Finite Group Databasebeta

← #29689096

Subgroups — #29689096

Order 15616.

1558 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840272C211yesmaximal, characteristic
2C2×#16839725C211yesmaximal, characteristic, complemented
3#380×D61C211yesmaximal, characteristic, complemented
4#21446258C211yesmaximal, characteristic, complemented
5#21446252C211yesmaximal, characteristic, complemented
6#21446250C211yesmaximal, characteristic, complemented
7#21446133C212yesmaximal, complemented
8#21446125C212yesmaximal, complemented
9#21445912C212yesmaximal, complemented
10#21445756C212yesmaximal, complemented
11C61×#1895C211yesmaximal, characteristic, complemented
12C22×#11758224C2211yescharacteristic
13C23×C4×D61C2211yescharacteristic
14C22×#11758194C2211yescharacteristic
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×#14519735C2211yescharacteristic
25C2×C4⋊C4×D61C2212yes—
26C2×C22⋊C4×D61—21no—
27C2×#14519702C2212yescomplemented
28C2×#14519700C2212yescomplemented
29C2×#14519698C2212yescomplemented
30#16839725—24no—
31C2×#14519689C2212yes—
32C2×#14519689C2212yes—
33#16839622C2211yescharacteristic
34#16839621—21no—
35#16839620—21no—
36#16839619C2212yes—
37#95×D61—21no—
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C22×M16×C61C2211yescharacteristic
42C2×#121×C61C2211yescharacteristic
43C2×#102×C61C2212yescomplemented
44C61×#380C2211yescharacteristic
45C61×#340C2212yescomplemented
46C24×D61D411yescharacteristic
47C2×#11758224—22no—
48C2×#11758224—42no—
49C2×#11758224—42no—
50C2×#11758224C2×C414yescomplemented
51C22×Dic122C2311yescharacteristic
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C23×Dic61—21no—
55C23×Dic61D411yescharacteristic
56C22×C4×D61—22no—
57C22×C4×D61—42no—
58C22×C4×D61—42no—
59C22×C4×D61C2×C414yes—
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D414yes—
63C22×C4×D61C2311yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C2×#11758194—22no—
66C2×#11758194—42no—
67C22×#92418C2312yes—
68#14519759—28no—
69#14519757—22no—
70#14519757C2×C414yes—
71#14519756—22no—
72#14519756C2×C414yescomplemented
73#14519748—22no—
74#14519748C2×C414yes—
75#14519745—28no—
76D4×Dic61—28no—
77#14519737—28no—
78#14519735—22no—
79#14519735C2×C414yescomplemented
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2312yes—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—22no—
85C4⋊C4×D61—42no—
86C4⋊C4×D61—42no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—22no—
93C22⋊C4×D61—44no—
94C2×C4×Dic61—21no—
95#14519702—42no—
96#14519700—42no—
97#14519698—42no—
98#14519689—24no—
99#14519689—24no—
100#14519684—21no—
101#14519684—22no—
102C22×D4×C61—21no—
103C22×D4×C61C2311yescharacteristic
104C23×C4×C61C2311yescharacteristic
105C2×M16×C61—22no—
106C2×M16×C61—42no—
107C22×C8×C61C2312yes—
108#121×C61—22no—
109#121×C61C2×C414yescomplemented
110C2×C4⋊C4×C61C2312yes—
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C22⋊C4×C61—21no—
113C2×C22⋊C4×C61—21no—
114#102×C61—42no—
115#95×C61—21no—
116C24×C61—21no—
117C24×C61C2×D411yescharacteristic
118C23×D61—22no—
119C23×D61—22no—
120C23×D61—42no—
121C23×D61—42no—
122C23×D61C22⋊C414yes—
123C23×D61C2×D411yescharacteristic
124C23×D61C2×D411yescharacteristic
125C2×D4×C61—24no—
