Finite Group Databasebeta

← #29601887

Subgroups — #29601887

Order 15616.

3686 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519733C211yesmaximal, characteristic, complemented
2C23×C22⋊Dic61C211yesmaximal, characteristic
3C2×#16839623C211yesmaximal, characteristic, complemented
4C2×#16839618C211yesmaximal, characteristic
5#21444062C214yesmaximal, complemented
6#21444056C214yesmaximal, complemented
7#21444041C211yesmaximal, characteristic, complemented
8#21444041C211yesmaximal, characteristic, complemented
9C2×C61×#379C211yesmaximal, characteristic
10C24×Dic61C2211yescharacteristic
11C24×Dic61C2211yescharacteristic
12C2×#14519733—24no—
13C2×#14519733—24no—
14C2×#14519733—24no—
15C22×#11758196C2212yescomplemented
16C22×C22⋊Dic61—24no—
17C22×C22⋊Dic61—24no—
18C22×C22⋊Dic61C418yescomplemented
19C22×C22⋊Dic61C2211yescharacteristic
20C22×C22⋊Dic61C2214yes—
21C22×C22⋊Dic61C2212yes—
22C22×C22⋊Dic61C2212yes—
23C22×C22⋊Dic61C2211yescharacteristic
24C22×C4×Dic61C2212yescomplemented
25#16839623—24no—
26#16839618—24no—
27C2×#14519684C2212yescomplemented
28C2×#14519684C2212yescomplemented
29C2×#14519684C2214yescomplemented
30C2×#14519684C2214yescomplemented
31C26×C61C2211yescharacteristic
32C22×C22⋊C4×C61C2211yescharacteristic
33C22×C22⋊C4×C61C2214yes—
34C22×C22⋊C4×C61C2211yescharacteristic
35C61×#379—24no—
36C23×Dic61—22no—
37C23×Dic61—22no—
38C23×Dic61—22no—
39C23×Dic61—22no—
40C23×Dic61—22no—
41C23×Dic61—22no—
42C23×Dic61—44no—
43C23×Dic61C2×C414yes—
44C23×Dic61C2×C414yes—
45C23×Dic61D414yes—
46C23×Dic61D414yes—
47C23×Dic61C2312yes—
48C23×Dic61C2312yes—
49C23×Dic61C2312yes—
50C23×Dic61C2311yescharacteristic
51C23×Dic61C2311yescharacteristic
52#14519733—232no—
53C2×#11758196—24no—
54C2×#11758196—24no—
55C2×#11758196—24no—
56C2×C22⋊Dic61—22no—
57C2×C22⋊Dic61—22no—
58C2×C22⋊Dic61—232no—
59C2×C22⋊Dic61—24no—
60C2×C22⋊Dic61—24no—
61C2×C22⋊Dic61—24no—
62C2×C22⋊Dic61—24no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61—28no—
67C2×C22⋊Dic61—28no—
68C2×C22⋊Dic61—44no—
69C2×C22⋊Dic61C2×C418yes—
70C2×C22⋊Dic61C2×C418yes—
71C2×C22⋊Dic61D414yes—
72C2×C4×Dic61—24no—
73C2×C4×Dic61—24no—
74C2×C4×Dic61—24no—
75#14519684—24no—
76#14519684—24no—
77#14519684—44no—
78#14519684—44no—
79C25×C61—22no—
80C25×C61—22no—
81C25×C61—28no—
82C25×C61—22no—
83C25×C61—22no—
84C25×C61—22no—
85C25×C61—22no—
86C25×C61—44no—
87C25×C61C2×C414yes—
88C25×C61C2312yes—
89C25×C61C2311yescharacteristic
90C23×C4×C61C2312yes—
91C23×C4×C61C2312yes—
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61—22no—
94C2×C22⋊C4×C61—22no—
95C2×C22⋊C4×C61—22no—
96C2×C22⋊C4×C61—28no—
97C2×C22⋊C4×C61—22no—
98C2×C22⋊C4×C61—28no—
99C2×C22⋊C4×C61—44no—
100C2×C22⋊C4×C61D414yes—
101C24×C61—24no—
102C24×C61—24no—
103C24×C61—216no—
104C24×C61—216no—
105C24×C61—28no—
106C24×C61—216no—
107C24×C61—22no—
108C24×C61—22no—
109C24×C61—22no—
110C24×C61—22no—
111C24×C61—24no—
112C24×C61—22no—
