Finite Group Databasebeta

← #29646038

Subgroups — #29646038

Order 15616.

1080 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840351C211yesmaximal, characteristic
2#21454847C211yesmaximal, characteristic, complemented
3#21454842C211yesmaximal, characteristic, complemented
4#21444188C211yesmaximal, characteristic, complemented
5#21444177C212yesmaximal, complemented
6#21444172C211yesmaximal, characteristic, complemented
7#21444170C211yesmaximal, characteristic, complemented
8#21444166C212yesmaximal, complemented
9#21444136C212yesmaximal, complemented
10#323×D61C212yesmaximal, complemented
11C61×#1473C211yesmaximal, characteristic, complemented
12#16840351—22no—
13#16840351—22no—
14#16840351—22no—
15#16840351—22no—
16C2×#14519772C411yescharacteristic
17C2×#14519765C2211yescharacteristic
18C2×#14519763C411yescharacteristic
19C2×#14519762C2211yescharacteristic
20C2×#14519759C411yescharacteristic
21C2×#14519755C411yescharacteristic
22C2×#14519752C411yescharacteristic
23C2×#14519752C411yescharacteristic
24C2×#14519750C2211yescharacteristic
25C2×#14519747C411yescharacteristic
26C2×#14519747C411yescharacteristic
27C2×C4×D244C2211yescharacteristic
28C2×#14519739C2211yescharacteristic
29C2×C22⋊C4×D61C2211yescharacteristic
30#16839835—22no—
31#16839832—22no—
32C8×D244—22no—
33#16839827—22no—
34#16839819C2211yescharacteristic, complemented
35#16839815C2211yescharacteristic, complemented
36#16839724C2212yescomplemented
37C22⋊C8×D61C2212yes—
38C22⋊C8×D61C2212yes—
39C22⋊C8×D61C2212yescomplemented
40#16839488C2212yescomplemented
41#16839486C2212yes—
42#16839485C2212yes—
43#16839485C2212yes—
44#16839485C2212yescomplemented
45#16839485C2212yescomplemented
46C2×#126×C61C2211yescharacteristic
47C61×#432C2211yescharacteristic, complemented
48C61×#431C2211yescharacteristic, complemented
49C61×#324C2212yescomplemented
50C61×#323C2212yes—
51C24×D61C2×C411yescharacteristic
52C22×Dic122C2×C411yescharacteristic
53C22×D244C2×C411yescharacteristic
54C22×D244C2×C411yescharacteristic
55C22×D244C2×C411yescharacteristic
56C22×D244C2×C411yescharacteristic
57C22×D244C2×C411yescharacteristic
58C22×C22⋊D61C2×C411yescharacteristic
59C22×C22⋊D61C2×C411yescharacteristic
60C22×C4×D61C2×C411yescharacteristic
61C22×C4×D61C2×C411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C22×C4×D61C2311yescharacteristic
64C2×#11758189—22no—
65C2×C8×D61—22no—
66C2×C8×D61—22no—
67C2×C8×D61—22no—
68#14519772—22no—
69#14519772—22no—
70#14519765—21no—
71#14519765—21no—
72#14519765—21no—
73#14519765—21no—
74#14519763—22no—
75#14519763—22no—
76#14519762—22no—
77#14519762—22no—
78#14519759—22no—
79#14519759—22no—
80#14519755—22no—
81#14519755—22no—
82#14519752—22no—
83#14519752—22no—
84#14519752—22no—
85#14519752—22no—
86#14519750—22no—
87#14519750—22no—
88#14519747—22no—
89#14519747—22no—
90#14519747—22no—
91#14519747—22no—
92C4×D244D412yes—
93C4×D244D412yes—
94C4×D244D412yes—
95C4×D244D412yes—
96#14519739—22no—
97#14519739—22no—
98C2×C4⋊Dic61C2×C411yescharacteristic
99C2×#11758196C2×C411yescharacteristic
100C2×#11758196C2×C411yescharacteristic
