Finite Group Databasebeta

← #28402061

Subgroups — #28402061

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×#16813650C211yesmaximal, characteristic, complemented
2C2×D8×D113C211yesmaximal, characteristic, complemented
3#20299245C211yesmaximal, characteristic, complemented
4#20299232C211yesmaximal, characteristic, complemented
5#20298986C211yesmaximal, characteristic, complemented
6#20298828C211yesmaximal, characteristic, complemented
7#20298812C211yesmaximal, characteristic, complemented
8#20298792C211yesmaximal, characteristic, complemented
9#20298750C211yesmaximal, characteristic, complemented
10#20298741C211yesmaximal, characteristic, complemented
11#20298733C211yesmaximal, characteristic, complemented
12#20298731C211yesmaximal, characteristic, complemented
13#20298709C211yesmaximal, characteristic, complemented
14#20298667C211yesmaximal, characteristic, complemented
15#469×C113C211yesmaximal, characteristic, complemented
16C2×#14268647C2211yescharacteristic, complemented
17C22×D452C2211yescharacteristic, complemented
18C2×D4×D113—21no—
19C2×D4×D113C2211yescharacteristic
20C2×D4×D113C2211yescharacteristic, complemented
21#16813650—24no—
22C2×#14268636C2211yescharacteristic, complemented
23C2×#14268636C2211yescharacteristic, complemented
24C2×#14268635C2211yescharacteristic, complemented
25C2×D904C2211yescharacteristic
26C2×D904C2211yescharacteristic, complemented
27C2×#14268632C2211yescharacteristic
28C2×#14268632C2211yescharacteristic, complemented
29C2×#14268632C2211yescharacteristic, complemented
30D8×D113—24no—
31C2×C8×D113C2211yescharacteristic, complemented
32#16813618C2211yescharacteristic, complemented
33#16813609C2211yescharacteristic, complemented
34#16813604—21no—
35#16813602C2211yescharacteristic, complemented
36#16813597C2211yescharacteristic
37#16813595—21no—
38#16813593—21no—
39#16813592—21no—
40#16813590—21no—
41#16813590—21no—
42#16813584C2211yescharacteristic, complemented
43#16813584C2211yescharacteristic, complemented
44C4×C22⋊D113C2211yescharacteristic
45C22⋊C4×D113—21no—
46#16813551C2211yescharacteristic, complemented
47#16813545C2211yescharacteristic, complemented
48#16813542C2211yescharacteristic, complemented
49#16813542C2211yescharacteristic, complemented
50#16813542C2211yescharacteristic, complemented
51C22⋊#11744751C2211yescharacteristic, complemented
52#16813534C2211yescharacteristic, complemented
53#16813532C2211yescharacteristic, complemented
54C2×D8×C113C2211yescharacteristic, complemented
55C2×M16×C113C2211yescharacteristic, complemented
56#121×C113C2211yescharacteristic, complemented
57#121×C113C2211yescharacteristic, complemented
58#106×C113C2211yescharacteristic, complemented
59#102×C113C2211yescharacteristic, complemented
60#102×C113C2211yescharacteristic, complemented
61C23×D113—21no—
62C23×D113—21no—
63C23×D113—21no—
64#14268647—22no—
65#14268647—41no—
66C2×Dic226C2311yescharacteristic
67C2×D452—22no—
68C2×D452—21no—
69C2×D452—21no—
70C2×D452—41no—
71C2×D452C2311yescharacteristic
72C2×D452C2311yescharacteristic
73C2×D452C2311yescharacteristic
74C2×C22⋊D113—21no—
75C2×C22⋊D113—21no—
76C2×C22⋊D113—21no—
77C2×C22⋊D113—21no—
78C2×C22⋊D113—21no—
79C2×C22⋊D113D411yescharacteristic
80D4×D113—22no—
81D4×D113—22no—
82D4×D113—41no—
83D4×D113—41no—
84D4×D113—44no—
85C2×C4×D113—21no—
86C2×C4×D113—21no—
87C2×C4×D113—21no—
88C2×C4×D113C2311yescharacteristic
89#14268636—22no—
90#14268636—22no—
91#14268635—41no—
92D904—22no—
93D904D414yescomplemented
94#14268632—22no—
95#14268632—22no—
96#14268632—41no—
97C2×#11744751C2311yescharacteristic
98C2×#11744751C2311yescharacteristic
99C8×D113—22no—
100C4⋊Dic113—21no—
101#14268626D411yescharacteristic
102C22⋊Dic113—21no—
103C22⋊Dic113—21no—
104C22⋊Dic113D411yescharacteristic
105#14268624—21no—
106#14268624—21no—
107#14268624—21no—
108#14268624—21no—
109#14268624D411yescharacteristic
110C4×Dic113C2311yescharacteristic
