Finite Group Databasebeta

← #29779551

Subgroups — #29779551

Order 15616.

1078 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454127C211yesmaximal, characteristic, complemented
2C2×#16839665C211yesmaximal, characteristic
3#21447235C212yesmaximal, complemented
4#21446963C212yesmaximal, complemented
5#21446518C212yesmaximal, complemented
6#21446513C212yesmaximal, complemented
7#21445906C211yesmaximal, characteristic, complemented
8#21445903C211yesmaximal, characteristic, complemented
9#21445750C211yesmaximal, characteristic
10#21445583C211yesmaximal, characteristic, complemented
11C61×#1786C211yesmaximal, characteristic
12C2×#14519768C2211yescharacteristic, complemented
13C22×C4⋊Dic61C2211yescharacteristic
14C22×C22⋊Dic61—21no—
15C22×C22⋊Dic61—21no—
16C22×C4×Dic61C2211yescharacteristic
17C2×#14519698C2212yes—
18C2×#14519698C2212yes—
19C2×C22⋊#92418C2211yescharacteristic
20C2×#14519687C2212yescomplemented
21#16839665—21no—
22#16839665—21no—
23#16839665C2212yes—
24#16839665C2212yes—
25#16839661C2211yescharacteristic, complemented
26#16839658C2211yescharacteristic, complemented
27#16839649—21no—
28#16839645—21no—
29#16839637—21no—
30#16839634—21no—
31#16839627C2211yescharacteristic, complemented
32#16839622C2211yescharacteristic
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
43C23×Dic61—21no—
44C23×Dic61D411yescharacteristic
45C23×Dic61D411yescharacteristic
46C22×#92418C2312yes—
47#14519768—42no—
48C2×C4⋊Dic61—22no—
49C2×C4⋊Dic61—22no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C4⋊Dic61C2311yescharacteristic
55C2×C4⋊Dic61C2311yescharacteristic
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×C22⋊Dic61—21no—
59C2×C22⋊Dic61—22no—
60C2×C22⋊Dic61—22no—
61C2×C22⋊Dic61—22no—
62C2×C22⋊Dic61—22no—
63C2×C22⋊Dic61—44no—
64C2×C22⋊Dic61—44no—
65C2×C4×Dic61—42no—
66C2×C4×Dic61D411yescharacteristic
67C2×C4×Dic61D411yescharacteristic
68C2×C4×Dic61C2311yescharacteristic
69C2×C4×Dic61C2311yescharacteristic
70#14519698—24no—
71#14519698—42no—
72C22⋊#92418—22no—
73#14519687—42no—
74#14519684—21no—
75#14519684—21no—
76#14519684—21no—
77#14519684—21no—
78#14519684—21no—
79C25×C61—21no—
80C22×D4×C61—24no—
81C22×D4×C61—21no—
82C22×D4×C61—22no—
83C22×D4×C61—22no—
84C22×D4×C61—42no—
85C22×D4×C61C2312yes—
86C23×C4×C61C2311yescharacteristic
87C22×C8×C61C2312yes—
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C22⋊C4×C61—21no—
91#102×C61—24no—
92#102×C61—42no—
93C22⋊C8×C61—22no—
94#95×C61—21no—
95C24×C61—24no—
96C24×C61—22no—
97C24×C61—42no—
98C24×C61—42no—
99C24×C61—42no—
100C24×C61—42no—
101C24×C61—42no—
102C24×C61—42no—
103C24×C61C2×D411yescharacteristic
104C2×D4×C61—24no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—48no—
111C2×D4×C61—48no—
112C2×D4×C61—42no—
113C2×D4×C61—42no—
114C2×D4×C61#4114yes—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61#4112yes—
121C22×C4×C61#4112yes—
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—22no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—41no—
129C22×Dic61—41no—
130C22×Dic61—41no—
131C22×Dic61—41no—
132C22×Dic61—42no—
133C22×Dic61—42no—
134C22×Dic61—42no—
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C22⋊C4×C61—42no—
145C22⋊Dic61—44no—
146C22⋊Dic61—44no—
147C22⋊Dic61—44no—
148C22⋊Dic61—44no—
149C22⋊Dic61—42no—
150C4⋊Dic61—22no—
151C4⋊Dic61—22no—
152C4⋊Dic61—22no—
153C4⋊Dic61—22no—
154C4⋊Dic61—42no—
155C4⋊Dic61—42no—
156C4⋊Dic61—42no—
157#11758196—42no—
158#11758196—42no—
159C4×Dic61—22no—
160C4×Dic61—22no—
161C4×Dic61—41no—
162C4×Dic61—41no—
163C2×#92418—22no—
164C2×#92418—22no—
165C2×#92418—22no—
166C23×C61—21no—
167C23×C61—21no—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—24no—
171C23×C61—48no—
172C23×C61—48no—
173C23×C61—48no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—84no—
