Finite Group Databasebeta

← #29689031

Subgroups — #29689031

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×#16840272C211yesmaximal, characteristic
2C2×C22⋊C8×D61C211yesmaximal, characteristic, complemented
3#21448198C211yesmaximal, characteristic, complemented
4#21446244C212yesmaximal, complemented
5#21446241C212yesmaximal, complemented
6#21446142C211yesmaximal, characteristic, complemented
7#21446131C211yesmaximal, characteristic, complemented
8#21446129C211yesmaximal, characteristic, complemented
9#21445913C212yesmaximal, complemented
10#21445756C212yesmaximal, complemented
11C61×#1875C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C23×C4×D61C2211yescharacteristic
14C22×C8×D61C2212yes—
15#16840272—24no—
16#16840272C418yescomplemented
17C2×#14519759—21no—
18C2×#14519757C2211yescharacteristic
19C2×#14519756C2211yescharacteristic
20C2×#14519748C2211yescharacteristic
21C2×#14519745—21no—
22C2×D4×Dic61—21no—
23C2×#14519737—21no—
24C2×#14519736C2211yescharacteristic
25C2×#14519735C2211yescharacteristic
26C2×#14519702C2212yescomplemented
27C2×#14519700C2212yescomplemented
28C2×#14519698C2212yescomplemented
29C2×C22⋊#92418C2211yescharacteristic
30C2×#14519689C2212yes—
31C22⋊C8×D61—24no—
32#16839623—21no—
33#16839619C2212yes—
34#16839619C2212yes—
35#16839617—21no—
36C4×#11758198—21no—
37C4×#11758198—21no—
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C2×#121×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C2×#102×C61C2212yescomplemented
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#340C2212yescomplemented
46C24×D61D411yescharacteristic
47C2×#11758224—22no—
48C2×#11758224—42no—
49C2×#11758224—42no—
50C2×#11758224C2×C414yes—
51C22×Dic122C2311yescharacteristic
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C23×Dic61—21no—
55C23×Dic61D411yescharacteristic
56C22×C4×D61—42no—
57C22×C4×D61—42no—
58C22×C4×D61C2×C414yes—
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D414yes—
62C22×C4×D61C2311yescharacteristic
63C22×C4×D61C2311yescharacteristic
64C22×#92418C2312yes—
65C2×C8×D61—24no—
66C2×C8×D61—42no—
67C2×C8×D61—42no—
68#14519759—28no—
69#14519757—22no—
70#14519757C2×C414yes—
71#14519756—22no—
72#14519756C2×C414yes—
73#14519748—22no—
74#14519748C2×C414yes—
75#14519745—28no—
76D4×Dic61—28no—
77#14519737—28no—
78#14519736—42no—
79#14519735—22no—
80#14519735C2×C414yes—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61C2312yes—
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—22no—
91C2×#11758198—22no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94#14519702—42no—
95#14519700—42no—
96#14519698—42no—
97C22⋊#92418—22no—
98#14519689—24no—
99#14519684—22no—
100#14519684—22no—
101C22×D4×C61—21no—
102C22×D4×C61C2311yescharacteristic
103C23×C4×C61C2311yescharacteristic
104C22×C8×C61C2312yes—
105#121×C61—22no—
106#121×C61C2×C414yes—
107#117×C61—42no—
108C2×C4⋊C4×C61C2312yes—
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C42×C61—21no—
113#102×C61—42no—
114C22⋊C8×C61—22no—
115C24×C61—21no—
116C24×C61C2×D411yescharacteristic
117C23×D61—24no—
118C23×D61—41no—
119C23×D61—41no—
120C23×D61—42no—
121C23×D61C22⋊C414yes—
122C23×D61C2×D411yescharacteristic
123C23×D61C2×D411yescharacteristic
124C2×D4×C61—24no—
125C2×D4×C61—21no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—44no—
129C2×D4×C61C22×C412yes—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—21no—
133C22×C4×C61—22no—
134C22×C4×C61C22×C412yes—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2411yescharacteristic
138C2×C22⋊D61—24no—
139C2×C22⋊D61—24no—
140C2×C22⋊D61—42no—
141C2×C22⋊D61—42no—
142C2×C22⋊D61—44no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—24no—
146C22×Dic61—22no—
147C22×Dic61—41no—
148C22×Dic61—41no—
149C22×Dic61—42no—
150C22×Dic61—42no—
151C22×Dic61C22⋊C414yes—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154#11758224—48no—
155#11758224—48no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—48no—
161C2×C4×D61—48no—
162C2×C4×D61—41no—
163C2×C4×D61—41no—
164C2×C4×D61—41no—
165C2×C4×D61—41no—
166C2×C4×D61C22⋊C414yes—
