Finite Group Databasebeta

← #29715702

Subgroups — #29715702

Order 15616.

1478 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840000C211yesmaximal, characteristic, complemented
2C2×#16839995C211yesmaximal, characteristic, complemented
3C2×C8×C22⋊D61C211yesmaximal, characteristic, complemented
4#21449158C212yesmaximal, complemented
5#21447132C212yesmaximal, complemented
6#21446819C212yesmaximal, complemented
7#21446686C211yesmaximal, characteristic, complemented
8#21446681C211yesmaximal, characteristic, complemented
9#21445848C211yesmaximal, characteristic, complemented
10#21445759C212yesmaximal, complemented
11C22×#135×C61C211yesmaximal, characteristic, complemented
12C22×C8×D61C2211yescharacteristic
13C2×#14519753C2212yes—
14C2×#14519748C2212yes—
15C2×C4×C22⋊D61C2211yescharacteristic
16#16840000—24no—
17#16839995—24no—
18C8×C22⋊D61—24no—
19C2×#14519711C2211yescharacteristic
20C2×#14519711C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519701C2212yes—
23C2×#14519701C2212yes—
24C2×#14519698C2212yes—
25C2×#14519698C2212yes—
26C2×C22⋊#92418C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28C2×C8×Dic61C2211yescharacteristic
29#16839676—22no—
30#16839670—22no—
31#16839669C2212yes—
32#16839665C2212yes—
33#16839641—22no—
34#16839635—22no—
35#16839622C2212yes—
36#16839525C2212yes—
37C22×#41×C61C2212yes—
38C2×#135×C61—24no—
39C2×#135×C61—24no—
40C2×#135×C61—24no—
41C22×Q16×C61C2211yescharacteristic
42C22×SD16×C61C2212yes—
43C22×D8×C61C2211yescharacteristic
44C23×C8×C61C2211yescharacteristic
45C22×C22⋊D61—22no—
46C22×C22⋊D61D411yescharacteristic
47C22×C4×D61C2311yescharacteristic
48C22×#92418C2311yescharacteristic
49C22×#92418C2311yescharacteristic
50C2×C8×D61—22no—
51C2×C8×D61—22no—
52C2×C8×D61—22no—
53#14519753—24no—
54#14519753—42no—
55#14519748—24no—
56#14519748—42no—
57C4×C22⋊D61—22no—
58C4×C22⋊D61—22no—
59C2×C4⋊Dic61C2312yes—
60C2×C4⋊Dic61C2312yes—
61C2×#11758196—22no—
62C2×#11758196D411yescharacteristic
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61—22no—
67C2×C22⋊Dic61D411yescharacteristic
68C2×#11758198—22no—
69C2×#11758198D411yescharacteristic
70C2×C4×Dic61C2311yescharacteristic
71#14519711—22no—
72#14519711—22no—
73#14519706—22no—
74#14519701—24no—
75#14519701—42no—
76#14519698—24no—
77#14519698—42no—
78C22⋊#92418—22no—
79#14519689—22no—
80C8×Dic61—22no—
81#14519684—22no—
82#14519684—22no—
83#14519684—22no—
84C2×#41×C61—24no—
85C2×#41×C61—24no—
86C2×#41×C61—24no—
87C2×#41×C61—24no—
88C2×#41×C61—42no—
89C2×#41×C61—42no—
90C22×Q8×C61C2312yes—
91C22×D4×C61—22no—
92C22×D4×C61C2312yes—
93C23×C4×C61—22no—
94C23×C4×C61C2311yescharacteristic
95#135×C61—232no—
96C2×Q16×C61—22no—
97C2×Q16×C61—22no—
98C2×Q16×C61D414yescomplemented
99C2×SD16×C61—24no—
100C2×SD16×C61—24no—
101C2×SD16×C61—24no—
102C2×D8×C61—22no—
103C2×D8×C61—22no—
104C2×D8×C61D414yescomplemented
105C22×C8×C61—22no—
106C22×C8×C61—22no—
107C22×C8×C61D414yescomplemented
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C24×C61—22no—
111C24×C61C2×D411yescharacteristic
112C23×D61C2×D411yescharacteristic