111C2×D4×C113—21no—
112C2×D4×C113—21no—
113C2×D4×C113C2311yescharacteristic
114C2×D4×C113C2311yescharacteristic
115C22×C4×C113C2311yescharacteristic
116D8×C113—22no—
117M16×C113—22no—
118C2×C8×C113C2311yescharacteristic
119C2×C8×C113C2311yescharacteristic
120C4⋊C4×C113C2311yescharacteristic
121C4⋊C4×C113C2311yescharacteristic
122C22⋊C4×C113—21no—
123C22⋊C4×C113—21no—
124C23×C113—21no—
125C23×C113—21no—
126C23×C113C2×D411yescharacteristic
127C22×D113—21no—
128C22×D113—21no—
129C22×D113—21no—
130C22×D113—42no—
131C22×D113—42no—
132C22×D113—42no—
133C22×D113—42no—
134C22×D113—42no—
135C22×D113—42no—
136C22×D113—81no—
137C22×D113—81no—
138C22×D113—81no—
139C22×D113C2×D411yescharacteristic
140D4×C113—21no—
141D4×C113—21no—
142D4×C113—21no—
143D4×C113—21no—
144D4×C113—42no—
145D4×C113—81no—
146C2×C4×C113—21no—
147C2×C4×C113—21no—
148C2×C4×C113—21no—
149C2×C4×C113—21no—
150C2×C4×C113C2×D411yescharacteristic
151C2×C4×C113C2411yescharacteristic
152C22⋊D113—22no—
153C22⋊D113—42no—
154C22⋊D113—42no—
155C22⋊D113—42no—
156C22⋊D113—42no—
157C22⋊D113—81no—
158C2×Dic113—21no—
159C2×Dic113C2×D411yescharacteristic
160C2×Dic113C2×D411yescharacteristic
161C2×Dic113C2×D411yescharacteristic
162D452—21no—
163D452—21no—
164D452—21no—
165D452—21no—
166D452—42no—
167D452—42no—
168D452C2×D412yes—
169D452C2×D412yescomplemented
170C4×D113—21no—
171C4×D113—21no—
172C4×D113—42no—
173C4×D113—42no—
174C4×D113—81no—
175Dic226—21no—
176Dic226—21no—
177C8×C113—21no—
178C8×C113C2×D412yescomplemented
179#11744751—21no—
180#11744751—21no—
181C22×C113—21no—
182C22×C113—21no—
183C22×C113—21no—
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—22no—
194C2×D113—22no—
195C2×D113—41no—
196C2×D113—41no—
197C2×D113—41no—
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—41no—
208C4×C113—41no—
209C4×C113C22×D411yescharacteristic
210C4×C113H(5,𝔽2)11yescharacteristic
211Dic113—21no—
212Dic113—21no—
213Dic113—21no—
214Dic113—41no—
215C2×C113—41no—
216C2×C113—81no—
217C2×C113—81no—
218C2×C113D4211yescharacteristic
219C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C113#57611yescharacteristic
221D113—22no—
222D113—42no—
223D113—42no—
224D113—81no—
225#3264—1131nomaximal, Sylow
226C113#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1131no—
228C22×D8—1131no—
229#546—1131no—
230D42—1131no—
231#496—1131no—
232#469D11311yescharacteristic, complemented
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×SD16—1131no—
250C2×SD16—1131no—
251C2×D8—1131no—
252C2×D8—1131no—
253C2×D8—1131no—
254C2×D8—1131no—
255C2×D8—1131no—
256C2×D8—2264no—
257C2×D8C2×D11311yescharacteristic, complemented
258C2×M16C2×D11311yescharacteristic, complemented
259C22×C8—1131no—
260#127—1131no—
261#124—1131no—
262#123—2261no—
263C22⋊Q8—1131no—
264#121—1131no—
265#121—2261no—
266#121—2261no—
267#121C2×D11311yescharacteristic, complemented
268#121C2×D11311yescharacteristic, complemented
269#120—2261no—
270#120—2261no—
271#120—2261no—
272C4×D4—1131no—
273C4×D4—1131no—
274C4×D4—1131no—
275C2×C22⋊C4—2261no—
276#106C2×D11311yescharacteristic, complemented
277C4⋊C8—1131no—
278#103—1131no—
279#102—1131no—
280#102—1131no—
281#102—1131no—
282#102C2×D11311yescharacteristic, complemented
283#102C2×D11311yescharacteristic, complemented
284C22⋊C8—1131no—
285C22⋊C8—1131no—
286#97—1131no—
287C24—2261no—
288C24—2261no—
289C24—2261no—
290#41—4521no—
291#41—2262no—
292C2×Q8—1131no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—1131no—
296C2×D4—1131no—
297C2×D4—1131no—
298C2×D4—1131no—
299C2×D4—4521no—
300C2×D4—4524no—
301C2×D4—4521no—
302C2×D4—4521no—
303C2×D4—2262no—
304C2×D4—2261no—
305C2×D4—2261no—
306C2×D4—2261no—
307C2×D4—2261no—
308C2×D4—2261no—
309C2×D4—2262no—
310C2×D4—2262no—
311C2×D4—2261no—
312C2×D4—2261no—
313C2×D4C22×D11311yescharacteristic
314C2×D4C22×D11311yescharacteristic
315C22×C4—1131no—
316C22×C4—2261no—
317C22×C4—2261no—
318C22×C4—2261no—
