Finite Group Databasebeta

← #29679831

Subgroups — #29679831

Order 15616.

1374 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840242C211yesmaximal, characteristic
2C2×#16839730C211yesmaximal, characteristic
3#21448412C211yesmaximal, characteristic
4#21445913C212yesmaximal, complemented
5#21445912C212yesmaximal, complemented
6#21445908C212yesmaximal, complemented
7#21445904C212yesmaximal, complemented
8#21445539C211yesmaximal, characteristic, complemented
9#21445535C211yesmaximal, characteristic, complemented
10C2×#16839524C211yesmaximal, characteristic, complemented
11C61×#1736C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14#16840242—24no—
15#16840242C418yescomplemented
16C2×C4×Dic122—21no—
17C2×C4×D244—21no—
18C2×#14519737—22no—
19C2×#14519736C2211yescharacteristic
20C2×#14519735C2212yes—
21C22×C4⋊Dic61C2211yescharacteristic
22C2×C4⋊C4×D61C2212yes—
23#16839730—24no—
24C2×#14519689C2212yes—
25C2×#14519689C2212yes—
26#16839667—21no—
27#16839663—21no—
28#16839641—22no—
29#16839619C2212yes—
30#16839619C2212yes—
31#16839525C2212yes—
32#16839525C2212yes—
33#16839524C2211yescharacteristic
34#16839524C2212yes—
35#16839524C2211yescharacteristic
36#16839524C2212yes—
37#16839524C2211yescharacteristic
38#16839524C2211yescharacteristic
39C22×M16×C61C2211yescharacteristic
40C2×#117×C61C2211yescharacteristic
41C22×C4⋊C4×C61C2211yescharacteristic
42C61×#340C2212yes—
43C61×#340C2212yes—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222C2×C414yes—
49C2×#11758222D414yes—
50C22×Dic122D411yescharacteristic
51C22×D244D411yescharacteristic
52C22×C22⋊D61D412yes—
53C23×Dic61—21no—
54C22×C4×D61—22no—
55C22×C4×D61C2312yes—
56C22×#92418—22no—
57C22×#92418—22no—
58C22×#92418—22no—
59C22×#92418C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61C4×Dic122—28no—
62C4×D244—28no—
63#14519737—216no—
64#14519736—22no—
65#14519736—22no—
66#14519735—24no—
67#14519735C2×C418yescomplemented
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61C2×C414yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2312yes—
74C2×#11758196—22no—
75C4⋊C4×D61—24no—
76C4⋊C4×D61C2×C418yescomplemented
77C2×C22⋊Dic61—21no—
78C2×#11758198—22no—
79C2×#11758198—22no—
80C2×#11758198—22no—
81C2×C4×Dic61—22no—
82C2×C4×Dic61—21no—
83#14519689—24no—
84#14519689—24no—
85#14519684—22no—
86#14519684—22no—
87C23×C4×C61—21no—
88C23×C4×C61C2311yescharacteristic
89C2×M16×C61—22no—
90C2×M16×C61—22no—
91C2×M16×C61—22no—
92C22×C8×C61C2312yes—
93#117×C61—22no—
94#117×C61C2×C414yes—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61C2312yes—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C42×C61—21no—
103C24×C61C2×D411yescharacteristic
104C23×D61C2×D412yes—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61—22no—
108C22×C4×C61—42no—
109C22×C4×C61—81no—
110C22×C4×C61—81no—
111C22×C4×C61C22×C412yes—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D412yes—
114C22×C4×C61#4112yes—
115C22×C4×C61#4112yes—
116C22×C4×C61#4112yes—
117C22×C4×C61#4112yes—
118C22×C4×C61C2411yescharacteristic
119C2×C22⋊D61—24no—
120C2×C22⋊D61—24no—
121C2×C22⋊D61C22⋊C418yes—
122C22×Dic61—22no—
123C22×Dic61—24no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—81no—
127C22×Dic61—81no—
128C22×Dic61C2×D412yes—
129#11758222—28no—
130#11758222—28no—
131#11758222—28no—
132#11758222—28no—
133C2×D244—22no—
134C2×D244—22no—
135C2×D244C22⋊C414yes—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—28no—
141C2×C4×D61—82no—
142C2×C4×D61—82no—
143C2×C4×D61C22×C414yes—
144C2×C4×D61C2×D414yes—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122C22⋊C414yes—
148M16×C61—44no—
149C2×C8×C61—22no—
150C2×C8×C61—22no—
151C2×C8×C61—22no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—42no—
157C4⋊C4×C61C22×C412yes—
158C4⋊C4×C61C22×C412yes—
159C22⋊C4×C61—24no—
160C42×C61—24no—
161C22⋊Dic61—42no—
162#11758198—28no—
163#11758198—82no—
164#11758198—82no—
165C4⋊Dic61—24no—
166C4⋊Dic61—22no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61C22×C414yes—
172#11758196—28no—
