Finite Group Databasebeta

← #29739064

Subgroups — #29739064

Order 15616.

678 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21451054C211yesmaximal, characteristic, complemented
2#21448792C211yesmaximal, characteristic
3#21447345C211yesmaximal, characteristic
4#21447342C211yesmaximal, characteristic
5#21447189C211yesmaximal, characteristic
6#21447189C211yesmaximal, characteristic
7#21446590C211yesmaximal, characteristic, complemented
8#21446589C211yesmaximal, characteristic, complemented
9#21446581C211yesmaximal, characteristic, complemented
10#21446535C211yesmaximal, characteristic, complemented
11#21446414C211yesmaximal, characteristic, complemented
12#21446393C211yesmaximal, characteristic, complemented
13#21445981C211yesmaximal, characteristic, complemented
14C4×#14519698C211yesmaximal, characteristic
15C61×#1804C211yesmaximal, characteristic
16C2×#14519711C2211yescharacteristic
17C2×#14519710C2211yescharacteristic
18C2×#14519710C2211yescharacteristic
19C2×#14519709C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×C4×#92418C2211yescharacteristic
27#16839704—21no—
28#16839694—21no—
29#16839665C2211yescharacteristic
30#16839665C2211yescharacteristic
31#16839661C2211yescharacteristic
32#16839661C2211yescharacteristic
33#16839655—21no—
34#16839655—21no—
35#16839649—21no—
36#16839645—21no—
37#16839631C2211yescharacteristic
38#16839628C2211yescharacteristic
39#16839627C2211yescharacteristic
40#16839625—21no—
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×C4⋊Dic61C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45C4×C22⋊Dic61—21no—
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C2×#102×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C61×#382C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×#92418C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61D411yescharacteristic
64C2×C4⋊Dic61D411yescharacteristic
65C2×C4⋊Dic61C2311yescharacteristic
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—21no—
70C2×#11758196—21no—
71C2×C22⋊Dic61—21no—
72C2×C22⋊Dic61—21no—
73C2×C22⋊Dic61—21no—
74C2×C22⋊Dic61—21no—
75C2×C4×Dic61—21no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61C2311yescharacteristic
79C2×C4×Dic61C2311yescharacteristic
80#14519711—41no—
81#14519710—22no—
82#14519710—22no—
83#14519709—41no—
84#14519708—41no—
85#14519708—41no—
86#14519706—22no—
87#14519706—22no—
88#14519698—22no—
89#14519698—22no—
90C4×#92418—22no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684D411yescharacteristic
101#14519684D411yescharacteristic
102C22×D4×C61C2311yescharacteristic
103C23×C4×C61—21no—
104C22×C8×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106C4×D4×C61—22no—
107C4×D4×C61Q814yes—
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C42×C61C2311yescharacteristic
113C4⋊C8×C61—22no—
114#102×C61—41no—
115#102×C61—41no—
116#95×C61—21no—
117#95×C61—21no—
118C24×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—21no—
124C2×D4×C61C2×Q812yes—
125C22×C4×C61—21no—
126C22×C4×C61—24no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—81no—
130C22×C4×C61—81no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C2×C8×C61—21no—
143C2×C8×C61—21no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61C2×Q812yes—
154C22⋊C4×C61—24no—
155C42×C61C2×Q812yes—
156C42×C61#4112yes—
157C22⋊Dic61—42no—
158C22⋊Dic61—42no—
159C22⋊Dic61—81no—
160C22⋊Dic61—81no—
161C4⋊Dic61—22no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—41no—
171C4⋊Dic61—42no—
172C4⋊Dic61—81no—
173#11758196—42no—
174#11758196—81no—
175C4×Dic61—21no—
176C4×Dic61—21no—
177C4×Dic61—21no—
