Finite Group Databasebeta

← #29779543

Subgroups — #29779543

Order 15616.

1122 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454169C211yesmaximal, characteristic, complemented
2C2×#16839665C211yesmaximal, characteristic
3#21447225C212yesmaximal, complemented
4#21446953C212yesmaximal, complemented
5#21446518C212yesmaximal, complemented
6#21446513C212yesmaximal, complemented
7#21445910C211yesmaximal, characteristic, complemented
8#21445907C211yesmaximal, characteristic, complemented
9#21445750C211yesmaximal, characteristic
10#21445583C211yesmaximal, characteristic, complemented
11C61×#1786C211yesmaximal, characteristic
12C2×#14519776C2211yescharacteristic, complemented
13C2×#14519754—21no—
14C22×C4⋊Dic61C2211yescharacteristic
15C22×C22⋊Dic61—21no—
16C22×C22⋊Dic61—21no—
17C22×C4×Dic61C2211yescharacteristic
18C2×#14519698C2212yes—
19C2×#14519698C2212yes—
20C2×C22⋊#92418C2211yescharacteristic
21C2×#14519687C2212yescomplemented
22#16839666—21no—
23#16839665—21no—
24#16839665—21no—
25#16839665C2212yes—
26#16839665C2212yes—
27#16839661C2211yescharacteristic, complemented
28#16839659—21no—
29#16839658C2211yescharacteristic, complemented
30#16839631C2211yescharacteristic, complemented
31#16839622C2211yescharacteristic
32#16839620—21no—
33#16839525C2212yescomplemented
34#16839525C2212yescomplemented
35#16839524C2212yescomplemented
36#16839524C2212yescomplemented
37#16839516C2212yescomplemented
38C23×D4×C61C2211yescharacteristic
39C2×#102×C61C2212yes—
40C2×#102×C61C2212yes—
41C2×C22⋊C8×C61C2211yescharacteristic
42C61×#380C2211yescharacteristic
43C22×Dic122—21no—
44C23×Dic61—21no—
45C23×Dic61D411yescharacteristic
46C23×Dic61D411yescharacteristic
47C22×#92418C2312yes—
48#14519776—42no—
49#14519754—44no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2312yes—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×C4⋊Dic61C2311yescharacteristic
57C2×#11758196—21no—
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×#11758196—21no—
61C2×C22⋊Dic61—21no—
62C2×C22⋊Dic61—22no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61—22no—
66C2×C22⋊Dic61—44no—
67C2×C22⋊Dic61—44no—
68C2×C4×Dic61—42no—
69C2×C4×Dic61D411yescharacteristic
70C2×C4×Dic61D411yescharacteristic
71C2×C4×Dic61C2311yescharacteristic
72C2×C4×Dic61C2311yescharacteristic
73#14519698—24no—
74#14519698—42no—
75C22⋊#92418—22no—
76#14519687—42no—
77#14519684—21no—
78#14519684—21no—
79#14519684—21no—
80C25×C61—21no—
81C22×D4×C61—24no—
82C22×D4×C61—21no—
83C22×D4×C61—22no—
84C22×D4×C61—22no—
85C22×D4×C61—42no—
86C22×D4×C61C2312yes—
87C23×C4×C61C2311yescharacteristic
88C22×C8×C61C2312yes—
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C22⋊C4×C61—21no—
92#102×C61—24no—
93#102×C61—42no—
94C22⋊C8×C61—22no—
95#95×C61—21no—
96C24×C61—24no—
97C24×C61—22no—
98C24×C61—42no—
99C24×C61—42no—
100C24×C61—42no—
101C24×C61—42no—
102C24×C61—42no—
103C24×C61—42no—
104C24×C61C2×D411yescharacteristic
105C2×D4×C61—24no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—48no—
112C2×D4×C61—48no—
113C2×D4×C61—42no—
114C2×D4×C61—42no—
115C2×D4×C61#4114yes—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61#4112yes—
122C22×C4×C61#4112yes—
123C22×C4×C61C2411yescharacteristic
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—41no—
132C22×Dic61—41no—
133C22×Dic61—42no—
134C22×Dic61—42no—
135C22×Dic61—42no—
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C2×Dic122—42no—
141C2×Dic122—42no—
142C2×Dic122—42no—
143C2×C8×C61—22no—
144C2×C8×C61—22no—
145C2×C8×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C22⋊C4×C61—42no—
149C22⋊Dic61—44no—
150C22⋊Dic61—44no—
151C22⋊Dic61—42no—
152C22⋊Dic61—44no—
153C22⋊Dic61—44no—
154C4⋊Dic61—22no—
155C4⋊Dic61—22no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—42no—
159C4⋊Dic61—42no—
160#11758196—42no—
161#11758196—42no—
162#11758196—42no—
163#11758196—42no—
164C4×Dic61—22no—
165C4×Dic61—22no—