101C2×C22⋊Dic61C2×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×Dic61C2×C411yescharacteristic
115C4⋊#92418—21no—
116C4⋊#92418—21no—
117C22⋊#92418C2312yes—
118C22⋊#92418C2312yes—
119#14519689—22no—
120#14519689—22no—
121#14519689—22no—
122#14519689—22no—
123#14519688—21no—
124C4×#92418—21no—
125#126×C61—21no—
126#126×C61—21no—
127#126×C61—21no—
128#126×C61—21no—
129C2×C4⋊C4×C61C2×C411yescharacteristic
130C2×C4⋊C4×C61C2×C411yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C22⋊C4×C61C2×C411yescharacteristic
133C2×C22⋊C4×C61C2×C411yescharacteristic
134C2×C22⋊C4×C61C2311yescharacteristic
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61—21no—
138C22⋊C8×C61C2312yes—
139C22⋊C8×C61C2312yes—
140#97×C61—21no—
141C4×C8×C61—21no—
142C24×C61C22×C411yescharacteristic
143C23×D61—22no—
144C23×D61—24no—
145C23×D61—81no—
146C23×D61—81no—
147C23×D61C22×C411yescharacteristic
148C23×D61C22×C411yescharacteristic
149C23×D61C22×C411yescharacteristic
150C23×D61C22×C411yescharacteristic
151C22×C4×C61C22×C411yescharacteristic
152C22×C4×C61C22×C411yescharacteristic
153C22×C4×C61C22×C411yescharacteristic
154C22×C4×C61C22×C411yescharacteristic
155C22×C4×C61C22×C411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C2×C22⋊D61—22no—
158C2×C22⋊D61—22no—
159C2×C22⋊D61—81no—
160C2×C22⋊D61—81no—
161C22×Dic61C22×C411yescharacteristic
162C22×Dic61C22×C411yescharacteristic
163C22×Dic61C22×C411yescharacteristic
164C22×Dic61C22×C411yescharacteristic
165C2×D244—24no—
166C2×D244—22no—
167C2×D244—24no—
168C2×D244—22no—
169C2×D244—22no—
170C2×D244—21no—
171C2×D244—21no—
172C2×D244—81no—
173C2×D244—81no—
174C2×D244—81no—
175C2×D244C22⋊C412yes—
176C2×D244C22⋊C412yes—
177C2×C4×D61—22no—
178C2×C4×D61—22no—
179C2×C4×D61—81no—
180C2×C4×D61—81no—
181C2×C4×D61—81no—
182C2×C4×D61—81no—
183C2×C4×D61C2×D412yes—
184C2×C4×D61C2×D412yes—
185C2×C4×D61C2×D412yes—
186C2×C4×D61C2×D412yes—
187C2×Dic122—21no—
188C2×Dic122—21no—
189C2×Dic122—81no—
190C2×C8×C61—22no—
191C2×C8×C61—22no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C22⋊C4×C61—21no—
199C22⋊C4×C61—21no—
200C22⋊C4×C61—21no—
201C22⋊C4×C61—21no—
202C22⋊C4×C61—21no—
203C22⋊C4×C61—21no—
204C42×C61C2×D411yescharacteristic
205C42×C61C2×D411yescharacteristic
206C42×C61C2×D411yescharacteristic
207C42×C61C2×D411yescharacteristic
208C22⋊Dic61—22no—
209#11758198—22no—
210#11758198—22no—
211#11758198—21no—
212#11758198—21no—
213#11758198—21no—
214#11758198—21no—
215#11758198—21no—
216#11758198—21no—
217#11758198—21no—
218#11758198—21no—
219#11758198—22no—
220#11758198—22no—
221#11758198C22⋊C412yes—
222#11758198C22⋊C412yes—
223#11758198C22⋊C412yes—
224#11758198C22⋊C412yes—
225C4⋊Dic61C22⋊C412yes—
226C4⋊Dic61C22⋊C412yes—
227#11758196—22no—
228#11758196—22no—
229C4×Dic61—22no—
230C2×#92418—22no—
231C2×#92418—22no—
232#11758189—82no—
233C8×D61—82no—
234C8×D61—82no—
235C8×D61—82no—
236C23×C61—21no—
237C23×C61—22no—
238C23×C61—41no—
239C23×C61—41no—
240C23×C61C23×C411yescharacteristic
241C22×D61—22no—
242C22×D61—21no—
243C22×D61—21no—
244C22×D61—22no—
245C22×D61—22no—
246C22×D61—41no—