178C4×Dic61—21no—
179C4×Dic61—81no—
180C4×Dic61—81no—
181C4×Dic61—81no—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418#4112yes—
187C2×#92418#4112yes—
188C23×C61—22no—
189C23×C61—41no—
190C23×C61—41no—
191C23×C61—41no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61C22×D411yescharacteristic
196D4×C61—22no—
197D4×C61—22no—
198D4×C61—41no—
199D4×C61—41no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—22no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—84no—
214C2×C4×C61C22⋊Q812yes—
215C2×C4×C61C22⋊Q812yes—
216C2×C4×C61C22×Q811yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61H(5,𝔽2)11yescharacteristic
221C2×C4×C612−1+411yescharacteristic
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61#13512yes—
246C8×C61—41no—
247C8×C61—41no—
248#92418—22no—
249#92418—22no—
250#26094—611nomaximal, Sylow
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—81no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61—81no—
258C22×C61C2×C22⋊Q811yescharacteristic
259C22×C61#54711yescharacteristic
260C22×C61#55011yescharacteristic
261C22×C61C2×#13511yescharacteristic
262C22×C61#57511yescharacteristic
263C22×C61#57611yescharacteristic
264C22×C61#57711yescharacteristic
265C4×C61—22no—
266C4×C61—22no—
267C4×C61—42no—
268C4×C61—81no—
269C4×C61—81no—
270C4×C61#53311yescharacteristic
271C4×C61#54411yescharacteristic
272C4×C61#55211yescharacteristic
273C4×C61#55311yescharacteristic
274Dic61—22no—
275Dic61—41no—
276Dic61—41no—
277Dic61—41no—
278Dic61—81no—
279Dic61—81no—
280Dic61—81no—
281Dic61—81no—
282#2570—611no—
283#2357—611no—
284#2093—611no—
285#2092—611no—
286#2065—611no—
287#2065—611no—
288#1949—611no—
289#1949—611no—
290#1948—611no—
291#1939—611no—
292#1921—611no—
293#1918—611no—
294#1846—611no—
295#1804D6111yescharacteristic
296C4×#102—611no—
297C2×C61—82no—
298C2×C61#244111yescharacteristic
299C2×C61#307611yescharacteristic
300C2×C61#308111yescharacteristic
301C2×C61#333711yescharacteristic
302C2×C61#335511yescharacteristic
303C2×C61#339611yescharacteristic
304C2×C61#340111yescharacteristic
305C2×C4×D4C2×D6111yescharacteristic
306C2×#107—611no—
307C2×#107—611no—
308C2×#107—611no—
309C2×#106—611no—
310C2×#106—611no—
311C2×#106—611no—
312C2×C4⋊C8—611no—
313C2×C4⋊C8—611no—
314C2×C4⋊C8C2×D6111yescharacteristic
315C2×#102—611no—
316C2×#102—611no—
317C2×#102C2×D6111yescharacteristic
318C2×#102C2×D6111yescharacteristic
319C2×C4×C8—611no—
320#400—1221no—
321#398—1221no—
322#390—611no—
323#390—611no—
324#389—611no—
325#389—611no—
326#388—1221no—
327#388—1221no—
328#387—1221no—
329#387—1221no—
330#384—611no—
331#382—611no—
332#382—611no—
333#382C2×D6111yescharacteristic
334#381—1221no—
335C4×C4⋊C4—611no—
336C4×C4⋊C4—611no—
337C4×C4⋊C4—611no—
338C4×C4⋊C4—611no—
339C4×C22⋊C4—1221no—
340#340—611no—
341#340—611no—
342#340—611no—
343#340—611no—
344#340C2×D6111yescharacteristic
345#340C2×D6111yescharacteristic
346#336—611no—
347#336—611no—
348C61#2609411yescharacteristic, complemented, Sylow
349C22×D4C22×D6111yescharacteristic
350C23×C4—21no—
351C22×C8—611no—
352C22×C8—611no—
353C22×C8C22×D6111yescharacteristic
354C22×C8C22×D6111yescharacteristic
355C4×D4—22no—
356C4×D4Dic12214yes—
357C2×C4⋊C4—611no—
358C2×C4⋊C4—611no—
359C2×C4⋊C4—611no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—1221no—
364C2×C4⋊C4—1221no—
365C2×C4⋊C4—1221no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4C22×D6111yescharacteristic
368C2×C4⋊C4C22×D6111yescharacteristic
369C2×C4⋊C4C22×D6111yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—1221no—
372C2×C22⋊C4—1221no—
373C2×C22⋊C4—1221no—