113#41×C61—216no—
114#41×C61—48no—
115#41×C61—48no—
116#41×C61—48no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61C2×D414yes—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—22no—
128C2×D4×C61—44no—
129C2×D4×C61—44no—
130C2×D4×C61—44no—
131C2×D4×C61C2×D414yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—22no—
137C22×C4×C61—44no—
138C22×C4×C61—44no—
139C22×C4×C61—44no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D412yes—
142C22×C4×C61#4112yes—
143C22×C4×C61C2411yescharacteristic
144C2×C22⋊D61—22no—
145C2×C22⋊D61—22no—
146C2×C22⋊D61—22no—
147C2×C22⋊D61—44no—
148C2×C22⋊D61—44no—
149C2×C22⋊D61—82no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161Q16×C61—28no—
162SD16×C61—216no—
163D8×C61—28no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—28no—
167C2×C8×C61C2×D412yes—
168C2×C8×C61C2×D412yes—
169C2×C8×C61C2×D412yes—
170C2×C8×C61C2×D412yes—
171C22⋊Dic61—22no—
172C22⋊Dic61—44no—
173C22⋊Dic61—44no—
174C22⋊Dic61—82no—
175C22⋊Dic61—82no—
176#11758198—22no—
177#11758198—44no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61—22no—
181C4⋊Dic61—22no—
182#11758196—22no—
183#11758196—44no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C8×D61—44no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—24no—
200C23×C61—42no—
201C23×C61—42no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61—42no—
205C23×C61—42no—
206C23×C61#12112yes—
207C23×C61C22×D411yescharacteristic
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61#13512yescomplemented
215Q8×C61—24no—
216Q8×C61—24no—
217Q8×C61—42no—
218Q8×C61—42no—
219D4×C61—24no—
220D4×C61—24no—
221D4×C61—416no—
222D4×C61—42no—
223D4×C61—42no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—24no—
231C2×C4×C61—416no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61#12112yes—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×C4×C61H(5,𝔽2)11yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C22⋊D61—28no—
242C22⋊D61—88no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61#13512yescomplemented
258C4×D61—41no—
259C4×D61—41no—
260C4×D61—41no—
261C4×D61—41no—
262C8×C61—22no—
263C8×C61#12014yes—
264#92418—41no—
265#92418—41no—
266#24266—611nomaximal, Sylow
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—44no—
275C22×C61—44no—
276C22×C61—44no—
277C22×C61—82no—
278C22×C61—82no—
279C22×C61—82no—
280C22×C61—82no—
281C22×C61C2×#12111yescharacteristic
282C22×C61D4211yescharacteristic
283C22×C61D4211yescharacteristic
284C22×C61C2×#13511yescharacteristic
285C22×C61C2×#13511yescharacteristic
286C22×C61#57611yescharacteristic
287C22×C61#57811yescharacteristic
288C2×D61—22no—
289C2×D61—22no—
290C2×D61—22no—
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—41no—
294C2×D61—41no—
295C4×C61—22no—
296C4×C61—44no—
297C4×C61C2×#12011yescharacteristic
298C4×C61#53511yescharacteristic
299C4×C61#54611yescharacteristic
300C4×C61#54611yescharacteristic
301Dic61—22no—
302Dic61—41no—
303Dic61—41no—
304Dic61—82no—
305Dic61—82no—
306C22×#135D6111yescharacteristic, complemented
307C2×#467—611no—
308C2×#466—611no—
309C2×C8×D4—611no—
