Finite Group Databasebeta

← #29725164

Subgroups — #29725164

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450739C211yesmaximal, characteristic
2#21448771C211yesmaximal, characteristic, complemented
3#21447390C211yesmaximal, characteristic, complemented
4#21447386C211yesmaximal, characteristic, complemented
5#21447235C211yesmaximal, characteristic
6#21446753C211yesmaximal, characteristic, complemented
7#21446581C211yesmaximal, characteristic, complemented
8#21446448C211yesmaximal, characteristic
9#21446443C211yesmaximal, characteristic
10#21446407C211yesmaximal, characteristic, complemented
11#21446405C211yesmaximal, characteristic, complemented
12#21445945C211yesmaximal, characteristic, complemented
13#21445817C211yesmaximal, characteristic
14#102×Dic61C211yesmaximal, characteristic
15C61×#2073C211yesmaximal, characteristic
16C2×#14519754—21no—
17C2×C4×Dic122C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×#14519709C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519698C2211yescharacteristic
23C2×#14519698C2211yescharacteristic
24C2×C8×Dic61C2211yescharacteristic
25#16839685—21no—
26#16839685—21no—
27#16839666—21no—
28#16839665C2211yescharacteristic
29#16839661C2211yescharacteristic
30#16839660C2211yescharacteristic
31#16839647—21no—
32#16839644—21no—
33#16839631C2211yescharacteristic
34#16839631C2211yescharacteristic
35#16839627C2211yescharacteristic
36#16839620—21no—
37C4⋊C4×Dic61C2211yescharacteristic
38C4⋊C4×Dic61C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40#16839613—21no—
41#16839525C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#102×C61C2211yescharacteristic
53C2×#102×C61C2211yescharacteristic
54C61×#390C2211yescharacteristic
55C61×#382C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122D411yescharacteristic
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519754—44no—
64C4×Dic122—22no—
65C4×Dic122—22no—
66D4×Dic61—22no—
67D4×Dic61—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196D411yescharacteristic
80C2×#11758196D411yescharacteristic
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88C2×C4×Dic61C2311yescharacteristic
89#14519709—22no—
90#14519708—22no—
91#14519706—22no—
92#14519698—22no—
93#14519698—22no—
94C8×Dic61—22no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102C22×D4×C61C2311yescharacteristic
103C22×C8×C61C2311yescharacteristic
104C22×C8×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C42×C61C2311yescharacteristic
111#102×C61—22no—
112#102×C61—22no—
113#95×C61—21no—
114#95×C61—21no—
115C24×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61#4112yes—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—42no—
133C22×Dic61—42no—
134C22×Dic61—42no—
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C2×Dic122—22no—
140C2×Dic122—22no—
141C2×Dic122—22no—
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×C61—21no—
154C22⋊C4×C61—42no—
155C22⋊C4×C61—81no—
156C42×C61—21no—
157C42×C61—21no—
158C22⋊Dic61—42no—
159C22⋊Dic61—42no—
160C22⋊Dic61—81no—
161C4⋊Dic61—21no—
162C4⋊Dic61—22no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—42no—
170C4⋊Dic61#4112yes—
171#11758196—22no—
172#11758196—22no—
173#11758196—42no—
174#11758196—42no—
175#11758196—42no—
176#11758196—81no—
177#11758196—81no—
178C4×Dic61—21no—
179C4×Dic61—21no—
180C4×Dic61—42no—
181C4×Dic61—81no—
182C4×Dic61—81no—
183C4×Dic61#4112yes—
184C4×Dic61#4112yes—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C23×C61—22no—
192C23×C61—41no—
193C23×C61—41no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61C22×D411yescharacteristic
199D4×C61—22no—
200D4×C61—22no—
201D4×C61—41no—
202D4×C61—41no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C612−1+411yescharacteristic
223C2×C4×C612−1+411yescharacteristic
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—22no—
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—84no—
245C2×Dic61#12312yes—
246C2×Dic61#12312yes—
247C2×Dic61#13512yes—
248C2×Dic61AGL(1,ℤ/8)12yes—
249Dic122—44no—
250C8×C61—41no—
251C8×C61—41no—
252#92418—41no—
253#92418—41no—
254#24991—611nomaximal, Sylow
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—81no—
259C22×C61—81no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61C2×#12311yescharacteristic
263C22×C61#54811yescharacteristic
264C22×C61#54811yescharacteristic
265C22×C61C2×#13511yescharacteristic
266C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C61#57711yescharacteristic
268C22×C61#57711yescharacteristic
269C4×C61—41no—
270C4×C61—41no—
271C4×C61—81no—
272C4×C61—81no—
273C4×C61—81no—
274C4×C61#52911yescharacteristic
275C4×C61#54111yescharacteristic
276C4×C61#55211yescharacteristic
277C4×C61#55611yescharacteristic
278Dic61—22no—
279Dic61—41no—
280Dic61—41no—
281Dic61—42no—
282Dic61—81no—
283Dic61—81no—
284Dic61—81no—
285Dic61#48214yes—
286#2538—611no—
287#2355—611no—
288#2101—611no—
289#2101—611no—
290#2073—611no—
291#2073D6111yescharacteristic
292#1984—611no—
293#1948—611no—
294#1926—611no—