247C22×D61—41no—
248C22×D61—41no—
249C22×D61—41no—
250C22×D61—44no—
251C22×D61—41no—
252C22×D61—41no—
253C22×D61—41no—
254C22×D61—41no—
255C22×D61—82no—
256C22×D61—82no—
257C22×D61—88no—
258C22×D61C2×C22⋊C412yes—
259C22×D61C2×C22⋊C412yes—
260C22×D61#13114yes—
261C2×C4×C61—22no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—21no—
265C2×C4×C61—22no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61C2×C22⋊C411yescharacteristic
271C2×C4×C61C2×C22⋊C411yescharacteristic
272C2×C4×C61C2×C22⋊C411yescharacteristic
273C2×C4×C61C2×C22⋊C411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C22×D411yescharacteristic
276C22⋊D61—82no—
277C22⋊D61—82no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61C2×C22⋊C412yes—
285C2×Dic61C2×C22⋊C412yes—
286D244—24no—
287D244—24no—
288D244—44no—
289D244—81no—
290D244—81no—
291D244—82no—
292D244—82no—
293C4×D61—42no—
294C4×D61—42no—
295C4×D61—42no—
296C4×D61—42no—
297C4×D61—82no—
298C4×D61—82no—
299Dic122—81no—
300Dic122—81no—
301C8×C61—42no—
302C8×C61—42no—
303#92418—42no—
304#92418—42no—
305#17386—611nomaximal, Sylow
306C22×C61—21no—
307C22×C61—21no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61—82no—
312C22×C61C22×C22⋊C411yescharacteristic
313C22×C61C2×#13112yes—
314C2×D61—22no—
315C2×D61—22no—
316C2×D61—22no—
317C2×D61—22no—
318C2×D61—22no—
319C2×D61—44no—
320C2×D61—41no—
321C2×D61—41no—
322C2×D61—41no—
323C2×D61—41no—
324C2×D61—41no—
325C2×D61—41no—
326C2×D61—88no—
327C2×D61—82no—
328C2×D61—82no—
329C2×D61—82no—
330C2×D61#40814yes—
331C4×C61—21no—
332C4×C61—21no—
333C4×C61—21no—
334C4×C61—21no—
335C4×C61—21no—
336C4×C61—21no—
337C4×C61—41no—
338C4×C61—41no—
339C4×C61—41no—
340Dic61—22no—
341Dic61—22no—
342Dic61—41no—
343Dic61—41no—
344C2×#538—611no—
345#2973—611no—
346#2972—611no—
347#1473—611no—
348#1473D6111yescharacteristic, complemented
349#1471—612no—
350#1469—611no—
351#1469—611no—
352#1468—612no—
353#1459—612no—
354C2×#323—612no—
355C2×C61—22no—
356C2×C61—81no—
357C2×C61C2×#40811yescharacteristic
358C2×C61#288311yescharacteristic
359C2×C61#288811yescharacteristic
360D61—24no—
361D61—42no—
362D61—81no—
363D61—81no—
364#538—1222no—
365#538—1222no—
366#538—1222no—
367#538—1222no—
368C2×#126—611no—
369C2×#126C2×D6111yescharacteristic
370C2×#124—611no—
371C2×#124—611no—
372C2×#123—611no—
373C2×#123—611no—
374C2×C22⋊Q8—611no—
375C2×#121—611no—
376C2×#121—611no—
377C2×#121—611no—
378C2×#120—611no—
379C2×#120—611no—
380C2×C4×D4—611no—
381C2×C4×D4—611no—
382C22×C22⋊C4—611no—
383#435—1222no—
384#435—1222no—
385C8×D4—1222no—
386C8×D4—1222no—
387#432—611no—
388#432C2×D6111yescharacteristic, complemented
389#431—611no—
390#431C2×D6111yescharacteristic, complemented
391#407—612no—
392C2×C22⋊C8—612no—
393C2×C22⋊C8—612no—
394C2×C22⋊C8—612no—
395#324—612no—
396#324C2×D6112yescomplemented
397#323—612no—
398#323—612no—
399#323—612no—
400#323—612no—
401#323—612no—
402#323C2×D6112yes—
403C61#1738611yescharacteristic, complemented, Sylow
404C25—611no—
405C22×Q8—611no—
406C22×D4—611no—
407C22×D4—611no—
408C22×D4—611no—
409C22×D4—611no—
410C22×D4—611no—
411C22×D4—611no—
