Finite Group Databasebeta

← #29606387

Subgroups — #29606387

Order 15616.

2142 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519754C211yesmaximal, characteristic
2C22×#14519733C211yesmaximal, characteristic
3C2×#16839666C212yesmaximal, complemented
4C2×#16839620C211yesmaximal, characteristic
5#21444071C212yesmaximal, complemented
6#21444070C212yesmaximal, complemented
7#21444068C211yesmaximal, characteristic
8#21444066C211yesmaximal, characteristic
9#21444066C211yesmaximal, characteristic
10#21444047C212yesmaximal, complemented
11C61×#1440C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C24×Dic61C2211yescharacteristic
14C2×#14519754—24no—
15C2×#14519754—24no—
16C2×#14519754C418yescomplemented
17C2×#14519733—28no—
18C2×#14519733—24no—
19C22×C4⋊Dic61C2211yescharacteristic
20C22×#11758196C2212yes—
21C22×#11758196C2212yes—
22C22×#11758196C2211yescharacteristic
23C22×#11758196C2211yescharacteristic
24C22×C22⋊Dic61C2211yescharacteristic
25C22×C22⋊Dic61C2211yescharacteristic
26C22×C22⋊Dic61C2211yescharacteristic
27C22×C4×Dic61C2212yes—
28#16839666—28no—
29#16839620—24no—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C2×#14519684C2212yes—
33C2×#14519684C2212yes—
34C2×#14519684C2212yes—
35C2×#14519684C2212yes—
36C2×#14519684C2212yes—
37C22×C22⋊C4×C61C2211yescharacteristic
38C22×C22⋊C4×C61C2211yescharacteristic
39C22×C22⋊C4×C61C2211yescharacteristic
40C2×#95×C61C2212yes—
41C2×#95×C61C2212yes—
42C22×Dic122—22no—
43C22×Dic122—22no—
44C22×Dic122—24no—
45C22×Dic122—24no—
46C22×Dic122C2×C414yes—
47C23×Dic61—24no—
48C23×Dic61—24no—
49C23×Dic61C2×C414yes—
50C23×Dic61D414yes—
51C23×Dic61D414yes—
52C23×Dic61C2312yes—
53C23×Dic61C2311yescharacteristic
54C23×Dic61C2311yescharacteristic
55C23×Dic61C2312yes—
56C23×Dic61C2311yescharacteristic
57C23×Dic61C2311yescharacteristic
58#14519754—232no—
59#14519733—232no—
60C2×C4⋊Dic61—22no—
61C2×C4⋊Dic61—22no—
62C2×C4⋊Dic61C2×C414yes—
63C2×#11758196—24no—
64C2×#11758196—24no—
65C2×#11758196—24no—
66C2×#11758196—22no—
67C2×#11758196—22no—
68C2×#11758196—22no—
69C2×#11758196—22no—
70C2×#11758196—24no—
71C2×#11758196—24no—
72C2×#11758196—24no—
73C2×#11758196C2×C414yes—
74C2×#11758196C2×C414yes—
75C2×C22⋊Dic61—22no—
76C2×C22⋊Dic61—22no—
77C2×C22⋊Dic61—22no—
78C2×C22⋊Dic61—24no—
79C2×C22⋊Dic61—22no—
80C2×C22⋊Dic61—42no—
81C2×C22⋊Dic61C2×C414yes—
82C2×C4×Dic61—24no—
83C2×C4×Dic61—24no—
84C2×C4×Dic61D418yes—
85#14519684—24no—
86#14519684—24no—
87#14519684—24no—
88#14519684—24no—
89#14519684—42no—
90#14519684—42no—
91#14519684—42no—
92C25×C61C2311yescharacteristic
93C23×C4×C61C2312yes—
94C23×C4×C61C2311yescharacteristic
95C23×C4×C61C2311yescharacteristic
96C23×C4×C61C2312yes—
97C2×C22⋊C4×C61—22no—
98C2×C22⋊C4×C61—22no—
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61—24no—
101C2×C22⋊C4×C61—22no—
102C2×C22⋊C4×C61—42no—
103C2×C22⋊C4×C61C2×C414yes—
104#95×C61—24no—
105#95×C61—42no—
106C24×C61—22no—
107C24×C61—22no—
108C24×C61—22no—
109C24×C61—22no—
110C24×C61—22no—
111C24×C61—22no—
112C24×C61C22×C412yes—
113C24×C61C2×D412yes—
114C24×C61C2×D412yes—
115C24×C61C2411yescharacteristic
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
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×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61C22×C412yes—
