Finite Group Databasebeta

← #29732192

Subgroups — #29732192

Order 15616.

1412 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840310C211yesmaximal, characteristic
2C2×#16839988C212yesmaximal, complemented
3#21454160C211yesmaximal, characteristic, complemented
4#21447153C211yesmaximal, characteristic, complemented
5#21447050C211yesmaximal, characteristic, complemented
6#21446960C211yesmaximal, characteristic, complemented
7#21446244C212yesmaximal, complemented
8#21446155C212yesmaximal, complemented
9#21446082C212yesmaximal, complemented
10#336×D61C211yesmaximal, characteristic, complemented
11C61×#2014C211yesmaximal, characteristic, complemented
12C22×#11758224—21no—
13C22×C8×D61C2212yes—
14#16840310—28no—
15C2×#14519777C2211yescharacteristic
16C2×#14519764—21no—
17C2×#14519753C2211yescharacteristic
18C2×#14519748—21no—
19C2×#14519748C2212yes—
20C2×D4×Dic61—21no—
21C2×D4×Dic61C2212yes—
22C2×C42×D61C2211yescharacteristic
23#16839988—28no—
24C2×#14519709C2212yes—
25C2×#14519701C2212yes—
26C2×#14519698C2212yes—
27C2×#14519698C2212yes—
28#16839669C2211yescharacteristic
29#16839661C2211yescharacteristic
30#16839660C2211yescharacteristic
31#16839645—21no—
32#16839644—21no—
33#16839636—21no—
34#16839624—21no—
35#16839619C2211yescharacteristic
36#16839525C2212yes—
37#16839524C2212yes—
38#16839517C2211yescharacteristic
39#16839516C2212yes—
40C22×SD16×C61C2212yes—
41C2×#127×C61C2211yescharacteristic
42C2×#102×C61C2212yes—
43C61×#391C2211yescharacteristic
44C61×#336C2211yescharacteristic
45C2×#11758224—28no—
46C2×#11758224—44no—
47C2×#11758224—44no—
48C22×Dic122—21no—
49C22×C22⋊D61—22no—
50C23×Dic61—22no—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C22×#92418C2312yes—
55C2×C8×D61—24no—
56C2×C8×D61—24no—
57C2×C8×D61—42no—
58#14519777—24no—
59#14519764—28no—
60#14519753—24no—
61#14519748—28no—
62#14519748—24no—
63#14519748D418yes—
64D4×Dic61—28no—
65D4×Dic61—24no—
66D4×Dic61—24no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2312yes—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×C22⋊Dic61—22no—
75C2×C22⋊Dic61—22no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×C4×Dic61D411yescharacteristic
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61C2311yescharacteristic
81C42×D61—24no—
82#14519709—24no—
83#14519701—24no—
84#14519698—24no—
85#14519698—42no—
86#14519684—21no—
87#14519684—21no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91C22×Q8×C61C2311yescharacteristic
92C22×D4×C61—21no—
93C22×D4×C61—21no—
94C22×D4×C61C2312yes—
95C2×SD16×C61—24no—
96C2×SD16×C61—24no—
97C2×SD16×C61—42no—
98C22×C8×C61C2312yes—
99#127×C61—24no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C42×C61C2311yescharacteristic
102#102×C61—24no—
103#95×C61—21no—
104#95×C61—21no—
105C24×C61—22no—
106C23×D61C2×D411yescharacteristic
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×D4×C61—24no—
112C2×D4×C61—24no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—42no—
119C2×D4×C61—42no—
120C2×D4×C61—42no—
121C2×D4×C61—42no—
122C2×D4×C61C2×D414yes—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C2×C22⋊D61—28no—
129C2×C22⋊D61—44no—
130C2×C22⋊D61—44no—
131C22×Dic61—28no—
132C22×Dic61—22no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—44no—
136C22×Dic61—44no—
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140#11758224—432no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—41no—
149C2×C4×D61SD1614yescomplemented
150C2×C4×D61C2×D414yes—
151C2×Dic122—24no—
152C2×Dic122—42no—
153C2×Dic122—42no—
154SD16×C61—48no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61#4114yescomplemented
158C4⋊C4×C61—22no—
159C42×C61—22no—
160C22⋊Dic61—28no—
161C22⋊Dic61—28no—
162#11758198—42no—
163#11758198—42no—
164C4⋊Dic61—24no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—42no—
170C4⋊Dic61C2×D414yes—
171#11758196—42no—
