Finite Group Databasebeta

← #29664024

Subgroups — #29664024

Order 15616.

766 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456396C211yesmaximal, characteristic, complemented
2#21455278C212yesmaximal, complemented
3#436×D61C211yesmaximal, characteristic, complemented
4#21445379C212yesmaximal, complemented
5#21445378C212yesmaximal, complemented
6#21445333C211yesmaximal, characteristic, complemented
7#21445332C211yesmaximal, characteristic, complemented
8#21445295C211yesmaximal, characteristic, complemented
9#21444573C211yesmaximal, characteristic, complemented
10#21444493C212yesmaximal, complemented
11C61×#1687C211yesmaximal, characteristic, complemented
12C2×#14519801—22no—
13C2×M16×D61—21no—
14#16840310C2212yes—
15C4×D4×D61C2211yescharacteristic
16#16840167C2211yescharacteristic, complemented
17#16840108C2212yescomplemented
18#16839995—21no—
19#16839995—21no—
20#16839976—22no—
21#16839975—22no—
22#16839968C2212yes—
23#16839967C2212yes—
24#16839954C2212yes—
25D8×Dic61—21no—
26D8×Dic61—21no—
27#16839841C2211yescharacteristic
28#16839840C2211yescharacteristic
29#16839839C2211yescharacteristic
30#16839838—21no—
31#16839837—21no—
32#16839784C2212yes—
33C4⋊C8×D61C2211yescharacteristic
34#16839760—22no—
35C22⋊C8×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839513C2211yescharacteristic, complemented
38#16839512C2212yescomplemented
39#16839509C2212yescomplemented
40#16839507C2212yescomplemented
41#16839506C2212yescomplemented
42C61×#493C2211yescharacteristic, complemented
43C61×#459C2212yes—
44C61×#436C2211yescharacteristic
45C61×#334C2211yescharacteristic, complemented
46C61×#331C2212yescomplemented
47C2×#11758224—22no—
48C2×D4×D61D411yescharacteristic
49C22×C4×D61—21no—
50#14519801—48no—
51C2×#11758190—22no—
52C2×#11758200—22no—
53C2×#11758199—22no—
54C2×#11758194—21no—
55C2×#11758189—22no—
56C2×#11758189—21no—
57C2×#11758189—21no—
58M16×D61—44no—
59C2×C8×D61—21no—
60C2×C8×D61D411yescharacteristic
61C2×C8×D61D411yescharacteristic
62#14519777C2312yes—
63#14519764—22no—
64#14519748—22no—
65#14519748—22no—
66C4×D244C2311yescharacteristic
67D4×Dic61—22no—
68D4×Dic61—22no—
69D4×Dic61D411yescharacteristic
70#14519739D411yescharacteristic
71C4×C22⋊D61—21no—
72#14519737—21no—
73C4⋊C4×D61D411yescharacteristic
74C22⋊C4×D61—21no—
75C42×D61C2311yescharacteristic
76#14519711—21no—
77#14519711—21no—
78C4⋊#92418C2312yes—
79C4⋊#92418C2311yescharacteristic
80#14519706—22no—
81#14519706—21no—
82#14519702—22no—
83#14519700—22no—
84#14519698—22no—
85#14519698—22no—
86#14519698—22no—
87C22⋊#92418—21no—
88#14519689—21no—
89#14519689—21no—
90C4×#92418C2311yescharacteristic
91C8×Dic61D411yescharacteristic
92C8×Dic61D411yescharacteristic
93C2×D8×C61—22no—
94C2×D8×C61—21no—
95C2×D8×C61—21no—
96C2×M16×C61—21no—
97#127×C61C2312yes—
98C4×D4×C61C2311yescharacteristic
99C4⋊C8×C61C2312yes—
100C4⋊C8×C61C2311yescharacteristic