138C22×C4×C61C22×C412yes—
139C22×C4×C61C2×D414yes—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—216no—
143C22×Dic61—28no—
144C22×Dic61—216no—
145C22×Dic61—28no—
146C22×Dic61—22no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61—21no—
152C22×Dic61—21no—
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×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61—22no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61C22⋊C4116yes—
168C22×Dic61C22×C412yes—
169C22×Dic61C22×C412yes—
170C22×Dic61C22×C412yes—
171C22×Dic61C22×C412yes—
172C22×Dic61C2×D414yes—
173C22×Dic61C2×D412yes—
174C22×Dic61C2×D412yes—
175C22×Dic61C2×D412yes—
176C22×Dic61C2×D412yes—
177C22×Dic61C2×D414yes—
178C22×Dic61#4114yes—
179C2×Dic122—28no—
180C2×Dic122—216no—
181C2×Dic122—44no—
182C2×Dic122—48no—
183C2×Dic122—44no—
184C22⋊C4×C61—28no—
185C22⋊C4×C61—28no—
186C22⋊C4×C61—44no—
187C22⋊Dic61—28no—
188C22⋊Dic61—28no—
189C22⋊Dic61—44no—
190C4⋊Dic61—28no—
191#11758196—216no—
192#11758196—28no—
193#11758196—28no—
194#11758196—48no—
195C4×Dic61—216no—
196C23×C61—24no—
197C23×C61—22no—
198C23×C61—22no—
199C23×C61—24no—
200C23×C61—22no—
201C23×C61—24no—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61—24no—
205C23×C61—24no—
206C23×C61—22no—
207C23×C61—24no—
208C23×C61—42no—
209C23×C61—42no—
210C23×C61—42no—
211C23×C61—42no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61—42no—
215C23×C61—42no—
216C23×C61C2×C22⋊C414yes—
217C23×C61C23×C411yescharacteristic
218C23×C61C22×D411yescharacteristic
219C23×C61C22×D411yescharacteristic
220C23×C61C22×D412yes—
221C23×C61C2×#4112yes—
222C23×C61H(5,𝔽2)12yes—
223C23×C612−1+412yes—
224C23×C612−1+412yes—
225C23×C612−1+412yes—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—41no—
234C2×C4×C61—42no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—42no—
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×Dic61—24no—
251C2×Dic61—22no—
252C2×Dic61—24no—
253C2×Dic61—22no—
254C2×Dic61—232no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—24no—
258C2×Dic61—24no—
259C2×Dic61—24no—
260C2×Dic61—24no—
261C2×Dic61—24no—
262C2×Dic61—41no—
263C2×Dic61—42no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—42no—
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×Dic61C2×C22⋊C414yes—
280C2×Dic61C2×C22⋊C414yes—
281C2×Dic61C4×D418yes—
282C2×Dic61#12014yes—
283C2×Dic61#12118yes—
284C2×Dic61#12314yes—
285Dic122—416no—
286#14258—611nomaximal, Sylow
287C22×C61—24no—
288C22×C61—24no—
289C22×C61—24no—
290C22×C61—44no—
291C22×C61—44no—
292C22×C61—44no—
293C22×C61—44no—
294C22×C61—44no—
295C22×C61—44no—
296C22×C61C22×C22⋊C411yescharacteristic
297C22×C61C2×C4×D412yes—
298C22×C61#51911yescharacteristic
299C22×C61#51911yescharacteristic
300C22×C61#52012yes—
301C22×C61C2×#12011yescharacteristic
302C22×C61C2×#12112yes—
303C22×C61C2×#12311yescharacteristic
304C22×C61#53511yescharacteristic
305C22×C61#53612yes—
306C22×C61#53612yes—
307C22×C61#53612yes—
308C22×C61#53611yescharacteristic
309C22×C61#53612yes—
310C22×C61#53712yes—
311C22×C61#53712yes—
312C22×C61#53712yes—
313C22×C61#54112yes—
314C22×C61#54112yes—
315C22×C61#54212yes—
316C22×C61#54212yes—
317C4×C61—42no—
318C4×C61—44no—
319C4×C61—42no—
320C4×C61—82no—
321Dic61—24no—
322Dic61—42no—
323Dic61—44no—
324Dic61—42no—
325Dic61—82no—
