Finite Group Databasebeta

← #29608053

Subgroups — #29608053

Order 15616.

1866 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839655C212yesmaximal, complemented
2C2×#16839622C212yesmaximal, complemented
3#21444076C212yesmaximal, complemented
4#21444064C212yesmaximal, complemented
5#21444060C211yesmaximal, characteristic
6#21444060C211yesmaximal, characteristic
7#21444056C212yesmaximal, complemented
8#21444053C212yesmaximal, complemented
9C2×C61×#386C211yesmaximal, characteristic
10C22×C4⋊Dic61C2212yes—
11C22×C4⋊Dic61C2212yes—
12C22×C22⋊Dic61C2212yes—
13C22×C22⋊Dic61C2212yes—
14C22×C22⋊Dic61C2211yescharacteristic
15C22×C22⋊Dic61C2212yes—
16C22×C22⋊Dic61C2211yescharacteristic
17C22×C4×Dic61C2212yes—
18#16839655—28no—
19#16839622—28no—
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2211yescharacteristic
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2211yescharacteristic
27C2×#14519684C2212yes—
28C23×D4×C61C2211yescharacteristic
29C24×C4×C61C2211yescharacteristic
30C22×C22⋊C4×C61C2212yes—
31C22×C22⋊C4×C61C2211yescharacteristic
32C61×#386C418yescomplemented
33C2×#95×C61C2212yes—
34C23×Dic61C2312yes—
35C23×Dic61C2312yes—
36C23×Dic61C2312yes—
37C23×Dic61C2312yes—
38C2×C4⋊Dic61—24no—
39C2×C4⋊Dic61—24no—
40C2×C4⋊Dic61—24no—
41C2×C4⋊Dic61—24no—
42C2×C4⋊Dic61—42no—
43C2×C4⋊Dic61—42no—
44C2×C22⋊Dic61—24no—
45C2×C22⋊Dic61—24no—
46C2×C22⋊Dic61—24no—
47C2×C22⋊Dic61—24no—
48C2×C22⋊Dic61—24no—
49C2×C22⋊Dic61—24no—
50C2×C22⋊Dic61—41no—
51C2×C22⋊Dic61—42no—
52C2×C22⋊Dic61—42no—
53C2×C22⋊Dic61—42no—
54C2×C22⋊Dic61—42no—
55C2×C22⋊Dic61—42no—
56C2×C22⋊Dic61—41no—
57C2×C4×Dic61—24no—
58C2×C4×Dic61—42no—
59C2×C4×Dic61—42no—
60#14519684—24no—
61#14519684—24no—
62#14519684—42no—
63#14519684—42no—
64#14519684—41no—
65#14519684—42no—
66#14519684—41no—
67#14519684D418yescomplemented
68C25×C61C2311yescharacteristic
69C25×C61C2311yescharacteristic
70C22×D4×C61—24no—
71C22×D4×C61—42no—
72C22×D4×C61—41no—
73C22×D4×C61—41no—
74C22×D4×C61C2×C414yes—
75C23×C4×C61—24no—
76C23×C4×C61—24no—
77C23×C4×C61—41no—
78C23×C4×C61—41no—
79C23×C4×C61C2×C414yes—
80C23×C4×C61C2312yes—
81C23×C4×C61C2311yescharacteristic
82C23×C4×C61C2312yes—
83C2×C22⋊C4×C61—24no—
84C2×C22⋊C4×C61—24no—
85C2×C22⋊C4×C61—42no—
86C2×C22⋊C4×C61C2×C418yes—
87C2×C22⋊C4×C61C2×C414yes—
88#95×C61C2×C418yes—
89C24×C61—22no—
90C24×C61—22no—
91C24×C61—22no—
92C24×C61—21no—
93C24×C61—21no—
94C24×C61—22no—
95C24×C61—22no—
96C24×C61—22no—
97C24×C61—22no—
98C24×C61—22no—
99C24×C61—22no—
100C24×C61—21no—
101C24×C61—21no—
102C24×C61—22no—
103C24×C61—22no—
104C24×C61C22×C412yes—
105C24×C61C22×C412yes—
106C24×C61#4114yes—
107C24×C61C2411yescharacteristic
108C2×D4×C61—28no—
109C2×D4×C61—44no—
110C2×D4×C61—44no—
111C2×D4×C61—48no—
112C2×D4×C61—48no—
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—21no—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—216no—
128C22×C4×C61—44no—
129C22×C4×C61—44no—
130C22×C4×C61—48no—
131C22×C4×C61—44no—
132C22×C4×C61C22×C412yes—
133C22×C4×C61C22×C414yes—
134C22×C4×C61C22×C414yes—
135C22×C4×C61C2×D414yes—
136C22×C4×C61#4114yes—
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×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×Dic61C2×D414yes—
164C22×Dic61C2×D414yes—
165C22⋊C4×C61—216no—
166C22⋊C4×C61—44no—
167C22⋊Dic61—44no—
168C22⋊Dic61—48no—
169C22⋊Dic61—48no—
170C22⋊Dic61—48no—
171C22⋊Dic61—44no—
172C4⋊Dic61—48no—
173C4⋊Dic61—48no—
174C4×Dic61—48no—
175C23×C61—24no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—24no—
179C23×C61—24no—
180C23×C61—24no—
181C23×C61—24no—
182C23×C61—24no—
183C23×C61—24no—
184C23×C61—24no—
185C23×C61—22no—
186C23×C61—24no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61—24no—
190C23×C61—42no—
191C23×C61—42no—
192C23×C61—42no—
193C23×C61—42no—
194C23×C61—41no—
