Finite Group Databasebeta

← #29647580

Subgroups — #29647580

Order 15616.

982 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459868C211yesmaximal, characteristic
2#21448462C211yesmaximal, characteristic, complemented
3#21444392C212yesmaximal, complemented
4#21444376C212yesmaximal, complemented
5#21444363C212yesmaximal, complemented
6#21444353C212yesmaximal, complemented
7#21444345C211yesmaximal, characteristic, complemented
8#21444332C211yesmaximal, characteristic, complemented
9#21444319C211yesmaximal, characteristic, complemented
10#21444306C211yesmaximal, characteristic, complemented
11C61×#2974C211yesmaximal, characteristic, complemented
12#16840482C411yescharacteristic, complemented
13#16840467—21no—
14#16840452C412yescomplemented
15#16840438—22no—
16D4×Dic122C411yescharacteristic, complemented
17#16840421C412yescomplemented
18#16840417C411yescharacteristic, complemented
19#16840416C411yescharacteristic, complemented
20#16840406—21no—
21#16840377—21no—
22#16840354—22no—
23#16840319C2211yescharacteristic
24#16840296C2211yescharacteristic
25#16840273C2212yes—
26C2×#14519756C2212yes—
27#16840241—21no—
28#16839665—21no—
29#16839661—21no—
30#16839655—22no—
31#16839625C2212yes—
32#16839622C2212yes—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839608C2211yescharacteristic
35#16839500C2211yescharacteristic
36#16839500C2212yes—
37#16839500C2212yes—
38#16839500C2212yes—
39#16839500C2212yes—
40#16839500C2212yes—
41#16839500C2211yescharacteristic
42#16839497C2211yescharacteristic
43#16839497C2212yes—
44#16839497C2211yescharacteristic
45C2×C4×D4×C61C2211yescharacteristic
46C61×#436—21no—
47C61×#435—22no—
48C8×D4×C61—21no—
49C61×#432C2211yescharacteristic
50C61×#431C2211yescharacteristic
51C61×#407C2212yes—
52C2×C22⋊C8×C61C2212yes—
53C2×#11758222C2×C412yes—
54C2×#11758224—21no—
55C22×Dic122C2×C412yes—
56#14519777C2×C411yescharacteristic
57#14519777C2×C411yescharacteristic
58#14519776—21no—
59#14519771—21no—
60#14519770—22no—
61#14519770—22no—
62#14519769—21no—
63#14519769C2×C411yescharacteristic
64#14519767—22no—
65#14519765C2×C411yescharacteristic
66#14519764—21no—
67#14519763—22no—
68#14519761—22no—
69#14519760—22no—
70#14519760—21no—
71#14519759—22no—
72#14519758—22no—
73#14519757—21no—
74#14519756—22no—
75#14519756—21no—
76#14519756—22no—
77#14519756C2×C412yes—
78#14519756C2×C412yes—
79#14519756C2×C412yes—
80#14519756C2311yescharacteristic
81#14519756C2312yes—
82#14519755—22no—
83#14519754—22no—
84#14519751C2×C412yes—
85#14519751C2311yescharacteristic
86#14519749—22no—
87#14519748—21no—
88#14519746—22no—
89C4×Dic122—21no—
90C4×Dic122C2×C412yes—
91C4×Dic122C2×C411yescharacteristic
92#14519742—21no—
93C4×D244C2×C411yescharacteristic
94D4×Dic61—21no—
95C4×C22⋊D61—22no—
96C4×C22⋊D61—21no—
97C4×C22⋊D61—22no—
98#14519737—22no—
99#14519736—21no—
100#14519736C2×C412yes—
101#14519734—21no—
102#14519733—22no—
103C2×C4⋊Dic61—21no—
104C2×C4⋊Dic61—22no—
105C2×C4⋊Dic61C2×C412yes—
106C2×#11758196—22no—
107C4⋊C4×D61—21no—
108C2×C22⋊Dic61—22no—
109C2×C22⋊Dic61—22no—
110C2×C22⋊Dic61—22no—
111C2×C4×Dic61—21no—
112C2×C4×Dic61—21no—
113#14519684—21no—
114#14519684—22no—
115#14519684C2311yescharacteristic
116#14519684C2311yescharacteristic
117#14519684C2312yes—
118C22×D4×C61D411yescharacteristic
119C23×C4×C61C2312yes—
120C2×M16×C61—22no—
121C22×C8×C61—22no—
122C4×D4×C61C2×C412yes—
123C4×D4×C61C2×C411yescharacteristic
124C4×D4×C61C2×C411yescharacteristic
125C4×D4×C61D411yescharacteristic
126C4×D4×C61D411yescharacteristic
127C4×D4×C61D412yes—
