Finite Group Databasebeta

← #28400197

Subgroups — #28400197

Order 14464.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16813651C211yesmaximal, characteristic, complemented
2C22×#14268630C211yesmaximal, characteristic, complemented
3#20299108C211yesmaximal, characteristic
4C22⋊Q8×D113C211yesmaximal, characteristic, complemented
5#20298857C211yesmaximal, characteristic, complemented
6#20298855C211yesmaximal, characteristic, complemented
7#20298843C211yesmaximal, characteristic, complemented
8#20298841C211yesmaximal, characteristic, complemented
9#20298839C211yesmaximal, characteristic, complemented
10#20298837C211yesmaximal, characteristic, complemented
11#20298825C211yesmaximal, characteristic, complemented
12#20298637C211yesmaximal, characteristic, complemented
13#20298605C211yesmaximal, characteristic, complemented
14#20298596C211yesmaximal, characteristic, complemented
15#465×C113C211yesmaximal, characteristic, complemented
16C2×#14268646C2211yescharacteristic
17C2×Q8×D113C2211yescharacteristic
18C22×C4×D113C2211yescharacteristic
19#16813651—24no—
20C2×Dic452C2211yescharacteristic, complemented
21C2×#14268637C2211yescharacteristic, complemented
22C2×#14268635C2211yescharacteristic, complemented
23C2×#14268630—24no—
24C2×#14268630C2211yescharacteristic
25C2×#14268630C2211yescharacteristic
26C2×#14268630C2211yescharacteristic
27C2×#14268630C2211yescharacteristic
28C22×#11744751C2211yescharacteristic
29#16813604—21no—
30#16813602C2211yescharacteristic
31#16813602C2211yescharacteristic
32#16813601C2211yescharacteristic
33#16813601C2211yescharacteristic
34#16813600—21no—
35#16813599—21no—
36#16813598C2211yescharacteristic
37#16813593C2211yescharacteristic
38#16813590—21no—
39D4×Dic113—21no—
40#16813582—21no—
41#16813580C2211yescharacteristic
42C4⋊C4×D113—21no—
43C4⋊C4×D113C2211yescharacteristic
44C22⋊C4×D113—21no—
45#16813556C2211yescharacteristic, complemented
46#16813555C2211yescharacteristic, complemented
47#16813553C2211yescharacteristic, complemented
48#16813547C2211yescharacteristic, complemented
49#16813547C2211yescharacteristic, complemented
50#16813546C2211yescharacteristic, complemented
51#16813545C2211yescharacteristic, complemented
52#16813545C2211yescharacteristic, complemented
53#16813543C2211yescharacteristic, complemented
54C2×SD16×C113C2211yescharacteristic, complemented
55C22×C8×C113C2211yescharacteristic
56C22⋊Q8×C113C2211yescharacteristic
57#121×C113C2211yescharacteristic
58#106×C113C2211yescharacteristic, complemented
59#103×C113C2211yescharacteristic, complemented
60#102×C113C2211yescharacteristic, complemented
61C23×D113D411yescharacteristic
62#14268646—22no—
63#14268646—22no—
64C2×Dic226—21no—
65C2×Dic226C2311yescharacteristic
66C2×Dic226C2311yescharacteristic
67Q8×D113—22no—
68Q8×D113—22no—
69C2×C22⋊D113—21no—
70C2×C22⋊D113—21no—
71C22×Dic113—21no—
72C22×Dic113D411yescharacteristic
73C2×C4×D113—22no—
74C2×C4×D113—22no—
75C2×C4×D113—21no—
76C2×C4×D113—21no—
77C2×C4×D113D411yescharacteristic
78C2×C4×D113D411yescharacteristic
79C2×C4×D113C2311yescharacteristic
80C2×C4×D113C2311yescharacteristic
81Dic452—22no—
82#14268637—22no—
83#14268635—22no—
84#14268630—22no—
85#14268630—22no—
86#14268630—22no—
87#14268630D414yescomplemented
88C2×#11744751—22no—
89C2×#11744751C2311yescharacteristic
90C2×#11744751C2311yescharacteristic
