Finite Group Databasebeta

← #29663579

Subgroups — #29663579

Order 15616.

722 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457135C211yesmaximal, characteristic, complemented
2#21456760C211yesmaximal, characteristic, complemented
3#21454766C211yesmaximal, characteristic, complemented
4#21445300C212yesmaximal, complemented
5#21445288C212yesmaximal, complemented
6#21445176C212yesmaximal, complemented
7#21445164C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444468C211yesmaximal, characteristic, complemented
10C2×#16839506C211yesmaximal, characteristic, complemented
11C61×#3151C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840134—21no—
14#16840108—21no—
15#16840052—21no—
16#16840044C2212yescomplemented
17#16839989—22no—
18#16839967—21no—
19#16839959C2212yes—
20#16839959C2212yes—
21#16839959C2212yescomplemented
22#16839913—22no—
23#16839892C2212yescomplemented
24#16839890—22no—
25C8×Dic122C2212yescomplemented
26#16839815C2211yescharacteristic
27#16839790—21no—
28C2×C4⋊#92418C2211yescharacteristic
29C2×C4⋊#92418C2211yescharacteristic
30#16839720—21no—
31#16839515C2212yescomplemented
32#16839513C2212yescomplemented
33#16839508C2211yescharacteristic, complemented
34#16839508C2211yescharacteristic, complemented
35#16839506C2212yes—
36#16839506C2212yes—
37#16839506C2211yescharacteristic, complemented
38#16839506C2211yescharacteristic, complemented
39C2×AGL(1,ℤ/8)×C61—21no—
40C2×AGL(1,ℤ/8)×C61—21no—
41C61×#532C2211yescharacteristic
42C2×#127×C61C2211yescharacteristic
43C61×#496—22no—
44C61×#493C2212yes—
45C61×#492C2212yes—
46C4×M16×C61C2211yescharacteristic
47C2×#11758200—22no—
48C2×#11758200—22no—
49C2×#11758200—22no—
50C22×#92418—21no—
51C22×#92418—21no—
52#14519777D411yescharacteristic
53#14519777D411yescharacteristic
54#14519769—21no—
55#14519756—21no—
56#14519756D412yes—
57C4×Dic122—21no—
58C4×Dic122C2312yes—
59#14519736—21no—
60#14519708—22no—
61C4⋊#92418—21no—
62C4⋊#92418—21no—
63C4⋊#92418C2311yescharacteristic
64C4⋊#92418C2311yescharacteristic
65C4⋊#92418C2312yes—
66C4⋊#92418C2311yescharacteristic
67C4⋊#92418C2311yescharacteristic
68#14519706—22no—
69#14519706—22no—
70#14519700—22no—
71#14519698—22no—
72#14519698—22no—
73#14519698—22no—
74C22⋊#92418—21no—
75#14519687—22no—
76C4×#92418—21no—
77C8×Dic61—22no—
78C2×#41×C61—21no—
79C22×D4×C61—21no—
80C22×D4×C61—21no—
81C22×D4×C61—21no—
82AGL(1,ℤ/8)×C61—44no—
83AGL(1,ℤ/8)×C61—44no—
84C2×SD16×C61—22no—
85C2×SD16×C61—22no—
86C2×D8×C61—22no—
87C2×D8×C61—22no—
88C2×M16×C61D411yescharacteristic
89C2×M16×C61D411yescharacteristic
90C4⋊Q8×C61C2311yescharacteristic
91#127×C61—21no—
92#127×C61—21no—
93#127×C61—21no—
94#127×C61C2312yes—
95#127×C61C2311yescharacteristic
96#124×C61—21no—
97#121×C61—21no—
98#121×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#97×C61D412yes—
103C4×C8×C61C2312yes—
104C24×C61—41no—
105#41×C61—42no—
106#41×C61—42no—
107C2×Q8×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61—41no—
115C2×D4×C61—41no—
116C2×D4×C61—41no—
117C2×D4×C61—41no—
118C2×D4×C61—42no—
119C2×D4×C61—42no—
120C2×D4×C61—81no—
121C2×D4×C61—81no—
122C2×D4×C61—82no—
123C22×C4×C61—41no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61#4111yescharacteristic
127C2×Dic122—21no—
128C2×Dic122C2×D412yes—
129SD16×C61—44no—
130SD16×C61—44no—
131D8×C61—44no—
132D8×C61—44no—
133M16×C61—22no—
134M16×C61SD1614yescomplemented
135C2×C8×C61C2×D412yes—
136C2×C8×C61C2×D412yes—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C22⋊C4×C61—41no—
140C22⋊C4×C61—41no—
141C42×C61C2×D411yescharacteristic
142C42×C61C2×D411yescharacteristic
143C42×C61#4111yescharacteristic
144C42×C61C2411yescharacteristic
145C22⋊Dic61—22no—
