Finite Group Databasebeta

← #29708774

Subgroups — #29708774

Order 15616.

1226 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839765C212yesmaximal
2#390×D61C211yesmaximal, characteristic
3#21448224C211yesmaximal, characteristic, complemented
4#21447236C212yesmaximal, complemented
5#21447235C212yesmaximal, complemented
6#21447224C211yesmaximal, characteristic, complemented
7#21447223C211yesmaximal, characteristic, complemented
8#21447222C211yesmaximal, characteristic, complemented
9#21445537C212yesmaximal, complemented
10#21445511C211yesmaximal, characteristic, complemented
11C61×#2071C211yesmaximal, characteristic, complemented
12C22×D4×D61C2211yescharacteristic
13C22×#11758189C2212yes—
14C2×#14519735C2212yes—
15C2×C4⋊C4×D61C2211yescharacteristic
16C2×C22⋊C4×D61—21no—
17C2×C22⋊C4×D61—21no—
18C2×C42×D61C2211yescharacteristic
19#16839765—28no—
20C2×#14519699C2212yes—
21C2×#14519698C2212yes—
22C2×#14519697C2212yes—
23#16839665C2211yescharacteristic
24#16839664C2211yescharacteristic
25#16839663C2211yescharacteristic
26#16839662—21no—
27#16839662—21no—
28#16839632C2211yescharacteristic, complemented
29#16839627C2212yescomplemented
30#16839619—21no—
31#16839619—21no—
32C4⋊C4×Dic61—21no—
33C4⋊C4×Dic61—21no—
34#16839525C2212yescomplemented
35#16839524C2212yescomplemented
36#16839524C2212yescomplemented
37#16839517C2211yescharacteristic, complemented
38#16839516C2212yescomplemented
39C2×#102×C61C2212yes—
40C61×#390C2211yescharacteristic
41C61×#384C2211yescharacteristic, complemented
42C61×#340C2212yescomplemented
43C61×#336C2211yescharacteristic, complemented
44C24×D61—21no—
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—21no—
47C2×D4×D61—24no—
48C2×D4×D61—24no—
49C2×D4×D61—22no—
50C2×D4×D61D414yes—
51C22×C4×D61—21no—
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61C2311yescharacteristic
55C2×#11758189—24no—
56C2×#11758189—24no—
57C2×#11758189—24no—
58C22×#92418C2312yes—
59#14519735—24no—
60#14519735—24no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×C4⋊Dic61C2312yes—
65C2×#11758196—21no—
66C4⋊C4×D61—24no—
67C2×C22⋊Dic61—21no—
68C2×C22⋊Dic61—21no—
69C2×#11758198—22no—
70C2×#11758198—21no—
71C2×#11758198—21no—
72C2×#11758198—21no—
73C2×#11758198—21no—
74C22⋊C4×D61—44no—
75C22⋊C4×D61—44no—
76C2×C4×Dic61—22no—
77C2×C4×Dic61D411yescharacteristic
78C2×C4×Dic61D411yescharacteristic
79C2×C4×Dic61C2311yescharacteristic
80C42×D61—24no—
81#14519699—24no—
82#14519698—24no—
83#14519697—24no—
84#14519684—22no—
85#14519684—22no—
86C22×D4×C61C2311yescharacteristic
87C22×C8×C61C2312yes—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2312yes—
92C2×C22⋊C4×C61—21no—
93C2×C22⋊C4×C61—21no—
94C2×C42×C61C2311yescharacteristic
95#102×C61—24no—
96C24×C61—21no—
97C23×D61—21no—
98C23×D61—24no—
99C23×D61—42no—
100C23×D61—42no—
101C23×D61—42no—
102C23×D61—42no—
103C23×D61—42no—
104C23×D61—42no—
105C23×D61C2×D411yescharacteristic
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—21no—
109C2×D4×C61C2×D412yes—
110C22×C4×C61—21no—
111C22×C4×C61—22no—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2411yescharacteristic