91C4⋊Dic113—21no—
92C4⋊Dic113—21no—
93C4⋊Dic113C2311yescharacteristic
94C4⋊Dic113C2311yescharacteristic
95#14268626—21no—
96#14268626—21no—
97#14268626—21no—
98#14268626—21no—
99C22⋊Dic113—21no—
100C22⋊Dic113—21no—
101C22⋊Dic113—21no—
102#14268624—21no—
103#14268624—21no—
104#14268624—21no—
105#14268624—21no—
106C4×Dic113—21no—
107C2×Q8×C113C2311yescharacteristic
108C2×D4×C113—21no—
109C2×D4×C113C2311yescharacteristic
110C22×C4×C113C2311yescharacteristic
111SD16×C113—22no—
112C2×C8×C113—22no—
113C2×C8×C113C2311yescharacteristic
114C2×C8×C113C2311yescharacteristic
115C4⋊C4×C113—21no—
116C4⋊C4×C113C2311yescharacteristic
117C4⋊C4×C113C2311yescharacteristic
118C22⋊C4×C113—21no—
119C22⋊C4×C113—21no—
120C23×C113—21no—
121C23×C113C2×D411yescharacteristic
122C22×D113—22no—
123C22×D113—22no—
124C22×D113—41no—
125C22×D113C2×D411yescharacteristic
126C22×D113C2×D411yescharacteristic
127Q8×C113—21no—
128Q8×C113—21no—
129D4×C113—21no—
130D4×C113—21no—
131D4×C113—42no—
132C2×C4×C113—21no—
133C2×C4×C113—21no—
134C2×C4×C113—21no—
135C2×C4×C113—21no—
136C2×C4×C113—21no—
137C2×C4×C113C2×D411yescharacteristic
138C2×C4×C113C2411yescharacteristic
139C22⋊D113—42no—
140C22⋊D113—42no—
141C2×Dic113—22no—
142C2×Dic113—21no—
143C2×Dic113—21no—
144C2×Dic113—21no—
145C2×Dic113—21no—
146C2×Dic113—42no—
147C2×Dic113C2×D411yescharacteristic
148C2×Dic113C2×D411yescharacteristic
149C4×D113—22no—
150C4×D113—21no—
151C4×D113—21no—
152C4×D113—22no—
153C4×D113—42no—
154C4×D113—42no—
155C4×D113C2×D412yes—
156C4×D113#4112yes—
157Dic226—21no—
158Dic226—21no—
159Dic226—21no—
160Dic226—21no—
161Dic226—42no—
162C8×C113—21no—
163C8×C113C2×D412yes—
164#11744751—21no—
165#11744751C2×D412yes—
166C22×C113—21no—
167C22×C113—21no—
168C22×C113—41no—
169C22×C113—41no—
170C22×C113—41no—
171C22×C113#13712yescomplemented
172C22×C113C22×D411yescharacteristic
173C2×D113—21no—
174C2×D113—21no—
175C2×D113—21no—
176C2×D113—21no—
177C2×D113—21no—
178C2×D113—44no—
179C2×D113#12112yes—
180C4×C113—21no—
181C4×C113—41no—
182C4×C113—41no—
183C4×C113—41no—
184C4×C113C22×D411yescharacteristic
185C4×C113C2×#4111yescharacteristic
186Dic113—21no—
187Dic113—41no—
188Dic113—41no—
189Dic113—41no—
190Dic113—41no—
191Dic113#12112yes—
192C2×C113—22no—
193C2×C113—81no—
194C2×C113C2×#12111yescharacteristic
195C2×C113C2×#13711yescharacteristic
196C2×C113#57511yescharacteristic
197D113—22no—
198D113—41no—
199#3056—1131nomaximal, Sylow
200C113#305611yescharacteristic, complemented, Sylow
201C2×#137—1131no—
202C22×M16—1131no—
203#525—1131no—
204C2×C22⋊Q8—1131no—
205#470—1131no—
206#470—1131no—
207#468—1131no—
208#468—1131no—
209#467—1131no—
210#467—1131no—
211#465—1131no—
212#465D11311yescharacteristic, complemented
213#428—1131no—
214#419—1131no—
215#417—1131no—
216C2×#41—1131no—
217C22×Q8—1131no—
218C23×C4—1131no—
219#137—2264no—
220C2×Q16—1131no—
221C2×Q16—1131no—
222C2×SD16—1131no—
223C2×SD16C2×D11311yescharacteristic, complemented
224C2×M16—1131no—
225C2×M16—1131no—
226C2×M16—1131no—
227C2×M16—1131no—
228C2×M16—2264no—
229C22×C8—1131no—
230C22×C8C2×D11311yescharacteristic
231#123—2261no—
232C22⋊Q8—1131no—
233C22⋊Q8—1131no—
234C22⋊Q8—1131no—
235C22⋊Q8—1131no—
236C22⋊Q8—1131no—
237C22⋊Q8—2261no—
238C22⋊Q8—2261no—
239C22⋊Q8C2×D11311yescharacteristic
240#121—1131no—
241#121C2×D11311yescharacteristic
242#120—2261no—
