Finite Group Databasebeta

← #29652682

Subgroups — #29652682

Order 15616.

1218 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459622C211yesmaximal, characteristic, complemented
2#21455079C212yesmaximal, complemented
3#21444792C212yesmaximal, complemented
4#21444775C212yesmaximal, complemented
5#21444758C212yesmaximal, complemented
6#21444741C212yesmaximal, complemented
7#21444338C212yesmaximal, complemented
8#21444157C211yesmaximal, characteristic, complemented
9C61×#2388C211yesmaximal, characteristic
10C2×#14519807—22no—
11#16840471—24no—
12#16840469—22no—
13D4×Dic122—22no—
14#16840393C2212yes—
15#16840388—21no—
16#16840388C2212yes—
17#16840319—21no—
18C2×#14519756—21no—
19C2×#14519756C2212yes—
20#16840237—21no—
21#16839924—24no—
22#16839918C2214yes—
23#16839913C2214yes—
24#16839763—22no—
25#16839726C2212yes—
26#16839498C2214yes—
27#16839498C2214yes—
28#16839488C2212yes—
29#16839486C2212yes—
30C22×#41×C61C2211yescharacteristic
31C2×C4×D4×C61—21no—
32C2×C4×D4×C61C2211yescharacteristic
33C61×#395C2211yescharacteristic
34C61×#394—21no—
35C61×#394C2211yescharacteristic
36C61×#393C2211yescharacteristic
37C61×#392—21no—
38C61×#392C2211yescharacteristic
39C4×C22⋊C4×C61—21no—
40C4×C22⋊C4×C61C2211yescharacteristic
41C22×Dic122—21no—
42C22×Dic122D412yescomplemented
43#14519807—48no—
44C2×#11758202—24no—
45C2×#11758200—24no—
46C2×#11758194—22no—
47#14519777—21no—
48#14519777—21no—
49#14519776—24no—
50#14519769—22no—
51#14519768—24no—
52#14519756—22no—
53#14519756—22no—
54#14519756—22no—
55#14519756—22no—
56#14519756—22no—
57#14519756—22no—
58#14519756D414yescomplemented
59#14519756C2314yes—
60#14519754—24no—
61#14519754—24no—
62#14519754—24no—
63#14519754—24no—
64C4×Dic122—22no—
65C4×Dic122—22no—
66D4×Dic61—24no—
67D4×Dic61—24no—
68#14519736—22no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61D412yescomplemented
71C2×#11758196—22no—
72C2×#11758196—22no—
73C2×C22⋊Dic61—22no—
74C2×C22⋊Dic61—22no—
75#14519701—24no—
76#14519698—24no—
77C22⋊#92418—22no—
78C22⋊#92418C2314yes—
79C2×#41×C61—22no—
80C2×#41×C61—22no—
81C2×#41×C61—44no—
82C22×Q8×C61C2311yescharacteristic
83C22×D4×C61—21no—
84C22×D4×C61—21no—
85C22×D4×C61D411yescharacteristic
86C22×D4×C61C2311yescharacteristic
87C23×C4×C61—21no—
88C23×C4×C61D411yescharacteristic
89C23×C4×C61C2311yescharacteristic
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61—44no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C22⋊C4×C61—21no—
96C2×C22⋊C4×C61—21no—
97C2×C22⋊C4×C61—21no—
98C2×C22⋊C4×C61—21no—
99C2×C22⋊C4×C61—21no—
100C2×C22⋊C4×C61D411yescharacteristic
101C2×C22⋊C4×C61C2311yescharacteristic
102C2×C42×C61—21no—
103C2×C42×C61D411yescharacteristic
104#95×C61—21no—
105#95×C61—21no—
106#95×C61C2311yescharacteristic
107#95×C61C2311yescharacteristic
108C24×C61—21no—
109C24×C61C2×D411yescharacteristic
110C24×C61C2×D411yescharacteristic
111#41×C61—28no—
112#41×C61—44no—
113#41×C61—44no—
114#41×C61—44no—
115C2×Q8×C61—22no—
116C2×Q8×C61—42no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—42no—
122C2×D4×C61—42no—
123C2×D4×C61—42no—
124C2×D4×C61—42no—
125C2×D4×C61—42no—
126C2×D4×C61—42no—
127C2×D4×C61—42no—
128C22×C4×C61—24no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—42no—
135C22×C4×C61—42no—
136C22×C4×C61—42no—
137C22×C4×C61—41no—
138C22×C4×C61—41no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—22no—
153C2×Dic122—44no—
154C2×Dic122—82no—
155C2×Dic122C2×D414yes—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—42no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—42no—
161C22⋊C4×C61—42no—
