Finite Group Databasebeta

← #29602724

Subgroups — #29602724

Order 15616.

2230 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519737C211yesmaximal, characteristic
2C2×#16839629C211yesmaximal, characteristic, complemented
3C2×#16839608C211yesmaximal, characteristic, complemented
4#21444085C212yesmaximal, complemented
5#21444058C212yesmaximal
6#21444057C211yesmaximal, characteristic, complemented
7#21444056C211yesmaximal, characteristic, complemented
8#21444055C211yesmaximal, characteristic, complemented
9#21444041C212yesmaximal, complemented
10#21444040C212yesmaximal, complemented
11C61×#1437C211yesmaximal, characteristic, complemented
12C23×C22⋊D61C2211yescharacteristic
13C24×Dic61C2211yescharacteristic
14C23×C4×D61C2211yescharacteristic
15C2×#14519737—24no—
16C2×#14519737—24no—
17C2×#14519737C418yescomplemented
18C22×#11758196C2211yescharacteristic
19C22×C22⋊Dic61C2212yes—
20C22×#11758198C2212yes—
21C22×#11758198C2212yes—
22C22×#11758198C2211yescharacteristic
23C22×C4×Dic61C2211yescharacteristic
24C22×C4×Dic61C2211yescharacteristic
25#16839629—24no—
26#16839608—24no—
27C2×#14519684C2212yes—
28C2×#14519684C2211yescharacteristic
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C2×#14519684C2212yes—
34C2×#14519684C2212yes—
35C22×C22⋊C4×C61C2212yes—
36C22×C22⋊C4×C61C2211yescharacteristic
37C22×C42×C61C2211yescharacteristic
38C2×#95×C61C2211yescharacteristic
39C2×#95×C61C2212yes—
40C24×D61C2311yescharacteristic
41C22×C22⋊D61—24no—
42C22×C22⋊D61—24no—
43C22×C22⋊D61—22no—
44C22×C22⋊D61—22no—
45C22×C22⋊D61C2×C414yes—
46C23×Dic61—22no—
47C23×Dic61—22no—
48C23×Dic61—42no—
49C23×Dic61—42no—
50C23×Dic61C2×C414yes—
51C23×Dic61C2312yes—
52C23×Dic61C2311yescharacteristic
53C23×Dic61C2312yes—
54C23×Dic61C2311yescharacteristic
55C23×Dic61C2311yescharacteristic
56C22×C4×D61—22no—
57C22×C4×D61—22no—
58C22×C4×D61—42no—
59C22×C4×D61—42no—
60C22×C4×D61C2×C414yes—
61#14519737—232no—
62C2×#11758196—22no—
63C2×#11758196—22no—
64C2×#11758196C2×C414yes—
65C2×C22⋊Dic61—24no—
66C2×C22⋊Dic61—24no—
67C2×C22⋊Dic61—42no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×#11758198—24no—
71C2×#11758198—24no—
72C2×#11758198—24no—
73C2×#11758198—24no—
74C2×#11758198—42no—
75C2×#11758198—42no—
76C2×#11758198C2×C414yes—
77C2×C4×Dic61—22no—
78C2×C4×Dic61—24no—
79C2×C4×Dic61C2×C414yes—
80C2×C4×Dic61D414yes—
81C2×C4×Dic61D414yes—
82#14519684—22no—
83#14519684—24no—
84#14519684—24no—
85#14519684—24no—
86#14519684—24no—
87#14519684—42no—
88#14519684—42no—
89#14519684—42no—
90C25×C61C2311yescharacteristic
91C23×C4×C61C2312yes—
92C23×C4×C61C2311yescharacteristic
93C23×C4×C61C2312yes—
94C23×C4×C61C2311yescharacteristic
95C2×C22⋊C4×C61—22no—
96C2×C22⋊C4×C61—22no—
97C2×C22⋊C4×C61—24no—
98C2×C22⋊C4×C61—24no—
99C2×C22⋊C4×C61—42no—
100C2×C22⋊C4×C61C2×C414yes—
101C2×C42×C61—22no—
102C2×C42×C61—24no—
103#95×C61—22no—
104#95×C61—42no—
105C24×C61—21no—
106C24×C61—21no—
107C24×C61—22no—
108C24×C61—22no—
