Finite Group Databasebeta

← #28402053

Subgroups — #28402053

Order 14464.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16813649C211yesmaximal, characteristic, complemented
2C2×D8×D113C211yesmaximal, characteristic, complemented
3#20299241C211yesmaximal, characteristic, complemented
4#20299233C211yesmaximal, characteristic, complemented
5#20298848C211yesmaximal, characteristic, complemented
6#20298824C211yesmaximal, characteristic, complemented
7#20298810C211yesmaximal, characteristic, complemented
8#20298790C211yesmaximal, characteristic, complemented
9#20298747C211yesmaximal, characteristic, complemented
10#20298742C211yesmaximal, characteristic, complemented
11#20298730C211yesmaximal, characteristic, complemented
12#20298730C211yesmaximal, characteristic, complemented
13#20298711C211yesmaximal, characteristic, complemented
14#20298671C211yesmaximal, characteristic, complemented
15#496×C113C211yesmaximal, characteristic, complemented
16C2×#14268648C2211yescharacteristic, complemented
17C2×D4×D113—21no—
18C2×D4×D113C2211yescharacteristic
19C2×D4×D113C2211yescharacteristic, complemented
20C2×D4×D113C2211yescharacteristic, complemented
21#16813649—24no—
22C2×#14268634C2211yescharacteristic, complemented
23C2×D904C2211yescharacteristic
24C2×#14268632C2211yescharacteristic
25C2×#14268632C2211yescharacteristic, complemented
26C2×#14268632C2211yescharacteristic, complemented
27D8×D113—24no—
28C2×#14268630C2211yescharacteristic, complemented
29C2×C8×D113C2211yescharacteristic, complemented
30#16813604—21no—
31#16813598C2211yescharacteristic, complemented
32#16813595—21no—
33#16813593—21no—
34#16813593C2211yescharacteristic
35#16813593C2211yescharacteristic, complemented
36#16813593C2211yescharacteristic, complemented
37#16813592—21no—
38#16813590—21no—
39#16813590—21no—
40C4×D452C2211yescharacteristic
41C4×D452C2211yescharacteristic, complemented
42C4×D452C2211yescharacteristic, complemented
43C22⋊C4×D113—21no—
44#16813554C2211yescharacteristic, complemented
45C4⋊#11744751C2211yescharacteristic, complemented
46#16813546C2211yescharacteristic, complemented
47#16813544C2211yescharacteristic, complemented
48#16813544C2211yescharacteristic, complemented
49#16813543C2211yescharacteristic, complemented
50#16813543C2211yescharacteristic, complemented
51#16813543C2211yescharacteristic, complemented
52#16813534C2211yescharacteristic, complemented
53#16813534C2211yescharacteristic, complemented
54C2×SD16×C113C2211yescharacteristic, complemented
55C2×SD16×C113C2211yescharacteristic, complemented
56C2×D8×C113C2211yescharacteristic, complemented
57C2×D8×C113C2211yescharacteristic, complemented
58#127×C113C2211yescharacteristic, complemented
59#124×C113C2211yescharacteristic, complemented
60#97×C113C2211yescharacteristic, complemented
61C23×D113—21no—
62C23×D113—21no—
63C23×D113—21no—
64#14268648—22no—
65#14268648—41no—
66C2×D452—21no—
67C2×D452D411yescharacteristic
68C2×D452C2311yescharacteristic
69C2×D452C2311yescharacteristic
70C2×C22⋊D113—21no—
71C2×C22⋊D113—21no—
72C2×C22⋊D113—21no—
73C2×C22⋊D113—21no—
74C2×C22⋊D113—21no—
