Finite Group Databasebeta

← #29713930

Subgroups — #29713930

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
1C2×#16840473C211yesmaximal, characteristic, complemented
2C2×C4×#11758200C211yesmaximal, characteristic, complemented
3#21450732C211yesmaximal, characteristic
4#21447017C212yesmaximal, complemented
5#21446823C212yesmaximal, complemented
6#21446781C211yesmaximal, characteristic, complemented
7#21446612C211yesmaximal, characteristic, complemented
8#21446133C212yesmaximal, complemented
9#21446010C212yesmaximal, complemented
10#21445825C211yesmaximal, characteristic, complemented
11C2×C61×#476C211yesmaximal, characteristic, complemented
12C22×#11758200C2211yescharacteristic
13#16840473—24no—
14#16840473—24no—
15C2×#14519768—21no—
16C2×#14519757C2212yes—
17C2×#14519753—22no—
18C2×#14519751—21no—
19C2×#14519748C2212yes—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×#14519739—21no—
23C2×C4⋊C4×D61C2212yes—
24C4×#11758200—24no—
25C2×#14519708C2211yescharacteristic
26C2×#14519702C2212yes—
27C2×#14519700C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519689C2212yes—
30C2×#14519689C2212yes—
31C2×C4×#92418C2211yescharacteristic
32#16839678—22no—
33#16839632C2211yescharacteristic
34#16839622—21no—
35#16839619C2212yes—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39C2×C4⋊Q8×C61C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C61×#476—24no—
42C2×#106×C61C2212yes—
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#102×C61C2212yes—
45C22×Dic122C2311yescharacteristic
46C22×D244D411yescharacteristic
47C22×C22⋊D61—22no—
48C22×C4×D61—22no—
49C22×C4×D61C2312yes—
50C2×#11758200—22no—
51C2×#11758200—22no—
52C2×#11758200—22no—
53C22×#92418C2311yescharacteristic
54C22×#92418C2311yescharacteristic
55#14519768—28no—
56#14519757—24no—
57#14519757—24no—
58#14519753—216no—
59#14519751—44no—
60#14519748—24no—
61#14519748—42no—
62C4×Dic122—22no—
63C4×Dic122D414yescomplemented
64C4×D244—22no—
65C4×D244—22no—
66#14519739—28no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—22no—
75C4⋊C4×D61—24no—
76C4⋊C4×D61—24no—
77C2×C22⋊Dic61—22no—
78C2×#11758198—22no—
79C2×#11758198—22no—
80C2×#11758198D412yes—
81C2×C4×Dic61—21no—
82#14519708—22no—
83#14519702—42no—
84#14519700—22no—
85#14519698—22no—
86#14519689—24no—
87#14519689—42no—
88C4×#92418—22no—
89#14519684—22no—
90C22×Q8×C61—21no—
91C22×D4×C61C2311yescharacteristic
92C23×C4×C61—21no—
93C22×C8×C61C2312yes—
94C4⋊Q8×C61—22no—
95C4⋊Q8×C61D414yescomplemented
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101C2×C22⋊C4×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#106×C61—24no—
104C4⋊C8×C61—22no—
105#102×C61—24no—
106C24×C61—21no—
107C23×D61C2×D412yes—
108C2×Q8×C61—24no—
109C2×Q8×C61—81no—
110C2×Q8×C61—81no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61#4112yes—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—44no—
126C2×C22⋊D61—44no—
127C2×C22⋊D61—82no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—22no—
131C22×Dic61C2×D412yes—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244SD1614yescomplemented
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—28no—
142C2×C4×D61—82no—
143C2×C4×D61—82no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122—21no—
148C2×Dic122—21no—
149C2×Dic122C2×D412yes—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61C2×D414yes—
158C22⋊C4×C61—42no—
159C42×C61C2×D412yes—
160C42×C61C2×D412yes—
161C22⋊Dic61—44no—
162#11758200—28no—
163#11758198—28no—
164#11758198—24no—
165#11758198—44no—
166C4⋊Dic61—28no—
167C4⋊Dic61—22no—
168C4⋊Dic61—21no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—42no—
172C4⋊Dic61—81no—
173C4⋊Dic61C2×D412yes—
174#11758196—28no—
175#11758196—24no—
176C4×Dic61—24no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418—21no—
182C2×#92418#4112yes—
