Finite Group Databasebeta

← #29689085

Subgroups — #29689085

Order 15616.

1810 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#121×D61C211yesmaximal, characteristic
2C2×#16839724C211yesmaximal, characteristic, complemented
3#21448198C211yesmaximal, characteristic, complemented
4#21446256C212yesmaximal, complemented
5#21446254C212yesmaximal, complemented
6#21446142C211yesmaximal, characteristic, complemented
7#21446141C211yesmaximal, characteristic, complemented
8#21446139C211yesmaximal, characteristic, complemented
9#21445904C212yesmaximal, complemented
10#21445748C212yesmaximal, complemented
11C61×#1875C211yesmaximal, characteristic, complemented
12C22×D4×D61—21no—
13C22×D4×D61C2211yescharacteristic
14C23×C4×D61C2211yescharacteristic
15C22×#11758189C2212yescomplemented
16C2×#14519752C2211yescharacteristic
17C2×#14519751C2211yescharacteristic
18C2×#14519750—21no—
19C2×#14519749—21no—
20C2×#14519748C2211yescharacteristic
21#121×D61—24no—
22#121×D61C418yescomplemented
23C2×#14519736C2211yescharacteristic, complemented
24C2×C4⋊C4×D61C2211yescharacteristic
25C2×C22⋊C4×D61—21no—
26C2×#14519699C2212yescomplemented
27C2×#14519698C2212yescomplemented
28C2×#14519697C2212yescomplemented
29#16839724—24no—
30C2×C22⋊#92418C2211yescharacteristic, complemented
31C2×#14519689C2212yes—
32#16839623—21no—
33#16839619C2212yes—
34#16839619C2212yescomplemented
35#16839617—21no—
36C4×#11758198—21no—
37C4×#11758198—21no—
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C2×#121×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic, complemented
43C2×#102×C61C2212yescomplemented
44C2×C22⋊C8×C61C2211yescharacteristic, complemented
45C61×#340C2212yescomplemented
46C24×D61—21no—
47C24×D61D411yescharacteristic
48C22×D244—21no—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C22×C22⋊D61—21no—
52C22×C22⋊D61—21no—
53C2×D4×D61—28no—
54C2×D4×D61—22no—
55C2×D4×D61—42no—
56C2×D4×D61—42no—
57C2×D4×D61—48no—
58C2×D4×D61C2×C414yes—
59C23×Dic61D411yescharacteristic
60C22×C4×D61—21no—
61C22×C4×D61—42no—
62C22×C4×D61—42no—
63C22×C4×D61C2×C414yes—
64C22×C4×D61D411yescharacteristic
65C22×C4×D61D411yescharacteristic
66C22×C4×D61D414yes—
67C22×C4×D61C2311yescharacteristic
68C22×C4×D61C2311yescharacteristic
69C2×#11758189—24no—
70C2×#11758189—42no—
71C2×#11758189—42no—
72C22×#92418C2312yes—
73#14519752—22no—
74#14519752C2×C414yes—
75#14519751—22no—
76#14519751C2×C414yes—
77#14519750—28no—
78#14519749—28no—
79#14519748—22no—
80#14519748C2×C414yes—
81#14519736—42no—
82C2×C4⋊Dic61C2312yes—
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C4⋊C4×D61—22no—
86C4⋊C4×D61C2×C414yes—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—22no—
92C2×#11758198—22no—
93C22⋊C4×D61—28no—
94C2×C4×Dic61—21no—
95#14519699—42no—
96#14519698—42no—
97#14519697—42no—
98C22⋊#92418—22no—
99#14519689—24no—
100#14519684—22no—
101#14519684—22no—
102C22×D4×C61—21no—
103C22×D4×C61C2311yescharacteristic
104C23×C4×C61C2311yescharacteristic
