Finite Group Databasebeta

← #29647595

Subgroups — #29647595

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#21459902C211yesmaximal, characteristic
2#21454906C211yesmaximal, characteristic, complemented
3#21448471C211yesmaximal, characteristic, complemented
4#21444395C212yesmaximal, complemented
5#21444377C212yesmaximal, complemented
6#21444364C212yesmaximal, complemented
7#21444356C212yesmaximal, complemented
8#21444344C211yesmaximal, characteristic, complemented
9#21444326C211yesmaximal, characteristic, complemented
10#21444315C211yesmaximal, characteristic, complemented
11C61×#1500C211yesmaximal, characteristic, complemented
12#16840497C411yescharacteristic, complemented
13#16840494C412yescomplemented
14Q8×Dic122C411yescharacteristic, complemented
15#16840469C411yescharacteristic, complemented
16#16840468C412yescomplemented
17#16840433C411yescharacteristic, complemented
18#16840421—21no—
19#16840416—21no—
20#16840410—22no—
21#16840383—21no—
22#16840378—22no—
23#16840312C2211yescharacteristic
24#16840307C2212yes—
25#16840298C2212yes—
26#16840242—21no—
27C2×C4×Dic122C2211yescharacteristic
28#16839892—21no—
29#16839891—22no—
30C8×Dic122—21no—
31#16839819C2211yescharacteristic
32#16839816C2212yes—
33#16839812C2212yes—
34#16839668—21no—
35#16839660—21no—
36#16839644—22no—
37#16839629C2212yes—
38#16839626C2212yes—
39C4×C4⋊Dic61C2211yescharacteristic
40#16839499C2212yes—
41#16839497C2212yes—
42#16839497C2212yes—
43#16839496C2212yes—
44#16839495C2212yes—
45#16839495C2212yes—
46C61×#529C2211yescharacteristic
47C61×#431C2211yescharacteristic
48C61×#376C2211yescharacteristic
49C61×#328C2211yescharacteristic
50C61×#328C2211yescharacteristic
51C61×#327C2211yescharacteristic
52C61×#327C2211yescharacteristic
53C2×#11758222—21no—
54C2×#11758222—21no—
55C2×#11758224—22no—
56C22×Dic122D411yescharacteristic
57#14519777—21no—
58#14519777C2×C411yescharacteristic
59#14519776—22no—
60#14519776C2×C412yes—
61#14519774C2×C411yescharacteristic
62#14519773C2×C412yes—
63#14519772—22no—
64#14519768C2×C412yes—
65#14519766C2×C412yes—
66#14519765—21no—
67#14519765—21no—
68#14519763—22no—
69#14519763—22no—
70#14519762—22no—
71#14519761—22no—
72#14519761—22no—
73#14519759—22no—
74#14519757—22no—
75#14519756—21no—
76#14519755—22no—
77#14519755—22no—
78#14519754—22no—
79#14519751—21no—
80#14519751—21no—
81#14519751C2312yes—
82#14519750—22no—
83#14519749—22no—
84Q8×Dic61C2×C412yes—
85C4×Dic122—21no—
86C4×Dic122C2×C412yes—
87C4×Dic122C2×C411yescharacteristic
88C4×Dic122C2×C411yescharacteristic
89C4×Dic122D411yescharacteristic
90C4×Dic122D412yes—
91C4×Dic122D411yescharacteristic
92C4×D244—21no—
93C4×D244—21no—
94C4×D244C2×C412yes—
95D4×Dic61C2×C412yes—
96#14519739C2×C412yes—
97#14519739C2×C412yes—
98#14519737—22no—
99#14519737—22no—
100#14519737—22no—
101#14519735—22no—
102C2×C4⋊Dic61—21no—
103C2×C4⋊Dic61—21no—
104C2×C4⋊Dic61D411yescharacteristic
105C2×#11758196—22no—
106C2×#11758196D412yes—
107C4⋊C4×D61—22no—
108C2×C4×Dic61—21no—
109C2×C4×Dic61C2312yes—
110C4⋊#92418—21no—
111#14519706—22no—
112#14519706—22no—
113C22⋊#92418C2312yes—
114#14519688—21no—
115#14519687—22no—
116C8×Dic61—22no—
117#14519684—22no—
118#14519684—22no—
119#14519684—22no—
120#14519684C2312yes—
121#126×C61—21no—
122#125×C61C2×C411yescharacteristic
123#124×C61C2×C411yescharacteristic
124#123×C61—21no—
125C22⋊Q8×C61C2311yescharacteristic
126#121×C61C2311yescharacteristic
127C4×Q8×C61C2×C411yescharacteristic
128C4×D4×C61—21no—
129C4×D4×C61C2×C411yescharacteristic
130#117×C61—21no—
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133C4⋊C8×C61—21no—
134C22⋊C8×C61C2311yescharacteristic
135C22⋊C8×C61C2311yescharacteristic
136C4×C8×C61—21no—
137#95×C61—21no—
138#95×C61C2311yescharacteristic
139#95×C61C2311yescharacteristic
140C2×Q8×C61C22×C411yescharacteristic
141C2×D4×C61—21no—
142C2×D4×C61C22×C411yescharacteristic
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C2×C22⋊D61—22no—
150C2×C22⋊D61—22no—
151C2×C22⋊D61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61C2×D412yes—
155C22×Dic61C2×D412yes—