115C2×C22⋊D61—24no—
116C2×C22⋊D61—42no—
117C2×C22⋊D61—42no—
118C22×Dic61—21no—
119C22×Dic61—22no—
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C22×Dic61C2×D411yescharacteristic
123D4×D61—216no—
124D4×D61—48no—
125C2×D244—22no—
126C2×D244—22no—
127C2×D244—21no—
128C2×D244C2×D412yes—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—41no—
134C2×C4×D61—41no—
135C2×C4×D61—42no—
136C2×C4×D61—42no—
137C2×C4×D61—42no—
138C2×C4×D61C2×D412yes—
139C2×C4×D61#4114yes—
140C2×C4×D61#4114yes—
141C2×C4×D61#4112yes—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—42no—
149C4⋊C4×C61—42no—
150C22⋊C4×C61—42no—
151C22⋊C4×C61—42no—
152C42×C61—22no—
153C22⋊Dic61—42no—
154C22⋊Dic61—42no—
155#11758198—42no—
156#11758198—42no—
157#11758198—44no—
158#11758198—42no—
159#11758198—42no—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—42no—
164C4⋊Dic61—42no—
165#11758196—42no—
166C4×Dic61—22no—
167C4×Dic61—22no—
168C4×Dic61—22no—
169C4×Dic61—44no—
170C2×#92418—22no—
171C2×#92418—22no—
172C2×#92418—22no—
173#11758189—48no—
174C23×C61—22no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61C22×D411yescharacteristic
182C22×D61—21no—
183C22×D61—21no—
184C22×D61—21no—
185C22×D61—21no—
186C22×D61—22no—
187C22×D61—48no—
188C22×D61—48no—
189C22×D61—41no—
190C22×D61—41no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—48no—
196C22×D61—84no—
197C22×D61—84no—
198C22×D61—84no—
199C22×D61—84no—
200C22×D61#12112yes—
201C22×D61#12312yes—
202C22×D61#12412yes—
203D4×C61—24no—
204D4×C61—42no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—42no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61AGL(1,ℤ/8)12yes—
216C2×C4×C61AGL(1,ℤ/8)12yes—
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4112yes—
219C2×C4×C61C2×#4112yes—
220C2×C4×C61C2×#4111yescharacteristic
221C22⋊D61—48no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—42no—
229C2×Dic61—41no—
230C2×Dic61—42no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—42no—
234C2×Dic61#12112yes—
235C2×Dic61#12312yes—
236C2×Dic61#12412yes—
237D244—24no—
238D244—42no—
239C4×D61—44no—
240C4×D61—44no—
241C4×D61—84no—
242C4×D61#12318yes—
243C4×D61#12618yes—
244C8×C61—42no—
245#92418—42no—
246#23628—611nomaximal, Sylow
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—81no—
251C22×C61—81no—
252C22×C61—81no—
253C22×C61—81no—
254C22×C61C2×#12111yescharacteristic
255C22×C61C2×#12311yescharacteristic
256C22×C61C2×#12411yescharacteristic
257C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C61#57511yescharacteristic
260C22×C61#57511yescharacteristic
261C2×D61—22no—
262C2×D61—22no—
263C2×D61—22no—
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—42no—
270C2×D61—816no—
271C2×D61—81no—
272C2×D61—81no—
273C2×D61—81no—
274C2×D61—81no—
275C2×D61#39714yes—
276C4×C61—22no—
277C4×C61—22no—
278C4×C61—81no—
279C4×C61—82no—
280C4×C61C2×#12312yes—
281C4×C61C2×#12612yes—
282Dic61—41no—
283Dic61—41no—
284Dic61—81no—
285Dic61—82no—
286Dic61#39714yes—
287C2×#418—612no—
288C2×#390—611no—
289#2285—611no—
290#2073—612no—
291#2073—612no—
292#2071—611no—
293#2071—611no—
294#2071—611no—
