Finite Group Databasebeta

← #29646201

Subgroups — #29646201

Order 15616.

1046 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840436C211yesmaximal, characteristic
2#21454472C212yesmaximal, complemented
3#21444187C212yesmaximal, complemented
4#21444170C212yesmaximal, complemented
5#21444166C212yesmaximal, complemented
6#21444158C211yesmaximal, characteristic, complemented
7#21444153C211yesmaximal, characteristic, complemented
8#21444144C211yesmaximal, characteristic, complemented
9#21444138C211yesmaximal, characteristic, complemented
10#323×D61C211yesmaximal, characteristic, complemented
11C61×#1458C211yesmaximal, characteristic, complemented
12#16840436—22no—
13#16840436—22no—
14#16840436—22no—
15#16840436—22no—
16C2×#14519772C2211yescharacteristic
17C2×#14519771C411yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×#14519762C411yescharacteristic
20C2×#14519762C2211yescharacteristic
21C2×#14519760C411yescharacteristic
22C2×#14519755C411yescharacteristic
23C2×#14519754C411yescharacteristic
24C2×#14519747C411yescharacteristic
25C2×C4×D244C411yescharacteristic
26C2×#14519737C2211yescharacteristic
27C2×#14519734C2211yescharacteristic
28C22×#11758198C411yescharacteristic
29C2×C22⋊C4×D61C2211yescharacteristic
30#16839822—22no—
31#16839821—22no—
32#16839819C2212yescomplemented
33#16839816C2212yescomplemented
34#16839812C2212yescomplemented
35#16839787—22no—
36#16839724C2212yes—
37C22⋊C8×D61C2212yes—
38#16839716—22no—
39#16839488C2212yescomplemented
40#16839486C2211yescharacteristic
41#16839486C2211yescharacteristic
42#16839485C2212yes—
43#16839485C2212yes—
44#16839485C2212yescomplemented
45#16839485C2212yescomplemented
46C2×#117×C61C2211yescharacteristic
47C61×#431C2212yescomplemented
48C61×#324C2212yescomplemented
49C61×#323C2211yescharacteristic
50C61×#323C2211yescharacteristic
51C24×D61C2×C411yescharacteristic
52C22×Dic122C2×C411yescharacteristic
53C22×D244C2×C411yescharacteristic
54C22×D244C2×C411yescharacteristic
55C22×C22⋊D61C2×C411yescharacteristic
56C23×Dic61C2×C411yescharacteristic
57C22×C4×D61C2×C411yescharacteristic
58C22×C4×D61C2×C411yescharacteristic
59C22×C4×D61C2311yescharacteristic
60C2×#11758189—22no—
61C2×C8×D61—22no—
62#14519772—22no—
63#14519772—22no—
64#14519771—22no—
65#14519771—22no—
66#14519765—22no—
67#14519765—22no—
68#14519762—22no—
69#14519762—22no—
70#14519762—22no—
71#14519762—22no—
72#14519760—22no—
73#14519760—22no—
74#14519755—22no—
75#14519755—22no—
76#14519754—22no—
77#14519754—22no—
78#14519747—22no—
79#14519747—22no—
80C4×D244—21no—
81C4×D244—21no—
82C4×D244—21no—
83C4×D244—21no—
84#14519737—22no—
85#14519737—22no—
86#14519734D412yes—
87#14519734D412yes—
88#14519734D412yes—
89#14519734D412yes—
90C2×C4⋊Dic61C2×C411yescharacteristic
91C2×#11758196C2×C411yescharacteristic
92C2×#11758196C2×C411yescharacteristic
93C2×#11758196C2×C411yescharacteristic
94C2×#11758196C2×C411yescharacteristic
95C2×#11758196C2×C411yescharacteristic
96C2×C22⋊Dic61C2×C411yescharacteristic
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×#11758198C2×C411yescharacteristic
102C2×#11758198C2×C411yescharacteristic
103C2×#11758198C2×C411yescharacteristic
104C2×#11758198C2×C411yescharacteristic
105C2×#11758198C2×C411yescharacteristic
