Finite Group Databasebeta

← #29711145

Subgroups — #29711145

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×#16840421C211yesmaximal, characteristic
3C2×#16839922C212yesmaximal, complemented
4C2×#16839901C211yesmaximal, characteristic, complemented
5#21446912C212yesmaximal, complemented
6#21446827C212yesmaximal, complemented
7#21446245C211yesmaximal, characteristic, complemented
8#21446163C212yesmaximal, complemented
9#21446112C211yesmaximal, characteristic, complemented
10#21445753C211yesmaximal, characteristic, complemented
11C61×#2409C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×Dic122C2211yescharacteristic
14C22×#11758202—28no—
15C22×#11758202—24no—
16C22×#11758202C2212yes—
17C22×#11758202C2212yes—
18C22×#11758201C2212yes—
19C23×#92418C2211yescharacteristic
20#16840421—24no—
21#16840421—24no—
22C2×#14519765C2211yescharacteristic
23C2×#14519763—21no—
24C2×#14519756C2211yescharacteristic
25C2×#14519755—21no—
26C2×#14519751C2211yescharacteristic
27C2×#14519749—21no—
28C2×C4×Dic122C2211yescharacteristic
29C2×#14519737—21no—
30#16839922—28no—
31#16839901—24no—
32C2×#14519701C2212yes—
33C2×#14519700C2212yes—
34C2×#14519700C2212yes—
35C2×#14519697C2212yes—
36C2×C22⋊#92418C2211yescharacteristic
37C2×C22⋊#92418C2212yes—
38#16839524C2212yes—
39#16839517C2212yes—
40C23×Q8×C61C2211yescharacteristic
41C2×C22⋊Q8×C61C2212yes—
42C2×C4×D4×C61C2211yescharacteristic
43C61×#395—21no—
44C61×#393—21no—
45C61×#391C2212yes—
46C61×#386—21no—
47C61×#385—21no—
48C61×#380C2211yescharacteristic
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—24no—
52C2×#11758222—42no—
53C22×Dic122—24no—
54C22×Dic122—21no—
55C22×Dic122—22no—
56C22×Dic122—42no—
57C22×Dic122D414yes—
58C22×Dic122C2311yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C22⋊D61—21no—
62C22×C22⋊D61—21no—
63C23×Dic61—21no—
64C22×C4×D61—21no—
65C2×#11758202—232no—
66C2×#11758202—24no—
67C2×#11758202—24no—
68C2×#11758202—24no—
69C2×#11758202—24no—
70C2×#11758202—24no—
71C2×#11758202D418yescomplemented
72C2×#11758201—24no—
73C2×#11758201—24no—
74C2×#11758201—42no—
75C22×#92418—22no—
76C22×#92418—24no—
77C22×#92418C2312yes—
78C22×#92418C2312yes—
79#14519765—22no—
80#14519765—22no—
81#14519763—28no—
82#14519756—22no—
83#14519756—22no—
84#14519755—28no—
85#14519751—22no—
86#14519751D414yes—
87#14519749—28no—
88C4×Dic122—22no—
89C4×Dic122—22no—
90#14519737—28no—
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×C22⋊Dic61—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×C4×Dic61—21no—
99#14519701—24no—
100#14519700—24no—
101#14519700—42no—
102#14519697—24no—
103C22⋊#92418—22no—
104C22⋊#92418—24no—
105C22×Q8×C61—21no—
106C22×Q8×C61—24no—
107C22×Q8×C61—22no—
108C22×Q8×C61—22no—
109C22×Q8×C61—42no—
110C22×Q8×C61C2312yes—
111C22×D4×C61D411yescharacteristic
112C23×C4×C61—21no—
113C23×C4×C61C2311yescharacteristic
114C23×C4×C61C2311yescharacteristic
115C22⋊Q8×C61—24no—
116C22⋊Q8×C61—42no—
117C4×D4×C61—22no—
118C4×D4×C61—22no—
119C2×C4⋊C4×C61—22no—
120C2×C4⋊C4×C61D411yescharacteristic
121C2×C4⋊C4×C61C2312yes—
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—22no—
124C2×C22⋊C4×C61D411yescharacteristic
125C2×C22⋊C4×C61D411yescharacteristic
126C2×C42×C61C2311yescharacteristic
127#95×C61—21no—
128#95×C61—22no—
129#95×C61—22no—
