Finite Group Databasebeta

← #29606690

Subgroups — #29606690

Order 15616.

1610 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519772C211yesmaximal, characteristic
2C2×#16839627C211yesmaximal, characteristic, complemented
3C2×#16839613C211yesmaximal, characteristic, complemented
4#21444072C211yesmaximal, characteristic, complemented
5#21444071C212yesmaximal, complemented
6#21444065C211yesmaximal, characteristic
7#21444063C212yesmaximal
8#21444061C211yesmaximal, characteristic
9#21444055C212yesmaximal
10#21444050C212yesmaximal, complemented
11C61×#1441C211yesmaximal, characteristic, complemented
12C2×#14519772—24no—
13C2×#14519772—24no—
14C2×#14519772C418yescomplemented
15C22×C4⋊Dic61C2211yescharacteristic
16C22×C4⋊Dic61C2211yescharacteristic
17C22×#11758196C2212yes—
18C22×#11758196C2211yescharacteristic
19C22×#11758198C2211yescharacteristic
20C22×#11758198C2211yescharacteristic
21C22×#11758198C2211yescharacteristic
22C22×#11758198C2212yes—
23C22×#11758198C2212yes—
24C22×C4×Dic61C2212yes—
25C22×C4×Dic61C2211yescharacteristic
26#16839627—24no—
27#16839613—24no—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2211yescharacteristic
33C2×#14519684C2212yes—
34C2×#14519684C2212yes—
35C22×C4⋊C4×C61C2211yescharacteristic
36C22×C4⋊C4×C61C2211yescharacteristic
37C2×#95×C61C2211yescharacteristic
38C2×#95×C61C2212yes—
39C2×#95×C61C2212yes—
40C24×D61C2311yescharacteristic
41C23×Dic61C2312yes—
42C23×Dic61C2311yescharacteristic
43C23×Dic61C2312yes—
44C23×Dic61C2311yescharacteristic
45C23×Dic61C2311yescharacteristic
46#14519772—232no—
47C2×C4⋊Dic61—22no—
48C2×C4⋊Dic61—22no—
49C2×C4⋊Dic61—22no—
50C2×C4⋊Dic61—42no—
51C2×C4⋊Dic61C2×C414yes—
52C2×#11758196—24no—
53C2×#11758196—22no—
54C2×#11758196—22no—
55C2×#11758196—24no—
56C2×#11758196—24no—
57C2×#11758196C2×C414yes—
58C2×#11758198—22no—
59C2×#11758198—22no—
60C2×#11758198—22no—
61C2×#11758198—22no—
62C2×#11758198—22no—
63C2×#11758198—22no—
64C2×#11758198—24no—
65C2×#11758198—24no—
66C2×#11758198—42no—
67C2×#11758198—42no—
68C2×#11758198—42no—
69C2×#11758198—42no—
70C2×#11758198C2×C414yes—
71C2×#11758198C2×C414yes—
72C2×#11758198C2×C414yes—
73C2×C4×Dic61—24no—
74C2×C4×Dic61—42no—
75C2×C4×Dic61—42no—
76C2×C4×Dic61C2×C414yes—
77C2×C4×Dic61D414yes—
78C2×C4×Dic61D414yes—
79#14519684—22no—
80#14519684—24no—
81#14519684—24no—
82#14519684—24no—
83#14519684—24no—
84#14519684—42no—
85#14519684—42no—
86C23×C4×C61C2312yes—
87C23×C4×C61C2311yescharacteristic
88C23×C4×C61C2311yescharacteristic
89C23×C4×C61C2312yes—
90C23×C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61—22no—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61—42no—
95C2×C4⋊C4×C61C2×C414yes—
96#95×C61—22no—
97#95×C61—42no—
98#95×C61—42no—
99C24×C61C2411yescharacteristic
100C23×D61—21no—
101C23×D61—21no—
102C23×D61—22no—
103C23×D61—22no—
104C23×D61—21no—
105C23×D61—21no—
106C23×D61—22no—
107C23×D61—22no—
108C23×D61—21no—
109C23×D61—21no—
110C23×D61C22×C412yes—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61C22×C412yes—
137C22×C4×C61C22×C412yes—
138C22×C4×C61C22×C412yes—
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2×D412yes—
