Finite Group Databasebeta

← #29685212

Subgroups — #29685212

Order 15616.

1218 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×#16840297C211yesmaximal, characteristic
2C2×#16839714C211yesmaximal, characteristic, complemented
3#21448224C211yesmaximal, characteristic, complemented
4#21446789C211yesmaximal, characteristic, complemented
5#21446635C211yesmaximal, characteristic, complemented
6#21446633C211yesmaximal, characteristic, complemented
7#21446256C212yesmaximal
8#21446251C212yesmaximal, complemented
9#21445638C212yesmaximal, complemented
10#21445633C212yesmaximal, complemented
11C61×#1992C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13#16840297—24no—
14#16840297C418yescomplemented
15C2×#14519772—21no—
16C2×#14519765C2211yescharacteristic
17C2×#14519762—21no—
18C2×#14519757C2211yescharacteristic
19C2×#14519752C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×#14519739—21no—
22C2×#14519739C2211yescharacteristic
23C2×#14519735—21no—
24C2×#14519735—21no—
25C2×C42×D61C2211yescharacteristic
26C2×#14519702C2212yes—
27C2×#14519700C2212yes—
28C2×#14519699C2212yes—
29C2×#14519697C2212yes—
30#16839714—24no—
31C2×#14519688C2211yescharacteristic
32C2×C8×Dic61C2212yes—
33#16839632C2211yescharacteristic
34#16839627—21no—
35#16839619—21no—
36#16839619C2212yes—
37C4⋊C4×Dic61—21no—
38C4⋊C4×Dic61C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#125×C61C2211yescharacteristic
42C2×#97×C61C2211yescharacteristic
43C61×#384C2211yescharacteristic
44C61×#340C2212yes—
45C61×#340C2212yes—
46C22×Dic122C2311yescharacteristic
47C22×D244—21no—
48C22×D244C2311yescharacteristic
49C22×C4×D61—21no—
50C22×C4×D61D411yescharacteristic
51C22×C4×D61D411yescharacteristic
52C22×C4×D61C2311yescharacteristic
53C2×#11758189—24no—
54C2×#11758189—24no—
55C2×#11758189—24no—
56C22×#92418C2312yes—
57#14519772—28no—
58#14519765—22no—
59#14519765C2×C414yes—
60#14519762—28no—
61#14519757—22no—
62#14519757C2×C414yes—
63#14519752—22no—
64#14519752C2×C414yes—
65#14519742—22no—
66#14519742C2×C414yes—
67#14519739—28no—
68#14519739—22no—
69#14519739C2×C414yes—
70#14519735—28no—
71#14519735—44no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61C2312yes—
76C2×#11758196—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—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—
90#14519702—24no—
91#14519700—24no—
92#14519699—24no—
93#14519697—24no—
94#14519688—22no—
95C8×Dic61—24no—
96#14519684—22no—
97#14519684—22no—
98C22×C8×C61C2312yes—
99#125×C61—22no—
100#125×C61C2×C414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61C2312yes—
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C42×C61C2311yescharacteristic
109#97×C61—22no—
110C23×D61—21no—
111C23×D61C2×D411yescharacteristic
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2411yescharacteristic
118C22×Dic61—21no—
119C22×Dic61—22no—
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C22×Dic61C2×D411yescharacteristic
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—24no—
126C2×D244—21no—
127C2×D244—42no—
128C2×D244—42no—
129C2×D244C22×C412yes—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—24no—
135C2×C4×D61—41no—
136C2×C4×D61—41no—
137C2×C4×D61—42no—
138C2×C4×D61—42no—
139C2×C4×D61C22⋊C414yes—
140C2×C4×D61C22⋊C414yes—
141C2×C4×D61C22×C412yes—
142C2×C4×D61C2×D412yes—
143C2×Dic122—22no—
144C2×Dic122—22no—
145C2×Dic122—21no—
146C2×Dic122C22×C412yes—
147C2×C8×C61—22no—
148C2×C8×C61—22no—
149C2×C8×C61#4114yes—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—42no—
157C4⋊C4×C61—42no—
158C4⋊C4×C61C22×C412yes—
159C4⋊C4×C61C22×C412yes—
160C42×C61C22×C412yes—
161C42×C61C2×D412yes—
162#11758198—24no—
163#11758198—24no—
164#11758198—24no—
165#11758198—24no—
166#11758198—24no—
167#11758198—44no—
168C4⋊Dic61—24no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—42no—
172C4⋊Dic61—42no—
173#11758196—24no—
174C4×Dic61—24no—
175C4×Dic61—44no—
