Finite Group Databasebeta

← #29689055

Subgroups — #29689055

Order 15616.

1590 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840274C211yesmaximal, characteristic
2C2×C22⋊C8×D61C211yesmaximal, characteristic, complemented
3#380×D61C211yesmaximal, characteristic, complemented
4#21446246C212yesmaximal, complemented
5#21446241C212yesmaximal, complemented
6#21446135C212yesmaximal, complemented
7#21446125C212yesmaximal, complemented
8#21445914C211yesmaximal, characteristic, complemented
9#21445760C211yesmaximal, characteristic, complemented
10#21445757C211yesmaximal, characteristic, complemented
11C61×#1825C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×C4×D61C2211yescharacteristic
14C22×C8×D61C2212yes—
15#16840274—24no—
16#16840274C418yescomplemented
17C2×#14519761—21no—
18C2×#14519757C2211yescharacteristic
19C2×#14519757C2211yescharacteristic
20C2×#14519752C2211yescharacteristic
21C2×#14519752C2211yescharacteristic
22C2×#14519747—21no—
23C2×C4×D244—21no—
24C2×C4×D244—21no—
25C2×C4⋊C4×D61C2212yes—
26C2×C22⋊C4×D61—21no—
27C2×#14519702C2212yescomplemented
28C2×#14519700C2212yescomplemented
29C2×#14519699C2212yescomplemented
30C2×#14519697C2212yescomplemented
31C2×C22⋊#92418C2211yescharacteristic
32C2×#14519689C2212yes—
33C22⋊C8×D61—24no—
34#16839622C2211yescharacteristic
35#16839621—21no—
36#16839620—21no—
37#16839619C2212yes—
38#95×D61—21no—
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C2×#117×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2211yescharacteristic
43C61×#380C2211yescharacteristic
44C61×#340C2212yescomplemented
45C61×#340C2212yescomplemented
46C24×D61D411yescharacteristic
47C2×#11758222—22no—
48C2×#11758222—42no—
49C2×#11758222—42no—
50C2×#11758222C2×C414yescomplemented
51C22×Dic122C2311yescharacteristic
52C22×D244—21no—
53C22×D244—21no—
54C22×D244C2311yescharacteristic
55C22×C22⋊D61—21no—
56C23×Dic61D411yescharacteristic
57C22×C4×D61—21no—
58C22×C4×D61—22no—
59C22×C4×D61—42no—
60C22×C4×D61—42no—
61C22×C4×D61C2×C414yes—
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C22×C4×D61D414yes—
65C22×C4×D61C2311yescharacteristic
66C22×C4×D61C2311yescharacteristic
67C22×#92418C2312yes—
68C2×C8×D61—24no—
69C2×C8×D61—42no—
70C2×C8×D61—42no—
71#14519761—28no—
72#14519757—22no—
73#14519757—22no—
74#14519757C2×C414yes—
75#14519757C2×C414yescomplemented
76#14519752—22no—
77#14519752—22no—
78#14519752C2×C414yes—
79#14519752C2×C414yescomplemented
80#14519747—28no—
81C4×D244—28no—
82C4×D244—28no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61C2312yes—
86C2×#11758196—22no—
87C4⋊C4×D61—42no—
88C4⋊C4×D61—42no—
89C2×C22⋊Dic61—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—22no—
95C22⋊C4×D61—44no—
96#14519702—42no—
97#14519700—42no—
98#14519699—42no—
99#14519697—42no—
100C22⋊#92418—22no—
101#14519689—24no—
102#14519684—21no—
103#14519684—22no—
104C23×C4×C61C2311yescharacteristic
105C22×C8×C61C2312yes—
106#117×C61—22no—
107#117×C61C2×C414yescomplemented
108C2×C4⋊C4×C61C2312yes—
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113C2×C42×C61—21no—
114C22⋊C8×C61—22no—
115#95×C61—21no—
116C24×C61C2×D411yescharacteristic
117C23×D61—24no—
118C23×D61—22no—
119C23×D61—22no—
120C23×D61—21no—
121C23×D61—42no—
122C23×D61C22⋊C414yes—
123C23×D61C2×D411yescharacteristic
124C23×D61C2×D411yescharacteristic
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—22no—
