Finite Group Databasebeta

← #29689389

Subgroups — #29689389

Order 15616.

1442 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840331C211yesmaximal, characteristic
2#21448428C211yesmaximal, characteristic, complemented
3#21446749C212yesmaximal, complemented
4#21446745C212yesmaximal, complemented
5#21446643C211yesmaximal, characteristic, complemented
6#21446635C211yesmaximal, characteristic, complemented
7#21446630C211yesmaximal, characteristic, complemented
8#21446625C211yesmaximal, characteristic, complemented
9#21445760C212yesmaximal, complemented
10#21445751C212yesmaximal, complemented
11C2×C61×#433C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×D244C2211yescharacteristic
14#16840331—24no—
15#16840331C418yescomplemented
16C2×#14519774C2211yescharacteristic
17C2×#14519774C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×#14519765C2211yescharacteristic
20C2×#14519752—22no—
21C2×#14519747—22no—
22C2×#14519702C2212yescomplemented
23C2×#14519702C2212yescomplemented
24C2×#14519699C2212yes—
25C2×#14519699C2212yes—
26C2×#14519699C2212yes—
27C2×#14519699C2212yes—
28C2×#14519699C2212yescomplemented
29C2×#14519699C2212yescomplemented
30#16839655—22no—
31#16839648—22no—
32#16839632C2211yescharacteristic, complemented
33#16839628C2211yescharacteristic, complemented
34#16839622C2212yes—
35C4×C4⋊Dic61C2212yescomplemented
36C2×#117×C61C2211yescharacteristic
37C61×#433—24no—
38C2×C4⋊C8×C61C2212yescomplemented
39C2×C22⋊C8×C61C2212yes—
40C2×#97×C61C2211yescharacteristic, complemented
41C2×C4×C8×C61C2211yescharacteristic, complemented
42C24×D61—21no—
43C2×#11758222—22no—
44C2×#11758222—42no—
45C2×#11758222—42no—
46C2×#11758222C2×C414yes—
47C22×Dic122C2311yescharacteristic
48C22×D244—21no—
49C22×D244—22no—
50C22×D244—22no—
51C22×D244—22no—
52C22×D244—42no—
53C22×D244—42no—
54C22×D244C2×C414yes—
55C22×D244C2311yescharacteristic
56C22×D244C2311yescharacteristic
57C22×D244C2311yescharacteristic
58C22×C22⋊D61—21no—
59C22×C4×D61—21no—
60#14519774—22no—
61#14519774—22no—
62#14519774C2×C414yes—
63#14519774C2×C414yes—
64#14519765—22no—
65#14519765—22no—
66#14519765C2×C414yes—
67#14519765C2×C414yes—
68#14519752—216no—
69#14519747—216no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2312yes—
73C2×C22⋊Dic61—22no—
74C2×#11758198—22no—
75C2×#11758198—22no—
76C2×C4×Dic61—22no—
77#14519702—42no—
78#14519702—42no—
79#14519699—42no—
80#14519699—42no—
81#14519699—42no—
82#14519699—42no—
83#14519699—42no—
84#14519699—42no—
85#14519684—22no—
86#14519684—22no—
87#14519684—22no—
88C23×C4×C61C2311yescharacteristic
89C22×C8×C61C2312yes—
90C22×C8×C61C2312yes—
91#117×C61C2×C414yes—
92#117×C61D414yes—
93C2×C4⋊C4×C61D412yes—
94C2×C22⋊C4×C61D412yes—
95C2×C42×C61C2311yescharacteristic
96C2×C42×C61C2311yescharacteristic
97C4⋊C8×C61—24no—
98C22⋊C8×C61—24no—
99#97×C61—22no—
100C4×C8×C61—22no—
101C24×C61C2×D411yescharacteristic
102C23×D61—21no—
103C23×D61—21no—
104C23×D61—21no—
105C23×D61—24no—
106C23×D61—42no—
107C23×D61—42no—
108C23×D61—82no—
