Finite Group Databasebeta

← #29606552

Subgroups — #29606552

Order 15616.

1926 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839633C212yesmaximal
2C2×#16839621C211yesmaximal, characteristic
3C2×#16839619C211yesmaximal, characteristic
4C2×#95×D61C211yesmaximal, characteristic
5#21444082C211yesmaximal, characteristic, complemented
6#21444076C211yesmaximal, characteristic, complemented
7#21444075C211yesmaximal, characteristic, complemented
8#21444070C212yesmaximal, complemented
9#21444069C212yesmaximal, complemented
10#21444040C212yesmaximal
11C61×#1444C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C23×C4×D61C2211yescharacteristic
14C23×C4×D61C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16C22×#11758196C2212yes—
17C22×#11758196C2212yes—
18C22×#11758198C2212yes—
19C22×#11758198C2212yes—
20#16839633—28no—
21#16839621C418yescomplemented
22#16839619—24no—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2211yescharacteristic
26C2×#14519684C2211yescharacteristic
27C2×#14519684C2211yescharacteristic
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2211yescharacteristic
33#95×D61—24no—
34C22×C4⋊C4×C61C2212yes—
35C22×C4⋊C4×C61C2211yescharacteristic
36C2×#95×C61C2211yescharacteristic
37C2×#95×C61C2212yes—
38C2×#95×C61C2211yescharacteristic
39C24×D61C2311yescharacteristic
40C23×Dic61C2312yes—
41C23×Dic61C2312yes—
42C23×Dic61C2311yescharacteristic
43C23×Dic61C2311yescharacteristic
44C23×Dic61C2311yescharacteristic
45C22×C4×D61—22no—
46C22×C4×D61—22no—
47C22×C4×D61—22no—
48C22×C4×D61—22no—
49C22×C4×D61—42no—
50C22×C4×D61—42no—
51C22×C4×D61—42no—
52C22×C4×D61—42no—
53C22×C4×D61—42no—
54C22×C4×D61—42no—
55C22×C4×D61C2×C414yes—
56C22×C4×D61D414yes—
57C22×C4×D61D414yes—
58C22×C4×D61Q814yes—
59C22×C4×D61Q814yes—
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61—42no—
62C2×#11758196—24no—
63C2×#11758196—42no—
64C2×#11758196—42no—
65C2×#11758196—42no—
66C2×#11758196—42no—
67C2×#11758196C2×C418yescomplemented
68C2×#11758198—24no—
69C2×#11758198—24no—
70C2×#11758198—24no—
71C2×#11758198—24no—
72C2×#11758198—24no—
73C2×#11758198C2×C418yes—
74#14519684—22no—
75#14519684—24no—
76#14519684—22no—
77#14519684—22no—
78#14519684—24no—
79#14519684—24no—
80#14519684—42no—
81#14519684—42no—
82#14519684—42no—
83#14519684C2×C414yescomplemented
84C23×C4×C61C2312yes—
85C23×C4×C61C2312yes—
86C23×C4×C61C2311yescharacteristic
87C23×C4×C61C2311yescharacteristic
88C23×C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61—24no—
91C2×C4⋊C4×C61—42no—
92C2×C4⋊C4×C61—42no—
93C2×C4⋊C4×C61—42no—
94#95×C61—22no—
95#95×C61—42no—
96#95×C61C2×C414yescomplemented
97C24×C61C2411yescharacteristic
98C23×D61—22no—
99C23×D61—22no—
100C23×D61—22no—
101C23×D61—22no—
102C23×D61C22×C412yes—
103C23×D61C2×D412yes—
104C23×D61C2×D412yes—
105C23×D61C2×Q812yes—
106C23×D61C2×Q812yes—
107C23×D61#4112yes—
108C23×D61#4112yes—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61C22×C414yes—
133C22×C4×C61C22×C412yes—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×Q812yes—
137C22×C4×C61C2×Q812yes—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—22no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61C22×C414yes—