130C24×C61C2×D411yescharacteristic
131C24×C61C2×D411yescharacteristic
132C23×D61—21no—
133C2×Q8×C61—22no—
134C2×Q8×C61—28no—
135C2×Q8×C61—22no—
136C2×Q8×C61—22no—
137C2×Q8×C61—22no—
138C2×Q8×C61—22no—
139C2×Q8×C61—44no—
140C2×Q8×C61—48no—
141C2×Q8×C61—44no—
142C2×Q8×C61—42no—
143C2×Q8×C61C2×D414yes—
144C2×D4×C61—22no—
145C2×D4×C61—22no—
146C2×D4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—22no—
153C22×C4×C61—22no—
154C22×C4×C61—22no—
155C22×C4×C61—44no—
156C22×C4×C61—42no—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D412yes—
162C22×C4×C61C2×D412yes—
163C22×C4×C61C2411yescharacteristic
164C2×C22⋊D61—24no—
165C2×C22⋊D61—24no—
166C2×C22⋊D61—44no—
167C2×C22⋊D61—42no—
168C2×C22⋊D61—42no—
169C22×Dic61—21no—
170C22×Dic61—24no—
171C22×Dic61—21no—
172C22×Dic61—21no—
173C22×Dic61—44no—
174#11758222—48no—
175#11758222—44no—
176#11758222—44no—
177C2×D244—22no—
178C2×D244—22no—
179C2×D244—21no—
180C2×D244C2×D412yes—
181C2×C4×D61—24no—
182C2×C4×D61—44no—
183C2×Dic122—21no—
184C2×Dic122—22no—
185C2×Dic122—22no—
186C2×Dic122—22no—
187C2×Dic122—28no—
188C2×Dic122—24no—
189C2×Dic122—21no—
190C2×Dic122—44no—
191C2×Dic122—48no—
192C2×Dic122—44no—
193C2×Dic122C2×D412yes—
194C2×Dic122C2×D414yes—
195C2×Dic122C2×D412yes—
196C4⋊C4×C61—22no—
197C4⋊C4×C61—22no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61—44no—
200C22⋊C4×C61—22no—
201C22⋊C4×C61—22no—
202C22⋊C4×C61—44no—
203C22⋊C4×C61—42no—
204C42×C61—21no—
205C42×C61—21no—
206C22⋊Dic61—24no—
207#11758202—216no—
208#11758202—216no—
209#11758201—48no—
210#11758198—24no—
211#11758198—24no—
212#11758198—24no—
213C4⋊Dic61—21no—
214C4⋊Dic61C2×D412yes—
215#11758196—24no—
216#11758196—24no—
217C4×Dic61—24no—
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—21no—
230C23×C61—21no—
231C23×C61—22no—
232C23×C61#12012yes—
233C23×C61#13512yes—
234C23×C61#13712yes—
235C23×C61C22×D411yescharacteristic
236C22×D61—22no—
237C22×D61—42no—
238C22×D61—41no—
239C22×D61—41no—
240C22×D61—42no—
241Q8×C61—24no—
242Q8×C61—24no—
243Q8×C61—48no—
244Q8×C61—42no—
245Q8×C61—42no—
246D4×C61—44no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—24no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—24no—
255C2×C4×C61—22no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—22no—
259C2×C4×C61—48no—
260C2×C4×C61—42no—
261C2×C4×C61—42no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—42no—
265C2×C4×C61—42no—
266C2×C4×C61—42no—
267C2×C4×C61—42no—
268C2×C4×C61#12012yes—
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D412yes—
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61H(5,𝔽2)12yes—
273C22⋊D61—48no—
274C22⋊D61—48no—
275C2×Dic61—22no—
276C2×Dic61—24no—
277C2×Dic61—22no—
278C2×Dic61—22no—
279C2×Dic61—48no—
280C2×Dic61—42no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283D244—24no—
284D244—41no—
285D244—41no—
286C4×D61—48no—
287Dic122—22no—
288Dic122—24no—
289Dic122—22no—
290Dic122—22no—
291Dic122—48no—
292Dic122—42no—
293Dic122#12014yes—
294#92418—24no—
295#92418—42no—
296#23859—611nomaximal, Sylow
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—24no—
300C22×C61—24no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61#45214yescomplemented
307C22×C61C2×#12011yescharacteristic
308C22×C61D4212yes—
309C22×C61C2×#13511yescharacteristic
310C22×C61C2×#13711yescharacteristic
