Finite Group Databasebeta

← #29647694

Subgroups — #29647694

Order 15616.

990 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459554C211yesmaximal, characteristic
2#21448704C211yesmaximal, characteristic, complemented
3#21444388C211yesmaximal, characteristic, complemented
4#21444380C212yesmaximal, complemented
5#21444363C211yesmaximal, characteristic, complemented
6#21444358C212yesmaximal, complemented
7#21444336C211yesmaximal, characteristic, complemented
8#21444328C212yesmaximal, complemented
9#21444311C211yesmaximal, characteristic, complemented
10#21444306C212yesmaximal, complemented
11C61×#2924C211yesmaximal, characteristic, complemented
12#16840475—21no—
13#16840460—21no—
14#16840456—21no—
15#16840436—22no—
16#16840413—21no—
17#16840402—22no—
18#16840390C411yescharacteristic, complemented
19#16840383C412yescomplemented
20#16840367C412yescomplemented
21#16840331C411yescharacteristic, complemented
22#16840312C2212yes—
23#16840296C2212yes—
24#16840273C2211yescharacteristic
25C2×#14519751C2211yescharacteristic
26#16840242C411yescharacteristic, complemented
27#16840236C411yescharacteristic, complemented
28#16839661—21no—
29#16839655—22no—
30#16839628C2212yescomplemented
31#16839622C2211yescharacteristic
32C4⋊C4×Dic61—21no—
33C4×C22⋊Dic61C2211yescharacteristic
34#16839608C2212yescomplemented
35#16839500C2211yescharacteristic, complemented
36#16839500C2212yescomplemented
37#16839500C2211yescharacteristic, complemented
38#16839497C2212yes—
39#16839497C2212yes—
40#16839497C2212yes—
41#16839497C2212yes—
42#16839497C2212yescomplemented
43#16839497C2211yescharacteristic, complemented
44#16839497C2211yescharacteristic, complemented
45C2×#117×C61C2211yescharacteristic
46C61×#433—22no—
47C61×#432C2212yescomplemented
48C61×#431C2212yescomplemented
49C61×#424—21no—
50C61×#407C2211yescharacteristic
51C2×C22⋊C8×C61C2211yescharacteristic
52C61×#405—21no—
53C2×#11758222—22no—
54C2×#11758222C2×C411yescharacteristic
55C22×D244C2×C411yescharacteristic
56C22×C22⋊D61—21no—
57#14519777C2×C412yes—
58#14519776—21no—
59#14519774C2×C412yes—
60#14519772—22no—
61#14519771—22no—
62#14519769C2×C412yes—
63#14519768—21no—
64#14519765—22no—
65#14519765C2×C412yes—
66#14519763—22no—
67#14519762—22no—
68#14519762—22no—
69#14519760—21no—
70#14519760—22no—
71#14519759—22no—
72#14519756C2×C411yescharacteristic
73#14519756C2311yescharacteristic
74#14519755—22no—
75#14519754—22no—
76#14519752—21no—
77#14519752—21no—
78#14519751—21no—
79#14519751—21no—
80#14519751—22no—
81#14519751C2×C412yes—
82#14519751C2×C411yescharacteristic
83#14519751C2312yes—
84#14519751C2311yescharacteristic
85#14519750—22no—
86#14519750—22no—
87#14519747—22no—
88#14519746—21no—
89C4×Dic122C2×C412yes—
90C4×D244—22no—
91C4×D244C2×C412yes—
92C4×D244C2×C412yes—
93C4×D244C2×C412yes—
94#14519739—21no—
95#14519739—21no—
96C4×C22⋊D61—21no—
97C4×C22⋊D61—21no—
98C4×C22⋊D61—22no—
99#14519737—22no—
100#14519736C2×C411yescharacteristic
101#14519735—21no—
102#14519734—22no—
103C2×C4⋊Dic61—21no—
