Finite Group Databasebeta

← #29609161

Subgroups — #29609161

Order 15616.

1558 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839665C211yesmaximal, characteristic
2#21444066C212yesmaximal, complemented
3#21444066C212yesmaximal, complemented
4#21444060C211yesmaximal, characteristic
5#21444056C212yesmaximal, complemented
6#21444056C212yesmaximal, complemented
7#21444053C212yesmaximal, complemented
8#21444045C212yesmaximal, complemented
9C61×#1438C211yesmaximal, characteristic
10C22×C4⋊Dic61C2211yescharacteristic
11C22×C22⋊Dic61C2211yescharacteristic
12C22×C22⋊Dic61C2212yes—
13C22×C22⋊Dic61C2211yescharacteristic
14C22×C22⋊Dic61C2212yes—
15C22×C22⋊Dic61C2212yes—
16C22×C4×Dic61C2212yes—
17C22×C4×Dic61C2211yescharacteristic
18#16839665C418yescomplemented
19C2×#14519684C2212yes—
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C23×D4×C61C2211yescharacteristic
28C22×C22⋊C4×C61C2211yescharacteristic
29C22×C22⋊C4×C61C2212yes—
30C22×C22⋊C4×C61C2211yescharacteristic
31C2×#95×C61C2212yes—
32C23×Dic61C2312yes—
33C23×Dic61C2312yes—
34C23×Dic61C2312yes—
35C23×Dic61C2312yes—
36C2×C4⋊Dic61—41no—
37C2×C4⋊Dic61—41no—
38C2×C4⋊Dic61C2×C414yes—
39C2×C22⋊Dic61—24no—
40C2×C22⋊Dic61—24no—
41C2×C22⋊Dic61—42no—
42C2×C22⋊Dic61—42no—
43C2×C22⋊Dic61—42no—
44C2×C22⋊Dic61—42no—
45C2×C22⋊Dic61—41no—
46C2×C22⋊Dic61—42no—
47C2×C22⋊Dic61—42no—
48C2×C22⋊Dic61—41no—
49C2×C22⋊Dic61—42no—
50C2×C22⋊Dic61C2×C418yes—
51C2×C22⋊Dic61C2×C418yes—
52C2×C4×Dic61—42no—
53C2×C4×Dic61—42no—
54C2×C4×Dic61—42no—
55C2×C4×Dic61—42no—
56C2×C4×Dic61C2×C414yes—
57#14519684—24no—
58#14519684—24no—
59#14519684—24no—
60#14519684—42no—
61#14519684—42no—
62#14519684—42no—
63#14519684—42no—
64#14519684—42no—
65C25×C61C2311yescharacteristic
66C25×C61C2311yescharacteristic
67C22×D4×C61—24no—
68C22×D4×C61—42no—
69C22×D4×C61—41no—
70C22×D4×C61—41no—
71C22×D4×C61C2×C414yes—
72C23×C4×C61C2312yes—
73C23×C4×C61C2312yes—
74C23×C4×C61C2311yescharacteristic
75C2×C22⋊C4×C61—24no—
76C2×C22⋊C4×C61—24no—
77C2×C22⋊C4×C61—41no—
78C2×C22⋊C4×C61—42no—
79C2×C22⋊C4×C61—42no—
80C2×C22⋊C4×C61—42no—
81C2×C22⋊C4×C61—41no—
82#95×C61—24no—
83C24×C61—22no—
84C24×C61—22no—
85C24×C61—22no—
86C24×C61—22no—
87C24×C61—22no—
88C24×C61—22no—
89C24×C61—22no—
90C24×C61—21no—
91C24×C61—21no—
92C24×C61—22no—
93C24×C61—22no—
94C24×C61—22no—
95C24×C61—21no—
96C24×C61—21no—
97C24×C61—22no—
98C24×C61—22no—
99C24×C61C22×C412yes—
100C24×C61C22×C412yes—
101C24×C61C2411yescharacteristic
102C2×D4×C61—28no—
103C2×D4×C61—44no—
104C2×D4×C61—44no—
105C2×D4×C61—48no—
106C2×D4×C61—48no—
107C22×C4×C61—22no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61C22×C412yes—
124C22×C4×C61#4114yes—
125C22×C4×C61#4114yes—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61C22×C414yes—
152C22×Dic61C22×C414yes—
153C22×Dic61#4114yes—
154C22⋊C4×C61—44no—
155C22⋊C4×C61—48no—
156C22⋊C4×C61—44no—
157C22⋊Dic61—48no—
158C22⋊Dic61—48no—
159C22⋊Dic61—44no—
160C22⋊Dic61—48no—
161C22⋊Dic61—44no—
162C4⋊Dic61—44no—
163C4×Dic61—44no—
164C4×Dic61—48no—
165C23×C61—22no—
166C23×C61—22no—
167C23×C61—24no—
168C23×C61—24no—
169C23×C61—24no—
170C23×C61—24no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—42no—
195C23×C61—42no—
196C23×C61—42no—