156#11758224—82no—
157#11758224—82no—
158#11758222—44no—
159#11758222—81no—
160#11758222—81no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244C22×C412yes—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×Dic122—21no—
168C2×Dic122—22no—
169C2×Dic122—42no—
170C2×Dic122C22⋊C411yescharacteristic
171C2×Dic122C22⋊C411yescharacteristic
172C2×Dic122C22⋊C412yes—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C22×C411yescharacteristic
180C4⋊C4×C61C22×C411yescharacteristic
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C42×C61—21no—
187C42×C61—42no—
188C42×C61C22×C411yescharacteristic
189C42×C61C22×C411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C22⋊Dic61—22no—
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199C4⋊Dic61—21no—
200C4⋊Dic61—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61—42no—
203C4⋊Dic61C22⋊C411yescharacteristic
204C4⋊Dic61C22⋊C411yescharacteristic
205C4⋊Dic61C22⋊C412yes—
206C4⋊Dic61C22×C412yes—
207#11758196—22no—
208#11758196—22no—
209#11758196—82no—
210#11758196C22⋊C412yes—
211#11758196C22⋊C412yes—
212#11758196C22⋊C412yes—
213#11758196C22⋊C412yes—
214C4×Dic61—22no—
215C4×Dic61—81no—
216C4×Dic61C22×C412yes—
217C4×Dic61C22×C412yes—
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220C2×#92418—22no—
221C23×C61—21no—
222C23×C61C22×D411yescharacteristic
223C22×D61—22no—
224Q8×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—42no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C22⋊D61—44no—
249C22⋊D61—42no—
250C22⋊D61—42no—
251C22⋊D61—82no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—82no—
260C2×Dic61C2×C22⋊C412yes—
261C2×Dic61C2×C22⋊C412yes—
262C2×Dic61C2×C22⋊C412yes—
263C2×Dic61C2×C22⋊C412yes—
264D244—42no—
265D244—42no—
266D244—81no—
267C4×D61—44no—
268C4×D61—82no—
269C4×D61—82no—
270Dic122—22no—
271Dic122—22no—
272Dic122—42no—
273Dic122—82no—
274Dic122—81no—
275C8×C61—41no—
276C8×C61—41no—
277#92418—42no—
278#17602—611nomaximal, Sylow
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—41no—
284C22×C61C22×C22⋊C411yescharacteristic
285C22×C61#57611yescharacteristic
286C22×C61#57711yescharacteristic
287C2×D61—42no—
288C2×D61—42no—
289C2×D61—42no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—81no—
299C4×C61—81no—
300C4×C61#42112yes—
301Dic61—22no—
302Dic61—22no—
303Dic61—22no—
304Dic61—22no—
305Dic61—42no—
306Dic61—82no—
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#1502—611no—
317#1500D6111yescharacteristic, complemented
318C2×C61—21no—
319C2×C61—21no—
320C2×C61—81no—
321C2×C61#288411yescharacteristic
322C2×C61#289711yescharacteristic
323C2×C61C2×#42111yescharacteristic
324D61—82no—
325#559—612no—
326#559—611no—
327Q82—611no—
328#554—612no—
329#554—611no—
330#550—611no—
331#548—1221no—
332#547—1221no—
333#547—1222no—
334#542—1221no—
335#542—1222no—
336#532—612no—
337#532—611no—
338#529—612no—
339#529C2×D6111yescharacteristic
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⋊C4—611no—
355#376C2×D6111yescharacteristic
356#328—612no—
357#328C2×D6111yescharacteristic
358#328C2×D6111yescharacteristic
359#327—612no—
360#327—612no—
361#327—612no—
362#327—612no—
363#327—612no—
364#327C2×D6111yescharacteristic
365#327C2×D6111yescharacteristic
366C61#1760211yescharacteristic, complemented, Sylow
367C2×#41—1221no—
368C2×#41—1222no—
369C2×#41—1221no—
370C22×Q8—611no—
371C4⋊Q8—612no—
372C4⋊Q8—611no—
373C4⋊Q8—1221no—
374C4⋊Q8—1222no—
375#127—612no—
376#127—611no—
377#126—21no—
378#126—1222no—
379#125—612no—
380#125C4×D6111yescharacteristic
381#124—612no—
382#124—1221no—
383#124—1222no—
384#124—1222no—
385#124—1221no—
386#124C4×D6111yescharacteristic
387#123—21no—
388#123—1222no—
389#123—1222no—
390#123—1222no—
391#123—1222no—
392C22⋊Q8—1222no—
393C22⋊Q8—1222no—
394C22⋊Q8—1222no—
395C22⋊Q8—1222no—
396C22⋊Q8—1221no—
397C22⋊Q8C22×D6111yescharacteristic
398#121—612no—
399#121—1222no—
400#121—1221no—
401#121—1222no—
402#121—1221no—
403#121C22×D6111yescharacteristic
404C4×Q8—611no—
405C4×Q8—611no—
406C4×Q8—612no—
407C4×Q8—611no—
408C4×Q8—611no—
409C4×Q8—612no—
410C4×Q8—612no—
411C4×Q8—1221no—
