Finite Group Databasebeta

← #29711148

Subgroups — #29711148

Order 15616.

1760 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
1C23×#11758202C211yesmaximal, characteristic, complemented
2C2×#16839924C212yesmaximal, complemented
3C2×#16839900C211yesmaximal, characteristic, complemented
4#21449184C211yesmaximal, characteristic, complemented
5#21446913C212yesmaximal, complemented
6#21446827C212yesmaximal, complemented
7#21446248C211yesmaximal, characteristic, complemented
8#21446160C212yesmaximal, complemented
9#21446112C211yesmaximal, characteristic, complemented
10#21445754C211yesmaximal, characteristic, complemented
11C2×C61×#548C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C22×#11758202—28no—
14C22×#11758202—24no—
15C22×#11758202C2212yes—
16C22×#11758202C2212yes—
17C22×#11758200C2212yes—
18C23×#92418C2211yescharacteristic
19C2×#14519756C2212yes—
20#16839924—28no—
21#16839900—24no—
22C2×#14519701C2212yes—
23C2×#14519701C2212yes—
24C2×#14519700C2212yes—
25C2×#14519698C2212yes—
26C2×C22⋊#92418C2211yescharacteristic
27C2×C22⋊#92418C2212yes—
28#16839682—21no—
29#16839674—21no—
30#16839668C2212yes—
31#16839642—21no—
32#16839634—21no—
33#16839622C2211yescharacteristic
34#16839524C2212yes—
35#16839517C2212yes—
36C22×#41×C61C2211yescharacteristic
37C23×Q8×C61C2211yescharacteristic
38C61×#548—24no—
39C61×#548—24no—
40C2×#124×C61—21no—
41C2×#124×C61C2211yescharacteristic
42C2×C22⋊Q8×C61—21no—
43C2×C22⋊Q8×C61C2211yescharacteristic
44C2×#121×C61—21no—
45C2×#121×C61C2211yescharacteristic
46C2×C4×Q8×C61C2211yescharacteristic
47C2×C4×D4×C61—21no—
48C2×C4×D4×C61C2211yescharacteristic
49C22×Dic122—22no—
50C22×Dic122—24no—
51C22×Dic122—22no—
52C22×Dic122—21no—
53C22×Dic122—42no—
54C22×Dic122C2312yes—
55C23×Dic61—21no—
56C2×#11758202—232no—
57C2×#11758202—24no—
58C2×#11758202—24no—
59C2×#11758202—24no—
60C2×#11758202—24no—
61C2×#11758202—24no—
62C2×#11758202D418yescomplemented
63C2×#11758200—24no—
64C2×#11758200—24no—
65C2×#11758200—42no—
66C22×#92418—22no—
67C22×#92418—24no—
68C22×#92418C2312yes—
69C22×#92418C2312yes—
70#14519756—24no—
71#14519756—42no—
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—22no—
74C2×C22⋊Dic61—22no—
75C2×C22⋊Dic61—21no—
76#14519701—24no—
77#14519701—42no—
78#14519700—24no—
79#14519698—24no—
80C22⋊#92418—22no—
81C22⋊#92418—24no—
82#14519684—21no—
83#14519684—22no—
84#14519684—22no—
85C2×#41×C61—22no—
86C2×#41×C61—22no—
87C2×#41×C61—24no—
88C2×#41×C61—42no—
89C22×Q8×C61—22no—
90C22×Q8×C61—21no—
91C22×Q8×C61—24no—
92C22×Q8×C61—42no—
93C22×Q8×C61D414yes—
94C22×Q8×C61C2311yescharacteristic
95C22×Q8×C61C2311yescharacteristic
96C22×D4×C61—21no—
97C22×D4×C61—21no—
98C22×D4×C61D411yescharacteristic
99C22×D4×C61C2311yescharacteristic
100C23×C4×C61—21no—
101C23×C4×C61—21no—
102C23×C4×C61C2311yescharacteristic
103C23×C4×C61C2311yescharacteristic
104#124×C61—28no—
105#124×C61—22no—
106#124×C61—22no—
107C22⋊Q8×C61—28no—