183C23×C61—22no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61—41no—
189C23×C61—41no—
190C23×C61C22×D411yescharacteristic
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—22no—
197C22×D61C2×SD1614yes—
198Q8×C61—84no—
199D4×C61—22no—
200D4×C61—22no—
201D4×C61—41no—
202D4×C61—41no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—24no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—84no—
212C2×C4×C61#12012yes—
213C2×C4×C61#12012yes—
214C2×C4×C61C2×SD1612yes—
215C2×C4×C61#13712yes—
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61H(5,𝔽2)11yescharacteristic
221C2×C4×C612−1+411yescharacteristic
222C22⋊D61—88no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235D244—28no—
236C4×D61—42no—
237C4×D61—42no—
238C4×D61—42no—
239C4×D61—42no—
240C4×D61—88no—
241Dic122—22no—
242Dic122—22no—
243Dic122—41no—
244Dic122—41no—
245C8×C61—42no—
246#92418—22no—
247#92418—41no—
248#24118—611nomaximal, Sylow
249C22×C61—42no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61—81no—
255C22×C61—81no—
256C22×C61C2×#12011yescharacteristic
257C22×C61#54611yescharacteristic
258C22×C61#54711yescharacteristic
259C22×C61C22×SD1611yescharacteristic
260C22×C61C2×#13711yescharacteristic
261C22×C61#57611yescharacteristic
262C22×C61#57711yescharacteristic
263C2×D61—24no—
264C2×D61—24no—
265C2×D61—24no—
266C2×D61—42no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—42no—
270C4×C61—22no—
271C4×C61—44no—
272C4×C61—81no—
273C4×C61#44814yes—
274C4×C61#52511yescharacteristic
275C4×C61#53711yescharacteristic
276C4×C61#54811yescharacteristic
277C4×C61#55011yescharacteristic
278Dic61—42no—
279Dic61—42no—
280Dic61—42no—
281Dic61—82no—
282C2×#554—611no—
283C2×#476D6111yescharacteristic, complemented
284C2×C4×SD16—611no—
285#2537—611no—
286#2035—612no—
287#2000—612no—
288#1990—611no—
289#1954—611no—
290#1874—612no—
291#1853—612no—
292#1803—611no—
293C2×C61—82no—
294C2×C61#240711yescharacteristic
295C2×C61C2×#44811yescharacteristic
296C2×C61#301211yescharacteristic
297C2×C61#329511yescharacteristic
298C2×C61#333111yescharacteristic
299C2×C61#333511yescharacteristic
300C2×C61#335711yescharacteristic
301D61—44no—
302C22×SD16—611no—
303#554—1224no—
304#554—1224no—
305C2×C4⋊Q8C2×D6111yescharacteristic
306C2×#125—1221no—
307C2×C22⋊Q8—612no—
308C2×C22⋊Q8—1222no—
309C2×#121—612no—
310C2×#121—1221no—
311C2×C4×Q8—611no—
312C2×C4×D4—611no—
313C2×C4×D4—1221no—
314C2×C4×D4C2×D6111yescharacteristic
315C22×C4⋊C4—612no—
316#476—24no—
317C4×SD16—1224no—
318C2×#106—611no—
319C2×#106C2×D6112yes—
320C2×C4⋊C8C2×D6111yescharacteristic
321C2×#103—611no—
322C2×#103—612no—
323C2×#102—611no—
324C2×#102C2×D6112yes—
325C2×C22⋊C8—612no—
326C2×C22⋊C8—612no—
327C2×C4×C8—611no—
328#394—1222no—
329#384—611no—
330#380—612no—
331#380—1221no—
332#340—612no—
333#340—612no—
334#340—612no—
335C61#2411811yescharacteristic, complemented, Sylow
336C22×Q8—21no—
337C22×Q8—611no—
338C22×D4—611no—
339C22×D4—1222no—
340C22×D4C22×D6111yescharacteristic
341C23×C4—21no—
342C23×C4—612no—
343C23×C4—1222no—
344C2×SD16—1222no—
345C2×SD16—1222no—
346C2×SD16—1222no—
347C22×C8—611no—
348C22×C8—611no—
349C22×C8C22×D6112yes—
350C4⋊Q8—22no—
351C4⋊Q8C22⋊D6114yescomplemented
352#125—1228no—
353C22⋊Q8—1224no—
354C22⋊Q8—12216no—
355C22⋊Q8—1224no—
356#121—1224no—
357#121—2442no—
358#121—2444no—
359C4×Q8—614no—
360C4×Q8—1222no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—1228no—
364C4×D4—1222no—
365C4×D4—1222no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—1224no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1224no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C4⋊C4C22×D6112yes—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—612no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1222no—
384C2×C42—1221no—
385C2×C42C22×D6111yescharacteristic
386#106—24no—
387#106—1222no—