106C2×#11758198C2×C411yescharacteristic
107C2×#11758198C2×C411yescharacteristic
108C2×#11758198C2×C411yescharacteristic
109C2×#11758198C2×C411yescharacteristic
110C2×#11758198C2311yescharacteristic
111C2×#11758198C2311yescharacteristic
112C22⋊C4×D61—22no—
113C22⋊C4×D61—22no—
114C2×C4×Dic61C2311yescharacteristic
115C4⋊#92418—22no—
116#14519706—22no—
117#14519706—22no—
118C22⋊#92418C2312yes—
119C22⋊#92418C2312yes—
120#14519689—22no—
121#14519689—22no—
122#14519688—22no—
123#14519687—22no—
124C8×Dic61—22no—
125C23×C4×C61C2×C411yescharacteristic
126#117×C61—22no—
127#117×C61—22no—
128C2×C4⋊C4×C61C2×C411yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C22⋊C4×C61C2×C411yescharacteristic
131C2×C22⋊C4×C61C2311yescharacteristic
132C2×C42×C61C2×C411yescharacteristic
133C2×C42×C61C2311yescharacteristic
134C4⋊C8×C61—22no—
135C22⋊C8×C61C2312yes—
136C22⋊C8×C61C2312yes—
137C4×C8×C61—22no—
138C24×C61C22×C411yescharacteristic
139C23×D61—22no—
140C23×D61—21no—
141C23×D61—21no—
142C23×D61—41no—
143C23×D61—42no—
144C23×D61—81no—
145C23×D61C22×C411yescharacteristic
146C23×D61C22×C411yescharacteristic
147C23×D61C22×C411yescharacteristic
148C22×C4×C61—22no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—41no—
152C22×C4×C61C22×C411yescharacteristic
153C22×C4×C61C22×C411yescharacteristic
154C22×C4×C61C22×C411yescharacteristic
155C22×C4×C61C22×C411yescharacteristic
156C22×C4×C61C22×C411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C2×C22⋊D61—22no—
159C2×C22⋊D61—22no—
160C2×C22⋊D61—41no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—81no—
164C22×Dic61C22×C411yescharacteristic
165C22×Dic61C22×C411yescharacteristic
166C22×Dic61C22×C411yescharacteristic
167C22×Dic61C22×C411yescharacteristic
168C22×Dic61C22×C411yescharacteristic
169C2×D244—21no—
170C2×D244—21no—
171C2×D244—22no—
172C2×D244—42no—
173C2×D244—81no—
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—21no—
177C2×C4×D61—21no—
178C2×C4×D61—42no—
179C2×C4×D61—81no—
180C2×C4×D61—81no—
181C2×C4×D61C2×D412yes—
182C2×C4×D61C2×D412yes—
183C2×Dic122—22no—
184C2×Dic122—81no—
185C2×C8×C61—22no—
186C2×C8×C61—22no—
187C4⋊C4×C61—22no—
188C4⋊C4×C61—22no—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—22no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61C2×D412yes—
194C42×C61C2×D412yes—
195C22⋊Dic61—22no—
196#11758198—21no—
197#11758198—21no—
198#11758198—21no—
199#11758198—21no—
200#11758198—21no—
201#11758198—21no—
202#11758198—22no—
203#11758198—22no—
204#11758198—22no—
205#11758198—21no—
206#11758198—21no—
207#11758198—22no—
208#11758198—22no—
209#11758198C22⋊C411yescharacteristic
210#11758198C22⋊C411yescharacteristic
211#11758198C22⋊C411yescharacteristic
212#11758198C22⋊C411yescharacteristic
213#11758198C22⋊C412yes—
214#11758198C22⋊C412yes—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217#11758196—22no—
218#11758196—22no—
219#11758196—22no—
220#11758196C22⋊C412yes—
221#11758196C22⋊C412yes—
222#11758196C22⋊C412yes—
223#11758196C22⋊C412yes—
224C4×Dic61C2×D412yes—
225C4×Dic61C2×D412yes—
226C2×#92418—22no—
227C2×#92418—22no—
228#11758189—44no—
229C8×D61—44no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61#13112yes—
235C23×C61C23×C411yescharacteristic
236C22×D61—21no—