412C22×D4—611no—
413C23×C4—611no—
414C23×C4—611no—
415C23×C4—611no—
416C23×C4—611no—
417C2×M16—1222no—
418C22×C8—1222no—
419C22×C8—1222no—
420C22×C8—1222no—
421#126—21no—
422#126—21no—
423#126—21no—
424#126—21no—
425#126—1222no—
426#126—1222no—
427#124—1222no—
428#124—1222no—
429#124—1221no—
430#124—1221no—
431#124—1221no—
432#124—1221no—
433#123—1222no—
434#123—1222no—
435#123—1222no—
436#123—1222no—
437C22⋊Q8—1222no—
438C22⋊Q8—1222no—
439#121—1222no—
440#121—1222no—
441#121—1222no—
442#121—1222no—
443#121—1222no—
444#121—1222no—
445#120—1222no—
446#120—1222no—
447#120—1222no—
448#120—1222no—
449C4×D4—612no—
450C4×D4—612no—
451C4×D4—612no—
452C4×D4—612no—
453C4×D4—1222no—
454C4×D4—1222no—
455C2×C4⋊C4—611no—
456C2×C4⋊C4—611no—
457C2×C4⋊C4—611no—
458C2×C4⋊C4C4×D6111yescharacteristic
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—1222no—
473C2×C22⋊C4—1222no—
474C2×C22⋊C4C4×D6111yescharacteristic
475C2×C22⋊C4C4×D6111yescharacteristic
476C2×C22⋊C4C22×D6111yescharacteristic
477C2×C42—611no—
478C2×C42C22×D6111yescharacteristic
479C4⋊C8—21no—
480C4⋊C8—21no—
481C4⋊C8—1221no—
482C4⋊C8—1221no—
483C22⋊C8—612no—
484C22⋊C8—612no—
485C22⋊C8—1222no—
486C22⋊C8—1222no—
487C22⋊C8—1222no—
488C22⋊C8—1222no—
489C22⋊C8C22×D6112yes—
490C22⋊C8C22×D6112yes—
491#97—21no—
492#97—1221no—
493C4×C8—21no—
494C4×C8—1221no—
495C24—611no—
496C24—611no—
497C24—611no—
498C24—611no—
499C24—1222no—
500C24—1224no—
501C24—4881no—
502C24—4881no—
503C24C2×C4×D6111yescharacteristic
504C2×Q8—1221no—
505C2×Q8—1221no—
506C2×Q8—4881no—
507C2×D4—612no—
508C2×D4—612no—
509C2×D4—1224no—
510C2×D4—1222no—
511C2×D4—1222no—
512C2×D4—1222no—
513C2×D4—1224no—
514C2×D4—1222no—
515C2×D4—1222no—
516C2×D4—1221no—
517C2×D4—1221no—
518C2×D4—4881no—
519C2×D4—4881no—
520C2×D4—4881no—
521C2×D4—4881no—
522C2×D4—4881no—
523C22×C4—611no—
524C22×C4—611no—
525C22×C4—612no—
526C22×C4—612no—
527C22×C4—612no—
528C22×C4—612no—
529C22×C4—611no—
530C22×C4—611no—
531C22×C4—1222no—
532C22×C4—1222no—
533C22×C4—4881no—
534C22×C4—4881no—
535C22×C4—4881no—
536C22×C4—4881no—
537C22×C4C2×C4×D6111yescharacteristic
538C22×C4C2×C4×D6111yescharacteristic
539C22×C4C2×C4×D6111yescharacteristic
540C22×C4C2×C4×D6111yescharacteristic
541C22×C4C2×C4×D6111yescharacteristic
542C22×C4C23×D6111yescharacteristic
543M16—4882no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—1222no—
547C2×C8—1222no—
548C2×C8—4882no—
549C2×C8—4882no—
550C2×C8—4882no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—612no—
558C4⋊C4—612no—
559C4⋊C4—1222no—
560C4⋊C4—1222no—
561C22⋊C4—21no—
562C22⋊C4—21no—
563C22⋊C4—21no—
564C22⋊C4—21no—
565C22⋊C4—21no—
566C22⋊C4—21no—
567C22⋊C4—612no—
568C22⋊C4—612no—
569C22⋊C4—612no—
570C22⋊C4—612no—
571C22⋊C4—1222no—
572C22⋊C4—1222no—
573C22⋊C4—1222no—
574C22⋊C4—1222no—
575C22⋊C4—1222no—
576C22⋊C4—1221no—
577C22⋊C4—1221no—
578C22⋊C4—1221no—
579C22⋊C4—1221no—
580C22⋊C4—1221no—
581C22⋊C4—1221no—
582C22⋊C4—1221no—
583C22⋊C4—1221no—
584C42—1222no—
585C42D4×D6111yescharacteristic
586C42D4×D6111yescharacteristic
587C42D4×D6111yescharacteristic
588C42D4×D6111yescharacteristic