195C23×C61—42no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—42no—
199C23×C61—41no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61—42no—
206C23×C61—41no—
207C23×C61—42no—
208C23×C61—42no—
209C23×C61—42no—
210C23×C61—41no—
211C23×C61—42no—
212C23×C61—42no—
213C23×C61—41no—
214C23×C61—42no—
215C23×C61—41no—
216C23×C61#11714yes—
217C23×C61C23×C411yescharacteristic
218C23×C61C22×D412yes—
219C23×C61C2×#4112yes—
220C23×C61C2×#4112yes—
221C23×C61H(5,𝔽2)12yes—
222C23×C61H(5,𝔽2)11yescharacteristic
223C23×C61H(5,𝔽2)11yescharacteristic
224C23×C61H(5,𝔽2)12yes—
225C23×C612−1+412yes—
226D4×C61—416no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—24no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—24no—
239C2×C4×C61—416no—
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×C61C4×D418yes—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—24no—
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—
264C2×Dic61—42no—
265C2×Dic61—42no—
266C2×Dic61—42no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275#14410—611nomaximal, Sylow
276C22×C61—24no—
277C22×C61—28no—
278C22×C61—44no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—44no—
282C22×C61—44no—
283C22×C61—44no—
284C22×C61—44no—
285C22×C61—42no—
286C22×C61—44no—
287C22×C61—44no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—82no—
291C22×C61—82no—
292C22×C61—82no—
293C22×C61—82no—
294C22×C61C2×#11711yescharacteristic
295C22×C61C2×C4×D412yes—
296C22×C61#51811yescharacteristic
297C22×C61#51811yescharacteristic
298C22×C61#52012yes—
299C22×C61#53512yes—
300C22×C61#53712yes—
301C22×C61#53812yes—
302C22×C61#53812yes—
303C22×C61#54012yes—
304C22×C61#54212yes—
305C22×C61D4211yescharacteristic
306C22×C61#54612yes—
307C22×C61#54612yes—
308C22×C61#54812yes—
309C22×C61#55111yescharacteristic
310C22×C61#55212yes—
311C22×C61#55312yes—
312C22×C61#56011yescharacteristic
313C22×C61#56212yes—
314C22×C61#56311yescharacteristic
315C4×C61—28no—
316C4×C61—42no—
317C4×C61—44no—
318Dic61—44no—
319Dic61—44no—
320Dic61—82no—
321Dic61—82no—
322C2×#388—612no—
323C2×#386D6111yescharacteristic
324C2×#380—612no—
325#1442—612no—
326#1439—612no—
327#1438—611no—
328#1438—611no—
329#1437—612no—
330#1436—612no—
331C2×C61—44no—
332C2×C61—82no—
333C2×C61#231211yescharacteristic
334C2×C61#232412yes—
335C2×C61#235212yes—
336C2×C61#235811yescharacteristic
337C2×C61#237611yescharacteristic
338C2×C61#267612yes—
339C2×C61#270212yes—
340C2×C61#270312yes—
341C2×C61#274112yes—
342C23×D4C2×D6111yescharacteristic
343C24×C4C2×D6111yescharacteristic
344C22×C4⋊C4—612no—
345C22×C4⋊C4—612no—
346C22×C22⋊C4—612no—
347C22×C22⋊C4—612no—
348C22×C22⋊C4—611no—
349C22×C22⋊C4—612no—
350C22×C22⋊C4—611no—
351C22×C22⋊C4C2×D6112yes—
352C22×C22⋊C4C2×D6111yescharacteristic
353C22×C42—612no—
354#388—1228no—
355#386Dic6118yescomplemented
356#380—1228no—
357C2×#95—612no—
358C2×#95—612no—
359C2×#95—612no—
360C2×#95—611no—
361C2×#95—612no—
362C2×#95—612no—
363C2×#95—612no—
364C2×#95—611no—
365C2×#95C2×D6112yes—
366C61#1441011yescharacteristic, complemented, Sylow
367C25C22×D6111yescharacteristic
368C25C22×D6111yescharacteristic
369C22×D4—24no—
370C22×D4—42no—
371C22×D4—41no—
372C22×D4—41no—
373C22×D4C2×Dic6114yes—
374C23×C4—24no—
375C23×C4—24no—
376C23×C4—41no—
377C23×C4—41no—
378C23×C4—612no—
379C23×C4—612no—
380C23×C4—612no—
381C23×C4—612no—
382C23×C4C2×Dic6114yes—
383C23×C4C22×D6112yes—
384C23×C4C22×D6111yescharacteristic
385C23×C4C22×D6112yes—
386C2×C4⋊C4—1224no—
387C2×C4⋊C4—1224no—
388C2×C4⋊C4—1224no—
389C2×C4⋊C4—1224no—
390C2×C4⋊C4—2442no—
391C2×C4⋊C4—2442no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—42no—
395C2×C22⋊C4—1224no—
396C2×C22⋊C4—1224no—
397C2×C22⋊C4—1224no—
398C2×C22⋊C4—1224no—
399C2×C22⋊C4—1224no—
400C2×C22⋊C4—1224no—
401C2×C22⋊C4—2441no—
402C2×C22⋊C4—2442no—