105C22×C8×C61C2312yes—
106#121×C61—22no—
107#121×C61C2×C414yes—
108#117×C61—42no—
109C2×C4⋊C4×C61C2312yes—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—21no—
114#102×C61—42no—
115C22⋊C8×C61—22no—
116C24×C61—21no—
117C24×C61C2×D411yescharacteristic
118C23×D61—24no—
119C23×D61—21no—
120C23×D61—24no—
121C23×D61—41no—
122C23×D61—41no—
123C23×D61—42no—
124C23×D61—42no—
125C23×D61—42no—
126C23×D61—44no—
127C23×D61—44no—
128C23×D61C22⋊C414yes—
129C23×D61C2×D411yescharacteristic
130C23×D61C2×D411yescharacteristic
131C2×D4×C61—24no—
132C2×D4×C61—21no—
133C2×D4×C61—22no—
134C2×D4×C61—22no—
135C2×D4×C61—44no—
136C2×D4×C61C22×C412yes—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61C22×C412yes—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2411yescharacteristic
145C2×C22⋊D61—24no—
146C2×C22⋊D61—24no—
147C2×C22⋊D61—24no—
148C2×C22⋊D61—42no—
149C2×C22⋊D61—42no—
150C2×C22⋊D61—44no—
151C2×C22⋊D61—44no—
152C22×Dic61—21no—
153C22×Dic61—22no—
154C22×Dic61—41no—
155C22×Dic61—41no—
156C22×Dic61C22⋊C414yes—
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159D4×D61—48no—
160D4×D61—48no—
161D4×D61—432no—
162C2×D244—22no—
163C2×D244—22no—
164C2×D244—24no—
165C2×D244—21no—
166C2×D244—44no—
167C2×D244C22×C412yes—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—24no—
173C2×C4×D61—48no—
174C2×C4×D61—48no—
175C2×C4×D61—41no—
176C2×C4×D61—41no—
177C2×C4×D61—41no—
178C2×C4×D61—41no—
179C2×C4×D61—44no—
180C2×C4×D61C22⋊C414yes—
181C2×C4×D61C22⋊C414yes—
182C2×C4×D61C22×C412yes—
183C2×C4×D61C22×C412yes—
184C2×C4×D61C2×D412yes—
185C2×C4×D61C2×D412yes—
186C2×C8×C61—22no—
187C2×C8×C61—22no—
188C2×C8×C61—22no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—22no—
191C4⋊C4×C61—22no—
192C4⋊C4×C61C22×C412yes—
193C22⋊C4×C61—24no—
194C22⋊C4×C61—42no—
195C42×C61—42no—
196C22⋊Dic61—24no—
197C22⋊Dic61—42no—
198#11758198—24no—
199#11758198—24no—
200#11758198—44no—
201#11758198—44no—
202C4⋊Dic61—21no—
203C4⋊Dic61—22no—
204C4⋊Dic61—22no—
205C4⋊Dic61C22×C412yes—
206#11758196—24no—
207C4×Dic61—42no—
208C2×#92418—22no—
209C2×#92418—22no—
210C2×#92418—22no—
211#11758189—48no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—22no—
215C23×C61—22no—
216C23×C61—22no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—42no—
220C23×C61—42no—
221C23×C61C2×C22⋊C412yes—
222C23×C61C22×D411yescharacteristic
223C22×D61—21no—
224C22×D61—21no—
225C22×D61—22no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—22no—
229C22×D61—22no—
230C22×D61—28no—
231C22×D61—22no—
232C22×D61—44no—
233C22×D61—44no—
234C22×D61—48no—
235C22×D61—48no—
236C22×D61—48no—
237C22×D61—48no—
238C22×D61—41no—
239C22×D61—41no—
240C22×D61—42no—
241C22×D61—42no—
242C22×D61—48no—
243C22×D61—88no—
244C22×D61—88no—
245C22×D61C2×C22⋊C412yes—
246C22×D61C2×C22⋊C412yes—
247C22×D61AGL(1,ℤ/8)14yes—
248D4×C61—24no—
249D4×C61—48no—
250D4×C61—42no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—24no—
254C2×C4×C61—24no—
255C2×C4×C61—22no—
