Finite Group Databasebeta

← #29689032

Subgroups — #29689032

Order 15616.

1518 conjugacy classes of subgroups.

The subgroup lattice has 644 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#21448201C211yesmaximal, characteristic, complemented
4#21446246C212yesmaximal, complemented
5#21446241C212yesmaximal, complemented
6#21446140C212yesmaximal, complemented
7#21446130C212yesmaximal, 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×#14519735C2212yes—
26C2×#14519702C2212yescomplemented
27C2×#14519700C2212yescomplemented
28C2×#14519699C2212yescomplemented
29C2×#14519697C2212yescomplemented
30C2×C22⋊#92418C2211yescharacteristic
31C2×#14519689C2212yes—
32C22⋊C8×D61—24no—
33#16839624—21no—
34#16839622C2211yescharacteristic
35#16839619C2212yes—
36#16839617—21no—
37C22⋊C4×Dic61—21no—
38#16839609—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×C414yes—
51C22×Dic122C2311yescharacteristic
52C22×D244—21no—
53C22×D244—21no—
54C22×D244C2311yescharacteristic
55C22×C22⋊D61—21no—
56C23×Dic61D411yescharacteristic
57C22×C4×D61—21no—
58C22×C4×D61—42no—
59C22×C4×D61—42no—
60C22×C4×D61C2×C414yes—
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D414yes—
64C22×C4×D61C2311yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C22×#92418C2312yes—
67C2×C8×D61—24no—
68C2×C8×D61—42no—
69C2×C8×D61—42no—
70#14519761—28no—
71#14519757—22no—
72#14519757—22no—
73#14519757C2×C414yes—
74#14519757C2×C414yes—
75#14519752—22no—
76#14519752—22no—
77#14519752C2×C414yes—
78#14519752C2×C414yes—
79#14519747—28no—
80C4×D244—28no—
81C4×D244—28no—
82#14519735—42no—
83#14519735—42no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2312yes—
87C2×C22⋊Dic61—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—22no—
93C2×#11758198—22no—
94C2×C4×Dic61—22no—
95#14519702—42no—
96#14519700—42no—
97#14519699—42no—
98#14519697—42no—
99C22⋊#92418—22no—
100#14519689—24no—
101#14519684—21no—
102#14519684—22no—
103C23×C4×C61C2311yescharacteristic
104C22×C8×C61C2312yes—
105#117×C61—22no—
106#117×C61C2×C414yes—
107C2×C4⋊C4×C61C2312yes—
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C42×C61—21no—
113C22⋊C8×C61—22no—
114#95×C61—21no—
115C24×C61C2×D411yescharacteristic
116C23×D61—24no—
117C23×D61—21no—
118C23×D61—42no—
119C23×D61—41no—
120C23×D61—41no—
121C23×D61C22⋊C414yes—
122C23×D61C2×D411yescharacteristic
123C23×D61C2×D411yescharacteristic
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—22no—
129C22×C4×C61C22×C412yes—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D412yes—
132C22×C4×C61C2411yescharacteristic
133C2×C22⋊D61—24no—
134C2×C22⋊D61—44no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C22×Dic61—42no—
138C22×Dic61C22⋊C414yes—
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141#11758222—44no—
142#11758222—44no—
143#11758222—48no—
144C2×D244—22no—
145C2×D244—22no—
146C2×D244—24no—
147C2×D244—24no—
148C2×D244—21no—
149C2×D244—42no—
150C2×D244—42no—
151C2×D244—44no—
152C2×D244C22×C412yes—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—24no—
158C2×C4×D61—44no—
159C2×C4×D61—44no—
160C2×C4×D61—48no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—44no—
164C2×C4×D61C22⋊C414yes—
165C2×C4×D61C22⋊C414yes—
166C2×C4×D61C22×C412yes—
167C2×C4×D61C22×C412yes—
168C2×C4×D61C2×D412yes—
169C2×C4×D61C2×D412yes—
170C2×Dic122—22no—
171C2×Dic122—22no—
172C2×Dic122—21no—
173C2×Dic122C22×C412yes—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61C22×C412yes—
182C4⋊C4×C61C22×C412yes—
183C22⋊C4×C61—24no—
184C22⋊C4×C61—42no—
185C42×C61—24no—
186C22⋊Dic61—42no—
187#11758198—24no—
188#11758198—24no—
189#11758198—24no—
190#11758198—24no—
191#11758198—44no—
192#11758198—44no—
193C4⋊Dic61—24no—
194C4⋊Dic61—24no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4×Dic61—44no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201C8×D61—48no—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61C2×C22⋊C412yes—
207C23×C61C22×D411yescharacteristic
208C22×D61—22no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—22no—
212C22×D61—22no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—28no—
