Finite Group Databasebeta

← #29663796

Subgroups — #29663796

Order 15616.

744 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456379C211yesmaximal, characteristic, complemented
2#21455612C211yesmaximal, characteristic, complemented
3#21455285C211yesmaximal, characteristic
4C2×#16839827C211yesmaximal, characteristic, complemented
5#21445241C212yesmaximal, complemented
6#21445231C212yesmaximal, complemented
7#21445197C212yesmaximal, complemented
8#21445163C212yesmaximal, complemented
9#21444574C211yesmaximal, characteristic, complemented
10#21444494C211yesmaximal, characteristic, complemented
11C61×#1550C211yesmaximal, characteristic, complemented
12C2×#14519805—21no—
13C22×C8×D61—21no—
14#16840312C2211yescharacteristic
15C2×C4×D244C2211yescharacteristic
16#16840158C2211yescharacteristic, complemented
17#16840136C2211yescharacteristic, complemented
18#16840112C2211yescharacteristic, complemented
19#16840110C2211yescharacteristic, complemented
20#16840059—22no—
21#16840049C2212yes—
22#16840039C2212yes—
23#16839993—21no—
24#16839993—21no—
25#16839978—22no—
26#16839968C2212yes—
27#16839958C2212yes—
28C4×#11758201—21no—
29C4×#11758201—21no—
30#16839827—21no—
31#16839827—21no—
32#16839827C2212yes—
33#16839827C2212yes—
34#16839805—21no—
35#16839786C2211yescharacteristic
36#16839766—21no—
37#16839764—21no—
38C2×#14519689—21no—
39C2×C4×#92418C2211yescharacteristic
40#16839512C2212yescomplemented
41#16839512C2212yescomplemented
42#16839507C2212yescomplemented
43#16839505C2212yescomplemented
44C61×#533C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#333C2212yescomplemented
48C61×#332C2212yescomplemented
49C2×#11758222—21no—
50C22×D244D411yescharacteristic
51C22×C4×D61—21no—
52#14519805—44no—
53C2×#11758190—22no—
54C2×#11758201—21no—
55C2×#11758201—21no—
56C2×D488—22no—
57C2×#11758194—21no—
58C22×#92418D411yescharacteristic
59C22×#92418D411yescharacteristic
60C2×C8×D61—22no—
61C2×C8×D61—22no—
62C2×C8×D61—44no—
63#14519777C2311yescharacteristic
64#14519774C2311yescharacteristic
65#14519765—21no—
66#14519751—21no—
67#14519751—21no—
68C4×D244—21no—
69C4×D244—21no—
70C4×D244—21no—
71C4×D244—21no—
72C4×D244D412yes—
73C4×D244C2312yes—
74C2×C4⋊Dic61D411yescharacteristic
75C2×#11758198—21no—
76#14519710—22no—
77#14519708—22no—
78#14519708—21no—
79#14519708—21no—
80C4⋊#92418—21no—
81C4⋊#92418C2312yes—
82#14519702—22no—
83#14519701—21no—
84#14519701—21no—
85#14519699—22no—
86#14519697—22no—
87#14519697—22no—
88#14519697—21no—
89#14519697—21no—
90#14519689—22no—
91#14519689—22no—
92C4×#92418D411yescharacteristic
93C4×#92418D411yescharacteristic
94C4×#92418C2311yescharacteristic
95C4×#92418C2311yescharacteristic
96C2×M16×C61—21no—
97C22×C8×C61—21no—
98C4⋊Q8×C61C2311yescharacteristic
99C4⋊Q8×C61C2311yescharacteristic
100#125×C61—21no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61—21no—
103C4×Q8×C61—21no—
104C4×Q8×C61—21no—
105#117×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—21no—
108C4⋊C8×C61—21no—
109C4⋊C8×C61C2312yes—
110C4⋊C8×C61C2312yes—
111C23×D61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61#4111yescharacteristic
117C2×C22⋊D61—41no—
118C22×Dic61—21no—
119#11758222—42no—
120#11758222—42no—
121C2×D244—21no—
122C2×D244—22no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244SD1614yescomplemented
127C2×D244C2×D412yes—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—42no—
131C2×C4×D61—42no—
132C2×C4×D61—41no—
133C2×Dic122—21no—
134M16×C61—42no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C2×C8×C61—42no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61—41no—
144C22⋊C4×C61—41no—
145C42×C61—41no—
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C42×C61#4111yescharacteristic
149C42×C61C2411yescharacteristic
150#11758201—42no—
151#11758201—42no—
152#11758198—22no—
153#11758198—22no—
154#11758198—41no—