113C24×C61—22no—
114C24×C61—24no—
115C24×C61—24no—
116C24×C61—28no—
117C24×C61—28no—
118C24×C61—48no—
119C24×C61—416no—
120C24×C61—416no—
121C24×C61—416no—
122C24×C61—416no—
123C24×C61—48no—
124C24×C61—48no—
125C24×C61—44no—
126C24×C61—44no—
127C24×C61—44no—
128C24×C61—44no—
129C24×C61C22×C414yes—
130C24×C61C22×C412yes—
131C24×C61C2×D412yes—
132C24×C61C2×D412yes—
133C24×C61C2×D412yes—
134C24×C61C2×D412yes—
135C24×C61#4118yes—
136C24×C61#4112yes—
137C24×C61#4112yes—
138C24×C61C2411yescharacteristic
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—24no—
142C22×C4×C61—24no—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—24no—
147C22×C4×C61—24no—
148C22×C4×C61C2×D414yes—
149C22×Dic61—28no—
150C22×Dic61—28no—
151C22×Dic61—28no—
152C22×Dic61—28no—
153C22×Dic61—28no—
154C22×Dic61—28no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—28no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61—22no—
168C22×Dic61—24no—
169C22×Dic61—24no—
170C22×Dic61—24no—
171C22×Dic61—24no—
172C22×Dic61—22no—
173C22×Dic61—22no—
174C22×Dic61—22no—
175C22×Dic61—24no—
176C22×Dic61—24no—
177C22×Dic61—48no—
178C22×Dic61—48no—
179C22×Dic61—48no—
180C22×Dic61C22⋊C4116yescomplemented
181C22×Dic61C22×C414yes—
182C22×Dic61C22×C412yes—
183C22×Dic61C22×C412yes—
184C22×Dic61C2×D414yes—
185C22×Dic61C2×D412yes—
186C22×Dic61C2×D412yes—
187C22×Dic61C2×D412yes—
188C22×Dic61C2×D412yes—
189C22⋊C4×C61—28no—
190C22⋊C4×C61—416no—
191C22⋊C4×C61—44no—
192C22⋊Dic61—28no—
193C22⋊Dic61—216no—
194C22⋊Dic61—216no—
195C22⋊Dic61—416no—
196C22⋊Dic61—44no—
197#11758196—216no—
198C4×Dic61—216no—
199C23×C61—24no—
200C23×C61—24no—
201C23×C61—24no—
202C23×C61—24no—
203C23×C61—24no—
204C23×C61—24no—
205C23×C61—28no—
206C23×C61—28no—
207C23×C61—28no—
208C23×C61—28no—
209C23×C61—232no—
210C23×C61—24no—
211C23×C61—24no—
212C23×C61—24no—
213C23×C61—24no—
214C23×C61—24no—
215C23×C61—24no—
216C23×C61—28no—
217C23×C61—28no—
218C23×C61—28no—
219C23×C61—28no—
220C23×C61—28no—
221C23×C61—28no—
222C23×C61—432no—
223C23×C61—432no—
224C23×C61—416no—
225C23×C61—432no—
226C23×C61—432no—
227C23×C61—432no—
228C23×C61—432no—
229C23×C61—42no—
230C23×C61—42no—
231C23×C61—42no—
232C23×C61—42no—
233C23×C61—44no—
234C23×C61—44no—
235C23×C61—44no—
236C23×C61—44no—
237C23×C61—44no—
238C23×C61—44no—
239C23×C61—416no—
240C23×C61C2×C22⋊C414yes—
241C23×C61#11718yescomplemented
242C23×C61C4×D414yes—
243C23×C61C4×D414yes—
244C23×C61#12014yes—
245C23×C61#12114yes—
246C23×C61#12114yes—
247C23×C61#12314yes—
248C23×C61C23×C411yescharacteristic
249C23×C61C22×D411yescharacteristic
250C23×C61C22×D411yescharacteristic
251C23×C61C22×D411yescharacteristic
252C23×C61C22×D411yescharacteristic
253C23×C61C2×#4114yes—
254C23×C61C2×#4111yescharacteristic
255C23×C61C2×#4111yescharacteristic
256C23×C61H(5,𝔽2)14yes—
257C2×C4×C61—24no—
258C2×C4×C61—28no—
259C2×C4×C61—42no—
260C2×C4×C61—42no—
261C2×C4×C61—44no—
262C2×C4×C61—42no—