130C22×C4×C61C22×C412yes—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2411yescharacteristic
134C2×C22⋊D61—24no—
135C2×C22⋊D61—44no—
136C22×Dic61—21no—
137C22×Dic61—22no—
138C22×Dic61—42no—
139C22×Dic61C22⋊C414yes—
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142#11758222—44no—
143#11758222—44no—
144#11758222—48no—
145C2×D244—22no—
146C2×D244—22no—
147C2×D244—24no—
148C2×D244—24no—
149C2×D244—21no—
150C2×D244—44no—
151C2×D244—42no—
152C2×D244—42no—
153C2×D244C22×C412yes—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—24no—
159C2×C4×D61—44no—
160C2×C4×D61—44no—
161C2×C4×D61—48no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61—44no—
165C2×C4×D61—44no—
166C2×C4×D61—44no—
167C2×C4×D61—44no—
168C2×C4×D61C22⋊C414yes—
169C2×C4×D61C22⋊C414yes—
170C2×C4×D61C22×C412yes—
171C2×C4×D61C22×C412yes—
172C2×C4×D61C2×D412yes—
173C2×C4×D61C2×D412yes—
174C2×Dic122—22no—
175C2×Dic122—22no—
176C2×Dic122—21no—
177C2×Dic122C22×C412yes—
178C2×C8×C61—22no—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61C22×C412yes—
186C4⋊C4×C61C22×C412yes—
187C22⋊C4×C61—24no—
188C22⋊C4×C61—42no—
189C42×C61—24no—
190C22⋊Dic61—42no—
191#11758198—24no—
192#11758198—24no—
193#11758198—24no—
194#11758198—24no—
195#11758198—44no—
196C4⋊Dic61—24no—
197C4⋊Dic61—24no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200#11758196—44no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—22no—
204C8×D61—48no—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61C2×C22⋊C412yes—
210C23×C61C22×D411yescharacteristic
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—22no—
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—28no—
217C22×D61—28no—
218C22×D61—22no—
219C22×D61—44no—
220C22×D61—48no—
221C22×D61—48no—
222C22×D61—42no—
223C22×D61—42no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61C2×C22⋊C412yes—
227C22×D61C2×C22⋊C412yes—
228C22×D61#13514yes—
229C22×D61AGL(1,ℤ/8)12yes—
230C22×D61#13712yes—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—42no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61C2×C22⋊C412yes—
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61H(5,𝔽2)12yes—
247C2×C4×C61H(5,𝔽2)12yes—
248C22⋊D61—48no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—44no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61C2×C22⋊C412yes—
258C2×Dic61C2×C22⋊C412yes—
259D244—22no—
260D244—22no—
261D244—48no—
262D244—48no—
263D244—42no—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—24no—
269C4×D61—24no—
270C4×D61—44no—
271C4×D61—44no—
272C4×D61—48no—
273C4×D61—88no—
274Dic122—22no—
275Dic122—22no—
276Dic122—42no—
277C8×C61—42no—
278#92418—42no—
279#21763—611nomaximal, Sylow
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61C22×C22⋊C411yescharacteristic
286C22×C61#53412yes—
287C22×C61C2×#13512yes—
288C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
289C22×C61C2×#13711yescharacteristic
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—24no—
295C2×D61—24no—
296C2×D61—24no—
297C2×D61—24no—
298C2×D61—416no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—82no—
303C2×D61—82no—
304C2×D61#41614yes—
305C2×D61#41714yes—
306C4×C61—22no—
307C4×C61—42no—
308C4×C61—42no—
309C4×C61—82no—
310C4×C61#51811yescharacteristic
311C4×C61#51811yescharacteristic
312C4×C61#53412yes—
313Dic61—22no—
314Dic61—22no—
315Dic61—42no—
316Dic61—82no—
317C2×#525—611no—
318C22×C22⋊C8—611no—
319C2×#380—611no—
320#1893—612no—