126C2×D4×C61—21no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—44no—
130C2×D4×C61C22×C412yes—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61C22×C412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2411yescharacteristic
139C2×C22⋊D61—24no—
140C2×C22⋊D61—24no—
141C2×C22⋊D61—44no—
142C2×C22⋊D61—42no—
143C2×C22⋊D61—42no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—24no—
147C22×Dic61—22no—
148C22×Dic61—42no—
149C22×Dic61—44no—
150C22×Dic61C22⋊C414yes—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153#11758224—48no—
154#11758224—44no—
155#11758224—44no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—48no—
161C2×C4×D61—44no—
162C2×C4×D61—44no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
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—
178M16×C61—44no—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C2×C8×C61—22no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61C22×C412yes—
186C22⋊C4×C61—24no—
187C22⋊C4×C61—42no—
188C22⋊Dic61—24no—
189C22⋊Dic61—24no—
190C22⋊Dic61—42no—
191#11758198—24no—
192#11758198—24no—
193#11758198—44no—
194C4⋊Dic61—21no—
195C4⋊Dic61—24no—
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198C4⋊Dic61C22×C412yes—
199#11758196—24no—
200#11758196—44no—
201C4×Dic61—24no—
202#11758194—44no—
203C2×#92418—22no—
204C2×#92418—22no—
205C2×#92418—22no—
206C23×C61—22no—
207C23×C61—21no—
208C23×C61—21no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—42no—
212C23×C61—41no—
213C23×C61—41no—
214C23×C61—42no—
215C23×C61C2×C22⋊C412yes—
216C23×C61C22×D411yescharacteristic
217C22×D61—22no—
218C22×D61—22no—
219C22×D61—22no—
220C22×D61—21no—
221C22×D61—21no—
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×D61AGL(1,ℤ/8)12yes—
231C22×D61#13712yes—
232D4×C61—24no—
233D4×C61—48no—
234D4×C61—41no—
235D4×C61—41no—
236C2×C4×C61—22no—
237C2×C4×C61—24no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
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—48no—
258C2×Dic61—42no—
259C2×Dic61—41no—
260C2×Dic61—41no—
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—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—44no—
273C4×D61—44no—
274C4×D61—88no—
275Dic122—24no—
276Dic122—41no—
277Dic122—41no—
278C8×C61—42no—
279#92418—42no—
280#21769—611nomaximal, Sylow
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—82no—
290C22×C61—82no—
291C22×C61C22×C22⋊C411yescharacteristic
292C22×C61#53412yes—
293C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C61C2×#13711yescharacteristic
295C22×C61#57512yes—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—24no—
299C2×D61—24no—
300C2×D61—22no—
301C2×D61—22no—
302C2×D61—22no—
303C2×D61—416no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61#41714yes—
307C4×C61—22no—
308C4×C61—42no—
309C4×C61—82no—
310C4×C61#51812yes—
311C4×C61#53411yescharacteristic
312C4×C61#53411yescharacteristic
313Dic61—22no—
314Dic61—22no—
315Dic61—42no—
316Dic61—42no—
317Dic61—82no—
318C2×#525—611no—
319C2×#407—611no—
320C2×#380—611no—
321#1895—611no—
322#1895D6111yescharacteristic, complemented
323#1894—611no—
324#1894—611no—
325#1874—612no—
326#1872—612no—
327#1825—612no—
328#1788—612no—
329C2×C61—42no—
330C2×C61—82no—
331C2×C61#230811yescharacteristic