166C4×Dic61—41no—
167C4×Dic61—41no—
168C2×#92418—22no—
169C2×#92418—22no—
170C2×#92418—22no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—21no—
175C23×C61—24no—
176C23×C61—48no—
177C23×C61—48no—
178C23×C61—48no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—84no—
186C23×C61—84no—
187C23×C61—84no—
188C23×C61—84no—
189C23×C61#12412yes—
190C23×C61AGL(1,ℤ/8)12yes—
191C23×C61#13712yes—
192C23×C61C22×D411yescharacteristic
193D4×C61—24no—
194D4×C61—24no—
195D4×C61—48no—
196D4×C61—42no—
197D4×C61—42no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—24no—
201C2×C4×C61—24no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61#12412yes—
209C2×C4×C61C2×#4112yes—
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61H(5,𝔽2)12yes—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—21no—
223C2×Dic61—44no—
224C2×Dic61—44no—
225C2×Dic61—41no—
226C2×Dic61—42no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—41no—
234C2×Dic61—84no—
235C2×Dic61AGL(1,ℤ/8)12yes—
236C2×Dic61AGL(1,ℤ/8)12yes—
237Dic122—84no—
238C8×C61—42no—
239#92418—42no—
240#29109—611nomaximal, Sylow
241C22×C61—22no—
242C22×C61—22no—
243C22×C61—22no—
244C22×C61—22no—
245C22×C61—22no—
246C22×C61—22no—
247C22×C61—44no—
248C22×C61—44no—
249C22×C61—44no—
250C22×C61—816no—
251C22×C61—82no—
252C22×C61—82no—
253C22×C61—82no—
254C22×C61—82no—
255C22×C61#48914yes—
256C22×C61#52511yescharacteristic
257C22×C61C2×#12411yescharacteristic
258C22×C61#53411yescharacteristic
259C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C61C2×#13711yescharacteristic
263C4×C61—22no—
264C4×C61—81no—
265C4×C61—81no—
266C4×C61#55112yes—
267C4×C61#55112yes—
268Dic61—22no—
269Dic61—22no—
270Dic61—41no—
271Dic61—41no—
272Dic61—81no—
273Dic61—82no—
274Dic61—81no—
275#2857—611no—
276C2×#390—611no—
277#2071—612no—
278#2023—612no—
279#1936—612no—
280#1936—612no—
281#1824—611no—
282#1824—611no—
283#1786—611no—
284#1786D6111yescharacteristic
285#1749—611no—
286C2×C61—24no—
287C2×C61—84no—
288C2×C61#256111yescharacteristic
289C2×C61#310911yescharacteristic
290C2×C61#311011yescharacteristic
291C2×C61C2×#48911yescharacteristic
292C2×C61#313811yescharacteristic
293C2×C61#319211yescharacteristic
294C2×C61#319311yescharacteristic
295C23×D4C2×D6111yescharacteristic
296C2×C4⋊Q8—611no—
297C2×C22⋊Q8—1221no—
298C22×C4⋊C4—611no—
299C22×C22⋊C4—1221no—
300C22×C22⋊C4—1221no—
301C22×C42—611no—
302C2×#102—612no—
303C2×#102—612no—
304C2×#102C2×D6112yes—
305C2×#102C2×D6112yes—
306C2×C22⋊C8—611no—
307C2×C22⋊C8C2×D6111yescharacteristic
308C2×#97—612no—
309#391—1221no—
310#390—612no—
311#390—612no—
312#390—1221no—
313#390—1221no—
314#389—611no—
315#389—611no—
316#389—1221no—
317#384—611no—
318#380—611no—
319#380—1221no—
320#380C2×D6111yescharacteristic
321#340—612no—
322#340—612no—
323#340—612no—
324#340—612no—
325#336—612no—
326C61#2910911yescharacteristic, complemented, Sylow
327C25—21no—
328C22×Q8—1221no—
329C22×D4—24no—
330C22×D4—22no—
331C22×D4—21no—
332C22×D4—22no—
333C22×D4—42no—
334C22×D4C22×D6112yes—
335C23×C4—611no—
336C23×C4—611no—
337C23×C4—1221no—
338C23×C4C22×D6111yescharacteristic
339C22×C8—612no—
340C22×C8C22×D6112yes—
341C4⋊Q8—2442no—
342C22⋊Q8—2444no—
343C2×C4⋊C4—612no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—611no—
346C2×C4⋊C4—1222no—
347C2×C4⋊C4—1222no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1221no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—1221no—
353C2×C4⋊C4—1221no—
354C2×C4⋊C4C22×D6111yescharacteristic
355C2×C4⋊C4C22×D6111yescharacteristic
356C2×C22⋊C4—21no—
357C2×C22⋊C4—1222no—
358C2×C22⋊C4—1222no—
359C2×C22⋊C4—1222no—
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1221no—
362C2×C22⋊C4—2444no—
363C2×C22⋊C4—2444no—
364C2×C42—611no—
365C2×C42—611no—
366C2×C42—611no—
367C2×C42—611no—
368C2×C42—2442no—
369#102—24no—
370#102—42no—
371#102—1224no—
372#102—2442no—
373C22⋊C8—22no—
374C22⋊C8—1222no—
375#97—2442no—
376#95—21no—
