Finite Group Databasebeta

← #28402268

Subgroups — #28402268

Order 14464.

546 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14268635C211yesmaximal, characteristic, complemented
2C2×SD16×D113C211yesmaximal, characteristic, complemented
3#20299261C211yesmaximal, characteristic, complemented
4D4×C22⋊D113C211yesmaximal, characteristic
5#20298969C211yesmaximal, characteristic, complemented
6#20298829C211yesmaximal, characteristic, complemented
7#20298826C211yesmaximal, characteristic, complemented
8#20298801C211yesmaximal, characteristic, complemented
9#20298797C211yesmaximal, characteristic, complemented
10#20298750C211yesmaximal, characteristic, complemented
11#20298739C211yesmaximal, characteristic, complemented
12#20298735C211yesmaximal, characteristic, complemented
13#20298689C211yesmaximal, characteristic, complemented
14#20298661C211yesmaximal, characteristic, complemented
15#450×C113C211yesmaximal, characteristic, complemented
16C22×Dic226C2211yescharacteristic, complemented
17C2×Q8×D113C2211yescharacteristic, complemented
18C22×C22⋊D113—21no—
19C2×D4×D113C2211yescharacteristic
20C2×#14268636C2211yescharacteristic, complemented
21C2×#14268636C2211yescharacteristic, complemented
22C2×#14268635—24no—
23C2×#14268635C2211yescharacteristic
24C2×#14268635C2211yescharacteristic
25C2×#14268635C2211yescharacteristic, complemented
26C2×#14268635C2211yescharacteristic, complemented
27C2×#14268634C2211yescharacteristic, complemented
28SD16×D113—24no—
29C22×#11744751C2211yescharacteristic, complemented
30C2×C8×D113C2211yescharacteristic, complemented
31#16813621—21no—
32#16813620C2211yescharacteristic, complemented
33#16813618C2211yescharacteristic
34#16813602C2211yescharacteristic, complemented
35#16813600—21no—
36#16813599—21no—
37#16813596C2211yescharacteristic
38#16813594—21no—
39#16813593C2211yescharacteristic
40#16813591—21no—
41#16813590—21no—
42#16813587C2211yescharacteristic, complemented
43D4×Dic113C2211yescharacteristic
44C4×C22⋊D113C2211yescharacteristic
45#16813582—21no—
46#16813553C2211yescharacteristic, complemented
47#16813551C2211yescharacteristic, complemented
48#16813547C2211yescharacteristic, complemented
49#16813547C2211yescharacteristic, complemented
50#16813546C2211yescharacteristic, complemented
51#16813542C2211yescharacteristic, complemented
52#16813534C2211yescharacteristic, complemented
53C8×Dic113C2211yescharacteristic, complemented
54C22×D4×C113C2211yescharacteristic
55C2×SD16×C113C2211yescharacteristic
56C2×SD16×C113C2211yescharacteristic, complemented
57C22⋊Q8×C113C2211yescharacteristic, complemented
58#102×C113C2211yescharacteristic
59#102×C113C2211yescharacteristic, complemented
60C22⋊C8×C113C2211yescharacteristic, complemented
61C23×D113—21no—
62C2×Dic226—22no—
63C2×Dic226—21no—
64C2×Dic226—21no—
65C2×Dic226—41no—
66C2×Dic226C2311yescharacteristic
67C2×Dic226C2311yescharacteristic
68Q8×D113—22no—
69Q8×D113—41no—
70C2×D452C2311yescharacteristic
71C2×C22⋊D113—21no—
72C2×C22⋊D113—21no—
73C2×C22⋊D113—21no—
