Finite Group Databasebeta

← #30609951

Subgroups — #30609951

Order 15616.

1322 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462498C212yesmaximal, complemented
2#21461072C212yesmaximal, complemented
3#21461070C212yesmaximal, complemented
4#21461068C212yesmaximal, complemented
5#21461065C212yesmaximal
6#21461053C212yesmaximal, complemented
7#21461051C212yesmaximal, complemented
8C2×#117×D61C211yesmaximal, characteristic
9#16840785—22no—
10#16840782C2214yescomplemented
11#16840780—22no—
12#16840779C2214yescomplemented
13#16840767—22no—
14#16840733C2214yes—
15C4×#11758213—22no—
16#16840608C2214yescomplemented
17#16840607C2214yescomplemented
18#16840606C2214yes—
19#16840605C2214yes—
20C23×C4×D61C411yescharacteristic
21C2×#14519736C2211yescharacteristic
22C2×#14519735C411yescharacteristic
23C2×#14519735C2211yescharacteristic
24C2×#14519734C411yescharacteristic
25C2×#14519734C2211yescharacteristic
26C2×#14519733C411yescharacteristic
27C2×#14519733C2211yescharacteristic
28#117×D61—22no—
29#117×D61—22no—
30#117×D61—22no—
31#117×D61—22no—
32C2×C4⋊C4×D61C411yescharacteristic
33C2×C4⋊C4×D61C2211yescharacteristic
34C2×C22⋊C4×D61C411yescharacteristic
35C2×C22⋊C4×D61C2211yescharacteristic
36C2×C42×D61C411yescharacteristic
37C2×C42×D61C2211yescharacteristic
38C2×#117×C61C411yescharacteristic
39C2×#11758213—24no—
40#14519844—24no—
41#14519844—24no—
42#14519834—24no—
43C22⋊#92420C2318yes—
44#14519832—24no—
45C4×#92420—24no—
46C24×D61C2×C411yescharacteristic
47C23×Dic61C2×C411yescharacteristic
48C22×C4×D61—24no—
49C22×C4×D61—21no—
50C22×C4×D61—21no—
51C22×C4×D61—21no—
52C22×C4×D61—21no—
53C22×C4×D61—24no—
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61C2×C411yescharacteristic
56C22×C4×D61C2×C411yescharacteristic
57C22×C4×D61C2×C411yescharacteristic
58C22×C4×D61C2×C411yescharacteristic
59C22×C4×D61C2×C411yescharacteristic
60C22×C4×D61C2×C411yescharacteristic
61C22×C4×D61C2311yescharacteristic
62#14519736—22no—
63#14519736—22no—
64#14519735—22no—
65#14519735—22no—
66#14519735—22no—
67#14519735—22no—
68#14519734—22no—
69#14519734—22no—
70#14519734—22no—
71#14519734—22no—
72#14519733—22no—
73#14519733—22no—
74#14519733—22no—
75#14519733—22no—
76C2×C4⋊Dic61C2×C411yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196C2×C411yescharacteristic
79C2×#11758196C2×C411yescharacteristic
80C2×#11758196C2×C411yescharacteristic
81C2×#11758196C2311yescharacteristic
82C4⋊C4×D61—22no—
83C4⋊C4×D61—22no—
84C4⋊C4×D61—22no—
85C4⋊C4×D61—22no—
86C2×C22⋊Dic61C2×C411yescharacteristic
87C2×C22⋊Dic61C2311yescharacteristic
88C2×#11758198C2×C411yescharacteristic
89C2×#11758198C2×C411yescharacteristic
90C2×#11758198C2×C411yescharacteristic
91C2×#11758198C2311yescharacteristic
92C22⋊C4×D61—22no—
93C22⋊C4×D61—22no—
94C22⋊C4×D61—22no—
95C22⋊C4×D61—22no—
96C2×C4×Dic61C2×C411yescharacteristic
97C2×C4×Dic61C2×C411yescharacteristic
98C2×C4×Dic61C2×C411yescharacteristic
99C2×C4×Dic61C2×C411yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101C2×C4×Dic61C2311yescharacteristic
102C42×D61—21no—
103C42×D61—21no—
104C42×D61—21no—
105C42×D61—21no—
106C42×D61D412yes—
107C42×D61D412yes—
108C42×D61D412yes—
109C42×D61D412yes—
110C23×C4×C61C2×C411yescharacteristic
111#117×C61—22no—
112#117×C61—22no—
113C2×C4⋊C4×C61C2×C411yescharacteristic
114C2×C4⋊C4×C61C2×C411yescharacteristic
115C2×C22⋊C4×C61C2×C411yescharacteristic
116C2×C22⋊C4×C61C2×C411yescharacteristic