109C23×D61—82no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61C22×C412yes—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2411yescharacteristic
118C2×C22⋊D61—24no—
119C2×C22⋊D61—42no—
120C2×C22⋊D61—42no—
121C22×Dic61—21no—
122C22×Dic61—22no—
123C22×Dic61—22no—
124#11758222—48no—
125#11758222—48no—
126C2×D244—22no—
127C2×D244—22no—
128C2×D244—22no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—24no—
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—28no—
137C2×D244—28no—
138C2×D244—48no—
139C2×D244—42no—
140C2×D244—42no—
141C2×D244—48no—
142C2×D244—82no—
143C2×D244—82no—
144C2×D244—82no—
145C2×D244—82no—
146C2×D244C22×C412yes—
147C2×D244C22×C412yes—
148C2×D244C22×C412yes—
149C2×C4×D61—24no—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×Dic122—22no—
153C2×Dic122—22no—
154C2×Dic122—21no—
155C2×Dic122C22×C412yes—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C2×C8×C61—22no—
160C2×C8×C61—22no—
161C2×C8×C61—22no—
162C4⋊C4×C61C22⋊C418yes—
163C22⋊C4×C61C22⋊C418yes—
164C42×C61C22×C412yes—
165C42×C61C22×C412yes—
166C42×C61C2×D412yes—
167C42×C61C2×D412yes—
168C22⋊Dic61—82no—
169#11758198—28no—
170#11758198—28no—
171C4⋊Dic61—22no—
172C4⋊Dic61—22no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61—82no—
176C4×Dic61—82no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61C2×C22⋊C412yes—
182C23×C61C22×D411yescharacteristic
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—48no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61—42no—
198C22×D61—42no—
199C22×D61—88no—
200C22×D61—84no—
201C22×D61—84no—
202C22×D61—88no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—24no—
210C2×C4×C61—24no—
211C2×C4×C61C2×C22⋊C414yes—
212C2×C4×C61C2×C22⋊C414yes—
213C2×C4×C61C2×C22⋊C412yes—
214C2×C4×C61C23×C411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61H(5,𝔽2)12yes—
217C2×C4×C61H(5,𝔽2)12yes—
218C22⋊D61—48no—
219C2×Dic61—22no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228D244—24no—
229D244—24no—
230D244—24no—
231D244—48no—
232D244—42no—
233D244—42no—
234D244—42no—
235D244—88no—
236D244—88no—
237C4×D61—48no—
238Dic122—24no—
239Dic122—42no—
240C8×C61—42no—
241C8×C61—42no—
242#21813—611nomaximal, Sylow
243C22×C61—22no—
244C22×C61—24no—
245C22×C61—41no—
246C22×C61—41no—
247C22×C61—42no—
248C22×C61C22×C22⋊C411yescharacteristic
249C22×C61#53511yescharacteristic
250C22×C61#53511yescharacteristic
251C22×C61#57612yes—
252C22×C61#57712yes—
253C2×D61—42no—
254C2×D61—42no—
255C2×D61—42no—
256C2×D61—44no—
257C2×D61—44no—
258C2×D61—44no—
259C2×D61—816no—
260C2×D61—81no—
261C2×D61—81no—
262C2×D61—81no—
263C2×D61—81no—
264C2×D61—81no—
265C2×D61—81no—
266C2×D61—82no—
267C2×D61—82no—
268C2×D61—82no—
269C4×C61—22no—
270C4×C61—24no—
271C4×C61—24no—
272C4×C61#51812yes—
273C4×C61#53512yes—
274Dic61—42no—
275Dic61—82no—
276Dic61—82no—
277C2×#535—611no—
278C2×#433D6111yescharacteristic, complemented
279#2314—611no—
280#1983—612no—
281#1982—612no—
282#1961—611no—
283#1959—611no—
284#1958—611no—
285#1957—611no—
286#1788—612no—
287#1786—612no—
288C2×C61—42no—
289C2×C61#230911yescharacteristic
290C2×C61#289712yes—
291C2×C61#322011yescharacteristic