263C2×C4×C61—42no—
264C2×C4×C61—42no—
265C2×C4×C61—44no—
266C2×C4×C61—44no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—22no—
279C2×Dic61—232no—
280C2×Dic61—24no—
281C2×Dic61—24no—
282C2×Dic61—24no—
283C2×Dic61—24no—
284C2×Dic61—28no—
285C2×Dic61—416no—
286C2×Dic61—42no—
287C2×Dic61—42no—
288C2×Dic61—44no—
289C2×Dic61—44no—
290C2×Dic61—44no—
291C2×Dic61—42no—
292C2×Dic61—42no—
293C2×Dic61—42no—
294C2×Dic61—44no—
295C2×Dic61—44no—
296C2×Dic61C2×C22⋊C414yes—
297C2×Dic61C2×C22⋊C414yes—
298#13868—611nomaximal, Sylow
299C22×C61—24no—
300C22×C61—28no—
301C22×C61—28no—
302C22×C61—28no—
303C22×C61—28no—
304C22×C61—28no—
305C22×C61—28no—
306C22×C61—24no—
307C22×C61—28no—
308C22×C61—28no—
309C22×C61—28no—
310C22×C61—464no—
311C22×C61—44no—
312C22×C61—44no—
313C22×C61—44no—
314C22×C61—48no—
315C22×C61—48no—
316C22×C61—48no—
317C22×C61—48no—
318C22×C61—44no—
319C22×C61#38618yes—
320C22×C61C22×C22⋊C411yescharacteristic
321C22×C61C2×#11712yes—
322C22×C61C2×C4×D411yescharacteristic
323C22×C61C2×C4×D411yescharacteristic
324C22×C61#51812yes—
325C22×C61C2×#12011yescharacteristic
326C22×C61C2×#12111yescharacteristic
327C22×C61C2×#12111yescharacteristic
328C22×C61C2×#12311yescharacteristic
329C22×C61#52512yes—
330C22×C61#52512yes—
331C22×C61#53512yes—
332C22×C61#53512yes—
333C22×C61#54614yes—
334C22×C61#54614yes—
335C22×C61#55114yes—
336C22×C61#55114yes—
337C4×C61—44no—
338C4×C61—82no—
339Dic61—24no—
340Dic61—24no—
341Dic61—44no—
342Dic61—44no—
343Dic61—82no—
344C22×#117—611no—
345C23×C22⋊C4—611no—
346C2×#381—611no—
347C2×#379—611no—
348C2×#379D6111yescharacteristic
349#1439—614no—
350#1437—614no—
351C22⋊#95—611no—
352C22⋊#95—611no—
353C2×C61—28no—
354C2×C61—48no—
355C2×C61#228011yescharacteristic
356C2×C61C2×#38611yescharacteristic
357C2×C61#230911yescharacteristic
358C2×C61#234212yes—
359C2×C61#234212yes—
360C2×C61#240412yes—
361C2×C61#241112yes—
362C2×C61#241112yes—
363C2×C61#248212yes—
364C26C2×D6111yescharacteristic
365C24×C4—611no—
366C24×C4—611no—
367C2×#117—1224no—
368C2×#117—1224no—
369C2×#117—1224no—
370C22×C4⋊C4—612no—
371C22×C22⋊C4—618no—
372C22×C22⋊C4—612no—
373C22×C22⋊C4—612no—
374C22×C22⋊C4—611no—
375C22×C22⋊C4—611no—
376C22×C22⋊C4—614no—
377C22×C22⋊C4—1224no—
378C22×C22⋊C4—1224no—
379C22×C22⋊C4C2×D6111yescharacteristic
380C22×C22⋊C4C2×D6114yes—
381C22×C22⋊C4C2×D6111yescharacteristic
382C22×C42—612no—
383#381—1224no—
384#379—24no—
385#379—1224no—
386C2×#95—612no—
387C2×#95—612no—
388C2×#95—614no—
389C2×#95—614no—
390C61#1386811yescharacteristic, complemented, Sylow
391C25—22no—
392C25—22no—
393C25—28no—
394C25—22no—
395C25—22no—
396C25—22no—
397C25—22no—
398C25—44no—
399C25C4×D6114yes—
400C25C22×D6112yes—
401C25C22×D6111yescharacteristic
402C23×C4—614no—
403C23×C4—614no—
404C23×C4—614no—
405C23×C4—611no—
406C23×C4—611no—
407C23×C4—612no—
408C23×C4—612no—
409C23×C4—614no—
410C23×C4—612no—
411C23×C4—1222no—
412C23×C4—1222no—