146#11758200—82no—
147#11758200—82no—
148#11758200—82no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61C2×D412yes—
153#11758196—22no—
154#11758196—22no—
155C4×Dic61—22no—
156C2×#92418—21no—
157C2×#92418—21no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418—22no—
161C2×#92418—42no—
162C2×#92418—81no—
163C23×C61—41no—
164C23×C61—42no—
165C23×C61—82no—
166C23×C61—82no—
167C23×C61—82no—
168C23×C61#12111yescharacteristic
169Q8×C61—41no—
170Q8×C61—41no—
171D4×C61—41no—
172D4×C61—42no—
173D4×C61—41no—
174D4×C61—42no—
175D4×C61—41no—
176D4×C61—42no—
177D4×C61—42no—
178D4×C61—41no—
179D4×C61—81no—
180D4×C61—82no—
181D4×C61—82no—
182D4×C61—82no—
183D4×C61—82no—
184D4×C61—82no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—82no—
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61#12111yescharacteristic
192C2×C4×C61#12111yescharacteristic
193C2×C4×C61C2×SD1612yes—
194C2×C4×C61AGL(1,ℤ/8)12yes—
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C2×#4111yescharacteristic
198C2×Dic61—22no—
199C2×Dic61—22no—
200Dic122—22no—
201Dic122—22no—
202Dic122—81no—
203C8×C61—22no—
204C8×C61C2×SD1614yes—
205#92418—41no—
206#92418—41no—
207#92418—42no—
208#92418—41no—
209#92418—41no—
210#19238—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—81no—
214C22×C61—82no—
215C22×C61—82no—
216C22×C61—81no—
217C22×C61—82no—
218C22×C61—81no—
219C22×C61—164no—
220C22×C61#46012yes—
221C22×C61C2×#12111yescharacteristic
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—81no—
226C4×C61#46012yes—
227C4×C61C22×SD1611yescharacteristic
228C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
229C4×C61#57611yescharacteristic
230C4×C61#57711yescharacteristic
231Dic61—42no—
232Dic61—42no—
233#3246—611no—
234#3208—611no—
235#3151D6111yescharacteristic
236#2962—611no—
237#1688—612no—
238#1686—612no—
239#1668—612no—
240#1666—612no—
241#1565—611no—
242#1540—611no—
243C2×#331—611no—
244C2×C61—22no—
245C2×C61—161no—
246C2×C61—162no—
247C2×C61C2×#46011yescharacteristic
248C2×C61#306211yescharacteristic
249C2×C61#306811yescharacteristic
250C2×AGL(1,ℤ/8)—21no—
251C2×AGL(1,ℤ/8)—21no—
252#533—611no—
253#532C2×D6111yescharacteristic
254C2×#127C2×D6111yescharacteristic
255#496—22no—
256#493C2×D6112yes—
257#492—1221no—
258#492C2×D6112yes—
259#486—1221no—
260#476—1221no—
261#475—612no—
262#465—1222no—
263#461—1221no—
264#460—612no—
265#460—612no—
266#460—612no—
267#450—1222no—
268#446—612no—
269#446—1222no—
270C8×Q8—612no—
271#431—611no—
272#424—1221no—
273C2×C4⋊C8—611no—
274C2×C4⋊C8—611no—
275#405—1221no—
276C4×M16C2×D6111yescharacteristic
277#335—612no—
278#334—612no—
279#332—611no—
280#332—611no—
281#331—612no—
282#331—612no—
283#331—611no—
284#331—611no—
285C61#1923811yescharacteristic, complemented, Sylow
286C2×#41—21no—
287C22×D4—21no—
288C22×D4—21no—
289C22×D4—21no—
290AGL(1,ℤ/8)—44no—
291AGL(1,ℤ/8)—44no—
292C2×SD16—22no—
293C2×SD16—22no—
294C2×SD16—1222no—
295C2×SD16—1222no—
296C2×SD16—1222no—
297C2×D8—22no—
298C2×D8—22no—
299C2×M16C22⋊D6111yescharacteristic
300C2×M16C22⋊D6111yescharacteristic
301C22×C8—1221no—
302C22×C8—1221no—
303C4⋊Q8—611no—
304C4⋊Q8—611no—
305C4⋊Q8C22×D6111yescharacteristic
306#127—21no—
307#127—21no—
308#127—21no—
309#127C22×D6112yes—
310#127C22×D6111yescharacteristic
311#125—1221no—
312#124—21no—
313C22⋊Q8—612no—
314C22⋊Q8—1221no—
315#121—21no—
316#121—21no—
317C4×Q8—612no—
318C4×Q8—1221no—
319C4×D4—21no—
320C4×D4—21no—
321#117—1221no—
322C2×C42C22×D6111yescharacteristic
323#106—1222no—
324C4⋊C8—611no—
325C4⋊C8—611no—
326C4⋊C8—612no—
327C4⋊C8—611no—
328C4⋊C8—611no—
329C4⋊C8—1222no—
330C4⋊C8—1222no—
331C4⋊C8—1221no—
332C4⋊C8—1221no—
333#103—1222no—
334#102—1222no—