256C2×C4×C61—42no—
257C2×C4×C61—42no—
258C2×C4×C61—42no—
259C2×C4×C61—42no—
260C2×C4×C61C2×C22⋊C412yes—
261C2×C4×C61C23×C411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61H(5,𝔽2)12yes—
264C2×C4×C61H(5,𝔽2)12yes—
265C22⋊D61—48no—
266C22⋊D61—48no—
267C22⋊D61—48no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—22no—
273C2×Dic61—44no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61C2×C22⋊C412yes—
279C2×Dic61C2×C22⋊C412yes—
280D244—24no—
281D244—48no—
282D244—42no—
283C4×D61—24no—
284C4×D61—24no—
285C4×D61—24no—
286C4×D61—24no—
287C4×D61—44no—
288C4×D61—44no—
289C4×D61—48no—
290C8×C61—42no—
291#92418—42no—
292#21767—611nomaximal, Sylow
293C22×C61—22no—
294C22×C61—22no—
295C22×C61—22no—
296C22×C61—42no—
297C22×C61—42no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—82no—
302C22×C61—82no—
303C22×C61C22×C22⋊C411yescharacteristic
304C22×C61#53411yescharacteristic
305C22×C61#53411yescharacteristic
306C22×C61C2×AGL(1,ℤ/8)12yes—
307C22×C61#57512yes—
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—24no—
311C2×D61—24no—
312C2×D61—22no—
313C2×D61—22no—
314C2×D61—22no—
315C2×D61—416no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61—42no—
319C2×D61—42no—
320C2×D61—42no—
321C2×D61—816no—
322C2×D61—82no—
323C2×D61—82no—
324C2×D61#41814yes—
325C4×C61—22no—
326C4×C61—42no—
327C4×C61—82no—
328C4×C61#51812yes—
329C4×C61#53412yes—
330Dic61—22no—
331Dic61—22no—
332Dic61—42no—
333Dic61—82no—
334C22×#121—611no—
335C2×#407—611no—
336#2280—611no—
337#1895—612no—
338#1895—612no—
339#1875—611no—
340#1875—611no—
341#1875—611no—
342#1875D6111yescharacteristic, complemented
343#1823—612no—
344#1786—612no—
345C2×C61—42no—
346C2×C61—82no—
347C2×C61#230811yescharacteristic
348C2×C61C2×#41811yescharacteristic
349C2×C61#289611yescharacteristic
350C2×C61#319412yes—
351C2×C61#319412yes—
352D61—24no—
353D61—42no—
354D61—82no—
355C23×D4—611no—
356C23×D4—1221no—
357C24×C4—611no—
358C22×M16—612no—
359C2×#121—611no—
360C2×#121—611no—
361C2×#121—618no—
362C2×#121—611no—
363C2×#121—1221no—
364C2×#121—1224no—
365C2×#121—1221no—
366C2×#121C2×D6111yescharacteristic
367C2×#117—611no—
368C2×#117C2×D6111yescharacteristic, complemented
369C22×C4⋊C4—611no—
370C22×C22⋊C4—1221no—
371C2×#102—612no—
372C2×#102—612no—
373C2×#102—612no—
374C2×#102C2×D6112yescomplemented
375#407—1224no—
376C2×C22⋊C8—611no—
377C2×C22⋊C8—612no—
378C2×C22⋊C8C2×D6111yescharacteristic, complemented
379#381—1221no—
380#380—612no—
381#380—612no—
382#379—1221no—
383C4×C22⋊C4—1221no—
384C4×C22⋊C4—1221no—
385#340—612no—
386#340—612no—
387#340—612no—
388#340C2×D6112yescomplemented
389C61#2176711yescharacteristic, complemented, Sylow
390C25—611no—
391C25—1221no—
392C22×D4—21no—
393C22×D4—614no—
394C22×D4—611no—
395C22×D4—1221no—
396C22×D4—1222no—
397C22×D4—1221no—
398C22×D4—1228no—
399C22×D4—1221no—
400C22×D4—1221no—
401C22×D4—2442no—
402C22×D4—2442no—
403C22×D4—2448no—
404C22×D4C22×D6111yescharacteristic
405C23×C4—611no—
