Finite Group Databasebeta

← #29664030

Subgroups — #29664030

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#21456399C211yesmaximal, characteristic, complemented
2#21455284C212yesmaximal, complemented
3#436×D61C211yesmaximal, characteristic, complemented
4#21445383C212yesmaximal, complemented
5#21445382C212yesmaximal, complemented
6#21445329C211yesmaximal, characteristic, complemented
7#21445299C211yesmaximal, characteristic, complemented
8#21445298C211yesmaximal, characteristic, complemented
9#21444573C211yesmaximal, characteristic, complemented
10#21444495C212yesmaximal, complemented
11C61×#1693C211yesmaximal, characteristic, complemented
12C2×#14519804—22no—
13C2×M16×D61—21no—
14#16840313C2212yes—
15C4×D4×D61C2211yescharacteristic
16#16840139C2211yescharacteristic, complemented
17#16840110C2212yescomplemented
18#16839999—21no—
19#16839999—21no—
20#16839980—22no—
21#16839979—22no—
22#16839964C2212yes—
23#16839958C2212yes—
24#16839957C2212yes—
25#16839844—21no—
26#16839844—21no—
27#16839841C2211yescharacteristic
28#16839840C2211yescharacteristic
29#16839839C2211yescharacteristic
30#16839838—21no—
31#16839837—21no—
32#16839784C2212yes—
33C4⋊C8×D61C2211yescharacteristic
34#16839762—22no—
35C22⋊C8×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839513C2211yescharacteristic, complemented
38#16839512C2212yescomplemented
39#16839508C2212yescomplemented
40#16839505C2212yescomplemented
41#16839505C2212yescomplemented
42C61×#500C2211yescharacteristic, complemented
43C61×#461C2212yes—
44C61×#436C2211yescharacteristic
45C61×#334C2211yescharacteristic, complemented
46C61×#332C2212yescomplemented
47C2×#11758226—22no—
48C2×D4×D61D411yescharacteristic
49C22×C4×D61—21no—
50#14519804—48no—
51C2×#11758201—22no—
52C2×D488—22no—
53C2×D488—21no—
54C2×D488—21no—
55C2×#11758199—22no—
56C2×#11758194—21no—
57C2×#11758189—22no—
58C2×#11758189—21no—
59C2×#11758189—21no—
60M16×D61—44no—
61C2×C8×D61—21no—
62C2×C8×D61D411yescharacteristic
63C2×C8×D61D411yescharacteristic
64#14519774C2312yes—
65#14519766—22no—
66#14519752—22no—
67#14519752—22no—
68C4×D244C2311yescharacteristic
69D4×Dic61D411yescharacteristic
70#14519739—22no—
71#14519739—22no—
72#14519739D411yescharacteristic
73C4×C22⋊D61—21no—
74#14519737—21no—
75C4⋊C4×D61D411yescharacteristic
76C22⋊C4×D61—21no—
77C42×D61C2311yescharacteristic
78C4⋊#92418C2312yes—
79C4⋊#92418C2311yescharacteristic
80#14519706—22no—
81#14519706—21no—
82#14519701—22no—
83#14519699—22no—
84#14519697—22no—
85#14519697—22no—
86#14519697—22no—
87C22⋊#92418—21no—
88#14519689—21no—
89#14519689—21no—
90C4×#92418C2311yescharacteristic
91C8×Dic61D411yescharacteristic
92C8×Dic61D411yescharacteristic
93C2×SD16×C61—22no—
94C2×M16×C61—21no—
95C4⋊Q8×C61C2312yes—
96C4×D4×C61C2311yescharacteristic
97#107×C61—21no—
98#107×C61—21no—
99C4⋊C8×C61C2312yes—
100C4⋊C8×C61C2311yescharacteristic
101#103×C61—22no—
102C22⋊C8×C61—21no—
103C4×C8×C61C2311yescharacteristic
104C23×D61—21no—
105C2×Q8×C61—22no—
106C2×D4×C61C2×D411yescharacteristic
107C22×C4×C61—21no—
108C2×C22⋊D61—21no—
109C22×Dic61—21no—
110#11758226—44no—
111#11758226—44no—
112D4×D61—24no—
113C2×D244—22no—
114C2×D244—22no—
115C2×D244—22no—
116C2×D244—42no—
