Finite Group Databasebeta

← #29647375

Subgroups — #29647375

Order 15616.

1036 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459114C211yesmaximal, characteristic
2#431×D61C211yesmaximal, characteristic, complemented
3#21448212C211yesmaximal, characteristic, complemented
4#21444393C214yesmaximal, complemented
5#21444384C211yesmaximal, characteristic, complemented
6#21444382C211yesmaximal, characteristic, complemented
7#21444379C211yesmaximal, characteristic, complemented
8#21444307C214yesmaximal, complemented
9C61×#1517C211yesmaximal, characteristic, complemented
10#16840319C2211yescharacteristic
11#16840317C411yescharacteristic, complemented
12#16840317C411yescharacteristic, complemented
13#16840310C411yescharacteristic, complemented
14#16840310C411yescharacteristic, complemented
15#16840294—21no—
16#16840294C412yescomplemented
17#16840293—21no—
18#16840293C2212yes—
19#16840291—21no—
20#16840291—21no—
21#16840272—21no—
22#16840272C2212yes—
23#16840269—21no—
24#41×Dic61—21no—
25C4×#11758224—21no—
26C4×#11758224C412yescomplemented
27C2×C42×D61C2211yescharacteristic
28#16839815C2211yescharacteristic
29#16839814—21no—
30#16839813—21no—
31#16839812C2212yes—
32C4⋊C8×D61—21no—
33C22⋊C8×D61C2212yes—
34C4×C8×D61—21no—
35#16839629C2212yes—
36#16839628C2211yescharacteristic
37#16839624—21no—
38#16839619—21no—
39C4⋊C4×Dic61—21no—
40#16839609C2212yes—
41#16839607—21no—
42#16839500C2214yes—
43#16839498C2214yes—
44#16839496C2214yes—
45C61×#532C2211yescharacteristic
46C61×#431C2211yescharacteristic
47C61×#382C2211yescharacteristic
48C61×#327C2214yes—
49C2×#11758224—21no—
50C2×#11758224C2×C412yes—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C2×C8×D61—22no—
55#14519777C2×C411yescharacteristic
56#14519777C2×C411yescharacteristic
57#14519770—22no—
58#14519770—22no—
59#14519769—21no—
60#14519768—21no—
61#14519768C2×C412yes—
62#14519764—21no—
63#14519764C2×C412yes—
64#14519759—22no—
65#14519759—22no—
66#14519758—22no—
67#14519757—21no—
68#14519757C2×C412yes—
69#14519756—21no—
70#14519756C2312yes—
71#14519753—21no—
72#14519753—21no—
73#14519748—21no—
74#14519748—21no—
75#14519748—21no—
76#14519748C2×C412yes—
77#14519745—22no—
78Q8×Dic61—21no—
79Q8×Dic61—21no—
80C4×Dic122—21no—
81C4×Dic122C2×C412yes—
82#14519742—21no—
83#14519742C2×C412yes—
84D4×Dic61—22no—
85D4×Dic61—22no—
86D4×Dic61—21no—
87D4×Dic61—21no—
88D4×Dic61—21no—
89D4×Dic61C2×C412yes—
90C4×C22⋊D61—22no—
91C4×C22⋊D61—22no—
92#14519737—22no—
93#14519737—22no—
94#14519736—21no—
95#14519736—22no—
96#14519735—21no—
97#14519735C2×C412yes—
98#14519733—22no—
99#14519733—22no—
100C2×C4⋊Dic61—21no—
101C2×#11758198—22no—
102C2×C4×Dic61—22no—
103C2×C4×Dic61—22no—
104C2×C4×Dic61D411yescharacteristic
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61C2311yescharacteristic
107C42×D61C2×C411yescharacteristic
108C42×D61C2×C411yescharacteristic
109C42×D61C2×C411yescharacteristic
110C42×D61C2×C411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61D411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C4⋊#92418—21no—
116#14519706—22no—
117C22⋊#92418C2312yes—
118#14519689—22no—
119C4×#92418—21no—
120C8×Dic61—22no—
121#14519684—21no—
122#14519684—22no—
123#14519684—22no—
124#14519684C2312yes—
125C2×#41×C61—21no—
126C4⋊Q8×C61C2×C411yescharacteristic
