Finite Group Databasebeta

← #29663197

Subgroups — #29663197

Order 15616.

660 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457135C211yesmaximal, characteristic, complemented
2#21456141C211yesmaximal, characteristic, complemented
3#21454531C211yesmaximal, characteristic, complemented
4#21445364C212yesmaximal, complemented
5#21445350C212yesmaximal, complemented
6#21445268C212yesmaximal, complemented
7#21445254C212yesmaximal, complemented
8C2×#16839514C211yesmaximal, characteristic, complemented
9#21444507C211yesmaximal, characteristic, complemented
10#21444496C211yesmaximal, characteristic, complemented
11C61×#3245C211yesmaximal, characteristic
12C2×#14519784—21no—
13#16840319C2211yescharacteristic
14#16840116—21no—
15#16840116—21no—
16#16840112C2211yescharacteristic
17#16840112C2211yescharacteristic
18#16840108C2211yescharacteristic
19#16840108C2211yescharacteristic
20#16840052C2212yes—
21#16840044—21no—
22#16839989—22no—
23#16839967C2212yes—
24#16839959—21no—
25#16839840—21no—
26#16839840C2212yes—
27D4×#92418—21no—
28D4×#92418C2212yes—
29#16839786C2211yescharacteristic
30C2×C4⋊#92418C2211yescharacteristic
31#16839758—21no—
32#16839758—21no—
33#16839729—21no—
34C2×C4×#92418C2211yescharacteristic
35#16839514C2212yes—
36#16839514C2212yes—
37#16839510C2212yes—
38#16839509C2212yes—
39#16839508C2212yes—
40#16839506C2212yes—
41C61×#532C2211yescharacteristic
42C61×#532C2211yescharacteristic
43C61×#474—21no—
44C61×#474C2212yes—
45C61×#466—22no—
46C61×#459—21no—
47C61×#459C2212yes—
48C2×C4⋊C8×C61C2211yescharacteristic
49C2×#11758200—22no—
50#14519784—44no—
51C2×#11758194—21no—
52C2×#11758194—22no—
53C22×#92418—21no—
54C22×#92418—22no—
55C22×#92418D411yescharacteristic
56C22×#92418D411yescharacteristic
57#14519777C2311yescharacteristic
58#14519777C2311yescharacteristic
59#14519769—21no—
60#14519756—21no—
61#14519756—21no—
62C4×Dic122—21no—
63C4×Dic122—21no—
64#14519736—21no—
65#14519708—22no—
66C4⋊#92418—21no—
67C4⋊#92418—21no—
68C4⋊#92418C2312yes—
69C4⋊#92418C2312yes—
70#14519701—21no—
71#14519701—21no—
72#14519701—21no—
73#14519701—21no—
74#14519700—22no—
75#14519698—22no—
76#14519698—21no—
77#14519698—21no—
78#14519698—21no—
79#14519698—21no—
80C22⋊#92418—22no—
81C22⋊#92418—22no—
82C4×#92418D411yescharacteristic
83C4×#92418D411yescharacteristic
84C4×#92418C2311yescharacteristic
85C4×#92418C2311yescharacteristic
86C2×#41×C61—21no—
87C2×#41×C61—21no—
88C2×D8×C61—22no—
89C22×C8×C61—21no—
90C4⋊Q8×C61D411yescharacteristic
91C4⋊Q8×C61C2311yescharacteristic
92#127×C61D411yescharacteristic
93#127×C61C2311yescharacteristic
94#124×C61—21no—
95#124×C61—21no—
96#121×C61—21no—
97#121×C61—21no—
98#121×C61—21no—
99#121×C61D412yes—
100C4×D4×C61—21no—
101C4×D4×C61—21no—
102C4×D4×C61—21no—
103C4×D4×C61C2312yes—
104C2×C42×C61C2311yescharacteristic
105#107×C61—22no—
106C4⋊C8×C61—21no—
107C4⋊C8×C61C2312yes—
108#102×C61—22no—
109#102×C61—22no—
110#41×C61—42no—
111#41×C61—42no—
112#41×C61—42no—
113#41×C61—42no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—41no—
123C2×D4×C61C2×D412yes—
124C22×C4×C61—22no—
125C22×C4×C61—41no—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61#4111yescharacteristic
129C2×Dic122—21no—
130C2×Dic122—21no—
131D8×C61—82no—
132C2×C8×C61—22no—
133C2×C8×C61—81no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61C2×D412yes—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—22no—
140C22⋊C4×C61—41no—
141C22⋊C4×C61—41no—
142C42×C61C2×D411yescharacteristic