104C2×C4⋊Dic61—22no—
105C2×C4⋊Dic61C2×C411yescharacteristic
106C4⋊C4×D61—21no—
107C2×C22⋊Dic61—21no—
108C2×C22⋊Dic61—21no—
109C2×#11758198—21no—
110C22⋊C4×D61—22no—
111C2×C4×Dic61—22no—
112C2×C4×Dic61—21no—
113#14519684—22no—
114#14519684—22no—
115#14519684C2311yescharacteristic
116#14519684C2312yes—
117#14519684C2311yescharacteristic
118C23×C4×C61C2311yescharacteristic
119C2×M16×C61—21no—
120C22×C8×C61—21no—
121#117×C61C2×C411yescharacteristic
122#117×C61C2×C411yescharacteristic
123#117×C61C2×C412yes—
124#117×C61D411yescharacteristic
125#117×C61D411yescharacteristic
126#117×C61D412yes—
127C2×C4⋊C4×C61D411yescharacteristic
128C2×C4⋊C4×C61D411yescharacteristic
129C2×C22⋊C4×C61D412yes—
130C2×C42×C61C2312yes—
131C4⋊C8×C61—22no—
132C4⋊C8×C61—22no—
133C22⋊C8×C61—21no—
134C22⋊C8×C61—21no—
135C22⋊C8×C61C2311yescharacteristic
136C22⋊C8×C61C2311yescharacteristic
137C22⋊C8×C61C2312yes—
138#97×C61—22no—
139C4×C8×C61—22no—
140C24×C61C2×D411yescharacteristic
141C23×D61—21no—
142C22×C4×C61—42no—
143C22×C4×C61C22×C411yescharacteristic
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C2×C22⋊D61—21no—
152C2×C22⋊D61—22no—
153C2×C22⋊D61—21no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61—21no—
156C2×C22⋊D61—81no—
157C2×C22⋊D61—81no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—22no—
161C22×Dic61—21no—
162#11758222—42no—
163#11758222—82no—
164#11758222—82no—
165C2×D244—21no—
166C2×D244—22no—
167C2×D244—22no—
168C2×D244—42no—
169C2×D244C22×C412yes—
170C2×D244C22×C411yescharacteristic
171C2×C4×D61—21no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×Dic122—22no—
176C2×Dic122C22×C411yescharacteristic
177M16×C61—42no—
178C2×C8×C61—21no—
179C2×C8×C61—22no—
180C2×C8×C61—21no—
181C2×C8×C61—42no—
182C4⋊C4×C61C22⋊C411yescharacteristic
183C4⋊C4×C61C22⋊C411yescharacteristic
184C4⋊C4×C61C22⋊C411yescharacteristic
185C4⋊C4×C61C22⋊C411yescharacteristic
186C4⋊C4×C61C22⋊C412yes—
187C4⋊C4×C61C22⋊C412yes—
188C22⋊C4×C61C22⋊C412yes—
189C22⋊C4×C61C22⋊C412yes—
190C22⋊C4×C61C22⋊C412yes—
191C22⋊C4×C61C22⋊C412yes—
192C42×C61C22×C412yes—
193C42×C61C22×C412yes—
194C42×C61C2×D412yes—
195C42×C61C2×D412yes—
196C22⋊Dic61—21no—
197C22⋊Dic61—21no—
198C22⋊Dic61—21no—
199C22⋊Dic61—42no—
200C22⋊Dic61—81no—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198—22no—
205#11758198—22no—
206#11758198—22no—
207#11758198—21no—
208#11758198—22no—
209#11758198—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—22no—
212C4⋊Dic61—42no—
213C4⋊Dic61—82no—
214C4⋊Dic61C22×C412yes—
215C4⋊Dic61C22×C411yescharacteristic
216C4⋊Dic61C22×C411yescharacteristic
217#11758196—22no—
218#11758196—22no—
219#11758196—22no—
220#11758196—21no—
221#11758196—21no—
222C4×Dic61—22no—
223C4×Dic61—21no—
224C4×Dic61—44no—
225C4×Dic61—42no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61C2×C22⋊C411yescharacteristic
231C23×C61C2×C22⋊C411yescharacteristic
232C23×C61C22×D411yescharacteristic
233C22×D61—22no—
234C22×D61—21no—
235C22×D61—42no—
236C22×D61—81no—
