Finite Group Databasebeta

← #29647604

Subgroups — #29647604

Order 15616.

938 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459896C211yesmaximal, characteristic
2#21454906C211yesmaximal, characteristic, complemented
3#21448465C211yesmaximal, characteristic, complemented
4#21444393C212yesmaximal, complemented
5#21444377C212yesmaximal, complemented
6#21444364C212yesmaximal, complemented
7#21444354C212yesmaximal, complemented
8#21444342C211yesmaximal, characteristic, complemented
9#21444327C211yesmaximal, characteristic, complemented
10#21444306C211yesmaximal, characteristic, complemented
11C61×#1502C211yesmaximal, characteristic, complemented
12#16840498C411yescharacteristic, complemented
13#16840495C412yescomplemented
14#16840492C411yescharacteristic, complemented
15#16840473C411yescharacteristic, complemented
16#16840471C412yescomplemented
17#16840432C411yescharacteristic, complemented
18#16840418—21no—
19#16840412—21no—
20#16840411—22no—
21#16840377—21no—
22#16840376—22no—
23#16840307C2212yes—
24#16840296C2211yescharacteristic
25#16840293C2212yes—
26#16840241—21no—
27C2×C4×Dic122C2211yescharacteristic
28#16839892—21no—
29#16839891—22no—
30C8×Dic122—21no—
31#16839819C2211yescharacteristic
32#16839816C2212yes—
33#16839812C2212yes—
34#16839667—21no—
35#16839658—21no—
36#16839646—22no—
37#16839629C2212yes—
38#16839626C2212yes—
39#16839608C2211yescharacteristic
40#16839500C2211yescharacteristic
41#16839500C2211yescharacteristic
42#16839498C2212yes—
43#16839497C2211yescharacteristic
44#16839497C2211yescharacteristic
45#16839496C2212yes—
46#16839496C2212yes—
47#16839495C2212yes—
48C61×#532C2211yescharacteristic
49C61×#431C2211yescharacteristic
50C4×C4⋊C4×C61C2211yescharacteristic
51C61×#327C2212yes—
52C61×#327C2212yes—
53C2×#11758224—22no—
54C2×#11758224—21no—
55C22×Dic122D411yescharacteristic
56#14519776—21no—
57#14519776C2×C412yes—
58#14519775—22no—
59#14519773C2×C412yes—
60#14519771—21no—
61#14519770—22no—
62#14519769C2×C411yescharacteristic
63#14519768C2×C412yes—
64#14519765C2×C411yescharacteristic
65#14519764—21no—
66#14519764C2×C412yes—
67#14519763—22no—
68#14519763—22no—
69#14519760—21no—
70#14519760—22no—
71#14519759—22no—
72#14519758—22no—
73#14519758—22no—
74#14519757—21no—
75#14519756—22no—
76#14519756C2311yescharacteristic
77#14519754—22no—
78#14519754—22no—
79#14519753—22no—
80#14519751C2311yescharacteristic
81#14519749—22no—
82#14519749—22no—
83#14519748—21no—
84C4×Dic122C2×C411yescharacteristic
85C4×Dic122C2×C412yes—
86C4×Dic122C2×C411yescharacteristic
87C4×Dic122C2×C411yescharacteristic
88C4×Dic122D411yescharacteristic
89C4×Dic122D412yes—
90C4×Dic122D411yescharacteristic
91#14519742—21no—
92#14519742C2×C412yes—
93C4×D244C2×C411yescharacteristic
94D4×Dic61—22no—
95D4×Dic61—21no—
96D4×Dic61C2×C412yes—
97D4×Dic61C2×C412yes—
98#14519739C2×C412yes—
99C4×C22⋊D61—21no—
100C4×C22⋊D61—22no—
101#14519737—22no—
102#14519734—21no—
103#14519733—22no—
104C2×C4⋊Dic61D411yescharacteristic
105C2×#11758196—22no—
106C2×#11758196—22no—
107C2×#11758196D412yes—
108C4⋊C4×D61—21no—
109C2×C4×Dic61—21no—
110C2×C4×Dic61C2312yes—
111C4⋊#92418—21no—
112#14519706—22no—
113#14519706—22no—
114C22⋊#92418C2312yes—
115#14519688—21no—
116#14519687—22no—
117C8×Dic61—22no—
118#14519684—21no—
119#14519684—22no—