172#11758196—42no—
173C4×Dic61—22no—
174C4×Dic61—22no—
175C4×Dic61—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C8×D61—48no—
180C23×C61—24no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61C22×D411yescharacteristic
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61#12012yes—
193C22×D61C2×SD1612yes—
194C22×D61AGL(1,ℤ/8)12yes—
195Q8×C61—42no—
196Q8×C61—42no—
197D4×C61—24no—
198D4×C61—24no—
199D4×C61—48no—
200D4×C61—48no—
201D4×C61—42no—
202D4×C61—42no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61C2×SD1612yes—
214C2×C4×C61#13512yes—
215C2×C4×C61C22×D412yes—
216C2×C4×C61C2×#4112yes—
217C2×C4×C61H(5,𝔽2)11yescharacteristic
218C2×C4×C612−1+411yescharacteristic
219C22⋊D61—416no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—24no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—416no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61#12012yes—
238C2×Dic61C2×SD1612yes—
239C4×D61—28no—
240C4×D61—24no—
241C4×D61—24no—
242C4×D61—44no—
243Dic122—48no—
244C8×C61—24no—
245#92418—42no—
246#25495—611nomaximal, Sylow
247C22×C61—44no—
248C22×C61—44no—
249C22×C61—44no—
250C22×C61—82no—
251C22×C61—82no—
252C22×C61—82no—
253C22×C61—82no—
254C22×C61C2×#12011yescharacteristic
255C22×C61#53211yescharacteristic
256C22×C61#53711yescharacteristic
257C22×C61C22×SD1611yescharacteristic
258C22×C61C2×#13511yescharacteristic
259C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C61#57511yescharacteristic
261C2×D61—22no—
262C2×D61—22no—
263C2×D61—22no—
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61#45014yes—
268C4×C61—22no—
269C4×C61—22no—
270C4×C61—81no—
271C4×C61—81no—
272C4×C61#54612yes—
273C4×C61#54712yes—
274Dic61—22no—
275Dic61—22no—
276Dic61—44no—
277Dic61—41no—
278Dic61—81no—
279Dic61—81no—
280C2×#532—611no—
281C2×#465—612no—
282#2855—611no—
283#2058—611no—
284#2041—611no—
285#2024—611no—
286#2014D6111yescharacteristic, complemented
287#1893—612no—
288#1877—612no—
289#1864—612no—
290C2×#336—611no—
291C2×C61—84no—
292C2×C61#243011yescharacteristic
293C2×C61C2×#45011yescharacteristic
294C2×C61#300211yescharacteristic
295C2×C61#316711yescharacteristic
296C2×C61#318011yescharacteristic
297C2×C61#324611yescharacteristic
298C2×C61#324711yescharacteristic
299D61—24no—
300C22×#41—1221no—
301C22×SD16C2×D6112yes—
302C23×C8—612no—
303#532—1228no—
304C2×C4⋊Q8—611no—
305C2×#127C2×D6111yescharacteristic
306C2×#124—1221no—
307C2×C22⋊Q8—611no—
308C2×#121—612no—
309C2×#121—1221no—
310C2×C4×D4—612no—
311C2×C4×D4—1221no—
312C22×C42—611no—
313#465—1228no—
314C2×#106—612no—
315C2×#103—612no—
316C2×#102—612no—
317C2×#102—612no—
318C2×#102C2×D6112yes—
319#391—611no—
320#391C2×D6111yescharacteristic
321#389—611no—
322#389—611no—
323#387—1221no—
324#387—1221no—
325#385—1221no—
326#381—1221no—
327#380—611no—
328#340—612no—
329#340—612no—
330#336—612no—
331#336—611no—
332#336C2×D6111yescharacteristic
333C61#2549511yescharacteristic, complemented, Sylow
334C2×#41—1228no—
335C2×#41—2444no—
336C2×#41—2444no—
337C22×Q8—1221no—
338C22×Q8C22×D6111yescharacteristic
339C22×D4—21no—
340C22×D4—21no—
341C22×D4—1222no—
342C22×D4C22×D6112yes—
343C23×C4—611no—
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—1222no—
347C2×SD16—24no—
348C2×SD16—24no—
349C2×SD16—42no—
350C22×C8—612no—
351C22×C8—1224no—
352C22×C8—1224no—
353C22×C8—2442no—
354C22×C8C22×D6112yes—
355C4⋊Q8—1224no—
356#127—24no—
357#124—1228no—
358C22⋊Q8—1224no—
359#121—618no—
360#121—1224no—
361#121—1228no—
362C4×D4—1224no—
363C4×D4—1228no—
364C4×D4—1224no—
365C2×C4⋊C4—612no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4C22×D6111yescharacteristic
373C2×C22⋊C4—1222no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1221no—
377C2×C42—611no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—1224no—
381C2×C42C22×D6111yescharacteristic
382#106—1224no—
383#103—1224no—
384#102—24no—
