Finite Group Databasebeta

← #29680739

Subgroups — #29680739

Order 15616.

1590 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519735C211yesmaximal, characteristic
2C2×#16839797C212yesmaximal, complemented
3C2×C22⋊C8×D61C211yesmaximal, characteristic
4#21448201C211yesmaximal, characteristic
5#21446019C212yesmaximal, complemented
6#21446017C212yesmaximal, complemented
7#21446014C211yesmaximal, characteristic, complemented
8#21446007C211yesmaximal, characteristic, complemented
9#21446006C211yesmaximal, characteristic, complemented
10#21445534C212yesmaximal, complemented
11C61×#1853C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C22×C8×D61C2212yes—
14C2×#14519735—28no—
15C2×#14519735—21no—
16C2×#14519735—21no—
17C2×#14519735C2211yescharacteristic
18C2×#14519735C2211yescharacteristic
19C2×#14519735C2212yes—
20C2×#14519735C2212yes—
21C22×C4⋊Dic61C2211yescharacteristic
22C22×#11758198—21no—
23C22×C4×Dic61—21no—
24#16839797—28no—
25C2×#14519710C2212yescomplemented
26C2×#14519710C2212yescomplemented
27C2×#14519709C2212yescomplemented
28C2×C22⋊#92418C2211yescharacteristic
29C2×#14519689C2212yes—
30C22⋊C8×D61C418yescomplemented
31#16839624—21no—
32#16839622C2211yescharacteristic
33#16839619C2211yescharacteristic
34#16839619C2211yescharacteristic
35#16839617—21no—
36C22⋊C4×Dic61—21no—
37#16839609—21no—
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C22×C4⋊C4×C61C2211yescharacteristic
42C2×#106×C61C2212yescomplemented
43C2×C22⋊C8×C61C2211yescharacteristic
44C61×#380C2211yescharacteristic
45C61×#340C2212yescomplemented
46C24×D61D411yescharacteristic
47C23×Dic61—21no—
48C23×Dic61D411yescharacteristic
49C22×C4×D61—24no—
50C22×C4×D61—24no—
51C22×C4×D61C2×C414yes—
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61Q814yes—
55C22×C4×D61C2311yescharacteristic
56C22×C4×D61C2311yescharacteristic
57C22×#92418C2312yes—
58C2×C8×D61—24no—
59C2×C8×D61—24no—
60C2×C8×D61C2×C418yescomplemented
61#14519735—28no—
62#14519735—28no—
63#14519735—24no—
64#14519735—24no—
65#14519735—24no—
66#14519735—24no—
67#14519735—24no—
68#14519735D418yes—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—24no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2312yes—
74C2×C22⋊Dic61—21no—
75C2×#11758198—28no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×C4×Dic61—28no—
83C2×C4×Dic61—22no—
84C2×C4×Dic61—22no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87#14519710—24no—
88#14519710—24no—
89#14519709—24no—
90C22⋊#92418C2×C414yes—
91#14519689C2×C418yescomplemented
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95C23×C4×C61—21no—
96C23×C4×C61C2311yescharacteristic
97C22×C8×C61C2312yes—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—24no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C4⋊C4×C61C2312yes—
103C2×C22⋊C4×C61—21no—
104#106×C61—24no—
105C22⋊C8×C61C2×C414yes—
106#95×C61—21no—
107C24×C61C2×D411yescharacteristic
108C23×D61—22no—
109C23×D61—22no—
110C23×D61—22no—
111C23×D61—22no—
112C23×D61—41no—
113C23×D61—41no—
114C23×D61C4⋊C414yes—
115C23×D61C2×D411yescharacteristic
116C23×D61C2×D411yescharacteristic
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—24no—
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×Q812yes—
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—24no—
132C22×Dic61—22no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—42no—
136C22×Dic61—42no—
137C22×Dic61C4⋊C414yes—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—216no—
147C2×C4×D61—216no—
148C2×C4×D61C4⋊C414yes—
149C2×C4×D61C4⋊C414yes—
150C2×C4×D61C22×C412yes—
151C2×C4×D61C22×C412yes—
152C2×C4×D61C2×D414yes—
153C2×C4×D61C2×Q812yes—
154C2×C4×D61C2×Q812yes—
155C2×C4×D61#4114yes—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61C22×C414yes—
159C4⋊C4×C61—24no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61C2×D414yes—
164C22⋊C4×C61—42no—
165C22⋊Dic61—42no—
166#11758198—28no—
167#11758198—28no—
168#11758198—42no—
169#11758198—42no—
170#11758198—42no—
171#11758198—42no—
172C4⋊Dic61—24no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61C2×D414yes—
177C4×Dic61—28no—
178C4×Dic61—28no—
179C4×Dic61—42no—
180C4×Dic61—42no—
181C2×#92418—22no—
182C2×#92418—22no—
183C2×#92418C22×C414yes—
184C8×D61—216no—
185C23×C61—21no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—21no—