176C4×Dic61C22⋊C414yes—
177C4×Dic61C22⋊C414yes—
178C4×Dic61C22×C412yes—
179C4×Dic61C2×D412yes—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418#4114yes—
183#11758189—216no—
184C23×C61C22×D411yescharacteristic
185C22×D61—21no—
186C22×D61—21no—
187C22×D61—21no—
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—22no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61C2×C22⋊C412yes—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61C2×C22⋊C412yes—
213C2×C4×C61C2×C22⋊C412yes—
214C2×C4×C61C23×C411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C2×#4112yes—
217C2×C4×C61C2×#4112yes—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—22no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61C2×C22⋊C412yes—
229C2×Dic61C2×C22⋊C412yes—
230C2×Dic61C2×C22⋊C412yes—
231C2×Dic61AGL(1,ℤ/8)12yes—
232C2×Dic61#13712yes—
233D244—24no—
234D244—48no—
235D244—42no—
236C4×D61—24no—
237C4×D61—24no—
238C4×D61—28no—
239C4×D61—44no—
240C4×D61—48no—
241Dic122—24no—
242Dic122—42no—
243C8×C61—24no—
244#92418—24no—
245#21403—611nomaximal, Sylow
246C22×C61C22×C22⋊C411yescharacteristic
247C22×C61#53211yescharacteristic
248C22×C61#53211yescharacteristic
249C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
250C22×C61C2×#13711yescharacteristic
251C22×C61#57511yescharacteristic
252C22×C61#57511yescharacteristic
253C2×D61—22no—
254C2×D61—22no—
255C2×D61—22no—
256C2×D61—41no—
257C2×D61—41no—
258C2×D61—41no—
259C2×D61—41no—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—42no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C2×D61—81no—
267C4×C61—22no—
268C4×C61—22no—
269C4×C61—42no—
270C4×C61—42no—
271C4×C61—82no—
272C4×C61C4×#4112yes—
273C4×C61#52512yes—
274Dic61—22no—
275Dic61—22no—
276Dic61—42no—
277Dic61—82no—
278Dic61#41714yes—
279C2×#529—611no—
280C2×C4×M16—611no—
281#2285—611no—
282#1992—611no—
283#1992D6111yescharacteristic, complemented
284#1959—611no—
285#1959—611no—
286#1895—612no—
287#1894—612no—
288C4×#103—612no—
289C4×#102—612no—
290C2×C61#229711yescharacteristic
291C2×C61C2×#41711yescharacteristic
292C2×C61#289611yescharacteristic
293C2×C61#317111yescharacteristic
294C2×C61#317111yescharacteristic
295C2×C61#317211yescharacteristic
296C2×C61#317211yescharacteristic
297D61—42no—
298D61—82no—
299C22×M16—612no—
300#529—618no—
301#529—1224no—
302C2×#126—1221no—
303C2×#125C2×D6111yescharacteristic
304C2×#124—611no—
305C2×#123—1221no—
306C2×C22⋊Q8—611no—
307C2×#121—611no—
308C2×C4×Q8—611no—
309C2×C4×D4—611no—
310C2×C4×D4—1221no—
311C2×#117—1221no—
312C2×#117—1221no—
313C22×C42—611no—
314C2×#103—612no—
315C2×#103—612no—
316C2×#102—612no—
317C2×#102—612no—
318C4×M16—1224no—
319C2×#97—611no—
320C2×#97C2×D6111yescharacteristic
321C2×C4×C8—612no—
322#384—611no—
323#384C2×D6111yescharacteristic
324#382—1221no—
325#380—612no—
326#380—1221no—
327C4×C4⋊C4—612no—
328C4×C4⋊C4—1221no—
329#340—612no—
330#340—612no—
331#340C2×D6112yes—
332#340C2×D6112yes—
333C61#2140311yescharacteristic, complemented, Sylow
334C22×Q8—611no—
335C22×D4—611no—
336C22×D4—1221no—
337C23×C4—611no—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—1221no—
341C2×M16—1224no—
342C2×M16—1224no—
343C2×M16—1224no—
344C22×C8—612no—
345C22×C8C22×D6112yes—
346#126—1228no—
347#125—22no—
348#125C4×D6114yes—
349#124—614no—
350#124—1222no—
351#123—1228no—
352C22⋊Q8—614no—
353C22⋊Q8—1222no—
354#121—614no—
355#121—1222no—
356C4×Q8—614no—
357C4×Q8—1222no—
358C4×D4—614no—
359C4×D4—1222no—
360C4×D4—1228no—
361#117—1228no—
362#117—2444no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4C22×D6112yes—
373C2×C4⋊C4C22×D6111yescharacteristic
374C2×C4⋊C4C22×D6111yescharacteristic
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4—1221no—
378C2×C22⋊C4—1221no—
379C2×C22⋊C4—1221no—
380C2×C22⋊C4—1222no—
381C2×C42—611no—
382C2×C42—611no—
383C2×C42—611no—
384C2×C42—614no—
385C2×C42—614no—
386C2×C42—1221no—
387C2×C42—1222no—