292C2×C61#322211yescharacteristic
293C2×C61#322412yes—
294D61—82no—
295D61—82no—
296D61—82no—
297C22×#41—611no—
298C23×D4—611no—
299#535—618no—
300#535—1224no—
301C2×#127—611no—
302C2×#127—611no—
303C2×#124—611no—
304C2×#124—611no—
305C2×#121—1222no—
306C2×#120—1222no—
307C2×#117C2×D6111yescharacteristic
308#433—24no—
309C2×C4⋊C8C2×D6112yescomplemented
310C2×#103—612no—
311C2×#103—612no—
312C2×#102—612no—
313C2×#102—612no—
314C2×#102—612no—
315C2×#102—612no—
316C2×#102—612no—
317C2×#102—612no—
318C2×C22⋊C8C2×D6112yes—
319C2×#97C2×D6111yescharacteristic, complemented
320C2×C4×C8C2×D6111yescharacteristic, complemented
321#388—1222no—
322#387—1222no—
323#384—611no—
324#382—611no—
325#380—612no—
326C4×C4⋊C4—612no—
327C61#2181311yescharacteristic, complemented, Sylow
328C25—1221no—
329C2×#41—614no—
330C2×#41—1222no—
331C2×#41—2442no—
332C2×#41—2442no—
333C22×Q8—611no—
334C22×D4—614no—
335C22×D4—611no—
336C22×D4—611no—
337C22×D4—611no—
338C22×D4—1221no—
339C22×D4—1222no—
340C22×D4—1222no—
341C22×D4—1221no—
342C22×D4—1222no—
343C22×D4—2442no—
344C22×D4—2442no—
345C23×C4—1221no—
346C23×C4C22×D6111yescharacteristic
347C22×C8C22×D6112yes—
348C22×C8C22×D6112yes—
349#127—614no—
350#127—614no—
351#127—1222no—
352#127—1222no—
353#124—614no—
354#124—614no—
355#124—1222no—
356#124—1222no—
357#121—12216no—
358#120—12216no—
359#117C4×D6114yes—
360#117C22⋊D6114yes—
361C2×C4⋊C4—612no—
362C2×C4⋊C4—612no—
363C2×C4⋊C4—1222no—
364C2×C4⋊C4D24412yes—
365C2×C22⋊C4—1222no—
366C2×C22⋊C4—1222no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4D24412yes—
369C2×C42—1222no—
370C2×C42C22×D6111yescharacteristic
371C2×C42C22×D6111yescharacteristic
372C4⋊C8—24no—
373#103—2442no—
374#103—2442no—
375#102—2442no—
376#102—2442no—
377#102—2442no—
378#102—2442no—
379#102—2442no—
380#102—2442no—
381C22⋊C8—24no—
382#97—22no—
383C4×C8—22no—
384#95—1222no—
385#95—1222no—
386#95—1222no—
387C24—1224no—
388C24—1221no—
389C24—1221no—
390C24—1221no—
391C24—4882no—
392C24—4882no—
393C24—2442no—
394C24—2442no—
395C24C2×D24411yescharacteristic
396#41—2448no—
397#41—2448no—
398C2×Q8—612no—
399C2×Q8—1221no—
400C2×Q8—1222no—
401C2×Q8—1222no—
402C2×D4—612no—
403C2×D4—612no—
404C2×D4—612no—
405C2×D4—1224no—
406C2×D4—1221no—
407C2×D4—1221no—
408C2×D4—1221no—
409C2×D4—1222no—
410C2×D4—1222no—
411C2×D4—1222no—
412C2×D4—1222no—
413C2×D4—1222no—
414C2×D4—1222no—
415C2×D4—1228no—
416C2×D4—1228no—
417C2×D4—1224no—
418C2×D4—4882no—
419C2×D4—4882no—
420C2×D4—4882no—
421C2×D4—4882no—
422C2×D4—2448no—
423C2×D4—2442no—
424C2×D4—2442no—
425C2×D4—2448no—
426C2×D4—2442no—
427C2×D4—2442no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—1221no—
431C22×C4—1224no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—2442no—
435C22×C4—2442no—
436C22×C4C2×C4×D6112yes—
437C22×C4C2×D24411yescharacteristic
438C22×C4C2×D24412yes—
439C22×C4C2×D24412yes—
440C22×C4C2×C22⋊D6112yes—
441C22×C4C23×D6111yescharacteristic
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C4⋊C4—1222no—
449C4⋊C4—1222no—
450C4⋊C4—1222no—
451C4⋊C4—1222no—
452C4⋊C4—4882no—
453C4⋊C4#1175819818yes—