237C22×D61—81no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—42no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61C2×C22⋊C412yes—
248C2×C4×C61C2×C22⋊C412yes—
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C412yes—
251C2×C4×C61C2×C22⋊C412yes—
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C23×C411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C22⋊D61—42no—
256C22⋊D61—41no—
257C22⋊D61—42no—
258C22⋊D61—41no—
259C22⋊D61—42no—
260C22⋊D61—81no—
261C22⋊D61—82no—
262C2×Dic61—22no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—22no—
266C2×Dic61—21no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—42no—
275C2×Dic61—81no—
276D244—42no—
277D244—41no—
278D244—42no—
279D244—82no—
280D244—82no—
281C4×D61—42no—
282C4×D61—82no—
283C4×D61—82no—
284C4×D61—82no—
285Dic122—41no—
286Dic122—82no—
287C8×C61—41no—
288C8×C61—41no—
289C8×C61—42no—
290#17610—611nomaximal, Sylow
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61C22×C22⋊C411yescharacteristic
302C22×C61#57611yescharacteristic
303C22×C61#57711yescharacteristic
304C2×D61—41no—
305C2×D61—42no—
306C2×D61—41no—
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—41no—
310C2×D61—162no—
311C4×C61—21no—
312C4×C61—21no—
313C4×C61—21no—
314C4×C61—22no—
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—22no—
318C4×C61#41012yes—
319Dic61—42no—
320Dic61—41no—
321Dic61—41no—
322Dic61—42no—
323Dic61—41no—
324Dic61—82no—
325Dic61—81no—
326#3494—611no—
327#2924D6111yescharacteristic, complemented
328#2346—611no—
329#1519—611no—
330#1517—612no—
331#1513—611no—
332#1512—612no—
333#1507—611no—
334#1505—612no—
335#1501—611no—
336#1500—612no—
337C2×C61—21no—
338C2×C61—21no—
339C2×C61—41no—
340C2×C61—41no—
341C2×C61C2×#41011yescharacteristic
342C2×C61#289711yescharacteristic
343C2×C61#290111yescharacteristic
344D61—81no—
345D61—82no—
346#555—1221no—
347#553—1221no—
348#553—1221no—
349#551—1222no—
350#547—1221no—
351#546—1222no—
352#544—611no—
353#542—612no—
354#540—612no—
355#535—611no—
356#532—612no—
357#529—612no—
358#525—611no—
359C2×#121—611no—
360#520—611no—
361#518—611no—
362C2×#117C2×D6111yescharacteristic
363#433—22no—
364#432C2×D6112yescomplemented
365#431C2×D6112yescomplemented
366#424—21no—
367#407C2×D6111yescharacteristic
368C2×C22⋊C8C2×D6111yescharacteristic
369#405—21no—
370#389—1221no—
371#388—1222no—
372#382—612no—
373#380—611no—
374C4×C4⋊C4—1221no—
375C4×C22⋊C4—611no—
376#376—612no—
377#328—611no—
378#328—612no—
379#328—611no—
380#327—612no—
381#327—612no—
382#327—612no—
383#327—612no—
384#327—611no—
385#327—611no—
386#327—612no—
387C61#1761011yescharacteristic, complemented, Sylow
388C2×#41—611no—
389C2×#41—1222no—
390C22×D4—611no—
391C22×D4—1221no—
392C23×C4C22×D6111yescharacteristic
393C2×M16—21no—
394C22×C8—21no—
395C4⋊Q8—612no—
396C4⋊Q8—1221no—
397#127—612no—
398#126—1222no—
399#126—1222no—
400#125—612no—
401#125—1221no—
402#124—612no—
403#124—1222no—
404#124—1222no—
405#123—1222no—
406#123—1221no—
407#123—1222no—
408#123—1222no—