413C23×C4—1222no—
414C23×C4—1222no—
415C23×C4—1222no—
416C23×C4—1222no—
417C23×C4—2444no—
418C23×C4C22×D6112yes—
419C23×C4C22×D6112yes—
420#117—12232no—
421C2×C4⋊C4—1224no—
422C2×C4⋊C4—1224no—
423C2×C4⋊C4—1224no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4—28no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4—28no—
431C2×C22⋊C4—44no—
432C2×C22⋊C4—618no—
433C2×C22⋊C4—618no—
434C2×C22⋊C4—614no—
435C2×C22⋊C4—1224no—
436C2×C22⋊C4—1224no—
437C2×C22⋊C4—1224no—
438C2×C22⋊C4—1224no—
439C2×C22⋊C4—12232no—
440C2×C22⋊C4—1222no—
441C2×C22⋊C4—1222no—
442C2×C22⋊C4—1228no—
443C2×C22⋊C4—1222no—
444C2×C22⋊C4—1222no—
445C2×C22⋊C4—1222no—
446C2×C22⋊C4—1228no—
447C2×C22⋊C4—2444no—
448C2×C22⋊C4C22⋊D6114yes—
449C2×C42—1224no—
450C2×C42—1224no—
451C2×C42—1224no—
452#95—1224no—
453#95—1224no—
454#95—2444no—
455#95—2444no—
456C24—24no—
457C24—24no—
458C24—216no—
459C24—216no—
460C24—22no—
461C24—22no—
462C24—22no—
463C24—22no—
464C24—24no—
465C24—22no—
466C24—22no—
467C24—24no—
468C24—24no—
469C24—28no—
470C24—216no—
471C24—28no—
472C24—28no—
473C24—48no—
474C24—416no—
475C24—416no—
476C24—416no—
477C24—416no—
478C24—48no—
479C24—48no—
480C24—44no—
481C24—44no—
482C24—44no—
483C24—44no—
484C24C2×C4×D6114yes—
485C24C2×C4×D6112yes—
486C24#1175822212yes—
487C24D4×D6112yes—
488C24D4×D6112yes—
489C24D4×D6112yes—
490C24#1175822418yes—
491C24#1175822412yes—
492C24C2×C22⋊D6112yes—
493C24C23×D6111yescharacteristic
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—612no—
504C22×C4—612no—
505C22×C4—612no—
506C22×C4—612no—
507C22×C4—612no—
508C22×C4—612no—
509C22×C4—6116no—
510C22×C4—614no—
511C22×C4—614no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1228no—
521C22×C4—1228no—
522C22×C4—1228no—
523C22×C4—1228no—
524C22×C4—1228no—
525C22×C4—1228no—
526C22×C4—1222no—
527C22×C4—1222no—
528C22×C4—1222no—
529C22×C4—1222no—
530C22×C4—1224no—
531C22×C4—1224no—
532C22×C4—1224no—
533C22×C4—1224no—
534C22×C4—1228no—
535C22×C4—1222no—
536C22×C4—1222no—
537C22×C4—1224no—
538C22×C4—1222no—
539C22×C4—1224no—
540C22×C4—2448no—
541C22×C4—2448no—
542C22×C4—2448no—
543C22×C4C2×C22⋊D6114yes—
544C4⋊C4—12216no—
545C22⋊C4—28no—
546C22⋊C4—416no—
547C22⋊C4—44no—
548C22⋊C4—12216no—
549C22⋊C4—12216no—
550C22⋊C4—1228no—
551C22⋊C4—24416no—
552C22⋊C4—2444no—
553C42—12216no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—28no—
561C23—28no—
562C23—28no—
563C23—28no—
564C23—232no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—28no—
572C23—28no—
573C23—28no—
574C23—28no—
575C23—28no—
576C23—28no—
577C23—432no—
578C23—432no—
579C23—432no—
580C23—432no—
581C23—416no—
582C23—432no—
583C23—432no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—416no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23C22⋊C4×D6114yes—
596C23#1451973318yescomplemented
597C23#1451973714yes—
598C23C4×C22⋊D6114yes—
599C23#1451974514yes—
600C23#1451974814yes—
601C23#1451975014yes—