173C4×Dic61—28no—
174C4×Dic61—42no—
175C2×#92418—21no—
176C2×#92418—21no—
177C2×#92418—21no—
178C2×#92418—21no—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418—44no—
182C23×C61—21no—
183C23×C61—21no—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61C2×C22⋊C412yes—
187C23×C61#12312yes—
188C23×C61#12312yes—
189C23×C61C22×D411yescharacteristic
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—22no—
196C22×D61C2×C22⋊C414yes—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—84no—
207C2×C4×C61C2×C22⋊C412yes—
208C2×C4×C61#11714yes—
209C2×C4×C61#11714yes—
210C2×C4×C61#12314yes—
211C2×C4×C61#12314yes—
212C2×C4×C61#12312yes—
213C2×C4×C61#12312yes—
214C2×C4×C61C23×C411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C22⋊D61—216no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—24no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—84no—
233C2×Dic61C2×C22⋊C414yes—
234D244—28no—
235C4×D61—24no—
236C4×D61—24no—
237C4×D61—24no—
238C4×D61—24no—
239C4×D61—88no—
240Dic122—28no—
241C8×C61—42no—
242#92418—41no—
243#92418—41no—
244#20703—611nomaximal, Sylow
245C22×C61—22no—
246C22×C61—22no—
247C22×C61—22no—
248C22×C61—22no—
249C22×C61—22no—
250C22×C61—22no—
251C22×C61#38114yes—
252C22×C61C22×C22⋊C411yescharacteristic
253C22×C61C2×#12311yescharacteristic
254C22×C61C2×#12311yescharacteristic
255C22×C61#57611yescharacteristic
256C22×C61#57711yescharacteristic
257C22×C61#57711yescharacteristic
258C22×C61#57811yescharacteristic
259C2×D61—24no—
260C2×D61—24no—
261C2×D61—24no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C4×C61—42no—
267C4×C61—42no—
268C4×C61—82no—
269C4×C61#38114yes—
270C4×C61C2×#11711yescharacteristic
271C4×C61C2×#11711yescharacteristic
272C4×C61C2×#12311yescharacteristic
273C4×C61C2×#12311yescharacteristic
274Dic61—24no—
275Dic61—44no—
276Dic61—81no—
277Dic61—81no—
278C2×#520—611no—
279C2×#408—611no—
280#2311—611no—
281#1825—612no—
282#1825—612no—
283#1824—612no—
284#1823—612no—
285#1736—611no—
286#1736D6111yescharacteristic, complemented
287#1735—611no—
288C2×#340—611no—
289C2×C61—24no—
290C2×C61C2×#38111yescharacteristic
291C2×C61#289711yescharacteristic
292C2×C61#289811yescharacteristic
293C2×C61#309611yescharacteristic
294C2×C61#309711yescharacteristic
295C2×C61#309811yescharacteristic
296C2×C61#309911yescharacteristic
297D61—44no—
298C22×#41—611no—
299C22×M16C2×D6111yescharacteristic
300C23×C8—611no—
301#520—618no—
302#520—1224no—
303C2×C4×Q8—1221no—
304C2×C4×D4—1221no—
305C2×C4×D4—1222no—
306C2×#117—611no—
307C2×#117—612no—
308C2×#117C2×D6111yescharacteristic
309C22×C4⋊C4—611no—
310C22×C4⋊C4—612no—
311C22×C4⋊C4C2×D6111yescharacteristic
312#408—1224no—
313C2×C22⋊C8—612no—
314C2×C22⋊C8—612no—
315#391—1221no—
316#390—1221no—
317#386—1222no—
318#380—612no—
319#380—612no—
320#340—612no—
321#340—612no—
322#340—611no—
323#340—612no—
324#340—611no—
325#340—612no—
326#340—611no—
327#340—611no—
328#340C2×D6112yes—
329#340C2×D6112yes—
330C61#2070311yescharacteristic, complemented, Sylow
331C2×#41—614no—
332C2×#41—614no—
333C2×#41—1222no—
334C2×#41—1222no—
335C2×#41—1222no—
336C2×#41—1222no—
337C22×Q8—611no—
338C22×D4—612no—
339C22×D4—611no—
340C23×C4—21no—
341C23×C4—612no—
342C23×C4—1222no—
343C23×C4—1221no—
344C23×C4C22×D6111yescharacteristic
345C2×M16—22no—
346C2×M16—22no—
347C2×M16—22no—
348C22×C8—611no—
349C22×C8—611no—
350C22×C8—1222no—
351C22×C8—1222no—
352C22×C8—1222no—
353C22×C8C22×D6112yes—
354C4×Q8—1228no—
355C4×D4—1228no—
356C4×D4—12216no—
357#117—22no—
358#117—618no—
359#117—1224no—
360#117—1222no—
361#117—1222no—
362#117C4×D6114yes—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—614no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—618no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—1224no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4C22×D6112yes—
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—1222no—
380C2×C22⋊C4—1222no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—1221no—
383C2×C42—21no—
384C2×C42—1222no—
385C2×C42—1221no—