109C24×C61—22no—
110C24×C61—22no—
111C24×C61—22no—
112C24×C61C22×C412yes—
113C24×C61#4114yes—
114C24×C61C2411yescharacteristic
115C23×D61—21no—
116C23×D61—21no—
117C23×D61—22no—
118C23×D61—22no—
119C23×D61—22no—
120C23×D61—22no—
121C23×D61—22no—
122C23×D61C22×C412yes—
123C23×D61#4114yes—
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×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61C22×C412yes—
144C22×C4×C61C22×C412yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D412yes—
147C22×C4×C61#4114yes—
148C2×C22⋊D61—216no—
149C2×C22⋊D61—216no—
150C2×C22⋊D61—28no—
151C2×C22⋊D61—28no—
152C2×C22⋊D61—28no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61—22no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61—28no—
168C22×Dic61—22no—
169C22×Dic61—22no—
170C22×Dic61—22no—
171C22×Dic61—22no—
172C22×Dic61—22no—
173C22×Dic61—22no—
174C22×Dic61—22no—
175C22×Dic61—48no—
176C22×Dic61—44no—
177C22×Dic61—48no—
178C22×Dic61—44no—
179C22×Dic61C22×C412yes—
180C22×Dic61C22×C412yes—
181C22×Dic61C22×C412yes—
182C22×Dic61C2×D412yes—
183C22×Dic61C2×D412yes—
184C22×Dic61C2×D412yes—
185C22×Dic61C2×D412yes—
186C22×Dic61#4114yes—
187C2×C4×D61—28no—
188C2×C4×D61—48no—
189C2×C4×D61—44no—
190C2×C4×D61—48no—
191C2×C4×D61—44no—
192C22⋊C4×C61—28no—
193C22⋊C4×C61—48no—
194C42×C61—28no—
195C22⋊Dic61—48no—
196#11758198—28no—
197#11758198—48no—
198#11758198—48no—
199#11758196—28no—
200C4×Dic61—28no—
201C4×Dic61C22⋊C4116yescomplemented
202C23×C61—22no—
203C23×C61—22no—
204C23×C61—24no—
205C23×C61—24no—
206C23×C61—24no—
207C23×C61—24no—
208C23×C61—24no—
209C23×C61—24no—
210C23×C61—24no—
211C23×C61—24no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—42no—
215C23×C61—42no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—42no—
219C23×C61—42no—
220C23×C61—42no—
221C23×C61—42no—
222C23×C61#11714yes—
223C23×C61C23×C411yescharacteristic
224C23×C61C22×D411yescharacteristic
225C23×C61C22×D411yescharacteristic
226C23×C61C2×#4112yes—
227C23×C61C2×#4112yes—
228C23×C61C2×#4112yes—
229C23×C61C2×#4112yes—
230C23×C61H(5,𝔽2)12yes—
231C23×C61H(5,𝔽2)12yes—
232C22×D61—22no—
233C22×D61—22no—
234C22×D61—24no—
235C22×D61—24no—
236C22×D61—24no—
237C22×D61—24no—
238C22×D61—24no—
239C22×D61—24no—
240C22×D61—24no—
241C22×D61—24no—
242C22×D61—41no—
243C22×D61—41no—
244C22×D61—42no—
245C22×D61—42no—
246C22×D61—41no—
247C22×D61—41no—
248C22×D61—42no—
249C22×D61—42no—
250C22×D61—42no—
251C22×D61—42no—
252C22×D61#11714yes—
253C2×C4×C61—24no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—24no—
257C2×C4×C61—22no—
258C2×C4×C61—24no—
259C2×C4×C61—24no—
260C2×C4×C61—24no—
261C2×C4×C61—24no—
262C2×C4×C61—42no—
263C2×C4×C61—41no—
264C2×C4×C61—42no—
265C2×C4×C61—41no—
266C2×C4×C61—42no—
267C2×C4×C61—42no—
268C2×C4×C61—42no—
269C2×C4×C61—42no—
270C2×C4×C61—42no—
271C2×C4×C61—42no—
272C2×C4×C61—42no—
273C2×C4×C61—42no—
274C2×C4×C61—42no—
275C2×C4×C61—42no—
276C2×C4×C61C2×C22⋊C414yes—
277C22⋊D61—232no—
