Finite Group Databasebeta

← #29713300

Subgroups — #29713300

Order 15616.

1352 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840312C211yesmaximal, characteristic
2C2×#16840087C212yesmaximal, complemented
3#21448302C211yesmaximal, characteristic, complemented
4#21447153C211yesmaximal, characteristic, complemented
5#21447148C211yesmaximal, characteristic, complemented
6#21447054C211yesmaximal, characteristic, complemented
7#21447054C211yesmaximal, characteristic, complemented
8#21446139C212yesmaximal, complemented
9#21446083C212yesmaximal, complemented
10#21446069C212yesmaximal, complemented
11C61×#1817C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13#16840312—28no—
14C2×#14519777C2211yescharacteristic
15C2×#14519774C2211yescharacteristic
16C2×#14519765—21no—
17C2×#14519751—21no—
18C2×#14519751C2212yes—
19C2×C4×D244—21no—
20C2×C4×D244C2212yes—
21C2×#14519736C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23#16840087—28no—
24C2×#14519711C2212yes—
25C2×#14519709C2212yes—
26C2×#14519699C2212yes—
27C2×#14519699C2212yes—
28C2×#14519699C2212yes—
29C2×#14519699C2212yes—
30#16839660C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839635—21no—
35#16839635—21no—
36C4×#11758196—21no—
37C4×#11758196—21no—
38#16839525C2212yes—
39#16839525C2212yes—
40C22×C42×C61C2211yescharacteristic
41C2×C22⋊C8×C61C2212yes—
42C61×#336C2211yescharacteristic
43C61×#336C2211yescharacteristic
44C61×#336C2212yes—
45C2×#11758222—28no—
46C2×#11758222—44no—
47C2×#11758222—44no—
48C22×Dic122—21no—
49C22×D244—21no—
50C22×D244—21no—
51C22×D244C2312yes—
52C22×C22⋊D61—22no—
53C22×C4×D61—22no—
54#14519777—24no—
55#14519774—24no—
56#14519765—28no—
57#14519751—28no—
58#14519751—24no—
59#14519751D418yes—
60C4×D244—28no—
61C4×D244—24no—
62C4×D244—24no—
63#14519736—22no—
64#14519736—22no—
65#14519736—24no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2312yes—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×#11758198—22no—
79C2×#11758198—22no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82#14519711—24no—
83#14519709—24no—
84#14519699—24no—
85#14519699—24no—
86#14519699—24no—
87#14519699—24no—
88#14519684—21no—
89#14519684—21no—
90#14519684—21no—
91#14519684—21no—
92C23×C4×C61D411yescharacteristic
93C23×C4×C61D411yescharacteristic
94C23×C4×C61C2311yescharacteristic
95C22×C8×C61C2312yes—
96C22×C8×C61C2312yes—
97C2×C42×C61—24no—
98C2×C42×C61D411yescharacteristic
99C2×C42×C61D411yescharacteristic
100C2×C42×C61C2311yescharacteristic
101C2×C42×C61C2311yescharacteristic
102C22⋊C8×C61—24no—
103C24×C61C2×D411yescharacteristic
104C23×D61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—41no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D414yes—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C2×C22⋊D61—28no—
122C2×C22⋊D61—44no—
123C2×C22⋊D61—44no—
124C22×Dic61—22no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129#11758222—432no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—24no—
135C2×D244—24no—
136C2×D244—22no—
137C2×D244—42no—
138C2×D244—42no—
139C2×D244—42no—
140C2×D244—42no—
141C2×D244C2×D414yes—
142C2×C4×D61—28no—
143C2×C4×D61—44no—
144C2×C4×D61—44no—
145C2×Dic122—24no—
146C2×Dic122—42no—
147C2×Dic122—42no—
148C2×C8×C61—22no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C42×C61—22no—
155C42×C61—22no—
156C42×C61—22no—
157C42×C61—22no—
158C22⋊Dic61—24no—
159C22⋊Dic61—42no—
160#11758198—28no—
161#11758198—28no—
162C4⋊Dic61—24no—
163C4⋊Dic61—22no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61C2×D414yes—
169C4⋊Dic61#4112yescomplemented
170C4⋊Dic61#4112yescomplemented
171#11758196—42no—
172#11758196—42no—
173#11758196—42no—
174#11758196—42no—
175C4×Dic61—24no—
176C4×Dic61—42no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—21no—
183C23×C61#12012yes—