141C22×C4×C61#4114yes—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—22no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61—22no—
164C22×Dic61—22no—
165C22×Dic61C22×C412yes—
166C22×Dic61C22×C412yes—
167C22×Dic61C22×C412yes—
168C22×Dic61C2×D412yes—
169C22×Dic61C2×D412yes—
170C22×Dic61C2×D412yes—
171C22×Dic61C2×D412yes—
172C22×Dic61#4114yes—
173C4⋊C4×C61—28no—
174C4⋊C4×C61—44no—
175#11758198—28no—
176#11758198—28no—
177#11758198—28no—
178#11758198—48no—
179#11758198—48no—
180C4⋊Dic61—28no—
181C4⋊Dic61—44no—
182#11758196—216no—
183#11758196—28no—
184C4×Dic61—48no—
185C4×Dic61C22⋊C4116yescomplemented
186C23×C61C23×C411yescharacteristic
187C23×C61C22×D411yescharacteristic
188C23×C61C22×D411yescharacteristic
189C23×C61C2×#4112yes—
190C23×C61C2×#4112yes—
191C23×C61H(5,𝔽2)11yescharacteristic
192C23×C61H(5,𝔽2)11yescharacteristic
193C23×C61H(5,𝔽2)12yes—
194C23×C612−1+412yes—
195C23×C612−1+411yescharacteristic
196C23×C612−1+411yescharacteristic
197C22×D61—22no—
198C22×D61—24no—
199C22×D61—24no—
200C22×D61—22no—
201C22×D61—22no—
202C22×D61—42no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—42no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—42no—
213C22×D61—42no—
214C22×D61—42no—
215C22×D61—42no—
216C22×D61—42no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61—42no—
220C2×C4×C61—24no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—24no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—42no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61C2×C22⋊C414yes—
251C2×Dic61—24no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—24no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—24no—
263C2×Dic61—24no—
264C2×Dic61—24no—
265C2×Dic61—24no—
266C2×Dic61—42no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—42no—
279C2×Dic61—42no—
280C2×Dic61C2×C22⋊C414yes—
281C2×Dic61C2×C22⋊C414yes—
282#14291—611nomaximal, Sylow
283C22×C61C22×C22⋊C411yescharacteristic
284C22×C61C4×#4112yes—
285C22×C61#52012yes—
286C22×C61#52011yescharacteristic
287C22×C61#52011yescharacteristic
288C22×C61#52512yes—
289C22×C61#53211yescharacteristic
290C22×C61#53211yescharacteristic
291C22×C61#53511yescharacteristic
292C22×C61#53511yescharacteristic
293C22×C61#53611yescharacteristic
294C22×C61#53611yescharacteristic
295C22×C61#53712yes—
296C22×C61#53711yescharacteristic
297C22×C61#53711yescharacteristic
298C22×C61#53812yes—
299C22×C61#53812yes—
300C22×C61#53912yes—
301C22×C61#53912yes—
302C22×C61#54112yes—
303C22×C61#54112yes—
304C22×C61#54212yes—
305C22×C61#54212yes—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—44no—
309C2×D61—44no—
310C2×D61—42no—
311C2×D61—82no—
312C2×D61—82no—
313C2×D61—81no—
314C2×D61—81no—
315C2×D61—81no—
316C2×D61—81no—
317C4×C61—24no—
318C4×C61—42no—
319C4×C61—42no—
320C4×C61—44no—
321C4×C61—82no—
322Dic61—24no—
323Dic61—24no—
324Dic61—44no—
325Dic61—42no—
326Dic61—82no—
327C22×#126—611no—
328C2×#382—611no—
329C2×C4×C4⋊C4—611no—
330#1441—612no—
331#1441—611no—
332#1441D6111yescharacteristic, complemented
333#1439—612no—
334#1439—611no—
335#1439—611no—
336#1437—612no—