120#14519684—22no—
121#14519684—22no—
122#14519684C2311yescharacteristic
123#14519684C2311yescharacteristic
124C2×#41×C61—21no—
125C4⋊Q8×C61C2×C411yescharacteristic
126#127×C61C2×C411yescharacteristic
127#124×C61—21no—
128#121×C61—21no—
129#121×C61C2312yes—
130C4×D4×C61—21no—
131C4×D4×C61C2×C412yes—
132C2×C4⋊C4×C61—21no—
133C2×C42×C61—21no—
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61—21no—
136C22⋊C8×C61C2311yescharacteristic
137C22⋊C8×C61C2311yescharacteristic
138C4×C8×C61—21no—
139#95×C61C2312yes—
140#41×C61—81no—
141#41×C61—81no—
142C2×Q8×C61—21no—
143C2×D4×C61—22no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61C22×C412yes—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C2×C22⋊D61—21no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61C2×D412yes—
160C22×Dic61C2×D412yes—
161#11758224—44no—
162#11758224—82no—
163#11758224—82no—
164C2×D244C22×C411yescharacteristic
165C2×C4×D61—21no—
166C2×C4×D61—22no—
167C2×Dic122—22no—
168C2×Dic122—42no—
169C2×Dic122C22⋊C411yescharacteristic
170C2×Dic122C22⋊C411yescharacteristic
171C2×Dic122C22⋊C412yes—
172C2×Dic122C22×C411yescharacteristic
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—42no—
177C4⋊C4×C61C22×C412yes—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C42×C61—81no—
181C42×C61C22×C411yescharacteristic
182C42×C61C22×C411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C22⋊Dic61—21no—
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190#11758198—21no—
191#11758198—21no—
192#11758198—21no—
193#11758198—21no—
194#11758198—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61C22⋊C411yescharacteristic
197C4⋊Dic61C22⋊C411yescharacteristic
198C4⋊Dic61C22⋊C412yes—
199C4⋊Dic61C22×C411yescharacteristic
200C4⋊Dic61C22×C411yescharacteristic
201#11758196—21no—
202#11758196—21no—
203#11758196—21no—
204#11758196—21no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—82no—
209#11758196C22⋊C412yes—
210#11758196C22⋊C412yes—
211#11758196C22⋊C412yes—
212#11758196C22⋊C412yes—
213C4×Dic61—21no—
214C4×Dic61—22no—
215C4×Dic61—42no—
216C4×Dic61C22×C412yes—
217C4×Dic61C22×C412yes—
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220C2×#92418—22no—
221C23×C61—22no—
222C23×C61C22×D411yescharacteristic
223C22×D61—21no—
224Q8×C61—81no—
225D4×C61—42no—
226D4×C61—42no—
227D4×C61—42no—
228D4×C61—42no—
229D4×C61—81no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—41no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—82no—
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C23×C411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C22⋊D61—44no—
246C22⋊D61—41no—
247C22⋊D61—41no—
248C22⋊D61—82no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—42no—
255C2×Dic61—44no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—82no—
263C2×Dic61C2×C22⋊C412yes—
264C2×Dic61C2×C22⋊C412yes—
265C2×Dic61C2×C22⋊C412yes—
266C2×Dic61C2×C22⋊C412yes—
267D244—41no—
268C4×D61—42no—
269C4×D61—82no—
270Dic122—22no—
271Dic122—22no—
272Dic122—42no—
273Dic122—41no—
274Dic122—82no—
275C8×C61—41no—
276C8×C61—41no—
277#92418—42no—
278#17602—611nomaximal, Sylow
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61C22×C22⋊C411yescharacteristic
285C22×C61#57611yescharacteristic
286C22×C61#57711yescharacteristic
287C2×D61—41no—
288C2×D61—41no—