184C23×C61C22×D411yescharacteristic
185C22×D61—24no—
186C22×D61—42no—
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—42no—
192C2×C4×C61—28no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—24no—
202C2×C4×C61—24no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—44no—
207C2×C4×C61#12012yes—
208C2×C4×C61#13512yes—
209C2×C4×C61AGL(1,ℤ/8)12yes—
210C2×C4×C61AGL(1,ℤ/8)12yes—
211C2×C4×C61C22×D412yes—
212C2×C4×C61C2×#4111yescharacteristic
213C2×C4×C61C2×#4111yescharacteristic
214C2×C4×C612−1+412yes—
215C22⋊D61—416no—
216C2×Dic61—24no—
217C2×Dic61—22no—
218C2×Dic61—22no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231D244—24no—
232D244—24no—
233D244—48no—
234D244—48no—
235D244—42no—
236D244—42no—
237C4×D61—416no—
238Dic122—48no—
239C8×C61—42no—
240C8×C61—42no—
241#24062—611nomaximal, Sylow
242C22×C61—22no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—41no—
246C22×C61—41no—
247C22×C61—41no—
248C22×C61—41no—
249C22×C61C2×#12011yescharacteristic
250C22×C61#52511yescharacteristic
251C22×C61#53611yescharacteristic
252C22×C61C2×#13511yescharacteristic
253C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
254C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C61#57511yescharacteristic
256C2×D61—44no—
257C2×D61—44no—
258C2×D61—44no—
259C2×D61—82no—
260C2×D61—82no—
261C2×D61—82no—
262C2×D61—82no—
263C4×C61—22no—
264C4×C61—22no—
265C4×C61—22no—
266C4×C61—22no—
267C4×C61—41no—
268C4×C61#54712yes—
269C4×C61#54712yes—
270Dic61—42no—
271Dic61—42no—
272Dic61—44no—
273Dic61—81no—
274Dic61—81no—
275C2×#532—611no—
276C2×#482—612no—
277#2296—611no—
278#2058—611no—
279#2057—611no—
280#2041—611no—
281#2041—611no—
282#1875—612no—
283#1864—612no—
284#1862—612no—
285#1817D6111yescharacteristic, complemented
286C2×C61—42no—
287C2×C61#240611yescharacteristic
288C2×C61#300211yescharacteristic
289C2×C61#300311yescharacteristic
290C2×C61#310611yescharacteristic
291C2×C61#311511yescharacteristic
292C2×C61#322811yescharacteristic
293C2×C61#323511yescharacteristic
294D61—84no—
295C22×#41—1221no—
296#532—1228no—
297C2×C4⋊Q8—611no—
298C2×#127—611no—
299C2×#124—1221no—
300C2×#121—612no—
301C2×#121—1221no—
302C2×C4×D4—612no—
303C2×C4×D4—1221no—
304C2×#117—611no—
305C2×#117—611no—
306C22×C42C2×D6111yescharacteristic
307#482—1228no—
308C2×#107—612no—
309C2×#106—612no—
310C2×#102—612no—
311C2×#102—612no—
312C2×#102—612no—
313C2×#102—612no—
314C2×C22⋊C8C2×D6112yes—
315#389—611no—
316#389—611no—
317#389—611no—
318#389—611no—
319#385—1221no—
320#385—1221no—
321C4×C4⋊C4—1221no—
322C4×C4⋊C4—1221no—
323#340—612no—
324#340—612no—
325#336C2×D6111yescharacteristic
326#336C2×D6111yescharacteristic
327#336C2×D6112yes—
328C61#2406211yescharacteristic, complemented, Sylow
329C2×#41—1228no—
330C2×#41—2444no—
331C2×#41—2444no—
332C22×Q8—1221no—
333C22×D4—612no—
334C22×D4—1222no—
335C22×D4—1221no—
336C22×D4—1221no—
337C23×C4—1222no—
338C23×C4D24411yescharacteristic
339C23×C4D24411yescharacteristic
340C23×C4C22×D6111yescharacteristic
341C22×C8C22×D6112yes—
342C22×C8C22×D6112yes—
343C4⋊Q8—1224no—
344#127—1224no—
345#124—1228no—
346#121—618no—
347#121—1228no—
348#121—1224no—
349C4×D4—1224no—
350C4×D4—1224no—
351C4×D4—1228no—
352#117—1222no—
353#117—1222no—
354#117—1224no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—611no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—611no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4—1221no—
362C2×C4⋊C4—1221no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4—1221no—
365C2×C22⋊C4—1221no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1221no—
369C2×C42—24no—
370C2×C42—1221no—
371C2×C42—1221no—
372C2×C42D24411yescharacteristic
373C2×C42D24411yescharacteristic
374C2×C42C22×D6111yescharacteristic
375C2×C42C22×D6111yescharacteristic