406C23×C4—611no—
407C23×C4—611no—
408C23×C4—614no—
409C23×C4—611no—
410C23×C4—611no—
411C23×C4—614no—
412C23×C4—1221no—
413C23×C4—2442no—
414C23×C4—2442no—
415C23×C4C22×D6111yescharacteristic
416C2×M16—1224no—
417C2×M16—2442no—
418C2×M16—2442no—
419C22×C8—612no—
420C22×C8C22×D6112yes—
421#121—22no—
422#121—614no—
423#121—614no—
424#121—614no—
425#121—1228no—
426#121—1222no—
427#121—1222no—
428#121—1228no—
429#121—1222no—
430#121C4×D6114yes—
431#117—42no—
432#117—2442no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—614no—
435C2×C4⋊C4—612no—
436C2×C4⋊C4—1222no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4C22×D6112yes—
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—1228no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1222no—
447C2×C22⋊C4—1222no—
448C2×C22⋊C4—1221no—
449C2×C42—21no—
450C2×C42—1221no—
451#102—42no—
452#102—2442no—
453#102—2442no—
454#102—2442no—
455C22⋊C8—22no—
456C22⋊C8—1224no—
457C22⋊C8—1222no—
458#95—1222no—
459#95—1222no—
460C24—21no—
461C24—611no—
462C24—611no—
463C24—614no—
464C24—1224no—
465C24—1224no—
466C24—1221no—
467C24—2441no—
468C24—2441no—
469C24—2442no—
470C24—2444no—
471C24—2442no—
472C24—2442no—
473C24—2444no—
474C24D4×D6111yescharacteristic
475C2×D4—24no—
476C2×D4—21no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—44no—
480C2×D4—612no—
481C2×D4—1224no—
482C2×D4—1224no—
483C2×D4—1224no—
484C2×D4—1224no—
485C2×D4—1221no—
486C2×D4—1222no—
487C2×D4—1222no—
488C2×D4—2444no—
489C2×D4—2448no—
490C2×D4—2448no—
491C2×D4—2442no—
492C2×D4—2442no—
493C2×D4—24432no—
494C2×D4—2444no—
495C2×D4—2444no—
496C2×D4C2×C4×D6112yes—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—614no—
504C22×C4—612no—
505C22×C4—612no—
506C22×C4—612no—
507C22×C4—612no—
508C22×C4—614no—
509C22×C4—614no—
510C22×C4—1224no—
511C22×C4—1222no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—1221no—
516C22×C4—1222no—
517C22×C4—2448no—
518C22×C4—2448no—
519C22×C4—2441no—
520C22×C4—2441no—
521C22×C4—2441no—
522C22×C4—2441no—
523C22×C4—2444no—
524C22×C4—2441no—
525C22×C4—2441no—
526C22×C4C2×C4×D6112yes—
527C22×C4D4×D6112yes—
528C22×C4D4×D6111yescharacteristic
529C22×C4C23×D6111yescharacteristic
530M16—2448no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—1222no—
535C2×C8—1222no—
536C2×C8—1222no—
537C4⋊C4—21no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—612no—
541C4⋊C4—1224no—
542C4⋊C4—1221no—
543C4⋊C4—1222no—
544C4⋊C4—1222no—
545C4⋊C4C2×C4×D6112yes—
546C22⋊C4—24no—
547C22⋊C4—42no—
548C22⋊C4—1224no—
549C22⋊C4—1224no—
550C22⋊C4—1224no—
551C22⋊C4—2444no—
552C22⋊C4—2444no—
553C22⋊C4—2442no—
554C42—42no—
555C42—2442no—
556C23—22no—
557C23—22no—
558C23—21no—
559C23—21no—
560C23—22no—
561C23—42no—
562C23—42no—
563C23—41no—
564C23—41no—
565C23—612no—
566C23—614no—
567C23—612no—
568C23—1228no—
569C23—1222no—
570C23—1221no—
571C23—1221no—
572C23—1222no—
573C23—1222no—
574C23—1221no—
575C23—1221no—
576C23—1222no—
577C23—4888no—
578C23—4888no—
