Finite Group Databasebeta

← #29689966

Subgroups — #29689966

Order 15616.

1130 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840316C211yesmaximal, characteristic
2C2×#16839784C211yesmaximal, characteristic
3#21448224C211yesmaximal, characteristic
4#21446769C212yesmaximal, complemented
5#21446740C212yesmaximal, complemented
6#21446251C212yesmaximal
7#21446249C212yesmaximal
8#21445825C211yesmaximal, characteristic, complemented
9#21445824C211yesmaximal, characteristic, complemented
10#21445821C211yesmaximal, characteristic, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13#16840316—24no—
14#16840316C418yescomplemented
15C2×#14519776C2211yescharacteristic
16C2×#14519775C2211yescharacteristic
17C2×#14519767—21no—
18C2×#14519757C2211yescharacteristic
19C2×#14519755—21no—
20C2×#14519753C2211yescharacteristic
21C2×C4×Dic122—21no—
22C2×C4×Dic122C2211yescharacteristic
23C2×#14519735—21no—
24C2×#14519734—21no—
25C2×C42×D61C2211yescharacteristic
26#16839784—24no—
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519706C2212yes—
29C2×#14519702C2212yes—
30C2×#14519701C2212yes—
31C2×#14519700C2212yes—
32#16839632C2211yescharacteristic
33#16839627—21no—
34#16839619—21no—
35#16839619C2212yes—
36C4⋊C4×Dic61—21no—
37C4⋊C4×Dic61C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C61×#384C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C2×#11758189—24no—
52C2×#11758189—42no—
53C2×#11758189—42no—
54C22×#92418C2312yes—
55#14519776—22no—
56#14519776C2×C414yes—
57#14519775—22no—
58#14519775C2×C414yes—
59#14519767—28no—
60#14519757—22no—
61#14519757C2×C414yes—
62#14519755—28no—
63#14519753—22no—
64#14519753C2×C414yes—
65C4×Dic122—22no—
66C4×Dic122—28no—
67C4×Dic122C2×C414yes—
68#14519735—44no—
69#14519734—28no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2312yes—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—22no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—22no—
85C2×C4×Dic61D411yescharacteristic
86C2×C4×Dic61D411yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88C42×D61C2×C414yes—
89C42×D61D414yes—
90C4⋊#92418—22no—
91#14519706—24no—
92#14519702—42no—
93#14519701—42no—
94#14519700—42no—
95#14519684—22no—
96#14519684—22no—
97C22×Q8×C61C2311yescharacteristic
98C22×C8×C61C2312yes—
99C4×Q8×C61—22no—
100C4×Q8×C61C2×C414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2312yes—
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C4⋊C8×C61—22no—
109#103×C61—42no—
110C23×D61C2×D411yescharacteristic
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61C22×C412yes—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—22no—
126C22×Dic61C2×D411yescharacteristic
127C22×Dic61C2×D411yescharacteristic
128C22×Dic61C2×D411yescharacteristic
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61C22⋊C414yes—
138C2×C4×D61C22⋊C414yes—
139C2×C4×D61C22×C412yes—
140C2×C4×D61C2×D412yes—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—24no—
147C2×Dic122—42no—
148C2×Dic122—42no—
149C2×Dic122C22×C412yes—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C4⋊C4×C61—24no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—42no—
158C4⋊C4×C61—42no—
159C4⋊C4×C61C22×C412yes—
160C42×C61—24no—
161C42×C61C22×C412yes—
162C42×C61C2×D412yes—
163#11758198—24no—
164#11758198—24no—
165#11758198—44no—
166C4⋊Dic61—21no—
167C4⋊Dic61—24no—
168C4⋊Dic61—22no—
169C4⋊Dic61—22no—
170C4⋊Dic61—42no—
171C4⋊Dic61—42no—
172C4⋊Dic61C22×C412yes—
173#11758196—24no—
174#11758196—24no—
175#11758196—24no—
176#11758196—24no—
177#11758196—24no—
178C4×Dic61—24no—
179C4×Dic61—44no—
180C4×Dic61C22⋊C414yes—
181C4×Dic61C22⋊C414yes—
182C4×Dic61C22×C412yes—
183C4×Dic61C2×D412yes—
184C2×#92418—22no—
185C2×#92418—22no—
186C2×#92418—22no—