143C42×C61C2×D411yescharacteristic
144C42×C61#4111yescharacteristic
145C42×C61C2411yescharacteristic
146C22⋊Dic61—41no—
147#11758200—82no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152#11758196—41no—
153#11758196—41no—
154C4×Dic61—41no—
155#11758194—44no—
156#11758194—42no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—22no—
161C2×#92418—44no—
162C2×#92418—81no—
163C2×#92418C2×D411yescharacteristic
164C2×#92418C2×D411yescharacteristic
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×D411yescharacteristic
167C23×C61—22no—
168C23×C61—41no—
169C23×C61#12111yescharacteristic
170Q8×C61—42no—
171Q8×C61—41no—
172Q8×C61—41no—
173D4×C61—22no—
174D4×C61—22no—
175D4×C61—42no—
176D4×C61—44no—
177D4×C61—41no—
178D4×C61—41no—
179D4×C61—41no—
180D4×C61—41no—
181D4×C61—42no—
182D4×C61—81no—
183D4×C61—82no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—22no—
187C2×C4×C61—41no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—41no—
191C2×C4×C61—82no—
192C2×C4×C61#12111yescharacteristic
193C2×C4×C61#12111yescharacteristic
194C2×C4×C61#12111yescharacteristic
195C2×C4×C61#13512yes—
196C2×C4×C61AGL(1,ℤ/8)12yes—
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C2×#4111yescharacteristic
200C2×Dic61—41no—
201C2×Dic61—41no—
202Dic122—41no—
203Dic122—41no—
204Dic122—81no—
205C8×C61—42no—
206#92418—21no—
207#92418—21no—
208#92418—21no—
209#92418—21no—
210#92418—42no—
211#92418—42no—
212#19201—611nomaximal, Sylow
213C22×C61—21no—
214C22×C61—21no—
215C22×C61—42no—
216C22×C61—42no—
217C22×C61—42no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61—81no—
221C22×C61#46312yes—
222C22×C61C2×#12111yescharacteristic
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—42no—
227C4×C61—81no—
228C4×C61#46312yes—
229C4×C61C2×#13511yescharacteristic
230C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
231C4×C61#57611yescharacteristic
232C4×C61#57811yescharacteristic
233Dic61—81no—
234Dic61—81no—
235#3246—611no—
236#3245D6111yescharacteristic
237#3131—611no—
238#2931—611no—
239#1699—612no—
240#1697—612no—
241#1683—612no—
242#1681—612no—
243C2×#335—611no—
244#1550—611no—
245#1547—611no—
246C2×C61—22no—
247C2×C61—82no—
248C2×C61—161no—
249C2×C61C2×#46311yescharacteristic
250C2×C61#306111yescharacteristic
251C2×C61#306711yescharacteristic
252C2×#131—1221no—
253#533—611no—
254#532C2×D6111yescharacteristic
255#532C2×D6111yescharacteristic
256#488—1221no—
257#488—1221no—
258#487—611no—
259#487—611no—
260#486—611no—
261#486—611no—
262#476—612no—
263#475—1221no—
264#474—21no—
265#474C2×D6112yes—
266#466—22no—
267#465—1222no—
268#461—612no—
269#460—1221no—
270#459—21no—
271#459C2×D6112yes—
272#436—612no—
273#436—1221no—
274C8×D4—612no—
275C8×D4—1221no—
276#423—611no—
277C2×C4⋊C8—611no—
278C2×C4⋊C8C2×D6111yescharacteristic
279#416—1221no—
280#416—1221no—
281#408—1221no—
282C2×C4×C8—611no—
283#335—612no—
284#335—612no—
285#333—612no—
286#332—612no—
287#332—612no—
288#331—612no—
289C61#1920111yescharacteristic, complemented, Sylow
290C2×#41—21no—
291C2×#41—21no—
292C2×SD16—1222no—
293C2×D8—22no—
294#131—2444no—
295C2×M16—1222no—
296C2×M16—1221no—
297C22×C8—21no—
298C22×C8—611no—
299C22×C8—611no—
300C22×C8—1222no—
301C22×C8—1221no—
302C4⋊Q8—611no—
303C4⋊Q8—611no—
304C4⋊Q8C22⋊D6111yescharacteristic
305C4⋊Q8C22×D6111yescharacteristic
306#127C22⋊D6111yescharacteristic
307#127C22×D6111yescharacteristic
308#125—1221no—
309#124—21no—
310#124—21no—
311C22⋊Q8—1221no—
312C22⋊Q8—1221no—