311C22×C61#57812yes—
312C2×D61—42no—
313C2×D61—42no—
314C2×D61—42no—
315C2×D61—82no—
316C2×D61—82no—
317C4×C61—22no—
318C4×C61—41no—
319C4×C61—44no—
320C4×C61—41no—
321C4×C61—41no—
322C4×C61—82no—
323C4×C61C2×#12011yescharacteristic
324C4×C61#53411yescharacteristic
325C4×C61D4212yes—
326Dic61—24no—
327Dic61—42no—
328Dic61—42no—
329C23×Q16—611no—
330C2×#548—611no—
331C2×#452—612no—
332C2×#448—611no—
333#2409D6111yescharacteristic, complemented
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#240311yescharacteristic
343C2×C61C2×#45211yescharacteristic
344C2×C61#301111yescharacteristic
345C2×C61#327012yes—
346C2×C61#328512yes—
347D61—82no—
348C22×#41—611no—
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—1224no—
358#548—1224no—
359C2×#124—611no—
360C2×#124—1221no—
361C2×C22⋊Q8—611no—
362C2×C22⋊Q8—1221no—
363C2×C22⋊Q8C2×D6112yes—
364C2×#121—611no—
365C2×#121—1221no—
366C2×C4×Q8—611no—
367C2×C4×D4—1221no—
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—21no—
378#393—21no—
379#391C2×D6112yes—
380#386—21no—
381#385—21no—
382#380C2×D6111yescharacteristic
383#340—612no—
384#336—612no—
385C61#2385911yescharacteristic, complemented, Sylow
386C2×#41—1222no—
387C2×#41—1222no—
388C2×#41—1224no—
389C2×#41—2442no—
390C22×Q8—21no—
391C22×Q8—24no—
392C22×Q8—22no—
393C22×Q8—22no—
394C22×Q8—42no—
395C22×Q8—614no—
396C22×Q8—611no—
397C22×Q8—611no—
398C22×Q8—1224no—
399C22×Q8—1222no—
400C22×Q8—1221no—
401C22×Q8—2442no—
402C22×Q8C22×D6112yes—
403C22×D4—611no—
404C22×D4—1221no—
405C22×D4—1221no—
406C22×D4C22⋊D6111yescharacteristic
407C23×C4—21no—
408C23×C4—1221no—
409C23×C4—1221no—
410C23×C4C22×D6111yescharacteristic
411C23×C4C22×D6111yescharacteristic
412C2×Q16—618no—
413C2×Q16—1224no—
414C2×Q16—1224no—
415C2×Q16—1224no—
416C2×Q16—1224no—
417C2×Q16—12232no—
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—1222no—
427#124—1222no—
428#124—1228no—
429C22⋊Q8—24no—
430C22⋊Q8—42no—
431C22⋊Q8—1228no—
432C22⋊Q8—1222no—
433C22⋊Q8—1222no—
434#121—614no—
435#121—1228no—
436#121—1222no—
437C4×Q8—1222no—
438C4×Q8—1222no—
439C4×D4—22no—
440C4×D4—22no—
441C4×D4—1228no—
442C2×C4⋊C4—22no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4C22⋊D6111yescharacteristic
447C2×C4⋊C4C22×D6112yes—
448C2×C22⋊C4—22no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4C22⋊D6111yescharacteristic
455C2×C22⋊C4C22⋊D6111yescharacteristic
456C2×C42—1221no—
457C2×C42C22×D6111yescharacteristic
458#103—1224no—
459#103—1224no—
460#103—2442no—
461#102—1224no—
462C22⋊C8—1224no—
463C22⋊C8—1222no—
464#95—21no—
465#95—22no—
466#95—22no—
467C24—1221no—
468C24C2×C22⋊D6111yescharacteristic
469C24C2×C22⋊D6111yescharacteristic
470#41—2448no—
471#41—2444no—
472#41—2444no—
473C2×Q8—22no—
474C2×Q8—28no—
475C2×Q8—22no—
476C2×Q8—22no—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×Q8—44no—
480C2×Q8—48no—
481C2×Q8—44no—
482C2×Q8—42no—
483C2×Q8—614no—
484C2×Q8—612no—
485C2×Q8—612no—
486C2×Q8—1228no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—1222no—
490C2×Q8—1224no—
491C2×Q8—1222no—
492C2×Q8—1222no—
493C2×Q8—2444no—
494C2×Q8—2444no—
495C2×Q8—2448no—
496C2×Q8D4×D6114yes—
497C2×D4—22no—
498C2×D4—22no—
499C2×D4—22no—
500C2×D4—612no—
501C2×D4—1224no—
502C2×D4—1224no—
503C2×D4—1222no—
504C2×D4—1221no—
505C2×D4—1222no—