167C2×C4×D61C22⋊C414yes—
168C2×C4×D61C22×C412yes—
169C2×C4×D61C22×C412yes—
170C2×C4×D61C2×D412yes—
171C2×C4×D61C2×D412yes—
172C2×Dic122—22no—
173C2×Dic122—22no—
174C2×Dic122—21no—
175C2×Dic122C22×C412yes—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C2×C8×C61—22no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61C22×C412yes—
183C22⋊C4×C61—24no—
184C22⋊C4×C61—42no—
185C42×C61—42no—
186C22⋊Dic61—24no—
187C22⋊Dic61—24no—
188C22⋊Dic61—42no—
189#11758198—24no—
190#11758198—24no—
191#11758198—44no—
192#11758198—44no—
193C4⋊Dic61—21no—
194C4⋊Dic61—24no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61C22×C412yes—
198#11758196—24no—
199C4×Dic61—24no—
200C4×Dic61—42no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—22no—
204C8×D61—48no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—22no—
208C23×C61—22no—
209C23×C61—22no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61C2×C22⋊C412yes—
215C23×C61C22×D411yescharacteristic
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—22no—
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—22no—
222C22×D61—22no—
223C22×D61—28no—
224C22×D61—44no—
225C22×D61—44no—
226C22×D61—48no—
227C22×D61—48no—
228C22×D61C2×C22⋊C412yes—
229C22×D61C2×C22⋊C412yes—
230C22×D61#13514yes—
231D4×C61—24no—
232D4×C61—48no—
233D4×C61—42no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—24no—
237C2×C4×C61—24no—
238C2×C4×C61—22no—
239C2×C4×C61—42no—
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—
249C22⋊D61—48no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—44no—
257C2×Dic61—48no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61C2×C22⋊C412yes—
265C2×Dic61C2×C22⋊C412yes—
266C4×D61—24no—
267C4×D61—24no—
268C4×D61—24no—
269C4×D61—24no—
270C4×D61—44no—
271C4×D61—44no—
272Dic122—24no—
273Dic122—42no—
274C8×C61—42no—
275#92418—42no—
276#21760—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—82no—
286C22×C61—82no—
287C22×C61C22×C22⋊C411yescharacteristic
288C22×C61#53411yescharacteristic
289C22×C61#53411yescharacteristic
290C22×C61C2×#13512yes—
291C22×C61#57512yes—
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—24no—
295C2×D61—24no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—416no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61#41614yes—
303C4×C61—22no—
304C4×C61—42no—
305C4×C61—82no—
306C4×C61#51812yes—
307C4×C61#53412yes—
308Dic61—22no—
309Dic61—22no—
310Dic61—42no—
311Dic61—42no—
312Dic61—82no—
313C2×#525—611no—
314C22×C22⋊C8—611no—
315#2280—611no—
316#1893—612no—
317#1892—612no—
318#1875—611no—
319#1875D6111yescharacteristic, complemented
320#1873—611no—
321#1873—611no—
322#1825—612no—
323#1788—612no—
324C2×C61—42no—
325C2×C61—82no—
326C2×C61#230811yescharacteristic
327C2×C61C2×#41611yescharacteristic
328C2×C61#289611yescharacteristic
329C2×C61#319112yes—
330C2×C61#319112yes—
331D61—24no—
332D61—42no—
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×#121C2×D6111yescharacteristic
343C2×#120—1221no—
344C2×C4×D4—1221no—
345C2×C4×D4—1221no—
346C2×#117—611no—
347C2×#117—611no—
348C2×#117C2×D6111yescharacteristic
349C2×#103—612no—
350C2×#103—612no—
351C2×#102—612no—
352C2×#102C2×D6112yescomplemented
353C2×C22⋊C8—611no—
354C2×C22⋊C8—612no—
355C2×C22⋊C8—1224no—
356C2×C22⋊C8C2×D6111yescharacteristic
357#381—1221no—
358#380—612no—
359#380—612no—
360#379—1221no—
361C4×C22⋊C4—1221no—
362C4×C22⋊C4—1221no—
363#340—612no—
364#340—612no—
365#340—612no—
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—21no—
375C22×D4—1221no—
376C22×D4—1221no—
377C22×D4C22×D6111yescharacteristic
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—22no—
400#121—614no—
401#121—1222no—
402#121C4×D6114yes—
403#120—1228no—
404C4×D4—1228no—
405C4×D4—1228no—
406#117—42no—
407#117—614no—
408#117—1222no—
409#117—2442no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4C22×D6112yes—
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—1221no—
427C2×C42—1221no—
428#103—2442no—
429#103—2442no—
430#102—42no—
431#102—2442no—
432C22⋊C8—22no—
433C22⋊C8—1224no—
434C22⋊C8—1222no—
435#95—1222no—
436#95—1222no—