332C2×C61C2×#41711yescharacteristic
333C2×C61#289611yescharacteristic
334C2×C61#319412yes—
335C2×C61#319612yes—
336D61—24no—
337D61—42no—
338C22×#41—611no—
339C24×C4—611no—
340C22×M16—611no—
341C22×M16C2×D6111yescharacteristic
342#525—618no—
343#525—1224no—
344C2×#123—1221no—
345C2×C22⋊Q8—611no—
346C2×C22⋊Q8—611no—
347C2×#121—611no—
348C2×#121C2×D6111yescharacteristic
349C2×#120—1221no—
350C2×C4×D4—1221no—
351C2×C4×D4—1221no—
352C2×#117—611no—
353C22×C4⋊C4—612no—
354C22×C22⋊C4—1221no—
355C2×#103—612no—
356C2×#103—612no—
357C2×#102—612no—
358C2×#102C2×D6112yescomplemented
359#407—1224no—
360C2×C22⋊C8—612no—
361C2×C22⋊C8—612no—
362#380—611no—
363#380—612no—
364#380—1221no—
365#380—1221no—
366#380C2×D6111yescharacteristic
367C2×#95—1221no—
368#340—612no—
369#340—612no—
370#340—612no—
371#340C2×D6112yescomplemented
372C61#2176911yescharacteristic, complemented, Sylow
373C25—611no—
374C2×#41—614no—
375C2×#41—1222no—
376C2×#41—2442no—
377C2×#41—2442no—
378C22×Q8—611no—
379C22×D4—21no—
380C22×D4—1221no—
381C22×D4—1221no—
382C22×D4C22×D6111yescharacteristic
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—614no—
387C23×C4—611no—
388C23×C4—611no—
389C23×C4—614no—
390C23×C4—1222no—
391C23×C4—1221no—
392C23×C4—2442no—
393C23×C4—2442no—
394C23×C4C22×D6111yescharacteristic
395C2×M16—22no—
396C2×M16—42no—
397C2×M16—1222no—
398C2×M16—2442no—
399C22×C8—612no—
400C22×C8C22×D6112yes—
401#123—1228no—
402C22⋊Q8—614no—
403C22⋊Q8—614no—
404C22⋊Q8—1222no—
405C22⋊Q8—1222no—
406#121—22no—
407#121—614no—
408#121—1222no—
409#121C4×D6114yescomplemented
410#120—1228no—
411C4×D4—1228no—
412C4×D4—1228no—
413#117—614no—
414#117—1222no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—612no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1222no—
420C2×C4⋊C4—2442no—
421C2×C4⋊C4—2442no—
422C2×C4⋊C4C22×D6112yes—
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1222no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—2444no—
433C2×C42—1221no—
434#103—2442no—
435#103—2442no—
436#102—42no—
437#102—2442no—
438C22⋊C8—1224no—
439C22⋊C8—1224no—
440#95—21no—
441#95—1222no—
442#95—1221no—
443C24—21no—
444C24—611no—
445C24—611no—
446C24—614no—
447C24—1222no—
448C24—1222no—
449C24—2442no—
450C24—2442no—
451C24D4×D6111yescharacteristic
452#41—2448no—
453#41—2444no—
454#41—2444no—
455C2×Q8—612no—
456C2×Q8—1221no—
457C2×Q8—1222no—
458C2×Q8—1222no—
459C2×D4—24no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—44no—
464C2×D4—1224no—
465C2×D4—1224no—
466C2×D4—2442no—
467C2×D4—2442no—
468C2×D4—2444no—
469C2×D4C2×C4×D6112yes—
470C22×C4—22no—
471C22×C4—22no—
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—1224no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1222no—
491C22×C4—2448no—
492C22×C4—2444no—
493C22×C4—2444no—
494C22×C4—2442no—
495C22×C4—2442no—
496C22×C4—2444no—
497C22×C4—2444no—
498C22×C4—2444no—
499C22×C4—2442no—
500C22×C4—2444no—
501C22×C4C2×C4×D6112yes—
502C22×C4D4×D6112yes—
503C22×C4D4×D6111yescharacteristic
504C22×C4C23×D6111yescharacteristic
505M16—44no—
506M16—2444no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—1222no—
511C2×C8—1222no—
512C2×C8—1222no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—612no—
517C4⋊C4—1224no—
518C4⋊C4—1221no—