197C23×C61—42no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61—42no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61C23×C411yescharacteristic
208C23×C61C2×#4112yes—
209C23×C61C2×#4112yes—
210C23×C61C2×#4112yes—
211C23×C61H(5,𝔽2)12yes—
212C23×C61H(5,𝔽2)11yescharacteristic
213C23×C61H(5,𝔽2)11yescharacteristic
214C23×C61H(5,𝔽2)12yes—
215C23×C61H(5,𝔽2)12yes—
216D4×C61—416no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61#11714yes—
236C2×Dic61—24no—
237C2×Dic61—24no—
238C2×Dic61—24no—
239C2×Dic61—24no—
240C2×Dic61—24no—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61—24no—
244C2×Dic61—24no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264#14503—611nomaximal, Sylow
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—44no—
268C22×C61—44no—
269C22×C61—44no—
270C22×C61—44no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—82no—
276C22×C61—82no—
277C22×C61—82no—
278C22×C61—82no—
279C22×C61—82no—
280C22×C61—82no—
281C22×C61—82no—
282C22×C61—82no—
283C22×C61C2×#11711yescharacteristic
284C22×C61C4×#4112yes—
285C22×C61#51812yes—
286C22×C61#51811yescharacteristic
287C22×C61#51811yescharacteristic
288C22×C61#53812yes—
289C22×C61#53812yes—
290C22×C61#53812yes—
291C22×C61#53812yes—
292C22×C61#54012yes—
293C22×C61#54012yes—
294C22×C61#55111yescharacteristic
295C22×C61#55111yescharacteristic
296C22×C61#55312yes—
297C22×C61#55312yes—
298C22×C61#55512yes—
299C22×C61#55912yes—
300C22×C61#55912yes—
301C22×C61#56012yes—
302C22×C61#56011yescharacteristic
303C22×C61#56011yescharacteristic
304C4×C61—24no—
305C4×C61—44no—
306C4×C61—82no—
307Dic61—44no—
308Dic61—44no—
309Dic61—44no—
310Dic61—82no—
311C2×#390—611no—
312#1440—612no—
313#1440—612no—
314#1438—611no—
315#1438D6111yescharacteristic
316#1437—612no—
317#1437—612no—
318#1436—612no—
319#1434—612no—
320C2×C61—82no—
321C2×C61—82no—
322C2×C61#231211yescharacteristic
323C2×C61#233312yes—
324C2×C61#235712yes—
325C2×C61#237211yescharacteristic
326C2×C61#238011yescharacteristic
327C2×C61#279712yes—
328C2×C61#279712yes—
329C2×C61#280312yes—
330C2×C61#280312yes—
331C23×D4C2×D6111yescharacteristic
332C22×C4⋊C4—611no—
333C22×C22⋊C4—612no—
334C22×C22⋊C4—612no—
335C22×C22⋊C4—611no—
336C22×C22⋊C4—612no—
337C22×C22⋊C4—611no—
338C22×C22⋊C4C2×D6111yescharacteristic
339C22×C22⋊C4C2×D6112yes—
340C22×C22⋊C4C2×D6111yescharacteristic
341C22×C42—611no—
342C22×C42—612no—
343#390—618no—
344C2×#95—612no—
345C2×#95—612no—
346C2×#95—612no—
347C2×#95—612no—
348C2×#95—612no—
349C2×#95—612no—
350C2×#95—612no—
351C2×#95—612no—
352C2×#95C2×D6112yes—
353C61#1450311yescharacteristic, complemented, Sylow
354C25C22×D6111yescharacteristic
355C25C22×D6111yescharacteristic
356C22×D4—24no—
357C22×D4—42no—
358C22×D4—41no—
359C22×D4—41no—
360C22×D4C4×D6114yes—
361C23×C4—612no—
362C23×C4—612no—
363C23×C4—612no—
364C23×C4—612no—
365C23×C4C22×D6112yes—
366C23×C4C22×D6112yes—
367C23×C4C22×D6111yescharacteristic
368C2×C4⋊C4—614no—
369C2×C4⋊C4—2441no—
370C2×C4⋊C4—2441no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—41no—
374C2×C22⋊C4—42no—
375C2×C22⋊C4—42no—
376C2×C22⋊C4—41no—
377C2×C22⋊C4—42no—
378C2×C22⋊C4—618no—
379C2×C22⋊C4—618no—
380C2×C22⋊C4—1224no—
381C2×C22⋊C4—1224no—
382C2×C22⋊C4—2442no—
383C2×C22⋊C4—2442no—
384C2×C22⋊C4—2442no—
385C2×C22⋊C4—2442no—
386C2×C22⋊C4—2441no—
387C2×C22⋊C4—2442no—
388C2×C22⋊C4—2442no—
389C2×C22⋊C4—2441no—