117C2×C42×C61C2×C411yescharacteristic
118C2×C42×C61C2×C411yescharacteristic
119C24×C61C22×C411yescharacteristic
120C23×D61—22no—
121C23×D61—21no—
122C23×D61—21no—
123C23×D61—22no—
124C23×D61—42no—
125C23×D61—42no—
126C23×D61C22×C411yescharacteristic
127C23×D61C22×C411yescharacteristic
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—22no—
132C22×C4×C61C22×C411yescharacteristic
133C22×C4×C61C22×C411yescharacteristic
134C22×C4×C61C22×C411yescharacteristic
135C22×C4×C61C22×C411yescharacteristic
136C22×C4×C61C22×C411yescharacteristic
137C22×C4×C61C22×C411yescharacteristic
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—22no—
141C22×Dic61M1614yescomplemented
142C22×Dic61C22×C411yescharacteristic
143C22×Dic61C22×C411yescharacteristic
144C22×Dic61C22×C411yescharacteristic
145C22×Dic61C22×C411yescharacteristic
146C22×Dic61C22×C411yescharacteristic
147C22×Dic61C2411yescharacteristic
148C2×C4×D61—28no—
149C2×C4×D61—28no—
150C2×C4×D61—28no—
151C2×C4×D61—28no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—22no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—22no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×C4×D61—42no—
173C2×C4×D61C22⋊C411yescharacteristic
174C2×C4×D61C22⋊C411yescharacteristic
175C2×C4×D61C22⋊C411yescharacteristic
176C2×C4×D61C22⋊C411yescharacteristic
177C2×C4×D61C22⋊C412yes—
178C2×C4×D61C22⋊C412yes—
179C2×C4×D61M1614yescomplemented
180C2×C4×D61C2×D412yes—
181C2×C4×D61C2×D412yes—
182C2×#92420—28no—
183#11758213—48no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61C22⋊C412yes—
191C42×C61C22⋊C412yes—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C4×Dic61C22⋊C412yes—
211C4×Dic61C22⋊C412yes—
212C4×Dic61C2×D412yes—
213C4×Dic61C2×D412yes—
214C4×Dic61C2×D412yes—
215C4×Dic61C2×D412yes—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61C2×M1612yes—
221C23×C61C23×C411yescharacteristic
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—24no—
227C22×D61—24no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—48no—
231C22×D61—48no—
232C22×D61—42no—
233C22×D61—42no—
234C22×D61—42no—
235C22×D61—42no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61C2×C22⋊C411yescharacteristic
239C22×D61C2×C22⋊C411yescharacteristic
240C22×D61C2×M1612yes—
241C2×C4×C61—24no—
242C2×C4×C61—24no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C2×C22⋊C412yes—
257C2×C4×C61C2×M1612yes—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—24no—
261C2×Dic61—24no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61C2×C22⋊C412yes—
269C2×Dic61C2×C22⋊C411yescharacteristic
270C2×Dic61C2×C22⋊C411yescharacteristic
271C2×Dic61C2×M1612yes—
272C2×Dic61C2×M1612yes—
273C2×Dic61C2×M1612yes—
274C2×Dic61C22×D412yes—
275C4×D61—24no—
276C4×D61—24no—
277C4×D61—24no—
278C4×D61—24no—
279C4×D61—24no—
280C4×D61—24no—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—22no—
284C4×D61—22no—
285C4×D61—24no—
286C4×D61—24no—
287C4×D61—42no—
288C4×D61—42no—
289C4×D61—42no—
290C4×D61—42no—
291C4×D61—42no—
292C4×D61—42no—
293#92420—48no—
294#17357—611nomaximal, Sylow
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—22no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61C22×C22⋊C411yescharacteristic
303C22×C61C22×M1611yescharacteristic
304C22×C61C22×M1611yescharacteristic
305C2×D61—24no—
306C2×D61—24no—
307C2×D61—21no—
308C2×D61—21no—
309C2×D61—21no—
310C2×D61—21no—
311C2×D61—22no—
312C2×D61—48no—
313C2×D61—48no—
