Finite Group Databasebeta

← #29606982

Subgroups — #29606982

Order 15616.

1558 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839699C211yesmaximal, characteristic
2C2×#16839658C211yesmaximal, characteristic, complemented
3C2×#16839609C211yesmaximal, characteristic
4#21444088C211yesmaximal, characteristic, complemented
5#21444071C212yesmaximal, complemented
6#21444063C212yesmaximal
7#21444061C211yesmaximal, characteristic
8#21444053C212yesmaximal, complemented
9#21444044C211yesmaximal, characteristic, complemented
10#21444040C212yesmaximal
11C61×#1442C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14C22×C4⋊Dic61C2212yes—
15C22×#11758196C2212yes—
16C22×#11758198C2211yescharacteristic
17C22×#11758198C2211yescharacteristic
18C22×#11758198C2212yes—
19C22×#11758198C2212yes—
20C22×C4×Dic61C2211yescharacteristic
21C22×C4×Dic61C2211yescharacteristic
22#16839699C418yescomplemented
23#16839658—24no—
24#16839609—24no—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2211yescharacteristic
28C2×#14519684C2211yescharacteristic
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C22×C4⋊C4×C61C2211yescharacteristic
34C2×#95×C61C2211yescharacteristic
35C2×#95×C61C2212yes—
36C2×#95×C61C2212yes—
37C2×#95×C61C2211yescharacteristic
38C24×D61C2311yescharacteristic
39C23×Dic61C2312yes—
40C23×Dic61C2311yescharacteristic
41C23×Dic61C2312yes—
42C23×Dic61C2311yescharacteristic
43C23×Dic61C2311yescharacteristic
44C22×C4×D61—22no—
45C22×C4×D61—22no—
46C22×C4×D61—42no—
47C22×C4×D61—42no—
48C22×C4×D61Q814yes—
49C2×C4⋊Dic61—22no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61—42no—
52C2×C4⋊Dic61—42no—
53C2×C4⋊Dic61C2×C414yescomplemented
54C2×C4⋊Dic61C2×C418yescomplemented
55C2×#11758196—24no—
56C2×#11758196—42no—
57C2×#11758196—42no—
58C2×#11758198—22no—
59C2×#11758198—22no—
60C2×#11758198—22no—
61C2×#11758198—22no—
62C2×#11758198—22no—
63C2×#11758198—24no—
64C2×#11758198—42no—
65C2×#11758198—42no—
66C2×#11758198—42no—
67C2×#11758198—42no—
68C2×#11758198C2×C414yes—
69C2×#11758198C2×C418yes—
70C2×C4×Dic61—22no—
71C2×C4×Dic61—42no—
72C2×C4×Dic61—42no—
73C2×C4×Dic61D414yes—
74#14519684—22no—
75#14519684—22no—
76#14519684—24no—
77#14519684—24no—
78#14519684—42no—
79#14519684—42no—
80#14519684—42no—
81#14519684—42no—
82C23×C4×C61C2312yes—
83C23×C4×C61C2312yes—
84C23×C4×C61C2311yescharacteristic
85C23×C4×C61C2311yescharacteristic
86C23×C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61—42no—
89#95×C61—22no—
90#95×C61—24no—
91#95×C61—42no—
92#95×C61C2×C414yescomplemented
93C24×C61C2411yescharacteristic
94C23×D61—21no—
95C23×D61—21no—
96C23×D61—21no—
97C23×D61—21no—
98C23×D61—22no—
99C23×D61—22no—
100C23×D61—22no—
101C23×D61—22no—
102C23×D61C22×C412yes—
103C23×D61C2×Q812yes—
104C23×D61#4112yes—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—21no—
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×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61C22×C414yes—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C2×D412yes—
131C22×C4×C61C2×Q812yes—
132C22×C4×C61#4112yes—
133C22×C4×C61#4112yes—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—21no—