390C2×C22⋊C4—2442no—
391C2×C42—614no—
392C2×C42—2442no—
393C2×C42—2442no—
394C2×C42—2442no—
395C2×C42—2442no—
396#95—24no—
397#95—1224no—
398#95—1224no—
399#95—1224no—
400#95—2442no—
401#95—2442no—
402#95—2442no—
403#95—2442no—
404#95—2442no—
405C24—22no—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—21no—
414C24—21no—
415C24—22no—
416C24—21no—
417C24—21no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24C2×C4×D6112yes—
422C24C2×C4×D6112yes—
423C24C23×D6111yescharacteristic
424C2×D4—28no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—48no—
428C2×D4—48no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—614no—
446C22×C4—614no—
447C22×C4—614no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4C2×C4×D6112yes—
474C22×C4#1175822414yes—
475C22×C4#1175822414yes—
476C4⋊C4—2444no—
477C22⋊C4—44no—
478C22⋊C4—48no—
479C22⋊C4—44no—
480C22⋊C4—2448no—
481C22⋊C4—2448no—
482C22⋊C4—2444no—
483C22⋊C4—2448no—
484C22⋊C4—2444no—
485C42—2444no—
486C42—2448no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C22×C4×D6111yescharacteristic
530C23#41×D6112yes—
531C23C2×#1175822412yes—
532C23C2×#1175822412yes—
533C23#1451982212yes—
534C23#1451982211yescharacteristic
535C23#1451982211yescharacteristic
536C23#1451982212yes—
537C23#1451982312yes—
538D4—416no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—1224no—
558C2×C4—1224no—
559C2×C4—1224no—
560C2×C4—1224no—
561C2×C4—1224no—
562C2×C4—1224no—
563C2×C4—1224no—
564C2×C4—1224no—
565C2×C4—1224no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4—2442no—
573C2×C4—2442no—
574C2×C4—2442no—
575C2×C4—2442no—
576C2×C4—2442no—
577C2×C4—2442no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4#1451973314yes—
586C22—42no—
587C22—42no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22C2×#1451973311yescharacteristic
605C22C4×#1175822412yes—
606C22#1684023412yes—
607C22#1684023511yescharacteristic
608C22#1684023511yescharacteristic
609C22#1684034712yes—
610C22#1684034712yes—
611C22#1684034712yes—
612C22#1684034812yes—
613C22#1684036212yes—
614C22#1684036512yes—
615C22#1684044011yescharacteristic
616C22#1684044311yescharacteristic
617C22#1684045912yes—
618C22#1684046412yes—
619C22#1684047612yes—
620C22#1684049412yes—
621C22#1684049812yes—
622C22#1684050012yes—
623C22#1684050111yescharacteristic
624C22#1684050211yescharacteristic
625C4—24no—
626C4—44no—
627C4—82no—
628C4—4882no—
629C4—2444no—
630C4—2444no—
631C4—2444no—
632C2—82no—
633C2—82no—
634C2#2144841511yescharacteristic
635C2#2144858012yes—
636C2#2144879112yes—
637C2#2144892811yescharacteristic
638C2#2144899411yescharacteristic
639C2#2145367812yes—
640C2#2145368312yes—
641C2#2145375412yes—
642C2#2145375812yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1438C2
socle, cosocleC24×C61C24
prime radicalC23×D4×C61C22
hypercenter#1438D61
hypocenterC61#14503
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. #21444066C2semidirect
3. #21444066C2semidirect
5. #21444056C2semidirect
6. #21444056C2semidirect
7. #21444053C2semidirect
8. #21444045C2semidirect
18. #16839665C4semidirect
353. C61#14503semidirect

Maximal subgroups

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

subgroupindex
C2×#168396652
#214440662
#214440602
#214440562
#214440532
#214440452
C61×#14382
#1450361