74C2×C22⋊D113—21no—
75C2×C22⋊D113—21no—
76C2×C22⋊D113—44no—
77C2×C22⋊D113D411yescharacteristic
78D4×D113—22no—
79D4×D113—41no—
80C22×Dic113—21no—
81C22×Dic113—21no—
82C2×C4×D113—21no—
83C2×C4×D113C2311yescharacteristic
84#14268636—22no—
85#14268636—41no—
86#14268635—22no—
87#14268635—22no—
88#14268635—22no—
89#14268635D414yescomplemented
90#14268634—22no—
91C2×#11744751—22no—
92C2×#11744751C2311yescharacteristic
93C2×#11744751C2311yescharacteristic
94C8×D113—22no—
95C4⋊Dic113—21no—
96C4⋊Dic113C2311yescharacteristic
97#14268626—21no—
98#14268626—21no—
99#14268626—21no—
100#14268626D411yescharacteristic
101C22⋊Dic113—21no—
102C22⋊Dic113—21no—
103C22⋊Dic113D411yescharacteristic
104#14268624—21no—
105#14268624—21no—
106#14268624D411yescharacteristic
107C4×Dic113—21no—
108C4×Dic113C2311yescharacteristic
109C24×C113—21no—
110C2×Q8×C113C2311yescharacteristic
111C2×D4×C113—22no—
112C2×D4×C113—21no—
113C2×D4×C113—41no—
114C2×D4×C113C2311yescharacteristic
115C2×D4×C113C2311yescharacteristic
116C22×C4×C113C2311yescharacteristic
117SD16×C113—22no—
118SD16×C113—41no—
119C2×C8×C113C2311yescharacteristic
120C2×C8×C113C2311yescharacteristic
121C4⋊C4×C113—21no—
122C4⋊C4×C113C2311yescharacteristic
123C22⋊C4×C113—21no—
124C23×C113—21no—
125C23×C113—21no—
126C23×C113—42no—
127C23×C113—42no—
128C23×C113—81no—
129C23×C113C2×D411yescharacteristic
130C22×D113—21no—
131C22×D113—42no—
132C22×D113—42no—
133C22×D113—81no—
134C22×D113C2×D411yescharacteristic
135Q8×C113—21no—
136Q8×C113—21no—
137D4×C113—21no—
138D4×C113—21no—
139D4×C113—21no—
140D4×C113—42no—
141D4×C113C2×D412yes—
142C2×C4×C113—21no—
143C2×C4×C113—21no—
144C2×C4×C113—21no—
145C2×C4×C113—21no—
146C2×C4×C113C2×D411yescharacteristic
147C2×C4×C113C2411yescharacteristic
148C22⋊D113—42no—
149C22⋊D113—42no—
150C22⋊D113—42no—
151C22⋊D113—42no—
152C22⋊D113—81no—
153C22⋊D113SD1614yescomplemented
154C2×Dic113—21no—
155C2×Dic113—21no—
156C2×Dic113—21no—
157C2×Dic113—42no—
158C2×Dic113—42no—
159C2×Dic113C2×D411yescharacteristic
160C2×Dic113C2×D411yescharacteristic
161C2×Dic113C2×D411yescharacteristic
162D452—21no—
163D452—21no—
164C4×D113—21no—
165C4×D113—21no—
166C4×D113—42no—
167Dic226—21no—
168Dic226—21no—
169Dic226—21no—
170Dic226—42no—
171Dic226—42no—
172Dic226C2×D412yescomplemented
173C8×C113—21no—
174C8×C113—21no—
175#11744751—21no—
176#11744751C2×D412yescomplemented
177C22×C113—21no—
178C22×C113—21no—
179C22×C113—22no—
180C22×C113—41no—
181C22×C113—41no—
182C22×C113—41no—
183C22×C113—41no—
184C22×C113—41no—
185C22×C113—84no—
186C22×C113C2×SD1612yescomplemented
187C22×C113C22×D411yescharacteristic
188C2×D113—21no—
189C2×D113—21no—
190C2×D113—41no—
191C2×D113—41no—
192C2×D113—41no—
193C2×D113—84no—
194C2×D113C2×SD1612yescomplemented
195C4×C113—21no—
196C4×C113—41no—
197C4×C113—41no—
198C4×C113C22×D411yescharacteristic
199C4×C113H(5,𝔽2)11yescharacteristic
200Dic113—22no—
201Dic113—21no—
202Dic113—21no—
203Dic113—41no—
204Dic113—41no—
205Dic113C2×SD1612yescomplemented
206C2×C113—22no—
207C2×C113—42no—