310#2407—612no—
311#2055—612no—
312#2000—612no—
313#1970—611no—
314#1969—611no—
315#1808—611no—
316#1788—612no—
317C2×C61—42no—
318C2×C61—84no—
319C2×C61#240811yescharacteristic
320C2×C61#305011yescharacteristic
321C2×C61#306111yescharacteristic
322C2×C61#328011yescharacteristic
323C2×C61#328311yescharacteristic
324C2×C61#328511yescharacteristic
325C2×C61#328711yescharacteristic
326D61—42no—
327C22×#41C2×D6112yes—
328C2×#135—24no—
329C2×#135—24no—
330C2×#135—24no—
331C22×Q16C2×D6111yescharacteristic
332C22×SD16C2×D6112yes—
333C22×D8C2×D6111yescharacteristic
334C23×C8—611no—
335C23×C8C2×D6111yescharacteristic
336C2×C22⋊Q8—612no—
337C2×#121—612no—
338C2×C4×D4—611no—
339#467—1224no—
340#466—1224no—
341C8×D4—1224no—
342C2×#107—611no—
343C2×#107—611no—
344C2×C4⋊C8—611no—
345C2×#103—612no—
346C2×#103—612no—
347C2×#102—612no—
348C2×#102—612no—
349C2×C22⋊C8—611no—
350C2×C22⋊C8—611no—
351C2×C4×C8—611no—
352#394—1222no—
353#392—1222no—
354#391—612no—
355#390—612no—
356#386—1222no—
357#385—1222no—
358#380—612no—
359#340—612no—
360C61#2426611yescharacteristic, complemented, Sylow
361C2×#41—24no—
362C2×#41—24no—
363C2×#41—24no—
364C2×#41—24no—
365C2×#41—42no—
366C2×#41—42no—
367C22×Q8C22×D6112yes—
368C22×D4—22no—
369C22×D4—611no—
370C22×D4—1222no—
371C22×D4C22×D6112yes—
372C23×C4—22no—
373C23×C4—611no—
374C23×C4C22×D6111yescharacteristic
375#135—232no—
376C2×Q16—22no—
377C2×Q16—22no—
378C2×Q16C22⋊D6114yescomplemented
379C2×SD16—24no—
380C2×SD16—24no—
381C2×SD16—24no—
382C2×D8—22no—
383C2×D8—22no—
384C2×D8C22⋊D6114yescomplemented
385C22×C8—22no—
386C22×C8—22no—
387C22×C8—611no—
388C22×C8—611no—
389C22×C8—1222no—
390C22×C8—1222no—
391C22×C8—1222no—
392C22×C8C22⋊D6114yescomplemented
393C22×C8C22×D6111yescharacteristic
394C22×C8C22×D6111yescharacteristic
395C22⋊Q8—1224no—
396C22⋊Q8—2442no—
397#121—1224no—
398#121—2442no—
399C4×D4—1222no—
400C4×D4—1222no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—612no—
403C2×C4⋊C4—612no—
404C2×C4⋊C4—1222no—
405C2×C22⋊C4—611no—
406C2×C22⋊C4—611no—
407C2×C22⋊C4—1222no—
408C2×C22⋊C4—1222no—
409C2×C22⋊C4—1222no—
410C2×C22⋊C4—1222no—
411C2×C22⋊C4—1222no—
412C2×C42—611no—
413#107—1222no—
414#107—1222no—
415C4⋊C8—1222no—
416#103—1224no—
417#103—2442no—
418#102—1224no—
419#102—2442no—
420C22⋊C8—1222no—
421C22⋊C8—1222no—
422C4×C8—1222no—
423#95—1222no—
424#95—1222no—
425#95—1222no—
426C24—22no—
427C24—611no—
428C24D4×D6111yescharacteristic
429#41—216no—
430#41—48no—
431#41—48no—
432#41—48no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8C2×C22⋊D6114yes—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—1222no—
448C2×D4—1222no—
449C2×D4—1222no—
450C2×D4—4882no—
451C2×D4—2444no—
452C2×D4—2444no—
453C2×D4C2×C22⋊D6114yes—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4D4×D6111yescharacteristic
474C22×C4#1175822412yes—
475C22×C4C2×C22⋊D6112yes—
476C22×C4C23×D6111yescharacteristic
477Q16—28no—
478SD16—216no—
479D8—28no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—28no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C2×C8—2444no—
490C2×C8C2×C22⋊D6112yes—