506C2×D4—2444no—
507C2×D4—2442no—
508C2×D4—2442no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—44no—
518C22×C4—42no—
519C22×C4—1224no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—1224no—
524C22×C4—2444no—
525C22×C4—2444no—
526C22×C4D4×D6112yes—
527C22×C4D4×D6112yes—
528C22×C4C2×C22⋊D6111yescharacteristic
529C22×C4C2×C22⋊D6111yescharacteristic
530C22×C4C2×C22⋊D6111yescharacteristic
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—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—44no—
547C4⋊C4—612no—
548C4⋊C4—1224no—
549C4⋊C4—1224no—
550C4⋊C4—1221no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—44no—
554C22⋊C4—42no—
555C22⋊C4—1224no—
556C22⋊C4—1224no—
557C22⋊C4—1224no—
558C22⋊C4—1224no—
559C42—21no—
560C42—21no—
561C42—1224no—
562C23—22no—
563C23—22no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—22no—
568C23—22no—
569C23—1222no—
570C23—2441no—
571C23—2441no—
572C23—2442no—
573C23—2442no—
574C23#1451974512yes—
575C23#1451979712yes—
576C23#1451980812yes—
577C23C22×C22⋊D6111yescharacteristic
578Q8—24no—
579Q8—24no—
580Q8—48no—
581Q8—42no—
582Q8—42no—
583Q8—614no—
584Q8—1222no—
585Q8—1224no—
586Q8—1222no—
587Q8—1222no—
588Q8—2448no—
589Q8—2442no—
590D4—44no—
591D4—1224no—
592D4—2448no—
593D4—2448no—
594D4—2441no—
595D4—2441no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—1222no—
618C2×C4—1224no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—2448no—
622C2×C4—2448no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2442no—
626C2×C4#1451974512yes—
627C2×C4C2×D4×D6111yescharacteristic
628C2×C4C2×D4×D6112yes—
629C2×C4C2×D4×D6111yescharacteristic
630C2×C4#1451982212yes—
631C8—1224no—
632C8—2442no—
633C22—22no—
634C22—24no—
635C22—22no—
636C22—24no—
637C22—41no—
638C22—42no—
639C22—41no—
640C22—41no—
641C22—42no—
642C22—4882no—
643C22—4882no—
644C22—2442no—
645C22—2442no—
646C22—2442no—
647C22#1683992214yescomplemented
648C22C2×#1451974511yescharacteristic
649C22D4×C22⋊D6112yes—
650C22C2×#1451979711yescharacteristic
651C22C2×#1451980811yescharacteristic
652C22#1684057712yes—
653C4—22no—
654C4—41no—
655C4—41no—
656C4—44no—
657C4—41no—
658C4—82no—
659C4—1224no—
660C4—2442no—
661C4—2442no—
662C4#120×D6111yescharacteristic
663C4#1684032211yescharacteristic
664C4#1684039412yes—
665C2—24no—
666C2—81no—
667C2—4882no—
668C2#2144913011yescharacteristic
669C2C2×#1683992211yescharacteristic
670C2#2145511211yescharacteristic
671C2#2145739012yes—
672C2#2145755612yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2409C2
socleC23×C61C22×D4
prime radicalC2×#16840421C2
hypercenter#2409D61
hypocenterC61#23859
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
1. C23×#11758202C2semidirect
3. C2×#16839922C2semidirect
4. C2×#16839901C2semidirect
5. #21446912C2semidirect
6. #21446827C2semidirect
7. #21446245C2semidirect
8. #21446163C2semidirect
9. #21446112C2semidirect
10. #21445753C2semidirect
11. C61×#2409C2semidirect
71. C2×#11758202D4semidirect
306. C22×C61#452semidirect
333. #2409D61semidirect
385. C61#23859semidirect
647. C22#16839922semidirect

Maximal subgroups

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

subgroupindex
C23×#117582022
C2×#168404212
C2×#168399222
C2×#168399012
#214469122
#214468272
#214462452
#214461632
#214461122
#214457532
C61×#24092
#2385961