162C22⋊C4×C61—42no—
163C22⋊C4×C61—42no—
164C22⋊C4×C61—42no—
165C42×C61—21no—
166C42×C61—21no—
167C42×C61—42no—
168C22⋊Dic61—24no—
169C22⋊Dic61—24no—
170C22⋊Dic61—24no—
171#11758202—48no—
172#11758200—48no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61—22no—
178C4⋊Dic61C2×D414yes—
179#11758196—24no—
180#11758196—24no—
181#11758196—24no—
182#11758196—24no—
183C4×Dic61—24no—
184#11758194—44no—
185C2×#92418—24no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—22no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—41no—
199C23×C61—41no—
200C23×C61—41no—
201C23×C61#12012yes—
202C23×C61#13712yescomplemented
203C23×C61C22×D411yescharacteristic
204Q8×C61—24no—
205Q8×C61—42no—
206D4×C61—28no—
207D4×C61—24no—
208D4×C61—42no—
209D4×C61—48no—
210D4×C61—48no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—24no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—42no—
220C2×C4×C61—41no—
221C2×C4×C61—44no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—42no—
227C2×C4×C61—44no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—81no—
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61#13712yescomplemented
237C2×C4×C61C22×D412yes—
238C2×Dic61—24no—
239C2×Dic61—24no—
240C2×Dic61—82no—
241C2×Dic61—82no—
242C2×Dic61—82no—
243C2×Dic61—82no—
244Dic122—44no—
245Dic122—44no—
246Dic122—82no—
247Dic122—82no—
248#92418—44no—
249#18167—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—42no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61—81no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61C2×#12011yescharacteristic
270C22×C61C2×#13711yescharacteristic
271C22×C61C2×#13711yescharacteristic
272C4×C61—22no—
273C4×C61—22no—
274C4×C61—21no—
275C4×C61—42no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—42no—
279C4×C61—81no—
280C4×C61B(4,𝔽2)12yes—
281Dic61—44no—
282Dic61—44no—
283#3501—611no—
284#3007—612no—
285#2388D6111yescharacteristic
286#1605—612no—
287#1602—612no—
288#1599—612no—
289#1596—612no—
290#1508—612no—
291#1465—611no—
292C2×C61—21no—
293C2×C61—21no—
294C2×C61—42no—
295C2×C61—42no—
296C2×C61—82no—
297C2×C61#300411yescharacteristic
298C2×C61#301811yescharacteristic
299C2×C61C2×B(4,𝔽2)11yescharacteristic
300C22×#41C2×D6111yescharacteristic
301C2×#137—1222no—
302#554—1222no—
303#554—1224no—
304D4×Q8—1222no—
305#544—612no—
306#543—612no—
307#543—1221no—
308#533—1221no—
309C2×C22⋊Q8—612no—
310C2×C22⋊Q8—1221no—
311#518—1221no—
312C2×C4×D4—21no—
313C2×C4×D4C2×D6111yescharacteristic
314#452—1224no—
315#451—614no—
316#450—614no—
317#417—1222no—
318#407—612no—
319#395C2×D6111yescharacteristic
320#394—21no—
321#394C2×D6111yescharacteristic
322#393C2×D6111yescharacteristic
323#392—21no—
324#392C2×D6111yescharacteristic
325C4×C22⋊C4—21no—
326C4×C22⋊C4C2×D6111yescharacteristic
327#328—614no—
328#328—614no—
329#324—612no—
330#323—612no—
331C61#1816711yescharacteristic, complemented, Sylow
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—44no—
335C22×Q8—612no—
336C22×Q8—1221no—
337C22×Q8C22×D6111yescharacteristic
338C22×D4—21no—
339C22×D4—21no—
340C22×D4C22⋊D6111yescharacteristic
341C22×D4C22×D6111yescharacteristic
342C23×C4—21no—
343C23×C4C22⋊D6111yescharacteristic
344C23×C4C22×D6111yescharacteristic
345#137—2448no—
346C2×Q16—1224no—
347C2×SD16—1224no—
348C2×M16—1222no—
349C4⋊Q8—1221no—
350C4⋊Q8—1221no—
351C4⋊Q8—1224no—
352#125—1222no—
353#125—1224no—
354C22⋊Q8—614no—
355C22⋊Q8—614no—
356C22⋊Q8—1224no—
357C22⋊Q8—1224no—
358C22⋊Q8—1224no—
359C22⋊Q8—1224no—
360C22⋊Q8—1222no—
361C22⋊Q8—1222no—
362C22⋊Q8—1222no—
363C22⋊Q8—1222no—
364C22⋊Q8—1222no—
365C22⋊Q8—1222no—
