Finite Group Databasebeta

← #29663815

Subgroups — #29663815

Order 15616.

716 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456374C211yesmaximal, characteristic, complemented
2#21455918C211yesmaximal, characteristic, complemented
3C2×D4×#92418C211yesmaximal, characteristic, complemented
4#21445272C212yesmaximal, complemented
5#21445254C212yesmaximal, complemented
6#21445196C212yesmaximal, complemented
7#21445158C212yesmaximal, complemented
8#21444580C211yesmaximal, characteristic, complemented
9#21444520C211yesmaximal, characteristic, complemented
10#21444514C211yesmaximal, characteristic, complemented
11C61×#3109C211yesmaximal, characteristic
12C23×#92418—21no—
13#16840319C2211yescharacteristic
14#16840158C2211yescharacteristic, complemented
15#16840134C2211yescharacteristic, complemented
16#16840114C2211yescharacteristic, complemented
17#16840108C2211yescharacteristic, complemented
18#16840097—21no—
19#16840042C2212yes—
20#16839989—21no—
21#16839989—21no—
22#16839967C2212yes—
23#16839913—21no—
24#16839862—22no—
25C4×#11758200—21no—
26C4×#11758200—21no—
27D4×#92418—21no—
28D4×#92418—21no—
29D4×#92418C2212yes—
30D4×#92418C2212yes—
31#16839819C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33C2×C22⋊#92418—21no—
34C2×C4×#92418C2211yescharacteristic
35#16839514C2212yescomplemented
36#16839512C2212yescomplemented
37#16839511C2212yescomplemented
38#16839509C2212yescomplemented
39#16839507C2212yescomplemented
40#16839506C2212yescomplemented
41C61×#532C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C61×#481—21no—
44C61×#476C2212yes—
45C61×#459C2212yes—
46C61×#447—21no—
47C61×#442—22no—
48C61×#432C2211yescharacteristic
49C2×#11758200—22no—
50C2×#11758200—21no—
51C2×#11758200—21no—
52C22×#92418—21no—
53C22×#92418—22no—
54C22×#92418—22no—
55C22×#92418—44no—
56C22×#92418D411yescharacteristic
57C22×#92418D411yescharacteristic
58#14519777C2311yescharacteristic
59#14519777C2311yescharacteristic
60#14519769—21no—
61#14519756—21no—
62#14519756—21no—
63C4×Dic122—21no—
64C4×Dic122—21no—
65#14519736—21no—
66#14519710—22no—
67#14519708—21no—
68#14519708—21no—
69C4⋊#92418—21no—
70C4⋊#92418C2312yes—
71C4⋊#92418C2312yes—
72#14519700—22no—
73#14519700—21no—
74#14519700—21no—
75#14519698—22no—
76#14519698—21no—
77#14519698—21no—
78C22⋊#92418—21no—
79C22⋊#92418—22no—
80C22⋊#92418—22no—
81#14519688—21no—
82C4×#92418D411yescharacteristic
83C4×#92418D411yescharacteristic
84C4×#92418C2311yescharacteristic
85C4×#92418C2311yescharacteristic
86C2×#41×C61—21no—
87C22×D4×C61D411yescharacteristic
88C23×C4×C61—21no—
89C2×D8×C61—22no—
90C4⋊Q8×C61C2311yescharacteristic
91#127×C61C2311yescharacteristic
92#124×C61—21no—
93#121×C61—21no—
94#121×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61D412yes—
100C4×D4×C61C2312yes—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#106×C61—22no—
105C4⋊C8×C61C2312yes—
106#102×C61—22no—
107#102×C61—22no—
108C22⋊C8×C61—21no—
109#97×C61—21no—
110C24×C61—21no—
111#41×C61—82no—
112C2×Q8×C61—21no—
113C2×D4×C61—24no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—21no—
118C2×D4×C61—41no—
119C2×D4×C61C2×D412yes—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61—41no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61#4111yescharacteristic
129C2×Dic122—21no—
130C2×Dic122—21no—
131D8×C61—44no—
132C2×C8×C61—22no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61C2×D412yes—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—41no—
140C22⋊C4×C61—41no—
141C42×C61C2×D411yescharacteristic
142C42×C61C2×D411yescharacteristic