376#107—1224no—
377#106—1224no—
378#102—1224no—
379#102—1224no—
380#102—1224no—
381#102—1224no—
382C22⋊C8—24no—
383#95—1221no—
384#95—1221no—
385#95—1221no—
386#95—1221no—
387C24—1222no—
388C24C2×D24411yescharacteristic
389#41—24432no—
390C2×Q8—1224no—
391C2×Q8—2442no—
392C2×Q8—2442no—
393C2×D4—614no—
394C2×D4—1224no—
395C2×D4—1224no—
396C2×D4—1228no—
397C2×D4—1222no—
398C2×D4—1222no—
399C2×D4—1222no—
400C2×D4—1222no—
401C2×D4—1222no—
402C2×D4—2442no—
403C2×D4—2442no—
404C2×D4—2444no—
405C2×D4—2444no—
406C2×D4—2442no—
407C2×D4—2442no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4—1228no—
418C22×C4—1221no—
419C22×C4—1221no—
420C22×C4—1222no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—2444no—
424C22×C4—2444no—
425C22×C4C2×D24411yescharacteristic
426C22×C4C2×D24411yescharacteristic
427C22×C4C2×D24411yescharacteristic
428C22×C4C2×D24411yescharacteristic
429C22×C4C2×D24411yescharacteristic
430C22×C4D4×D6114yes—
431C22×C4C23×D6111yescharacteristic
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C4⋊C4—614no—
439C4⋊C4—612no—
440C4⋊C4—612no—
441C4⋊C4—1224no—
442C4⋊C4—1222no—
443C4⋊C4—1221no—
444C4⋊C4—1221no—
445C4⋊C4—1222no—
446C4⋊C4—1222no—
447C4⋊C4—2442no—
448C4⋊C4—2442no—
449C4⋊C4—2442no—
450C4⋊C4—2442no—
451C22⋊C4—1228no—
452C22⋊C4—1224no—
453C22⋊C4—1228no—
454C22⋊C4—2442no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—1224no—
460C42—2442no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—1224no—
468C23—2442no—
469C23—2442no—
470C23—2442no—
471C23—2442no—
472C23—2442no—
473C23—2442no—
474C23#1451974712yes—
475C23C22×D24411yescharacteristic
476Q8—2448no—
477D4—1224no—
478D4—1224no—
479D4—2448no—
480D4—2448no—
481D4—24416no—
482D4—2442no—
483D4—2442no—
484C2×C4—28no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—44no—
499C2×C4—1224no—
500C2×C4—1222no—
501C2×C4—1222no—
502C2×C4—24416no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2441no—
515C2×C4#1451974712yes—
516C2×C4#1451979912yes—
517C2×C4#1451980512yes—
518C2×C4#1451980512yes—
519C2×C4C2×D4×D6112yes—
520C2×C4#41×D6111yescharacteristic
521C2×C4#41×D6111yescharacteristic
522C2×C4#1451982412yes—
523C8—42no—
524C8—42no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—4882no—
533C22—4882no—
534C22—4882no—
535C22—4882no—
536C22—2444no—
537C22—2444no—
538C22—2444no—
539C22C2×#1451974711yescharacteristic
540C22#1684027411yescharacteristic
541C22#1684033811yescharacteristic
542C22C2×#1451979911yescharacteristic
543C22C2×#1451980511yescharacteristic
544C22C2×#1451980511yescharacteristic
545C22#1684056211yescharacteristic
546C4—22no—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—41no—
551C4—4881no—
552C4—4881no—
553C4—2442no—
554C4—2442no—
555C4—2444no—
556C4#1684041012yes—
557C4#1684041012yes—
558C2—42no—
559C2—4884no—
560C2#2144915611yescharacteristic
561C2#2145504111yescharacteristic
562C2#2145504911yescharacteristic
563C2#2145588011yescharacteristic
564C2#2145597111yescharacteristic
565C2#2145695711yescharacteristic
566C2#2145701911yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1817C2
socleC23×C61C22×D4
prime radicalC2×#16840312C2
hypercenter#1817D61
hypocenterC61#24062
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#16840087C2semidirect
3. #21448302C2semidirect
4. #21447153C2semidirect
5. #21447148C2semidirect
6. #21447054C2semidirect
7. #21447054C2semidirect
8. #21446139C2semidirect
9. #21446083C2semidirect
10. #21446069C2semidirect
11. C61×#1817C2semidirect
169. C4⋊Dic61#41semidirect
170. C4⋊Dic61#41semidirect
285. #1817D61semidirect
328. C61#24062semidirect

Maximal subgroups

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

subgroupindex
C2×#168403122
C2×#168400872
#214483022
#214471532
#214471482
#214470542
#214461392
#214460832
#214460692
C61×#18172
#2406261