75C2×C22⋊D113—21no—
76D4×D113—22no—
77D4×D113—22no—
78D4×D113—22no—
79D4×D113—41no—
80D4×D113—41no—
81D4×D113—41no—
82D4×D113—44no—
83C2×C4×D113—21no—
84C2×C4×D113—21no—
85C2×C4×D113—21no—
86C2×C4×D113C2311yescharacteristic
87C2×C4×D113C2311yescharacteristic
88#14268634—22no—
89D904D414yescomplemented
90#14268632—22no—
91#14268632—22no—
92#14268632—22no—
93#14268630—22no—
94C2×#11744751C2311yescharacteristic
95C2×#11744751C2311yescharacteristic
96C8×D113—22no—
97C4⋊Dic113D411yescharacteristic
98C4⋊Dic113C2311yescharacteristic
99C4⋊Dic113C2311yescharacteristic
100#14268626—21no—
101C22⋊Dic113—21no—
102C22⋊Dic113—21no—
103C22⋊Dic113—21no—
104#14268624—21no—
105#14268624—21no—
106#14268624—21no—
107#14268624D411yescharacteristic
108#14268624D411yescharacteristic
109C2×Q8×C113C2311yescharacteristic
110C2×D4×C113—21no—
111C2×D4×C113—21no—
112C2×D4×C113C2311yescharacteristic
113C2×D4×C113C2311yescharacteristic
114C2×D4×C113C2311yescharacteristic
115SD16×C113—22no—
116SD16×C113—41no—
117D8×C113—22no—
118D8×C113—41no—
119C2×C8×C113C2311yescharacteristic
120C2×C8×C113C2311yescharacteristic
121C22⋊C4×C113—21no—
122C22⋊C4×C113—21no—
123C42×C113C2311yescharacteristic
124C23×C113—21no—
125C23×C113—21no—
126C23×C113—21no—
127C22×D113—21no—
128C22×D113—21no—
129C22×D113—42no—
130C22×D113—42no—
131C22×D113—42no—
132C22×D113—42no—
133C22×D113—42no—
134C22×D113—42no—
135C22×D113—81no—
136C22×D113—81no—
137C22×D113—81no—
138C22×D113C2×D411yescharacteristic
139C22×D113C2×D411yescharacteristic
140Q8×C113—21no—
141Q8×C113—21no—
142D4×C113—21no—
143D4×C113—21no—
144D4×C113—21no—
145D4×C113—21no—
146D4×C113—21no—
147D4×C113—21no—
148D4×C113—42no—
149D4×C113—81no—
150C2×C4×C113—21no—
151C2×C4×C113C2×D411yescharacteristic
152C2×C4×C113C2×D411yescharacteristic
153C2×C4×C113C2411yescharacteristic
154C22⋊D113—42no—
155C22⋊D113—42no—
156C22⋊D113—42no—
157C22⋊D113—42no—
158C22⋊D113—42no—
159C22⋊D113—81no—
160C2×Dic113—21no—
161C2×Dic113—21no—
162C2×Dic113C2×D411yescharacteristic
163C2×Dic113C2×D411yescharacteristic
164D452—22no—
165D452—21no—
166D452—21no—
167D452—42no—
168D452C2×D412yes—
169D452C2×D412yescomplemented
170C4×D113—21no—
171C4×D113—21no—
172C4×D113—21no—
173C4×D113—21no—
174C4×D113—42no—
175C4×D113—42no—
176C4×D113—81no—
177C8×C113—21no—
178C8×C113C2×D412yescomplemented
179#11744751—21no—
180#11744751—21no—
181C22×C113—41no—
182C22×C113—41no—
183C22×C113—41no—
184C22×C113—41no—
185C22×C113—41no—
186C22×C113—41no—
187C22×C113—41no—
188C22×C113—41no—
189C22×C113—41no—
190C22×C113C22×D411yescharacteristic
191C2×D113—21no—
192C2×D113—21no—
193C2×D113—21no—
194C2×D113—22no—
195C2×D113—22no—
196C2×D113—21no—
197C2×D113—21no—
198C2×D113—41no—
199C2×D113—41no—
200C2×D113—41no—
201C2×D113—41no—
202C2×D113—84no—
203C2×D113—84no—
204C2×D113—84no—
205C2×D113AGL(1,ℤ/8)12yescomplemented
206C4×C113—21no—
207C4×C113—21no—
208C4×C113—41no—
209C4×C113C22×D411yescharacteristic
210C4×C113H(5,𝔽2)11yescharacteristic
211Dic113—21no—
212Dic113—21no—
213Dic113—41no—
214Dic113—41no—
215C2×C113—81no—
216C2×C113—81no—
217C2×C113—81no—