366C4×Q8—1222no—
367C4×Q8—1222no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—44no—
371C4×D4—1224no—
372C4×D4—1224no—
373#117—1222no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—612no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—1222no—
386C2×C22⋊C4—1222no—
387C2×C22⋊C4C22⋊D6111yescharacteristic
388C2×C22⋊C4C22×D6111yescharacteristic
389C2×C42—21no—
390C2×C42C22⋊D6111yescharacteristic
391#103—1224no—
392#102—1224no—
393C22⋊C8—614no—
394C22⋊C8—1222no—
395#95—21no—
396#95—21no—
397#95C22×D6111yescharacteristic
398#95C22×D6111yescharacteristic
399C24—21no—
400C24C2×C22⋊D6111yescharacteristic
401C24C2×C22⋊D6111yescharacteristic
402#41—28no—
403#41—44no—
404#41—44no—
405#41—44no—
406C2×Q8—22no—
407C2×Q8—42no—
408C2×Q8—614no—
409C2×Q8—1222no—
410C2×Q8—1222no—
411C2×Q8—1222no—
412C2×Q8—4882no—
413C2×Q8—2444no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4D4×D6112yes—
441C22×C4D4×D6112yes—
442C22×C4C2×C22⋊D6111yescharacteristic
443C22×C4C2×C22⋊D6111yescharacteristic
444C22×C4C2×C22⋊D6111yescharacteristic
445C22×C4C2×C22⋊D6111yescharacteristic
446C22×C4C23×D6111yescharacteristic
447Q16—2448no—
448SD16—2448no—
449M16—2444no—
450C2×C8—1224no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—614no—
454C4⋊C4—1224no—
455C4⋊C4—1222no—
456C4⋊C4—1224no—
457C4⋊C4—1222no—
458C4⋊C4—1222no—
459C4⋊C4—1222no—
460C4⋊C4—1224no—
461C4⋊C4—1222no—
462C4⋊C4—1224no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—1224no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C42—21no—
474C42—21no—
475C42—42no—
476C42—1224no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—22no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23#1451974512yes—
493C23#1451980712yescomplemented
494C23C22×C22⋊D6111yescharacteristic
495Q8—24no—
496Q8—42no—
497Q8—4882no—
498Q8—4882no—
499Q8—2444no—
500Q8—2444no—
501D4—28no—
502D4—24no—
503D4—42no—
504D4—48no—
505D4—48no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—44no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—81no—
529C2×C4—1224no—
530C2×C4—1224no—
531C2×C4—4882no—
532C2×C4—4882no—
533C2×C4—4882no—
534C2×C4—4882no—
535C2×C4#1451974511yescharacteristic
536C2×C4#1451974511yescharacteristic
537C2×C4#1451980712yescomplemented
538C2×C4C2×D4×D6112yes—
539C8—2444no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—41no—
550C22—41no—
551C22—42no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×#1451974511yescharacteristic
560C22C2×#1451980711yescharacteristic
561C22C2×#1451980711yescharacteristic
562C4—22no—
563C4—21no—
564C4—22no—
565C4—42no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—81no—
570C4—2444no—
571C4—2444no—
572C4#1683994812yes—
573C2—21no—
574C2—21no—
575C2—42no—
576C2—42no—
577C2—82no—
578C2#2145505211yescharacteristic
579C2#2145516711yescharacteristic
580C2C2×#1683994811yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#2388C2
socleC22×C61C2×#120
hypercenter#2388D61
hypocenterC61#18167
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. #21459622C2semidirect
2. #21455079C2semidirect
3. #21444792C2semidirect
4. #21444775C2semidirect
5. #21444758C2semidirect
6. #21444741C2semidirect
7. #21444338C2semidirect
8. #21444157C2semidirect
42. C22×Dic122D4semidirect
58. #14519756D4semidirect
70. C2×C4⋊Dic61D4semidirect
202. C23×C61#137semidirect
236. C2×C4×C61#137semidirect
331. C61#18167semidirect
493. C23#14519807semidirect
537. C2×C4#14519807semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214550792
#214447922
#214447752
#214447582
#214447412
#214443382
#214441572
C61×#23882
#1816761