Finite Group Databasebeta

← #29653854

Subgroups — #29653854

Order 15616.

1408 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456576C212yesmaximal, complemented
2#21451624C211yesmaximal, characteristic, complemented
3#21444839C214yesmaximal, complemented
4#21444771C214yesmaximal, complemented
5#21444387C212yesmaximal, complemented
6C2×#16839486C211yesmaximal, characteristic, complemented
7C2×C61×#534C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic, complemented
9#16840092—22no—
10#16840078C2214yescomplemented
11#16839989—24no—
12#16839913—22no—
13#16839900C2214yescomplemented
14C2×C22⋊#92418C2212yescomplemented
15#16839698C2212yescomplemented
16#16839685—22no—
17#16839674C2212yescomplemented
18#16839635—21no—
19#16839622C2211yescharacteristic, complemented
20#16839618—21no—
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
32C22×Dic122C2311yescharacteristic, complemented
33C2×#11758200—24no—
34C22×#92418—22no—
35#14519756—22no—
36#14519756—22no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61C2311yescharacteristic, complemented
39C2×#11758196—21no—
40C2×#11758196—22no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—22no—
44C2×C22⋊Dic61—22no—
45#14519708—24no—
46#14519700—24no—
47#14519698—24no—
48C22⋊#92418—22no—
49C22⋊#92418C2314yescomplemented
50#14519684—22no—
51#14519684C2312yescomplemented
52C25×C61—21no—
53C25×C61D411yescharacteristic
54C22×D4×C61—24no—
55C22×D4×C61—22no—
56C22×D4×C61—24no—
57C22×D4×C61—22no—
58C22×D4×C61—21no—
59C22×D4×C61—44no—
60C22×D4×C61—44no—
61C22×D4×C61D412yes—
62C23×C4×C61—21no—
63C23×C4×C61D411yescharacteristic
64C23×C4×C61C2311yescharacteristic
65#123×C61—28no—
66#123×C61—28no—
67#121×C61—24no—
68#121×C61—22no—
69#121×C61—22no—
70#121×C61—24no—
71#121×C61D414yes—
72#121×C61C2314yes—
73#120×C61—28no—
74#120×C61—28no—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61D412yes—
77C2×C22⋊C4×C61—22no—
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61—22no—
81C2×C22⋊C4×C61D412yes—
82C2×C22⋊C4×C61C2312yes—
83C24×C61—21no—
84C24×C61—21no—
85C24×C61—22no—
86C24×C61—24no—
87C24×C61—22no—
88C24×C61—44no—
89C24×C61—81no—
90C24×C61—81no—
91C24×C61—82no—
92C24×C61—82no—
93C24×C61SD1614yescomplemented
94C24×C61C2×D412yes—
95C24×C61C2×D411yescharacteristic
96C2×D4×C61—28no—
97C2×D4×C61—24no—
98C2×D4×C61—24no—
99C2×D4×C61—24no—
100C2×D4×C61—22no—
101C2×D4×C61—48no—
102C2×D4×C61—416no—
103C2×D4×C61—48no—
104C2×D4×C61—48no—
105C2×D4×C61—44no—
106C2×D4×C61—44no—
107C2×D4×C61—82no—
108C2×D4×C61—82no—
109C2×D4×C61C2×D414yes—
110C22×C4×C61—24no—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—21no—
114C22×C4×C61—44no—
115C22×C4×C61—81no—
116C22×C4×C61—81no—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61—21no—
125C2×Dic122—22no—
126C2×Dic122—81no—
127C4⋊C4×C61—24no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61C2×D414yes—
130C22⋊C4×C61—28no—
131C22⋊C4×C61—24no—
132C22⋊C4×C61—24no—
133C22⋊C4×C61—24no—
134C22⋊C4×C61—24no—
135C22⋊Dic61—41no—
136C22⋊Dic61—41no—
137C22⋊Dic61—42no—
138C22⋊Dic61—82no—
139C22⋊Dic61—82no—
140#11758200—84no—
141C4⋊Dic61—22no—
142C4⋊Dic61—81no—
143#11758196—42no—
144#11758196—82no—
145C2×#92418—24no—
146C2×#92418—82no—
147C23×C61—22no—
148C23×C61—24no—
149C23×C61—22no—
150C23×C61—24no—
151C23×C61—24no—
152C23×C61—41no—
153C23×C61—41no—
154C23×C61—42no—
155C23×C61—42no—
156C23×C61—42no—
157C23×C61—42no—