189C23×C61C2×C4⋊C412yes—
190C23×C61#13512yes—
191C23×C61#13512yes—
192C23×C61C22×D411yescharacteristic
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—21no—
202C22×D61—21no—
203C22×D61—28no—
204C22×D61—28no—
205C22×D61—44no—
206C22×D61—44no—
207C22×D61—44no—
208C22×D61—44no—
209C22×D61—44no—
210C22×D61C2×C4⋊C412yes—
211C22×D61C2×C4⋊C412yes—
212C22×D61#12012yes—
213C22×D61#12312yes—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—24no—
217C2×C4×C61—24no—
218C2×C4×C61—24no—
219C2×C4×C61—48no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61C2×C4⋊C412yes—
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61C22×D412yes—
231C2×C4×C61C22×Q811yescharacteristic
232C2×C4×C61C2×#4112yes—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—28no—
236C2×Dic61—24no—
237C2×Dic61—48no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61C2×C4⋊C412yes—
247C2×Dic61C2×C4⋊C412yes—
248C2×Dic61#12012yes—
249C2×Dic61#12312yes—
250C4×D61—28no—
251C4×D61—24no—
252C4×D61—24no—
253C4×D61—28no—
254C4×D61C4×D418yes—
255C4×D61C22⋊Q818yes—
256C8×C61—24no—
257#92418—24no—
258#20843—611nomaximal, Sylow
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—22no—
262C22×C61—22no—
263C22×C61—22no—
264C22×C61—22no—
265C22×C61#42714yescomplemented
266C22×C61C22×C4⋊C411yescharacteristic
267C22×C61C2×#12011yescharacteristic
268C22×C61C2×#12311yescharacteristic
269C22×C61C2×#13511yescharacteristic
270C22×C61C2×#13511yescharacteristic
271C22×C61#57511yescharacteristic
272C22×C61#57511yescharacteristic
273C2×D61—22no—
274C2×D61—22no—
275C2×D61—22no—
276C2×D61—22no—
277C2×D61—22no—
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—22no—
281C2×D61—22no—
282C2×D61—416no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—41no—
286C2×D61—41no—
287C2×D61#38514yes—
288C4×C61—22no—
289C4×C61—44no—
290C4×C61—81no—
291C4×C61—81no—
292C4×C61C2×C4×D412yes—
293C4×C61C2×C22⋊Q812yes—
294Dic61—22no—
295Dic61—44no—
296Dic61—81no—
297Dic61—81no—
298Dic61#38514yes—
299C22×#117—611no—
300C2×#427—612no—
301C22×C22⋊C8—611no—
302#2280—611no—
303#1854—612no—
304#1854—612no—
305#1853—611no—
306#1853D6111yescharacteristic, complemented
307#1852—611no—
308#1852—611no—
309#1735—612no—
310C2×C61—24no—
311C2×C61C2×#38511yescharacteristic
312C2×C61C2×#42711yescharacteristic
313C2×C61#291111yescharacteristic
314C2×C61#300211yescharacteristic
315C2×C61#300211yescharacteristic
316C2×C61#309111yescharacteristic
317C2×C61#309111yescharacteristic
318D61—24no—
319D61—42no—
320C24×C4—611no—
321C23×C8—612no—
322C2×#117—612no—
323C2×#117—612no—
324C2×#117—611no—
325C2×#117—611no—
326C2×#117—1228no—
327C2×#117—1221no—
328C2×#117—1221no—
329C22×C4⋊C4—611no—
330C22×C4⋊C4C2×D6111yescharacteristic
331C22×C22⋊C4—1221no—
332C22×C42—1221no—
333#427—1228no—
334C2×#107—612no—
335C2×#107—612no—
336C2×#106—612no—
337C2×#106C2×D6112yescomplemented
338C2×C22⋊C8—611no—
339C2×C22⋊C8—612no—
340C2×C22⋊C8—618no—
341C2×C22⋊C8C2×D6111yescharacteristic
342#381—1221no—
343#380—611no—
344#380—611no—
345#380—611no—
346#380C2×D6111yescharacteristic
347#379—1221no—
348C4×C22⋊C4—1221no—
349C4×C22⋊C4—1221no—
350#340—612no—
351#340—612no—
352#340—612no—
353#340C2×D6112yescomplemented
354C61#2084311yescharacteristic, complemented, Sylow
355C25—611no—
356C23×C4—21no—
357C23×C4—611no—
358C23×C4—611no—
359C23×C4—614no—
360C23×C4—611no—
361C23×C4—611no—
362C23×C4—614no—
363C23×C4—611no—
364C23×C4—1224no—
365C23×C4—1224no—
366C23×C4—1221no—
367C23×C4C22×D6111yescharacteristic
368C22×C8—612no—
369C22×C8—618no—
370C22×C8—1224no—
371C22×C8—1224no—
372C22×C8C22×D6112yes—
373#117—618no—
374#117—1224no—
375#117—1224no—
376#117—1228no—
377#117—1228no—
378#117—1224no—
379#117—1224no—
380#117—1224no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—612no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—1224no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C4⋊C4C22×D6112yes—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—1222no—
393C2×C22⋊C4—1222no—
394C2×C22⋊C4—1228no—
395C2×C22⋊C4—1221no—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C42—1222no—
401C2×C42—1222no—
402C2×C42—1228no—
403C2×C42—1221no—
404C2×C42—1221no—
405#107—1224no—
406#107—1224no—
407#106—24no—
408#106—1224no—
409C22⋊C8—614no—
410C22⋊C8—618no—
411C22⋊C8C4×D6114yes—
412#95—21no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416C24—611no—