143C42×C61#4111yescharacteristic
144C42×C61C2411yescharacteristic
145C22⋊Dic61—41no—
146#11758200—42no—
147#11758200—44no—
148#11758200—42no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153#11758196—41no—
154#11758196—41no—
155C4×Dic61—41no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418—41no—
160C2×#92418—44no—
161C2×#92418—42no—
162C2×#92418—44no—
163C2×#92418C2×D411yescharacteristic
164C2×#92418C2×D411yescharacteristic
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×D411yescharacteristic
167C23×C61—24no—
168C23×C61—22no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61—41no—
172C23×C61#12111yescharacteristic
173Q8×C61—81no—
174D4×C61—24no—
175D4×C61—22no—
176D4×C61—42no—
177D4×C61—41no—
178D4×C61—41no—
179D4×C61—81no—
180D4×C61—82no—
181D4×C61#13514yes—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—22no—
187C2×C4×C61—44no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—41no—
192C2×C4×C61—82no—
193C2×C4×C61#12111yescharacteristic
194C2×C4×C61#12111yescharacteristic
195C2×C4×C61#12111yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C2×#4111yescharacteristic
199C2×Dic61—41no—
200C2×Dic61—41no—
201Dic122—42no—
202Dic122—41no—
203Dic122—41no—
204C8×C61—42no—
205#92418—21no—
206#92418—21no—
207#92418—21no—
208#92418—42no—
209#92418—42no—
210#92418AGL(1,ℤ/8)12yes—
211#19257—611nomaximal, Sylow
212C22×C61—21no—
213C22×C61—21no—
214C22×C61—24no—
215C22×C61—48no—
216C22×C61—42no—
217C22×C61—42no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61—81no—
221C22×C61#47112yes—
222C22×C61C2×#12111yescharacteristic
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—21no—
227C4×C61—42no—
228C4×C61—81no—
229C4×C61C2×#13512yes—
230C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
231C4×C61#57511yescharacteristic
232Dic61—81no—
233Dic61—81no—
234#3167—611no—
235#3110—611no—
236#3109D6111yescharacteristic
237C2×C8×D4—611no—
238#1683—612no—
239#1681—612no—
240#1671—612no—
241#1665—612no—
242#1568—611no—
243#1552—611no—
244#1551—611no—
245C2×C61—22no—
246C2×C61—44no—
247C2×C61—161no—
248C2×C61#303611yescharacteristic
249C2×C61#305011yescharacteristic
250C2×C61C2×#47111yescharacteristic
251C23×C8—1221no—
252#533—611no—
253#532C2×D6111yescharacteristic
254C2×C4×D4C2×D6111yescharacteristic
255#498—611no—
256#492—611no—
257#487—611no—
258#486—611no—
259#484—1221no—
260#481—21no—
261#476C2×D6112yes—
262#474—612no—
263#465—1221no—
264#465—1221no—
265#461—612no—
266#459C2×D6112yes—
267#450—1221no—
268#447—21no—
269#442—22no—
270#440—1222no—
271C4×SD16—1221no—
272C4×SD16—1221no—
273C8×D4—612no—
274C8×D4—612no—
275C8×D4—1221no—
276C8×D4—1221no—
277#432—611no—
278#432C2×D6111yescharacteristic
279C2×C4⋊C8—611no—
280C2×C22⋊C8—1221no—
281C2×C4×C8—611no—
282#335—612no—
283#334—612no—
284#333—612no—
285#332—612no—
286#332—612no—
287#331—612no—
288C61#1925711yescharacteristic, complemented, Sylow
289C2×#41—21no—
290C22×D4C22⋊D6111yescharacteristic
291C23×C4—21no—
292C2×SD16—1221no—
293C2×SD16—1221no—
294C2×SD16—1222no—
295C2×D8—22no—
296C22×C8—611no—
297C22×C8—611no—
298C22×C8—1222no—
299C22×C8—1222no—
300C22×C8—1221no—
301C22×C8—2444no—
302C4⋊Q8—611no—
303C4⋊Q8—611no—
304C4⋊Q8C22×D6111yescharacteristic
305#127C22×D6111yescharacteristic
306#125—1221no—
307#124—21no—
308C22⋊Q8—1221no—
309C22⋊Q8—1221no—
310#121—21no—
311#121—21no—
312C4×Q8—1221no—
313C4×Q8—1221no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4C22⋊D6112yes—
319C4×D4C22×D6112yes—
320#117—1221no—
321C2×C4⋊C4C22⋊D6111yescharacteristic
322C2×C22⋊C4—21no—