579C23—2444no—
580C23—2444no—
581C23—2448no—
582C23—2448no—
583C23—2448no—
584C23—2448no—
585C23—2448no—
586C23—2442no—
587C23—2442no—
588C23—2441no—
589C23—2441no—
590C23C22⋊C4×D6112yes—
591C23C2×D4×D6111yescharacteristic
592D4—24no—
593D4—48no—
594D4—42no—
595D4—1224no—
596D4—2448no—
597D4—2448no—
598D4—2448no—
599D4—2448no—
600D4—2442no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—612no—
611C2×C4—612no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1224no—
617C2×C4—1224no—
618C2×C4—1224no—
619C2×C4—1224no—
620C2×C4—1222no—
621C2×C4—2444no—
622C2×C4—2444no—
623C2×C4—2448no—
624C2×C4—2444no—
625C2×C4—2442no—
626C2×C4—2442no—
627C2×C4—2442no—
628C2×C4—2442no—
629C2×C4C22⋊C4×D6112yes—
630C2×C4C22×C4×D6111yescharacteristic
631C2×C4C2×D4×D6111yescharacteristic
632C2×C4#1451982212yes—
633C2×C4#1451982312yes—
634C8—42no—
635C8—2442no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—82no—
645C22—82no—
646C22—614no—
647C22—1222no—
648C22—1224no—
649C22—1224no—
650C22—1222no—
651C22—1222no—
652C22—1222no—
653C22—1222no—
654C22—48816no—
655C22—4882no—
656C22—4882no—
657C22—24416no—
658C22—2442no—
659C22—2442no—
660C22—2442no—
661C22—2442no—
662C22—2442no—
663C22C2×C22⋊C4×D6111yescharacteristic
664C22#1684032211yescharacteristic
665C22#1684032311yescharacteristic
666C22AGL(1,ℤ/8)×D6112yes—
667C22#1684055912yes—
668C4—22no—
669C4—42no—
670C4—82no—
671C4—1222no—
672C4—1222no—
673C4—4882no—
674C4—2442no—
675C4#1684023412yes—
676C4#1684032012yes—
677C2—42no—
678C2—82no—
679C2—1224no—
680C2—4882no—
681C2—2442no—
682C2#2144839311yescharacteristic
683C2#418×D6111yescharacteristic
684C2#2145433411yescharacteristic
685C2#2145663112yes—
686C2#2145663712yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1875C2
socleC23×C61C22×D4
prime radicalC2×#121×D61C2
cosocleC22×C4×C61C24
hypercenter#1875D61
hypocenterC61#21767
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
2. C2×#16839724C2semidirect
3. #21448198C2semidirect
4. #21446256C2semidirect
5. #21446254C2semidirect
6. #21446142C2semidirect
7. #21446141C2semidirect
8. #21446139C2semidirect
9. #21445904C2semidirect
10. #21445748C2semidirect
11. C61×#1875C2semidirect
15. C22×#11758189C22semidirect
22. #121×D61C4semidirect
23. C2×#14519736C22semidirect
26. C2×#14519699C22semidirect
27. C2×#14519698C22semidirect
28. C2×#14519697C22semidirect
30. C2×C22⋊#92418C22semidirect
34. #16839619C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
42. C2×#117×C61C22semidirect
43. C2×#102×C61C22semidirect
44. C2×C22⋊C8×C61C22semidirect
45. C61×#340C22semidirect
342. #1875D61semidirect
368. C2×#117C2×D61semidirect
374. C2×#102C2×D61semidirect
378. C2×C22⋊C8C2×D61semidirect
388. #340C2×D61semidirect
389. C61#21767semidirect

Maximal subgroups

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

subgroupindex
C2×#121×D612
C2×#168397242
#214481982
#214462562
#214462542
#214461422
#214461412
#214461392
#214459042
#214457482
C61×#18752
#2176761