128C2×C4⋊C4×C61D411yescharacteristic
129C2×C22⋊C4×C61D412yes—
130C2×C42×C61C2311yescharacteristic
131C4⋊C8×C61—21no—
132C4⋊C8×C61—21no—
133C22⋊C8×C61—22no—
134C22⋊C8×C61—22no—
135C22⋊C8×C61C2311yescharacteristic
136C22⋊C8×C61C2312yes—
137C22⋊C8×C61C2311yescharacteristic
138#97×C61—21no—
139C4×C8×C61—21no—
140C24×C61C2×D412yes—
141C2×D4×C61—42no—
142C2×D4×C61C22⋊C411yescharacteristic
143C2×D4×C61C22⋊C411yescharacteristic
144C2×D4×C61C22⋊C412yes—
145C22×C4×C61—82no—
146C22×C4×C61C22×C412yes—
147C22×C4×C61C22×C412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D412yes—
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—22no—
155C2×C22⋊D61—21no—
156C2×C22⋊D61—22no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—22no—
162#11758224—44no—
163#11758222—82no—
164C2×D244C22×C411yescharacteristic
165C2×C4×D61—21no—
166C2×C4×D61—22no—
167C2×Dic122—21no—
168C2×Dic122—22no—
169C2×Dic122—82no—
170C2×Dic122C22×C411yescharacteristic
171C2×Dic122C22×C412yes—
172M16×C61—82no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—22no—
176C2×C8×C61—82no—
177C4⋊C4×C61C22⋊C411yescharacteristic
178C4⋊C4×C61C22⋊C412yes—
179C4⋊C4×C61C22⋊C411yescharacteristic
180C22⋊C4×C61C22⋊C412yes—
181C22⋊C4×C61C22⋊C412yes—
182C22⋊C4×C61C22⋊C412yes—
183C22⋊C4×C61C22⋊C412yes—
184C42×C61C22×C411yescharacteristic
185C42×C61C22×C411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—21no—
193C22⋊Dic61—22no—
194C22⋊Dic61—82no—
195C22⋊Dic61—82no—
196#11758198—21no—
197#11758198—21no—
198#11758198—22no—
199#11758198—21no—
200#11758198—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—22no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—42no—
206C4⋊Dic61—82no—
207C4⋊Dic61C22×C411yescharacteristic
208C4⋊Dic61C22×C412yes—
209C4⋊Dic61C22×C411yescharacteristic
210#11758196—22no—
211#11758196—22no—
212#11758196—21no—
213#11758196—22no—
214#11758196—21no—
215#11758196—22no—
216#11758196—21no—
217#11758196—21no—
218#11758196—22no—
219C4×Dic61—22no—
220C4×Dic61—22no—
221C4×Dic61—21no—
222C4×Dic61—42no—
223C4×Dic61—81no—
224C23×C61—22no—
225C23×C61—22no—
226C23×C61—42no—
227C23×C61—42no—
228C23×C61C2×C22⋊C412yes—
229C23×C61C2×C22⋊C412yes—
230C23×C61C22×D411yescharacteristic
231C22×D61—21no—
232D4×C61—22no—
233D4×C61—22no—
234D4×C61—42no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—41no—
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C412yes—
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C412yes—
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C22⋊D61—44no—
253C22⋊D61—41no—
254C22⋊D61—41no—
255C22⋊D61—82no—
256C2×Dic61—21no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—44no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—42no—
265C2×Dic61—41no—
266C2×Dic61—42no—
267C2×Dic61—41no—
268C2×Dic61—42no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—82no—
272D244—41no—
273C4×D61—42no—
274C4×D61—82no—
275Dic122—42no—
276Dic122—41no—
277Dic122—42no—
278Dic122—82no—
279C8×C61—41no—
280C8×C61—42no—
281C8×C61—41no—
282#17600—611nomaximal, Sylow
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—22no—
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—22no—
291C22×C61—82no—
292C22×C61—82no—
293C22×C61C22×C22⋊C411yescharacteristic
294C22×C61#57711yescharacteristic
295C22×C61#57811yescharacteristic
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—41no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—22no—
302C4×C61—22no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61#42112yes—
307Dic61—41no—
308Dic61—42no—
309Dic61—41no—