187#11758189—48no—
188C23×C61C22×D411yescharacteristic
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—21no—
195C22×D61C2×C22⋊C412yes—
196Q8×C61—22no—
197Q8×C61—22no—
198Q8×C61—41no—
199Q8×C61—41no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61C2×C22⋊C412yes—
212C2×C4×C61C2×C22⋊C412yes—
213C2×C4×C61#13712yes—
214C2×C4×C61#13712yes—
215C2×C4×C61C23×C411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61H(5,𝔽2)11yescharacteristic
218C2×C4×C61H(5,𝔽2)11yescharacteristic
219C2×C4×C612−1+411yescharacteristic
220C2×C4×C612−1+411yescharacteristic
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61C2×C22⋊C412yes—
235C2×Dic61C2×C22⋊C412yes—
236C2×Dic61C2×C22⋊C412yes—
237C2×Dic61AGL(1,ℤ/8)12yes—
238C2×Dic61#13712yes—
239C4×D61—22no—
240C4×D61—22no—
241C4×D61—22no—
242C4×D61—22no—
243C4×D61—28no—
244C4×D61—28no—
245Dic122—22no—
246Dic122—22no—
247Dic122—41no—
248Dic122—41no—
249Dic122—48no—
250C8×C61—42no—
251#92418—42no—
252#21894—611nomaximal, Sylow
253C22×C61C22×C22⋊C411yescharacteristic
254C22×C61#53711yescharacteristic
255C22×C61#53711yescharacteristic
256C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
257C22×C61C2×#13711yescharacteristic
258C22×C61C2×#13711yescharacteristic
259C22×C61C2×#13711yescharacteristic
260C2×D61—22no—
261C2×D61—22no—
262C2×D61—22no—
263C2×D61—41no—
264C2×D61—41no—
265C2×D61—41no—
266C2×D61—41no—
267C4×C61—22no—
268C4×C61—42no—
269C4×C61—42no—
270C4×C61—82no—
271C4×C61#41914yes—
272C4×C61#52011yescharacteristic
273C4×C61#52011yescharacteristic
274C4×C61#53511yescharacteristic
275C4×C61#53611yescharacteristic
276Dic61—22no—
277Dic61—22no—
278Dic61—42no—
279Dic61—42no—
280Dic61—82no—
281Dic61#41714yes—
282C2×#533—611no—
283C2×#423—611no—
284#2285—611no—
285#1988—612no—
286#1981—612no—
287#1894—612no—
288#1894—612no—
289#1803—611no—
290#1803—611no—
291#1803—611no—
292#1803D6111yescharacteristic, complemented
293C2×C61#231111yescharacteristic
294C2×C61C2×#41911yescharacteristic
295C2×C61C2×#41711yescharacteristic
296C2×C61#325411yescharacteristic
297C2×C61#325411yescharacteristic
298C2×C61#325511yescharacteristic
299C2×C61#325511yescharacteristic
300D61—42no—
301C22×M16—612no—
302#533—618no—
303#533—1224no—
304C2×C4⋊Q8—611no—
305C2×C4⋊Q8—611no—
306C2×#125—1221no—
307C2×C22⋊Q8—611no—
308C2×C22⋊Q8—611no—
309C2×C22⋊Q8—1221no—
310C2×C4×Q8—611no—
311C2×C4×Q8—1221no—
312C2×C4×Q8C2×D6111yescharacteristic
313C2×#117—1221no—
314C2×#117—1221no—
315C22×C42—611no—
316#423—1224no—
317C2×C4⋊C8—612no—
318C2×C4⋊C8—611no—
319C2×C4⋊C8C2×D6111yescharacteristic
320C2×#103—612no—
321C2×#103—612no—
322C2×#103—612no—
323C2×#103C2×D6112yes—
324#384—611no—
325#384C2×D6111yescharacteristic
326#382—1221no—
327#380—612no—
328#380—1221no—
329C4×C4⋊C4—612no—
330C4×C4⋊C4—1221no—
331#340—612no—
332#340—612no—
333#340—612no—
334#340C2×D6112yes—
335C61#2189411yescharacteristic, complemented, Sylow
336C22×Q8—611no—
337C22×Q8—1221no—
338C22×Q8C22×D6111yescharacteristic
339C23×C4—611no—
340C23×C4—611no—
341C23×C4—611no—
342C2×M16—1224no—
343C2×M16—2442no—
344C2×M16—2442no—
345C22×C8—612no—
346C22×C8C22×D6112yes—
347C4⋊Q8—614no—
348C4⋊Q8—614no—
349C4⋊Q8—1222no—
350C4⋊Q8—1222no—
351#125—1228no—
352C22⋊Q8—614no—
353C22⋊Q8—614no—
354C22⋊Q8—1222no—
355C22⋊Q8—1228no—
356C22⋊Q8—1222no—
357C4×Q8—22no—
358C4×Q8—614no—
359C4×Q8—1228no—
360C4×Q8—1222no—
361C4×Q8C4×D6114yes—
362#117—1228no—
363#117—2444no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4C22×D6112yes—
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C22⋊C4—1221no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1222no—
382C2×C42—21no—
383C2×C42—611no—
384C2×C42—611no—
385C2×C42—611no—
386C2×C42—614no—
387C2×C42—614no—
388C2×C42—1221no—
