Finite Group Databasebeta

← #29689076

Subgroups — #29689076

Order 15616.

1486 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×#16840269C211yesmaximal, characteristic
2C2×#16839724C211yesmaximal, characteristic, complemented
3#21448201C211yesmaximal, characteristic, complemented
4#21446254C212yesmaximal, complemented
5#21446249C212yesmaximal, complemented
6#21446138C212yesmaximal, complemented
7#21446128C212yesmaximal, complemented
8#21445915C211yesmaximal, characteristic, complemented
9#21445911C211yesmaximal, characteristic, complemented
10#21445759C211yesmaximal, characteristic, complemented
11C61×#1788C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C22×#11758189C2212yes—
14#16840269—24no—
15#16840269C418yescomplemented
16C2×#14519758—21no—
17C2×#14519753C2211yescharacteristic
18C2×#14519753C2211yescharacteristic
19C2×#14519748C2211yescharacteristic
20C2×#14519748C2211yescharacteristic
21C2×#14519745—21no—
22C2×C4×C22⋊D61—21no—
23C2×C4×C22⋊D61—21no—
24C2×#14519736C2211yescharacteristic
25C2×#14519735C2212yes—
26C2×#14519701C2212yescomplemented
27C2×#14519698C2212yescomplemented
28#16839724—24no—
29C2×C22⋊#92418C2211yescharacteristic
30C2×#14519689C2212yes—
31#16839624—21no—
32#16839622C2211yescharacteristic
33#16839619C2212yes—
34#16839617—21no—
35C22⋊C4×Dic61—21no—
36#16839609—21no—
37#16839525C2212yescomplemented
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C22×#41×C61C2211yescharacteristic
42C2×#103×C61C2212yescomplemented
43C2×#102×C61C2212yescomplemented
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#380C2211yescharacteristic
46C24×D61D411yescharacteristic
47C22×C22⋊D61—21no—
48C22×C22⋊D61—21no—
49C23×Dic61D411yescharacteristic
50C22×C4×D61—42no—
51C22×C4×D61—42no—
52C22×C4×D61C2×C414yes—
53C22×C4×D61D411yescharacteristic
54C22×C4×D61D411yescharacteristic
55C22×C4×D61D414yes—
56C22×C4×D61C2311yescharacteristic
57C22×C4×D61C2311yescharacteristic
58C2×#11758189—24no—
59C2×#11758189—42no—
60C2×#11758189—42no—
61C22×#92418C2312yes—
62#14519758—28no—
63#14519753—22no—
64#14519753—22no—
65#14519753C2×C414yes—
66#14519753C2×C414yes—
67#14519748—22no—
68#14519748—22no—
69#14519748C2×C414yes—
70#14519748C2×C414yes—
71#14519745—28no—
72C4×C22⋊D61—28no—
73C4×C22⋊D61—28no—
74#14519736—22no—
75#14519736C2×C414yes—
76#14519735—42no—
77#14519735—42no—
78C2×C4⋊Dic61C2312yes—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—22no—
90C2×#11758198—22no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—22no—
93#14519701—42no—
94#14519698—42no—
95C22⋊#92418—22no—
96#14519689—24no—
97#14519684—21no—
98#14519684—22no—
99C2×#41×C61—22no—
100C2×#41×C61—42no—
101C2×#41×C61—42no—
102C2×#41×C61C2×C414yes—
103C22×Q8×C61C2311yescharacteristic
104C22×D4×C61—21no—
105C22×D4×C61C2311yescharacteristic
106C23×C4×C61—21no—
107C23×C4×C61C2311yescharacteristic
108C22×C8×C61C2312yes—
109C2×C4⋊C4×C61C2312yes—
110C2×C22⋊C4×C61—21no—
111#103×C61—42no—
112#102×C61—42no—
113C22⋊C8×C61—22no—
114#95×C61—21no—
115C24×C61—21no—
116C24×C61C2×D411yescharacteristic
117C23×D61—42no—
118C23×D61—42no—
119C23×D61—41no—
120C23×D61—41no—
121C23×D61C22⋊C414yes—
122C23×D61C2×D411yescharacteristic
123C23×D61C2×D411yescharacteristic
124#41×C61—44no—
125#41×C61—44no—
126#41×C61—48no—
127C2×Q8×C61—21no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61C22×C412yes—
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—24no—
141C22×C4×C61—22no—
142C22×C4×C61—44no—
143C22×C4×C61C22×C412yes—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2411yescharacteristic
147C2×C22⋊D61—24no—
148C2×C22⋊D61—24no—
149C2×C22⋊D61—42no—
150C2×C22⋊D61—44no—
151C2×C22⋊D61—42no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—22no—
155C22×Dic61—42no—
156C22×Dic61C22⋊C414yes—
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—44no—
164C2×C4×D61—44no—
165C2×C4×D61—48no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
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×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C22⋊C4×C61—42no—
180C22⋊Dic61—24no—
181C22⋊Dic61—24no—
182C22⋊Dic61—24no—
183C22⋊Dic61—42no—