295#1926—611no—
296#1920—611no—
297#1920—611no—
298#1834—611no—
299#1802—611no—
300C4×#102—611no—
301C2×C61—82no—
302C2×C61#242111yescharacteristic
303C2×C61C2×#48211yescharacteristic
304C2×C61#309911yescharacteristic
305C2×C61#336011yescharacteristic
306C2×C61#336011yescharacteristic
307C2×C61#336811yescharacteristic
308C2×C61#337311yescharacteristic
309C2×C22⋊Q8—1221no—
310C2×C4×Q8—611no—
311C2×C4×D4—611no—
312C2×#106—611no—
313C2×#106—611no—
314C2×C4⋊C8—611no—
315C2×#102—611no—
316C2×#102—611no—
317C2×#102C2×D6111yescharacteristic
318C2×#102C2×D6111yescharacteristic
319C2×C4×C8—611no—
320#396—1221no—
321#396—1221no—
322#391—1221no—
323#390—611no—
324#390C2×D6111yescharacteristic
325#389—611no—
326#389—611no—
327#387—1221no—
328#387—1221no—
329#384—611no—
330#384—611no—
331#382—611no—
332#382C2×D6111yescharacteristic
333#380—1221no—
334C4×C4⋊C4—611no—
335C4×C4⋊C4—611no—
336C4×C4⋊C4—611no—
337C4×C4⋊C4—1221no—
338#340—611no—
339#340—611no—
340#340—611no—
341#340—611no—
342#340—611no—
343#340—611no—
344#340—611no—
345#340—611no—
346#340C2×D6111yescharacteristic
347#340C2×D6111yescharacteristic
348#336—611no—
349#336—611no—
350#336—611no—
351#336C2×D6111yescharacteristic
352C61#2499111yescharacteristic, complemented, Sylow
353C22×Q8—611no—
354C22×D4C22×D6111yescharacteristic
355C23×C4—1221no—
356C22×C8—611no—
357C22×C8—611no—
358C22×C8C22×D6111yescharacteristic
359C22×C8C22×D6111yescharacteristic
360C22⋊Q8—2444no—
361C4×Q8—1222no—
362C4×Q8—1222no—
363C4×D4—1222no—
364C4×D4—1222no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C42—611no—
387C2×C42—611no—
388C2×C42—1221no—
389C2×C42—1221no—
390C2×C42—1221no—
391C2×C42C22×D6111yescharacteristic
392#106—1222no—
393#106—1222no—
394C4⋊C8—1222no—
395#102—22no—
396#102—22no—
397#102—1222no—
398#102—1222no—
399C4×C8—1222no—
400#95—21no—
401#95—21no—
402#95—1221no—
403#95—1221no—
404#95—1221no—
405#95—1221no—
406#95—1221no—
407#95—1221no—
408#95—1221no—
409C24—21no—
410C2×Q8—1222no—
411C2×Q8—1222no—
412C2×Q8—1222no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4#1175822412yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—2442no—
431C22×C4—2442no—
432C22×C4—2442no—
433C22×C4D4×D6111yescharacteristic
434C22×C4D4×D6111yescharacteristic
435C22×C4C23×D6111yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—612no—
455C4⋊C4—1222no—
456C4⋊C4—1222no—
457C4⋊C4—1222no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—4881no—
466C4⋊C4—4881no—
467C4⋊C4—2442no—
468C4⋊C4—2442no—
469C4⋊C4—2442no—
470C4⋊C4—2442no—
471C22⋊C4—42no—
472C22⋊C4—81no—
473C22⋊C4—4881no—
474C22⋊C4—2442no—
475C22⋊C4—2442no—
476C42—21no—
477C42—21no—
478C42—612no—
479C42—612no—
480C42—1221no—
481C42—1221no—
482C42—4881no—
483C42—4881no—
484C42—2442no—
485C23—22no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23C2×D4×D6111yescharacteristic
493Q8—2444no—
494D4—22no—
495D4—22no—
496D4—41no—
497D4—41no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—612no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1222no—
526C2×C4—4884no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4#41×D6111yescharacteristic
539C2×C4#41×D6111yescharacteristic
540C2×C4#41×D6111yescharacteristic
541C2×C4C2×#1175822411yescharacteristic
542C2×C4#1451982411yescharacteristic
543C2×C4#1451982411yescharacteristic
544C8—41no—
545C8—41no—
546C8—2441no—
547C8—2441no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22#123×D6111yescharacteristic
556C22#1684041811yescharacteristic
557C22#1684042011yescharacteristic
558C22#135×D6111yescharacteristic
559C22AGL(1,ℤ/8)×D6111yescharacteristic
560C22#1684056911yescharacteristic
561C22#1684057211yescharacteristic
562C4—41no—
563C4—41no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—614no—
568C4—1222no—
569C4—4881no—
570C4—4881no—
571C4—4881no—
572C4—2442no—
573C4—2441no—
574C4—2441no—
575C4#1684028911yescharacteristic
576C4#1684036911yescharacteristic
577C4#1684044911yescharacteristic
578C4#1684047911yescharacteristic
579C2—82no—
580C2#2144931511yescharacteristic
581C2#482×D6111yescharacteristic
582C2#2145581111yescharacteristic
583C2#2145846311yescharacteristic
584C2#2145846511yescharacteristic
585C2#2145854511yescharacteristic
586C2#2145860611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2073C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2073D61
hypocenterC61#24991
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2073

Complements

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

NH ≅ G/Ntype
2. #21448771C2semidirect
3. #21447390C2semidirect
4. #21447386C2semidirect
6. #21446753C2semidirect
7. #21446581C2semidirect
10. #21446407C2semidirect
11. #21446405C2semidirect
12. #21445945C2semidirect
352. C61#24991semidirect

Maximal subgroups

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

subgroupindex
#214507392
#214487712
#214473902
#214473862
#214472352
#214467532
#214465812
#214464482
#214464432
#214464072
#214464052
#214459452
#214458172
#102×Dic612
C61×#20732
#2499161