Finite Group Databasebeta

← #29713306

Subgroups — #29713306

Order 15616.

1144 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840314C211yesmaximal, characteristic
2C2×#16840093C211yesmaximal, characteristic
3C2×#16840090C211yesmaximal, characteristic
4C2×#16839661C211yesmaximal, characteristic
5#21447151C212yesmaximal, complemented
6#21447064C212yesmaximal, complemented
7#21446126C211yesmaximal, characteristic
8#21446089C212yesmaximal, complemented
9#21446088C212yesmaximal, complemented
10#21445885C211yesmaximal, characteristic
11C61×#1872C211yesmaximal, characteristic
12#16840314—24no—
13#16840314—24no—
14C2×#14519776C2211yescharacteristic
15C2×#14519775C2211yescharacteristic
16C2×#14519767—21no—
17C2×#14519756C2211yescharacteristic
18C2×#14519754—21no—
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519742—21no—
21C2×#14519742C2211yescharacteristic
22C2×#14519733—21no—
23C22×C4⋊Dic61—21no—
24C22×C4⋊Dic61C2211yescharacteristic
25C22×C4×Dic61C2211yescharacteristic
26#16840093—24no—
27#16840090—24no—
28C2×#14519709C2212yes—
29C2×#14519708C2212yes—
30C2×#14519702C2212yes—
31C2×#14519701C2212yes—
32C2×#14519700C2212yes—
33C2×C22⋊#92418C2211yescharacteristic
34#16839661—21no—
35#16839661—21no—
36#16839661C2212yes—
37#16839661C2212yes—
38C2×#14519684—21no—
39#16839524C2212yes—
40#16839516C2212yes—
41#16839516C2212yes—
42C2×C22⋊Q8×C61C2211yescharacteristic
43C22×C4⋊C4×C61C2211yescharacteristic
44C2×#103×C61C2212yes—
45C2×C22⋊C8×C61C2211yescharacteristic
46C61×#340C2212yes—
47C22×Dic122—21no—
48C22×Dic122C2311yescharacteristic
49C23×Dic61—21no—
50C23×Dic61D411yescharacteristic
51C23×Dic61D411yescharacteristic
52C22×#92418C2312yes—
53#14519776—22no—
54#14519776—22no—
55#14519775—22no—
56#14519775—22no—
57#14519767—28no—
58#14519756—22no—
59#14519756D414yes—
60#14519754—28no—
61Q8×Dic61—22no—
62Q8×Dic61—22no—
63#14519742—28no—
64#14519742—22no—
65#14519742—22no—
66#14519733—28no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—22no—
73C2×C4⋊Dic61—44no—
74C2×C4⋊Dic61C2312yes—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61—22no—
83C2×C4×Dic61—22no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61D411yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90#14519709—24no—
91#14519708—24no—
92#14519702—24no—
93#14519701—24no—
94#14519700—24no—
95C22⋊#92418—22no—
96#14519684—22no—
97#14519684—22no—
98C22×Q8×C61C2311yescharacteristic
99C23×C4×C61—21no—
100C23×C4×C61C2311yescharacteristic
101C22×C8×C61C2312yes—
102C22⋊Q8×C61—22no—
103C22⋊Q8×C61D414yes—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—24no—
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2312yes—
109C2×C22⋊C4×C61—21no—
110#103×C61—24no—
111C22⋊C8×C61—22no—
112C24×C61C2×D411yescharacteristic
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61C2×D412yes—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—24no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—41no—
137C22×Dic61—42no—
138C22×Dic61—42no—
139C22×Dic61—42no—
140C22×Dic61SD1614yes—
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—24no—
148C2×Dic122—21no—
149C2×Dic122—42no—
150C2×Dic122—42no—
151C2×Dic122C2×D412yes—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61C2×D412yes—
160C4⋊C4×C61#4114yes—
161C22⋊C4×C61—24no—
162C22⋊Dic61—24no—
163C4⋊Dic61—24no—
164C4⋊Dic61—21no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—42no—
168C4⋊Dic61—44no—
169C4⋊Dic61—44no—
170C4⋊Dic61C2×D412yes—
171#11758196—24no—
172#11758196—24no—
173#11758196—24no—
174#11758196—24no—
175#11758196—24no—
176C4×Dic61—24no—
177C4×Dic61—24no—
178C4×Dic61—22no—
