Finite Group Databasebeta

← #29736188

Subgroups — #29736188

Order 15616.

1338 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
1C22×#14519775C211yesmaximal, characteristic
2C2×#16840111C212yesmaximal, complemented
3#21454169C211yesmaximal, characteristic
4#21447064C212yesmaximal, complemented
5#21447063C212yesmaximal, complemented
6#21446619C212yesmaximal, complemented
7#21445755C211yesmaximal, characteristic
8#21445754C211yesmaximal, characteristic
9#21445753C211yesmaximal, characteristic
10C22⋊C8×Dic61C211yesmaximal, characteristic
11C61×#1787C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C2×#14519776C2211yescharacteristic
14C2×#14519775—28no—
15C2×#14519775—21no—
16C2×#14519775—21no—
17C2×#14519775C2212yes—
18C2×#14519775C2212yes—
19C2×#14519754—21no—
20C22×#11758196—21no—
21C22×#11758196—21no—
22C22×C4×Dic61C2211yescharacteristic
23#16840111—28no—
24C2×#14519702C2212yes—
25C2×#14519702C2212yes—
26C2×#14519701C2212yes—
27C2×#14519701C2212yes—
28C2×#14519700C2212yes—
29C2×#14519700C2212yes—
30C2×C22⋊#92418C2211yescharacteristic
31C2×C8×Dic61C2212yes—
32#16839666—21no—
33#16839661C2211yescharacteristic
34#16839659—21no—
35#16839658C2211yescharacteristic
36#16839631C2211yescharacteristic
37#16839622C2211yescharacteristic
38#16839620—21no—
39#16839516C2212yes—
40C23×Q8×C61C2211yescharacteristic
41C2×#103×C61C2212yes—
42C2×#103×C61C2212yes—
43C2×C22⋊C8×C61C2211yescharacteristic
44C61×#380C2211yescharacteristic
45C22×Dic122—21no—
46C22×Dic122—24no—
47C22×Dic122—22no—
48C22×Dic122—22no—
49C22×Dic122—21no—
50C22×Dic122—42no—
51C22×Dic122C2312yes—
52C23×Dic61—21no—
53C23×Dic61D411yescharacteristic
54C23×Dic61D411yescharacteristic
55C22×#92418C2312yes—
56#14519776—24no—
57#14519775—28no—
58#14519775—28no—
59#14519775—24no—
60#14519775—24no—
61#14519775—24no—
62#14519775D418yes—
63#14519754—44no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×#11758196—28no—
68C2×#11758196—28no—
69C2×#11758196—22no—
70C2×#11758196—22no—
71C2×#11758196—22no—
72C2×#11758196—22no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×C22⋊Dic61—21no—
78C2×C4×Dic61—24no—
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61C2311yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83#14519702—24no—
84#14519702—42no—
85#14519701—24no—
86#14519701—42no—
87#14519700—24no—
88#14519700—42no—
89C22⋊#92418—22no—
90C8×Dic61—24no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94C22×Q8×C61—24no—
95C22×Q8×C61—21no—
96C22×Q8×C61—22no—
97C22×Q8×C61—22no—
98C22×Q8×C61—42no—
99C22×Q8×C61C2312yes—
100C23×C4×C61—21no—
101C23×C4×C61C2311yescharacteristic
102C22×C8×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C22⋊C4×C61—21no—
106#103×C61—24no—
107#103×C61—42no—
108C22⋊C8×C61—22no—
109#95×C61—21no—
110C24×C61C2×D411yescharacteristic
111C2×Q8×C61—24no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—48no—
118C2×Q8×C61—48no—
119C2×Q8×C61—42no—
120C2×Q8×C61—42no—
121C2×Q8×C61C2×D414yes—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—24no—
125C22×C4×C61—22no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—42no—
129C22×C4×C61—42no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61#4112yes—
132C22×C4×C61#4112yes—
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—24no—
139C22×Dic61—22no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—42no—
143C22×Dic61—42no—
144C22×Dic61SD1614yes—
145C22×Dic61Q1614yes—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C2×Dic122—22no—
151C2×Dic122—22no—
152C2×Dic122—22no—
153C2×Dic122—22no—
154C2×Dic122—24no—
155C2×Dic122—22no—
156C2×Dic122—48no—
157C2×Dic122—48no—
158C2×Dic122—42no—
159C2×Dic122—42no—
160C2×Dic122—42no—
161C2×Dic122—42no—
162C2×Dic122—42no—
163C2×Dic122C2×D414yes—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C22⋊C4×C61—42no—