162C22×Dic61C22×C412yes—
163C22×Dic61C2×D412yes—
164C22×Dic61C2×D412yes—
165C22×Dic61C2×Q812yes—
166C22×Dic61C2×Q812yes—
167C2×C4×D61—28no—
168C2×C4×D61—28no—
169C2×C4×D61—48no—
170C2×C4×D61—44no—
171C2×C4×D61—44no—
172C2×C4×D61—48no—
173C2×C4×D61—48no—
174C2×C4×D61—48no—
175C2×C4×D61—44no—
176C2×C4×D61—44no—
177C2×C4×D61—44no—
178C2×C4×D61—44no—
179C2×C4×D61—48no—
180C2×C4×D61—48no—
181C2×C4×D61C4⋊C4116yescomplemented
182C4⋊C4×C61—44no—
183C4⋊C4×C61—48no—
184#11758198—216no—
185#11758198—48no—
186C4⋊Dic61—44no—
187#11758196—48no—
188#11758196—48no—
189C23×C61C23×C411yescharacteristic
190C23×C61C22×D411yescharacteristic
191C23×C61C22×D411yescharacteristic
192C23×C61C22×Q811yescharacteristic
193C23×C61C22×Q811yescharacteristic
194C23×C61C2×#4111yescharacteristic
195C23×C61C2×#4111yescharacteristic
196C23×C61H(5,𝔽2)12yes—
197C23×C61H(5,𝔽2)12yes—
198C23×C61H(5,𝔽2)12yes—
199C23×C612−1+412yes—
200C22×D61—24no—
201C22×D61—24no—
202C22×D61—22no—
203C22×D61—22no—
204C22×D61—24no—
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—24no—
208C22×D61—24no—
209C22×D61—24no—
210C22×D61—24no—
211C22×D61—24no—
212C22×D61—24no—
213C22×D61—24no—
214C22×D61—24no—
215C22×D61—24no—
216C22×D61C2×C4⋊C414yes—
217C22×D61C4×D414yes—
218C22×D61C4×Q814yes—
219C22×D61#12114yes—
220C22×D61C22⋊Q814yes—
221C22×D61#12514yes—
222C22×D61C4⋊Q814yes—
223C2×C4×C61—24no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—24no—
233C2×C4×C61—24no—
234C2×C4×C61—42no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61C2×C4⋊C414yes—
252C2×Dic61—24no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—24no—
256C2×Dic61—24no—
257C2×Dic61—24no—
258C2×Dic61—24no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—24no—
262C2×Dic61—24no—
263C2×Dic61—42no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61—42no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—42no—
279C2×Dic61—42no—
280C2×Dic61C2×C4⋊C414yes—
281C4×D61—416no—
282C4×D61—416no—
283C4×D61—416no—
284#14275—611nomaximal, Sylow
285C22×C61C22×C4⋊C411yescharacteristic
286C22×C61C2×C4×D411yescharacteristic
287C22×C61C2×C4×Q811yescharacteristic
288C22×C61#51812yes—
289C22×C61#52012yes—
290C22×C61C2×#12111yescharacteristic
291C22×C61C2×C22⋊Q811yescharacteristic
292C22×C61C2×#12511yescharacteristic
293C22×C61C2×C4⋊Q811yescharacteristic
294C22×C61#53411yescharacteristic
295C22×C61#53412yes—
296C22×C61#53411yescharacteristic
297C22×C61#53511yescharacteristic
298C22×C61#53611yescharacteristic
299C22×C61#53712yes—
300C22×C61#53812yes—
301C22×C61#53812yes—
302C22×C61#53912yes—
303C22×C61#53912yes—
304C22×C61#54312yes—
305C22×C61#54312yes—
306C22×C61#54412yes—
307C22×C61#54412yes—
308C2×D61—24no—
309C2×D61—24no—
310C2×D61—28no—
311C2×D61—28no—
312C2×D61—24no—
313C2×D61—24no—
314C2×D61—24no—
315C2×D61—24no—
316C2×D61—28no—
317C2×D61—28no—
318C2×D61#38918yes—
319C4×C61—24no—
320C4×C61—44no—
321C4×C61—42no—
322C4×C61—42no—
323C4×C61—82no—
324Dic61—24no—
325Dic61—44no—
326Dic61—42no—
327Dic61—42no—
328Dic61—82no—
329C2×#385—612no—
330C2×#380—611no—
331C2×#380—611no—
332C22×#95—611no—
333#1444—611no—
334#1444D6111yescharacteristic, complemented
335#1442—611no—
336#1442—611no—