589C23—21no—
590C23—22no—
591C23—41no—
592C23—41no—
593C23—612no—
594C23—614no—
595C23—612no—
596C23—1222no—
597C23—1222no—
598C23—1222no—
599C23—1221no—
600C23—1221no—
601C23—4882no—
602C23—4882no—
603C23—4888no—
604C23—2441no—
605C23—2441no—
606C23—2441no—
607C23—2441no—
608C23—2441no—
609C23—2441no—
610C23—2441no—
611C23—2441no—
612C23—2444no—
613C23C22×C4×D6111yescharacteristic
614Q8—4881no—
615Q8—4881no—
616D4—1224no—
617D4—1224no—
618D4—4881no—
619D4—4881no—
620D4—4882no—
621D4—4882no—
622D4—4882no—
623D4—4882no—
624D4—2444no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—22no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—612no—
635C2×C4—612no—
636C2×C4—1221no—
637C2×C4—1221no—
638C2×C4—4882no—
639C2×C4—4882no—
640C2×C4—2442no—
641C2×C4—2442no—
642C2×C4—2442no—
643C2×C4—2442no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4C22⋊C4×D6111yescharacteristic
649C2×C4C22⋊C4×D6111yescharacteristic
650C2×C4C22⋊C4×D6111yescharacteristic
651C2×C4C22⋊C4×D6111yescharacteristic
652C2×C4C2×D4×D6111yescharacteristic
653C2×C4C2×D4×D6111yescharacteristic
654C8—42no—
655C8—42no—
656C8—2442no—
657C8—2442no—
658C22—21no—
659C22—21no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—82no—
664C22—614no—
665C22—1222no—
666C22—1222no—
667C22—1222no—
668C22—1222no—
669C22—1222no—
670C22—4882no—
671C22—4882no—
672C22—4882no—
673C22—4888no—
674C22—2444no—
675C22—2441no—
676C22—2441no—
677C22—2441no—
678C22—2441no—
679C22—2441no—
680C22—2441no—
681C22C2×C22⋊C4×D6111yescharacteristic
682C22#131×D6112yes—
683C4—21no—
684C4—21no—
685C4—21no—
686C4—21no—
687C4—21no—
688C4—21no—
689C4—41no—
690C4—41no—
691C4—41no—
692C4—1222no—
693C4—1222no—
694C4—2441no—
695C4—2441no—
696C2—22no—
697C2—81no—
698C2—1224no—
699C2—4881no—
700C2—4881no—
701C2—2442no—
702C2#408×D6111yescharacteristic
703C2#2145427811yescharacteristic
704C2#2145429611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1473C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840351C2
cosocleC22×C4×C61C24
hypercenter#1473D61
hypocenterC61#17386
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1473

Complements

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

NH ≅ G/Ntype
2. #21454847C2semidirect
3. #21454842C2semidirect
4. #21444188C2semidirect
5. #21444177C2semidirect
6. #21444172C2semidirect
7. #21444170C2semidirect
8. #21444166C2semidirect
9. #21444136C2semidirect
10. #323×D61C2semidirect
11. C61×#1473C2semidirect
34. #16839819C22semidirect
35. #16839815C22semidirect
36. #16839724C22semidirect
39. C22⋊C8×D61C22semidirect
40. #16839488C22semidirect
44. #16839485C22semidirect
45. #16839485C22semidirect
47. C61×#432C22semidirect
48. C61×#431C22semidirect
49. C61×#324C22semidirect
348. #1473D61semidirect
388. #432C2×D61semidirect
390. #431C2×D61semidirect
396. #324C2×D61semidirect
403. C61#17386semidirect

Maximal subgroups

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

subgroupindex
C2×#168403512
#214548472
#214548422
#214441882
#214441772
#214441722
#214441702
#214441662
#214441362
#323×D612
C61×#14732
#1738661