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61—22no—
110#121×C61—28no—
111#121×C61—22no—
112#121×C61D414yes—
113C4×Q8×C61—22no—
114C4×Q8×C61—22no—
115C4×D4×C61—28no—
116C4×D4×C61—22no—
117C4×D4×C61—22no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61D411yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C22⋊C4×C61—21no—
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61D411yescharacteristic
127C2×C22⋊C4×C61D411yescharacteristic
128C2×C42×C61—21no—
129C2×C42×C61C2311yescharacteristic
130C24×C61—21no—
131C24×C61C2×D411yescharacteristic
132C24×C61C2×D411yescharacteristic
133#41×C61—48no—
134#41×C61—44no—
135#41×C61—44no—
136C2×Q8×C61—24no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—22no—
140C2×Q8×C61—28no—
141C2×Q8×C61—22no—
142C2×Q8×C61—22no—
143C2×Q8×C61—44no—
144C2×Q8×C61—44no—
145C2×Q8×C61—48no—
146C2×Q8×C61C2×D412yes—
147C2×Q8×C61C2×D414yes—
148C2×Q8×C61C2×D412yes—
149C2×D4×C61—24no—
150C2×D4×C61—22no—
151C2×D4×C61—24no—
152C2×D4×C61—22no—
153C2×D4×C61—22no—
154C2×D4×C61—22no—
155C2×D4×C61—21no—
156C2×D4×C61—22no—
157C2×D4×C61—44no—
158C2×D4×C61—42no—
159C2×D4×C61—42no—
160C2×D4×C61C2×D412yes—
161C22×C4×C61—21no—
162C22×C4×C61—22no—
163C22×C4×C61—22no—
164C22×C4×C61—24no—
165C22×C4×C61—24no—
166C22×C4×C61—21no—
167C22×C4×C61—21no—
168C22×C4×C61—21no—
169C22×C4×C61—21no—
170C22×C4×C61—22no—
171C22×C4×C61—22no—
172C22×C4×C61—44no—
173C22×C4×C61—44no—
174C22×C4×C61C2×D411yescharacteristic
175C22×C4×C61C2×D411yescharacteristic
176C22×C4×C61C2×D411yescharacteristic
177C22×C4×C61C2×D411yescharacteristic
178C22×C4×C61C2×D412yes—
179C22×C4×C61C2×D412yes—
180C22×C4×C61C2411yescharacteristic
181C22×Dic61—22no—
182C22×Dic61—22no—
183C22×Dic61—44no—
184C22×Dic61—42no—
185C2×Dic122—28no—
186C2×Dic122—22no—
187C2×Dic122—22no—
188C2×Dic122—22no—
189C2×Dic122—22no—
190C2×Dic122—22no—
191C2×Dic122—48no—
192C2×Dic122—44no—
193C2×Dic122—44no—
194C2×Dic122—42no—
195C2×Dic122C2×D414yes—
196C4⋊C4×C61—24no—
197C4⋊C4×C61—24no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61C2×D412yes—
201C22⋊C4×C61—24no—
202C22⋊C4×C61—24no—
203C22⋊C4×C61—24no—
204C22⋊C4×C61—22no—
205C22⋊C4×C61—24no—
206C22⋊C4×C61—22no—
207C42×C61—24no—
208C42×C61—21no—
209C42×C61—21no—
210C22⋊Dic61—44no—
211C22⋊Dic61—42no—
212#11758202—216no—
213#11758202—216no—
214#11758200—48no—
215C4⋊Dic61—22no—
216C4⋊Dic61—22no—
217#11758196—44no—
218C2×#92418—28no—
219C2×#92418—22no—
220C2×#92418—22no—
221C2×#92418—22no—
222C2×#92418—22no—
223C2×#92418—22no—
224C2×#92418C2×D414yes—
225C23×C61—22no—
226C23×C61—22no—
227C23×C61—21no—
228C23×C61—22no—
229C23×C61—22no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—22no—
233C23×C61—42no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61—42no—
237C23×C61#12012yes—
238C23×C61#13512yes—
239C23×C61#13712yes—
240C23×C61C22×D411yescharacteristic
241Q8×C61—22no—
242Q8×C61—24no—
243Q8×C61—22no—
244Q8×C61—22no—
245Q8×C61—48no—