314C2×D61—42no—
315C2×D61—41no—
316C2×D61—41no—
317C2×D61—42no—
318C2×D61#41012yes—
319C2×D61#41312yes—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—22no—
323C4×C61—22no—
324C4×C61—22no—
325C4×C61—41no—
326C4×C61—41no—
327Dic61—22no—
328Dic61—22no—
329Dic61—22no—
330Dic61—41no—
331Dic61—41no—
332Dic61#40712yes—
333Dic61#40712yes—
334C22×#117—611no—
335#2920—612no—
336#1464—612no—
337#1464—612no—
338#1463—612no—
339#1462—612no—
340#1458—612no—
341#1458—612no—
342C2×C61—21no—
343C2×C61—21no—
344C2×C61—42no—
345C2×C61C2×#40711yescharacteristic
346C2×C61C2×#41011yescharacteristic
347C2×C61C2×#41311yescharacteristic
348D61—22no—
349D61—22no—
350D61—42no—
351C24×C4—611no—
352C2×#117—611no—
353C2×#117—611no—
354C2×#117—611no—
355C2×#117—611no—
356C2×#117—611no—
357C2×#117—611no—
358C2×#117—611no—
359C2×#117—1222no—
360C2×#117—1222no—
361C2×#117—1222no—
362C2×#117—1222no—
363C2×#117#692311yescharacteristic
364C22×C4⋊C4—611no—
365C22×C4⋊C4—611no—
366C22×C22⋊C4—611no—
367C22×C22⋊C4—611no—
368C22×C42—611no—
369C22×C42—611no—
370#432—614no—
371#432—1222no—
372#431—614no—
373#431—1222no—
374#423—1222no—
375#407—614no—
376C4×M16—1222no—
377#324—614no—
378#324—614no—
379#323—614no—
380#323—614no—
381C61#1735711yescharacteristic, complemented, Sylow
382C25—611no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—611no—
387C23×C4—611no—
388C23×C4—611no—
389C23×C4—611no—
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—1221no—
393C23×C4—1221no—
394C23×C4—1221no—
395C23×C4—1221no—
396C23×C4—1224no—
397C23×C4—1224no—
398C23×C4C2×#692311yescharacteristic
399C2×M16—1224no—
400#117—22no—
401#117—22no—
402#117—1222no—
403#117—1222no—
404#117—1222no—
405#117—1222no—
406#117—1222no—
407#117—1222no—
408#117—1222no—
409#117—1222no—
410#117—1222no—
411#117—1222no—
412#117—1222no—
413#117—1222no—
414#117—1222no—
415#117—1222no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—1222no—
423C2×C4⋊C4—1222no—
424C2×C4⋊C4—1222no—
425C2×C4⋊C4—1222no—
426C2×C4⋊C4C2×#692311yescharacteristic
427C2×C4⋊C4C2×#692311yescharacteristic
428C2×C22⋊C4—611no—
429C2×C22⋊C4—611no—
430C2×C22⋊C4—611no—
431C2×C22⋊C4—611no—
432C2×C22⋊C4—611no—
433C2×C22⋊C4—611no—
434C2×C22⋊C4—1222no—
435C2×C22⋊C4—1222no—
436C2×C22⋊C4—1222no—
437C2×C22⋊C4—1222no—
438C2×C22⋊C4C2×#692311yescharacteristic
439C2×C22⋊C4C2×#692311yescharacteristic
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—612no—
444C2×C42—612no—
445C2×C42—612no—
446C2×C42—612no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—1221no—
451C2×C42—1221no—
452C2×C42—1221no—
453C2×C42—1221no—
454C2×C42C2×#692311yescharacteristic
455C2×C42C2×#692311yescharacteristic
456C4⋊C8—1224no—
457C4⋊C8—1224no—
458C22⋊C8—618no—
459C22⋊C8—1224no—
460#97—1224no—
461C4×C8—1224no—
462C24—611no—
463C24—611no—
464C24—1221no—
465C24—1221no—
466C24—1222no—
467C24—1222no—
468C24—2442no—
469C24—2442no—
470C24C22×#692311yescharacteristic
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—611no—
476C22×C4—614no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—612no—
485C22×C4—612no—
486C22×C4—614no—
487C22×C4—612no—
488C22×C4—612no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1222no—
501C22×C4—1228no—
502C22×C4—1228no—
503C22×C4—1228no—
504C22×C4—1228no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1222no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1222no—
515C22×C4—2442no—
516C22×C4—2442no—
517C22×C4—2442no—