101#102×C61—22no—
102C22⋊C8×C61—21no—
103C4×C8×C61C2311yescharacteristic
104C23×D61—21no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61C2×D411yescharacteristic
109C22×C4×C61—21no—
110C2×C22⋊D61—21no—
111C2×C22⋊D61—42no—
112C22×Dic61—21no—
113C22×Dic61—42no—
114#11758224—44no—
115#11758224—44no—
116D4×D61—24no—
117C2×D244C2×D411yescharacteristic
118C2×C4×D61—21no—
119C2×C4×D61—21no—
120C2×C4×D61—21no—
121C2×C4×D61—42no—
122C2×C4×D61—42no—
123C2×C4×D61C2×D411yescharacteristic
124C2×C4×D61C2×D411yescharacteristic
125C2×C4×D61#4111yescharacteristic
126C2×Dic122—22no—
127D8×C61—42no—
128D8×C61—44no—
129D8×C61—42no—
130M16×C61—42no—
131C2×C8×C61—21no—
132C2×C8×C61—22no—
133C2×C8×C61C2×D411yescharacteristic
134C2×C8×C61C2×D411yescharacteristic
135C4⋊C4×C61C2×D411yescharacteristic
136C22⋊C4×C61—21no—
137C42×C61C2411yescharacteristic
138C22⋊Dic61—21no—
139C22⋊Dic61—42no—
140C22⋊Dic61—42no—
141#11758200—44no—
142#11758199—44no—
143#11758198—21no—
144#11758198—21no—
145C4⋊Dic61—22no—
146C4⋊Dic61—22no—
147C4⋊Dic61C2×D411yescharacteristic
148#11758196—21no—
149C4×Dic61C2×D411yescharacteristic
150C4×Dic61C2×D411yescharacteristic
151C4×Dic61#4111yescharacteristic
152#11758194—42no—
153C2×#92418—21no—
154C2×#92418—22no—
155C2×#92418C2×D411yescharacteristic
156C2×#92418C2×D411yescharacteristic
157#11758190—44no—
158#11758189—42no—
159#11758189—44no—
160#11758189—42no—
161C8×D61—22no—
162C8×D61—41no—
163C8×D61—42no—
164C23×C61—21no—
165C23×C61—42no—
166C22×D61—21no—
167C22×D61—42no—
168C22×D61—42no—
169C22×D61—42no—
170C22×D61#12111yescharacteristic
171D4×C61—22no—
172D4×C61—42no—
173D4×C61—42no—
174D4×C61—42no—
175D4×C61—42no—
176D4×C61—82no—
177C2×C4×C61—21no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C2×#4111yescharacteristic
183C22⋊D61—42no—
184C22⋊D61—84no—
185C2×Dic61—21no—
186C2×Dic61—42no—
187C2×Dic61—42no—
188C2×Dic61—84no—
189C2×Dic61#12111yescharacteristic
190C2×Dic61#12111yescharacteristic
191C2×Dic61#12111yescharacteristic
192D244—22no—
193C4×D61—21no—
194C4×D61—21no—
195C4×D61—41no—
196C4×D61—42no—
197C4×D61—42no—
198C4×D61—41no—
199C4×D61—41no—
200C4×D61AGL(1,ℤ/8)14yes—
201Dic122—42no—
202Dic122—42no—
203C8×C61—21no—
204C8×C61—41no—
205C8×C61—42no—
206C8×C61AGL(1,ℤ/8)12yescomplemented
207#92418—21no—
208#92418—21no—
209#92418—41no—
210#92418—42no—
211#19280—611nomaximal, Sylow
212C22×C61—41no—
213C22×C61—41no—
214C22×C61—41no—
215C22×C61—82no—
216C22×C61—82no—
217C22×C61—82no—
218C22×C61C2×#12111yescharacteristic
219C2×D61—21no—
220C2×D61—21no—
221C2×D61—41no—
222C2×D61—41no—
223C2×D61—41no—
224C2×D61—84no—
225C2×D61#46912yes—
226C4×C61—21no—
227C4×C61—41no—
228C4×C61C2×AGL(1,ℤ/8)12yes—
229C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
230C4×C61#57511yescharacteristic
231Dic61—21no—
232Dic61—21no—
233Dic61—41no—
234Dic61—82no—
235Dic61#46912yes—
236#3169—611no—