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×D61—42no—
124C2×C4×D61C2×D411yescharacteristic
125C2×C4×D61C2×D411yescharacteristic
126C2×C4×D61#4111yescharacteristic
127SD16×C61—44no—
128M16×C61—42no—
129C2×C8×C61—21no—
130C2×C8×C61—22no—
131C2×C8×C61C2×D411yescharacteristic
132C2×C8×C61C2×D411yescharacteristic
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61C2×D411yescharacteristic
136C22⋊C4×C61—21no—
137C42×C61C2411yescharacteristic
138C22⋊Dic61—21no—
139#11758201—44no—
140#11758199—44no—
141#11758198—21no—
142#11758198—21no—
143#11758198—42no—
144#11758198—42no—
145C4⋊Dic61C2×D411yescharacteristic
146#11758196—21no—
147C4×Dic61C2×D411yescharacteristic
148C4×Dic61C2×D411yescharacteristic
149C4×Dic61#4111yescharacteristic
150#11758194—42no—
151C2×#92418—21no—
152C2×#92418—22no—
153C2×#92418C2×D411yescharacteristic
154C2×#92418C2×D411yescharacteristic
155D488—42no—
156D488—44no—
157D488—42no—
158#11758189—42no—
159#11758189—44no—
160#11758189—42no—
161C8×D61—22no—
162C8×D61—41no—
163C8×D61—42no—
164C23×C61—21no—
165C22×D61—21no—
166C22×D61—42no—
167C22×D61—42no—
168C22×D61—42no—
169C22×D61—42no—
170C22×D61#12111yescharacteristic
171Q8×C61—42no—
172Q8×C61—42no—
173D4×C61—22no—
174C2×C4×C61—21no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—42no—
178C2×C4×C61C22×D411yescharacteristic
179C2×C4×C61C22×D411yescharacteristic
180C2×C4×C61C2×#4111yescharacteristic
181C22⋊D61—42no—
182C2×Dic61—21no—
183C2×Dic61—42no—
184C2×Dic61#12111yescharacteristic
185C2×Dic61#12111yescharacteristic
186C2×Dic61#12111yescharacteristic
187D244—22no—
188D244—42no—
189D244—42no—
190D244—42no—
191D244—42no—
192D244—82no—
193D244—84no—
194C4×D61—21no—
195C4×D61—21no—
196C4×D61—41no—
197C4×D61—42no—
198C4×D61—42no—
199C4×D61—41no—
200C4×D61—41no—
201C4×D61—84no—
202C4×D61AGL(1,ℤ/8)14yes—
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×C61C2×#12111yescharacteristic
216C2×D61—21no—
217C2×D61—21no—
218C2×D61—41no—
219C2×D61—41no—
220C2×D61—41no—
221C2×D61—84no—
222C2×D61—82no—
223C2×D61—82no—
224C2×D61—82no—
225C2×D61#46912yes—
226C4×C61—21no—
227C4×C61—41no—
228C4×C61—82no—
229C4×C61C2×AGL(1,ℤ/8)12yes—
230C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
231C4×C61#57511yescharacteristic
232Dic61—21no—
233Dic61—21no—
234Dic61—41no—
235Dic61#46912yes—
236#3169—611no—
237#3036—612no—
238C2×#436—611no—
239#1701—612no—
240#1701—612no—
241#1693—611no—
242#1693D6111yescharacteristic, complemented
243#1687—611no—
244#1687—611no—
245#1566—611no—
246#1547—612no—
247C2×C61—81no—
248C2×C61#303611yescharacteristic
249C2×C61C2×#46911yescharacteristic
250C2×C61#306711yescharacteristic
251D61—22no—
252D61—81no—
253D61—162no—
254C2×AGL(1,ℤ/8)—1222no—
255C22×M16—1221no—
256#532—612no—
257C2×C4×D4—611no—
258#500C2×D6111yescharacteristic, complemented
259#493—611no—
260#486—612no—
261#466—1221no—
262#466—1221no—
263#463—1222no—
264#463—1222no—
265#461—612no—
266#461C2×D6112yes—
267#459—612no—
268#459—612no—
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#332C2×D6112yescomplemented
287#331—612no—
288#331—612no—
289C61#1928011yescharacteristic, complemented, Sylow
290C2×#41—1222no—
291C22×D4—611no—
292C23×C4—1221no—