403C2×C22⋊C4—2442no—
404C2×C22⋊C4—2442no—
405C2×C22⋊C4—2442no—
406C2×C22⋊C4—2441no—
407C2×C22⋊C4—2442no—
408C2×C22⋊C4C2×Dic6118yes—
409C2×C22⋊C4C2×Dic6114yes—
410C2×C42—1224no—
411C2×C42—2442no—
412C2×C42—2442no—
413#95—618no—
414#95—1224no—
415#95—1224no—
416#95—2442no—
417#95—2442no—
418#95—2441no—
419#95—2441no—
420#95—2442no—
421#95C2×Dic6118yes—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—21no—
426C24—21no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—21no—
433C24—21no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24#1175822214yes—
438C24C22×Dic6112yes—
439C24C22×Dic6112yes—
440C24C23×D6111yescharacteristic
441C2×D4—28no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—48no—
445C2×D4—48no—
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—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—216no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—48no—
464C22×C4—44no—
465C22×C4—614no—
466C22×C4—614no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
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—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4D4×D6114yes—
494C22×C4#1175822414yes—
495C22×C4C22×Dic6112yes—
496C22×C4C22×Dic6114yes—
497C22×C4C22×Dic6114yes—
498C4⋊C4—2448no—
499C4⋊C4—2448no—
500C22⋊C4—216no—
501C22⋊C4—44no—
502C22⋊C4—2444no—
503C22⋊C4—2448no—
504C22⋊C4—2448no—
505C22⋊C4—2448no—
506C22⋊C4—2444no—
507C42—2448no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—41no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—41no—
549C23#1451973614yes—
550C23C23×Dic6111yescharacteristic
551C23C2×D4×D6112yes—
552C23C2×#1175822412yes—
553C23C2×#1175822212yes—
554C23#1451982211yescharacteristic
555C23#1451982211yescharacteristic
556C23#1451982212yes—
557C23#1451982312yes—
558C23#1451982412yes—
559D4—416no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—416no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—1224no—
580C2×C4—1224no—
581C2×C4—1224no—
582C2×C4—1224no—
583C2×C4—1224no—
584C2×C4—1224no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4—2442no—
607C2×C4D4×Dic6118yes—
608C22—28no—
609C22—24no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22—42no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22C2×#1451973611yescharacteristic
627C22C2×D4×Dic6112yes—
628C22#1684023711yescharacteristic
629C22#1684023711yescharacteristic
630C22#1684024512yes—
631C22#1684032812yes—
632C22#1684034112yes—
633C22#1684035012yes—
634C22#1684035012yes—
635C22#1684036512yes—
636C22#1684037912yes—
637C22#1684039411yescharacteristic
638C22#1684039912yes—
639C22#1684040812yes—
640C22#1684042112yes—
641C22#1684044311yescharacteristic
642C22#1684045212yes—
643C22#1684045812yes—
644C22#1684050211yescharacteristic
645C22#1684050912yes—
646C22#1684051511yescharacteristic
647C4—28no—
648C4—44no—
649C4—42no—
650C4—4882no—
651C4—4882no—
652C4—2444no—
653C4—2444no—
654C2—44no—
655C2—82no—
656C2#2144841711yescharacteristic
657C2#2144851412yes—
658C2#2144874812yes—
659C2#2144880311yescharacteristic
660C2#2144896711yescharacteristic
661C2#2145212412yes—
662C2#2145245112yes—
663C2#2145246612yes—
664C2#2145293812yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#386C2
socle, cosocleC24×C61C24
prime radicalC23×D4×C61C22
hypercenterC2×#386D61
hypocenterC61#14410
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
1. C2×#16839655C2semidirect
2. C2×#16839622C2semidirect
3. #21444076C2semidirect
4. #21444064C2semidirect
7. #21444056C2semidirect
8. #21444053C2semidirect
32. C61×#386C4semidirect
67. #14519684D4semidirect
355. #386Dic61semidirect
366. C61#14410semidirect

Maximal subgroups

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

subgroupindex
C2×#168396552
C2×#168396222
#214440762
#214440642
#214440602
#214440562
#214440532
C2×C61×#3862
#1441061