246Q8×C61—42no—
247Q8×C61#12014yes—
248D4×C61—24no—
249D4×C61—48no—
250D4×C61—44no—
251D4×C61—48no—
252D4×C61—41no—
253D4×C61—41no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—24no—
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—24no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—22no—
263C2×C4×C61—22no—
264C2×C4×C61—21no—
265C2×C4×C61—22no—
266C2×C4×C61—21no—
267C2×C4×C61—21no—
268C2×C4×C61—22no—
269C2×C4×C61—48no—
270C2×C4×C61—48no—
271C2×C4×C61—42no—
272C2×C4×C61—41no—
273C2×C4×C61—41no—
274C2×C4×C61—41no—
275C2×C4×C61—41no—
276C2×C4×C61—42no—
277C2×C4×C61#12012yes—
278C2×C4×C61C22×D411yescharacteristic
279C2×C4×C61C22×D412yes—
280C2×C4×C61C22×D411yescharacteristic
281C2×C4×C61H(5,𝔽2)12yes—
282C2×Dic61—24no—
283C2×Dic61—48no—
284C2×Dic61—42no—
285C2×Dic61—42no—
286C2×Dic61—42no—
287C2×Dic61—42no—
288C2×Dic61—42no—
289Dic122—24no—
290Dic122—24no—
291Dic122—48no—
292Dic122—42no—
293Dic122—42no—
294#92418—24no—
295#92418—42no—
296#23859—611nomaximal, Sylow
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—24no—
300C22×C61—24no—
301C22×C61—42no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61—42no—
305C22×C61—41no—
306C22×C61—42no—
307C22×C61—42no—
308C22×C61—42no—
309C22×C61—82no—
310C22×C61—82no—
311C22×C61#45214yescomplemented
312C22×C61C2×#12011yescharacteristic
313C22×C61D4212yes—
314C22×C61C2×#13511yescharacteristic
315C22×C61C2×#13711yescharacteristic
316C22×C61#57812yes—
317C4×C61—22no—
318C4×C61—24no—
319C4×C61—41no—
320C4×C61—42no—
321C4×C61—41no—
322C4×C61—42no—
323C4×C61—41no—
324C4×C61C2×#12011yescharacteristic
325C4×C61#53411yescharacteristic
326C4×C61D4212yes—
327Dic61—44no—
328Dic61—82no—
329C23×Q16—611no—
330C2×#548D6111yescharacteristic
331C2×#452—612no—
332C2×#448—611no—
333#2409—611no—
334#2015—612no—
335#2001—612no—
336#1893—611no—
337#1878—612no—
338#1869—611no—
339#1787—611no—
340C2×C61—24no—
341C2×C61—81no—
342C2×C61—82no—
343C2×C61#240311yescharacteristic
344C2×C61C2×#45211yescharacteristic
345C2×C61#301111yescharacteristic
346C2×C61#327012yes—
347C2×C61#328512yes—
348C22×#41C2×D6111yescharacteristic
349C23×Q8—611no—
350C23×Q8C2×D6111yescharacteristic
351C22×Q16—612no—
352C22×Q16—612no—
353C22×Q16—1228no—
354C22×Q16—1224no—
355C22×SD16—612no—
356C23×C8—611no—
357#548—24no—
358#548—24no—
359C2×#124—21no—
360C2×#124C2×D6111yescharacteristic
361C2×C22⋊Q8—21no—
362C2×C22⋊Q8—612no—
363C2×C22⋊Q8C2×D6111yescharacteristic
364C2×#121—21no—
365C2×#121C2×D6111yescharacteristic
366C2×C4×Q8C2×D6111yescharacteristic
367C2×C4×D4—21no—
368C2×C4×D4C2×D6111yescharacteristic
369#452—1228no—
370#448—1224no—
371C2×#103—612no—
372C2×#103—612no—
373C2×#103—612no—
374C2×#102—612no—
375C2×C22⋊C8—611no—
376C2×C22⋊C8—612no—
377#395—1221no—
378#393—1221no—
379#391—612no—
380#386—1221no—
381#385—1221no—
382#380—611no—
383#340—612no—
384#336—612no—
385C61#2385911yescharacteristic, complemented, Sylow
386C2×#41—22no—
387C2×#41—22no—