374C2×C22⋊C4—1221no—
375C2×C42—611no—
376C2×C42—611no—
377C2×C42—1221no—
378C2×C42—1221no—
379C2×C42—1221no—
380C2×C42C22×D6111yescharacteristic
381#107—1222no—
382#107—1222no—
383#107—2441no—
384#106—2441no—
385#106—2441no—
386#106—2441no—
387C4⋊C8—22no—
388C4⋊C8—1222no—
389C4⋊C8—1222no—
390#102—41no—
391#102—41no—
392#102—1222no—
393#102—1222no—
394C4×C8—1222no—
395#95—21no—
396#95—21no—
397#95—611no—
398#95—611no—
399#95—1221no—
400#95—1221no—
401#95—1221no—
402#95—1221no—
403#95—1221no—
404#95—1221no—
405#95—1221no—
406#95—1221no—
407#95—1221no—
408C24—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4C2×Dic12212yes—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4D4×D6111yescharacteristic
430C22×C4D4×D6111yescharacteristic
431C22×C4C23×D6111yescharacteristic
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—612no—
439C2×C8—612no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—1222no—
450C4⋊C4—1221no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—4881no—
459C4⋊C4—4881no—
460C4⋊C4—2441no—
461C4⋊C4—2442no—
462C4⋊C4—2442no—
463C4⋊C4C2×Dic12212yes—
464C22⋊C4—24no—
465C22⋊C4—4881no—
466C22⋊C4—4881no—
467C22⋊C4—2442no—
468C22⋊C4—2442no—
469C42—1221no—
470C42—1221no—
471C42—1221no—
472C42—1221no—
473C42—4881no—
474C42—4881no—
475C42—4881no—
476C42C2×Dic12212yes—
477C42#1175822412yes—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23C2×D4×D6111yescharacteristic
486D4—22no—
487D4—22no—
488D4—41no—
489D4—41no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—84no—
504C2×C4—612no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1221no—
509C2×C4—1221no—
510C2×C4—1221no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1221no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4#1451975412yes—
529C2×C4#1451975412yes—
530C2×C4C22×Dic12211yescharacteristic
531C2×C4#41×D6111yescharacteristic
532C2×C4#41×D6111yescharacteristic
533C2×C4C2×#1175822411yescharacteristic
534C2×C4#1451982211yescharacteristic
535C2×C4#1451982511yescharacteristic
536C8—41no—
537C8—41no—
538C8—1222no—
539C8—1222no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22C2×#1451975411yescharacteristic
548C22#1684041411yescharacteristic
549C22#1684043011yescharacteristic
550C22#135×D6111yescharacteristic
551C22#1684055911yescharacteristic
552C22#1684056611yescharacteristic
553C22#1684057211yescharacteristic
554C4—22no—
555C4—22no—
556C4—42no—
557C4—81no—
558C4—81no—
559C4—1222no—
560C4—4881no—
561C4—4881no—
562C4—4881no—
563C4—4881no—
564C4—2441no—
565C4—2441no—
566C4—2441no—
567C4#1684031511yescharacteristic
568C4#1684039311yescharacteristic
569C4#1684044611yescharacteristic
570C4#1684045911yescharacteristic
571C2—82no—
572C2#2144953311yescharacteristic
573C2#2145560311yescharacteristic
574C2#2145565211yescharacteristic
575C2#2145820511yescharacteristic
576C2#2145842211yescharacteristic
577C2#2145882611yescharacteristic
578C2#2145888311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1804C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1804D61
hypocenterC61#26094
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
1. #21451054C2semidirect
7. #21446590C2semidirect
8. #21446589C2semidirect
9. #21446581C2semidirect
10. #21446535C2semidirect
11. #21446414C2semidirect
12. #21446393C2semidirect
13. #21445981C2semidirect
348. C61#26094semidirect

Maximal subgroups

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

subgroupindex
#214510542
#214487922
#214473452
#214473422
#214471892
#214465902
#214465892
#214465812
#214465352
#214464142
#214463932
#214459812
C4×#145196982
C61×#18042
#2609461