Finite Group Databasebeta

← #29652383

Subgroups — #29652383

Order 15616.

1654 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1654 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455069C212yesmaximal, complemented
2#21449138C211yesmaximal, characteristic, complemented
3#21444771C214yesmaximal, complemented
4#21444737C214yesmaximal, complemented
5#21444348C212yesmaximal, complemented
6#21444146C211yesmaximal, characteristic, complemented
7C2×C61×#534C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic, complemented
9C2×#14519803—22no—
10C2×#14519746—21no—
11#16839905—24no—
12#16839900C2214yescomplemented
13#16839895C2214yescomplemented
14#16839761—22no—
15#16839726C2212yescomplemented
16#16839674C2212yescomplemented
17#16839671C2212yescomplemented
18#16839641—21no—
19#16839638—22no—
20#16839625C2211yescharacteristic, complemented
21#16839496C2218yescomplemented
22#16839486C2214yescomplemented
23C23×D4×C61—21no—
24C61×#534—24no—
25C61×#534—22no—
26C61×#534C2214yes—
27C2×#123×C61—22no—
28C2×#121×C61—21no—
29C2×#121×C61C2212yes—
30C2×#120×C61—22no—
31C22×C22⋊C4×C61C2211yescharacteristic
32C2×#11758222—22no—
33C2×#11758222—22no—
34C2×#11758222—44no—
35C22×Dic122C2311yescharacteristic, complemented
36C22×D244C2311yescharacteristic, complemented
37C22×C22⋊D61—21no—
38C22×C22⋊D61—22no—
39C22×C4×D61—21no—
40#14519803—48no—
41C2×#11758200—24no—
42C2×#11758199—24no—
43C2×#11758194—22no—
44#14519746—44no—
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61—22no—
47C2×#11758198—22no—
48#14519700—24no—
49#14519697—24no—
50C22⋊#92418—22no—
51C22⋊#92418C2314yescomplemented
52#14519684—21no—
53#14519684—22no—
54#14519684C2312yescomplemented
55C25×C61—21no—
56C25×C61D411yescharacteristic
57C22×D4×C61—24no—
58C22×D4×C61—22no—
59C22×D4×C61—24no—
60C22×D4×C61—22no—
61C22×D4×C61—21no—
62C22×D4×C61—48no—
63C22×D4×C61D412yescomplemented
64C23×C4×C61—21no—
65C23×C4×C61D411yescharacteristic
66C23×C4×C61C2311yescharacteristic
67#123×C61—28no—
68#123×C61—28no—
69#121×C61—24no—
70#121×C61—22no—
71#121×C61—22no—
72#121×C61—24no—
73#121×C61D414yescomplemented
74#121×C61C2314yes—
75#120×C61—28no—
76#120×C61—28no—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61D412yescomplemented
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61—22no—
81C2×C22⋊C4×C61—22no—
82C2×C22⋊C4×C61—22no—
83C2×C22⋊C4×C61D412yes—
84C2×C22⋊C4×C61C2312yes—
85C24×C61—21no—
86C24×C61—21no—
87C24×C61—24no—
88C24×C61—24no—
89C24×C61—22no—
90C24×C61—44no—
91C24×C61—42no—
92C24×C61—81no—
93C24×C61—82no—
94C24×C61—82no—
95C24×C61C2×D411yescharacteristic
96C24×C61C2×D412yes—
97C23×D61—21no—
98C2×D4×C61—28no—
99C2×D4×C61—24no—
100C2×D4×C61—24no—
101C2×D4×C61—24no—
102C2×D4×C61—22no—
103C2×D4×C61—44no—
104C2×D4×C61—48no—
105C2×D4×C61—416no—
106C2×D4×C61—48no—
107C2×D4×C61—48no—
108C2×D4×C61—48no—
109C2×D4×C61—82no—
110C2×D4×C61C2×D414yes—
111C22×C4×C61—24no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61—21no—
115C22×C4×C61—42no—
116C22×C4×C61—44no—
117C22×C4×C61—81no—
118C22×C4×C61C2×D412yes—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2411yescharacteristic
123C2×C22⋊D61—24no—
124C2×C22⋊D61—42no—
125C2×C22⋊D61—41no—
126C2×C22⋊D61—41no—
127C2×C22⋊D61—44no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—82no—
130C22×Dic61—21no—
131C22×Dic61—22no—
132#11758222—28no—
133#11758222—44no—
134#11758222—44no—
135#11758222—44no—
136C2×D244—22no—
137C2×D244—42no—
138C2×C4×D61—42no—
139C2×C4×D61—42no—
140C2×C4×D61—42no—
141C2×Dic122—22no—
142C2×Dic122—42no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61C2×D414yes—
146C22⋊C4×C61—28no—
147C22⋊C4×C61—24no—
148C22⋊C4×C61—24no—
149C22⋊C4×C61—24no—
150C22⋊C4×C61—24no—
151C22⋊Dic61—44no—
152C22⋊Dic61—42no—
153#11758200—48no—
154#11758199—48no—
155#11758198—82no—
156#11758194—44no—
157C2×#92418—24no—
158C23×C61—24no—
159C23×C61—24no—
160C23×C61—22no—
161C23×C61—22no—
162C23×C61—22no—
163C23×C61—22no—
164C23×C61—24no—
165C23×C61—41no—
166C23×C61—42no—
167C23×C61—42no—
168C23×C61—42no—
169C23×C61—44no—
170C23×C61—48no—
171C23×C61—44no—
172C23×C61—42no—
173C23×C61—44no—
174C23×C61—81no—
175C23×C61—81no—
176C23×C61—88no—
177C23×C61—84no—
178C23×C61—84no—
179C23×C61—816no—
180C23×C61#12011yescharacteristic
181C23×C61#12011yescharacteristic
182C23×C61AGL(1,ℤ/8)14yescomplemented
183C23×C61C22×D411yescharacteristic
184C22×D61—22no—
185C22×D61—42no—
186C22×D61—41no—
187C22×D61—41no—
188C22×D61—42no—
189D4×C61—44no—
190D4×C61—416no—
191D4×C61—48no—
192D4×C61—48no—
193D4×C61—82no—
194D4×C61—84no—
195C2×C4×C61—24no—
196C2×C4×C61—22no—
197C2×C4×C61—24no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61—44no—
202C2×C4×C61—48no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—81no—
206C2×C4×C61#12012yes—
207C2×C4×C61C22×D412yes—
208C22⋊D61—44no—
209C22⋊D61—44no—
210C22⋊D61—88no—
211C2×Dic61—22no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217D244—24no—
218D244—42no—
219C4×D61—48no—
220Dic122—24no—
221Dic122—42no—
222#92418—44no—
223#18132—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—24no—
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—44no—
233C22×C61—44no—
234C22×C61—48no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—44no—