278C2×Dic61—22no—
279C2×Dic61—22no—
280C2×Dic61—24no—
281C2×Dic61—24no—
282C2×Dic61—24no—
283C2×Dic61—22no—
284C2×Dic61—24no—
285C2×Dic61—22no—
286C2×Dic61—22no—
287C2×Dic61—24no—
288C2×Dic61—24no—
289C2×Dic61—24no—
290C2×Dic61—24no—
291C2×Dic61—416no—
292C2×Dic61—42no—
293C2×Dic61—42no—
294C2×Dic61—41no—
295C2×Dic61—42no—
296C2×Dic61—41no—
297C2×Dic61—42no—
298C2×Dic61—42no—
299C2×Dic61—42no—
300C2×Dic61—42no—
301C2×Dic61—42no—
302C2×Dic61—42no—
303C2×Dic61—42no—
304C2×Dic61—42no—
305C2×Dic61—42no—
306C2×Dic61C2×C22⋊C414yes—
307C2×Dic61C2×C22⋊C414yes—
308C4×D61—416no—
309#13959—611nomaximal, Sylow
310C22×C61—28no—
311C22×C61—24no—
312C22×C61—42no—
313C22×C61—44no—
314C22×C61—44no—
315C22×C61—44no—
316C22×C61—44no—
317C22×C61—44no—
318C22×C61—42no—
319C22×C61C22×C22⋊C411yescharacteristic
320C22×C61C2×#11711yescharacteristic
321C22×C61C2×#11711yescharacteristic
322C22×C61C4×#4112yes—
323C22×C61#51812yes—
324C22×C61#52511yescharacteristic
325C22×C61#52511yescharacteristic
326C22×C61#52512yes—
327C22×C61#52512yes—
328C22×C61#53212yes—
329C22×C61#53511yescharacteristic
330C22×C61#53512yes—
331C22×C61#53511yescharacteristic
332C22×C61#55112yes—
333C22×C61#55112yes—
334C22×C61#55112yes—
335C22×C61#55112yes—
336C22×C61#55312yes—
337C22×C61#55312yes—
338C22×C61#55312yes—
339C22×C61#55312yes—
340C2×D61—28no—
341C2×D61—24no—
342C2×D61—42no—
343C2×D61—44no—
344C2×D61—44no—
345C2×D61—44no—
346C2×D61—44no—
347C2×D61—44no—
348C2×D61—42no—
349C4×C61—24no—
350C4×C61—42no—
351C4×C61—44no—
352C4×C61—82no—
353Dic61—24no—
354Dic61—24no—
355Dic61—42no—
356Dic61—44no—
357Dic61—82no—
358C22×C4×D4—611no—
359C2×#383—611no—
360C2×#376—611no—
361#1445—612no—
362#1438—612no—
363#1437—611no—
364#1437—611no—
365#1437—611no—
366#1437D6111yescharacteristic, complemented
367C22⋊#95—612no—
368C22⋊#95—612no—
369C2×C61—44no—
370C2×C61#228011yescharacteristic
371C2×C61#229711yescharacteristic
372C2×C61#230911yescharacteristic
373C2×C61#235312yes—
374C2×C61#235312yes—
375C2×C61#259812yes—
376C2×C61#259812yes—
377C2×C61#260312yes—
378C2×C61#260312yes—
379D61—44no—
380C23×D4—611no—
381C24×C4—611no—
382C24×C4—611no—
383C2×C4×D4—618no—
384C2×C4×D4—1224no—
385C2×C4×D4—1224no—
386C22×C4⋊C4—611no—
387C22×C22⋊C4—611no—
388C22×C22⋊C4—612no—
389C22×C22⋊C4—612no—
390C22×C22⋊C4—612no—
391C22×C22⋊C4C2×D6112yes—
392C22×C22⋊C4C2×D6111yescharacteristic
393C22×C42—611no—
394C22×C42—611no—
395C22×C42C2×D6111yescharacteristic
396#383—1224no—
397#376—1224no—
398C2×#95—612no—
399C2×#95—612no—
400C2×#95—611no—
401C2×#95—612no—
402C2×#95—612no—
403C2×#95—612no—
404C2×#95—612no—
405C2×#95—612no—
406C2×#95C2×D6111yescharacteristic
407C2×#95C2×D6112yes—
408C61#1395911yescharacteristic, complemented, Sylow
409C25—611no—
410C25C22×D6111yescharacteristic
411C22×D4—614no—
412C22×D4—1222no—
413C22×D4—1222no—
414C22×D4—1224no—
415C22×D4—1224no—
416C23×C4—611no—
417C23×C4—611no—
418C23×C4—612no—
419C23×C4—611no—
420C23×C4—614no—