137C22×Dic61—22no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
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×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61C22×C414yes—
158C22×Dic61C22×C412yes—
159C22×Dic61C2×D412yes—
160C22×Dic61C2×D412yes—
161C22×Dic61C2×Q812yes—
162C22×Dic61#4112yes—
163C2×C4×D61—28no—
164C2×C4×D61—48no—
165C2×C4×D61—44no—
166C2×C4×D61—44no—
167C2×C4×D61—48no—
168C4⋊C4×C61—44no—
169#11758198—28no—
170#11758198—28no—
171#11758198—48no—
172#11758198—48no—
173C4⋊Dic61—28no—
174C4⋊Dic61—48no—
175#11758196—48no—
176C4×Dic61—44no—
177C4×Dic61—44no—
178C23×C61C23×C411yescharacteristic
179C23×C61C22×D411yescharacteristic
180C23×C61C22×Q811yescharacteristic
181C23×C61C2×#4111yescharacteristic
182C23×C61C2×#4111yescharacteristic
183C23×C61C2×#4111yescharacteristic
184C23×C61C2×#4111yescharacteristic
185C23×C61H(5,𝔽2)12yes—
186C23×C61H(5,𝔽2)12yes—
187C23×C61H(5,𝔽2)12yes—
188C23×C612−1+412yes—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—24no—
192C22×D61—24no—
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—22no—
197C22×D61—24no—
198C22×D61—24no—
199C22×D61—24no—
200C22×D61—24no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—42no—
206C22×D61—42no—
207C22×D61—42no—
208C22×D61—42no—
209C22×D61—42no—
210C22×D61—42no—
211C22×D61C4×Q814yes—
212C2×C4×C61—24no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—24no—
216C2×C4×C61—24no—
217C2×C4×C61—24no—
218C2×C4×C61—24no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
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—42no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61C4×D414yes—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—24no—
244C2×Dic61—24no—
245C2×Dic61—24no—
246C2×Dic61—22no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—41no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—41no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61C22⋊Q814yes—
270C4×D61—416no—
271#14317—611nomaximal, Sylow
272C22×C61C2×C4×D411yescharacteristic
273C22×C61C2×C4×Q811yescharacteristic
274C22×C61C4×#4111yescharacteristic
275C22×C61#51812yes—
276C22×C61#52012yes—
277C22×C61C2×C22⋊Q811yescharacteristic
278C22×C61#52511yescharacteristic
279C22×C61#52911yescharacteristic
280C22×C61#53311yescharacteristic
281C22×C61#53411yescharacteristic
282C22×C61#53511yescharacteristic
283C22×C61#53511yescharacteristic
284C22×C61#53611yescharacteristic
285C22×C61#53812yes—
286C22×C61#53812yes—
287C22×C61#53912yes—
288C22×C61#53912yes—
289C22×C61#53912yes—
290C22×C61#54012yes—
291C22×C61#54012yes—
292C22×C61#54212yes—
293C22×C61#54312yes—
294C22×C61#54412yes—
295C2×D61—28no—
296C2×D61—24no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—44no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—44no—
305C2×D61—44no—
306C4×C61—24no—
307C4×C61—44no—
308C4×C61—42no—
309C4×C61—42no—
310C4×C61—82no—
311Dic61—24no—
312Dic61—44no—
313Dic61—42no—
314Dic61—42no—
315Dic61—82no—
316C2×#399—611no—
317C2×#389—611no—
318C2×C4×C22⋊C4—611no—
319#1445—611no—
320#1442D6111yescharacteristic, complemented