386C22⋊C8—1224no—
387C22⋊C8—1224no—
388#95—1222no—
389#95—1222no—
390C24—612no—
391C24D4×D6111yescharacteristic
392#41—1228no—
393#41—1228no—
394#41—1228no—
395#41—1228no—
396C2×Q8—614no—
397C2×Q8—1222no—
398C2×Q8—1222no—
399C2×D4—618no—
400C2×D4—614no—
401C2×D4—1224no—
402C2×D4—1224no—
403C2×D4—1222no—
404C2×D4—1222no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—612no—
412C22×C4—614no—
413C22×C4—614no—
414C22×C4—1228no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1224no—
420C22×C4—1222no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—4882no—
424C22×C4—4882no—
425C22×C4—4881no—
426C22×C4—4881no—
427C22×C4C2×C4×D6112yes—
428C22×C4#1175822212yes—
429C22×C4#1175822212yes—
430C22×C4D4×D6112yes—
431C22×C4D4×D6111yescharacteristic
432C22×C4#1175822412yes—
433C22×C4#1175822612yes—
434C22×C4C23×D6111yescharacteristic
435M16—44no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—2444no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—614no—
452C4⋊C4—1228no—
453C4⋊C4—1222no—
454C4⋊C4—1224no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4C2×C4×D6112yes—
460C4⋊C4C2×C4×D6112yes—
461C22⋊C4—24no—
462C22⋊C4—1228no—
463C22⋊C4—4882no—
464C22⋊C4—4882no—
465C22⋊C4—2442no—
466C42—24no—
467C42—1228no—
468C42—2442no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—614no—
474C23—1222no—
475C23—1222no—
476C23—1222no—
477C23—1222no—
478C23—1222no—
479C23—1222no—
480C23C22⋊C4×D6112yes—
481C23#1451975912yes—
482C23#1451976212yes—
483C23C2×D4×D6111yescharacteristic
484Q8—1228no—
485D4—12216no—
486D4—1228no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—84no—
497C2×C4—614no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—1224no—
501C2×C4—1224no—
502C2×C4—1224no—
503C2×C4—1224no—
504C2×C4—1224no—
505C2×C4—4888no—
506C2×C4—4884no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4C22⋊C4×D6112yes—
516C2×C4#1451973414yes—
517C2×C4#1451973514yes—
518C2×C4#1451975914yes—
519C2×C4#1451975912yes—
520C2×C4#1451976214yes—
521C2×C4#1451976212yes—
522C2×C4C22×C4×D6111yescharacteristic
523C2×C4C2×D4×D6111yescharacteristic
524C2×C4C2×#1175822411yescharacteristic
525C2×C4C2×#1175822211yescharacteristic
526C2×C4C2×#1175822211yescharacteristic
527C2×C4C2×#1175822611yescharacteristic
528C8—42no—
529C8—2441no—
530C8—2441no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—1224no—
538C22—1224no—
539C22—1224no—
540C22—2442no—
541C22—2442no—
542C22—2442no—
543C22—2442no—
544C22#1683962414yes—
545C22C2×C22⋊C4×D6111yescharacteristic
546C22C2×#1451975911yescharacteristic
547C22C2×#1451976211yescharacteristic
548C22#1684056611yescharacteristic
549C22#1684057111yescharacteristic
550C22#1684057211yescharacteristic
551C22#1684057811yescharacteristic
552C4—42no—
553C4—42no—
554C4—82no—
555C4—1224no—
556C4—4881no—
557C4—4881no—
558C4—2444no—
559C4#1683962414yes—
560C4C2×#1451973411yescharacteristic
561C4C2×#1451973511yescharacteristic
562C4C2×#1451975911yescharacteristic
563C4C2×#1451976211yescharacteristic
564C2—24no—
565C2—2444no—
566C2C2×#1683962411yescharacteristic
567C2#2145434311yescharacteristic
568C2#2145435011yescharacteristic
569C2#2145577511yescharacteristic
570C2#2145578611yescharacteristic
571C2#2145580111yescharacteristic
572C2#2145581211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1736C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenter#1736D61
hypocenterC61#20703
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #21445913C2semidirect
5. #21445912C2semidirect
6. #21445908C2semidirect
7. #21445904C2semidirect
8. #21445539C2semidirect
9. #21445535C2semidirect
10. C2×#16839524C2semidirect
11. C61×#1736C2semidirect
15. #16840242C4semidirect
67. #14519735C2×C4semidirect
76. C4⋊C4×D61C2×C4semidirect
286. #1736D61semidirect
330. C61#20703semidirect

Maximal subgroups

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

subgroupindex
C2×#168402422
C2×#168397302
#214484122
#214459132
#214459122
#214459082
#214459042
#214455392
#214455352
C2×#168395242
C61×#17362
#2070361