158C23×C61—48no—
159C23×C61—44no—
160C23×C61—44no—
161C23×C61—44no—
162C23×C61—81no—
163C23×C61—81no—
164C23×C61—84no—
165C23×C61—84no—
166C23×C61—84no—
167C23×C61—84no—
168C23×C61—816no—
169C23×C61#12011yescharacteristic
170C23×C61#12011yescharacteristic
171C23×C61C2×SD1614yes—
172C23×C61C2×SD1612yes—
173C23×C61AGL(1,ℤ/8)12yes—
174C23×C61C22×D411yescharacteristic
175D4×C61—44no—
176D4×C61—416no—
177D4×C61—44no—
178D4×C61—44no—
179D4×C61—48no—
180D4×C61—82no—
181D4×C61—84no—
182C2×C4×C61—22no—
183C2×C4×C61—24no—
184C2×C4×C61—42no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—42no—
188C2×C4×C61—48no—
189C2×C4×C61—42no—
190C2×C4×C61—44no—
191C2×C4×C61—42no—
192C2×C4×C61—81no—
193C2×C4×C61#12012yes—
194C2×C4×C61C22×D412yes—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—42no—
202Dic122—44no—
203#92418—44no—
204#18285—611nomaximal, Sylow
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—24no—
212C22×C61—42no—
213C22×C61—42no—
214C22×C61—44no—
215C22×C61—48no—
216C22×C61—42no—
217C22×C61—42no—
218C22×C61—44no—
219C22×C61—44no—
220C22×C61—44no—
221C22×C61—82no—
222C22×C61—84no—
223C22×C61—88no—
224C22×C61—816no—
225C22×C61#45012yes—
226C22×C61#45012yes—
227C22×C61C2×#12011yescharacteristic
228C22×C61C22×SD1611yescharacteristic
229C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
230C4×C61—22no—
231C4×C61—22no—
232C4×C61—44no—
233C4×C61—42no—
234Dic61—82no—
235Dic61—81no—
236Dic61—81no—
237C2×#534D6111yescharacteristic
238#3187—612no—
239#2626—611no—
240#1613—614no—
241#1601—614no—
242#1519—612no—
243C2×#323—611no—
244C2×C61—21no—
245C2×C61—22no—
246C2×C61—21no—
247C2×C61—22no—
248C2×C61—44no—
249C2×C61—84no—
250C2×C61C2×#45011yescharacteristic
251C2×C61#301911yescharacteristic
252C2×C61#302611yescharacteristic
253C23×D4—21no—
254#534—24no—
255#534—22no—
256#534C2×D6114yes—
257C2×#123—22no—
258C2×C22⋊Q8—611no—
259C2×#121—21no—
260C2×#121C2×D6112yes—
261C2×#120—22no—
262C22×C22⋊C4C2×D6111yescharacteristic
263#483—1222no—
264#481—614no—
265#465—1224no—
266#450—1222no—
267#448—614no—
268C2×C22⋊C8—612no—
269#399—612no—
270#396—1222no—
271#393—612no—
272#385—1221no—
273#380—611no—
274#379—1221no—
275#327—618no—
276#323—614no—
277C61#1828511yescharacteristic, complemented, Sylow
278C25—21no—
279C25C22⋊D6111yescharacteristic
280C22×Q8—611no—
281C22×D4—22no—
282C22×D4—24no—
283C22×D4—22no—
284C22×D4—24no—
285C22×D4—21no—
286C22×D4—44no—
287C22×D4—44no—
288C22×D4C22⋊D6112yes—
289C23×C4—21no—
290C23×C4C22⋊D6111yescharacteristic
291C23×C4C22×D6111yescharacteristic
292C2×SD16—1224no—
293C22×C8—1222no—
294#123—28no—
295#123—28no—
296C22⋊Q8—1222no—
297C22⋊Q8—1222no—
298#121—24no—
299#121—24no—
300#121—22no—
301#121—22no—
302#121C22⋊D6114yes—
303#121C22×D6114yes—
304#120—28no—
305#120—28no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4—1221no—
309C2×C4⋊C4—1222no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4C22⋊D6112yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—1222no—
317C2×C22⋊C4—1222no—
318C2×C22⋊C4—1221no—
319C2×C22⋊C4—1221no—
320C2×C22⋊C4C22⋊D6112yes—
321C2×C22⋊C4C22×D6112yes—
322#106—1224no—
323#103—1224no—
324#102—1224no—
325C22⋊C8—614no—
326C22⋊C8—1222no—
327#95—612no—
328#95—1222no—
329C24—21no—
330C24—21no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—44no—
335C24—81no—
336C24—81no—
337C24—82no—
338C24—82no—
339C24#1175820014yescomplemented
340C24C2×C22⋊D6112yes—