184#11758198—24no—
185#11758198—24no—
186#11758198—44no—
187#11758198—44no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61C22×C412yes—
193C4⋊Dic61C22×C412yes—
194#11758196—24no—
195#11758196—24no—
196C4×Dic61—24no—
197C4×Dic61—44no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—22no—
201#11758189—48no—
202C23×C61—22no—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—42no—
208C23×C61—41no—
209C23×C61—42no—
210C23×C61—41no—
211C23×C61C2×C22⋊C412yes—
212C23×C61C22×D411yescharacteristic
213C22×D61—22no—
214C22×D61—22no—
215C22×D61—22no—
216C22×D61—21no—
217C22×D61—22no—
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—44no—
222C22×D61—44no—
223C22×D61—44no—
224C22×D61—48no—
225C22×D61—48no—
226C22×D61C2×C22⋊C412yes—
227C22×D61C2×C22⋊C412yes—
228Q8×C61—22no—
229Q8×C61—22no—
230Q8×C61—42no—
231D4×C61—22no—
232D4×C61—22no—
233D4×C61—48no—
234D4×C61—42no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—24no—
239C2×C4×C61—22no—
240C2×C4×C61—48no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61C2×C22⋊C412yes—
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61H(5,𝔽2)12yes—
249C2×C4×C61H(5,𝔽2)12yes—
250C22⋊D61—48no—
251C22⋊D61—48no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—44no—
257C2×Dic61—41no—
258C2×Dic61—42no—
259C2×Dic61—41no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61C2×C22⋊C412yes—
264C2×Dic61C2×C22⋊C412yes—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—24no—
270C4×D61—24no—
271C4×D61—44no—
272C4×D61—44no—
273C8×C61—42no—
274#92418—42no—
275#21766—611nomaximal, Sylow
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—82no—
285C22×C61—82no—
286C22×C61C22×C22⋊C411yescharacteristic
287C22×C61#53412yes—
288C22×C61#57511yescharacteristic
289C22×C61#57512yes—
290C22×C61#57511yescharacteristic
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—416no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C4×C61—22no—
303C4×C61—42no—
304C4×C61—82no—
305C4×C61#51811yescharacteristic
306C4×C61#51811yescharacteristic
307C4×C61#53412yes—
308Dic61—22no—
309Dic61—22no—
310Dic61—42no—
311Dic61—42no—
312Dic61—82no—
313C2×#525—611no—
314C2×#407—611no—
315#2280—611no—
316#1895—612no—
317#1894—612no—
318#1875—612no—
319#1873—612no—
320#1825—611no—
321#1825—611no—
322#1788—611no—
323#1788D6111yescharacteristic, complemented
324C2×C61—42no—
325C2×C61—82no—
326C2×C61#230811yescharacteristic
327C2×C61#289611yescharacteristic
328C2×C61#289611yescharacteristic
329C2×C61#319712yes—
330C2×C61#319712yes—
331D61—42no—
332D61—42no—
333C22×#41C2×D6111yescharacteristic
334C24×C4—611no—
335C22×M16—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—611no—
347C2×#117—612no—
348C2×#103—612no—
349C2×#103C2×D6112yescomplemented
350C2×#102—612no—
351C2×#102C2×D6112yescomplemented
352#407—1224no—
353C2×C22⋊C8—611no—
354C2×C22⋊C8—612no—
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#340—612no—
366#340—612no—
367C61#2176611yescharacteristic, complemented, Sylow
368C25—611no—
369C2×#41—22no—
370C2×#41—42no—
371C2×#41—42no—
372C2×#41C4×D6114yes—
373C22×Q8C22×D6111yescharacteristic
374C22×D4—21no—
375C22×D4—1221no—
376C22×D4—1221no—
377C22×D4C22×D6111yescharacteristic
378C23×C4—21no—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—614no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—614no—
386C23×C4—2442no—
387C23×C4—2442no—
388C23×C4C22×D6111yescharacteristic
389C2×M16—1224no—
390C2×M16—2442no—
391C2×M16—2442no—
392C22×C8—612no—
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—614no—
407#117—1222no—
408#117—2442no—
409#117—2442no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4C22×D6112yes—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—1221no—
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—1221no—
426C2×C42—1222no—
427#103—42no—
428#103—2442no—
429#102—42no—
430#102—2442no—
431C22⋊C8—22no—
432C22⋊C8—1224no—
433C22⋊C8—1222no—
434#95—21no—
435#95—1222no—
436#95—1221no—
437C24—21no—
438C24—611no—
439C24—611no—
440C24—614no—
441C24—2442no—
442C24—2441no—
443C24—2442no—
444C24—2441no—
445C24D4×D6111yescharacteristic
446#41—44no—
447#41—44no—