310Dic61—42no—
311Dic61—41no—
312Dic61—81no—
313Dic61—82no—
314Dic61—81no—
315#3520—611no—
316#2974D6111yescharacteristic, complemented
317#2319—611no—
318#1520—612no—
319#1516—612no—
320#1513—612no—
321#1511—612no—
322#1509—611no—
323#1506—611no—
324#1503—611no—
325#1500—611no—
326C2×C61—21no—
327C2×C61—21no—
328C2×C61—42no—
329C2×C61—42no—
330C2×C61#288311yescharacteristic
331C2×C61#289811yescharacteristic
332C2×C61C2×#42111yescharacteristic
333D61—81no—
334#556—611no—
335#554—1221no—
336#552—612no—
337#551—1222no—
338D4×Q8—611no—
339#548—612no—
340#547—611no—
341#547—611no—
342#546—1221no—
343#542—1221no—
344#539—1222no—
345#533—611no—
346#529—611no—
347#525—612no—
348C2×C22⋊Q8—612no—
349#520—1221no—
350C2×C4×D4C2×D6111yescharacteristic
351#436—21no—
352#435—22no—
353C8×D4—21no—
354#432C2×D6111yescharacteristic
355#431C2×D6111yescharacteristic
356#407C2×D6112yes—
357C2×C22⋊C8C2×D6112yes—
358#390—1221no—
359#389—1221no—
360#388—1222no—
361#381—612no—
362#380—612no—
363C4×C4⋊C4—611no—
364#376—611no—
365#328—611no—
366#328—611no—
367#328—612no—
368#328—612no—
369#328—612no—
370#328—612no—
371#328—612no—
372#327—611no—
373#327—611no—
374#327—612no—
375C61#1760011yescharacteristic, complemented, Sylow
376C2×#41—612no—
377C2×#41—1221no—
378C22×Q8—612no—
379C22×D4D24411yescharacteristic
380C23×C4C22×D6112yes—
381C2×M16—22no—
382C22×C8—22no—
383C4⋊Q8—611no—
384C4⋊Q8—611no—
385C4⋊Q8—1221no—
386#126—1222no—
387#126—1222no—
388#126—1221no—
389#125—611no—
390#125—1222no—
391#125—1221no—
392#124—611no—
393#124—1221no—
394#124—1222no—
395#123—1222no—
396#123—1221no—
397#123—1222no—
398#123—1222no—
399#123—1222no—
400C22⋊Q8—612no—
401C22⋊Q8—612no—
402C22⋊Q8—612no—
403C22⋊Q8—611no—
404C22⋊Q8—612no—
405C22⋊Q8—1222no—
406C22⋊Q8—1221no—
407C22⋊Q8—1222no—
408C22⋊Q8—1222no—
409C22⋊Q8—1221no—
410C22⋊Q8—1222no—
411#121—612no—
412#121—611no—
413#121—1222no—
414#121—1221no—
415#120—1222no—
416C4×Q8—611no—
417C4×Q8—612no—
418C4×Q8—1221no—
419C4×Q8—1221no—
420C4×D4—611no—
421C4×D4—1221no—
422C4×D4—1222no—
423C4×D4—1221no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4C4×D6111yescharacteristic
427C4×D4C4×D6111yescharacteristic
428C4×D4C4×D6112yes—
429C4×D4C22⋊D6111yescharacteristic
430C4×D4C22⋊D6112yes—
431C4×D4C22⋊D6111yescharacteristic
432#117—612no—
433#117—1221no—
434#117—1222no—
435#117—1221no—
436C2×C4⋊C4—612no—
437C2×C4⋊C4—1222no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1222no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4D24411yescharacteristic
442C2×C22⋊C4—1222no—
443C2×C22⋊C4—1222no—
444C2×C22⋊C4—1222no—
445C2×C22⋊C4D24412yes—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42C22×D6111yescharacteristic
449C4⋊C8—21no—
450C4⋊C8—21no—
451C22⋊C8—22no—
452C22⋊C8—22no—
453C22⋊C8C22×D6111yescharacteristic
454C22⋊C8C22×D6112yes—
455C22⋊C8C22×D6111yescharacteristic
456#97—21no—
457C4×C8—21no—
458#95—611no—
459#95—611no—
460#95—612no—
461#95—1221no—
462#95—1222no—
463C24C2×D24412yes—
464#41—4882no—
465#41—2444no—
466C2×Q8—611no—
467C2×Q8—612no—
468C2×Q8—1222no—
469C2×Q8—1221no—
470C2×Q8—4882no—
471C2×D4—42no—
472C2×D4—611no—
473C2×D4—1222no—
474C2×D4—1221no—
475C2×D4—1222no—
476C2×D4#1175819811yescharacteristic
477C2×D4#1175819811yescharacteristic
478C2×D4#1175819812yes—
479C22×C4—82no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1222no—
487C22×C4C2×C4×D6112yes—
488C22×C4C2×C4×D6112yes—
489C22×C4C2×D24411yescharacteristic
490C22×C4C2×D24411yescharacteristic
491C22×C4C2×D24412yes—
492C22×C4C2×C22⋊D6112yes—
493C22×C4C2×C22⋊D6112yes—