518C22×C4—2442no—
519C22×C4C22×#692311yescharacteristic
520C22×C4C22×#692311yescharacteristic
521C22×C4C22×#692311yescharacteristic
522C22×C4C22×#692311yescharacteristic
523C22×C4C22×#692311yescharacteristic
524C22×C4C22×#692311yescharacteristic
525M16—2448no—
526C2×C8—1228no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—1222no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1222no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4—1222no—
542C22⋊C4—1222no—
543C42—21no—
544C42—21no—
545C42—612no—
546C42—612no—
547C42—612no—
548C42—612no—
549C42—612no—
550C42—612no—
551C42—1221no—
552C42—1221no—
553C42—1221no—
554C42—1221no—
555C42—1221no—
556C42—1221no—
557C42C22⋊#692312yes—
558C42C22⋊#692312yes—
559C23—21no—
560C23—21no—
561C23—41no—
562C23—41no—
563C23—611no—
564C23—611no—
565C23—612no—
566C23—1222no—
567C23—1222no—
568C23—1224no—
569C23—1224no—
570C23—1221no—
571C23—1221no—
572C23—1221no—
573C23—1221no—
574C23—2441no—
575C23—2441no—
576C23—2442no—
577C23—2442no—
578C23—2448no—
579C23—2448no—
580C23—2442no—
581C23—2442no—
582C23D61⋊M1612yes—
583C23C23×#692311yescharacteristic
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—611no—
597C2×C4—611no—
598C2×C4—612no—
599C2×C4—612no—
600C2×C4—612no—
601C2×C4—612no—
602C2×C4—612no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1224no—
608C2×C4—1224no—
609C2×C4—1224no—
610C2×C4—1224no—
611C2×C4—1224no—
612C2×C4—1224no—
613C2×C4—1224no—
614C2×C4—1224no—
615C2×C4—1224no—
616C2×C4—1224no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—2442no—
622C2×C4—2442no—
623C2×C4—2442no—
624C2×C4—2442no—
625C2×C4—2442no—
626C2×C4—2442no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4C2×C22⋊#692311yescharacteristic
632C2×C4C2×C22⋊#692311yescharacteristic
633C2×C4C2×C22⋊#692312yes—
634C2×C4C2×C22⋊#692312yes—
635C2×C4D61⋊M1612yes—
636C8—2448no—
637C22—21no—
638C22—21no—
639C22—22no—
640C22—42no—
641C22—42no—
642C22—41no—
643C22—41no—
644C22—612no—
645C22—612no—
646C22—1224no—
647C22—1224no—
648C22—1221no—
649C22—1221no—
650C22—1221no—
651C22—1221no—
652C22—1222no—
653C22—2442no—
654C22—2442no—
655C22—2448no—
656C22—2448no—
657C22—2441no—
658C22—2441no—
659C22C22×C22⋊#692311yescharacteristic
660C22C2×D61⋊M1611yescharacteristic
661C22C2×D61⋊M1611yescharacteristic
662C4—21no—
663C4—21no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—41no—
668C4—41no—
669C4—612no—
670C4—612no—
671C4—1222no—
672C4—1222no—
673C4—1222no—
674C4—2441no—
675C4—2441no—
676C2—21no—
677C2—21no—
678C2—42no—
679C2—1222no—
680C2—1222no—
681C2—2442no—
682C2D61⋊#40711yescharacteristic
683C2D61⋊#41011yescharacteristic
684C2D61⋊#41311yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#117×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC24×D61C2×C4
cosocleC22×Dic61C24
hypercenterC2×#117#6923
hypocenterC61#17357
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #21462498C2semidirect
2. #21461072C2semidirect
3. #21461070C2semidirect
4. #21461068C2semidirect
6. #21461053C2semidirect
7. #21461051C2semidirect
10. #16840782C22semidirect
12. #16840779C22semidirect
16. #16840608C22semidirect
17. #16840607C22semidirect
141. C22×Dic61M16semidirect
179. C2×C4×D61M16semidirect
381. C61#17357semidirect

Maximal subgroups

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

subgroupindex
#214624982
#214610722
#214610702
#214610682
#214610652
#214610532
#214610512
C2×#117×D612
#1735761