341C24C2×C22⋊D6111yescharacteristic
342C2×Q8—1222no—
343C2×Q8—4881no—
344C2×D4—28no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—22no—
349C2×D4—48no—
350C2×D4—416no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—82no—
356C2×D4—82no—
357C2×D4C2×C22⋊D6114yes—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—44no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—1222no—
366C22×C4—1221no—
367C22×C4—1221no—
368C22×C4C2×C22⋊D6112yes—
369C22×C4C2×C22⋊D6112yes—
370C22×C4C2×C22⋊D6112yes—
371C22×C4C2×C22⋊D6111yescharacteristic
372C22×C4C23×D6111yescharacteristic
373SD16—4884no—
374C2×C8—1224no—
375C2×C8—4882no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—1222no—
379C4⋊C4—4882no—
380C4⋊C4—4881no—
381C4⋊C4—2442no—
382C4⋊C4C2×C22⋊D6114yes—
383C22⋊C4—28no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—4882no—
389C22⋊C4—4882no—
390C22⋊C4—2441no—
391C22⋊C4—2441no—
392C22⋊C4—2442no—
393C23—22no—
394C23—24no—
395C23—22no—
396C23—24no—
397C23—24no—
398C23—41no—
399C23—41no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—48no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—81no—
409C23—81no—
410C23—84no—
411C23—84no—
412C23—84no—
413C23—84no—
414C23—816no—
415C23#1451974611yescharacteristic
416C23#1451974611yescharacteristic
417C23C2×#1175820014yes—
418C23C2×#1175820012yes—
419C23#1451980312yes—
420C23C22×C22⋊D6111yescharacteristic
421Q8—2444no—
422D4—44no—
423D4—416no—
424D4—44no—
425D4—44no—
426D4—48no—
427D4—82no—
428D4—84no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—48no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—81no—
440C2×C4—2442no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4—2442no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2442no—
447C2×C4#1451974612yes—
448C2×C4C22×C22⋊D6112yes—
449C8—2444no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—24no—
457C22—42no—
458C22—42no—
459C22—44no—
460C22—48no—
461C22—42no—
462C22—42no—
463C22—44no—
464C22—44no—
465C22—44no—
466C22—82no—
467C22—84no—
468C22—88no—
469C22—816no—
470C22#1683991312yes—
471C22#1683991312yes—
472C22C2×#1451974611yescharacteristic
473C22C22×#1175820011yescharacteristic
474C22C2×#1451980311yescharacteristic
475C4—22no—
476C4—22no—
477C4—42no—
478C4—44no—
479C4—4882no—
480C4—4881no—
481C4—4881no—
482C2—21no—
483C2—22no—
484C2—21no—
485C2—22no—
486C2—44no—
487C2—84no—
488C2C2×#1683991311yescharacteristic
489C2#2145517611yescharacteristic
490C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#534C2
socleC22×C61C2×#120
hypercenterC2×#534D61
hypocenterC61#18285
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. #21456576C2semidirect
2. #21451624C2semidirect
3. #21444839C2semidirect
4. #21444771C2semidirect
5. #21444387C2semidirect
6. C2×#16839486C2semidirect
8. C2×#14519756C22semidirect
10. #16840078C22semidirect
13. #16839900C22semidirect
14. C2×C22⋊#92418C22semidirect
15. #16839698C22semidirect
17. #16839674C22semidirect
19. #16839622C22semidirect
21. #16839496C22semidirect
22. #16839486C22semidirect
32. C22×Dic122C23semidirect
38. C2×C4⋊Dic61C23semidirect
49. C22⋊#92418C23semidirect
51. #14519684C23semidirect
93. C24×C61SD16semidirect
277. C61#18285semidirect
339. C24#11758200semidirect

Maximal subgroups

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

subgroupindex
#214565762
#214516242
#214448392
#214447712
#214443872
C2×#168394862
C2×C61×#5342
#1828561