409#123—1222no—
410C22⋊Q8—611no—
411C22⋊Q8—611no—
412C22⋊Q8—1222no—
413C22⋊Q8—1222no—
414#121—612no—
415#121—611no—
416#121—612no—
417#121—611no—
418#121—1221no—
419#121—1222no—
420#121—1221no—
421#121—1221no—
422#121—1221no—
423#121—1222no—
424#121—1222no—
425#120—1221no—
426#120—1222no—
427C4×Q8—612no—
428C4×D4—612no—
429C4×D4—612no—
430C4×D4—612no—
431C4×D4—1221no—
432C4×D4—1222no—
433C4×D4—1221no—
434C4×D4—1221no—
435C4×D4—1221no—
436C4×D4—1222no—
437C4×D4—1222no—
438#117—611no—
439#117—1221no—
440#117—1222no—
441#117C4×D6111yescharacteristic
442#117C4×D6111yescharacteristic
443#117C4×D6112yes—
444#117C22⋊D6112yes—
445#117C22⋊D6111yescharacteristic
446#117C22⋊D6111yescharacteristic
447C2×C4⋊C4—611no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1222no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4D24411yescharacteristic
452C2×C4⋊C4D24411yescharacteristic
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1222no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4D24412yes—
458C2×C42—1222no—
459C2×C42—1221no—
460C2×C42C22×D6112yes—
461C4⋊C8—22no—
462C4⋊C8—22no—
463C22⋊C8—21no—
464C22⋊C8—21no—
465C22⋊C8C22×D6111yescharacteristic
466C22⋊C8C22×D6111yescharacteristic
467C22⋊C8C22×D6112yes—
468#97—22no—
469C4×C8—22no—
470#95—611no—
471#95—612no—
472#95—611no—
473#95—1222no—
474#95—1222no—
475C24—1221no—
476C24C2×D24411yescharacteristic
477#41—4882no—
478#41—4882no—
479#41—2442no—
480C2×Q8—611no—
481C2×Q8—1222no—
482C2×D4—612no—
483C2×D4—611no—
484C2×D4—1221no—
485C2×D4—1222no—
486C2×D4—1221no—
487C2×D4—1222no—
488C2×D4—1221no—
489C2×D4—1222no—
490C2×D4—1222no—
491C2×D4—1221no—
492C2×D4—4881no—
493C2×D4—4881no—
494C2×D4—2442no—
495C22×C4—42no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1222no—
504C22×C4C2×C4×D6111yescharacteristic
505C22×C4C2×C4×D6111yescharacteristic
506C22×C4C2×D24412yes—
507C22×C4C2×D24411yescharacteristic
508C22×C4C2×D24412yes—
509C22×C4C2×C22⋊D6111yescharacteristic
510C22×C4C2×C22⋊D6111yescharacteristic
511C22×C4C23×D6111yescharacteristic
512M16—42no—
513C2×C8—21no—
514C2×C8—22no—
515C2×C8—21no—
516C2×C8—42no—
517C4⋊C4—612no—
518C4⋊C4—611no—
519C4⋊C4—611no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1222no—
525C4⋊C4—1221no—
526C4⋊C4—1222no—
527C4⋊C4—4882no—
528C4⋊C4—2442no—
529C4⋊C4#1175819811yescharacteristic
530C4⋊C4#1175819811yescharacteristic
531C4⋊C4#1175819811yescharacteristic
532C4⋊C4#1175819811yescharacteristic
533C4⋊C4#1175819812yes—
534C4⋊C4#1175819812yes—
535C22⋊C4—1221no—
536C22⋊C4—1221no—
537C22⋊C4—1221no—
538C22⋊C4—1222no—
539C22⋊C4—1221no—
540C22⋊C4—1222no—
541C22⋊C4—1222no—
542C22⋊C4—1222no—
543C22⋊C4—1222no—
544C22⋊C4—1222no—
545C22⋊C4—1222no—
546C22⋊C4—1221no—
547C22⋊C4—4881no—
548C22⋊C4—2442no—
549C22⋊C4#1175819812yes—
550C22⋊C4#1175819812yes—
551C22⋊C4#1175819812yes—
552C22⋊C4#1175819812yes—
553C42—1221no—
554C42—1222no—
555C42—2442no—
556C42—2444no—
557C42C2×C4×D6112yes—
558C42C2×C4×D6112yes—
559C42C2×C22⋊D6112yes—
560C42C2×C22⋊D6112yes—
561C23—21no—