323C2×C42C22×D6111yescharacteristic
324#107—1222no—
325#106—22no—
326#106—1221no—
327#106—1221no—
328C4⋊C8—612no—
329C4⋊C8—612no—
330C4⋊C8—1221no—
331C4⋊C8C22×D6112yes—
332#103—1221no—
333#103—1221no—
334#103—1222no—
335#102—22no—
336#102—22no—
337#102—1221no—
338#102—1221no—
339#102—1222no—
340C22⋊C8—21no—
341C22⋊C8—1222no—
342C22⋊C8—1222no—
343C22⋊C8—1221no—
344#97—21no—
345#97—1221no—
346C4×C8—611no—
347C4×C8—611no—
348C4×C8—611no—
349C4×C8—611no—
350C24—21no—
351#41—82no—
352C2×Q8—21no—
353C2×Q8—1221no—
354C2×Q8—1221no—
355C2×D4—24no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—41no—
361C2×D4C2×C22⋊D6112yes—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4D4×D6111yescharacteristic
369C22×C4#1175822411yescharacteristic
370C22×C4C2×C22⋊D6111yescharacteristic
371SD16—2442no—
372SD16—2442no—
373SD16—2444no—
374D8—44no—
375C2×C8—22no—
376C2×C8—611no—
377C2×C8—611no—
378C2×C8—611no—
379C2×C8—611no—
380C2×C8—1222no—
381C2×C8—1222no—
382C2×C8—1222no—
383C2×C8—2444no—
384C2×C8—2441no—
385C2×C8—2444no—
386C2×C8—2442no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—1221no—
391C4⋊C4—1221no—
392C4⋊C4—1221no—
393C4⋊C4—1221no—
394C4⋊C4—2441no—
395C4⋊C4—2441no—
396C4⋊C4C2×C22⋊D6112yes—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—2441no—
402C42—2441no—
403C42D4×D6111yescharacteristic
404C42#1175822411yescharacteristic
405C42C2×C22⋊D6111yescharacteristic
406C42C23×D6111yescharacteristic
407C23—24no—
408C23—22no—
409C23—42no—
410C23—42no—
411C23—41no—
412C23#1451974911yescharacteristic
413Q8—81no—
414Q8—2442no—
415Q8—2441no—
416Q8—2441no—
417D4—24no—
418D4—22no—
419D4—42no—
420D4—41no—
421D4—41no—
422D4—81no—
423D4—82no—
424D4#1451979714yes—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—82no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4#1451974911yescharacteristic
439C2×C4#1451974911yescharacteristic
440C2×C4#1451974911yescharacteristic
441C2×C4C2×D4×D6111yescharacteristic
442C2×C4C22×C22⋊D6111yescharacteristic
443C2×C4C2×#1175822411yescharacteristic
444C8—42no—
445C8—612no—
446C8—1221no—
447C8—1221no—
448C8—1221no—
449C8—2442no—
450C8—2442no—
451C22—21no—
452C22—21no—
453C22—24no—
454C22—48no—
455C22—42no—
456C22—42no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22#1684002312yes—
461C22C2×#1451974911yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—81no—
468C4—4881no—
469C4—4881no—
470C4C2×#1451979712yes—
471C4AGL(1,ℤ/8)×D6111yescharacteristic
472C4#1684055811yescharacteristic
473C2—22no—
474C2—44no—
475C2—161no—
476C2#2145528011yescharacteristic
477C2#2145539711yescharacteristic
478C2C2×#1684002311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3109C2
socleC22×C61C2×#121
hypercenter#3109D61
hypocenterC61#19257
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
1. #21456374C2semidirect
2. #21455918C2semidirect
3. C2×D4×#92418C2semidirect
4. #21445272C2semidirect
5. #21445254C2semidirect
6. #21445196C2semidirect
7. #21445158C2semidirect
8. #21444580C2semidirect
9. #21444520C2semidirect
10. #21444514C2semidirect
14. #16840158C22semidirect
15. #16840134C22semidirect
16. #16840114C22semidirect
17. #16840108C22semidirect
35. #16839514C22semidirect
36. #16839512C22semidirect
37. #16839511C22semidirect
38. #16839509C22semidirect
39. #16839507C22semidirect
40. #16839506C22semidirect
288. C61#19257semidirect

Maximal subgroups

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

subgroupindex
#214563742
#214559182
C2×D4×#924182
#214452722
#214452542
#214451962
#214451582
#214445802
#214445202
#214445142
C61×#31092
#1925761