321#1892—612no—
322#1874—612no—
323#1872—612no—
324#1825—611no—
325#1825D6111yescharacteristic, complemented
326#1788—611no—
327#1788—611no—
328C2×C61—42no—
329C2×C61#230811yescharacteristic
330C2×C61C2×#41611yescharacteristic
331C2×C61C2×#41711yescharacteristic
332C2×C61#318812yes—
333C2×C61#319012yes—
334D61—24no—
335D61—24no—
336D61—82no—
337C22×#41—611no—
338C24×C4—611no—
339C23×C8—612no—
340#525—618no—
341#525—1224no—
342C2×#123—1221no—
343C2×C22⋊Q8—611no—
344C2×C22⋊Q8—611no—
345C2×#121—611no—
346C2×#121—611no—
347C2×#120—1221no—
348C2×C4×D4—1221no—
349C2×C4×D4—1221no—
350C2×#117C2×D6111yescharacteristic
351C22×C4⋊C4—612no—
352C22×C22⋊C4—1221no—
353C2×#103—612no—
354C2×#103—612no—
355C2×#102—612no—
356C2×#102—612no—
357C2×C22⋊C8—611no—
358C2×C22⋊C8—612no—
359C2×C22⋊C8—1224no—
360C2×C22⋊C8C2×D6111yescharacteristic
361#380—611no—
362#380—612no—
363#380—1221no—
364#380—1221no—
365#380C2×D6111yescharacteristic
366C2×#95—1221no—
367#340—612no—
368#340—612no—
369#340C2×D6112yescomplemented
370#340C2×D6112yescomplemented
371C61#2176311yescharacteristic, complemented, Sylow
372C25—611no—
373C2×#41—614no—
374C2×#41—1222no—
375C2×#41—2442no—
376C2×#41—2442no—
377C22×Q8—611no—
378C22×D4—611no—
379C22×D4—1221no—
380C22×D4—1221no—
381C22×D4—1221no—
382C23×C4—611no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—614no—
386C23×C4—611no—
387C23×C4—611no—
388C23×C4—614no—
389C23×C4—1222no—
390C23×C4—1221no—
391C23×C4—2442no—
392C23×C4—2442no—
393C23×C4C22×D6111yescharacteristic
394C22×C8—612no—
395C22×C8—1224no—
396C22×C8—2442no—
397C22×C8—2442no—
398C22×C8C22×D6112yes—
399#123—1228no—
400C22⋊Q8—614no—
401C22⋊Q8—614no—
402C22⋊Q8—1222no—
403C22⋊Q8—1222no—
404#121—614no—
405#121—614no—
406#121—1222no—
407#121—1222no—
408#120—1228no—
409C4×D4—1228no—
410C4×D4—1228no—
411#117—22no—
412#117C4×D6114yescomplemented
413C2×C4⋊C4—612no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1222no—
417C2×C4⋊C4—2442no—
418C2×C4⋊C4—2442no—
419C2×C4⋊C4C22×D6112yes—
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1222no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—2444no—
431C2×C42—21no—
432#103—2442no—
433#103—2442no—
434#102—2442no—
435#102—2442no—
436C22⋊C8—22no—
437C22⋊C8—1224no—
438C22⋊C8—1222no—
439#95—21no—
440#95—1222no—
441#95—1221no—
442C24—611no—
443C24—611no—
444C24—614no—
445C24—1222no—
446C24—1224no—
447C24—1222no—
448C24—1221no—
449C24—2442no—
450C24D4×D6111yescharacteristic
451#41—2444no—
452#41—2444no—
453#41—2448no—
454C2×Q8—612no—
455C2×Q8—1221no—
456C2×Q8—1222no—
457C2×Q8—1222no—
458C2×D4—612no—
459C2×D4—1224no—
460C2×D4—1224no—
461C2×D4—1224no—
462C2×D4—1221no—
463C2×D4—1222no—
464C2×D4—1222no—
465C2×D4—2444no—
466C2×D4—2442no—
467C2×D4—2442no—
468C2×D4—2444no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—614no—
477C22×C4—612no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—612no—
481C22×C4—614no—
482C22×C4—614no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1221no—
488C22×C4—1224no—
489C22×C4—1222no—
490C22×C4—2444no—
491C22×C4—2444no—
492C22×C4—2448no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4—2444no—
496C22×C4—2444no—
497C22×C4—2444no—
498C22×C4—2444no—
499C22×C4—2442no—
500C22×C4C2×C4×D6112yes—
501C22×C4D4×D6112yes—
502C22×C4D4×D6111yescharacteristic
503C22×C4C23×D6111yescharacteristic