127#127×C61C2×C411yescharacteristic
128#124×C61—21no—
129#121×C61—21no—
130#121×C61C2312yes—
131C4×D4×C61—21no—
132C4×D4×C61C2×C412yes—
133C2×C4⋊C4×C61—21no—
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61—21no—
136C22⋊C8×C61C2312yes—
137C4×C8×C61—21no—
138#95×C61—21no—
139#95×C61C2312yes—
140C23×D61C2×D411yescharacteristic
141#41×C61—42no—
142#41×C61—42no—
143C2×Q8×C61—21no—
144C2×D4×C61—22no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61C22×C412yes—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161#11758224—44no—
162#11758224—44no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61C22⋊C411yescharacteristic
168C2×C4×D61C22⋊C411yescharacteristic
169C2×C4×D61C22⋊C411yescharacteristic
170C2×C4×D61C22⋊C411yescharacteristic
171C2×C4×D61C22⋊C411yescharacteristic
172C2×C4×D61C22⋊C411yescharacteristic
173C2×C4×D61C22⋊C411yescharacteristic
174C2×C4×D61C22⋊C411yescharacteristic
175C2×C4×D61C22×C411yescharacteristic
176C2×C4×D61C22×C411yescharacteristic
177C2×C4×D61C2×D411yescharacteristic
178C2×C4×D61C2×D411yescharacteristic
179C2×Dic122—21no—
180C2×Dic122C22×C412yes—
181C2×C8×C61—22no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—42no—
184C4⋊C4×C61C22×C412yes—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C42×C61C22×C411yescharacteristic
188C42×C61C22×C411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—44no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—22no—
203C4⋊Dic61—22no—
204C4⋊Dic61—42no—
205C4⋊Dic61C22×C412yes—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61—44no—
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22×C411yescharacteristic
223C4×Dic61C22×C411yescharacteristic
224C4×Dic61C2×D411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226C2×#92418—22no—
227C8×D61—44no—
228C23×C61—22no—
229C23×C61C22×D411yescharacteristic
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—22no—
233C22×D61—41no—
234C22×D61C2×C22⋊C411yescharacteristic
235C22×D61C2×C22⋊C411yescharacteristic
236Q8×C61—42no—
237D4×C61—42no—
238D4×C61—42no—
239D4×C61—42no—
240D4×C61—42no—
241D4×C61—42no—
242C2×C4×C61—22no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—44no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61#13512yes—
254C2×C4×C61#13512yes—
255C2×C4×C61C23×C411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C22⋊D61—44no—
258C22⋊D61—44no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—41no—
265C2×Dic61—44no—
266C2×Dic61—44no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61C2×C22⋊C411yescharacteristic
271C2×Dic61C2×C22⋊C411yescharacteristic
272C2×Dic61C2×C22⋊C411yescharacteristic
273C2×Dic61C2×C22⋊C411yescharacteristic
274C2×Dic61C2×C22⋊C411yescharacteristic
275C2×Dic61C2×C22⋊C411yescharacteristic
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—22no—
284C4×D61—41no—
285C4×D61—41no—
286C4×D61—41no—
287C4×D61—41no—
288Dic122—42no—
289Dic122—42no—
290C8×C61—42no—
291#92418—42no—
292#17573—611nomaximal, Sylow
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—42no—
298C22×C61C22×C22⋊C411yescharacteristic
299C22×C61C2×#13511yescharacteristic
300C22×C61C2×#13511yescharacteristic
301C2×D61—21no—
302C2×D61—21no—
303C2×D61—21no—
304C2×D61—21no—
305C2×D61—21no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61#42112yes—