388C2×#41—24no—
389C2×#41—42no—
390C22×Q8—21no—
391C22×Q8—22no—
392C22×Q8—24no—
393C22×Q8—42no—
394C22×Q8—612no—
395C22×Q8—1224no—
396C22×Q8—1221no—
397C22×Q8—1222no—
398C22×Q8—1222no—
399C22×Q8—2442no—
400C22×Q8C22⋊D6114yes—
401C22×Q8C22×D6111yescharacteristic
402C22×Q8C22×D6111yescharacteristic
403C22×D4—21no—
404C22×D4—21no—
405C22×D4C22⋊D6111yescharacteristic
406C22×D4C22×D6111yescharacteristic
407C23×C4—21no—
408C23×C4—21no—
409C23×C4—1221no—
410C23×C4C22×D6111yescharacteristic
411C23×C4C22×D6111yescharacteristic
412C2×Q16—618no—
413C2×Q16—1224no—
414C2×Q16—1224no—
415C2×Q16—12232no—
416C2×Q16—1224no—
417C2×Q16—1224no—
418C2×Q16—1224no—
419C2×SD16—1224no—
420C2×SD16—1224no—
421C2×SD16—2442no—
422C22×C8—612no—
423C22×C8—612no—
424C22×C8—1224no—
425C22×C8—1222no—
426#124—28no—
427#124—22no—
428#124—22no—
429C22⋊Q8—28no—
430C22⋊Q8—22no—
431C22⋊Q8—22no—
432C22⋊Q8—1224no—
433C22⋊Q8—2442no—
434#121—28no—
435#121—22no—
436#121C22⋊D6114yes—
437C4×Q8—22no—
438C4×Q8—22no—
439C4×D4—28no—
440C4×D4—22no—
441C4×D4—22no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—612no—
445C2×C4⋊C4—1222no—
446C2×C4⋊C4C22⋊D6111yescharacteristic
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C22⋊C4—21no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—21no—
451C2×C22⋊C4—21no—
452C2×C22⋊C4—1222no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4C22⋊D6111yescharacteristic
455C2×C22⋊C4C22⋊D6111yescharacteristic
456C2×C42—21no—
457C2×C42C22×D6111yescharacteristic
458#103—1224no—
459#103—1224no—
460#103—2442no—
461#102—1224no—
462C22⋊C8—1224no—
463C22⋊C8—1222no—
464#95—1221no—
465#95—1222no—
466#95—1222no—
467C24—21no—
468C24C2×C22⋊D6111yescharacteristic
469C24C2×C22⋊D6111yescharacteristic
470#41—48no—
471#41—44no—
472#41—44no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—22no—
476C2×Q8—24no—
477C2×Q8—28no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—44no—
481C2×Q8—44no—
482C2×Q8—48no—
483C2×Q8—614no—
484C2×Q8—1228no—
485C2×Q8—1222no—
486C2×Q8—1222no—
487C2×Q8—1222no—
488C2×Q8—1222no—
489C2×Q8—1222no—
490C2×Q8—2444no—
491C2×Q8—2448no—
492C2×Q8—2444no—
493C2×Q8—2442no—
494C2×Q8D4×D6114yes—
495C2×Q8C2×C22⋊D6112yes—
496C2×Q8C2×C22⋊D6112yes—
497C2×D4—24no—
498C2×D4—22no—
499C2×D4—24no—
500C2×D4—22no—
501C2×D4—22no—
502C2×D4—22no—
503C2×D4—21no—
504C2×D4—22no—
505C2×D4—44no—
506C2×D4—42no—
507C2×D4—42no—
508C2×D4C2×C22⋊D6112yes—
509C22×C4—21no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—24no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—24no—
516C22×C4—22no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—22no—
520C22×C4—44no—
521C22×C4—44no—
522C22×C4—1222no—
523C22×C4—1222no—
524C22×C4—2444no—
525C22×C4—2442no—
526C22×C4C2×C22⋊D6111yescharacteristic
527C22×C4C2×C22⋊D6111yescharacteristic
528C22×C4C2×C22⋊D6111yescharacteristic
529C22×C4C2×C22⋊D6112yes—
530C22×C4C2×C22⋊D6112yes—
531C22×C4C2×C22⋊D6111yescharacteristic
