Finite Group Databasebeta

← #29713941

Subgroups — #29713941

Order 15616.

1174 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840486C211yesmaximal, characteristic, complemented
2C2×C4×#11758200C211yesmaximal, characteristic, complemented
3#21450732C211yesmaximal, characteristic
4#21447024C212yesmaximal, complemented
5#21446823C212yesmaximal, complemented
6#21446781C211yesmaximal, characteristic, complemented
7#21446612C211yesmaximal, characteristic, complemented
8#21446138C212yesmaximal, complemented
9#21446015C212yesmaximal, complemented
10#21445825C211yesmaximal, characteristic, complemented
11C2×C61×#474C211yesmaximal, characteristic, complemented
12C22×#11758200C2211yescharacteristic
13#16840486—24no—
14#16840486—24no—
15C2×#14519772—22no—
16C2×#14519766—21no—
17C2×#14519757C2212yes—
18C2×#14519751—21no—
19C2×#14519748C2212yes—
20C2×Q8×Dic61—21no—
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×#14519735C2212yes—
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×#474—24no—
42C2×#107×C61C2212yes—
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×#102×C61C2212yes—
45C22×Dic122C2311yescharacteristic
46C22×D244D411yescharacteristic
47C22×C22⋊D61—22no—
48C22×C4×D61C2312yes—
49C2×#11758200—22no—
50C2×#11758200—22no—
51C2×#11758200—22no—
52C22×#92418C2311yescharacteristic
53C22×#92418C2311yescharacteristic
54#14519772—216no—
55#14519766—28no—
56#14519757—24no—
57#14519757—24no—
58#14519751—44no—
59#14519748—24no—
60#14519748—42no—
61Q8×Dic61—28no—
62C4×Dic122—22no—
63C4×Dic122D414yescomplemented
64C4×D244—22no—
65C4×D244—22no—
66#14519735—24no—
67#14519735—24no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×C22⋊Dic61—22no—
76C2×#11758198—22no—
77C2×#11758198—22no—
78C2×#11758198—22no—
79C2×#11758198D412yes—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—22no—
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#107×C61—24no—
104C4⋊C8×C61—22no—
105#102×C61—24no—
106C24×C61—21no—
107C23×D61C2×D412yes—
108C2×Q8×C61—24no—
109C2×Q8×C61—42no—
110C2×Q8×C61—42no—
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×D244—22no—
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×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122C2×D412yes—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61C2×D414yes—
155C22⋊C4×C61—42no—
156C42×C61C2×D412yes—
157C42×C61C2×D412yes—
158C22⋊Dic61—44no—
159#11758200—28no—
160#11758198—28no—
161#11758198—28no—
162#11758198—24no—
163#11758198—44no—
164C4⋊Dic61—28no—
165C4⋊Dic61—22no—
166C4⋊Dic61—21no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—42no—
170C4⋊Dic61—81no—
171C4⋊Dic61C2×D412yes—
172#11758196—24no—
173C4×Dic61—28no—
174C4×Dic61—24no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418—21no—
180C2×#92418#4112yes—
181C23×C61—22no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61C22×D411yescharacteristic
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196Q8×C61—48no—
197D4×C61—22no—
198D4×C61—22no—
199D4×C61—41no—
200D4×C61—41no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—24no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—84no—
210C2×C4×C61#12012yes—
211C2×C4×C61#12012yes—
212C2×C4×C61#13512yes—
213C2×C4×C61AGL(1,ℤ/8)12yes—
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61H(5,𝔽2)11yescharacteristic
219C2×C4×C612−1+411yescharacteristic
220C22⋊D61—88no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233D244—44no—