237#3036—612no—
238C2×#436—611no—
239#1701—612no—
240#1701—612no—
241#1693—611no—
242#1693—611no—
243#1687—611no—
244#1687D6111yescharacteristic, complemented
245#1566—611no—
246#1547—612no—
247C2×C61—81no—
248C2×C61—162no—
249C2×C61#303611yescharacteristic
250C2×C61C2×#46911yescharacteristic
251C2×C61#306711yescharacteristic
252D61—22no—
253D61—81no—
254C2×AGL(1,ℤ/8)—1222no—
255C22×M16—1221no—
256#532—612no—
257C2×C4×D4—611no—
258#500—611no—
259#493C2×D6111yescharacteristic, complemented
260#486—612no—
261#466—1221no—
262#466—1221no—
263#463—1222no—
264#463—1222no—
265#461—612no—
266#461—612no—
267#459—612no—
268#459C2×D6112yes—
269C4×D8—1221no—
270C4×D8—1221no—
271#436—611no—
272#436—611no—
273#436—611no—
274#436—1221no—
275#436—1221no—
276#436C2×D6111yescharacteristic
277#423—612no—
278C2×C4⋊C8—611no—
279#417—1222no—
280C2×C22⋊C8—1221no—
281C2×C4×C8—611no—
282#334—612no—
283#334—611no—
284#334C2×D6111yescharacteristic, complemented
285#332—612no—
286#332—612no—
287#331—612no—
288#331C2×D6112yescomplemented
289C61#1928011yescharacteristic, complemented, Sylow
290C2×#41—1222no—
291C22×D4—611no—
292C23×C4—1221no—
293AGL(1,ℤ/8)—2448no—
294C2×SD16—1222no—
295C2×SD16—1222no—
296C2×D8—22no—
297C2×D8—21no—
298C2×D8—21no—
299C2×D8—1222no—
300C2×M16—21no—
301C2×M16—1221no—
302C2×M16—1222no—
303C2×M16—1221no—
304C2×M16—1221no—
305C2×M16—2444no—
306C22×C8—611no—
307C22×C8—611no—
308C22×C8—1221no—
309C4⋊Q8—612no—
310#127C22×D6112yes—
311#124—1222no—
312#121—1222no—
313#121—1222no—
314C4×D4—611no—
315C4×D4—611no—
316C4×D4—611no—
317C4×D4—1221no—
318C4×D4—1221no—
319C4×D4—1222no—
320C4×D4—1222no—
321C4×D4C22×D6111yescharacteristic
322C2×C4⋊C4—611no—
323C2×C22⋊C4—1221no—
324C2×C42—611no—
325#107—1221no—
326#107—1221no—
327C4⋊C8—611no—
328C4⋊C8—612no—
329C4⋊C8—1221no—
330C4⋊C8—1222no—
331C4⋊C8C22×D6111yescharacteristic
332C4⋊C8C22×D6112yes—
333#103—1222no—
334#103—1222no—
335#102—22no—
336#102—1222no—
337#102—1222no—
338#102—1222no—
339C22⋊C8—21no—
340C22⋊C8—1221no—
341C22⋊C8—1221no—
342C22⋊C8—1221no—
343C4×C8—611no—
344C4×C8—611no—
345C4×C8—611no—
346C4×C8C22×D6111yescharacteristic
347C24—1221no—
348#41—2444no—
349#41—2444no—
350C2×Q8—1222no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—611no—
355C2×D4—1224no—
356C2×D4—1221no—
357C2×D4—2442no—
358C2×D4D4×D6111yescharacteristic
359C22×C4—21no—
360C22×C4—611no—
361C22×C4—611no—
362C22×C4—611no—
363C22×C4—1221no—
364C22×C4—1221no—
365C22×C4—1221no—
366C22×C4—1221no—
367C22×C4—2442no—
368C22×C4—2442no—
369C22×C4—2442no—
370SD16—2444no—
371SD16—2444no—
372D8—42no—
373D8—44no—
374D8—42no—
375D8—2444no—
376M16—42no—
377M16—2442no—
378M16—2444no—
379M16—2442no—
380M16—2442no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—611no—
384C2×C8—611no—
385C2×C8—1222no—
386C2×C8—1221no—
387C2×C8—1222no—
388C2×C8—2441no—
389C2×C8—2442no—
390C2×C8D4×D6111yescharacteristic