218C2×C113D4211yescharacteristic
219C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C113#57611yescharacteristic
221D113—22no—
222D113—41no—
223D113—42no—
224D113—42no—
225#3264—1131nomaximal, Sylow
226C113#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1131no—
228C22×D8—1131no—
229#546—1131no—
230D42—1131no—
231#496D11311yescharacteristic, complemented
232#469—1131no—
233#465—1131no—
234#463—1131no—
235#459—1131no—
236#450—1131no—
237#449—1131no—
238#447—1131no—
239#447—1131no—
240#442—1131no—
241#435—1131no—
242C2×#41—1131no—
243C22×D4—1131no—
244C22×D4—1131no—
245C22×D4—1131no—
246C22×D4—2261no—
247AGL(1,ℤ/8)—2264no—
248C2×SD16—1131no—
249C2×SD16C2×D11311yescharacteristic, complemented
250C2×SD16C2×D11311yescharacteristic, complemented
251C2×D8—1131no—
252C2×D8—1131no—
253C2×D8—1131no—
254C2×D8—1131no—
255C2×D8—2264no—
256C2×D8C2×D11311yescharacteristic, complemented
257C2×D8C2×D11311yescharacteristic, complemented
258C2×M16—1131no—
259C22×C8—1131no—
260#127C2×D11311yescharacteristic, complemented
261#124C2×D11311yescharacteristic, complemented
262#123—2261no—
263C22⋊Q8—1131no—
264#121—1131no—
265#121—1131no—
266#121—1131no—
267#121—2261no—
268#121—2261no—
269#120—2261no—
270#120—2261no—
271#120—2261no—
272C4×D4—1131no—
273C4×D4—1131no—
274C4×D4—1131no—
275C2×C22⋊C4—2261no—
276#106—1131no—
277C4⋊C8—1131no—
278#103—1131no—
279#102—1131no—
280#102—1131no—
281#102—1131no—
282#102—1131no—
283#102—1131no—
284C22⋊C8—1131no—
285C22⋊C8—1131no—
286#97C2×D11311yescharacteristic, complemented
287C24—2261no—
288C24—2261no—
289C24—2261no—
290#41—4521no—
291#41—2262no—
292C2×Q8C22×D11311yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—1131no—
296C2×D4—1131no—
297C2×D4—1131no—
298C2×D4—4524no—
299C2×D4—4521no—
300C2×D4—4521no—
301C2×D4—4521no—
302C2×D4—2261no—
303C2×D4—2261no—
304C2×D4—2261no—
305C2×D4—2261no—
306C2×D4—2262no—
307C2×D4—2262no—
308C2×D4—2262no—
309C2×D4—2261no—
310C2×D4—2261no—
311C2×D4—2261no—
312C2×D4C22×D11311yescharacteristic
313C2×D4C22×D11311yescharacteristic
314C2×D4C22×D11311yescharacteristic
315C22×C4—1131no—
316C22×C4—1131no—
317C22×C4—2261no—
318C22×C4—2261no—
319C22×C4—2261no—
320SD16—22no—
321SD16—41no—
322SD16—2262no—
323D8—22no—
324D8—41no—
325D8—1134no—
326D8—2262no—
327D8—2262no—
328D8—2262no—
329M16—2262no—
330C2×C8—1131no—
331C2×C8—1131no—
332C2×C8—2262no—
333C2×C8C22×D11311yescharacteristic
334C2×C8C22×D11311yescharacteristic
335C4⋊C4—1131no—
336C4⋊C4—1131no—
337C4⋊C4—1131no—
338C4⋊C4—2261no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—1131no—
342C22⋊C4—1131no—
343C22⋊C4—2261no—
344C22⋊C4—2261no—
345C22⋊C4—2261no—
346C22⋊C4—2261no—
347C22⋊C4—2261no—
348C22⋊C4—2261no—
349C42C22×D11311yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—1131no—
354C23—1131no—
355C23—9041no—
356C23—9041no—
357C23—9041no—
358C23—4522no—
359C23—4522no—
360C23—4522no—
361C23—4522no—
362C23—4522no—
363C23—4522no—
364C23—2261no—
365C23—2261no—
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—81no—
376D4—1132no—
377D4—1132no—
378D4—9041no—
379D4—4522no—
380D4—4522no—
381D4—4522no—
382D4—4522no—