385#102—1224no—
386#102—2442no—
387#95—21no—
388#95—21no—
389#95—1221no—
390#95—1221no—
391#95—1221no—
392#95—1221no—
393#95—1221no—
394C24—22no—
395C24—611no—
396#41—24432no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—1224no—
402C2×Q8—2442no—
403C2×Q8—2442no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—1228no—
416C2×D4—2444no—
417C2×D4—2444no—
418C2×D4D4×D6114yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—614no—
424C22×C4—614no—
425C22×C4—611no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1228no—
433C22×C4—1222no—
434C22×C4—1222no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—2441no—
438C22×C4—2444no—
439C22×C4—2444no—
440C22×C4D4×D6111yescharacteristic
441C22×C4D4×D6111yescharacteristic
442C22×C4C23×D6111yescharacteristic
443SD16—48no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—1222no—
447C2×C8—1222no—
448C2×C8—1222no—
449C2×C8—2448no—
450C2×C8#1175822414yescomplemented
451C4⋊C4—22no—
452C4⋊C4—614no—
453C4⋊C4—1224no—
454C4⋊C4—1222no—
455C4⋊C4—1222no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—2442no—
459C4⋊C4—2442no—
460C4⋊C4—2442no—
461C22⋊C4—1228no—
462C22⋊C4—1228no—
463C22⋊C4—2442no—
464C22⋊C4—2442no—
465C42—22no—
466C42—1222no—
467C42—1222no—
468C42—1222no—
469C23—24no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—612no—
477C23—612no—
478C23—612no—
479C23—1221no—
480C23—1221no—
481C23—1221no—
482C23—1221no—
483C23C2×D4×D6111yescharacteristic
484Q8—42no—
485Q8—42no—
486Q8—2448no—
487D4—24no—
488D4—24no—
489D4—48no—
490D4—48no—
491D4—42no—
492D4—42no—
493D4—24416no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—612no—
505C2×C4—612no—
506C2×C4—1228no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1224no—
512C2×C4—1224no—
513C2×C4—1221no—
514C2×C4—1224no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—2444no—
518C2×C4—24416no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4SD16×D6112yes—
528C2×C4#1451979812yes—
529C2×C4C2×D4×D6112yes—
530C2×C4C2×#1175822412yes—
531C2×C4#1451982211yescharacteristic
532C2×C4#1451982411yescharacteristic
533C8—24no—
534C8—2442no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—614no—
543C22—1222no—
544C22—1222no—
545C22—1222no—
546C22—1222no—
547C22—1222no—
548C22—1222no—
549C22#120×D6111yescharacteristic
550C22#1684031011yescharacteristic
551C22#1684034111yescharacteristic
552C22C2×SD16×D6111yescharacteristic
553C22C2×#1451979811yescharacteristic
554C22AGL(1,ℤ/8)×D6111yescharacteristic
555C22#1684055811yescharacteristic
556C4—22no—
557C4—22no—
558C4—81no—
559C4—81no—
560C4—1222no—
561C4—1222no—
562C4—4881no—
563C4—4881no—
564C4—2444no—
565C4—2441no—
566C4#1684039912yes—
567C4#1684041212yes—
568C2—84no—
569C2—1224no—
570C2#2144942211yescharacteristic
571C2#450×D6111yescharacteristic
572C2#2145503811yescharacteristic
573C2#2145637611yescharacteristic
574C2#2145651411yescharacteristic
575C2#2145713311yescharacteristic
576C2#2145714311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2014C2
socleC23×C61C22×D4
prime radicalC2×#16840310C2
hypercenter#2014D61
hypocenterC61#25495
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2014

Complements

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

NH ≅ G/Ntype
2. C2×#16839988C2semidirect
3. #21454160C2semidirect
4. #21447153C2semidirect
5. #21447050C2semidirect
6. #21446960C2semidirect
7. #21446244C2semidirect
8. #21446155C2semidirect
9. #21446082C2semidirect
10. #336×D61C2semidirect
11. C61×#2014C2semidirect
149. C2×C4×D61SD16semidirect
157. C2×C8×C61#41semidirect
286. #2014D61semidirect
333. C61#25495semidirect
450. C2×C8#11758224semidirect

Maximal subgroups

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

subgroupindex
C2×#168403102
C2×#168399882
#214541602
#214471532
#214470502
#214469602
#214462442
#214461552
#214460822
#336×D612
C61×#20142
#2549561