321#1441—612no—
322#1439—612no—
323#1439—611no—
324#1436—612no—
325#1434—611no—
326C22⋊#95—612no—
327C2×C61C4×C22⋊Q811yescharacteristic
328C2×C61#232211yescharacteristic
329C2×C61#232811yescharacteristic
330C2×C61#232912yes—
331C2×C61#233512yes—
332C2×C61#262611yescharacteristic
333C2×C61#262811yescharacteristic
334C2×C61#263311yescharacteristic
335C2×C61#263511yescharacteristic
336C2×C61#264912yes—
337C2×C61#266712yes—
338D61—44no—
339C24×C4—611no—
340C22×C4⋊C4—611no—
341C22×C4⋊C4—612no—
342C22×C4⋊C4—612no—
343C22×C4⋊C4C2×D6111yescharacteristic
344C22×C22⋊C4—611no—
345C22×C22⋊C4—612no—
346C22×C22⋊C4—611no—
347C22×C22⋊C4—612no—
348C22×C42—611no—
349C22×C42—611no—
350#399—618no—
351#389—1224no—
352C4×C22⋊C4—1224no—
353C2×#95—612no—
354C2×#95—612no—
355C2×#95—611no—
356C2×#95—611no—
357C2×#95—612no—
358C2×#95—612no—
359C2×#95—612no—
360C2×#95—612no—
361C2×#95C2×D6112yes—
362C2×#95C2×D6111yescharacteristic
363C2×#95C2×D6112yes—
364C2×#95C2×D6111yescharacteristic
365C61#1431711yescharacteristic, complemented, Sylow
366C25—611no—
367C23×C4—611no—
368C23×C4—612no—
369C23×C4—611no—
370C23×C4—611no—
371C23×C4—614no—
372C23×C4—612no—
373C23×C4—1222no—
374C23×C4—1222no—
375C23×C4—2442no—
376C23×C4—2442no—
377C23×C4C22×D6112yes—
378C23×C4C22×D6112yes—
379C23×C4C22×D6111yescharacteristic
380C23×C4C22×D6111yescharacteristic
381C23×C4C22×D6111yescharacteristic
382C2×C4⋊C4—22no—
383C2×C4⋊C4—42no—
384C2×C4⋊C4—614no—
385C2×C4⋊C4—618no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1224no—
389C2×C4⋊C4—2442no—
390C2×C4⋊C4—2442no—
391C2×C4⋊C4—2442no—
392C2×C4⋊C4—2442no—
393C2×C22⋊C4—614no—
394C2×C22⋊C4—618no—
395C2×C22⋊C4—1222no—
396C2×C22⋊C4—1222no—
397C2×C22⋊C4—1222no—
398C2×C22⋊C4—1222no—
399C2×C22⋊C4—1222no—
400C2×C22⋊C4—1224no—
401C2×C22⋊C4—2442no—
402C2×C22⋊C4—2442no—
403C2×C22⋊C4—2442no—
404C2×C22⋊C4—2442no—
405C2×C42—614no—
406C2×C42—1222no—
407C2×C42—2442no—
408C2×C42—2442no—
409#95—22no—
410#95—24no—
411#95—42no—
412#95—1222no—
413#95—1224no—
414#95—1224no—
415#95—1222no—
416#95—2442no—
417#95—2442no—
418#95—2442no—
419#95—2442no—
420#95C4×D6114yescomplemented
421C24—612no—
422C24—612no—
423C24—612no—
424C24—1221no—
425C24—1221no—
426C24—1221no—
427C24—1221no—
428C24—1222no—
429C24—1222no—
430C24—1222no—
431C24—1222no—
432C24C23×D6111yescharacteristic
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—612no—
459C22×C4—612no—
460C22×C4—614no—
461C22×C4—612no—
462C22×C4—1228no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1221no—
466C22×C4—1222no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—2448no—
487C22×C4—2444no—
488C22×C4—2444no—
489C22×C4—2448no—
490C22×C4C2×C4×D6114yes—
491C22×C4C2×C4×D6112yes—
492C22×C4D4×D6112yes—
493C22×C4#1175822412yes—
494C22×C4Q8×D6112yes—
495C22×C4#1175822612yes—
496C4⋊C4—44no—
497C4⋊C4—1228no—
498C4⋊C4—2448no—
499C4⋊C4—2448no—
500C22⋊C4—1228no—
501C22⋊C4—1228no—
502C22⋊C4—2448no—
503C22⋊C4—2448no—
504C42—2444no—
505C42—2444no—
506C23—614no—
507C23—1222no—
508C23—1222no—
509C23—1224no—
510C23—1224no—
511C23—1222no—
512C23—1222no—
513C23—1222no—
514C23—1222no—
515C23—1224no—
516C23—1224no—