491C2×C8C2×C22⋊D6112yes—
492C2×C8C2×C22⋊D6112yes—
493C2×C8C2×C22⋊D6112yes—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—2444no—
500C22⋊C4—1222no—
501C22⋊C4—1222no—
502C22⋊C4—4882no—
503C22⋊C4—4882no—
504C22⋊C4—2444no—
505C22⋊C4—2444no—
506C22⋊C4—2444no—
507C42—1221no—
508C42—1221no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—24no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—612no—
523C23—1221no—
524C23—1221no—
525C23—1221no—
526C23—1221no—
527C23—1221no—
528C23—1221no—
529C23#1451974812yes—
530C23C2×D4×D6111yescharacteristic
531Q8—24no—
532Q8—24no—
533Q8—42no—
534Q8—42no—
535D4—24no—
536D4—24no—
537D4—416no—
538D4—42no—
539D4—42no—
540D4—1228no—
541D4—4888no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—416no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—612no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4—2442no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4#1451974812yes—
572C2×C4C22×C22⋊D6111yescharacteristic
573C2×C4C22×C22⋊D6111yescharacteristic
574C2×C4C22×C22⋊D6111yescharacteristic
575C2×C4C2×#1175822411yescharacteristic
576C2×C4#1451982211yescharacteristic
577C2×C4#1451982211yescharacteristic
578C8—22no—
579C8—2441no—
580C8—2441no—
581C8#1451974614yes—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—1222no—
597C22—1222no—
598C22—1222no—
599C22—2441no—
600C22—2441no—
601C22—2441no—
602C22—2441no—
603C22C2×#1451974811yescharacteristic
604C22D4×C22⋊D6111yescharacteristic
605C22D4×C22⋊D6111yescharacteristic
606C22#135×D6111yescharacteristic
607C22#135×D6111yescharacteristic
608C22#1684056411yescharacteristic
609C22#1684057611yescharacteristic
610C4—22no—
611C4—44no—
612C4—1222no—
613C4—4882no—
614C4—4882no—
615C4—2441no—
616C4—2441no—
617C4C2×#1451974611yescharacteristic
618C4#1684032911yescharacteristic
619C4#1684040611yescharacteristic
620C4#1684040611yescharacteristic
621C2—42no—
622C2—84no—
623C2—2442no—
624C2#2144916911yescharacteristic
625C2#2145539611yescharacteristic
626C2#2145548511yescharacteristic
627C2#2145749811yescharacteristic
628C2#2145753511yescharacteristic
629C2#2145755411yescharacteristic
630C2#2145757311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC22×#135×C61C2
socleC23×C61C22×D4
hypercenterC22×#135D61
hypocenterC61#24266
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16840000C2semidirect
2. C2×#16839995C2semidirect
3. C2×C8×C22⋊D61C2semidirect
4. #21449158C2semidirect
5. #21447132C2semidirect
6. #21446819C2semidirect
7. #21446686C2semidirect
8. #21446681C2semidirect
9. #21445848C2semidirect
10. #21445759C2semidirect
11. C22×#135×C61C2semidirect
98. C2×Q16×C61D4semidirect
104. C2×D8×C61D4semidirect
107. C22×C8×C61D4semidirect
214. C22×D61#135semidirect
257. C2×Dic61#135semidirect
306. C22×#135D61semidirect
360. C61#24266semidirect
378. C2×Q16C22⋊D61semidirect
384. C2×D8C22⋊D61semidirect
392. C22×C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400002
C2×#168399952
C2×C8×C22⋊D612
#214491582
#214471322
#214468192
#214466862
#214466812
#214458482
#214457592
C22×#135×C612
#2426661