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—1222no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—2448no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—1224no—
516C4⋊C4—1224no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—2444no—
520C4⋊C4C2×C4×D6112yes—
521C4⋊C4C2×C4×D6112yes—
522C22⋊C4—24no—
523C22⋊C4—42no—
524C22⋊C4—1224no—
525C22⋊C4—1224no—
526C22⋊C4—1224no—
527C22⋊C4—1224no—
528C22⋊C4—2444no—
529C22⋊C4—2442no—
530C42—24no—
531C23—22no—
532C23—21no—
533C23—21no—
534C23—22no—
535C23—612no—
536C23—612no—
537C23—614no—
538C23—612no—
539C23—612no—
540C23—1228no—
541C23—1228no—
542C23—1222no—
543C23—1221no—
544C23—1221no—
545C23—1222no—
546C23—1222no—
547C23—1222no—
548C23—2444no—
549C23—2448no—
550C23—2448no—
551C23—2442no—
552C23—2441no—
553C23—2441no—
554C23—2442no—
555C23C22⋊C4×D6112yes—
556C23C2×D4×D6111yescharacteristic
557Q8—1222no—
558Q8—1222no—
559Q8—2442no—
560D4—1222no—
561D4—1222no—
562D4—2448no—
563D4—2448no—
564D4—2448no—
565D4—2442no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—612no—
579C2×C4—612no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1224no—
587C2×C4—1224no—
588C2×C4—1222no—
589C2×C4—4888no—
590C2×C4—2444no—
591C2×C4—2444no—
592C2×C4—2448no—
593C2×C4—2444no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4C22⋊C4×D6112yes—
599C2×C4C22×C4×D6111yescharacteristic
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4#1451982312yes—
602C2×C4#1451982312yes—
603C8—42no—
604C8—2442no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—42no—
609C22—42no—
610C22—614no—
611C22—614no—
612C22—1222no—
613C22—1222no—
614C22—1224no—
615C22—1224no—
616C22—1224no—
617C22—1224no—
618C22—1222no—
619C22—1222no—
620C22—4882no—
621C22—4882no—
622C22—24416no—
623C22—2442no—
624C22—2442no—
625C22—2442no—
626C22C2×C22⋊C4×D6111yescharacteristic
627C22#1684032312yes—
628C22#135×D6112yes—
629C22AGL(1,ℤ/8)×D6111yescharacteristic
630C22#137×D6111yescharacteristic
631C4—22no—
632C4—42no—
633C4—42no—
634C4—82no—
635C4—1222no—
636C4—1222no—
637C4—4882no—
638C4—2442no—
639C4#1684023611yescharacteristic
640C4#1684023611yescharacteristic
641C4#1684032312yes—
642C2—42no—
643C2—1224no—
644C2—1224no—
645C2—4882no—
646C2#2144839611yescharacteristic
647C2#416×D6111yescharacteristic
648C2#417×D6111yescharacteristic
649C2#2145658812yes—
650C2#2145660412yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1825C2
socleC23×C61C22×D4
prime radicalC2×#16840274C2
cosocleC22×C4×C61C24
hypercenter#1825D61
hypocenterC61#21763
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C8×D61C2semidirect
3. #380×D61C2semidirect
4. #21446246C2semidirect
5. #21446241C2semidirect
6. #21446135C2semidirect
7. #21446125C2semidirect
8. #21445914C2semidirect
9. #21445760C2semidirect
10. #21445757C2semidirect
11. C61×#1825C2semidirect
16. #16840274C4semidirect
27. C2×#14519702C22semidirect
28. C2×#14519700C22semidirect
29. C2×#14519699C22semidirect
30. C2×#14519697C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
44. C61×#340C22semidirect
45. C61×#340C22semidirect
50. C2×#11758222C2×C4semidirect
75. #14519757C2×C4semidirect
79. #14519752C2×C4semidirect
107. #117×C61C2×C4semidirect
325. #1825D61semidirect
369. #340C2×D61semidirect
370. #340C2×D61semidirect
371. C61#21763semidirect
412. #117C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402742
C2×C22⋊C8×D612
#380×D612
#214462462
#214462412
#214461352
#214461252
#214459142
#214457602
#214457572
C61×#18252
#2176361