454C22⋊C4—1228no—
455C22⋊C4—1228no—
456C22⋊C4—4882no—
457C22⋊C4#1175819818yes—
458C42—4882no—
459C42C2×C4×D6112yes—
460C42C2×C4×D6112yes—
461C42C2×C22⋊D6112yes—
462C42C2×C22⋊D6112yes—
463C23—22no—
464C23—22no—
465C23—21no—
466C23—21no—
467C23—1222no—
468C23—1222no—
469C23—1222no—
470C23—4888no—
471C23—4888no—
472C23—4884no—
473C23—4884no—
474C23—2448no—
475C23—2442no—
476C23—2442no—
477C23—2442no—
478C23—2442no—
479C23—2442no—
480C23—2442no—
481C23—2441no—
482C23—2441no—
483C23—2441no—
484C23—2441no—
485C23—2441no—
486C23—2441no—
487C23C2×#1175819812yes—
488C23C22×D24411yescharacteristic
489Q8—1224no—
490Q8—2442no—
491D4—1224no—
492D4—1224no—
493D4—1224no—
494D4—4888no—
495D4—4888no—
496D4—2448no—
497D4—2448no—
498D4—2442no—
499D4—2442no—
500D4—2442no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—1222no—
510C2×C4—2448no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4C2×#1175819814yes—
520C2×C4C2×#1175819814yes—
521C2×C4C2×#1175819812yes—
522C2×C4C22×C4×D6111yescharacteristic
523C2×C4C22×C22⋊D6111yescharacteristic
524C2×C4#1451982312yes—
525C2×C4#1451982312yes—
526C8—42no—
527C8—42no—
528C22—24no—
529C22—22no—
530C22—42no—
531C22—41no—
532C22—41no—
533C22—48816no—
534C22—4882no—
535C22—4882no—
536C22—4882no—
537C22—4881no—
538C22—4881no—
539C22—4881no—
540C22—4881no—
541C22—4881no—
542C22—4881no—
543C22—2444no—
544C22—2444no—
545C22—2444no—
546C22—2442no—
547C22—2442no—
548C22—2442no—
549C22C22×#1175819811yescharacteristic
550C22#1684033111yescharacteristic
551C22#1684033111yescharacteristic
552C22#1684056712yes—
553C22#1684057412yes—
554C4—24no—
555C4—24no—
556C4—22no—
557C4—4882no—
558C4—4882no—
559C4—2442no—
560C4#1684023612yes—
561C4#1684032612yes—
562C2—42no—
563C2—4882no—
564C2—4882no—
565C2—4882no—
566C2#2144839911yescharacteristic
567C2#2145434612yes—
568C2#2145687111yescharacteristic
569C2#2145689311yescharacteristic
570C2#2145691512yes—
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#433C2
socleC23×C61C22×D4
prime radicalC2×#16840331C2
cosocleC22×C4×C61C24
hypercenterC2×#433D61
hypocenterC61#21813
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
2. #21448428C2semidirect
3. #21446749C2semidirect
4. #21446745C2semidirect
5. #21446643C2semidirect
6. #21446635C2semidirect
7. #21446630C2semidirect
8. #21446625C2semidirect
9. #21445760C2semidirect
10. #21445751C2semidirect
11. C2×C61×#433C2semidirect
15. #16840331C4semidirect
22. C2×#14519702C22semidirect
23. C2×#14519702C22semidirect
28. C2×#14519699C22semidirect
29. C2×#14519699C22semidirect
32. #16839632C22semidirect
33. #16839628C22semidirect
35. C4×C4⋊Dic61C22semidirect
38. C2×C4⋊C8×C61C22semidirect
40. C2×#97×C61C22semidirect
41. C2×C4×C8×C61C22semidirect
278. C2×#433D61semidirect
309. C2×C4⋊C8C2×D61semidirect
319. C2×#97C2×D61semidirect
320. C2×C4×C8C2×D61semidirect
327. C61#21813semidirect

Maximal subgroups

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

subgroupindex
C2×#168403312
#214484282
#214467492
#214467452
#214466432
#214466352
#214466302
#214466252
#214457602
#214457512
C2×C61×#4332
#2181361