238C22×C61—44no—
239C22×C61—44no—
240C22×C61—82no—
241C22×C61—88no—
242C22×C61—84no—
243C22×C61—816no—
244C22×C61B(4,𝔽2)12yes—
245C22×C61C2×#12011yescharacteristic
246C22×C61C2×AGL(1,ℤ/8)12yes—
247C2×D61—44no—
248C2×D61—42no—
249C2×D61—82no—
250C2×D61—81no—
251C2×D61—81no—
252C4×C61—22no—
253C4×C61—22no—
254C4×C61—44no—
255C4×C61—42no—
256Dic61—42no—
257Dic61—82no—
258C2×#534D6111yescharacteristic, complemented
259#3006—612no—
260#2404—611no—
261#1601—614no—
262#1595—614no—
263#1510—612no—
264#1462—611no—
265C2×C61—22no—
266C2×C61—22no—
267C2×C61—41no—
268C2×C61—44no—
269C2×C61—84no—
270C2×C61#300311yescharacteristic
271C2×C61#301911yescharacteristic
272C2×C61C2×B(4,𝔽2)11yescharacteristic
273D61—82no—
274C22×#41—611no—
275C23×D4—21no—
276C2×AGL(1,ℤ/8)—1222no—
277#534—24no—
278#534—22no—
279#534C2×D6114yes—
280C2×#123—22no—
281C2×#121—21no—
282C2×#121C2×D6112yes—
283C2×#120—22no—
284C2×#120—1221no—
285C22×C22⋊C4C2×D6111yescharacteristic
286#449—1224no—
287#448—614no—
288#447—614no—
289#417—1222no—
290#407—612no—
291#393—612no—
292#392—612no—
293#386—1221no—
294#386—1222no—
295#381—611no—
296#327—618no—
297#323—614no—
298C61#1813211yescharacteristic, complemented, Sylow
299C25—21no—
300C25C22⋊D6111yescharacteristic
301C2×#41—1222no—
302C2×#41—1222no—
303C2×#41—2444no—
304C22×Q8—611no—
305C22×D4—22no—
306C22×D4—24no—
307C22×D4—22no—
308C22×D4—24no—
309C22×D4—21no—
310C22×D4—48no—
311C22×D4—611no—
312C22×D4—1221no—
313C22×D4—1222no—
314C22×D4C22⋊D6112yescomplemented
315C23×C4—21no—
316C23×C4—1221no—
317C23×C4C22⋊D6111yescharacteristic
318C23×C4C22×D6111yescharacteristic
319AGL(1,ℤ/8)—2448no—
320C2×SD16—1224no—
321C2×D8—1224no—
322C2×M16—1222no—
323#123—28no—
324#123—28no—
325#121—24no—
326#121—24no—
327#121—22no—
328#121—22no—
329#121C22⋊D6114yescomplemented
330#121C22×D6114yes—
331#120—28no—
332#120—28no—
333#120—2444no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4C22⋊D6112yescomplemented
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1221no—
342C2×C22⋊C4—1222no—
343C2×C22⋊C4C22⋊D6112yes—
344C2×C22⋊C4C22×D6112yes—
345#103—1224no—
346#102—1224no—
347C22⋊C8—614no—
348C22⋊C8—1222no—
349#95—612no—
350#95—1221no—
351#95—1222no—
352C24—21no—
353C24—21no—
354C24—24no—
355C24—24no—
356C24—22no—
357C24—44no—
358C24—42no—
359C24—81no—
360C24—82no—
361C24—82no—
362C24—1221no—
363C24C2×C22⋊D6111yescharacteristic
364C24C2×C22⋊D6112yes—
365#41—1228no—
366#41—2444no—
367#41—2444no—
368#41—2444no—
369C2×Q8—1222no—
370C2×Q8—2442no—
371C2×D4—28no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—44no—
377C2×D4—48no—
378C2×D4—416no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—82no—
383C2×D4—1224no—
384C2×D4—1222no—
385C2×D4—4882no—
386C2×D4—2442no—
387C2×D4—2444no—
388C2×D4—2444no—
389C2×D4—2442no—
390C2×D4—2441no—
391C2×D4—2441no—
392C2×D4C2×C22⋊D6114yes—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—81no—
400C22×C4—1221no—
401C22×C4—1222no—
402C22×C4—2442no—
403C22×C4—2442no—
404C22×C4—2442no—
405C22×C4C2×C22⋊D6112yes—
406C22×C4C2×C22⋊D6112yes—
407C22×C4C2×C22⋊D6112yes—
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C23×D6111yescharacteristic
410SD16—2448no—
411D8—2448no—
412M16—2444no—
413C2×C8—1224no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4C2×C22⋊D6114yes—
417C22⋊C4—28no—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—4882no—
423C22⋊C4—2444no—
424C22⋊C4—2442no—
425C23—24no—
426C23—24no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—24no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—44no—
437C23—48no—
438C23—44no—
439C23—42no—
440C23—44no—
441C23—81no—
442C23—81no—
443C23—88no—
444C23—84no—
445C23—84no—
446C23—816no—
447C23—1222no—
448C23—2442no—
449C23—2441no—
450C23—2441no—
451C23—2442no—
452C23#1451974611yescharacteristic
453C23#1451974611yescharacteristic
454C23#1451980314yescomplemented
455C23C22×C22⋊D6111yescharacteristic
456Q8—1224no—
457Q8—2442no—
458D4—44no—
459D4—416no—
460D4—48no—
461D4—48no—
462D4—82no—
463D4—84no—
464D4—1224no—
465D4—4888no—
466D4—2442no—
467D4—2444no—
468D4—2444no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—44no—
476C2×C4—48no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—81no—
480C2×C4—1222no—
481C2×C4—2448no—
482C2×C4—2442no—
483C2×C4—2442no—
484C2×C4—2442no—
485C2×C4—2441no—
486C2×C4—2441no—
487C2×C4#1451974612yes—
488C2×C4C22×C22⋊D6112yes—
489C8—2444no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—24no—
497C22—21no—
498C22—44no—
499C22—44no—
500C22—48no—
501C22—42no—
502C22—42no—
503C22—44no—
504C22—44no—
505C22—44no—
506C22—82no—
507C22—88no—
508C22—84no—
509C22—816no—
510C22—4882no—
511C22—4881no—
512C22—4881no—
513C22—2444no—
514C22—2442no—
515C22#1683994912yes—
516C22C2×#1451974611yescharacteristic
517C22C2×#1451980312yes—
518C4—22no—
519C4—22no—
520C4—42no—
521C4—44no—
522C4—4882no—
523C4—2442no—
524C2—22no—
525C2—22no—
526C2—41no—
527C2—44no—
528C2—84no—
529C2—4882no—
530C2#2145504611yescharacteristic
531C2#2145517611yescharacteristic
532C2C2×#1683994911yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#534C2
socleC22×C61C2×#120
hypercenterC2×#534D61
hypocenterC61#18132
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#534