448#41—48no—
449C2×Q8—21no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8C2×C4×D6112yes—
453C2×D4—24no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—44no—
458C2×D4—1224no—
459C2×D4—1224no—
460C2×D4—2442no—
461C2×D4—2444no—
462C2×D4—2442no—
463C2×D4C2×C4×D6112yes—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—44no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—614no—
473C22×C4—612no—
474C22×C4—612no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—614no—
478C22×C4—614no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1222no—
486C22×C4—2444no—
487C22×C4—2444no—
488C22×C4—2448no—
489C22×C4—2442no—
490C22×C4—2442no—
491C22×C4—2442no—
492C22×C4C2×C4×D6112yes—
493C22×C4D4×D6112yes—
494C22×C4D4×D6111yescharacteristic
495C22×C4C23×D6111yescharacteristic
496M16—2448no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—1222no—
501C2×C8—1222no—
502C2×C8—1222no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—612no—
506C4⋊C4—612no—
507C4⋊C4—1224no—
508C4⋊C4—1224no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C22⋊C4—42no—
514C22⋊C4—1224no—
515C22⋊C4—1224no—
516C22⋊C4—1224no—
517C22⋊C4—1224no—
518C22⋊C4—1224no—
519C22⋊C4—2444no—
520C22⋊C4—2444no—
521C22⋊C4—2442no—
522C42—1224no—
523C42—2444no—
524C23—21no—
525C23—22no—
526C23—21no—
527C23—22no—
528C23—22no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—612no—
534C23—612no—
535C23—1221no—
536C23—1221no—
537C23—1222no—
538C23—1222no—
539C23—1221no—
540C23—1221no—
541C23—1222no—
542C23—1222no—
543C23—2444no—
544C23—2444no—
545C23—2444no—
546C23—2448no—
547C23—2448no—
548C23C22⋊C4×D6112yes—
549C23C2×D4×D6111yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—42no—
553D4—22no—
554D4—22no—
555D4—48no—
556D4—42no—
557D4—2448no—
558D4—2448no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—48no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1222no—
577C2×C4—1224no—
578C2×C4—1224no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—2444no—
582C2×C4—2444no—
583C2×C4—2444no—
584C2×C4—2442no—
585C2×C4—2442no—
586C2×C4—2442no—
587C2×C4—2441no—
588C2×C4—2442no—
589C2×C4—2441no—
590C2×C4C22⋊C4×D6112yes—
591C2×C4C22×C4×D6111yescharacteristic
592C2×C4C2×D4×D6111yescharacteristic
593C2×C4#1451982212yes—
594C2×C4#1451982212yes—
595C8—42no—
596C8—2442no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—82no—
606C22—82no—
607C22—1222no—
608C22—1222no—
609C22—1222no—
610C22—1222no—
611C22—1222no—
612C22—1222no—
613C22—24416no—
614C22—2442no—
615C22—2442no—
616C22—2442no—
617C22—2442no—
618C22C2×C22⋊C4×D6111yescharacteristic
619C22#1684032212yes—
620C22#1684055811yescharacteristic
621C22#1684055912yes—
622C22#1684056111yescharacteristic
623C4—22no—
624C4—42no—
625C4—82no—
626C4—1222no—
627C4—1222no—
628C4—4882no—
629C4—2442no—
630C4—2442no—
631C4#1684023511yescharacteristic
632C4#1684023511yescharacteristic
633C4#1684032212yes—
634C2—42no—
635C2—82no—
636C2—2442no—
637C2—2442no—
638C2#2144839511yescharacteristic
639C2#2145433411yescharacteristic
640C2#2145433511yescharacteristic
641C2#2145665812yes—
642C2#2145666012yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1788C2
socleC23×C61C22×D4
prime radicalC2×#16840269C2
cosocleC22×C4×C61C24
hypercenter#1788D61
hypocenterC61#21766
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
2. C2×#16839724C2semidirect
3. #21448201C2semidirect
4. #21446254C2semidirect
5. #21446249C2semidirect
6. #21446138C2semidirect
7. #21446128C2semidirect
8. #21445915C2semidirect
9. #21445911C2semidirect
10. #21445759C2semidirect
11. C61×#1788C2semidirect
15. #16840269C4semidirect
26. C2×#14519701C22semidirect
27. C2×#14519698C22semidirect
37. #16839525C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
42. C2×#103×C61C22semidirect
43. C2×#102×C61C22semidirect
323. #1788D61semidirect
349. C2×#103C2×D61semidirect
351. C2×#102C2×D61semidirect
367. C61#21766semidirect

Maximal subgroups

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

subgroupindex
C2×#168402692
C2×#168397242
#214482012
#214462542
#214462492
#214461382
#214461282
#214459152
#214459112
#214457592
C61×#17882
#2176661