179C4×Dic61—22no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—21no—
183C4×Dic61—21no—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—22no—
187C23×C61—21no—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61#12012yes—
192C23×C61C2×SD1612yes—
193C23×C61#13712yes—
194C23×C61C22×D411yescharacteristic
195Q8×C61—24no—
196Q8×C61—42no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—24no—
200C2×C4×C61—24no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—24no—
204C2×C4×C61—48no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—42no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61#12012yes—
212C2×C4×C61C22×D411yescharacteristic
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C2×#4112yes—
215C2×C4×C612−1+412yes—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—28no—
223C2×Dic61—21no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—44no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—84no—
237C2×Dic61C2×SD1612yes—
238C2×Dic61C2×SD1612yes—
239C2×Dic61#13712yes—
240Dic122—24no—
241Dic122—48no—
242Dic122—42no—
243C8×C61—42no—
244#92418—42no—
245#24063—611nomaximal, Sylow
246C22×C61—22no—
247C22×C61—22no—
248C22×C61—22no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61—22no—
252C22×C61#44814yes—
253C22×C61C2×#12011yescharacteristic
254C22×C61#52511yescharacteristic
255C22×C61#53611yescharacteristic
256C22×C61C22×SD1611yescharacteristic
257C22×C61C2×#13711yescharacteristic
258C22×C61C2×#13711yescharacteristic
259C22×C61#57511yescharacteristic
260C4×C61—22no—
261C4×C61—44no—
262C4×C61—42no—
263C4×C61—42no—
264C4×C61#54712yes—
265C4×C61#54712yes—
266Dic61—22no—
267Dic61—22no—
268Dic61—22no—
269Dic61—42no—
270Dic61—42no—
271Dic61—41no—
272Dic61—82no—
273C2×#533—611no—
274C2×#483—611no—
275C2×#483—611no—
276C2×#389—611no—
277#2058—612no—
278#2043—612no—
279#1872—611no—
280#1872D6111yescharacteristic
281#1865—612no—
282#1865—612no—
283#1817—611no—
284C2×C61—24no—
285C2×C61#240611yescharacteristic
286C2×C61C2×#44811yescharacteristic
287C2×C61#300711yescharacteristic
288C2×C61#310111yescharacteristic
289C2×C61#311311yescharacteristic
290C2×C61#322911yescharacteristic
291C2×C61#323611yescharacteristic
292#533—1224no—
293#533—1224no—
294C2×C4⋊Q8—611no—
295C2×C4⋊Q8—611no—
296C2×#125—1221no—
297C2×C22⋊Q8—611no—
298C2×C22⋊Q8—1221no—
299C2×C22⋊Q8C2×D6111yescharacteristic
300C2×C4×Q8—611no—
301C2×C4×Q8—611no—
302C2×C4×Q8—1221no—
303C2×#117—1221no—
304C22×C4⋊C4—611no—
305C22×C4⋊C4—1221no—
306C22×C4⋊C4C2×D6111yescharacteristic
307C22×C42—611no—
308#483—1224no—
309#483—1224no—
310C2×#106—612no—
311C2×#106—612no—
312C2×#103—612no—
313C2×#103—612no—
314C2×#103—612no—
315C2×#103C2×D6112yes—
316C2×C22⋊C8—611no—
317C2×C22⋊C8C2×D6111yescharacteristic
318#389—612no—
319#389—612no—
320#389—1221no—
321#389—1221no—
322C2×#95—1221no—
323#340—612no—
324#340C2×D6112yes—
325#336—612no—
326#336—612no—
327C61#2406311yescharacteristic, complemented, Sylow
328C22×Q8—611no—
329C22×Q8—1221no—
330C22×Q8C22×D6111yescharacteristic
331C23×C4—21no—
332C23×C4—611no—
333C23×C4—611no—
334C23×C4—1221no—
335C23×C4C22×D6111yescharacteristic
336C22×C8—612no—
337C22×C8C22×D6112yes—
338C4⋊Q8—1222no—
339C4⋊Q8—1222no—
340C4⋊Q8—1222no—
341C4⋊Q8—1222no—
342#125—1228no—
343C22⋊Q8—22no—
344C22⋊Q8—614no—
345C22⋊Q8—1228no—
346C22⋊Q8—1222no—
347C22⋊Q8D24414yes—
348C4×Q8—1222no—
349C4×Q8—1228no—
350C4×Q8—1222no—
351C4×Q8—1222no—
352C4×Q8—1222no—
353#117—1228no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—611no—
358C2×C4⋊C4—612no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—2444no—
371C2×C4⋊C4C22×D6111yescharacteristic