216C22×D61—22no—
217C22×D61—44no—
218C22×D61—44no—
219C22×D61—48no—
220C22×D61—48no—
221C22×D61—42no—
222C22×D61—42no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61C2×C22⋊C412yes—
226C22×D61C2×C22⋊C412yes—
227C22×D61#13514yes—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—24no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—42no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61C2×C22⋊C412yes—
241C2×C4×C61C23×C411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61H(5,𝔽2)12yes—
244C2×C4×C61H(5,𝔽2)12yes—
245C22⋊D61—48no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—44no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61C2×C22⋊C412yes—
255C2×Dic61C2×C22⋊C412yes—
256D244—22no—
257D244—22no—
258D244—48no—
259D244—48no—
260D244—42no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—24no—
266C4×D61—24no—
267C4×D61—44no—
268C4×D61—44no—
269C4×D61—48no—
270Dic122—22no—
271Dic122—22no—
272Dic122—42no—
273C8×C61—42no—
274#92418—42no—
275#21760—611nomaximal, Sylow
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61C22×C22⋊C411yescharacteristic
282C22×C61#53412yes—
283C22×C61C2×#13512yes—
284C22×C61#57511yescharacteristic
285C22×C61#57511yescharacteristic
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—24no—
289C2×D61—24no—
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—416no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—82no—
300C2×D61—82no—
301C2×D61#41614yes—
302C4×C61—22no—
303C4×C61—42no—
304C4×C61—42no—
305C4×C61—82no—
306C4×C61#51811yescharacteristic
307C4×C61#51811yescharacteristic
308C4×C61#53412yes—
309Dic61—22no—
310Dic61—22no—
311Dic61—42no—
312Dic61—82no—
313C2×#525—611no—
314C22×C22⋊C8—611no—
315#2280—611no—
316#1893—612no—
317#1892—612no—
318#1875—612no—
319#1873—612no—
320#1825—611no—
321#1825D6111yescharacteristic, complemented
322#1788—611no—
323#1788—611no—
324C2×C61—42no—
325C2×C61#230811yescharacteristic
326C2×C61C2×#41611yescharacteristic
327C2×C61#289611yescharacteristic
328C2×C61#319112yes—
329C2×C61#319112yes—
330D61—24no—
331D61—42no—
332D61—82no—
333C22×#41—611no—
334C24×C4—611no—
335C23×C8—612no—
336#525—618no—
337#525—1224no—
338C2×#123—1221no—
339C2×C22⋊Q8—611no—
340C2×C22⋊Q8—611no—
341C2×#121—611no—
342C2×#121—611no—
343C2×#120—1221no—
344C2×C4×D4—1221no—
345C2×C4×D4—1221no—
346C2×#117—612no—
347C2×#117C2×D6111yescharacteristic
348C2×#103—612no—
349C2×#103—612no—
350C2×#102—612no—
351C2×#102—612no—
352C2×C22⋊C8—611no—
353C2×C22⋊C8—612no—
354C2×C22⋊C8—1224no—
355C2×C22⋊C8C2×D6111yescharacteristic
356#381—1221no—
357#380—611no—
358#380—612no—
359#380C2×D6111yescharacteristic
360#379—1221no—
361C4×C22⋊C4—1221no—
362C4×C22⋊C4—1221no—
363#340—612no—
364#340—612no—
365#340C2×D6112yescomplemented
366#340C2×D6112yescomplemented
367C61#2176011yescharacteristic, complemented, Sylow
368C25—611no—
369C2×#41—614no—
370C2×#41—1222no—
371C2×#41—2442no—
372C2×#41—2442no—
373C22×Q8—611no—
374C22×D4—611no—
375C22×D4—1221no—
376C22×D4—1221no—
377C22×D4—1221no—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—614no—
382C23×C4—611no—
383C23×C4—611no—
384C23×C4—614no—
385C23×C4—1221no—
386C23×C4—2442no—
387C23×C4—2442no—
388C23×C4C22×D6111yescharacteristic
389C22×C8—612no—
390C22×C8—1224no—
391C22×C8—2442no—
392C22×C8—2442no—
393C22×C8C22×D6112yes—
394#123—1228no—
395C22⋊Q8—614no—
396C22⋊Q8—614no—
397C22⋊Q8—1222no—
398C22⋊Q8—1222no—
399#121—614no—
400#121—614no—
401#121—1222no—
402#121—1222no—
403#120—1228no—
404C4×D4—1228no—
405C4×D4—1228no—
406#117—22no—
407#117—2442no—
408#117—2442no—
409#117C4×D6114yes—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4C22×D6112yes—
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1222no—
424C2×C22⋊C4—1221no—
425C2×C42—21no—
426C2×C42—1222no—
427#103—2442no—
428#103—2442no—
429#102—2442no—
430#102—2442no—
431C22⋊C8—22no—
432C22⋊C8—1224no—
433C22⋊C8—1222no—
434#95—21no—
435#95—1222no—
436#95—1221no—
437C24—611no—
438C24—611no—
439C24—614no—
440C24—1224no—
441C24—1221no—
442C24—2442no—