337#1436—612no—
338C2×C61#229711yescharacteristic
339C2×C61#231111yescharacteristic
340C2×C61#231111yescharacteristic
341C2×C61#233412yes—
342C2×C61#233412yes—
343C2×C61#263612yes—
344C2×C61#263712yes—
345C2×C61#266211yescharacteristic
346C2×C61#266211yescharacteristic
347C2×C61#266411yescharacteristic
348C2×C61#266411yescharacteristic
349D61—82no—
350C2×#126—618no—
351C2×#126—1224no—
352C2×#126—1224no—
353C22×C4⋊C4—611no—
354C22×C4⋊C4—611no—
355C22×C4⋊C4—612no—
356C22×C4⋊C4—611no—
357C22×C4⋊C4C2×D6111yescharacteristic
358C22×C4⋊C4C2×D6111yescharacteristic
359C22×C22⋊C4—611no—
360C22×C22⋊C4—611no—
361C22×C22⋊C4—611no—
362C22×C22⋊C4—612no—
363C22×C22⋊C4—612no—
364C22×C42—611no—
365C22×C42—612no—
366#382—1224no—
367C4×C4⋊C4—1224no—
368C2×#95—612no—
369C2×#95—612no—
370C2×#95—612no—
371C2×#95—611no—
372C2×#95—612no—
373C2×#95—612no—
374C2×#95—612no—
375C2×#95C2×D6111yescharacteristic
376C2×#95C2×D6112yes—
377C2×#95C2×D6112yes—
378C61#1429111yescharacteristic, complemented, Sylow
379C25—611no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—611no—
383C23×C4—612no—
384C23×C4—612no—
385C23×C4C22×D6112yes—
386C23×C4C22×D6111yescharacteristic
387C23×C4C22×D6111yescharacteristic
388C23×C4C22×D6112yes—
389C23×C4C22×D6111yescharacteristic
390#126—12232no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—42no—
395C2×C4⋊C4—614no—
396C2×C4⋊C4—614no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4—1222no—
399C2×C4⋊C4—1222no—
400C2×C4⋊C4—1222no—
401C2×C4⋊C4—1224no—
402C2×C4⋊C4—1224no—
403C2×C4⋊C4—1224no—
404C2×C4⋊C4—1222no—
405C2×C4⋊C4—2442no—
406C2×C4⋊C4C4×D6114yes—
407C2×C22⋊C4—614no—
408C2×C22⋊C4—614no—
409C2×C22⋊C4—614no—
410C2×C22⋊C4—1222no—
411C2×C22⋊C4—1222no—
412C2×C22⋊C4—1222no—
413C2×C22⋊C4—1222no—
414C2×C22⋊C4—1222no—
415C2×C22⋊C4—1222no—
416C2×C22⋊C4—1224no—
417C2×C22⋊C4—1224no—
418C2×C22⋊C4—2442no—
419C2×C22⋊C4—2442no—
420C2×C22⋊C4—2442no—
421C2×C22⋊C4—2442no—
422C2×C42—614no—
423C2×C42—614no—
424C2×C42—614no—
425C2×C42—1224no—
426C2×C42—2442no—
427C2×C42—2442no—
428#95—22no—
429#95—42no—
430#95—42no—
431#95—1224no—
432#95—1224no—
433#95—1224no—
434#95—1222no—
435#95—1224no—
436#95—2442no—
437#95—2442no—
438C24—612no—
439C24—1221no—
440C24—1221no—
441C24—1222no—
442C24—1222no—
443C24—1222no—
444C24—1221no—
445C24—1221no—
446C24—1222no—
447C24—1221no—
448C24—1221no—
449C24C23×D6111yescharacteristic
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—612no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—612no—
481C22×C4—612no—
482C22×C4—614no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1222no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4—1222no—
503C22×C4—1222no—
504C22×C4—1222no—
505C22×C4—1222no—
506C22×C4C2×C4×D6112yes—
507C22×C4C2×C4×D6112yes—
508C22×C4C2×C4×D6112yes—
509C22×C4D4×D6112yes—
510C22×C4D4×D6112yes—
511C22×C4#1175822614yes—
512C4⋊C4—28no—
513C4⋊C4—44no—
514C4⋊C4—1228no—
515C4⋊C4—1228no—
516C4⋊C4—12216no—
517C4⋊C4—2444no—
518C22⋊C4—1228no—
519C22⋊C4—1228no—
520C22⋊C4—1228no—
521C22⋊C4—2448no—
522C22⋊C4—2448no—
523C42—6116no—
524C42—2448no—
525C23—1222no—
526C23—1224no—