494C22×C4C23×D6111yescharacteristic
495M16—82no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—22no—
499C2×C8—82no—
500C4⋊C4—611no—
501C4⋊C4—612no—
502C4⋊C4—611no—
503C4⋊C4—1221no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1222no—
511C4⋊C4—1221no—
512C4⋊C4—1222no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—4882no—
517C4⋊C4—2442no—
518C4⋊C4#1175819811yescharacteristic
519C4⋊C4#1175819812yes—
520C4⋊C4#1175819811yescharacteristic
521C22⋊C4—1221no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—1221no—
528C22⋊C4—1221no—
529C22⋊C4—1221no—
530C22⋊C4—1222no—
531C22⋊C4—1221no—
532C22⋊C4—4882no—
533C22⋊C4—4882no—
534C22⋊C4#1175819812yes—
535C22⋊C4#1175819812yes—
536C22⋊C4#1175819812yes—
537C22⋊C4#1175819812yes—
538C42—1222no—
539C42—1222no—
540C42—1221no—
541C42—4881no—
542C42—2442no—
543C42C2×C4×D6111yescharacteristic
544C42C2×C4×D6111yescharacteristic
545C42C2×C22⋊D6111yescharacteristic
546C42C2×C22⋊D6111yescharacteristic
547C23—22no—
548C23—22no—
549C23—42no—
550C23—42no—
551C23—1221no—
552C23C2×#1175819812yes—
553C23C2×#1175819812yes—
554C23C22×D24411yescharacteristic
555Q8—4882no—
556Q8—2441no—
557Q8—2442no—
558Q8—2442no—
559D4—22no—
560D4—22no—
561D4—42no—
562D4—4882no—
563D4—2441no—
564D4—2444no—
565D4—2441no—
566D4—2441no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—1221no—
577C2×C4—1222no—
578C2×C4—1221no—
579C2×C4—1222no—
580C2×C4—1221no—
581C2×C4—4882no—
582C2×C4—4882no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2442no—
587C2×C4—2441no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2444no—
593C2×C4—2442no—
594C2×C4C2×#1175819811yescharacteristic
595C2×C4C2×#1175819812yes—
596C2×C4C2×#1175819811yescharacteristic
597C2×C4C2×#1175819811yescharacteristic
598C2×C4C2×#1175819812yes—
599C2×C4C2×#1175819811yescharacteristic
600C2×C4C22×C4×D6111yescharacteristic
601C2×C4C22×C22⋊D6111yescharacteristic
602C8—41no—
603C8—42no—
604C8—41no—
605C22—21no—
606C22—21no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—82no—
614C22—82no—
615C22—2441no—
616C22—2441no—
617C22—2441no—
618C22C22×#1175819811yescharacteristic
619C22#1684057411yescharacteristic
620C22#1684057911yescharacteristic
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—22no—
627C4—22no—
628C4—4881no—
629C4—4882no—
630C4—4881no—
631C4—2441no—
632C4—2441no—
633C4—2442no—
634C4—2442no—
635C4—2441no—
636C4#1683977812yes—
637C2—21no—
638C2—21no—
639C2—42no—
640C2—42no—
641C2—4881no—
642C2#2145427911yescharacteristic
643C2#2145435311yescharacteristic
644C2C2×#1683977811yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2974C2
socleC22×C61C22×C22⋊C4
prime radical#21459868C2
cosocleC22×C4×C61C24
hypercenter#2974D61
hypocenterC61#17600
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2974

Complements

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

NH ≅ G/Ntype
2. #21448462C2semidirect
3. #21444392C2semidirect
4. #21444376C2semidirect
5. #21444363C2semidirect
6. #21444353C2semidirect
7. #21444345C2semidirect
8. #21444332C2semidirect
9. #21444319C2semidirect
10. #21444306C2semidirect
11. C61×#2974C2semidirect
12. #16840482C4semidirect
14. #16840452C4semidirect
16. D4×Dic122C4semidirect
17. #16840421C4semidirect
18. #16840417C4semidirect
19. #16840416C4semidirect
316. #2974D61semidirect
375. C61#17600semidirect

Maximal subgroups

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

subgroupindex
#214598682
#214484622
#214443922
#214443762
#214443632
#214443532
#214443452
#214443322
#214443192
#214443062
C61×#29742
#1760061