Finite Group Databasebeta

← #29652673

Subgroups — #29652673

Order 15616.

1498 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459583C211yesmaximal, characteristic, complemented
2#21455071C212yesmaximal, complemented
3#21444787C212yesmaximal, complemented
4#21444770C212yesmaximal, complemented
5#21444753C212yesmaximal, complemented
6#21444736C212yesmaximal, complemented
7#21444334C212yesmaximal, complemented
8#21444157C211yesmaximal, characteristic, complemented
9C61×#2388C211yesmaximal, characteristic, complemented
10C2×#14519822—21no—
11C2×#14519802—22no—
12#16840416—22no—
13#16840407—24no—
14D4×D244—22no—
15#16840343C2212yes—
16#16840324—21no—
17#16840324C2212yes—
18#16840312—21no—
19C2×#14519751—21no—
20C2×#14519751C2212yes—
21#16839907—24no—
22#16839901C2214yescomplemented
23#16839896C2214yescomplemented
24#16839763—22no—
25#16839726C2212yescomplemented
26#16839495C2214yescomplemented
27#16839495C2214yescomplemented
28#16839488C2212yescomplemented
29#16839486C2212yescomplemented
30C22×#41×C61C2211yescharacteristic, complemented
31C2×C4×D4×C61—21no—
32C2×C4×D4×C61C2211yescharacteristic
33C61×#395C2211yescharacteristic, complemented
34C61×#394—21no—
35C61×#394C2211yescharacteristic, complemented
36C61×#393C2211yescharacteristic, complemented
37C61×#392—21no—
38C61×#392C2211yescharacteristic, complemented
39C4×C22⋊C4×C61—21no—
40C4×C22⋊C4×C61C2211yescharacteristic, complemented
41#14519822—44no—
42#14519822—44no—
43C2×#11758222—22no—
44C2×#11758224—24no—
45C22×D244—21no—
46C22×D244D412yescomplemented
47C22×C22⋊D61—22no—
48C22×C22⋊D61—22no—
49C2×D4×D61—24no—
50#14519802—48no—
51C2×#11758201—24no—
52C2×#11758199—24no—
53C2×#11758194—22no—
54#14519777—21no—
55#14519774—21no—
56#14519765—22no—
57#14519761—24no—
58#14519761—24no—
59#14519757—24no—
60#14519752—24no—
61#14519751—22no—
62#14519751—22no—
63#14519751—22no—
64#14519751—22no—
65#14519751—22no—
66#14519751—22no—
67#14519751D414yescomplemented
68#14519751C2314yes—
69#14519747—24no—
70#14519747—24no—
71C4×D244—22no—
72C4×D244—22no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61D412yescomplemented
75C2×#11758198—22no—
76C2×#11758198—22no—
77#14519701—24no—
78#14519698—24no—
79C22⋊#92418—22no—
80C22⋊#92418C2314yes—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61—44no—
84C22×Q8×C61C2311yescharacteristic
85C22×D4×C61—21no—
86C22×D4×C61—21no—
87C22×D4×C61D411yescharacteristic
88C22×D4×C61C2311yescharacteristic
89C23×C4×C61—21no—
90C23×C4×C61D411yescharacteristic
91C23×C4×C61C2311yescharacteristic
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61—44no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2311yescharacteristic
97C2×C22⋊C4×C61—21no—
98C2×C22⋊C4×C61—21no—
99C2×C22⋊C4×C61—21no—
100C2×C22⋊C4×C61—21no—
101C2×C22⋊C4×C61—21no—
102C2×C22⋊C4×C61D411yescharacteristic
103C2×C22⋊C4×C61C2311yescharacteristic
104C2×C42×C61—21no—
105C2×C42×C61D411yescharacteristic
106#95×C61—21no—
107#95×C61—21no—
108#95×C61C2311yescharacteristic
109#95×C61C2311yescharacteristic
110C24×C61—21no—
111C24×C61C2×D411yescharacteristic
112C24×C61C2×D411yescharacteristic
113C23×D61—22no—
114C23×D61—22no—
115#41×C61—28no—
116#41×C61—44no—
117#41×C61—44no—
118#41×C61—44no—
119C2×Q8×C61—22no—
120C2×Q8×C61—42no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—42no—
126C2×D4×C61—42no—
127C2×D4×C61—42no—
128C2×D4×C61—42no—
129C2×D4×C61—42no—
130C2×D4×C61—42no—
131C2×D4×C61—42no—
132C22×C4×C61—24no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—42no—
139C22×C4×C61—42no—
140C22×C4×C61—42no—
141C22×C4×C61—41no—
142C22×C4×C61—41no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2411yescharacteristic
150C2×C22⋊D61—24no—
151C2×C22⋊D61—24no—
152C2×C22⋊D61—24no—
153C2×C22⋊D61—44no—
154C2×C22⋊D61—44no—
155C2×C22⋊D61—44no—
156C2×C22⋊D61—44no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159#11758224—416no—