527C23—1224no—
528C23—1222no—
529C23—1222no—
530C23—2442no—
531C23—2441no—
532C23—2441no—
533C23—2441no—
534C23—2441no—
535C23—2441no—
536C23—2441no—
537C23—2442no—
538C23—2441no—
539C23—2441no—
540C23—2442no—
541C23—2442no—
542C23—2442no—
543C23—2442no—
544C23—2442no—
545C23—2442no—
546C23—2442no—
547C23—2442no—
548C23C22×C4×D6111yescharacteristic
549C23C2×D4×D6111yescharacteristic
550C23C2×D4×D6111yescharacteristic
551C23#41×D6112yes—
552C23C2×#1175822612yes—
553C23#1451982212yes—
554C23#1451982211yescharacteristic
555C23#1451982311yescharacteristic
556C23#1451982412yes—
557C23#1451982411yescharacteristic
558C23#1451982511yescharacteristic
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—614no—
590C2×C4—614no—
591C2×C4—1224no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1224no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1224no—
603C2×C4—1224no—
604C2×C4—1224no—
605C2×C4—1224no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4—2442no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2442no—
617C2×C4—2442no—
618C2×C4—2442no—
619C2×C4—2442no—
620C2×C4C22⋊C4×D6114yes—
621C22—4881no—
622C22—4881no—
623C22—4882no—
624C22—4881no—
625C22—4881no—
626C22—4882no—
627C22—2442no—
628C22—2444no—
629C22—2444no—
630C22—2442no—
631C22—2442no—
632C22C2×C22⋊C4×D6111yescharacteristic
633C22C4×#1175822612yes—
634C22#1684024112yes—
635C22#1684024211yescharacteristic
636C22#1684024311yescharacteristic
637C22#1684027512yes—
638C22#1684031111yescharacteristic
639C22#1684031311yescharacteristic
640C22#1684032711yescharacteristic
641C22#1684032811yescharacteristic
642C22#1684033411yescharacteristic
643C22#1684033511yescharacteristic
644C22#1684033912yes—
645C22#1684034111yescharacteristic
646C22#1684034211yescharacteristic
647C22#1684034412yes—
648C22#1684035212yes—
649C22#1684035312yes—
650C22#1684036112yes—
651C22#1684037112yes—
652C22#1684037512yes—
653C22#1684038012yes—
654C22#1684038612yes—
655C4—24no—
656C4—44no—
657C4—42no—
658C4—42no—
659C4—82no—
660C4—1224no—
661C4—1224no—
662C4—4882no—
663C4—2442no—
664C4—2444no—
665C2—4882no—
666C2#2144830811yescharacteristic
667C2#2144840611yescharacteristic
668C2#2144840811yescharacteristic
669C2#2144859312yes—
670C2#2144859712yes—
671C2#2145172312yes—
672C2#2145173412yes—
673C2#2145196311yescharacteristic
674C2#2145196411yescharacteristic
675C2#2145198511yescharacteristic
676C2#2145198611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1441C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenter#1441D61
hypocenterC61#14291
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
2. C2×#16839627C2semidirect
3. C2×#16839613C2semidirect
4. #21444072C2semidirect
5. #21444071C2semidirect
10. #21444050C2semidirect
11. C61×#1441C2semidirect
14. C2×#14519772C4semidirect
185. C4×Dic61C22⋊C4semidirect
332. #1441D61semidirect
378. C61#14291semidirect

Maximal subgroups

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

subgroupindex
C22×#145197722
C2×#168396272
C2×#168396132
#214440722
#214440712
#214440652
#214440632
#214440612
#214440552
#214440502
C61×#14412
#1429161