181C23×C61—84no—
182C23×C61—84no—
183C23×C61—84no—
184C23×C61#12412yes—
185C23×C61AGL(1,ℤ/8)12yes—
186C23×C61AGL(1,ℤ/8)12yes—
187C23×C61C22×D411yescharacteristic
188D4×C61—24no—
189D4×C61—24no—
190D4×C61—48no—
191D4×C61—42no—
192D4×C61—42no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—24no—
196C2×C4×C61—24no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61#12412yes—
204C2×C4×C61C2×#4112yes—
205C2×C4×C61C2×#4111yescharacteristic
206C2×C4×C61C2×#4111yescharacteristic
207C2×C4×C61H(5,𝔽2)12yes—
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—21no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—44no—
221C2×Dic61—44no—
222C2×Dic61—41no—
223C2×Dic61—42no—
224C2×Dic61—41no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—84no—
232C8×C61—42no—
233#92418—42no—
234#29110—611nomaximal, Sylow
235C22×C61—22no—
236C22×C61—22no—
237C22×C61—22no—
238C22×C61—22no—
239C22×C61—22no—
240C22×C61—22no—
241C22×C61—44no—
242C22×C61—44no—
243C22×C61—44no—
244C22×C61—816no—
245C22×C61—82no—
246C22×C61—82no—
247C22×C61—82no—
248C22×C61—82no—
249C22×C61#49014yes—
250C22×C61#52511yescharacteristic
251C22×C61C2×#12411yescharacteristic
252C22×C61#53411yescharacteristic
253C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
254C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
255C22×C61#57511yescharacteristic
256C22×C61#57511yescharacteristic
257C4×C61—22no—
258C4×C61—81no—
259C4×C61—81no—
260C4×C61#55112yes—
261C4×C61#55112yes—
262Dic61—41no—
263Dic61—41no—
264Dic61—41no—
265Dic61—41no—
266Dic61—81no—
267Dic61—82no—
268Dic61—81no—
269#2851—611no—
270C2×#390—611no—
271#2073—612no—
272#2025—612no—
273#1936—612no—
274#1936—612no—
275#1823—611no—
276#1823—611no—
277#1786—611no—
278#1786D6111yescharacteristic
279#1749—611no—
280C2×C61—24no—
281C2×C61—84no—
282C2×C61#256111yescharacteristic
283C2×C61#311411yescharacteristic
284C2×C61#311411yescharacteristic
285C2×C61C2×#49011yescharacteristic
286C2×C61#314011yescharacteristic
287C2×C61#319411yescharacteristic
288C2×C61#319411yescharacteristic
289C23×D4C2×D6111yescharacteristic
290C2×#125—611no—
291C22×C4⋊C4—611no—
292C22×C22⋊C4—1221no—
293C22×C22⋊C4—1221no—
294C22×C42—611no—
295C2×#102—612no—
296C2×#102—612no—
297C2×#102C2×D6112yes—
298C2×#102C2×D6112yes—
299C2×C22⋊C8—611no—
300C2×C22⋊C8C2×D6111yescharacteristic
301C2×#97—612no—
302#390—612no—
303#390—612no—
304#390—1221no—
305#390—1221no—
306#389—611no—
307#389—611no—
308#387—1221no—
309#387—1221no—
310#385—1221no—
311#385—1221no—
312#382—611no—
313#380—611no—
314#380C2×D6111yescharacteristic
315#340—612no—
316#340—612no—
317#340—612no—
318#340—612no—
319#336—612no—
320C61#2911011yescharacteristic, complemented, Sylow
321C25—21no—
322C22×D4—24no—
323C22×D4—22no—
324C22×D4—21no—
325C22×D4—22no—
326C22×D4—42no—
327C22×D4C22×D6112yes—
328C23×C4—611no—
329C23×C4—611no—
330C23×C4—1221no—
331C23×C4C22×D6111yescharacteristic
332C22×C8—612no—
333C22×C8C22×D6112yes—
334#125—2442no—
335C2×C4⋊C4—612no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—1222no—
339C2×C4⋊C4—1222no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4C22×D6111yescharacteristic
346C2×C4⋊C4C22×D6111yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—1222no—
349C2×C22⋊C4—1222no—
350C2×C22⋊C4—1222no—
351C2×C22⋊C4—1222no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—2444no—
354C2×C22⋊C4—2444no—
355C2×C42—611no—
356C2×C42—611no—
357C2×C42—611no—
358C2×C42—611no—
359C2×C42—2442no—
360#102—24no—
361#102—42no—
362#102—1224no—
363#102—2442no—
364C22⋊C8—22no—
365C22⋊C8—1222no—
366#97—2442no—
367#95—21no—
368#95—1221no—
369#95—1221no—
370#95—1221no—
371#95—1221no—
372#95—1221no—
373C24—24no—
374C24—22no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24D4×D6111yescharacteristic