295#2071D6111yescharacteristic, complemented
296#1736—612no—
297#1728—611no—
298C2×C61—82no—
299C2×C61C2×#39711yescharacteristic
300C2×C61#303611yescharacteristic
301C2×C61#303611yescharacteristic
302C2×C61#309411yescharacteristic
303C2×C61#309411yescharacteristic
304C2×C61#313811yescharacteristic
305C2×C61#314011yescharacteristic
306D61—24no—
307D61—82no—
308C23×D4—611no—
309C22×M16—612no—
310C2×#117—612no—
311C22×C4⋊C4—611no—
312C22×C22⋊C4—1221no—
313C22×C22⋊C4—1221no—
314C22×C42—611no—
315#418—1228no—
316C2×#102—612no—
317C2×#102—612no—
318C2×#102—612no—
319C2×#102C2×D6112yes—
320#390—611no—
321#390—611no—
322#390—611no—
323#390—1221no—
324#390—1221no—
325#390C2×D6111yescharacteristic
326#384—611no—
327#384C2×D6111yescharacteristic, complemented
328#382—612no—
329#380—1221no—
330#380—1221no—
331C4×C4⋊C4—1221no—
332C4×C4⋊C4—1221no—
333#340—612no—
334#340—612no—
335#340—612no—
336#340C2×D6112yescomplemented
337#336—612no—
338#336—611no—
339#336C2×D6111yescharacteristic, complemented
340C61#2362811yescharacteristic, complemented, Sylow
341C25—1221no—
342C22×D4—614no—
343C22×D4—611no—
344C22×D4—1224no—
345C22×D4—1224no—
346C22×D4—1222no—
347C22×D4—1221no—
348C22×D4C22×D6111yescharacteristic
349C23×C4—611no—
350C23×C4—611no—
351C23×C4—611no—
352C23×C4—1221no—
353C2×M16—1224no—
354C2×M16—1224no—
355C2×M16—1224no—
356C22×C8—612no—
357C22×C8C22×D6112yes—
358#117—1224no—
359#117—1224no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—612no—
364C2×C4⋊C4—1224no—
365C2×C4⋊C4—1221no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4C22×D6111yescharacteristic
369C2×C4⋊C4C22×D6112yes—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—1221no—
373C2×C22⋊C4—1221no—
374C2×C22⋊C4—1222no—
375C2×C22⋊C4—1221no—
376C2×C22⋊C4—1221no—
377C2×C22⋊C4—1221no—
378C2×C22⋊C4—1221no—
379C2×C22⋊C4—2444no—
380C2×C22⋊C4—2444no—
381C2×C42—611no—
382C2×C42—611no—
383C2×C42—611no—
384C2×C42—1224no—
385C2×C42—1222no—
386C2×C42C22×D6111yescharacteristic
387#102—24no—
388#102—1224no—
389#102—1224no—
390#102—1224no—
391#95—1222no—
392#95—1222no—
393C24—21no—
394C24—611no—
395C24—1224no—
396C24—1221no—
397C24—2442no—
398C24—2442no—
399C24—2442no—
400C24—2442no—
401C24—2442no—
402C24—2442no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—612no—
407C2×D4—12216no—
408C2×D4—1224no—
409C2×D4—1222no—
410C2×D4—1222no—
411C2×D4—1221no—
412C2×D4—2448no—
413C2×D4—2442no—
414C2×D4—2442no—
415C2×D4D4×D6112yes—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—611no—
419C22×C4—612no—
420C22×C4—614no—
421C22×C4—614no—
422C22×C4—612no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1221no—
431C22×C4—2441no—
432C22×C4—2441no—
433C22×C4—2442no—
434C22×C4—2442no—
435C22×C4—2442no—
436C22×C4D4×D6111yescharacteristic
437C22×C4D4×D6111yescharacteristic
438C22×C4C23×D6111yescharacteristic
439M16—2448no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—1222no—
444C2×C8—1222no—
445C2×C8—1222no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—1222no—
454C4⋊C4—2442no—
455C4⋊C4—2442no—
456C4⋊C4—2442no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—2442no—
460C22⋊C4—2442no—