Complements

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

NH ≅ G/Ntype
1. #21455069C2semidirect
2. #21449138C2semidirect
3. #21444771C2semidirect
4. #21444737C2semidirect
5. #21444348C2semidirect
6. #21444146C2semidirect
7. C2×C61×#534C2semidirect
8. C22×#11758222C22semidirect
12. #16839900C22semidirect
13. #16839895C22semidirect
15. #16839726C22semidirect
16. #16839674C22semidirect
17. #16839671C22semidirect
20. #16839625C22semidirect
21. #16839496C22semidirect
22. #16839486C22semidirect
35. C22×Dic122C23semidirect
36. C22×D244C23semidirect
51. C22⋊#92418C23semidirect
54. #14519684C23semidirect
63. C22×D4×C61D4semidirect
73. #121×C61D4semidirect
78. C2×C4⋊C4×C61D4semidirect
182. C23×C61AGL(1,ℤ/8)semidirect
258. C2×#534D61semidirect
298. C61#18132semidirect
314. C22×D4C22⋊D61semidirect
329. #121C22⋊D61semidirect
335. C2×C4⋊C4C22⋊D61semidirect
454. C23#14519803semidirect

Maximal subgroups

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

subgroupindex
#214550692
#214491382
#214447712
#214447372
#214443482
#214441462
C2×C61×#5342
#1813261