437C24—21no—
438C24—611no—
439C24—611no—
440C24—614no—
441C24—1224no—
442C24—2441no—
443C24—2441no—
444C24—2442no—
445C24D4×D6111yescharacteristic
446#41—2448no—
447#41—2448no—
448C2×Q8—612no—
449C2×Q8—1221no—
450C2×Q8—1222no—
451C2×Q8—1222no—
452C2×D4—24no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—44no—
457C2×D4—1224no—
458C2×D4—1224no—
459C2×D4—2444no—
460C2×D4—2442no—
461C2×D4—2442no—
462C2×D4C2×C4×D6112yes—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—614no—
470C22×C4—612no—
471C22×C4—612no—
472C22×C4—612no—
473C22×C4—612no—
474C22×C4—614no—
475C22×C4—614no—
476C22×C4—1224no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1222no—
484C22×C4—2448no—
485C22×C4—2448no—
486C22×C4—2441no—
487C22×C4—2441no—
488C22×C4—2441no—
489C22×C4—2441no—
490C22×C4—2441no—
491C22×C4—2441no—
492C22×C4—2442no—
493C22×C4—2442no—
494C22×C4C2×C4×D6112yes—
495C22×C4D4×D6112yes—
496C22×C4D4×D6111yescharacteristic
497C22×C4C23×D6111yescharacteristic
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—1222no—
502C2×C8—1222no—
503C2×C8—1222no—
504C2×C8—2448no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—612no—
509C4⋊C4—1224no—
510C4⋊C4—1221no—
511C4⋊C4—1224no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4C2×C4×D6112yes—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—1224no—
518C22⋊C4—1224no—
519C22⋊C4—1224no—
520C22⋊C4—1224no—
521C22⋊C4—2444no—
522C22⋊C4—2444no—
523C22⋊C4—2442no—
524C42—42no—
525C42—1224no—
526C42—2442no—
527C23—22no—
528C23—22no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—42no—
533C23—42no—
534C23—41no—
535C23—41no—
536C23—612no—
537C23—614no—
538C23—612no—
539C23—1228no—
540C23—1222no—
541C23—1221no—
542C23—1221no—
543C23—1222no—
544C23—1222no—
545C23—1221no—
546C23—1221no—
547C23—2444no—
548C23—2444no—
549C23—2448no—
550C23—2448no—
551C23C22⋊C4×D6112yes—
552C23C2×D4×D6111yescharacteristic
553Q8—1224no—
554Q8—2442no—
555D4—24no—
556D4—48no—
557D4—42no—
558D4—2448no—
559D4—2448no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1224no—
576C2×C4—1224no—
577C2×C4—1224no—
578C2×C4—1224no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—2444no—
582C2×C4—2444no—
583C2×C4—2444no—
584C2×C4—2448no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4C22⋊C4×D6112yes—
592C2×C4C22×C4×D6111yescharacteristic
593C2×C4C2×D4×D6111yescharacteristic
594C2×C4#1451982212yes—
595C2×C4#1451982312yes—
596C8—42no—
597C8—2442no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—82no—
607C22—82no—
608C22—614no—
609C22—1222no—
610C22—1224no—
611C22—1224no—
612C22—1222no—
613C22—1222no—
614C22—1222no—
615C22—1222no—
616C22—24416no—
617C22—2442no—
618C22—2442no—
619C22C2×C22⋊C4×D6111yescharacteristic
620C22#1684032211yescharacteristic
621C22#1684032311yescharacteristic
622C22#135×D6112yes—
623C22#1684055812yes—
624C4—22no—
625C4—42no—
626C4—82no—
627C4—1222no—
628C4—1222no—
629C4—4882no—
630C4—2442no—
631C4—2442no—
632C4#1684023412yes—
633C4#1684032012yes—
634C2—42no—
635C2—82no—
636C2—1224no—
637C2—2442no—
638C2#2144839311yescharacteristic
639C2#416×D6111yescharacteristic
640C2#2145433311yescharacteristic
641C2#2145660912yes—
642C2#2145661012yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1875C2
socleC23×C61C22×D4
prime radicalC2×#16840272C2
cosocleC22×C4×C61C24
hypercenter#1875D61
hypocenterC61#21760
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×C22⋊C8×D61C2semidirect
3. #21448198C2semidirect
4. #21446244C2semidirect
5. #21446241C2semidirect
6. #21446142C2semidirect
7. #21446131C2semidirect
8. #21446129C2semidirect
9. #21445913C2semidirect
10. #21445756C2semidirect
11. C61×#1875C2semidirect
16. #16840272C4semidirect
26. C2×#14519702C22semidirect
27. C2×#14519700C22semidirect
28. C2×#14519698C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
43. C2×#102×C61C22semidirect
45. C61×#340C22semidirect
319. #1875D61semidirect
352. C2×#102C2×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×#168402722
C2×C22⋊C8×D612
#214481982
#214462442
#214462412
#214461422
#214461312
#214461292
#214459132
#214457562
C61×#18752
#2176061