517C23—1224no—
518C23—1224no—
519C23—2441no—
520C23—2441no—
521C23—2441no—
522C23—2441no—
523C23—2442no—
524C23—2442no—
525C23—2442no—
526C23—2442no—
527C23—2442no—
528C23—2442no—
529C23C22×C4×D6111yescharacteristic
530C23C2×D4×D6111yescharacteristic
531C23C2×Q8×D6111yescharacteristic
532C23#41×D6111yescharacteristic
533C23#41×D6111yescharacteristic
534C23C2×#1175822411yescharacteristic
535C23C2×#1175822611yescharacteristic
536C23#1451982212yes—
537C23#1451982312yes—
538C23#1451982312yes—
539C23#1451982412yes—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—614no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1224no—
572C2×C4—1224no—
573C2×C4—1222no—
574C2×C4—1224no—
575C2×C4—1224no—
576C2×C4—1224no—
577C2×C4—1224no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—24416no—
582C2×C4—2442no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2441no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2441no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4#1451973914yes—
599C22—1228no—
600C22—1224no—
601C22—2442no—
602C22—2444no—
603C22—2442no—
604C22—2442no—
605C22—2442no—
606C22—2444no—
607C22—2442no—
608C22—2442no—
609C22—2444no—
610C22C2×#1451973911yescharacteristic
611C22C4×Q8×D6111yescharacteristic
612C22C4×#1175822411yescharacteristic
613C22#1684023612yes—
614C22#1684024112yes—
615C22C22⋊Q8×D6111yescharacteristic
616C22#1684027211yescharacteristic
617C22#1684029711yescharacteristic
618C22#1684031711yescharacteristic
619C22#1684032211yescharacteristic
620C22#1684033011yescharacteristic
621C22#1684033011yescharacteristic
622C22#1684033411yescharacteristic
623C22#1684034612yes—
624C22#1684034812yes—
625C22#1684035312yes—
626C22#1684035612yes—
627C22#1684036112yes—
628C22#1684036512yes—
629C22#1684036812yes—
630C22#1684037812yes—
631C22#1684038712yes—
632C22#1684039212yes—
633C4—24no—
634C4—42no—
635C4—44no—
636C4—42no—
637C4—82no—
638C4—1224no—
639C4—4882no—
640C4—2442no—
641C4—2444no—
642C4—2442no—
643C2—2444no—
644C2#2144834911yescharacteristic
645C2#2144848711yescharacteristic
646C2#2144854211yescharacteristic
647C2#2144854812yes—
648C2#2144861412yes—
649C2#2145162311yescharacteristic
650C2#2145164711yescharacteristic
651C2#2145169211yescharacteristic
652C2#2145171111yescharacteristic
653C2#2145182112yes—
654C2#2145201812yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1442C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenter#1442D61
hypocenterC61#14317
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1442

Complements

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

NH ≅ G/Ntype
2. C2×#16839658C2semidirect
4. #21444088C2semidirect
5. #21444071C2semidirect
8. #21444053C2semidirect
9. #21444044C2semidirect
11. C61×#1442C2semidirect
22. #16839699C4semidirect
53. C2×C4⋊Dic61C2×C4semidirect
54. C2×C4⋊Dic61C2×C4semidirect
92. #95×C61C2×C4semidirect
320. #1442D61semidirect
365. C61#14317semidirect
420. #95C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396992
C2×#168396582
C2×#168396092
#214440882
#214440712
#214440632
#214440612
#214440532
#214440442
#214440402
C61×#14422
#1431761