421C23×C4—614no—
422C23×C4—612no—
423C23×C4—1222no—
424C23×C4—1222no—
425C23×C4—1222no—
426C23×C4—1222no—
427C23×C4—2442no—
428C23×C4—2442no—
429C23×C4—2442no—
430C23×C4—2442no—
431C23×C4C22×D6112yes—
432C23×C4C22×D6111yescharacteristic
433C23×C4C22×D6112yes—
434C23×C4C22×D6111yescharacteristic
435C4×D4—12232no—
436C2×C4⋊C4—614no—
437C2×C4⋊C4—1222no—
438C2×C4⋊C4—1222no—
439C2×C22⋊C4—22no—
440C2×C22⋊C4—22no—
441C2×C22⋊C4—24no—
442C2×C22⋊C4—24no—
443C2×C22⋊C4—42no—
444C2×C22⋊C4—614no—
445C2×C22⋊C4—1222no—
446C2×C22⋊C4—1222no—
447C2×C22⋊C4—1224no—
448C2×C22⋊C4—1224no—
449C2×C22⋊C4—1224no—
450C2×C22⋊C4—1224no—
451C2×C22⋊C4—1224no—
452C2×C22⋊C4—1224no—
453C2×C22⋊C4—2442no—
454C2×C22⋊C4—2442no—
455C2×C22⋊C4—2442no—
456C2×C22⋊C4C4×D6114yes—
457C2×C42—22no—
458C2×C42—24no—
459C2×C42—614no—
460C2×C42—614no—
461C2×C42—614no—
462C2×C42—1224no—
463C2×C42—1222no—
464#95—22no—
465#95—42no—
466#95—1224no—
467#95—1224no—
468#95—1224no—
469#95—1224no—
470#95—1222no—
471#95—2442no—
472#95—2442no—
473#95—2442no—
474C24—21no—
475C24—21no—
476C24—22no—
477C24—22no—
478C24—22no—
479C24—22no—
480C24—22no—
481C24—612no—
482C24—614no—
483C24—1221no—
484C24—1221no—
485C24—1222no—
486C24—1222no—
487C24—1222no—
488C24—1222no—
489C24—1222no—
490C24C2×C4×D6112yes—
491C24#1175822214yes—
492C24C23×D6111yescharacteristic
493C2×D4—1228no—
494C2×D4—1228no—
495C2×D4—12216no—
496C2×D4—12216no—
497C2×D4—1228no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—612no—
518C22×C4—612no—
519C22×C4—612no—
520C22×C4—612no—
521C22×C4—612no—
522C22×C4—612no—
523C22×C4—612no—
524C22×C4—614no—
525C22×C4—1228no—
526C22×C4—1228no—
527C22×C4—1222no—
528C22×C4—1222no—
529C22×C4—1222no—
530C22×C4—1222no—
531C22×C4—1222no—
532C22×C4—1221no—
533C22×C4—1221no—
534C22×C4—1222no—
535C22×C4—1222no—
536C22×C4—1222no—
537C22×C4—1222no—
538C22×C4—1222no—
539C22×C4—1222no—
540C22×C4—1222no—
541C22×C4—1222no—
542C22×C4—1222no—
543C22×C4—1222no—
544C22×C4—1222no—
545C22×C4—1222no—
546C22×C4—1222no—
547C22×C4—1222no—
548C22×C4—2448no—
549C22×C4—2444no—
550C22×C4—2448no—
551C22×C4—2444no—
552C22×C4—2448no—
553C22×C4—2444no—
554C22×C4—2448no—
555C22×C4—2444no—
556C22×C4C2×C4×D6112yes—
557C22×C4C2×C4×D6112yes—
558C22×C4D4×D6112yes—
559C22×C4D4×D6112yes—
560C22×C4#1175822414yes—
561C4⋊C4—1228no—
562C22⋊C4—28no—
563C22⋊C4—48no—
564C22⋊C4—1228no—
565C22⋊C4—2448no—
566C22⋊C4—2448no—
567C22⋊C4—2448no—
568C42—28no—
569C42—6116no—
570C42—1228no—
571C23—24no—
572C23—24no—
573C23—22no—
574C23—22no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—24no—
579C23—24no—
580C23—24no—
581C23—42no—
582C23—42no—
583C23—41no—
584C23—41no—
585C23—42no—
586C23—41no—
587C23—42no—
588C23—41no—
589C23—42no—
590C23—42no—
591C23—614no—
592C23—1224no—
593C23—1224no—
594C23—1222no—
595C23—1222no—
596C23—1224no—
597C23—1224no—
598C23—1224no—
599C23—1224no—
600C23—1224no—
601C23—1224no—
602C23—2442no—
603C23—2442no—
604C23—2441no—
605C23—2441no—