160D4×D61—416no—
161#11758222—44no—
162#11758222—44no—
163C2×D244—22no—
164C2×D244—22no—
165C2×D244—22no—
166C2×D244—22no—
167C2×D244—44no—
168C2×D244—82no—
169C2×D244C2×D414yes—
170C2×C4×D61—24no—
171C2×Dic122—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—42no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—42no—
177C22⋊C4×C61—42no—
178C22⋊C4×C61—42no—
179C22⋊C4×C61—42no—
180C22⋊C4×C61—42no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—42no—
184#11758201—48no—
185#11758199—48no—
186#11758198—24no—
187#11758198—24no—
188#11758198—24no—
189#11758198—24no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61C2×D414yes—
194#11758194—44no—
195C2×#92418—24no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—22no—
201C23×C61—41no—
202C23×C61—41no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61#12012yes—
212C23×C61AGL(1,ℤ/8)12yescomplemented
213C23×C61C22×D411yescharacteristic
214C22×D61—24no—
215C22×D61—44no—
216C22×D61—44no—
217C22×D61—82no—
218C22×D61—82no—
219C22×D61—82no—
220C22×D61—82no—
221Q8×C61—24no—
222Q8×C61—42no—
223D4×C61—28no—
224D4×C61—24no—
225D4×C61—42no—
226D4×C61—48no—
227D4×C61—48no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—24no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—42no—
237C2×C4×C61—41no—
238C2×C4×C61—44no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—42no—
244C2×C4×C61—44no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—81no—
251C2×C4×C61#12011yescharacteristic
252C2×C4×C61#12011yescharacteristic
253C2×C4×C61AGL(1,ℤ/8)12yescomplemented
254C2×C4×C61C22×D412yes—
255C22⋊D61—48no—
256C22⋊D61—48no—
257C22⋊D61—44no—
258C22⋊D61—44no—
259C2×Dic61—24no—
260C2×Dic61—44no—
261C2×Dic61—44no—
262D244—44no—
263D244—44no—
264D244—44no—
265D244—82no—
266D244—82no—
267C4×D61—48no—
268Dic122—44no—
269#92418—44no—
270#18166—611nomaximal, Sylow
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—22no—
274C22×C61—22no—
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—42no—
283C22×C61—81no—
284C22×C61—81no—
285C22×C61—81no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61C2×#12011yescharacteristic
291C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
293C2×D61—44no—
294C2×D61—44no—
295C2×D61—44no—
296C2×D61—164no—
297C2×D61—164no—
298C4×C61—22no—
299C4×C61—22no—
300C4×C61—21no—
301C4×C61—42no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—42no—
305C4×C61—81no—
306C4×C61B(4,𝔽2)12yes—
307Dic61—44no—
308#3496—611no—
309#3006—612no—
310#2388D6111yescharacteristic, complemented
311#1604—612no—
312#1601—612no—
313#1598—612no—
314#1595—612no—
315#1507—612no—
316#1465—611no—
317C2×C61—21no—
318C2×C61—21no—
319C2×C61—42no—
320C2×C61—42no—
321C2×C61—82no—
322C2×C61#300311yescharacteristic
323C2×C61#301811yescharacteristic
324C2×C61C2×B(4,𝔽2)11yescharacteristic
325D61—84no—
326C2×H(5,𝔽2)—1221no—
327C22×#41C2×D6111yescharacteristic, complemented
328C2×AGL(1,ℤ/8)—1222no—
329#547—1222no—
330#546—1224no—
331D42—1222no—
332#537—612no—
333#534—612no—
334#534—1221no—
335#532—1221no—
336C2×#121—612no—
337C2×#121—1221no—
338C2×C4×D4—21no—
339C2×C4×D4C2×D6111yescharacteristic
340#449—1224no—
341#448—614no—
342#447—614no—
343#417—1222no—
344#407—612no—
345#395C2×D6111yescharacteristic, complemented
346#394—21no—
347#394C2×D6111yescharacteristic, complemented
348#393C2×D6111yescharacteristic, complemented
349#392—21no—
350#392C2×D6111yescharacteristic, complemented
351C4×C22⋊C4—21no—
352C4×C22⋊C4C2×D6111yescharacteristic, complemented
353#327—614no—
354#327—614no—
355#324—612no—
356#323—612no—
357C61#1816611yescharacteristic, complemented, Sylow
358H(5,𝔽2)—2444no—
359H(5,𝔽2)—2444no—
360C2×#41—22no—
361C2×#41—22no—
362C2×#41—44no—
363C2×#41—1224no—
364C2×#41—1222no—
365C22×Q8C22×D6111yescharacteristic
366C22×D4—21no—
367C22×D4—21no—
368C22×D4—612no—