562C23—21no—
563C23—41no—
564C23—41no—
565C23—1222no—
566C23—1221no—
567C23—4881no—
568C23—4881no—
569C23—2442no—
570C23C2×#1175819811yescharacteristic
571C23C2×#1175819811yescharacteristic
572C23C22×D24411yescharacteristic
573Q8—4882no—
574Q8—2441no—
575D4—4882no—
576D4—4882no—
577D4—4881no—
578D4—4882no—
579D4—2442no—
580D4—2441no—
581D4—2442no—
582D4—2442no—
583D4—2441no—
584D4—2442no—
585D4—2441no—
586D4—2442no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—1222no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1222no—
601C2×C4—4882no—
602C2×C4—4882no—
603C2×C4—4882no—
604C2×C4—4881no—
605C2×C4—2442no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2442no—
609C2×C4—2442no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2442no—
614C2×C4C2×#1175819812yes—
615C2×C4C2×#1175819812yes—
616C2×C4C2×#1175819812yes—
617C2×C4C2×#1175819812yes—
618C2×C4C2×#1175819811yescharacteristic
619C2×C4C2×#1175819811yescharacteristic
620C2×C4C22×C4×D6111yescharacteristic
621C2×C4C22×C22⋊D6111yescharacteristic
622C8—41no—
623C8—41no—
624C8—42no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—81no—
634C22—81no—
635C22—9762no—
636C22—2441no—
637C22—2442no—
638C22—2442no—
639C22—2441no—
640C22—2441no—
641C22—2442no—
642C22C22×#1175819811yescharacteristic
643C22#1684056711yescharacteristic
644C22#1684057411yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—4881no—
653C4—4882no—
654C4—2442no—
655C4—2442no—
656C4—2441no—
657C4—2441no—
658C4—2441no—
659C4#1683973512yes—
660C2—21no—
661C2—21no—
662C2—41no—
663C2—41no—
664C2—4882no—
665C2—4881no—
666C2C2×#1683973511yescharacteristic
667C2#2145434611yescharacteristic
668C2#2145436411yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2924C2
socleC22×C61C22×C22⋊C4
prime radical#21459554C2
cosocleC22×C4×C61C24
hypercenter#2924D61
hypocenterC61#17610
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#117 ◁ #2924

Complements

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

NH ≅ G/Ntype
2. #21448704C2semidirect
3. #21444388C2semidirect
4. #21444380C2semidirect
5. #21444363C2semidirect
6. #21444358C2semidirect
7. #21444336C2semidirect
8. #21444328C2semidirect
9. #21444311C2semidirect
10. #21444306C2semidirect
11. C61×#2924C2semidirect
18. #16840390C4semidirect
19. #16840383C4semidirect
20. #16840367C4semidirect
21. #16840331C4semidirect
26. #16840242C4semidirect
27. #16840236C4semidirect
30. #16839628C22semidirect
34. #16839608C22semidirect
35. #16839500C22semidirect
36. #16839500C22semidirect
37. #16839500C22semidirect
42. #16839497C22semidirect
43. #16839497C22semidirect
44. #16839497C22semidirect
47. C61×#432C22semidirect
48. C61×#431C22semidirect
327. #2924D61semidirect
364. #432C2×D61semidirect
365. #431C2×D61semidirect
387. C61#17610semidirect

Maximal subgroups

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

subgroupindex
#214595542
#214487042
#214443882
#214443802
#214443632
#214443582
#214443362
#214443282
#214443112
#214443062
C61×#29242
#1761061