237C22×D61—21no—
238C22×D61—22no—
239C22×D61—21no—
240C22×D61—21no—
241C22×D61—21no—
242C22×D61—21no—
243C22×D61—21no—
244C22×D61—21no—
245C22×D61—41no—
246C22×D61—42no—
247C22×D61—42no—
248C22×D61—42no—
249C22×D61—42no—
250C22×D61—41no—
251C22×D61—41no—
252C22×D61—41no—
253C22×D61—41no—
254C22×D61—81no—
255C22×D61—81no—
256C22×D61—84no—
257C22×D61—84no—
258C22×D61C2×C22⋊C411yescharacteristic
259C22×D61C2×C22⋊C411yescharacteristic
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—21no—
265C2×C4×C61—21no—
266C2×C4×C61—21no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—42no—
270C2×C4×C61—42no—
271C2×C4×C61—41no—
272C2×C4×C61—41no—
273C2×C4×C61C2×C22⋊C411yescharacteristic
274C2×C4×C61C2×C22⋊C411yescharacteristic
275C2×C4×C61C2×C22⋊C412yes—
276C2×C4×C61C22×D412yes—
277C22⋊D61—44no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—81no—
287C2×Dic61—81no—
288C2×Dic61C2×C22⋊C412yes—
289C2×Dic61C2×C22⋊C411yescharacteristic
290C2×Dic61C2×C22⋊C411yescharacteristic
291C2×Dic61C2×C22⋊C412yes—
292D244—42no—
293D244—42no—
294D244—82no—
295C4×D61—24no—
296C4×D61—42no—
297C4×D61—81no—
298C4×D61—81no—
299C4×D61—81no—
300C4×D61—81no—
301Dic122—82no—
302C8×C61—42no—
303C8×C61—42no—
304#92418—42no—
305#92418—42no—
306#17412—611nomaximal, Sylow
307C22×C61—21no—
308C22×C61—21no—
309C22×C61—22no—
310C22×C61—42no—
311C22×C61—42no—
312C22×C61—41no—
313C22×C61—41no—
314C22×C61C22×C22⋊C411yescharacteristic
315C22×C61C2×#13111yescharacteristic
316C22×C61#56811yescharacteristic
317C2×D61—21no—
318C2×D61—21no—
319C2×D61—21no—
320C2×D61—21no—
321C2×D61—21no—
322C2×D61—42no—
323C2×D61—42no—
324C2×D61—41no—
325C2×D61—41no—
326C2×D61—41no—
327C2×D61—41no—
328C2×D61—41no—
329C2×D61—41no—
330C2×D61—84no—
331C2×D61—84no—
332C2×D61—81no—
333C2×D61—81no—
334C2×D61—81no—
335C2×D61—81no—
336C2×D61#41012yes—
337C4×C61—21no—
338C4×C61—21no—
339C4×C61—22no—
340C4×C61—22no—
341C4×C61—41no—
342C4×C61—41no—
343C4×C61—41no—
344Dic61—22no—
345Dic61—22no—
346Dic61—21no—
347Dic61—21no—
348Dic61—41no—
349Dic61—41no—
350C2×#551—611no—
351#2924—612no—
352#1473—612no—
353#1469—612no—
354#1468—612no—
355#1466—611no—
356#1464—611no—
357#1462—611no—
358#1460—611no—
359#1458D6111yescharacteristic, complemented
360C2×#323—611no—
361C2×C61—21no—
362C2×C61—21no—
363C2×C61—42no—
364C2×C61#288011yescharacteristic
365C2×C61C2×#41011yescharacteristic
366C2×C61#288811yescharacteristic
367D61—22no—
368D61—41no—
369D61—81no—
370D61—81no—
371#551—1222no—
372#551—1222no—
373#551—1222no—
374#551—1222no—
375C2×#126—611no—
376C2×#126—611no—
377C2×#124—611no—
378C2×#123—611no—
379C2×#123—611no—
380C2×#123—611no—
381C2×C22⋊Q8—611no—
382C2×C22⋊Q8—611no—
383C2×#120—611no—
384C2×C4×D4—611no—
385C2×C4×D4—611no—
386C2×#117—611no—
387C2×#117C2×D6111yescharacteristic
388C22×C22⋊C4—611no—
389C22×C22⋊C4—611no—
390#433—1222no—
391#433—1222no—
392#432—612no—
393#432—612no—
394#431—612no—
395#431C2×D6112yescomplemented
396#424—1222no—
397#407—612no—
398C2×C22⋊C8—612no—
399#405—1222no—
400#324—612no—
401#324C2×D6112yescomplemented
402#323—611no—
403#323—611no—
404#323—612no—
405#323—612no—
406#323—612no—
407#323—612no—
408#323C2×D6111yescharacteristic
409#323C2×D6111yescharacteristic
410C61#1741211yescharacteristic, complemented, Sylow