369C22×D4—1221no—
370C22×D4—1224no—
371C22×D4—1222no—
372C22×D4—1222no—
373C22×D4C22⋊D6111yescharacteristic
374C22×D4C22×D6111yescharacteristic
375C23×C4—21no—
376C23×C4C22⋊D6111yescharacteristic
377C23×C4C22×D6111yescharacteristic
378AGL(1,ℤ/8)—2448no—
379C2×SD16—1224no—
380C2×D8—1224no—
381C2×M16—1222no—
382C4⋊Q8—1221no—
383#127—1221no—
384#124—1222no—
385#123—1224no—
386#123—1224no—
387C22⋊Q8—1224no—
388#121—614no—
389#121—614no—
390#121—1224no—
391#121—1222no—
392#121—1222no—
393#121—1222no—
394#121—1222no—
395#121—1222no—
396#121—1222no—
397#120—1224no—
398#120—1224no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—44no—
402C4×D4—1222no—
403C4×D4—1222no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—612no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—1222no—
414C2×C22⋊C4—1222no—
415C2×C22⋊C4C22⋊D6111yescharacteristic
416C2×C22⋊C4C22×D6111yescharacteristic
417C2×C42—21no—
418C2×C42C22⋊D6111yescharacteristic
419#103—1224no—
420#102—1224no—
421C22⋊C8—614no—
422C22⋊C8—1222no—
423#95—21no—
424#95—21no—
425#95C22×D6111yescharacteristic
426#95C22×D6111yescharacteristic
427C24—21no—
428C24—1222no—
429C24—1222no—
430C24C2×C22⋊D6111yescharacteristic
431C24C2×C22⋊D6111yescharacteristic
432#41—28no—
433#41—44no—
434#41—44no—
435#41—44no—
436#41—2444no—
437#41—24416no—
438#41—2444no—
439C2×Q8—22no—
440C2×Q8—42no—
441C2×Q8—1222no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—614no—
454C2×D4—1222no—
455C2×D4—1222no—
456C2×D4—1222no—
457C2×D4—1224no—
458C2×D4—1224no—
459C2×D4—1224no—
460C2×D4—1222no—
461C2×D4—4882no—
462C2×D4—2444no—
463C2×D4—24416no—
464C2×D4—2444no—
465C2×D4—2444no—
466C2×D4—2444no—
467C2×D4—2444no—
468C22×C4—24no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—1224no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4D4×D6112yes—
483C22×C4D4×D6112yes—
484C22×C4C2×C22⋊D6111yescharacteristic
485C22×C4C2×C22⋊D6111yescharacteristic
486C22×C4C2×C22⋊D6111yescharacteristic
487C22×C4C2×C22⋊D6111yescharacteristic
488C22×C4C23×D6111yescharacteristic
489SD16—2448no—
490D8—2448no—
491M16—2444no—
492C2×C8—1224no—
493C4⋊C4—22no—
494C4⋊C4—42no—
495C4⋊C4—614no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—1224no—
507C22⋊C4—1224no—
508C22⋊C4—1224no—
509C22⋊C4—1224no—
510C42—21no—
511C42—21no—
512C42—42no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—1224no—
529C23—4882no—
530C23—4882no—
531C23—4882no—
532C23—4882no—
533C23—2444no—
534C23—2444no—
535C23#1451974512yes—
536C23#1451980212yescomplemented
537C23C22×C22⋊D6111yescharacteristic
538Q8—24no—
539Q8—42no—
540Q8—2444no—
541D4—28no—
542D4—24no—
543D4—42no—
544D4—48no—
545D4—48no—
546D4—4882no—
547D4—4882no—
548D4—2444no—
549D4—2444no—
550D4—2448no—
551D4—2448no—
552D4—2444no—
553D4—2444no—
554D4—2444no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—81no—
578C2×C4—1224no—
579C2×C4—2448no—
580C2×C4—2444no—
581C2×C4—2444no—
582C2×C4#1451974511yescharacteristic
583C2×C4#1451974511yescharacteristic
584C2×C4#1451980212yescomplemented
585C2×C4C2×D4×D6112yes—
586C8—2444no—
587C22—21no—
588C22—21no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—41no—
597C22—41no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—9764no—
607C22—9764no—
608C22—2444no—
609C22—2444no—
610C22—2444no—
611C22C2×#1451974511yescharacteristic
612C22C2×#1451980211yescharacteristic
613C22C2×#1451980211yescharacteristic
614C4—22no—
615C4—21no—
616C4—22no—
617C4—42no—
618C4—41no—
619C4—42no—
620C4—41no—
621C4—81no—
622C4—2444no—
623C4#1683994812yes—
624C2—21no—
625C2—21no—
626C2—42no—
627C2—42no—
628C2—82no—
629C2—4884no—
630C2#2145504511yescharacteristic
631C2#2145516711yescharacteristic
632C2C2×#1683994811yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2388C2
socleC22×C61C2×#120
hypercenter#2388D61
hypocenterC61#18166
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2388