337#1441—612no—
338#1441—612no—
339C22⋊#95—612no—
340C2×C61C2×#38911yescharacteristic
341C2×C61#231511yescharacteristic
342C2×C61#231711yescharacteristic
343C2×C61#232512yes—
344C2×C61#232912yes—
345C2×C61#261711yescharacteristic
346C2×C61#261811yescharacteristic
347C2×C61#262611yescharacteristic
348C2×C61#262711yescharacteristic
349C2×C61#264612yes—
350C2×C61#265912yes—
351D61—28no—
352C24×C4—611no—
353C24×C4—611no—
354C24×C4—611no—
355C22×C4⋊C4—612no—
356C22×C4⋊C4—612no—
357C22×C4⋊C4—611no—
358C22×C4⋊C4C2×D6112yes—
359C22×C4⋊C4C2×D6111yescharacteristic
360C22×C22⋊C4—612no—
361C22×C22⋊C4—612no—
362#385—1228no—
363#380—618no—
364#380—1224no—
365C2×#95—611no—
366C2×#95—611no—
367C2×#95—611no—
368C2×#95—612no—
369C2×#95—612no—
370C2×#95—612no—
371C2×#95—611no—
372C2×#95—612no—
373C2×#95—612no—
374C2×#95—612no—
375C2×#95—1224no—
376C2×#95C2×D6111yescharacteristic
377C2×#95C2×D6112yes—
378C2×#95C2×D6111yescharacteristic
379C61#1427511yescharacteristic, complemented, Sylow
380C25—611no—
381C23×C4—611no—
382C23×C4—614no—
383C23×C4—614no—
384C23×C4—614no—
385C23×C4—612no—
386C23×C4—611no—
387C23×C4—611no—
388C23×C4—614no—
389C23×C4—614no—
390C23×C4—612no—
391C23×C4—1222no—
392C23×C4—1222no—
393C23×C4—1222no—
394C23×C4—1222no—
395C23×C4—2442no—
396C23×C4—2442no—
397C23×C4—2442no—
398C23×C4—2442no—
399C23×C4—2442no—
400C23×C4—2442no—
401C23×C4C22×D6112yes—
402C23×C4C22×D6112yes—
403C23×C4C22×D6111yescharacteristic
404C23×C4C22×D6111yescharacteristic
405C23×C4C22×D6111yescharacteristic
406C2×C4⋊C4—22no—
407C2×C4⋊C4—24no—
408C2×C4⋊C4—42no—
409C2×C4⋊C4—42no—
410C2×C4⋊C4—42no—
411C2×C4⋊C4—618no—
412C2×C4⋊C4—1224no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4—2442no—
415C2×C4⋊C4—2442no—
416C2×C4⋊C4—2442no—
417C2×C4⋊C4—2442no—
418C2×C4⋊C4—2442no—
419C2×C22⋊C4—618no—
420C2×C22⋊C4—1224no—
421C2×C22⋊C4—1224no—
422C2×C22⋊C4—1224no—
423C2×C22⋊C4—1224no—
424C2×C22⋊C4—1224no—
425#95—22no—
426#95—42no—
427#95—614no—
428#95—1222no—
429#95—1222no—
430#95—1224no—
431#95—1224no—
432#95—1222no—
433#95—1224no—
434#95—2442no—
435#95—2442no—
436#95—2442no—
437#95C4×D6114yescomplemented
438C24—612no—
439C24—612no—
440C24—612no—
441C24—612no—
442C24—612no—
443C24—612no—
444C24—612no—
445C24—1222no—
446C24—1222no—
447C24—1222no—
448C24—1222no—
449C24C23×D6111yescharacteristic
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—612no—
474C22×C4—612no—
475C22×C4—612no—
476C22×C4—6116no—
477C22×C4—614no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—1228no—
481C22×C4—1228no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4—1222no—
503C22×C4—1222no—
504C22×C4—1222no—
505C22×C4—2448no—
506C22×C4—2444no—
507C22×C4—2444no—
508C22×C4—2448no—
509C22×C4—2448no—
510C22×C4—2448no—
511C22×C4—2444no—
512C22×C4—2444no—
513C22×C4—2444no—
514C22×C4—2444no—
515C22×C4—2448no—
516C22×C4—2448no—
517C22×C4C2×C4×D6114yes—
518C22×C4C2×C4×D6112yes—
519C22×C4D4×D6112yes—
520C22×C4D4×D6112yes—
521C22×C4Q8×D6112yes—
522C22×C4Q8×D6112yes—
523C4⋊C4—44no—
524C4⋊C4—48no—
525C4⋊C4—2448no—
526C4⋊C4—2444no—
527C4⋊C4—2448no—
528C22⋊C4—12216no—
529C22⋊C4—2448no—
530C23—614no—
531C23—614no—
532C23—614no—
533C23—614no—
534C23—614no—
535C23—614no—