411C25—611no—
412C22×Q8—611no—
413C22×D4—611no—
414C22×D4—611no—
415C22×D4—611no—
416C23×C4—611no—
417C23×C4—611no—
418C23×C4—611no—
419C23×C4—611no—
420C23×C4C4×D6111yescharacteristic
421C2×M16—1222no—
422C22×C8—1222no—
423#126—1222no—
424#126—1222no—
425#126—1222no—
426#126—1222no—
427#124—1222no—
428#124—1222no—
429#123—1222no—
430#123—1222no—
431#123—1222no—
432#123—1222no—
433#123—1222no—
434#123—1222no—
435C22⋊Q8—1222no—
436C22⋊Q8—1222no—
437C22⋊Q8—1222no—
438C22⋊Q8—1222no—
439#120—1222no—
440#120—1222no—
441C4×D4—1221no—
442C4×D4—1221no—
443C4×D4—1221no—
444C4×D4—1221no—
445C4×D4—1222no—
446C4×D4—1222no—
447#117—22no—
448#117—22no—
449#117—612no—
450#117—612no—
451#117—612no—
452#117—612no—
453C2×C4⋊C4—611no—
454C2×C4⋊C4—611no—
455C2×C4⋊C4—611no—
456C2×C4⋊C4—611no—
457C2×C4⋊C4—611no—
458C2×C4⋊C4—611no—
459C2×C4⋊C4C4×D6111yescharacteristic
460C2×C4⋊C4C22×D6111yescharacteristic
461C2×C22⋊C4—611no—
462C2×C22⋊C4—611no—
463C2×C22⋊C4—611no—
464C2×C22⋊C4—611no—
465C2×C22⋊C4—611no—
466C2×C22⋊C4—611no—
467C2×C22⋊C4—611no—
468C2×C22⋊C4—611no—
469C2×C22⋊C4—611no—
470C2×C22⋊C4—611no—
471C2×C22⋊C4—611no—
472C2×C22⋊C4—611no—
473C2×C22⋊C4—1222no—
474C2×C22⋊C4—1222no—
475C2×C22⋊C4—1221no—
476C2×C22⋊C4—1221no—
477C2×C22⋊C4—1221no—
478C2×C22⋊C4—1221no—
479C2×C22⋊C4C4×D6111yescharacteristic
480C2×C22⋊C4C22×D6111yescharacteristic
481C2×C42—611no—
482C2×C42C4×D6111yescharacteristic
483C2×C42C22×D6111yescharacteristic
484C4⋊C8—22no—
485C4⋊C8—1222no—
486C4⋊C8—1222no—
487C4⋊C8—1222no—
488C22⋊C8—612no—
489C22⋊C8—612no—
490C22⋊C8—1222no—
491C22⋊C8—1222no—
492C22⋊C8C22×D6112yes—
493C22⋊C8C22×D6112yes—
494#97—1222no—
495#97—1222no—
496C4×C8—22no—
497C4×C8—1222no—
498C24—611no—
499C24—611no—
500C24—611no—
501C24—1221no—
502C24—1221no—
503C24—1222no—
504C24—4881no—
505C24—2441no—
506C24—2442no—
507C24C2×C4×D6111yescharacteristic
508C2×Q8—1222no—
509C2×Q8—4881no—
510C2×D4—1221no—
511C2×D4—1221no—
512C2×D4—1222no—
513C2×D4—1222no—
514C2×D4—1222no—
515C2×D4—4881no—
516C2×D4—2442no—
517C2×D4—2441no—
518C22×C4—22no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—41no—
522C22×C4—611no—
523C22×C4—611no—
524C22×C4—611no—
525C22×C4—612no—
526C22×C4—612no—
527C22×C4—611no—
528C22×C4—611no—
529C22×C4—1221no—
530C22×C4—1221no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—1221no—
534C22×C4—1221no—
535C22×C4—4881no—
536C22×C4—4881no—
537C22×C4—4881no—
538C22×C4—2442no—
539C22×C4C2×C4×D6111yescharacteristic
540C22×C4C2×C4×D6111yescharacteristic
541C22×C4C2×C4×D6111yescharacteristic
542C22×C4C2×C4×D6111yescharacteristic
543C22×C4C2×C4×D6111yescharacteristic
544C22×C4C23×D6111yescharacteristic
545M16—2444no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—1222no—
549C2×C8—1222no—
550C2×C8—2444no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—612no—
554C4⋊C4—612no—
555C4⋊C4—612no—
556C4⋊C4—612no—
557C4⋊C4—1222no—
558C4⋊C4—1222no—
559C4⋊C4—1222no—
560C4⋊C4—1221no—
561C4⋊C4—1221no—
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—611no—
565C22⋊C4—611no—
566C22⋊C4—611no—
567C22⋊C4—611no—
568C22⋊C4—612no—
569C22⋊C4—612no—
570C22⋊C4—1222no—
571C22⋊C4—1221no—
572C22⋊C4—1221no—
573C22⋊C4—1222no—
574C22⋊C4—1222no—
575C22⋊C4—1221no—