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4#1175822414yes—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—611no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—1221no—
404C22×C4—1221no—
405C22×C4—1222no—
406C22×C4—2441no—
407C22×C4—2441no—
408C22×C4—2441no—
409C22×C4—2441no—
410C22×C4—2442no—
411C22×C4—2442no—
412C22×C4—2442no—
413C22×C4#1175822212yes—
414C22×C4D4×D6111yescharacteristic
415C22×C4#1175822412yes—
416C22×C4C23×D6111yescharacteristic
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—1222no—
421C2×C8—1222no—
422C2×C8—1222no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—1222no—
426C4⋊C4—1222no—
427C4⋊C4—1222no—
428C4⋊C4—1222no—
429C4⋊C4—2442no—
430C4⋊C4—2442no—
431C4⋊C4—2442no—
432C4⋊C4—2442no—
433C4⋊C4—2442no—
434C22⋊C4—42no—
435C22⋊C4—2444no—
436C22⋊C4—2444no—
437C22⋊C4—2444no—
438C22⋊C4—2444no—
439C22⋊C4—2442no—
440C42—1222no—
441C42—1222no—
442C42—2441no—
443C42—2441no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—24no—
449C23—48no—
450C23—48no—
451C23—48no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—84no—
459C23—84no—
460C23—84no—
461C23—84no—
462C23#1451976312yes—
463C23#1451980112yes—
464C23#1451980412yes—
465C23C2×D4×D6111yescharacteristic
466D4—24no—
467D4—24no—
468D4—48no—
469D4—42no—
470D4—42no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—1221no—
482C2×C4—1221no—
483C2×C4—1221no—
484C2×C4—1221no—
485C2×C4—1221no—
486C2×C4—1221no—
487C2×C4—1221no—
488C2×C4—1221no—
489C2×C4—1221no—
490C2×C4—1221no—
491C2×C4—1221no—
492C2×C4—1221no—
493C2×C4—4884no—
494C2×C4—2441no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2442no—
498C2×C4—2441no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4—2441no—
502C2×C4—2441no—
503C2×C4—2444no—
504C2×C4—2444no—
505C2×C4#1451976312yes—
506C2×C4C2×#1175822412yes—
507C2×C4C2×#1175822411yescharacteristic
508C2×C4C2×#1175822211yescharacteristic
509C2×C4#1451982212yes—
510C8—42no—
511C8—2442no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—816no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22#1684012514yes—
527C22#1684026811yescharacteristic
528C22C2×#1451976311yescharacteristic
529C22#1684032211yescharacteristic
530C22C2×#1451980111yescharacteristic
531C22C2×#1451980411yescharacteristic
532C22#1684055811yescharacteristic
533C22#1684055811yescharacteristic
534C4—22no—
535C4—81no—
536C4—81no—
537C4—4882no—
538C4—4881no—
539C4—4881no—
540C4—2441no—
541C4—2441no—
542C4—2441no—
543C4—2441no—
544C4#1684043812yes—
545C4#1684044312yes—
546C2—24no—
547C2—84no—
548C2#2145097411yescharacteristic
549C2#2145595211yescharacteristic
550C2#2145595811yescharacteristic
551C2C2×#1684012511yescharacteristic
552C2#2145620311yescharacteristic
553C2#2145663011yescharacteristic
554C2#2145663611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1786C2
socleC23×C61C22×D4
prime radicalC23×D4×C61C22
hypercenter#1786D61
hypocenterC61#29110
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. #21454127C2semidirect
3. #21447235C2semidirect
4. #21446963C2semidirect
5. #21446518C2semidirect
6. #21446513C2semidirect
7. #21445906C2semidirect
8. #21445903C2semidirect
10. #21445583C2semidirect
12. C2×#14519768C22semidirect
20. C2×#14519687C22semidirect
25. #16839661C22semidirect
26. #16839658C22semidirect
31. #16839627C22semidirect
33. #16839525C22semidirect
34. #16839525C22semidirect
35. #16839524C22semidirect
36. #16839524C22semidirect
37. #16839516C22semidirect
320. C61#29110semidirect

Maximal subgroups

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

subgroupindex
#214541272
C2×#168396652
#214472352
#214469632
#214465182
#214465132
#214459062
#214459032
#214457502
#214455832
C61×#17862
#2911061