155#11758198—41no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61C2×D412yes—
160#11758194—42no—
161C2×#92418—22no—
162C2×#92418—22no—
163C2×#92418—41no—
164C2×#92418C2×D411yescharacteristic
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×D411yescharacteristic
167C2×#92418C2×D411yescharacteristic
168D488—44no—
169#11758190—44no—
170C8×D61—44no—
171C8×D61—44no—
172C23×C61#12111yescharacteristic
173C22×D61—22no—
174C22×D61—24no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—41no—
178Q8×C61—41no—
179Q8×C61—41no—
180Q8×C61—81no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61#12111yescharacteristic
187C2×C4×C61#12111yescharacteristic
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61C2×SD1612yes—
190C2×C4×C61#13512yes—
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C22⋊D61—82no—
195C2×Dic61—22no—
196C2×Dic61—42no—
197C2×Dic61—41no—
198D244—22no—
199D244—24no—
200D244—41no—
201D244—41no—
202D244—41no—
203D244—41no—
204D244—81no—
205D244C2×SD1614yes—
206C4×D61—44no—
207C4×D61—42no—
208C4×D61—42no—
209C4×D61—82no—
210Dic122—41no—
211Dic122—41no—
212C8×C61—42no—
213C8×C61—42no—
214#92418—21no—
215#92418—21no—
216#92418—21no—
217#92418—42no—
218#92418AGL(1,ℤ/8)12yescomplemented
219#19254—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61#46512yes—
223C22×C61C2×#12111yescharacteristic
224C2×D61—24no—
225C2×D61—48no—
226C2×D61—42no—
227C2×D61—42no—
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—81no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—81no—
235C4×C61—81no—
236C4×C61#46512yes—
237C4×C61C22×SD1611yescharacteristic
238C4×C61C2×#13511yescharacteristic
239C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
240C4×C61#57511yescharacteristic
241Dic61—42no—
242Dic61—81no—
243#3167—611no—
244#3077—611no—
245#3036—611no—
246C2×C8×D4—611no—
247#1679—612no—
248#1677—612no—
249#1671—612no—
250#1665—612no—
251#1566—611no—
252#1550D6111yescharacteristic, complemented
253#1547—611no—
254C2×C61—22no—
255C2×C61#303611yescharacteristic
256C2×C61C2×#46511yescharacteristic
257C2×C61#306711yescharacteristic
258D61—44no—
259D61—161no—
260C2×AGL(1,ℤ/8)—1221no—
261C23×C8—1221no—
262#533C2×D6111yescharacteristic
263#532—611no—
264C2×C4×D4—611no—
265#498—611no—
266#492—611no—
267#487—611no—
268#486—611no—
269#478—1222no—
270#476—612no—
271#474—612no—
272#465—1221no—
273#465—1221no—
274#463—1222no—
275#461—612no—
276#459—612no—
277C4×SD16—1221no—
278C4×SD16—1221no—
279C8×D4—612no—
280C8×D4—612no—
281C8×D4—1221no—
282C8×D4—1221no—
283#428—1221no—
284#423—611no—
285#423C2×D6111yescharacteristic
286C2×C4⋊C8C2×D6111yescharacteristic
287#418—1221no—
288#417—1221no—
289C2×C22⋊C8—1221no—
290C2×C4×C8—611no—
291#334—612no—
292#334—612no—
293#333C2×D6112yescomplemented
294#332—612no—
295#332C2×D6112yescomplemented
296#331—612no—
297C61#1925411yescharacteristic, complemented, Sylow
298C2×#41—1221no—
299C22×D4—611no—
300C23×C4—1221no—
301AGL(1,ℤ/8)—2444no—
302C2×SD16—1222no—
303C2×SD16—1221no—
304C2×SD16—1221no—
305C2×D8—1222no—
306C2×M16—21no—
307C2×M16—1221no—
308C22×C8—21no—
309C22×C8—611no—
310C22×C8—611no—
311C22×C8—1222no—
312C22×C8—1222no—
313C22×C8—2444no—
314C4⋊Q8—611no—
315C4⋊Q8C22×D6111yescharacteristic
316C4⋊Q8C22×D6111yescharacteristic
317#127—611no—
318#125—21no—
319#124—1221no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322#121—1221no—
323#121—1221no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×D4—612no—
327C4×D4—612no—
328C4×D4—1221no—
329C4×D4—1221no—
330C4×D4—1221no—
331C4×D4—1221no—
332#117—21no—
333C2×C4⋊C4—611no—
334C2×C22⋊C4—1221no—
335C2×C42C22×D6111yescharacteristic
336#107—1222no—
337#106—1222no—
338#106—1221no—
339#106—1221no—
340C4⋊C8—21no—
341C4⋊C8—21no—
342C4⋊C8—612no—
343C4⋊C8—1221no—
344C4⋊C8C22×D6112yes—
345C4⋊C8C22×D6112yes—
346#103—1222no—
347#103—1221no—