606C23—2442no—
607C23—2441no—
608C23—2442no—
609C23—2441no—
610C23—2442no—
611C23—2442no—
612C23#1451973414yes—
613C23C22×C4×D6111yescharacteristic
614C23C2×D4×D6111yescharacteristic
615C23C2×D4×D6111yescharacteristic
616C23#41×D6112yes—
617C23#41×D6112yes—
618C23C2×#1175822412yes—
619C23C2×#1175822212yes—
620C23#1451982212yes—
621C23#1451982312yes—
622D4—12232no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—614no—
647C2×C4—614no—
648C2×C4—1222no—
649C2×C4—1222no—
650C2×C4—1224no—
651C2×C4—1224no—
652C2×C4—1224no—
653C2×C4—1222no—
654C2×C4—1224no—
655C2×C4—1222no—
656C2×C4—1222no—
657C2×C4—1224no—
658C2×C4—1224no—
659C2×C4—1224no—
660C2×C4—1224no—
661C2×C4—24416no—
662C2×C4—24416no—
663C2×C4—2442no—
664C2×C4—2442no—
665C2×C4—2442no—
666C2×C4—2442no—
667C2×C4—2442no—
668C2×C4—2442no—
669C2×C4—2442no—
670C2×C4—2442no—
671C2×C4—2442no—
672C2×C4—2442no—
673C2×C4—2441no—
674C2×C4—2442no—
675C2×C4—2442no—
676C2×C4—2441no—
677C2×C4C22⋊C4×D6114yes—
678C22—24no—
679C22—28no—
680C22—44no—
681C22—42no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—42no—
687C22—1224no—
688C22—1228no—
689C22—2444no—
690C22—2442no—
691C22—2444no—
692C22—2444no—
693C22—2444no—
694C22—2444no—
695C22—2442no—
696C22C2×C22⋊C4×D6111yescharacteristic
697C22#117×D6111yescharacteristic
698C22C2×#1451973411yescharacteristic
699C22C4×#1175822412yes—
700C22#1684023412yes—
701C22#1684026811yescharacteristic
702C22#1684027011yescharacteristic
703C22#1684027112yes—
704C22#1684027112yes—
705C22#1684030712yes—
706C22#1684032511yescharacteristic
707C22#1684032812yes—
708C22#1684033011yescharacteristic
709C22#1684043512yes—
710C22#1684043612yes—
711C22#1684043812yes—
712C22#1684044212yes—
713C22#1684045812yes—
714C22#1684045912yes—
715C22#1684046112yes—
716C22#1684046312yes—
717C4—24no—
718C4—44no—
719C4—42no—
720C4—82no—
721C4—1224no—
722C4—1224no—
723C4—4882no—
724C4—2444no—
725C4—2442no—
726C2—44no—
727C2—2444no—
728C2#2144820011yescharacteristic
729C2#2144830411yescharacteristic
730C2#2144839811yescharacteristic
731C2#2144875312yes—
732C2#2144875412yes—
733C2#2145138112yes—
734C2#2145138312yes—
735C2#2145144212yes—
736C2#2145144412yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1437C2
socle, cosocleC24×C61C24
prime radicalC23×C22⋊D61C22
hypercenter#1437D61
hypocenterC61#13959
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. C2×#16839629C2semidirect
3. C2×#16839608C2semidirect
4. #21444085C2semidirect
6. #21444057C2semidirect
7. #21444056C2semidirect
8. #21444055C2semidirect
9. #21444041C2semidirect
10. #21444040C2semidirect
11. C61×#1437C2semidirect
17. C2×#14519737C4semidirect
201. C4×Dic61C22⋊C4semidirect
366. #1437D61semidirect
408. C61#13959semidirect

Maximal subgroups

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

subgroupindex
C22×#145197372
C2×#168396292
C2×#168396082
#214440852
#214440582
#214440572
#214440562
#214440552
#214440412
#214440402
C61×#14372
#1395961