602C23#1451975914yes—
603C23C22×C4×D6111yescharacteristic
604C23C2×D4×D6111yescharacteristic
605C23C2×D4×D6111yescharacteristic
606C23C2×D4×D6111yescharacteristic
607C23C22×C22⋊D6111yescharacteristic
608C23C2×#1175822414yes—
609C23C2×#1175822411yescharacteristic
610C23C2×#1175822211yescharacteristic
611C23#1451982214yes—
612C2×C4—24no—
613C2×C4—28no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—614no—
623C2×C4—614no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1222no—
631C2×C4—1222no—
632C2×C4—1222no—
633C2×C4—1222no—
634C2×C4—1222no—
635C2×C4—1222no—
636C2×C4—12232no—
637C2×C4—1224no—
638C2×C4—1224no—
639C2×C4—1224no—
640C2×C4—1224no—
641C2×C4—1228no—
642C2×C4—2442no—
643C2×C4—2442no—
644C2×C4—2442no—
645C2×C4—2444no—
646C2×C4—2444no—
647C2×C4—24416no—
648C2×C4—2442no—
649C2×C4—2442no—
650C2×C4—2444no—
651C2×C4—2444no—
652C2×C4—2444no—
653C22—24no—
654C22—28no—
655C22—28no—
656C22—24no—
657C22—28no—
658C22—28no—
659C22—28no—
660C22—28no—
661C22—28no—
662C22—28no—
663C22—28no—
664C22—464no—
665C22—44no—
666C22—48no—
667C22—48no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—48no—
672C22—48no—
673C22#1683963818yes—
674C22C2×C22⋊C4×D6111yescharacteristic
675C22C2×#1451973312yes—
676C22C2×#1451973711yescharacteristic
677C22C2×C4×C22⋊D6111yescharacteristic
678C22#1684023512yes—
679C22C2×#1451974511yescharacteristic
680C22C2×#1451974811yescharacteristic
681C22C2×#1451975011yescharacteristic
682C22C2×#1451975911yescharacteristic
683C22#1684026812yes—
684C22#1684026812yes—
685C22#1684032712yes—
686C22#1684032712yes—
687C22#1684039914yes—
688C22#1684040614yes—
689C22#1684043814yes—
690C22#1684044314yes—
691C4—44no—
692C4—82no—
693C4—1224no—
694C4—1224no—
695C4—4882no—
696C4—2444no—
697C4—2444no—
698C2—28no—
699C2—48no—
700C2#2144819711yescharacteristic
701C2C2×#1683963811yescharacteristic
702C2#2144840011yescharacteristic
703C2#2144866412yes—
704C2#2144867012yes—
705C2#2144913512yes—
706C2#2144920012yes—
707C2#2144920612yes—
708C2#2145009012yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#379C2
socle, cosocleC24×C61C24
prime radicalC26×C61C22
hypercenterC2×#379D61
hypocenterC61#13868
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
1. C22×#14519733C2semidirect
3. C2×#16839623C2semidirect
5. #21444062C2semidirect
6. #21444056C2semidirect
7. #21444041C2semidirect
8. #21444041C2semidirect
15. C22×#11758196C22semidirect
18. C22×C22⋊Dic61C4semidirect
24. C22×C4×Dic61C22semidirect
27. C2×#14519684C22semidirect
28. C2×#14519684C22semidirect
29. C2×#14519684C22semidirect
30. C2×#14519684C22semidirect
180. C22×Dic61C22⋊C4semidirect
241. C23×C61#117semidirect
390. C61#13868semidirect
596. C23#14519733semidirect

Maximal subgroups

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

subgroupindex
C22×#145197332
C23×C22⋊Dic612
C2×#168396232
C2×#168396182
#214440622
#214440562
#214440412
C2×C61×#3792
#1386861