319C22×C4C22×D11311yescharacteristic
320SD16—4521no—
321SD16—2262no—
322SD16—2262no—
323D8—22no—
324D8—1134no—
325D8—4521no—
326D8—2262no—
327D8—2262no—
328D8—2262no—
329M16—22no—
330C2×C8—1131no—
331C2×C8—1131no—
332C2×C8—2262no—
333C2×C8C22×D11311yescharacteristic
334C2×C8C22×D11311yescharacteristic
335C4⋊C4—1131no—
336C4⋊C4—2261no—
337C4⋊C4C22×D11311yescharacteristic
338C4⋊C4C22×D11311yescharacteristic
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—
349C42—1131no—
350C23—21no—
351C23—21no—
352C23—1131no—
353C23—9041no—
354C23—9041no—
355C23—9041no—
356C23—4522no—
357C23—4522no—
358C23—4522no—
359C23—4522no—
360C23—4522no—
361C23—4522no—
362C23—2261no—
363C23—2261no—
364C23—2261no—
365C23D4×D11311yescharacteristic
366Q8—2261no—
367Q8—2261no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—81no—
374D4—1132no—
375D4—1132no—
376D4—9041no—
377D4—4522no—
378D4—4522no—
379D4—4522no—
380D4—4522no—
381D4—4522no—
382D4—4522no—
383D4—2262no—
384D4—2261no—
385D4—2261no—
386D4—2261no—
387D4—2261no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—1131no—
393C2×C4—1131no—
394C2×C4—1131no—
395C2×C4—9041no—
396C2×C4—4522no—
397C2×C4—4522no—
398C2×C4—2261no—
399C2×C4—2261no—
400C2×C4—2261no—
401C2×C4D4×D11311yescharacteristic
402C2×C4C23×D11311yescharacteristic
403C8—21no—
404C8—2261no—
405C8—2261no—
406C8D4×D11312yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
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—4521no—
425C22—4521no—
426C22—4521no—
427C22—2262no—
428C22—2262no—
429C22—2261no—
430C22—2261no—
431C22C2×D4×D11311yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—4521no—
436C4—2261no—
437C4—2261no—
438C4—2261no—
439C4C2×D4×D11311yescharacteristic
440C4#1681366811yescharacteristic
441C2—41no—
442C2—81no—
443C2—81no—
444C2—9041no—
445C2—4522no—
446C2—4522no—
447C2—2262no—
448C2#2029923211yescharacteristic
449C2AGL(1,ℤ/8)×D11311yescharacteristic
450C2#2029940211yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#469×C113C2
socleC22×C113C22×D4
hypercenter#469D113
hypocenterC113#3264
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#16813650C2semidirect
2. C2×D8×D113C2semidirect
3. #20299245C2semidirect
4. #20299232C2semidirect
5. #20298986C2semidirect
6. #20298828C2semidirect
7. #20298812C2semidirect
8. #20298792C2semidirect
9. #20298750C2semidirect
10. #20298741C2semidirect
11. #20298733C2semidirect
12. #20298731C2semidirect
13. #20298709C2semidirect
14. #20298667C2semidirect
15. #469×C113C2semidirect
16. C2×#14268647C22semidirect
17. C22×D452C22semidirect
20. C2×D4×D113C22semidirect
22. C2×#14268636C22semidirect
23. C2×#14268636C22semidirect
24. C2×#14268635C22semidirect
26. C2×D904C22semidirect
28. C2×#14268632C22semidirect
29. C2×#14268632C22semidirect
31. C2×C8×D113C22semidirect
32. #16813618C22semidirect
33. #16813609C22semidirect
35. #16813602C22semidirect
42. #16813584C22semidirect
43. #16813584C22semidirect
46. #16813551C22semidirect
47. #16813545C22semidirect
48. #16813542C22semidirect
49. #16813542C22semidirect
50. #16813542C22semidirect
51. C22⋊#11744751C22semidirect
52. #16813534C22semidirect
53. #16813532C22semidirect
54. C2×D8×C113C22semidirect
55. C2×M16×C113C22semidirect
56. #121×C113C22semidirect
57. #121×C113C22semidirect
58. #106×C113C22semidirect
59. #102×C113C22semidirect
60. #102×C113C22semidirect
93. D904D4semidirect
169. D452C2×D4semidirect
178. C8×C113C2×D4semidirect
205. C2×D113AGL(1,ℤ/8)semidirect
226. C113#3264semidirect
232. #469D113semidirect
257. C2×D8C2×D113semidirect
258. C2×M16C2×D113semidirect
267. #121C2×D113semidirect
268. #121C2×D113semidirect
276. #106C2×D113semidirect
282. #102C2×D113semidirect
283. #102C2×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×#168136502
C2×D8×D1132
#202992452
#202992322
#202989862
#202988282
#202988122
#202987922
#202987502
#202987412
#202987332
#202987312
#202987092
#202986672
#469×C1132
#3264113