576C22⋊C4—1221no—
577C22⋊C4—1222no—
578C22⋊C4—1222no—
579C22⋊C4—1222no—
580C22⋊C4—1221no—
581C22⋊C4—1221no—
582C22⋊C4—1221no—
583C22⋊C4—1221no—
584C42—21no—
585C42—21no—
586C42—612no—
587C42—612no—
588C42D4×D6112yes—
589C42D4×D6112yes—
590C23—21no—
591C23—21no—
592C23—41no—
593C23—41no—
594C23—611no—
595C23—611no—
596C23—1222no—
597C23—1221no—
598C23—1221no—
599C23—1221no—
600C23—1221no—
601C23—1221no—
602C23—1221no—
603C23—1221no—
604C23—1221no—
605C23—4881no—
606C23—4881no—
607C23—4884no—
608C23—4884no—
609C23—2441no—
610C23—2442no—
611C23—2442no—
612C23—2441no—
613C23—2441no—
614C23—2441no—
615C23—2441no—
616C23—2442no—
617C23—2442no—
618C23#1451978312yes—
619C23C22×C4×D6111yescharacteristic
620Q8—4882no—
621D4—4882no—
622D4—2442no—
623D4—2442no—
624D4—2444no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—612no—
639C2×C4—611no—
640C2×C4—611no—
641C2×C4—612no—
642C2×C4—1224no—
643C2×C4—1221no—
644C2×C4—1221no—
645C2×C4—1221no—
646C2×C4—1221no—
647C2×C4—4881no—
648C2×C4—4881no—
649C2×C4—4881no—
650C2×C4—4881no—
651C2×C4—4881no—
652C2×C4—4881no—
653C2×C4—2442no—
654C2×C4—2441no—
655C2×C4—2441no—
656C2×C4—2441no—
657C2×C4—2441no—
658C2×C4C22⋊C4×D6111yescharacteristic
659C2×C4C22⋊C4×D6111yescharacteristic
660C2×C4C22⋊C4×D6112yes—
661C2×C4C2×D4×D6112yes—
662C8—42no—
663C8—42no—
664C8—2442no—
665C8—2442no—
666C22—21no—
667C22—21no—
668C22—22no—
669C22—42no—
670C22—42no—
671C22—41no—
672C22—41no—
673C22—612no—
674C22—1221no—
675C22—1221no—
676C22—1221no—
677C22—1221no—
678C22—1221no—
679C22—4881no—
680C22—4881no—
681C22—4881no—
682C22—4881no—
683C22—4884no—
684C22—4884no—
685C22—2442no—
686C22—2442no—
687C22—2441no—
688C22—2441no—
689C22—2441no—
690C22—2441no—
691C22—2441no—
692C22—2441no—
693C22C2×C22⋊C4×D6111yescharacteristic
694C22C2×#1451978311yescharacteristic
695C22#1684052811yescharacteristic
696C4—21no—
697C4—21no—
698C4—22no—
699C4—22no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—1222no—
704C4—1222no—
705C4—1221no—
706C4—1221no—
707C4—2441no—
708C4—2441no—
709C2—21no—
710C2—21no—
711C2—42no—
712C2—1222no—
713C2—4881no—
714C2—4881no—
715C2—2441no—
716C2#2145426511yescharacteristic
717C2#410×D6111yescharacteristic
718C2#2145429611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1458C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519747C4
cosocleC22×C4×C61C24
hypercenter#1458D61
hypocenterC61#17412
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #21454472C2semidirect
3. #21444187C2semidirect
4. #21444170C2semidirect
5. #21444166C2semidirect
6. #21444158C2semidirect
7. #21444153C2semidirect
8. #21444144C2semidirect
9. #21444138C2semidirect
10. #323×D61C2semidirect
11. C61×#1458C2semidirect
32. #16839819C22semidirect
33. #16839816C22semidirect
34. #16839812C22semidirect
39. #16839488C22semidirect
44. #16839485C22semidirect
45. #16839485C22semidirect
47. C61×#431C22semidirect
48. C61×#324C22semidirect
359. #1458D61semidirect
395. #431C2×D61semidirect
401. #324C2×D61semidirect
410. C61#17412semidirect

Maximal subgroups

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

subgroupindex
C2×#168404362
#214544722
#214441872
#214441702
#214441662
#214441582
#214441532
#214441442
#214441382
#323×D612
C61×#14582
#1741261