208C2×C113—81no—
209C2×C113D4211yescharacteristic
210C2×C113C22×SD1611yescharacteristic
211C2×C113#57711yescharacteristic
212D113—22no—
213D113—81no—
214D4×SD16—1131nomaximal, Sylow
215C113D4×SD1611yescharacteristic, complemented, Sylow
216C22×SD16—1131no—
217C22×SD16—1131no—
218D4×Q8—1131no—
219D42—1131no—
220#492—1131no—
221#465—1131no—
222#465—1131no—
223#461—1131no—
224#460—1131no—
225#450—1131no—
226#450D11311yescharacteristic, complemented
227#448—1131no—
228#448—1131no—
229C4×SD16—1131no—
230C8×D4—1131no—
231C22×Q8—1131no—
232C22×Q8—1131no—
233C22×D4—1131no—
234C22×D4—2261no—
235C22×D4C2×D11311yescharacteristic
236C2×SD16—1131no—
237C2×SD16—1131no—
238C2×SD16—1131no—
239C2×SD16—1131no—
240C2×SD16—1131no—
241C2×SD16—1131no—
242C2×SD16—1131no—
243C2×SD16—2264no—
244C2×SD16—2264no—
245C2×SD16C2×D11311yescharacteristic
246C2×SD16C2×D11311yescharacteristic, complemented
247C22×C8—1131no—
248C22×C8—1131no—
249C4⋊Q8—1131no—
250C4⋊Q8—2261no—
251#127—1131no—
252C22⋊Q8—1131no—
253C22⋊Q8—2261no—
254C22⋊Q8—2261no—
255C22⋊Q8C2×D11311yescharacteristic, complemented
256#121—1131no—
257#121—1131no—
258#121—2261no—
259#120—2261no—
260#120—2261no—
261C4×Q8—1131no—
262C4×D4—1131no—
263C4×D4—1131no—
264C4×D4—2261no—
265#106—1131no—
266C4⋊C8—1131no—
267#103—1131no—
268#103—1131no—
269#103—1131no—
270#102—1131no—
271#102C2×D11311yescharacteristic
272#102C2×D11311yescharacteristic, complemented
273C22⋊C8—1131no—
274C22⋊C8C2×D11311yescharacteristic, complemented
275C4×C8—1131no—
276C24—21no—
277C24—2261no—
278C2×Q8—1131no—
279C2×Q8—1131no—
280C2×Q8—4521no—
281C2×Q8—4521no—
282C2×Q8—2261no—
283C2×Q8—2262no—
284C2×Q8—2262no—
285C2×Q8—2261no—
286C2×Q8C22×D11311yescharacteristic
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—1131no—
291C2×D4—1131no—
292C2×D4—4524no—
293C2×D4—4521no—
294C2×D4—2261no—
295C2×D4—2261no—
296C2×D4—2261no—
297C2×D4—2261no—
298C2×D4—2261no—
299C2×D4—2262no—
300C2×D4C22×D11311yescharacteristic
301C2×D4C22×D11311yescharacteristic
302C22×C4—1131no—
303C22×C4—2261no—
304C22×C4—2261no—
305C22×C4—2261no—
306C22×C4C22×D11311yescharacteristic
307SD16—22no—
308SD16—41no—
309SD16—1134no—
310SD16—4521no—
311SD16—2262no—
312SD16—2262no—
313SD16—2262no—
314SD16—2262no—
315SD16—2262no—
316C2×C8—1131no—
317C2×C8—1131no—
318C2×C8—2262no—
319C2×C8—2262no—
320C2×C8C22×D11311yescharacteristic
321C2×C8C22×D11311yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—1131no—
324C4⋊C4—1131no—
325C4⋊C4—2261no—
326C4⋊C4—2261no—
327C4⋊C4—2261no—
328C4⋊C4—2261no—
329C4⋊C4C22×D11311yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—1131no—
332C22⋊C4—1131no—
333C22⋊C4—2261no—
334C22⋊C4—2261no—
335C22⋊C4—2261no—
336C22⋊C4—2261no—
337C42—1131no—
338C42—2261no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—1131no—
345C23—9041no—
346C23—4522no—
347C23—4522no—
348C23—2261no—
349C23D4×D11311yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—1132no—
353Q8—4522no—
354Q8—4522no—
355Q8—2261no—
356Q8—2261no—
357Q8—2261no—
358D4—21no—
359D4—21no—