170C22⋊Dic61—42no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—42no—
174#11758196—28no—
175#11758196—28no—
176#11758196—28no—
177#11758196—28no—
178#11758196—42no—
179#11758196—42no—
180#11758196—42no—
181#11758196—42no—
182C4×Dic61—22no—
183C4×Dic61—22no—
184C4×Dic61—22no—
185C4×Dic61C2×D414yes—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61#12412yes—
194C23×C61C2×SD1612yes—
195C23×C61C2×Q1612yes—
196C23×C61C22×D411yescharacteristic
197Q8×C61—24no—
198Q8×C61—24no—
199Q8×C61—48no—
200Q8×C61—42no—
201Q8×C61—42no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—24no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—48no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61#12412yes—
217C2×C4×C61C22×D412yes—
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61H(5,𝔽2)12yes—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—28no—
226C2×Dic61—28no—
227C2×Dic61—24no—
228C2×Dic61—48no—
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×Dic61#12012yes—
239C2×Dic61C2×SD1612yes—
240C2×Dic61C2×SD1612yes—
241C2×Dic61C2×Q1612yes—
242C2×Dic61C2×Q1612yes—
243C2×Dic61#13712yes—
244C2×Dic61#13712yes—
245Dic122—24no—
246Dic122—24no—
247Dic122—48no—
248Dic122—42no—
249Dic122—42no—
250Dic122—84no—
251C8×C61—42no—
252#92418—42no—
253#25856—611nomaximal, Sylow
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61#48714yescomplemented
261C22×C61C2×#12011yescharacteristic
262C22×C61C2×#12411yescharacteristic
263C22×C61#53511yescharacteristic
264C22×C61C22×SD1611yescharacteristic
265C22×C61C22×Q1611yescharacteristic
266C22×C61C2×#13711yescharacteristic
267C22×C61C2×#13711yescharacteristic
268C4×C61—22no—
269C4×C61—44no—
270C4×C61—81no—
271C4×C61—81no—
272C4×C61#54612yes—
273C4×C61#54612yes—
274Dic61—22no—
275Dic61—22no—
276Dic61—44no—
277Dic61—81no—
278Dic61—81no—
279Dic61#44814yes—
280Dic61#45114yes—
281C22×C4⋊Q8—611no—
282C2×#487—612no—
283#2857—611no—
284#2043—612no—
285#2043—612no—
286#1956—612no—
287#1787—611no—
288#1787—611no—
289#1787—611no—
290#1787D6111yescharacteristic
291C4×C22⋊C8—611no—
292C2×C61—24no—
293C2×C61#243511yescharacteristic
294C2×C61C2×#44811yescharacteristic
295C2×C61C2×#45111yescharacteristic
296C2×C61C2×#48711yescharacteristic
297C2×C61#313911yescharacteristic
298C2×C61#320611yescharacteristic
299C2×C61#320911yescharacteristic
300C23×Q8—611no—
301C23×Q8C2×D6111yescharacteristic
302C2×C4⋊Q8—611no—
303C2×C4⋊Q8—612no—
304C2×C4⋊Q8—612no—
305C2×C4⋊Q8—1221no—
306C2×C4⋊Q8—1221no—
307C2×C4⋊Q8—1228no—
308C2×C22⋊Q8—1221no—
309C22×C4⋊C4—1221no—
310C22×C4⋊C4—1221no—
311C22×C42—611no—
312#487—1228no—
313C2×#103—612no—
314C2×#103—612no—
315C2×#103—612no—
316C2×#103—612no—
317C2×#103—612no—
318C2×#103—612no—
319C2×#103C2×D6112yes—
320C2×#103C2×D6112yes—
321C2×C22⋊C8—611no—
322C2×C22⋊C8C2×D6111yescharacteristic
323C2×C4×C8—612no—
324#391—1221no—
325#389—611no—
326#389—611no—
327#389—1221no—
328#384—611no—
329#380—611no—
330#380—1221no—
331#380C2×D6111yescharacteristic
332#336—612no—
333C61#2585611yescharacteristic, complemented, Sylow
334C22×Q8—24no—
335C22×Q8—21no—
336C22×Q8—22no—
337C22×Q8—22no—
338C22×Q8—42no—
339C22×Q8—612no—
340C22×Q8—1221no—
341C22×Q8—1224no—
342C22×Q8—1221no—
343C22×Q8—1222no—
344C22×Q8—1222no—
345C22×Q8—2442no—
346C22×Q8C22×D6112yes—
347C23×C4—21no—
348C23×C4—611no—
349C23×C4—611no—
350C23×C4—1221no—
351C23×C4C22×D6111yescharacteristic
352C22×C8—612no—
353C22×C8C22×D6112yes—
354C4⋊Q8—618no—
355C4⋊Q8—1224no—
356C4⋊Q8—1224no—
357C4⋊Q8—1224no—
358C4⋊Q8—1228no—
359C4⋊Q8—1228no—
360C4⋊Q8—1224no—
361C22⋊Q8—2444no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—1222no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1228no—
369C2×C4⋊C4—1228no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4C22×D6111yescharacteristic
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—1221no—