Complements

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

NH ≅ G/Ntype
1. #21459583C2semidirect
2. #21455071C2semidirect
3. #21444787C2semidirect
4. #21444770C2semidirect
5. #21444753C2semidirect
6. #21444736C2semidirect
7. #21444334C2semidirect
8. #21444157C2semidirect
9. C61×#2388C2semidirect
22. #16839901C22semidirect
23. #16839896C22semidirect
25. #16839726C22semidirect
26. #16839495C22semidirect
27. #16839495C22semidirect
28. #16839488C22semidirect
29. #16839486C22semidirect
30. C22×#41×C61C22semidirect
33. C61×#395C22semidirect
35. C61×#394C22semidirect
36. C61×#393C22semidirect
38. C61×#392C22semidirect
40. C4×C22⋊C4×C61C22semidirect
46. C22×D244D4semidirect
67. #14519751D4semidirect
74. C2×C4⋊Dic61D4semidirect
212. C23×C61AGL(1,ℤ/8)semidirect
253. C2×C4×C61AGL(1,ℤ/8)semidirect
310. #2388D61semidirect
327. C22×#41C2×D61semidirect
345. #395C2×D61semidirect
347. #394C2×D61semidirect
348. #393C2×D61semidirect
350. #392C2×D61semidirect
352. C4×C22⋊C4C2×D61semidirect
357. C61#18166semidirect
536. C23#14519802semidirect
584. C2×C4#14519802semidirect

Maximal subgroups

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

subgroupindex
#214595832
#214550712
#214447872
#214447702
#214447532
#214447362
#214443342
#214441572
C61×#23882
#1816661