309C4×C61—21no—
310C4×C61—21no—
311C4×C61—21no—
312C4×C61—21no—
313C4×C61—42no—
314C4×C61—82no—
315C4×C61#41612yes—
316C4×C61#41612yes—
317Dic61—21no—
318Dic61—21no—
319Dic61—21no—
320Dic61—21no—
321Dic61—21no—
322Dic61—42no—
323Dic61—42no—
324Dic61—82no—
325Dic61#41012yes—
326#3444—611no—
327C2×#431—611no—
328#2283—611no—
329#1521—614no—
330#1518—611no—
331#1518—611no—
332#1517—611no—
333#1517D6111yescharacteristic, complemented
334#1501—614no—
335C2×C61—21no—
336C2×C61—21no—
337C2×C61—82no—
338C2×C61C2×#41011yescharacteristic
339C2×C61C2×#41611yescharacteristic
340C2×C61C2×#42111yescharacteristic
341D61—22no—
342D61—41no—
343#533—611no—
344#533—611no—
345#533—611no—
346#532—611no—
347#532—611no—
348#532C2×D6111yescharacteristic
349#529—612no—
350#529—612no—
351#529—1221no—
352#529—1221no—
353#529—1221no—
354#529—1221no—
355#525—612no—
356#525—1221no—
357#525—1221no—
358C4×#41—612no—
359C4×#41—1221no—
360C4×#41—1221no—
361C22×C42—611no—
362#431—611no—
363#431—612no—
364#431—1221no—
365#431—1221no—
366#431C2×D6111yescharacteristic
367C2×C4⋊C8—1221no—
368C2×C22⋊C8—612no—
369C2×C4×C8—1221no—
370#383—612no—
371#382—611no—
372#382C2×D6111yescharacteristic
373#381—1221no—
374#380—1221no—
375C4×C4⋊C4—1221no—
376C4×C22⋊C4—612no—
377#376—1221no—
378#328—614no—
379#328—614no—
380#327—614no—
381#327C2×D6114yes—
382C61#1757311yescharacteristic, complemented, Sylow
383C2×#41—21no—
384C2×#41—612no—
385C2×#41—1221no—
386C23×C4—611no—
387C23×C4—611no—
388C23×C4—611no—
389C22×C8—1222no—
390C4⋊Q8—611no—
391C4⋊Q8—611no—
392C4⋊Q8C4×D6111yescharacteristic
393#127C4×D6111yescharacteristic
394#126—1222no—
395#126—1222no—
396#125—612no—
397#125—1221no—
398#125—1221no—
399#124—21no—
400#124—612no—
401#124—1221no—
402#123—1222no—
403#123—1222no—
404#123—1222no—
405C22⋊Q8—612no—
406C22⋊Q8—612no—
407C22⋊Q8—1221no—
408C22⋊Q8—1221no—
409C22⋊Q8—1221no—
410C22⋊Q8—1221no—
411#121—21no—
412#121—612no—
413#121—1221no—
414#121—1221no—
415#121—1221no—
416#121C22×D6112yes—
417#120—1222no—
418C4×Q8—612no—
419C4×Q8—612no—
420C4×Q8—1221no—
421C4×Q8—1221no—
422C4×Q8—1221no—
423C4×Q8—1221no—
424C4×D4—21no—
425C4×D4—612no—
426C4×D4—1221no—
427C4×D4—1221no—
428C4×D4—1221no—
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—1222no—
432C4×D4—1222no—
433C4×D4—1222no—
434C4×D4—1222no—
435C4×D4C4×D6112yes—
436#117—612no—
437#117—1222no—
438#117—1222no—
439#117—1222no—
440#117—1221no—
441#117—1221no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—1221no—
444C2×C22⋊C4—1222no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—611no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42—611no—
455C2×C42—611no—
456C2×C42—1222no—
457C2×C42—1222no—
458C2×C42C22×D6111yescharacteristic
459C4⋊C8—21no—
460C4⋊C8—1222no—
461C4⋊C8—1221no—
462C22⋊C8—612no—
463C22⋊C8—1222no—
464C22⋊C8C22×D6112yes—
465C4×C8—21no—
466C4×C8—1222no—
467C4×C8—1221no—
468#95—21no—
469#95—612no—
470#95—1222no—
471#95—1222no—
472#95—1221no—
473#95C22×D6112yes—
474C24—611no—
475#41—42no—
476#41—42no—
477#41—2444no—
478#41—2444no—
479C2×Q8—21no—
480C2×Q8—612no—
481C2×Q8—1221no—
482C2×D4—22no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—1222no—
486C2×D4—1222no—
487C2×D4C2×C4×D6112yes—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—1222no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—1222no—