412C4×Q8C4×D6111yescharacteristic
413C4×D4—21no—
414C4×D4—612no—
415C4×D4—612no—
416C4×D4—612no—
417C4×D4—612no—
418C4×D4—1221no—
419C4×D4—1222no—
420C4×D4—1222no—
421C4×D4—1222no—
422C4×D4—1221no—
423C4×D4C4×D6111yescharacteristic
424#117—21no—
425#117—1222no—
426C2×C4⋊C4—612no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—1222no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1222no—
431C2×C4⋊C4—1221no—
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—21no—
448#95—612no—
449#95—1222no—
450#95—1222no—
451#95—1222no—
452#95C22×D6111yescharacteristic
453#95C22×D6111yescharacteristic
454#41—4881no—
455#41—4881no—
456#41—4882no—
457#41—4882no—
458#41—2444no—
459C2×Q8—611no—
460C2×Q8—611no—
461C2×Q8—612no—
462C2×Q8—1222no—
463C2×Q8—1221no—
464C2×Q8—2442no—
465C2×Q8C2×C4×D6111yescharacteristic
466C2×D4—21no—
467C2×D4—612no—
468C2×D4—1222no—
469C2×D4—1222no—
470C2×D4—1222no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4C2×C4×D6111yescharacteristic
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—612no—
478C22×C4—612no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4D4×D6111yescharacteristic
485C22×C4D4×D6111yescharacteristic
486C22×C4C23×D6111yescharacteristic
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—1222no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—612no—
495C4⋊C4—611no—
496C4⋊C4—612no—
497C4⋊C4—612no—
498C4⋊C4—612no—
499C4⋊C4—611no—
500C4⋊C4—612no—
501C4⋊C4—612no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—1221no—
507C4⋊C4—4882no—
508C4⋊C4—2442no—
509C4⋊C4C2×C4×D6111yescharacteristic
510C4⋊C4C2×C4×D6111yescharacteristic
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C42—21no—
524C42—42no—
525C42—612no—
526C42—612no—
527C42—612no—
528C42—612no—
529C42—1222no—
530C42—4881no—
531C42C2×C4×D6111yescharacteristic
532C42C2×C4×D6111yescharacteristic
533C42D4×D6111yescharacteristic
534C42D4×D6111yescharacteristic
535C23—21no—
536C23—1222no—
537C23C2×D4×D6111yescharacteristic
538Q8—41no—
539Q8—1222no—
540Q8—1222no—
541Q8—4881no—
542Q8—4882no—
543Q8—2442no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—4881no—
548D4—4882no—
549D4—2442no—
550D4—2444no—
551D4—2442no—
552D4—2442no—
553D4—2442no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—612no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—612no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—4882no—
576C2×C4—4882no—
577C2×C4—4882no—
578C2×C4—2444no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4—2442no—
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—41no—
595C22—41no—
596C22—41no—
597C22—2442no—
598C22—2442no—
599C22—2442no—
600C22C2×C22⋊C4×D6111yescharacteristic
601C22#1684056411yescharacteristic
602C22#1684056911yescharacteristic
603C4—21no—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—81no—
612C4—81no—
613C4—1222no—
614C4—1222no—
615C4—1222no—
616C4—1222no—
617C4—4882no—
618C4—2442no—
619C4#1683977712yes—
620C2—21no—
621C2—21no—
622C2—81no—
623C2—4882no—
624C2#2145428111yescharacteristic
625C2#2145434111yescharacteristic
626C2C2×#1683977711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1500C2
socleC22×C61C22×C22⋊C4
prime radical#21459902C2
cosocleC22×C4×C61C24
hypercenter#1500D61
hypocenterC61#17602
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1500

Complements

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

NH ≅ G/Ntype
2. #21454906C2semidirect
3. #21448471C2semidirect
4. #21444395C2semidirect
5. #21444377C2semidirect
6. #21444364C2semidirect
7. #21444356C2semidirect
8. #21444344C2semidirect
9. #21444326C2semidirect
10. #21444315C2semidirect
11. C61×#1500C2semidirect
12. #16840497C4semidirect
13. #16840494C4semidirect
14. Q8×Dic122C4semidirect
15. #16840469C4semidirect
16. #16840468C4semidirect
17. #16840433C4semidirect
317. #1500D61semidirect
366. C61#17602semidirect

Maximal subgroups

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

subgroupindex
#214599022
#214549062
#214484712
#214443952
#214443772
#214443642
#214443562
#214443442
#214443262
#214443152
C61×#15002
#1760261