388C2×C42C22×D6111yescharacteristic
389#103—1224no—
390#103—1224no—
391#102—1224no—
392#102—1224no—
393#97—22no—
394#97—1222no—
395C4×C8—1224no—
396#95—1222no—
397#95—1222no—
398C24—611no—
399C24—1221no—
400C2×Q8—612no—
401C2×Q8—1221no—
402C2×Q8—1222no—
403C2×Q8—1222no—
404C2×D4—612no—
405C2×D4—1224no—
406C2×D4—1221no—
407C2×D4—1222no—
408C2×D4—1222no—
409C2×D4—2442no—
410C2×D4—2442no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—612no—
418C22×C4—612no—
419C22×C4—614no—
420C22×C4—614no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1224no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1221no—
427C22×C4—1222no—
428C22×C4—2441no—
429C22×C4—2441no—
430C22×C4—2442no—
431C22×C4—2442no—
432C22×C4D4×D6111yescharacteristic
433C22×C4D4×D6111yescharacteristic
434C22×C4C23×D6111yescharacteristic
435M16—12216no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—614no—
439C2×C8—1222no—
440C2×C8—1222no—
441C2×C8#1175822614yes—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—1224no—
451C4⋊C4—1224no—
452C4⋊C4—1222no—
453C4⋊C4—1222no—
454C4⋊C4—2442no—
455C4⋊C4—2442no—
456C4⋊C4C2×C4×D6112yes—
457C4⋊C4C2×C4×D6112yes—
458C22⋊C4—1224no—
459C22⋊C4—1224no—
460C22⋊C4—1224no—
461C22⋊C4—1224no—
462C22⋊C4—1224no—
463C22⋊C4—2444no—
464C42—612no—
465C42—612no—
466C42—614no—
467C42—614no—
468C42—1224no—
469C42—2444no—
470C42C2×C4×D6112yes—
471C42D4×D6112yes—
472C23—612no—
473C23—1221no—
474C23—1221no—
475C23—1221no—
476C23—1221no—
477C23—1221no—
478C23—1221no—
479C23—1222no—
480C23—2441no—
481C23—2441no—
482C23—2441no—
483C23—2441no—
484C23—2441no—
485C23—2441no—
486C23C2×D4×D6111yescharacteristic
487Q8—1224no—
488Q8—2442no—
489D4—1224no—
490D4—2448no—
491D4—2442no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—612no—
506C2×C4—612no—
507C2×C4—612no—
508C2×C4—612no—
509C2×C4—612no—
510C2×C4—1228no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1224no—
516C2×C4—1224no—
517C2×C4—1222no—
518C2×C4—2444no—
519C2×C4—2448no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4C22⋊C4×D6112yes—
526C2×C4C22⋊C4×D6112yes—
527C2×C4C22×C4×D6111yescharacteristic
528C2×C4C2×D4×D6111yescharacteristic
529C2×C4#41×D6112yes—
530C2×C4C2×#1175822612yes—
531C8—24no—
532C8—1224no—
533C22—1222no—
534C22—1222no—
535C22—1222no—
536C22—4881no—
537C22—4881no—
538C22—4881no—
539C22—4881no—
540C22—2442no—
541C22—2442no—
542C22—2442no—
543C22—2441no—
544C22—2441no—
545C22—2441no—
546C22—2441no—
547C22C2×C22⋊C4×D6111yescharacteristic
548C22#1684031111yescharacteristic
549C22#1684031311yescharacteristic
550C22AGL(1,ℤ/8)×D6111yescharacteristic
551C22#137×D6111yescharacteristic
552C22#1684055811yescharacteristic
553C22#1684056111yescharacteristic
554C4—22no—
555C4—22no—
556C4—42no—
557C4—42no—
558C4—82no—
559C4—614no—
560C4—1222no—
561C4—1222no—
562C4—4882no—
563C4—2442no—
564C4C4×#1175822612yes—
565C4#1684027512yes—
566C2—4882no—
567C2—2442no—
568C2#2144830811yescharacteristic
569C2#417×D6111yescharacteristic
570C2#2145433511yescharacteristic
571C2#2145641311yescharacteristic
572C2#2145641611yescharacteristic
573C2#2145642411yescharacteristic
574C2#2145642711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1992C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
cosocleC22×C4×C61C24
hypercenter#1992D61
hypocenterC61#21403
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
2. C2×#16839714C2semidirect
3. #21448224C2semidirect
4. #21446789C2semidirect
5. #21446635C2semidirect
6. #21446633C2semidirect
8. #21446251C2semidirect
9. #21445638C2semidirect
10. #21445633C2semidirect
11. C61×#1992C2semidirect
14. #16840297C4semidirect
283. #1992D61semidirect
333. C61#21403semidirect

Maximal subgroups

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

subgroupindex
C2×#168402972
C2×#168397142
#214482242
#214467892
#214466352
#214466332
#214462562
#214462512
#214456382
#214456332
C61×#19922
#2140361