510C22×C4—2442no—
511C22×C4—2442no—
512C22×C4D4×D6111yescharacteristic
513C22×C4D4×D6111yescharacteristic
514C22×C4C23×D6111yescharacteristic
515C2×C8—22no—
516C2×C8—1222no—
517C2×C8—2444no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—612no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—2442no—
531C4⋊C4C2×C4×D6112yes—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4—2444no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—611no—
548C42—611no—
549C42—611no—
550C42—611no—
551C42—611no—
552C42—611no—
553C42—611no—
554C42—1222no—
555C42—1222no—
556C42—1222no—
557C42—2444no—
558C42C2×C4×D6111yescharacteristic
559C42C2×C4×D6111yescharacteristic
560C42D4×D6111yescharacteristic
561C42D4×D6111yescharacteristic
562C23—22no—
563C23—611no—
564C23—611no—
565C23—1222no—
566C23—1221no—
567C23—1221no—
568C23—2441no—
569C23C2×D4×D6111yescharacteristic
570Q8—42no—
571Q8—2442no—
572Q8—2442no—
573D4—42no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—2444no—
579D4—2444no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—611no—
588C2×C4—611no—
589C2×C4—611no—
590C2×C4—611no—
591C2×C4—611no—
592C2×C4—611no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2444no—
611C2×C4—2441no—
612C2×C4—2444no—
613C2×C4—2442no—
614C2×C4—2442no—
615C2×C4—2442no—
616C2×C4C22⋊C4×D6111yescharacteristic
617C2×C4C22⋊C4×D6111yescharacteristic
618C2×C4C22⋊C4×D6111yescharacteristic
619C2×C4C22⋊C4×D6111yescharacteristic
620C2×C4#1451979612yes—
621C2×C4#1451979812yes—
622C2×C4C22×C4×D6111yescharacteristic
623C2×C4C2×D4×D6111yescharacteristic
624C8—42no—
625C8—2442no—
626C22—21no—
627C22—21no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—612no—
632C22—1221no—
633C22—1221no—
634C22—1221no—
635C22—1221no—
636C22—1221no—
637C22—2442no—
638C22—2442no—
639C22C2×C22⋊C4×D6111yescharacteristic
640C22C2×#1451979611yescharacteristic
641C22C2×#1451979811yescharacteristic
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—42no—
647C4—82no—
648C4—612no—
649C4—1221no—
650C4—1221no—
651C4—1221no—
652C4—1221no—
653C4—1221no—
654C4—4882no—
655C4—2442no—
656C4—2442no—
657C4#1683975512yes—
658C4#1683975512yes—
659C2—21no—
660C2—21no—
661C2—82no—
662C2—1222no—
663C2—2441no—
664C2#410×D6111yescharacteristic
665C2C2×#1683975511yescharacteristic
666C2#421×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1517C2
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC22×C4×C61C24
hypercenter#1517D61
hypocenterC61#17573
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1517

Complements

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

NH ≅ G/Ntype
2. #431×D61C2semidirect
3. #21448212C2semidirect
4. #21444393C2semidirect
5. #21444384C2semidirect
6. #21444382C2semidirect
7. #21444379C2semidirect
8. #21444307C2semidirect
9. C61×#1517C2semidirect
11. #16840317C4semidirect
12. #16840317C4semidirect
13. #16840310C4semidirect
14. #16840310C4semidirect
16. #16840294C4semidirect
26. C4×#11758224C4semidirect
333. #1517D61semidirect
382. C61#17573semidirect

Maximal subgroups

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

subgroupindex
#214591142
#431×D612
#214482122
#214443932
#214443842
#214443822
#214443792
#214443072
C61×#15172
#1757361