372C2×C4⋊C4C22×D6112yes—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—1221no—
375C2×C42—611no—
376C2×C42—611no—
377C2×C42—611no—
378C2×C42—611no—
379C2×C42—1221no—
380C2×C42—1221no—
381C2×C42—1222no—
382C2×C42—1222no—
383#106—1224no—
384#106—1224no—
385#103—24no—
386#103—1224no—
387#103—1224no—
388#103—1224no—
389C22⋊C8—22no—
390C22⋊C8—1222no—
391#95—1222no—
392#95—1222no—
393C24D4×D6111yescharacteristic
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—612no—
398C2×Q8—1224no—
399C2×Q8—1221no—
400C2×Q8—1222no—
401C2×Q8—1222no—
402C2×Q8—2442no—
403C2×Q8—2442no—
404C2×Q8C2×D24412yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—614no—
417C22×C4—611no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—2441no—
426C22×C4—2442no—
427C22×C4—2442no—
428C22×C4—2442no—
429C22×C4C2×D24412yes—
430C22×C4D4×D6112yes—
431C22×C4D4×D6111yescharacteristic
432C22×C4C23×D6111yescharacteristic
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—1222no—
437C2×C8—1222no—
438C2×C8—1222no—
439C4⋊C4—24no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—612no—
444C4⋊C4—1224no—
445C4⋊C4—1224no—
446C4⋊C4—1224no—
447C4⋊C4—1224no—
448C4⋊C4—1224no—
449C4⋊C4—1221no—
450C4⋊C4—1224no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—2444no—
454C4⋊C4—2444no—
455C4⋊C4—2442no—
456C4⋊C4C2×D24412yes—
457C4⋊C4#1175822414yes—
458C22⋊C4—24no—
459C22⋊C4—1224no—
460C42—1224no—
461C42—1224no—
462C42—1221no—
463C42—1221no—
464C42—1221no—
465C42—1221no—
466C42—1222no—
467C42—1222no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23#1451974712yes—
473C23SD16×D6112yes—
474C23#1451980912yes—
475C23C2×D4×D6111yescharacteristic
476Q8—24no—
477Q8—42no—
478Q8—1224no—
479Q8—2448no—
480Q8—2442no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—48no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—612no—
496C2×C4—612no—
497C2×C4—612no—
498C2×C4—1221no—
499C2×C4—1221no—
500C2×C4—1221no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1228no—
505C2×C4—1221no—
506C2×C4—1222no—
507C2×C4—1222no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—4884no—
511C2×C4—2444no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4#1451974712yes—
520C2×C4C2×D4×D6111yescharacteristic
521C2×C4C22×D24411yescharacteristic
522C2×C4C2×#1175822412yes—
523C2×C4#1451982412yes—
524C8—42no—
525C8—2442no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22#1683990214yes—
533C22C2×#1451974711yescharacteristic
534C22#1684026811yescharacteristic
535C22#1684033411yescharacteristic
536C22C2×SD16×D6111yescharacteristic
537C22#137×D6111yescharacteristic
538C22C2×#1451980911yescharacteristic
539C22#1684055811yescharacteristic
540C4—22no—
541C4—42no—
542C4—44no—
543C4—42no—
544C4—1222no—
545C4—1222no—
546C4—1222no—
547C4—4882no—
548C4—2442no—
549C4—2441no—
550C4—2442no—
551C4#1684041212yes—
552C4#1684041612yes—
553C2—24no—
554C2#2144915311yescharacteristic
555C2C2×#1683990211yescharacteristic
556C2#2145508011yescharacteristic
557C2#2145582211yescharacteristic
558C2#2145594411yescharacteristic
559C2#2145696111yescharacteristic
560C2#2145702411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1872C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1872D61
hypocenterC61#24063
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
5. #21447151C2semidirect
6. #21447064C2semidirect
8. #21446089C2semidirect
9. #21446088C2semidirect
327. C61#24063semidirect

Maximal subgroups

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

subgroupindex
C2×#168403142
C2×#168400932
C2×#168400902
C2×#168396612
#214471512
#214470642
#214461262
#214460892
#214460882
#214458852
C61×#18722
#2406361