532C22×C4C23×D6111yescharacteristic
533Q16—12216no—
534Q16—12216no—
535SD16—2448no—
536C2×C8—614no—
537C2×C8—1228no—
538C2×C8—1222no—
539C2×C8—1222no—
540C2×C8—1222no—
541C2×C8—1222no—
542C2×C8—1222no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—21no—
546C4⋊C4—22no—
547C4⋊C4—1222no—
548C4⋊C4—1222no—
549C4⋊C4—2444no—
550C4⋊C4C2×C22⋊D6112yes—
551C22⋊C4—24no—
552C22⋊C4—24no—
553C22⋊C4—24no—
554C22⋊C4—22no—
555C22⋊C4—24no—
556C22⋊C4—22no—
557C22⋊C4—2444no—
558C22⋊C4—2442no—
559C42—24no—
560C42—21no—
561C42—21no—
562C23—22no—
563C23—22no—
564C23—21no—
565C23—22no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—22no—
570C23—41no—
571C23—41no—
572C23—42no—
573C23—42no—
574C23#1451974612yes—
575C23#1451979712yes—
576C23#1451980712yes—
577C23C22×C22⋊D6111yescharacteristic
578Q8—22no—
579Q8—24no—
580Q8—22no—
581Q8—22no—
582Q8—48no—
583Q8—42no—
584Q8—1224no—
585Q8—1224no—
586Q8—2448no—
587Q8—2442no—
588Q8—2442no—
589Q8#1451974514yes—
590D4—24no—
591D4—48no—
592D4—44no—
593D4—48no—
594D4—41no—
595D4—41no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—1224no—
620C2×C4—2442no—
621C2×C4—2442no—
622C2×C4—2442no—
623C2×C4—2448no—
624C2×C4—2442no—
625C2×C4—2442no—
626C2×C4#1451974612yes—
627C2×C4C2×D4×D6112yes—
628C2×C4C22×C22⋊D6111yescharacteristic
629C2×C4C22×C22⋊D6111yescharacteristic
630C2×C4#1451982212yes—
631C8—1224no—
632C8—2442no—
633C22—22no—
634C22—24no—
635C22—22no—
636C22—24no—
637C22—41no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—41no—
642C22—41no—
643C22—42no—
644C22—42no—
645C22—82no—
646C22—82no—
647C22#1683992414yescomplemented
648C22C2×#1451974611yescharacteristic
649C22D4×C22⋊D6112yes—
650C22C2×#1451979711yescharacteristic
651C22C2×#1451980711yescharacteristic
652C22#1684057712yes—
653C4—22no—
654C4—24no—
655C4—41no—
656C4—41no—
657C4—42no—
658C4—41no—
659C4—42no—
660C4—4882no—
661C4—2444no—
662C4C2×#1451974511yescharacteristic
663C4#1684032111yescharacteristic
664C4D4×C22⋊D6112yes—
665C2—24no—
666C2—81no—
667C2—82no—
668C2#2144912911yescharacteristic
669C2C2×#1683992411yescharacteristic
670C2#2145511411yescharacteristic
671C2#2145738512yes—
672C2#2145755612yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#548C2
socleC23×C61C22×D4
hypercenterC2×#548D61
hypocenterC61#23859
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
1. C23×#11758202C2semidirect
2. C2×#16839924C2semidirect
3. C2×#16839900C2semidirect
4. #21449184C2semidirect
5. #21446913C2semidirect
6. #21446827C2semidirect
7. #21446248C2semidirect
8. #21446160C2semidirect
9. #21446112C2semidirect
10. #21445754C2semidirect
62. C2×#11758202D4semidirect
311. C22×C61#452semidirect
385. C61#23859semidirect
647. C22#16839924semidirect

Maximal subgroups

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

subgroupindex
C23×#117582022
C2×#168399242
C2×#168399002
#214491842
#214469132
#214468272
#214462482
#214461602
#214461122
#214457542
C2×C61×#5482
#2385961