313#121—21no—
314#121—21no—
315#121—21no—
316#121C22⋊D6112yes—
317C4×Q8—1221no—
318C4×Q8—1221no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4C22×D6112yes—
323#117—1221no—
324C2×C42C22×D6111yescharacteristic
325#107—22no—
326#106—1222no—
327C4⋊C8—21no—
328C4⋊C8—612no—
329C4⋊C8—612no—
330C4⋊C8—1221no—
331C4⋊C8—1221no—
332C4⋊C8C22×D6112yes—
333#103—1221no—
334#103—1221no—
335#103—1221no—
336#103—1221no—
337#103—1222no—
338#102—22no—
339#102—22no—
340#102—1221no—
341#102—1221no—
342#102—1221no—
343#102—1221no—
344#102—1222no—
345C22⋊C8—1222no—
346C22⋊C8—1222no—
347C4×C8—611no—
348C4×C8—611no—
349C4×C8—611no—
350C4×C8—611no—
351#41—42no—
352#41—42no—
353#41—42no—
354#41—42no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—1221no—
358C2×Q8—1221no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—41no—
366C2×D4C2×C22⋊D6112yes—
367C22×C4—22no—
368C22×C4—41no—
369C22×C4D4×D6111yescharacteristic
370C22×C4#1175822411yescharacteristic
371C22×C4C2×C22⋊D6111yescharacteristic
372SD16—4882no—
373D8—82no—
374M16—2444no—
375M16—2442no—
376C2×C8—22no—
377C2×C8—81no—
378C2×C8—611no—
379C2×C8—611no—
380C2×C8—611no—
381C2×C8—611no—
382C2×C8—1222no—
383C2×C8—1222no—
384C2×C8—1222no—
385C2×C8—1222no—
386C2×C8—4881no—
387C2×C8—2444no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—1221no—
392C4⋊C4—1221no—
393C4⋊C4—1221no—
394C4⋊C4—1221no—
395C4⋊C4—2441no—
396C4⋊C4—2441no—
397C4⋊C4C2×C22⋊D6112yes—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—2441no—
403C42—2441no—
404C42D4×D6111yescharacteristic
405C42#1175822411yescharacteristic
406C42C2×C22⋊D6111yescharacteristic
407C42C23×D6111yescharacteristic
408C23—22no—
409C23—41no—
410C23#1451974911yescharacteristic
411Q8—42no—
412Q8—41no—
413Q8—41no—
414Q8—4881no—
415Q8—2441no—
416Q8—2441no—
417D4—22no—
418D4—22no—
419D4—42no—
420D4—44no—
421D4—42no—
422D4—41no—
423D4—41no—
424D4—41no—
425D4—41no—
426D4—81no—
427D4—82no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—41no—
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×C4#1451979612yes—
442C2×C4#1451980112yes—
443C2×C4C2×D4×D6111yescharacteristic
444C2×C4C22×C22⋊D6111yescharacteristic
445C2×C4C2×#1175822411yescharacteristic
446C8—42no—
447C8—1221no—
448C8—1221no—
449C8—1221no—
450C8—1221no—
451C8—2442no—
452C8—2442no—
453C22—21no—
454C22—21no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22#1683997512yes—
462C22C2×#1451974911yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—81no—
468C4—4881no—
469C4—4881no—
470C4#1683997512yes—
471C4C2×#1451979611yescharacteristic
472C4C2×#1451980111yescharacteristic
473C4#1684056511yescharacteristic
474C4#1684057711yescharacteristic
475C2—22no—
476C2—82no—
477C2—161no—
478C2C2×#1683997511yescharacteristic
479C2#2145548611yescharacteristic
480C2#2145553611yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3245C2
socleC22×C61C2×#121
hypercenter#3245D61
hypocenterC61#19201
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #21457135C2semidirect
2. #21456141C2semidirect
3. #21454531C2semidirect
4. #21445364C2semidirect
5. #21445350C2semidirect
6. #21445268C2semidirect
7. #21445254C2semidirect
8. C2×#16839514C2semidirect
9. #21444507C2semidirect
10. #21444496C2semidirect
289. C61#19201semidirect

Maximal subgroups

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

subgroupindex
#214571352
#214561412
#214545312
#214453642
#214453502
#214452682
#214452542
C2×#168395142
#214445072
#214444962
C61×#32452
#1920161