234C4×D61—42no—
235C4×D61—42no—
236C4×D61—42no—
237C4×D61—42no—
238Dic122—22no—
239Dic122—22no—
240Dic122—41no—
241Dic122—41no—
242C8×C61—42no—
243#92418—22no—
244#92418—41no—
245#24119—611nomaximal, Sylow
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61—81no—
252C22×C61—81no—
253C22×C61C2×#12011yescharacteristic
254C22×C61#54611yescharacteristic
255C22×C61#54711yescharacteristic
256C22×C61C2×#13511yescharacteristic
257C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C61#57711yescharacteristic
259C22×C61#57811yescharacteristic
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—42no—
267C4×C61—22no—
268C4×C61—44no—
269C4×C61—81no—
270C4×C61#44914yes—
271C4×C61#52511yescharacteristic
272C4×C61#53711yescharacteristic
273C4×C61#54811yescharacteristic
274C4×C61#55011yescharacteristic
275Dic61—42no—
276Dic61—42no—
277Dic61—42no—
278Dic61—82no—
279C2×#556—611no—
280C2×#474D6111yescharacteristic, complemented
281C2×C4×SD16—611no—
282#2537—611no—
283#2036—612no—
284#2000—612no—
285#1990—611no—
286#1954—611no—
287#1875—612no—
288#1854—612no—
289#1803—611no—
290C2×C61—82no—
291C2×C61#240711yescharacteristic
292C2×C61C2×#44911yescharacteristic
293C2×C61#301311yescharacteristic
294C2×C61#330211yescharacteristic
295C2×C61#332411yescharacteristic
296C2×C61#334011yescharacteristic
297C2×C61#335311yescharacteristic
298D61—82no—
299C22×SD16—611no—
300#556—1224no—
301#556—1224no—
302C2×C4⋊Q8C2×D6111yescharacteristic
303C2×#126—1222no—
304C2×#124—1221no—
305C2×C22⋊Q8—612no—
306C2×#121—612no—
307C2×#121—1221no—
308C2×C4×Q8—611no—
309C2×C4×Q8—1221no—
310C2×C4×D4—611no—
311C2×C4×D4C2×D6111yescharacteristic
312C2×#117—612no—
313#474—24no—
314C4×SD16—1224no—
315C2×#107C2×D6112yes—
316C2×#106—611no—
317C2×C4⋊C8C2×D6111yescharacteristic
318C2×#103—611no—
319C2×#103—612no—
320C2×#102—611no—
321C2×#102C2×D6112yes—
322C2×C22⋊C8—612no—
323C2×C22⋊C8—612no—
324C2×C4×C8—611no—
325#394—1222no—
326#384—611no—
327#380—612no—
328#380—1221no—
329#340—612no—
330#340—612no—
331#340—612no—
332C61#2411911yescharacteristic, complemented, Sylow
333C22×Q8—21no—
334C22×Q8—611no—
335C22×D4—611no—
336C22×D4—1222no—
337C22×D4C22×D6111yescharacteristic
338C23×C4—21no—
339C23×C4—612no—
340C2×SD16—1222no—
341C2×SD16—1222no—
342C2×SD16—1222no—
343C22×C8—611no—
344C22×C8—611no—
345C22×C8C22×D6112yes—
346C4⋊Q8—22no—
347C4⋊Q8C22⋊D6114yescomplemented
348#126—12216no—
349#124—1228no—
350C22⋊Q8—1224no—
351C22⋊Q8—1224no—
352#121—1224no—
353#121—2442no—
354#121—2444no—
355C4×Q8—614no—
356C4×Q8—1228no—
357C4×Q8—1222no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—1222no—
361C4×D4—1222no—
362#117—1224no—
363#117—1224no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4C22×D6111yescharacteristic
373C2×C4⋊C4C22×D6112yes—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—612no—
376C2×C22⋊C4—1222no—
377C2×C22⋊C4—1222no—
378C2×C22⋊C4—1222no—
379C2×C22⋊C4—1222no—
380C2×C42—1222no—
381C2×C42—1221no—
382C2×C42C22×D6111yescharacteristic
383#107—24no—
384#106—1222no—
385C4⋊C8—22no—
386#103—1222no—
387#103—2442no—
388#102—24no—
389#102—1222no—
390C22⋊C8—1224no—
391C22⋊C8—2442no—
392C4×C8—1222no—
393#95—1222no—
394C24—21no—
395C24—612no—
396C2×Q8—24no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—612no—
400C2×Q8—1221no—
401C2×Q8—1221no—
402C2×Q8—1221no—
403C2×Q8—1221no—
404C2×Q8—1221no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—1222no—