377#95—1221no—
378#95—1221no—
379#95—1221no—
380C24—24no—
381C24—22no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24D4×D6111yescharacteristic
389C2×Q8—2442no—
390C2×Q8—2442no—
391C2×Q8—2442no—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4#1175822414yes—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—611no—
408C22×C4—611no—
409C22×C4—611no—
410C22×C4—611no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—1221no—
416C22×C4—1221no—
417C22×C4—1222no—
418C22×C4—2441no—
419C22×C4—2441no—
420C22×C4—2442no—
421C22×C4—2442no—
422C22×C4—2442no—
423C22×C4#1175822212yes—
424C22×C4D4×D6111yescharacteristic
425C22×C4#1175822412yes—
426C22×C4C23×D6111yescharacteristic
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—1222no—
431C2×C8—1222no—
432C2×C8—1222no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—1222no—
436C4⋊C4—1222no—
437C4⋊C4—1222no—
438C4⋊C4—1222no—
439C4⋊C4—2442no—
440C4⋊C4—2442no—
441C4⋊C4—2442no—
442C4⋊C4—2442no—
443C4⋊C4—2442no—
444C4⋊C4—2442no—
445C22⋊C4—42no—
446C22⋊C4—2444no—
447C22⋊C4—2444no—
448C22⋊C4—2444no—
449C22⋊C4—2444no—
450C22⋊C4—2442no—
451C42—1222no—
452C42—1222no—
453C42—2441no—
454C42—2441no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—24no—
460C23—48no—
461C23—48no—
462C23—48no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—84no—
470C23—84no—
471C23—84no—
472C23—84no—
473C23#1451976312yes—
474C23#1451980112yes—
475C23#1451980612yes—
476C23C2×D4×D6111yescharacteristic
477Q8—4884no—
478D4—24no—
479D4—24no—
480D4—48no—
481D4—42no—
482D4—42no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—612no—
494C2×C4—612no—
495C2×C4—1221no—
496C2×C4—1221no—
497C2×C4—1221no—
498C2×C4—1221no—
499C2×C4—1221no—
500C2×C4—1221no—
501C2×C4—1221no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—4884no—
506C2×C4—2444no—
507C2×C4—2444no—
508C2×C4—2441no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4—2442no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4#1451976312yes—
518C2×C4C2×#1175822412yes—
519C2×C4C2×#1175822411yescharacteristic
520C2×C4C2×#1175822211yescharacteristic
521C2×C4#1451982212yes—
522C8—42no—
523C8—2442no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—816no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22#1684012014yes—
539C22#1684026811yescharacteristic
540C22C2×#1451976311yescharacteristic
541C22#1684032211yescharacteristic
542C22AGL(1,ℤ/8)×D6111yescharacteristic
543C22AGL(1,ℤ/8)×D6111yescharacteristic
544C22C2×#1451980111yescharacteristic
545C22C2×#1451980611yescharacteristic
546C4—22no—
547C4—81no—
548C4—81no—
549C4—1222no—
550C4—1222no—
551C4—4881no—
552C4—4882no—
553C4—4881no—
554C4—2441no—
555C4—2441no—
556C4#1684043812yes—
557C4#1684044312yes—
558C2—24no—
559C2—84no—
560C2#2145097411yescharacteristic
561C2#2145590511yescharacteristic
562C2#2145591311yescharacteristic
563C2C2×#1684012011yescharacteristic
564C2#2145619111yescharacteristic
565C2#2145661711yescharacteristic
566C2#2145662211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1786C2
socleC23×C61C22×D4
prime radicalC23×D4×C61C22
hypercenter#1786D61
hypocenterC61#29109
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
1. #21454169C2semidirect
3. #21447225C2semidirect
4. #21446953C2semidirect
5. #21446518C2semidirect
6. #21446513C2semidirect
7. #21445910C2semidirect
8. #21445907C2semidirect
10. #21445583C2semidirect
12. C2×#14519776C22semidirect
21. C2×#14519687C22semidirect
27. #16839661C22semidirect
29. #16839658C22semidirect
30. #16839631C22semidirect
33. #16839525C22semidirect
34. #16839525C22semidirect
35. #16839524C22semidirect
36. #16839524C22semidirect
37. #16839516C22semidirect
326. C61#29109semidirect

Maximal subgroups

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

subgroupindex
#214541692
C2×#168396652
#214472252
#214469532
#214465182
#214465132
#214459102
#214459072
#214457502
#214455832
C61×#17862
#2910961