289C2×D61—41no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—42no—
296C4×C61—82no—
297C4×C61#42112yes—
298Dic61—22no—
299Dic61—22no—
300Dic61—22no—
301Dic61—22no—
302Dic61—41no—
303Dic61—41no—
304Dic61—41no—
305Dic61—81no—
306Dic61—81no—
307#3522—611no—
308#2984—611no—
309#2320—611no—
310#1521—612no—
311#1517—612no—
312#1514—612no—
313#1512—612no—
314#1509—611no—
315#1505—611no—
316#1502D6111yescharacteristic, complemented
317#1500—611no—
318C2×C61—21no—
319C2×C61—21no—
320C2×C61—82no—
321C2×C61#288411yescharacteristic
322C2×C61#289711yescharacteristic
323C2×C61C2×#42111yescharacteristic
324D61—81no—
325#559—611no—
326#559—612no—
327Q82—611no—
328#554—611no—
329#554—612no—
330#550—611no—
331#548—1221no—
332#547—1221no—
333#547—1222no—
334#542—1221no—
335#542—1222no—
336#532—612no—
337#532C2×D6111yescharacteristic
338#529—612no—
339#529—611no—
340#520—1221no—
341C2×C4×Q8—611no—
342#446—1221no—
343#446—1222no—
344C8×Q8—1221no—
345#432—611no—
346#432—612no—
347#431—612no—
348#431C2×D6111yescharacteristic
349#391—1221no—
350#389—1221no—
351#387—1222no—
352#383—612no—
353#382—612no—
354C4×C4⋊C4C2×D6111yescharacteristic
355#376—611no—
356#328—611no—
357#328—611no—
358#328—612no—
359#327—612no—
360#327—612no—
361#327—611no—
362#327—611no—
363#327—612no—
364#327C2×D6112yes—
365#327C2×D6112yes—
366C61#1760211yescharacteristic, complemented, Sylow
367C2×#41—21no—
368C2×#41—1222no—
369C2×#41—1221no—
370C22×Q8—611no—
371C4⋊Q8—612no—
372C4⋊Q8—1221no—
373C4⋊Q8—1222no—
374C4⋊Q8C4×D6111yescharacteristic
375#127—612no—
376#127C4×D6111yescharacteristic
377#126—1222no—
378#126—1221no—
379#125—612no—
380#125—611no—
381#124—21no—
382#124—612no—
383#124—611no—
384#124—1222no—
385#124—1222no—
386#124—1221no—
387#123—1222no—
388#123—1222no—
389#123—1222no—
390#123—1222no—
391#123—1221no—
392C22⋊Q8—611no—
393C22⋊Q8—1222no—
394C22⋊Q8—1222no—
395C22⋊Q8—1221no—
396C22⋊Q8—1222no—
397C22⋊Q8—1222no—
398#121—21no—
399#121—611no—
400#121—1221no—
401#121—1222no—
402#121—1222no—
403#121C22×D6112yes—
404C4×Q8—611no—
405C4×Q8—611no—
406C4×Q8—612no—
407C4×Q8—611no—
408C4×Q8—611no—
409C4×Q8—612no—
410C4×Q8—612no—
411C4×Q8—611no—
412C4×Q8—1221no—
413C4×D4—21no—
414C4×D4—612no—
415C4×D4—612no—
416C4×D4—612no—
417C4×D4—611no—
418C4×D4—1222no—
419C4×D4—1222no—
420C4×D4—1221no—
421C4×D4—1222no—
422C4×D4—1221no—
423C4×D4C4×D6112yes—
424#117—1222no—
425#117—1221no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—612no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1222no—
431C2×C4⋊C4—1222no—
432C2×C42—21no—
433C2×C42—612no—
434C2×C42—1221no—
435C2×C42C22×D6111yescharacteristic
436C4⋊C8—21no—
437C4⋊C8—1222no—
438C4⋊C8—1222no—
439C4⋊C8—1221no—
440C22⋊C8—612no—
441C22⋊C8C22×D6111yescharacteristic
442C22⋊C8C22×D6111yescharacteristic
443#97—1222no—
444#97—1221no—
445C4×C8—21no—
446C4×C8—1222no—
447#95—611no—
448#95—611no—
449#95—1222no—
450#95—1222no—
451#95—1222no—
452#95—1221no—
453#95C22×D6112yes—
454#41—81no—
455#41—81no—
456#41—4882no—
457#41—4882no—
458#41—2444no—
459C2×Q8—21no—
460C2×Q8—611no—
461C2×Q8—611no—
462C2×Q8—612no—
463C2×Q8—611no—
464C2×Q8—1222no—
465C2×Q8—2442no—
466C2×D4—22no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—611no—
470C2×D4—1222no—
471C2×D4—1222no—
472C2×D4—1221no—