391C2×C8D4×D6111yescharacteristic
392C4⋊C4—611no—
393C4⋊C4—1221no—
394C4⋊C4—1222no—
395C4⋊C4—1222no—
396C4⋊C4D4×D6111yescharacteristic
397C22⋊C4—21no—
398C22⋊C4—1221no—
399C22⋊C4—1221no—
400C22⋊C4—1221no—
401C22⋊C4—2442no—
402C22⋊C4—2442no—
403C42—611no—
404C42—611no—
405C42—611no—
406C42C23×D6111yescharacteristic
407C23—21no—
408C23—42no—
409C23—611no—
410C23—1221no—
411C23—2442no—
412C23—2442no—
413C23—2442no—
414Q8—2442no—
415Q8—2442no—
416D4—22no—
417D4—42no—
418D4—42no—
419D4—42no—
420D4—42no—
421D4—82no—
422D4—1222no—
423D4—4884no—
424D4—2442no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—611no—
429C2×C4—614no—
430C2×C4—611no—
431C2×C4—611no—
432C2×C4—1221no—
433C2×C4—1221no—
434C2×C4—1221no—
435C2×C4—4884no—
436C2×C4—2441no—
437C2×C4—2442no—
438C2×C4—2442no—
439C2×C4—2442no—
440C2×C4—2442no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4C2×D4×D6111yescharacteristic
444C2×C4C2×D4×D6111yescharacteristic
445C2×C4#41×D6111yescharacteristic
446C8—21no—
447C8—42no—
448C8—41no—
449C8—1221no—
450C8—1221no—
451C8—2442no—
452C8—2441no—
453C8#1451980112yescomplemented
454C22—41no—
455C22—41no—
456C22—41no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—612no—
461C22—1221no—
462C22—1221no—
463C22—4884no—
464C22—2441no—
465C22—2441no—
466C22—2441no—
467C22#121×D6111yescharacteristic
468C4—21no—
469C4—41no—
470C4—612no—
471C4—1221no—
472C4—1221no—
473C4—4882no—
474C4—2441no—
475C4AGL(1,ℤ/8)×D6112yes—
476C4C2×#1451980111yescharacteristic
477C4#1684055811yescharacteristic
478C2—81no—
479C2—162no—
480C2—1222no—
481C2—4881no—
482C2#2145527811yescharacteristic
483C2#469×D6111yescharacteristic
484C2#2145553711yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1687C2
socleC22×C61C2×#121
hypercenter#1687D61
hypocenterC61#19280
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1687

Complements

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

NH ≅ G/Ntype
1. #21456396C2semidirect
2. #21455278C2semidirect
3. #436×D61C2semidirect
4. #21445379C2semidirect
5. #21445378C2semidirect
6. #21445333C2semidirect
7. #21445332C2semidirect
8. #21445295C2semidirect
9. #21444573C2semidirect
10. #21444493C2semidirect
11. C61×#1687C2semidirect
16. #16840167C22semidirect
17. #16840108C22semidirect
37. #16839513C22semidirect
38. #16839512C22semidirect
39. #16839509C22semidirect
40. #16839507C22semidirect
41. #16839506C22semidirect
42. C61×#493C22semidirect
45. C61×#334C22semidirect
46. C61×#331C22semidirect
206. C8×C61AGL(1,ℤ/8)semidirect
244. #1687D61semidirect
259. #493C2×D61semidirect
284. #334C2×D61semidirect
288. #331C2×D61semidirect
289. C61#19280semidirect
453. C8#14519801semidirect

Maximal subgroups

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

subgroupindex
#214563962
#214552782
#436×D612
#214453792
#214453782
#214453332
#214453322
#214452952
#214445732
#214444932
C61×#16872
#1928061