326Dic61#38618yes—
327C22×C22⋊Q8—611no—
328C22×#117—611no—
329C2×#391—612no—
330C2×#380—611no—
331#1441—612no—
332#1441—612no—
333#1440—611no—
334#1440—611no—
335#1440—611no—
336#1440D6111yescharacteristic
337C4⋊#95—612no—
338C2×C61—44no—
339C2×C61C2×#38611yescharacteristic
340C2×C61#231011yescharacteristic
341C2×C61#231111yescharacteristic
342C2×C61#232312yes—
343C2×C61#232312yes—
344C2×C61#261612yes—
345C2×C61#262012yes—
346C2×C61#262112yes—
347C2×C61#262512yes—
348C23×Q8—611no—
349C24×C4—611no—
350C2×C22⋊Q8—618no—
351C2×C22⋊Q8—1224no—
352C2×C22⋊Q8—1224no—
353C2×#117—1228no—
354C2×#117—1224no—
355C22×C4⋊C4—611no—
356C22×C4⋊C4—611no—
357C22×C4⋊C4—611no—
358C22×C4⋊C4—612no—
359C22×C4⋊C4—612no—
360C22×C22⋊C4—611no—
361C22×C22⋊C4—611no—
362C22×C22⋊C4—611no—
363C22×C22⋊C4C2×D6111yescharacteristic
364C22×C22⋊C4C2×D6111yescharacteristic
365C22×C22⋊C4C2×D6111yescharacteristic
366C22×C42—612no—
367#391—1228no—
368#380—1224no—
369C2×#95—612no—
370C2×#95—612no—
371C2×#95—612no—
372C2×#95—612no—
373C2×#95—612no—
374C2×#95—612no—
375C2×#95—612no—
376C2×#95C2×D6112yes—
377C2×#95C2×D6112yes—
378C61#1425811yescharacteristic, complemented, Sylow
379C25C22×D6111yescharacteristic
380C22×Q8—614no—
381C22×Q8—1222no—
382C22×Q8—1222no—
383C22×Q8—1224no—
384C22×Q8—1224no—
385C23×C4—611no—
386C23×C4—614no—
387C23×C4—614no—
388C23×C4—614no—
389C23×C4—611no—
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—612no—
393C23×C4—612no—
394C23×C4—1224no—
395C23×C4—1224no—
396C23×C4C22×D6112yes—
397C23×C4C22×D6111yescharacteristic
398C23×C4C22×D6111yescharacteristic
399C23×C4C22×D6112yes—
400C22⋊Q8—12232no—
401#117—12232no—
402C2×C4⋊C4—614no—
403C2×C4⋊C4—614no—
404C2×C4⋊C4—614no—
405C2×C4⋊C4—1224no—
406C2×C4⋊C4—1224no—
407C2×C4⋊C4—1222no—
408C2×C4⋊C4—1222no—
409C2×C4⋊C4—1222no—
410C2×C4⋊C4—1222no—
411C2×C4⋊C4—1222no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1224no—
414C2×C4⋊C4—1224no—
415C2×C4⋊C4—1224no—
416C2×C4⋊C4—1224no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—22no—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—24no—
421C2×C22⋊C4—22no—
422C2×C22⋊C4—42no—
423C2×C22⋊C4—614no—
424C2×C22⋊C4—1224no—
425C2×C22⋊C4—1222no—
426C2×C22⋊C4—1222no—
427C2×C22⋊C4—1222no—
428C2×C22⋊C4—1222no—
429C2×C22⋊C4—2442no—
430C2×C22⋊C4C4×D6114yes—
431C2×C42—618no—
432C2×C42—1224no—
433C2×C42—1224no—
434#95—24no—
435#95—42no—
436#95—1224no—
437#95—1224no—
438#95—1224no—
439#95—1224no—
440#95—2442no—
441#95—2442no—
442#95—2442no—
443C24—22no—
444C24—22no—
445C24—22no—
446C24—22no—
447C24—22no—
448C24—22no—
449C24C2×C4×D6112yes—
450C24D4×D6112yes—
451C24D4×D6112yes—
452C24C23×D6111yescharacteristic
453C2×Q8—1228no—
454C2×Q8—12216no—
455C2×Q8—2444no—
456C2×Q8—2448no—
457C2×Q8—2444no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
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—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—612no—
480C22×C4—614no—
481C22×C4—612no—
482C22×C4—612no—
483C22×C4—614no—
484C22×C4—612no—
485C22×C4—612no—
486C22×C4—612no—
487C22×C4—6116no—
488C22×C4—612no—
489C22×C4—612no—
490C22×C4—614no—
491C22×C4—1228no—
492C22×C4—12216no—
493C22×C4—12216no—
494C22×C4—1228no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1222no—
505C22×C4—1222no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—1222no—