360D4—21no—
361D4—42no—
362D4—1134no—
363D4—9041no—
364D4—4522no—
365D4—4522no—
366D4—4522no—
367D4—4522no—
368D4—2261no—
369D4—2261no—
370D4D4×D11312yes—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—1131no—
376C2×C4—1131no—
377C2×C4—1131no—
378C2×C4—4522no—
379C2×C4—4522no—
380C2×C4—4522no—
381C2×C4—2261no—
382C2×C4—2261no—
383C2×C4—2261no—
384C2×C4—2261no—
385C2×C4—2261no—
386C2×C4D4×D11311yescharacteristic
387C2×C4C23×D11311yescharacteristic
388C8—21no—
389C8—21no—
390C8—1132no—
391C8—2261no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—84no—
401C22—1132no—
402C22—9044no—
403C22—4521no—
404C22—4521no—
405C22—4521no—
406C22—2261no—
407C22—2261no—
408C22SD16×D11312yescomplemented
409C22C2×D4×D11311yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—1132no—
414C4—4521no—
415C4—4521no—
416C4—2261no—
417C4—2261no—
418C4—2262no—
419C4C2×D4×D11311yescharacteristic
420C4#1681366711yescharacteristic
421C2—22no—
422C2—42no—
423C2—81no—
424C2—9041no—
425C2—2262no—
426C2#2029923011yescharacteristic
427C2C2×SD16×D11311yescharacteristic
428C2#2029940811yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#450×C113C2
socleC22×C113C22×D4
prime radicalD4×C22⋊D113C2
hypercenter#450D113
hypocenterC113D4×SD16
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×#14268635C2semidirect
2. C2×SD16×D113C2semidirect
3. #20299261C2semidirect
5. #20298969C2semidirect
6. #20298829C2semidirect
7. #20298826C2semidirect
8. #20298801C2semidirect
9. #20298797C2semidirect
10. #20298750C2semidirect
11. #20298739C2semidirect
12. #20298735C2semidirect
13. #20298689C2semidirect
14. #20298661C2semidirect
15. #450×C113C2semidirect
16. C22×Dic226C22semidirect
17. C2×Q8×D113C22semidirect
20. C2×#14268636C22semidirect
21. C2×#14268636C22semidirect
25. C2×#14268635C22semidirect
26. C2×#14268635C22semidirect
27. C2×#14268634C22semidirect
29. C22×#11744751C22semidirect
30. C2×C8×D113C22semidirect
32. #16813620C22semidirect
34. #16813602C22semidirect
42. #16813587C22semidirect
46. #16813553C22semidirect
47. #16813551C22semidirect
48. #16813547C22semidirect
49. #16813547C22semidirect
50. #16813546C22semidirect
51. #16813542C22semidirect
52. #16813534C22semidirect
53. C8×Dic113C22semidirect
56. C2×SD16×C113C22semidirect
57. C22⋊Q8×C113C22semidirect
59. #102×C113C22semidirect
60. C22⋊C8×C113C22semidirect
89. #14268635D4semidirect
153. C22⋊D113SD16semidirect
172. Dic226C2×D4semidirect
176. #11744751C2×D4semidirect
186. C22×C113C2×SD16semidirect
194. C2×D113C2×SD16semidirect
205. Dic113C2×SD16semidirect
215. C113D4×SD16semidirect
226. #450D113semidirect
246. C2×SD16C2×D113semidirect
255. C22⋊Q8C2×D113semidirect
272. #102C2×D113semidirect
274. C22⋊C8C2×D113semidirect
408. C22SD16×D113semidirect

Maximal subgroups

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

subgroupindex
C22×#142686352
C2×SD16×D1132
#202992612
D4×C22⋊D1132
#202989692
#202988292
#202988262
#202988012
#202987972
#202987502
#202987392
#202987352
#202986892
#202986612
#450×C1132
D4×SD16113