461C22⋊C4—2444no—
462C22⋊C4—2442no—
463C22⋊C4—2442no—
464C22⋊C4—2442no—
465C22⋊C4—2442no—
466C42—22no—
467C42—1222no—
468C42—1222no—
469C42—1222no—
470C42—2444no—
471C23—22no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—612no—
479C23—612no—
480C23—612no—
481C23—1221no—
482C23—1221no—
483C23—1221no—
484C23—1221no—
485C23—1222no—
486C23—4884no—
487C23—4884no—
488C23—4884no—
489C23—4884no—
490C23—2448no—
491C23—2448no—
492C23—2448no—
493C23—2441no—
494C23—2441no—
495C23—2441no—
496C23—2441no—
497C23—2441no—
498C23—2441no—
499C23C2×D4×D6111yescharacteristic
500D4—24no—
501D4—42no—
502D4—1224no—
503D4—2448no—
504D4—2442no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—612no—
516C2×C4—612no—
517C2×C4—612no—
518C2×C4—618no—
519C2×C4—618no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—4884no—
527C2×C4—2444no—
528C2×C4—2444no—
529C2×C4—2442no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4#1451980112yes—
536C2×C4#1451980412yes—
537C2×C4C2×D4×D6111yescharacteristic
538C2×C4#41×D6112yes—
539C2×C4#41×D6112yes—
540C2×C4#41×D6111yescharacteristic
541C8—42no—
542C8—2442no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—614no—
551C22—1222no—
552C22—1222no—
553C22—1222no—
554C22—1222no—
555C22—1222no—
556C22—1222no—
557C22—48816no—
558C22—4881no—
559C22—4881no—
560C22—4881no—
561C22—4881no—
562C22—2442no—
563C22—2442no—
564C22—2442no—
565C22#121×D6111yescharacteristic
566C22#123×D6111yescharacteristic
567C22#124×D6111yescharacteristic
568C22C2×#1451980111yescharacteristic
569C22C2×#1451980411yescharacteristic
570C22#1684055811yescharacteristic
571C22#1684056111yescharacteristic
572C4—22no—
573C4—22no—
574C4—81no—
575C4—82no—
576C4—614no—
577C4—4881no—
578C4—4882no—
579C4—2441no—
580C4—2441no—
581C4#123×D6112yes—
582C4#126×D6112yes—
583C2—82no—
584C2—1224no—
585C2—4882no—
586C2#397×D6111yescharacteristic
587C2#2145527811yescharacteristic
588C2#2145528411yescharacteristic
589C2#2145575411yescharacteristic
590C2#2145575611yescharacteristic
591C2#2145619011yescharacteristic
592C2#2145620411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2071C2
socleC23×C61C22×D4
prime radicalC22×D4×D61C22
hypercenter#2071D61
hypocenterC61#23628
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
3. #21448224C2semidirect
4. #21447236C2semidirect
5. #21447235C2semidirect
6. #21447224C2semidirect
7. #21447223C2semidirect
8. #21447222C2semidirect
9. #21445537C2semidirect
10. #21445511C2semidirect
11. C61×#2071C2semidirect
28. #16839632C22semidirect
29. #16839627C22semidirect
34. #16839525C22semidirect
35. #16839524C22semidirect
36. #16839524C22semidirect
37. #16839517C22semidirect
38. #16839516C22semidirect
41. C61×#384C22semidirect
42. C61×#340C22semidirect
43. C61×#336C22semidirect
295. #2071D61semidirect
327. #384C2×D61semidirect
336. #340C2×D61semidirect
339. #336C2×D61semidirect
340. C61#23628semidirect

Maximal subgroups

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

subgroupindex
C2×#168397652
#390×D612
#214482242
#214472362
#214472352
#214472242
#214472232
#214472222
#214455372
#214455112
C61×#20712
#2362861