379C2×C42—611no—
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—611no—
383C2×C42—1224no—
384#103—24no—
385#103—42no—
386#103—1224no—
387#103—1224no—
388#103—1224no—
389#103—2442no—
390#103—2442no—
391#103—2442no—
392C22⋊C8—22no—
393C22⋊C8—1222no—
394C4×C8—1224no—
395#95—21no—
396#95—1221no—
397#95—1221no—
398#95—1221no—
399C24D4×D6111yescharacteristic
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—48no—
407C2×Q8—48no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—614no—
411C2×Q8—1224no—
412C2×Q8—1222no—
413C2×Q8—1222no—
414C2×Q8—1222no—
415C2×Q8—1222no—
416C2×Q8—1222no—
417C2×Q8—2448no—
418C2×Q8—2448no—
419C2×Q8—2442no—
420C2×Q8—2442no—
421C2×Q8—2442no—
422C2×Q8—2442no—
423C2×Q8—2442no—
424C2×Q8D4×D6114yes—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—611no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—614no—
438C22×C4—614no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1224no—
444C22×C4—1222no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—2442no—
448C22×C4—2442no—
449C22×C4#1175822212yes—
450C22×C4D4×D6111yescharacteristic
451C22×C4#1175822412yes—
452C22×C4C23×D6111yescharacteristic
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—1222no—
457C2×C8—1222no—
458C2×C8—1222no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—1228no—
462C4⋊C4—1228no—
463C4⋊C4—1228no—
464C4⋊C4—1228no—
465C4⋊C4—1222no—
466C4⋊C4—1222no—
467C4⋊C4—2442no—
468C4⋊C4—2442no—
469C4⋊C4—2442no—
470C4⋊C4—2442no—
471C4⋊C4—2442no—
472C22⋊C4—42no—
473C22⋊C4—2442no—
474C42—614no—
475C42—1222no—
476C42—1222no—
477C42—1222no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23#1451976312yes—
483C23SD16×D6112yes—
484C23Q16×D6112yes—
485C23C2×D4×D6111yescharacteristic
486Q8—24no—
487Q8—24no—
488Q8—48no—
489Q8—42no—
490Q8—42no—
491Q8—1224no—
492Q8—1224no—
493Q8—4884no—
494Q8—2448no—
495Q8—2442no—
496Q8—2442no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—612no—
512C2×C4—612no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—612no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—1228no—
520C2×C4—1228no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1224no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2448no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4#1451976312yes—
536C2×C4C2×D4×D6112yes—
537C2×C4C2×#1175822411yescharacteristic
538C2×C4C2×#1175822211yescharacteristic
539C2×C4#1451982212yes—
540C8—42no—
541C8—2442no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22#1684011114yescomplemented
549C22#120×D6111yescharacteristic
550C22C2×#1451976311yescharacteristic
551C22#1684032711yescharacteristic
552C22C2×SD16×D6111yescharacteristic
553C22C2×Q16×D6111yescharacteristic
554C22#137×D6111yescharacteristic
555C22#137×D6111yescharacteristic
556C4—22no—
557C4—44no—
558C4—81no—
559C4—81no—
560C4—614no—
561C4—614no—
562C4—1222no—
563C4—1222no—
564C4—4881no—
565C4—4881no—
566C4—2444no—
567C4#1684039912yes—
568C4#1684040212yes—
569C2—24no—
570C2#2144946811yescharacteristic
571C2#448×D6111yescharacteristic
572C2#451×D6111yescharacteristic
573C2C2×#1684011111yescharacteristic
574C2#2145619711yescharacteristic
575C2#2145674211yescharacteristic
576C2#2145676511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1787C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1787D61
hypocenterC61#25856
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
2. C2×#16840111C2semidirect
4. #21447064C2semidirect
5. #21447063C2semidirect
6. #21446619C2semidirect
260. C22×C61#487semidirect
333. C61#25856semidirect
548. C22#16840111semidirect

Maximal subgroups

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

subgroupindex
C22×#145197752
C2×#168401112
#214541692
#214470642
#214470632
#214466192
#214457552
#214457542
#214457532
C22⋊C8×Dic612
C61×#17872
#2585661