388C4⋊C8—22no—
389#103—1222no—
390#103—2442no—
391#102—24no—
392#102—1222no—
393C22⋊C8—1224no—
394C22⋊C8—2442no—
395C4×C8—1222no—
396#95—1222no—
397C24—21no—
398C24—612no—
399C2×Q8—24no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×Q8—612no—
403C2×Q8—1221no—
404C2×Q8—1221no—
405C2×Q8—1221no—
406C2×Q8—1221no—
407C2×Q8—1221no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—614no—
414C2×D4—1222no—
415C2×D4—1222no—
416C2×D4—4882no—
417C2×D4—2444no—
418C2×D4—2444no—
419C2×D4#1175822612yes—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—612no—
426C22×C4—1228no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—4882no—
437C22×C4—4882no—
438C22×C4D4×D6111yescharacteristic
439C22×C4D4×D6111yescharacteristic
440C22×C4C23×D6111yescharacteristic
441SD16—1228no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—612no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—612no—
456C4⋊C4—1228no—
457C4⋊C4—1228no—
458C4⋊C4—1221no—
459C4⋊C4—1222no—
460C4⋊C4—1224no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—4881no—
464C4⋊C4—2442no—
465C4⋊C4C2×C22⋊D6114yes—
466C22⋊C4—42no—
467C22⋊C4—1228no—
468C22⋊C4—1224no—
469C22⋊C4—2444no—
470C22⋊C4—2444no—
471C42—1224no—
472C42D4×D6112yes—
473C42C2×C22⋊D6112yes—
474C23—22no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—614no—
482C23—1222no—
483C23—1222no—
484C23—1222no—
485C23—1222no—
486C23—1222no—
487C23—1222no—
488C23C2×D4×D6111yescharacteristic
489Q8—84no—
490Q8—1222no—
491Q8—1222no—
492Q8—2441no—
493Q8—2441no—
494D4—22no—
495D4—22no—
496D4—41no—
497D4—41no—
498D4—1228no—
499D4—4888no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—84no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—4888no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4#1451974512yes—
527C2×C4#1451974512yes—
528C2×C4SD16×D6112yes—
529C2×C4#1451980612yes—
530C2×C4C2×D4×D6111yescharacteristic
531C2×C4C22×C22⋊D6111yescharacteristic
532C2×C4#41×D6111yescharacteristic
533C2×C4C2×#1175822611yescharacteristic
534C2×C4#1451982311yescharacteristic
535C2×C4#1451982511yescharacteristic
536C8—42no—
537C8—1222no—
538C8—2441no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—1224no—
547C22—1224no—
548C22—1224no—
549C22—2442no—
550C22—2442no—
551C22—2442no—
552C22—2442no—
553C22C2×#1451974511yescharacteristic
554C22#1684040311yescharacteristic
555C22#1684041411yescharacteristic
556C22C2×SD16×D6111yescharacteristic
557C22C2×#1451980611yescharacteristic
558C22#1684056611yescharacteristic
559C22#1684057211yescharacteristic
560C4—22no—
561C4—44no—
562C4—81no—
563C4—4882no—
564C4—2442no—
565C4—2442no—
566C4—2442no—
567C4#1683989914yes—
568C4#1684027511yescharacteristic
569C4#1684034211yescharacteristic
570C4#1684042311yescharacteristic
571C4#1684043111yescharacteristic
572C2—82no—
573C2—2444no—
574C2#2144916611yescharacteristic
575C2C2×#1683989911yescharacteristic
576C2#2145512211yescharacteristic
577C2#2145764011yescharacteristic
578C2#2145813211yescharacteristic
579C2#2145817311yescharacteristic
580C2#2145844811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#476C2
socleC23×C61C22×D4
prime radical#21450732C2
hypercenterC2×#476D61
hypocenterC61#24118
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×#16840473C2semidirect
2. C2×C4×#11758200C2semidirect
4. #21447017C2semidirect
5. #21446823C2semidirect
6. #21446781C2semidirect
7. #21446612C2semidirect
8. #21446133C2semidirect
9. #21446010C2semidirect
10. #21445825C2semidirect
11. C2×C61×#476C2semidirect
63. C4×Dic122D4semidirect
95. C4⋊Q8×C61D4semidirect
134. C2×D244SD16semidirect
283. C2×#476D61semidirect
335. C61#24118semidirect
351. C4⋊Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404732
C2×C4×#117582002
#214507322
#214470172
#214468232
#214467812
#214466122
#214461332
#214460102
#214458252
C2×C61×#4762
#2411861