348#103—1221no—
349#102—1221no—
350#102—1221no—
351#102—1222no—
352#102—1222no—
353#102—1222no—
354C22⋊C8—1222no—
355C22⋊C8—1222no—
356C4×C8—611no—
357C4×C8—611no—
358C4×C8—611no—
359C4×C8—611no—
360C24—1221no—
361#41—2442no—
362#41—2442no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—1221no—
366C2×D4—612no—
367C2×D4—614no—
368C2×D4—1221no—
369C2×D4—1221no—
370C2×D4—1221no—
371C2×D4—1221no—
372C2×D4—1222no—
373C2×D4—2441no—
374C22×C4—1221no—
375C22×C4—1222no—
376C22×C4—1222no—
377C22×C4—2442no—
378C22×C4—2442no—
379C22×C4—2441no—
380C22×C4#1175822211yescharacteristic
381C22×C4D4×D6111yescharacteristic
382C22×C4D4×D6111yescharacteristic
383SD16—2444no—
384SD16—2442no—
385SD16—2442no—
386D8—2444no—
387M16—42no—
388M16—2442no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—42no—
392C2×C8—611no—
393C2×C8—611no—
394C2×C8—611no—
395C2×C8—611no—
396C2×C8—1222no—
397C2×C8—1222no—
398C2×C8—2444no—
399C2×C8—2444no—
400C2×C8—2441no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—612no—
408C4⋊C4—1221no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C22⋊C4—41no—
412C22⋊C4—1222no—
413C22⋊C4—1222no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C42—41no—
417C42#1175822211yescharacteristic
418C42D4×D6111yescharacteristic
419C42D4×D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—1224no—
422C23—1222no—
423C23—2442no—
424C23—2442no—
425C23—2441no—
426C23#1451975011yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—81no—
430Q8—2441no—
431Q8—2441no—
432D4—614no—
433D4—1224no—
434D4—1222no—
435D4—4881no—
436D4—4882no—
437D4—2441no—
438D4—2441no—
439D4—2441no—
440D4—2441no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—1222no—
447C2×C4—4882no—
448C2×C4—2444no—
449C2×C4—2441no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4#1451975011yescharacteristic
454C2×C4#1451975011yescharacteristic
455C2×C4#1451975011yescharacteristic
456C2×C4SD16×D6112yes—
457C2×C4#1451980012yes—
458C2×C4C2×D4×D6111yescharacteristic
459C2×C4C2×D4×D6111yescharacteristic
460C2×C4C2×#1175822211yescharacteristic
461C8—42no—
462C8—42no—
463C8—612no—
464C8—1221no—
465C8—1221no—
466C8—1221no—
467C8—2442no—
468C22—21no—
469C22—21no—
470C22—1224no—
471C22—4881no—
472C22—4881no—
473C22—4881no—
474C22—2448no—
475C22—2442no—
476C22—2442no—
477C22#1683999412yes—
478C22C2×#1451975011yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—81no—
483C4—81no—
484C4—4881no—
485C4—2442no—
486C4#1683999412yes—
487C4C2×SD16×D6111yescharacteristic
488C4C2×#1451980011yescharacteristic
489C4AGL(1,ℤ/8)×D6111yescharacteristic
490C4#1684056111yescharacteristic
491C2—22no—
492C2—9761no—
493C2—2444no—
494C2#2145528211yescharacteristic
495C2C2×#1683999411yescharacteristic
496C2#2145553811yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1550C2
socleC22×C61C2×#121
prime radical#21455285C2
hypercenter#1550D61
hypocenterC61#19254
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
1. #21456379C2semidirect
2. #21455612C2semidirect
4. C2×#16839827C2semidirect
5. #21445241C2semidirect
6. #21445231C2semidirect
7. #21445197C2semidirect
8. #21445163C2semidirect
9. #21444574C2semidirect
10. #21444494C2semidirect
11. C61×#1550C2semidirect
16. #16840158C22semidirect
17. #16840136C22semidirect
18. #16840112C22semidirect
19. #16840110C22semidirect
40. #16839512C22semidirect
41. #16839512C22semidirect
42. #16839507C22semidirect
43. #16839505C22semidirect
47. C61×#333C22semidirect
48. C61×#332C22semidirect
126. C2×D244SD16semidirect
218. #92418AGL(1,ℤ/8)semidirect
252. #1550D61semidirect
293. #333C2×D61semidirect
295. #332C2×D61semidirect
297. C61#19254semidirect

Maximal subgroups

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

subgroupindex
#214563792
#214556122
#214552852
C2×#168398272
#214452412
#214452312
#214451972
#214451632
#214445742
#214444942
C61×#15502
#1925461