293AGL(1,ℤ/8)—2448no—
294C2×SD16—22no—
295C2×SD16—1222no—
296C2×D8—1222no—
297C2×D8—1222no—
298C2×D8—1221no—
299C2×D8—1221no—
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⋊Q8C22×D6112yes—
310#127—612no—
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—21no—
326#107—21no—
327C4⋊C8—611no—
328C4⋊C8—612no—
329C4⋊C8—1221no—
330C4⋊C8—1222no—
331C4⋊C8C22×D6111yescharacteristic
332C4⋊C8C22×D6112yes—
333#103—22no—
334#103—1222no—
335#102—1222no—
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—22no—
351C2×D4—611no—
352C2×D4—1224no—
353C2×D4—1221no—
354C2×D4—1222no—
355C2×D4—1222no—
356C2×D4—1222no—
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—44no—
371SD16—2444no—
372D8—2442no—
373D8—2444no—
374D8—2444no—
375D8—2442no—
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—22no—
393C4⋊C4—22no—
394C4⋊C4—611no—
395C4⋊C4—1221no—
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—611no—
409C23—1221no—
410C23—2442no—
411C23—2442no—
412C23—2442no—
413C23—2442no—
414Q8—42no—
415Q8—42no—
416D4—22no—
417D4—1222no—
418D4—4882no—
419D4—4884no—
420D4—2442no—
421D4—2442no—
422D4—2442no—
423D4—2442no—
424D4—2442no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—611no—
430C2×C4—614no—
431C2×C4—611no—
432C2×C4—611no—
433C2×C4—1221no—
434C2×C4—1221no—
435C2×C4—1221no—
436C2×C4—4884no—
437C2×C4—2441no—
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#1451980412yescomplemented
454C22—41no—
455C22—41no—
456C22—41no—
457C22—612no—
458C22—1221no—
459C22—1221no—
460C22—4884no—
461C22—4882no—
462C22—4882no—
463C22—4882no—
464C22—2441no—
465C22—2441no—
466C22—2441no—
467C22#121×D6111yescharacteristic
468C4—21no—
469C4—41no—
470C4—82no—
471C4—612no—
472C4—1221no—
473C4—1221no—
474C4—2441no—
475C4AGL(1,ℤ/8)×D6112yes—
476C4C2×#1451980411yescharacteristic
477C4#1684056111yescharacteristic
478C2—81no—
479C2—1222no—
480C2—9762no—
481C2—4881no—
482C2#2145528411yescharacteristic
483C2#469×D6111yescharacteristic
484C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1693C2
socleC22×C61C2×#121
hypercenter#1693D61
hypocenterC61#19280
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1693

Complements

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

NH ≅ G/Ntype
1. #21456399C2semidirect
2. #21455284C2semidirect
3. #436×D61C2semidirect
4. #21445383C2semidirect
5. #21445382C2semidirect
6. #21445329C2semidirect
7. #21445299C2semidirect
8. #21445298C2semidirect
9. #21444573C2semidirect
10. #21444495C2semidirect
11. C61×#1693C2semidirect
16. #16840139C22semidirect
17. #16840110C22semidirect
37. #16839513C22semidirect
38. #16839512C22semidirect
39. #16839508C22semidirect
40. #16839505C22semidirect
41. #16839505C22semidirect
42. C61×#500C22semidirect
45. C61×#334C22semidirect
46. C61×#332C22semidirect
206. C8×C61AGL(1,ℤ/8)semidirect
242. #1693D61semidirect
258. #500C2×D61semidirect
284. #334C2×D61semidirect
286. #332C2×D61semidirect
289. C61#19280semidirect
453. C8#14519804semidirect

Maximal subgroups

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

subgroupindex
#214563992
#214552842
#436×D612
#214453832
#214453822
#214453292
#214452992
#214452982
#214445732
#214444952
C61×#16932
#1928061