443C24—2441no—
444C24—2441no—
445C24D4×D6111yescharacteristic
446#41—2444no—
447#41—2444no—
448#41—2448no—
449C2×Q8—612no—
450C2×Q8—1221no—
451C2×Q8—1222no—
452C2×Q8—1222no—
453C2×D4—612no—
454C2×D4—1224no—
455C2×D4—1224no—
456C2×D4—1224no—
457C2×D4—1221no—
458C2×D4—1222no—
459C2×D4—1222no—
460C2×D4—2444no—
461C2×D4—2442no—
462C2×D4—2442no—
463C2×D4—2444no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—614no—
472C22×C4—612no—
473C22×C4—612no—
474C22×C4—612no—
475C22×C4—612no—
476C22×C4—614no—
477C22×C4—614no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1221no—
483C22×C4—1224no—
484C22×C4—1222no—
485C22×C4—2444no—
486C22×C4—2444no—
487C22×C4—2448no—
488C22×C4—2442no—
489C22×C4—2442no—
490C22×C4—2444no—
491C22×C4—2442no—
492C22×C4C2×C4×D6112yes—
493C22×C4D4×D6112yes—
494C22×C4D4×D6111yescharacteristic
495C22×C4C23×D6111yescharacteristic
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—1222no—
500C2×C8—1222no—
501C2×C8—1222no—
502C2×C8—2448no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—1224no—
508C4⋊C4—1224no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4C2×C4×D6112yes—
512C4⋊C4C2×C4×D6112yes—
513C22⋊C4—24no—
514C22⋊C4—42no—
515C22⋊C4—1224no—
516C22⋊C4—1224no—
517C22⋊C4—1224no—
518C22⋊C4—1224no—
519C22⋊C4—2444no—
520C22⋊C4—2444no—
521C22⋊C4—2442no—
522C42—24no—
523C42—2444no—
524C23—22no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—612no—
529C23—614no—
530C23—612no—
531C23—1228no—
532C23—1221no—
533C23—1221no—
534C23—1222no—
535C23—1222no—
536C23—1221no—
537C23—1221no—
538C23—1222no—
539C23—1222no—
540C23—2444no—
541C23—2444no—
542C23—2448no—
543C23—2448no—
544C23—2442no—
545C23—2441no—
546C23—2441no—
547C23—2442no—
548C23C22⋊C4×D6112yes—
549C23C2×D4×D6111yescharacteristic
550Q8—1222no—
551Q8—1222no—
552Q8—2442no—
553D4—1222no—
554D4—1222no—
555D4—2448no—
556D4—2448no—
557D4—2448no—
558D4—2442no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—612no—
572C2×C4—612no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1224no—
580C2×C4—1224no—
581C2×C4—1222no—
582C2×C4—2444no—
583C2×C4—2444no—
584C2×C4—2448no—
585C2×C4—2444no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4C22⋊C4×D6112yes—
591C2×C4C22×C4×D6111yescharacteristic
592C2×C4C2×D4×D6111yescharacteristic
593C2×C4#1451982312yes—
594C2×C4#1451982312yes—
595C8—42no—
596C8—2442no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—42no—
601C22—42no—
602C22—614no—
603C22—1222no—
604C22—1224no—
605C22—1224no—
606C22—1222no—
607C22—1222no—
608C22—1222no—
609C22—1222no—
610C22—4882no—
611C22—4882no—
612C22—24416no—
613C22—2442no—
614C22—2442no—
615C22—2442no—
616C22—2442no—
617C22—2442no—
618C22C2×C22⋊C4×D6111yescharacteristic
619C22#1684032312yes—
620C22#135×D6112yes—
621C22#1684055811yescharacteristic
622C22#1684056111yescharacteristic
623C4—22no—
624C4—42no—
625C4—42no—
626C4—82no—
627C4—1222no—
628C4—1222no—
629C4—4882no—
630C4—2442no—
631C4#1684023611yescharacteristic
632C4#1684023611yescharacteristic
633C4#1684032312yes—
634C2—42no—
635C2—1224no—
636C2—4882no—
637C2—2442no—
638C2#2144839611yescharacteristic
639C2#416×D6111yescharacteristic
640C2#2145433511yescharacteristic
641C2#2145661012yes—
642C2#2145661412yes—
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#21760
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. #21448201C2semidirect
4. #21446246C2semidirect
5. #21446241C2semidirect
6. #21446140C2semidirect
7. #21446130C2semidirect
8. #21445914C2semidirect
9. #21445760C2semidirect
10. #21445757C2semidirect
11. C61×#1825C2semidirect
16. #16840274C4semidirect
26. C2×#14519702C22semidirect
27. C2×#14519700C22semidirect
28. C2×#14519699C22semidirect
29. C2×#14519697C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
44. C61×#340C22semidirect
45. C61×#340C22semidirect
321. #1825D61semidirect
365. #340C2×D61semidirect
366. #340C2×D61semidirect
367. C61#21760semidirect

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
#214482012
#214462462
#214462412
#214461402
#214461302
#214459142
#214457602
#214457572
C61×#18252
#2176061