383D4—4522no—
384D4—4522no—
385D4—2262no—
386D4—2261no—
387D4—2261no—
388C2×C4—21no—
389C2×C4—1131no—
390C2×C4—1131no—
391C2×C4—9041no—
392C2×C4—4522no—
393C2×C4—4522no—
394C2×C4—2261no—
395C2×C4—2261no—
396C2×C4—2261no—
397C2×C4—2261no—
398C2×C4—2261no—
399C2×C4—2261no—
400C2×C4D4×D11311yescharacteristic
401C2×C4D4×D11311yescharacteristic
402C2×C4C23×D11311yescharacteristic
403C8—21no—
404C8—2261no—
405C8—2261no—
406C8D4×D11312yescomplemented
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—1132no—
417C22—9044no—
418C22—9044no—
419C22—9044no—
420C22—4521no—
421C22—4521no—
422C22—4521no—
423C22—4521no—
424C22—2261no—
425C22—2261no—
426C22—2262no—
427C22—2261no—
428C22—2262no—
429C22—2261no—
430C22—2261no—
431C22C2×D4×D11311yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—4521no—
436C4—4521no—
437C4—2261no—
438C4—2261no—
439C4C2×D4×D11311yescharacteristic
440C4#1681366711yescharacteristic
441C2—81no—
442C2—81no—
443C2—81no—
444C2—4522no—
445C2—4521no—
446C2—4522no—
447C2—2262no—
448C2#2029923311yescharacteristic
449C2AGL(1,ℤ/8)×D11311yescharacteristic
450C2#2029940211yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#496×C113C2
socleC22×C113C22×D4
hypercenter#496D113
hypocenterC113#3264
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#16813649C2semidirect
2. C2×D8×D113C2semidirect
3. #20299241C2semidirect
4. #20299233C2semidirect
5. #20298848C2semidirect
6. #20298824C2semidirect
7. #20298810C2semidirect
8. #20298790C2semidirect
9. #20298747C2semidirect
10. #20298742C2semidirect
11. #20298730C2semidirect
12. #20298730C2semidirect
13. #20298711C2semidirect
14. #20298671C2semidirect
15. #496×C113C2semidirect
16. C2×#14268648C22semidirect
19. C2×D4×D113C22semidirect
20. C2×D4×D113C22semidirect
22. C2×#14268634C22semidirect
25. C2×#14268632C22semidirect
26. C2×#14268632C22semidirect
28. C2×#14268630C22semidirect
29. C2×C8×D113C22semidirect
31. #16813598C22semidirect
35. #16813593C22semidirect
36. #16813593C22semidirect
41. C4×D452C22semidirect
42. C4×D452C22semidirect
44. #16813554C22semidirect
45. C4⋊#11744751C22semidirect
46. #16813546C22semidirect
47. #16813544C22semidirect
48. #16813544C22semidirect
49. #16813543C22semidirect
50. #16813543C22semidirect
51. #16813543C22semidirect
52. #16813534C22semidirect
53. #16813534C22semidirect
54. C2×SD16×C113C22semidirect
55. C2×SD16×C113C22semidirect
56. C2×D8×C113C22semidirect
57. C2×D8×C113C22semidirect
58. #127×C113C22semidirect
59. #124×C113C22semidirect
60. #97×C113C22semidirect
89. D904D4semidirect
169. D452C2×D4semidirect
178. C8×C113C2×D4semidirect
205. C2×D113AGL(1,ℤ/8)semidirect
226. C113#3264semidirect
231. #496D113semidirect
249. C2×SD16C2×D113semidirect
250. C2×SD16C2×D113semidirect
256. C2×D8C2×D113semidirect
257. C2×D8C2×D113semidirect
260. #127C2×D113semidirect
261. #124C2×D113semidirect
286. #97C2×D113semidirect
406. C8D4×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136492
C2×D8×D1132
#202992412
#202992332
#202988482
#202988242
#202988102
#202987902
#202987472
#202987422
#202987302
#202987112
#202986712
#496×C1132
#3264113