—
518C42—612no—
519C42—612no—
520C42—1222no—
521C42—1222no—
522C42—1222no—
523C42C2×C4×D6111yescharacteristic
524C42D4×D6111yescharacteristic
525C42D4×D6111yescharacteristic
526C42C23×D6111yescharacteristic
527C23—21no—
528C23—612no—
529C23C22⋊C4×D6111yescharacteristic
530Q8—21no—
531Q8—21no—
532Q8—1222no—
533Q8—1222no—
534Q8—1222no—
535Q8—2444no—
536Q8—2442no—
537D4—21no—
538D4—42no—
539D4—1222no—
540D4—2442no—
541D4—2444no—
542D4#1451978512yes—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—612no—
556C2×C4—612no—
557C2×C4—612no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—2442no—
565C2×C4—2444no—
566C2×C4—2444no—
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—42no—
575C8—2441no—
576C8—2441no—
577C8—2441no—
578C8—2441no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—1222no—
586C22—1222no—
587C22—1222no—
588C22C2×C22⋊C4×D6111yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—1222no—
596C4—1222no—
597C4—1222no—
598C4—2442no—
599C4—2442no—
600C4#1683977712yes—
601C4C2×#1451978511yescharacteristic
602C4#1684052811yescharacteristic
603C4#1684057211yescharacteristic
604C4#1684057811yescharacteristic
605C2—41no—
606C2—42no—
607C2—2442no—
608C2#2145426911yescharacteristic
609C2#2145435011yescharacteristic
610C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1489C2
socleC22×C61C22×C22⋊C4
prime radical#21459434C2
cosocleC42×C61C24
hypercenter#1489D61
hypocenterC61#17523
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1489

Complements

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

NH ≅ G/Ntype
2. #21454945C2semidirect
3. #21454464C2semidirect
4. #21444290C2semidirect
5. #21444273C2semidirect
6. #21444250C2semidirect
7. #21444241C2semidirect
8. #21444220C2semidirect
9. #21444203C2semidirect
10. #21444201C2semidirect
11. C61×#1489C2semidirect
305. #1489D61semidirect
358. C61#17523semidirect

Maximal subgroups

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

subgroupindex
#214594342
#214549452
#214544642
#214442902
#214442732
#214442502
#214442412
#214442202
#214442032
#214442012
C61×#14892
#1752361