417C24—611no—
418C24—614no—
419C24—1222no—
420C24—1222no—
421C24—1222no—
422C24—1222no—
423C24—2441no—
424C24—2441no—
425C24D4×D6111yescharacteristic
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—611no—
435C22×C4—612no—
436C22×C4—614no—
437C22×C4—612no—
438C22×C4—614no—
439C22×C4—611no—
440C22×C4—614no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—614no—
444C22×C4—614no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—12216no—
448C22×C4—12216no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1224no—
456C22×C4—1222no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—2442no—
460C22×C4—2442no—
461C22×C4C2×C4×D6112yes—
462C22×C4D4×D6111yescharacteristic
463C22×C4Q8×D6112yes—
464C22×C4C23×D6111yescharacteristic
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—614no—
468C2×C8—12216no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8C2×C4×D6114yes—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—614no—
477C4⋊C4—1224no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4D4×D6114yes—
482C22⋊C4—42no—
483C22⋊C4—1228no—
484C22⋊C4—1228no—
485C22⋊C4—2442no—
486C22⋊C4—2442no—
487C22⋊C4—2442no—
488C22⋊C4—2442no—
489C22⋊C4—2442no—
490C42—1228no—
491C42—1228no—
492C42—2442no—
493C42—2442no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—612no—
499C23—612no—
500C23—612no—
501C23—612no—
502C23—1228no—
503C23—1221no—
504C23—1221no—
505C23—1221no—
506C23—1221no—
507C23—1221no—
508C23—1221no—
509C23—1221no—
510C23—1221no—
511C23—1221no—
512C23—1221no—
513C23—1228no—
514C23—2444no—
515C23—2444no—
516C23—2444no—
517C23—2444no—
518C23—2444no—
519C23C4⋊C4×D6112yes—
520C23#1451979812yes—
521C23#1451979812yes—
522C23C2×D4×D6111yescharacteristic
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—618no—
541C2×C4—618no—
542C2×C4—612no—
543C2×C4—1228no—
544C2×C4—1228no—
545C2×C4—1228no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1224no—
549C2×C4—1224no—
550C2×C4—1224no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2448no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4C4⋊C4×D6112yes—
561C2×C4C22×C4×D6111yescharacteristic
562C2×C4C2×D4×D6112yes—
563C2×C4C2×Q8×D6111yescharacteristic
564C2×C4#41×D6112yes—
565C8—24no—
566C8—1224no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—614no—
574C22—1222no—
575C22—1222no—
576C22—1222no—
577C22—1222no—
578C22—1222no—
579C22—1222no—
580C22—1222no—
581C22—1222no—
582C22—1222no—
583C22—24416no—
584C22—2441no—
585C22—2441no—
586C22—2441no—
587C22—2441no—
588C22#1683979714yescomplemented
589C22C2×C4⋊C4×D6111yescharacteristic
590C22#120×D6111yescharacteristic
591C22#123×D6111yescharacteristic
592C22C2×#1451979811yescharacteristic
593C22C2×#1451979811yescharacteristic
594C22#1684055811yescharacteristic
595C22#1684056111yescharacteristic
596C4—22no—
597C4—44no—
598C4—81no—
599C4—81no—
600C4—614no—
601C4—1222no—
602C4—4881no—
603C4—4881no—
604C4—2444no—
605C4C4×D4×D6112yes—
606C4C22⋊Q8×D6112yes—
607C2—24no—
608C2—1224no—
609C2—2442no—
610C2#385×D6111yescharacteristic
611C2C2×#1683979711yescharacteristic
612C2#2145440611yescharacteristic
613C2#2145503611yescharacteristic
614C2#2145503811yescharacteristic
615C2#2145572911yescharacteristic
616C2#2145573311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1853C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1853D61
hypocenterC61#20843
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
2. C2×#16839797C2semidirect
5. #21446019C2semidirect
6. #21446017C2semidirect
7. #21446014C2semidirect
8. #21446007C2semidirect
9. #21446006C2semidirect
10. #21445534C2semidirect
11. C61×#1853C2semidirect
25. C2×#14519710C22semidirect
26. C2×#14519710C22semidirect
27. C2×#14519709C22semidirect
30. C22⋊C8×D61C4semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
42. C2×#106×C61C22semidirect
45. C61×#340C22semidirect
60. C2×C8×D61C2×C4semidirect
91. #14519689C2×C4semidirect
265. C22×C61#427semidirect
306. #1853D61semidirect
337. C2×#106C2×D61semidirect
353. #340C2×D61semidirect
354. C61#20843semidirect
588. C22#16839797semidirect

Maximal subgroups

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

subgroupindex
C22×#145197352
C2×#168397972
C2×C22⋊C8×D612
#214482012
#214460192
#214460172
#214460142
#214460072
#214460062
#214455342
C61×#18532
#2084361