519C4⋊C4—1224no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—2444no—
523C4⋊C4C2×C4×D6112yes—
524C22⋊C4—24no—
525C22⋊C4—42no—
526C22⋊C4—1224no—
527C22⋊C4—1224no—
528C22⋊C4—1224no—
529C22⋊C4—1224no—
530C22⋊C4—2444no—
531C22⋊C4—2442no—
532C42—1224no—
533C23—21no—
534C23—21no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—612no—
543C23—612no—
544C23—612no—
545C23—612no—
546C23—1228no—
547C23—1222no—
548C23—1222no—
549C23—1221no—
550C23—1221no—
551C23—1222no—
552C23—1222no—
553C23—2444no—
554C23—2444no—
555C23—2448no—
556C23—2448no—
557C23C22⋊C4×D6112yes—
558C23C2×D4×D6111yescharacteristic
559Q8—1224no—
560Q8—2441no—
561Q8—2441no—
562D4—24no—
563D4—48no—
564D4—41no—
565D4—41no—
566D4—2448no—
567D4—2448no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—612no—
578C2×C4—612no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1224no—
582C2×C4—1224no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—4888no—
590C2×C4—2444no—
591C2×C4—2444no—
592C2×C4—2444no—
593C2×C4—2448no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2442no—
600C2×C4C22⋊C4×D6112yes—
601C2×C4C22×C4×D6111yescharacteristic
602C2×C4C2×D4×D6111yescharacteristic
603C2×C4#1451982212yes—
604C2×C4#1451982312yes—
605C8—42no—
606C8—2442no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—82no—
616C22—82no—
617C22—614no—
618C22—1224no—
619C22—1222no—
620C22—1224no—
621C22—1222no—
622C22—1222no—
623C22—1222no—
624C22—1222no—
625C22—24416no—
626C22—2442no—
627C22—2442no—
628C22C2×C22⋊C4×D6111yescharacteristic
629C22#1684032012yes—
630C22AGL(1,ℤ/8)×D6111yescharacteristic
631C22#137×D6111yescharacteristic
632C22#1684055812yes—
633C4—22no—
634C4—42no—
635C4—82no—
636C4—1222no—
637C4—1222no—
638C4—4882no—
639C4—2442no—
640C4—2442no—
641C4#1684023412yes—
642C4#1684032211yescharacteristic
643C4#1684032311yescharacteristic
644C2—42no—
645C2—82no—
646C2—1224no—
647C2—2442no—
648C2#2144839311yescharacteristic
649C2#417×D6111yescharacteristic
650C2#2145433311yescharacteristic
651C2#2145662812yes—
652C2#2145664412yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1895C2
socleC23×C61C22×D4
prime radicalC2×#16840272C2
cosocleC22×C4×C61C24
hypercenter#1895D61
hypocenterC61#21769
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1895

Complements

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

NH ≅ G/Ntype
2. C2×#16839725C2semidirect
3. #380×D61C2semidirect
4. #21446258C2semidirect
5. #21446252C2semidirect
6. #21446250C2semidirect
7. #21446133C2semidirect
8. #21446125C2semidirect
9. #21445912C2semidirect
10. #21445756C2semidirect
11. C61×#1895C2semidirect
16. #16840272C4semidirect
27. C2×#14519702C22semidirect
28. C2×#14519700C22semidirect
29. C2×#14519698C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
43. C2×#102×C61C22semidirect
45. C61×#340C22semidirect
50. C2×#11758224C2×C4semidirect
72. #14519756C2×C4semidirect
79. #14519735C2×C4semidirect
109. #121×C61C2×C4semidirect
322. #1895D61semidirect
358. C2×#102C2×D61semidirect
371. #340C2×D61semidirect
372. C61#21769semidirect
409. #121C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402722
C2×#168397252
#380×D612
#214462582
#214462522
#214462502
#214461332
#214461252
#214459122
#214457562
C61×#18952
#2176961