510C22×C4—1222no—
511C22×C4—1222no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—1222no—
515C22×C4—1222no—
516C22×C4—1222no—
517C22×C4—1222no—
518C22×C4C2×C4×D6112yes—
519C22×C4C2×C4×D6112yes—
520C22×C4D4×D6114yes—
521C4⋊C4—1228no—
522C4⋊C4—1228no—
523C4⋊C4—1228no—
524C4⋊C4—12216no—
525C4⋊C4—2448no—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—44no—
529C22⋊C4—1228no—
530C22⋊C4—1228no—
531C22⋊C4—2444no—
532C42—12216no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—22no—
538C23—24no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23—22no—
544C23—24no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23C22⋊C4×D6114yes—
554C23C22×C4×D6111yescharacteristic
555C23C2×D4×D6111yescharacteristic
556C23C2×D4×D6111yescharacteristic
557C23C2×D4×D6112yes—
558C23#41×D6112yes—
559C23#1451982212yes—
560C23#1451982412yes—
561C23#1451982412yes—
562C23#1451982412yes—
563Q8—24416no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
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—614no—
589C2×C4—618no—
590C2×C4—614no—
591C2×C4—618no—
592C2×C4—614no—
593C2×C4—614no—
594C2×C4—12232no—
595C2×C4—1224no—
596C2×C4—1222no—
597C2×C4—1224no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1224no—
602C2×C4—1224no—
603C2×C4—1224no—
604C2×C4—1224no—
605C2×C4—1224no—
606C2×C4—2442no—
607C2×C4—2442no—
608C2×C4—2442no—
609C2×C4—2442no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4—2441no—
614C2×C4—2442no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2442no—
618C2×C4—2442no—
619C2×C4—2441no—
620C2×C4—2442no—
621C2×C4—2442no—
622C2×C4—2442no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22C2×C22⋊C4×D6111yescharacteristic
633C22C4×D4×D6112yes—
634C22#1684023911yescharacteristic
635C22#1684023911yescharacteristic
636C22#1684024112yes—
637C22#120×D6111yescharacteristic
638C22#121×D6112yes—
639C22#123×D6111yescharacteristic
640C22#1684032711yescharacteristic
641C22#1684033212yes—
642C22#1684033412yes—
643C22#1684033412yes—
644C22#1684033412yes—
645C22#1684033411yescharacteristic
646C22#1684033912yes—
647C22#1684033912yes—
648C22#1684034112yes—
649C22#1684036912yes—
650C22#1684036912yes—
651C22#1684037612yes—
652C22#1684038012yes—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—82no—
657C4—618no—
658C4—1224no—
659C4—4882no—
660C4—2444no—
661C4—2442no—
662C4—2442no—
663C2—44no—
664C2#386×D6111yescharacteristic
665C2#2144840411yescharacteristic
666C2#2144840611yescharacteristic
667C2#2144849412yes—
668C2#2144849812yes—
669C2#2145154712yes—
670C2#2145157512yes—
671C2#2145158412yes—
672C2#2145161512yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×#1440C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenter#1440D61
hypocenterC61#14258
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ #1440

Complements

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

NH ≅ G/Ntype
3. C2×#16839666C2semidirect
5. #21444071C2semidirect
6. #21444070C2semidirect
10. #21444047C2semidirect
16. C2×#14519754C4semidirect
378. C61#14258semidirect

Maximal subgroups

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

subgroupindex
C22×#145197542
C22×#145197332
C2×#168396662
C2×#168396202
#214440712
#214440702
#214440682
#214440662
#214440472
C61×#14402
#1425861