473C2×D4C2×C4×D6112yes—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—612no—
477C22×C4—612no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4D4×D6111yescharacteristic
485C22×C4D4×D6111yescharacteristic
486C22×C4C23×D6111yescharacteristic
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—1222no—
490C4⋊C4—21no—
491C4⋊C4—42no—
492C4⋊C4—612no—
493C4⋊C4—611no—
494C4⋊C4—612no—
495C4⋊C4—612no—
496C4⋊C4—612no—
497C4⋊C4—611no—
498C4⋊C4—612no—
499C4⋊C4—611no—
500C4⋊C4—611no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—4882no—
510C4⋊C4C2×C4×D6112yes—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1221no—
519C22⋊C4—1221no—
520C22⋊C4—1221no—
521C22⋊C4—1221no—
522C22⋊C4—1221no—
523C42—81no—
524C42—612no—
525C42—612no—
526C42—612no—
527C42—612no—
528C42—1222no—
529C42—1221no—
530C42—2442no—
531C42C2×C4×D6111yescharacteristic
532C42C2×C4×D6111yescharacteristic
533C42D4×D6111yescharacteristic
534C42D4×D6111yescharacteristic
535C23—22no—
536C23—1221no—
537C23C2×D4×D6111yescharacteristic
538Q8—81no—
539Q8—1222no—
540Q8—1222no—
541Q8—4882no—
542Q8—2442no—
543Q8—2441no—
544D4—42no—
545D4—42no—
546D4—42no—
547D4—42no—
548D4—81no—
549D4—4882no—
550D4—2441no—
551D4—2444no—
552D4—2441no—
553D4—2441no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—82no—
563C2×C4—612no—
564C2×C4—612no—
565C2×C4—612no—
566C2×C4—612no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—4882no—
573C2×C4—4882no—
574C2×C4—2442no—
575C2×C4—2444no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2442no—
583C2×C4C22⋊C4×D6111yescharacteristic
584C2×C4C22⋊C4×D6111yescharacteristic
585C2×C4C22⋊C4×D6111yescharacteristic
586C2×C4C22⋊C4×D6111yescharacteristic
587C2×C4C22×C4×D6111yescharacteristic
588C2×C4C2×D4×D6111yescharacteristic
589C8—41no—
590C8—41no—
591C8—2442no—
592C22—21no—
593C22—21no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—2441no—
598C22—2441no—
599C22—2441no—
600C22C2×C22⋊C4×D6111yescharacteristic
601C22#1684056311yescharacteristic
602C22#1684056811yescharacteristic
603C4—21no—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—42no—
609C4—82no—
610C4—1222no—
611C4—1222no—
612C4—1222no—
613C4—1222no—
614C4—4881no—
615C4—4881no—
616C4—2441no—
617C4—2441no—
618C4—2441no—
619C4#1683977712yes—
620C2—21no—
621C2—21no—
622C2—82no—
623C2—4881no—
624C2#2145428111yescharacteristic
625C2#2145434011yescharacteristic
626C2C2×#1683977711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1502C2
socleC22×C61C22×C22⋊C4
prime radical#21459896C2
cosocleC22×C4×C61C24
hypercenter#1502D61
hypocenterC61#17602
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #21454906C2semidirect
3. #21448465C2semidirect
4. #21444393C2semidirect
5. #21444377C2semidirect
6. #21444364C2semidirect
7. #21444354C2semidirect
8. #21444342C2semidirect
9. #21444327C2semidirect
10. #21444306C2semidirect
11. C61×#1502C2semidirect
12. #16840498C4semidirect
13. #16840495C4semidirect
14. #16840492C4semidirect
15. #16840473C4semidirect
16. #16840471C4semidirect
17. #16840432C4semidirect
316. #1502D61semidirect
366. C61#17602semidirect

Maximal subgroups

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

subgroupindex
#214598962
#214549062
#214484652
#214443932
#214443772
#214443642
#214443542
#214443422
#214443272
#214443062
C61×#15022
#1760261