536C23—614no—
537C23—1224no—
538C23—1224no—
539C23—1222no—
540C23—1224no—
541C23—1222no—
542C23—1222no—
543C23—1222no—
544C23—1224no—
545C23—1224no—
546C23—1224no—
547C23—1224no—
548C23—1224no—
549C23—1224no—
550C23—1224no—
551C23—1224no—
552C23—1224no—
553C23C22×C4×D6111yescharacteristic
554C23C2×D4×D6111yescharacteristic
555C23C2×D4×D6111yescharacteristic
556C23C2×Q8×D6111yescharacteristic
557C23C2×Q8×D6111yescharacteristic
558C23#41×D6111yescharacteristic
559C23#41×D6111yescharacteristic
560C23#1451982212yes—
561C23#1451982212yes—
562C23#1451982312yes—
563C23#1451982512yes—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—614no—
593C2×C4—1224no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1224no—
597C2×C4—1224no—
598C2×C4—1224no—
599C2×C4—1224no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1224no—
603C2×C4—1224no—
604C2×C4—24416no—
605C2×C4—24416no—
606C2×C4—24416no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4—2442no—
614C2×C4—2442no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2442no—
620C2×C4—2442no—
621C2×C4—2442no—
622C2×C4—2442no—
623C2×C4—2442no—
624C2×C4C4⋊C4×D6114yes—
625C22—618no—
626C22—1228no—
627C22—1224no—
628C22—1224no—
629C22—1228no—
630C22—1228no—
631C22—1224no—
632C22—1224no—
633C22—1228no—
634C22—1224no—
635C22—1224no—
636C22C2×C4⋊C4×D6111yescharacteristic
637C22C4×D4×D6111yescharacteristic
638C22C4×Q8×D6111yescharacteristic
639C22#1684023412yes—
640C22#1684024312yes—
641C22#121×D6111yescharacteristic
642C22C22⋊Q8×D6111yescharacteristic
643C22#125×D6111yescharacteristic
644C22C4⋊Q8×D6111yescharacteristic
645C22#1684032212yes—
646C22#1684032211yescharacteristic
647C22#1684032311yescharacteristic
648C22#1684032511yescharacteristic
649C22#1684033711yescharacteristic
650C22#1684034212yes—
651C22#1684034412yes—
652C22#1684034812yes—
653C22#1684035812yes—
654C22#1684036012yes—
655C22#1684038712yes—
656C22#1684038712yes—
657C22#1684038912yes—
658C22#1684038912yes—
659C4—24no—
660C4—42no—
661C4—44no—
662C4—42no—
663C4—82no—
664C4—1224no—
665C4—4882no—
666C4—2442no—
667C4—2444no—
668C4—2442no—
669C2—1228no—
670C2#389×D6111yescharacteristic
671C2#2144843011yescharacteristic
672C2#2144844011yescharacteristic
673C2#2144851612yes—
674C2#2144855312yes—
675C2#2145155211yescharacteristic
676C2#2145156211yescharacteristic
677C2#2145162511yescharacteristic
678C2#2145163111yescharacteristic
679C2#2145178812yes—
680C2#2145193212yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1444C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenter#1444D61
hypocenterC61#14275
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
5. #21444082C2semidirect
6. #21444076C2semidirect
7. #21444075C2semidirect
8. #21444070C2semidirect
9. #21444069C2semidirect
11. C61×#1444C2semidirect
21. #16839621C4semidirect
67. C2×#11758196C2×C4semidirect
83. #14519684C2×C4semidirect
96. #95×C61C2×C4semidirect
181. C2×C4×D61C4⋊C4semidirect
334. #1444D61semidirect
379. C61#14275semidirect
437. #95C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396332
C2×#168396212
C2×#168396192
C2×#95×D612
#214440822
#214440762
#214440752
#214440702
#214440692
#214440402
C61×#14442
#1427561