389C2×C42—1222no—
390C2×C42C22×D6111yescharacteristic
391C4⋊C8—22no—
392C4⋊C8—1224no—
393C4⋊C8—1222no—
394#103—42no—
395#103—2442no—
396#103—2442no—
397#103—2442no—
398#95—1222no—
399#95—1222no—
400C24—611no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—612no—
407C2×Q8—1224no—
408C2×Q8—1221no—
409C2×Q8—1221no—
410C2×Q8—1221no—
411C2×Q8—1221no—
412C2×Q8—1221no—
413C2×Q8—2442no—
414C2×Q8—2442no—
415C2×Q8C2×C4×D6112yes—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—612no—
423C22×C4—612no—
424C22×C4—614no—
425C22×C4—614no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1222no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1222no—
437C22×C4D4×D6111yescharacteristic
438C22×C4D4×D6111yescharacteristic
439C22×C4C23×D6111yescharacteristic
440M16—2448no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—1222no—
445C2×C8—1222no—
446C2×C8—1222no—
447C4⋊C4—21no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—612no—
454C4⋊C4—1224no—
455C4⋊C4—1224no—
456C4⋊C4—1224no—
457C4⋊C4—1224no—
458C4⋊C4—1224no—
459C4⋊C4—1221no—
460C4⋊C4—1224no—
461C4⋊C4—1222no—
462C4⋊C4—1222no—
463C4⋊C4—2442no—
464C4⋊C4—2442no—
465C4⋊C4C2×C4×D6112yes—
466C22⋊C4—1224no—
467C22⋊C4—1224no—
468C22⋊C4—2444no—
469C42—24no—
470C42—612no—
471C42—612no—
472C42—614no—
473C42—614no—
474C42—1224no—
475C42—2444no—
476C42C2×C4×D6112yes—
477C42D4×D6112yes—
478C23—612no—
479C23—1221no—
480C23—1221no—
481C23—1221no—
482C23—1221no—
483C23—1221no—
484C23—1221no—
485C23C2×D4×D6111yescharacteristic
486Q8—22no—
487Q8—22no—
488Q8—41no—
489Q8—41no—
490Q8—1222no—
491Q8—1222no—
492Q8—2448no—
493Q8—2441no—
494Q8—2441no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—612no—
507C2×C4—612no—
508C2×C4—612no—
509C2×C4—612no—
510C2×C4—612no—
511C2×C4—1228no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1228no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4C22⋊C4×D6112yes—
531C2×C4C22⋊C4×D6112yes—
532C2×C4#1451980612yes—
533C2×C4#1451980912yes—
534C2×C4C22×C4×D6111yescharacteristic
535C2×C4C2×D4×D6111yescharacteristic
536C2×C4#1451982211yescharacteristic
537C2×C4#1451982311yescharacteristic
538C2×C4#1451982411yescharacteristic
539C2×C4#1451982511yescharacteristic
540C8—42no—
541C8—2442no—
542C22—1222no—
543C22—1222no—
544C22—1222no—
545C22—2441no—
546C22—2441no—
547C22—2441no—
548C22—2441no—
549C22C2×C22⋊C4×D6111yescharacteristic
550C22#1684034111yescharacteristic
551C22#1684034211yescharacteristic
552C22AGL(1,ℤ/8)×D6111yescharacteristic
553C22#137×D6111yescharacteristic
554C22C2×#1451980611yescharacteristic
555C22C2×#1451980911yescharacteristic
556C4—22no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—614no—
561C4—1222no—
562C4—1222no—
563C4—4882no—
564C4—2442no—
565C4—2442no—
566C4#1683976914yes—
567C4#1684024211yescharacteristic
568C4#1684024311yescharacteristic
569C4#1684032811yescharacteristic
570C4#1684033511yescharacteristic
571C2—2442no—
572C2#2144840811yescharacteristic
573C2C2×#1683976911yescharacteristic
574C2#417×D6111yescharacteristic
575C2#2145720411yescharacteristic
576C2#2145720511yescharacteristic
577C2#2145721111yescharacteristic
578C2#2145721211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1803D61
hypocenterC61#21894
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
4. #21446769C2semidirect
5. #21446740C2semidirect
8. #21445825C2semidirect
9. #21445824C2semidirect
10. #21445821C2semidirect
11. C61×#1803C2semidirect
14. #16840316C4semidirect
292. #1803D61semidirect
335. C61#21894semidirect

Maximal subgroups

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

subgroupindex
C2×#168403162
C2×#168397842
#214482242
#214467692
#214467402
#214462512
#214462492
#214458252
#214458242
#214458212
C61×#18032
#2189461