411C2×D4—1222no—
412C2×D4—1222no—
413C2×D4—4882no—
414C2×D4—2444no—
415C2×D4—2444no—
416C2×D4#1175822612yes—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—612no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4D4×D6111yescharacteristic
433C22×C4D4×D6111yescharacteristic
434C22×C4C23×D6111yescharacteristic
435SD16—1228no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—612no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—612no—
450C4⋊C4—1228no—
451C4⋊C4—1221no—
452C4⋊C4—1222no—
453C4⋊C4—1224no—
454C4⋊C4—1222no—
455C4⋊C4—1222no—
456C4⋊C4—4881no—
457C4⋊C4—2442no—
458C4⋊C4C2×C22⋊D6114yes—
459C22⋊C4—42no—
460C22⋊C4—1228no—
461C22⋊C4—1228no—
462C22⋊C4—1224no—
463C22⋊C4—2444no—
464C22⋊C4—2444no—
465C42—1228no—
466C42—1224no—
467C42D4×D6112yes—
468C42C2×C22⋊D6112yes—
469C23—22no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—1222no—
477C23—1222no—
478C23—1222no—
479C23—1222no—
480C23—1222no—
481C23—1222no—
482C23—1222no—
483C23C2×D4×D6111yescharacteristic
484Q8—48no—
485Q8—1222no—
486Q8—1222no—
487Q8—2441no—
488Q8—2441no—
489D4—22no—
490D4—22no—
491D4—41no—
492D4—41no—
493D4—4888no—
494D4—2444no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—84no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1222no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4—2442no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4#1451974512yes—
521C2×C4#1451974512yes—
522C2×C4#1451979612yes—
523C2×C4#1451980112yes—
524C2×C4C2×D4×D6111yescharacteristic
525C2×C4C22×C22⋊D6111yescharacteristic
526C2×C4#41×D6111yescharacteristic
527C2×C4C2×#1175822611yescharacteristic
528C2×C4#1451982311yescharacteristic
529C2×C4#1451982511yescharacteristic
530C8—42no—
531C8—1222no—
532C8—2441no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—2442no—
541C22—2442no—
542C22—2442no—
543C22—2442no—
544C22—2442no—
545C22—2442no—
546C22—2442no—
547C22C2×#1451974511yescharacteristic
548C22#1684040311yescharacteristic
549C22#1684041411yescharacteristic
550C22C2×#1451979611yescharacteristic
551C22C2×#1451980111yescharacteristic
552C22#1684057111yescharacteristic
553C22#1684057811yescharacteristic
554C4—22no—
555C4—44no—
556C4—81no—
557C4—4882no—
558C4—2442no—
559C4—2442no—
560C4—2442no—
561C4#1683990414yes—
562C4#1684027511yescharacteristic
563C4#1684034211yescharacteristic
564C4#1684042311yescharacteristic
565C4#1684043111yescharacteristic
566C2—82no—
567C2—4882no—
568C2#2144916611yescharacteristic
569C2C2×#1683990411yescharacteristic
570C2#2145513311yescharacteristic
571C2#2145773311yescharacteristic
572C2#2145804711yescharacteristic
573C2#2145822911yescharacteristic
574C2#2145840011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#474C2
socleC23×C61C22×D4
prime radical#21450732C2
hypercenterC2×#474D61
hypocenterC61#24119
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#16840486C2semidirect
2. C2×C4×#11758200C2semidirect
4. #21447024C2semidirect
5. #21446823C2semidirect
6. #21446781C2semidirect
7. #21446612C2semidirect
8. #21446138C2semidirect
9. #21446015C2semidirect
10. #21445825C2semidirect
11. C2×C61×#474C2semidirect
63. C4×Dic122D4semidirect
95. C4⋊Q8×C61D4semidirect
280. C2×#474D61semidirect
332. C61#24119semidirect
347. C4⋊Q8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404862
C2×C4×#117582002
#214507322
#214470242
#214468232
#214467812
#214466122
#214461382
#214460152
#214458252
C2×C61×#4742
#2411961