335#102—1222no—
336#102—1222no—
337C22⋊C8—1221no—
338#97—1222no—
339#97C22⋊D6112yes—
340C4×C8—1222no—
341C4×C8—1221no—
342C4×C8C22×D6112yes—
343C24—41no—
344#41—42no—
345#41—42no—
346C2×Q8—21no—
347C2×Q8—612no—
348C2×Q8—1221no—
349C2×D4—21no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—21no—
354C2×D4—22no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—81no—
362C2×D4—81no—
363C2×D4—82no—
364C22×C4—41no—
365C22×C4D4×D6111yescharacteristic
366C22×C4#1175822411yescharacteristic
367C22×C4C2×C22⋊D6111yescharacteristic
368SD16—44no—
369SD16—44no—
370SD16—4882no—
371SD16—4882no—
372SD16—4882no—
373D8—44no—
374D8—44no—
375M16—22no—
376M16#1175820014yescomplemented
377C2×C8—1221no—
378C2×C8—1221no—
379C2×C8—1222no—
380C2×C8—1221no—
381C2×C8—1221no—
382C2×C8—4881no—
383C2×C8—2442no—
384C2×C8C2×C22⋊D6112yes—
385C2×C8C2×C22⋊D6112yes—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—612no—
389C4⋊C4—1222no—
390C4⋊C4—1222no—
391C4⋊C4—1221no—
392C4⋊C4—1221no—
393C4⋊C4—1221no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—1222no—
397C42—1222no—
398C42D4×D6111yescharacteristic
399C42#1175822411yescharacteristic
400C42C2×C22⋊D6111yescharacteristic
401C42C23×D6111yescharacteristic
402C23—41no—
403C23—42no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23#1451974811yescharacteristic
408Q8—41no—
409Q8—41no—
410Q8—1222no—
411Q8—1222no—
412Q8—4881no—
413D4—41no—
414D4—42no—
415D4—41no—
416D4—42no—
417D4—41no—
418D4—42no—
419D4—42no—
420D4—41no—
421D4—81no—
422D4—82no—
423D4—82no—
424D4—82no—
425D4—82no—
426D4—82no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—82no—
432C2×C4—1222no—
433C2×C4—1222no—
434C2×C4#1451974811yescharacteristic
435C2×C4#1451974811yescharacteristic
436C2×C4#1451974811yescharacteristic
437C2×C4C2×#1175820012yes—
438C2×C4#1451980312yes—
439C2×C4C2×D4×D6111yescharacteristic
440C2×C4C22×C22⋊D6111yescharacteristic
441C2×C4C2×#1175822411yescharacteristic
442C8—22no—
443C8—2441no—
444C8—2441no—
445C8—2442no—
446C8—2441no—
447C8—2441no—
448C8C2×#1175820014yes—
449C22—21no—
450C22—21no—
451C22—82no—
452C22—81no—
453C22—81no—
454C22—82no—
455C22—82no—
456C22—81no—
457C22—164no—
458C22#1683995912yes—
459C22C2×#1451974811yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—81no—
464C4—2442no—
465C4—2442no—
466C4#1683995912yes—
467C4C22×#1175820011yescharacteristic
468C4C2×#1451980311yescharacteristic
469C4#1684056611yescharacteristic
470C4#1684057211yescharacteristic
471C2—22no—
472C2—162no—
473C2—161no—
474C2C2×#1683995911yescharacteristic
475C2#2145549311yescharacteristic
476C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3151C2
socleC22×C61C2×#121
hypercenter#3151D61
hypocenterC61#19238
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3151

Complements

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

NH ≅ G/Ntype
1. #21457135C2semidirect
2. #21456760C2semidirect
3. #21454766C2semidirect
4. #21445300C2semidirect
5. #21445288C2semidirect
6. #21445176C2semidirect
7. #21445164C2semidirect
8. #21444571C2semidirect
9. #21444468C2semidirect
10. C2×#16839506C2semidirect
16. #16840044C22semidirect
21. #16839959C22semidirect
23. #16839892C22semidirect
25. C8×Dic122C22semidirect
31. #16839515C22semidirect
32. #16839513C22semidirect
33. #16839508C22semidirect
34. #16839508C22semidirect
37. #16839506C22semidirect
38. #16839506C22semidirect
134. M16×C61SD16semidirect
285. C61#19238semidirect
376. M16#11758200semidirect

Maximal subgroups

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

subgroupindex
#214571352
#214567602
#214547662
#214453002
#214452882
#214451762
#214451642
#214445712
#214444682
C2×#168395062
C61×#31512
#1923861