243C4×D4—2261no—
244C4×D4—2261no—
245#117—1131no—
246C2×C4⋊C4—1131no—
247C2×C4⋊C4—2261no—
248C2×C22⋊C4—2261no—
249#107—1131no—
250#107—1131no—
251#106—1131no—
252#106C2×D11311yescharacteristic, complemented
253#103—1131no—
254#103—1131no—
255#103—1131no—
256#103—1131no—
257#103—1131no—
258#103C2×D11311yescharacteristic, complemented
259#102—1131no—
260#102C2×D11311yescharacteristic, complemented
261C24—1131no—
262#41—2262no—
263#41—2262no—
264C2×Q8—1131no—
265C2×Q8—1131no—
266C2×Q8—2261no—
267C2×Q8—2262no—
268C2×Q8—2262no—
269C2×Q8C22×D11311yescharacteristic
270C2×D4—21no—
271C2×D4—2261no—
272C2×D4—2261no—
273C2×D4C22×D11311yescharacteristic
274C22×C4—1131no—
275C22×C4—1131no—
276C22×C4—1131no—
277C22×C4—1131no—
278C22×C4—1131no—
279C22×C4—2262no—
280C22×C4—2262no—
281C22×C4—2261no—
282C22×C4—2261no—
283C22×C4—2261no—
284C22×C4C22×D11311yescharacteristic
285Q16—2262no—
286Q16—2262no—
287SD16—22no—
288SD16—2262no—
289M16—1134no—
290M16—2262no—
291M16—2262no—
292M16—2262no—
293C2×C8—22no—
294C2×C8—1131no—
295C2×C8—1131no—
296C2×C8—2262no—
297C2×C8C22×D11311yescharacteristic
298C2×C8C22×D11311yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—1131no—
301C4⋊C4—1131no—
302C4⋊C4—2261no—
303C4⋊C4—2261no—
304C4⋊C4—2261no—
305C4⋊C4—2261no—
306C4⋊C4—2261no—
307C4⋊C4—2261no—
308C4⋊C4C22×D11311yescharacteristic
309C4⋊C4C22×D11311yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—2261no—
313C22⋊C4—2261no—
314C22⋊C4—2261no—
315C22⋊C4—2261no—
316C22⋊C4—2261no—
317C22⋊C4—2261no—
318C22⋊C4—2261no—
319C42—2261no—
320C23—21no—
321C23—1131no—
322C23—1131no—
323C23—4521no—
324C23—2262no—
325C23—2262no—
326C23D4×D11311yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—4522no—
330Q8—2261no—
331Q8—2261no—
332Q8—2261no—
333Q8—2261no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—4522no—
338D4—4522no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—1131no—
345C2×C4—1132no—
346C2×C4—1132no—
347C2×C4—1131no—
348C2×C4—4522no—
349C2×C4—4522no—
350C2×C4—4522no—
351C2×C4—2262no—
352C2×C4—2262no—
353C2×C4—2262no—
354C2×C4—2261no—
355C2×C4—2261no—
356C2×C4—2261no—
357C2×C4—2261no—
358C2×C4—2261no—
359C2×C4—2261no—
360C2×C4D4×D11311yescharacteristic
361C2×C4C23×D11311yescharacteristic
362C8—21no—
363C8—1132no—
364C8—2261no—
365C8D4×D11312yes—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—1132no—
372C22—4524no—
373C22—2261no—
374C22—2261no—
375C22—2261no—
376C22—2261no—
377C22—2261no—
378C22#1681365112yescomplemented
379C22C2×D4×D11311yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—1132no—
385C4—4521no—
386C4—4521no—
387C4—4521no—
388C4—4521no—
389C4—2261no—
390C4C2×D4×D11311yescharacteristic
391C4#41×D11311yescharacteristic
392C2—22no—
393C2—81no—
394C2—4521no—
395C2—2262no—
396C2#121×D11311yescharacteristic
397C2C2×#1681365111yescharacteristic
398C2#2029939511yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#465×C113C2
socleC22×C113C22×D4
prime radical#20299108C2
hypercenter#465D113
hypocenterC113#3056
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#16813651C2semidirect
2. C22×#14268630C2semidirect
4. C22⋊Q8×D113C2semidirect
5. #20298857C2semidirect
6. #20298855C2semidirect
7. #20298843C2semidirect
8. #20298841C2semidirect
9. #20298839C2semidirect
10. #20298837C2semidirect
11. #20298825C2semidirect
12. #20298637C2semidirect
13. #20298605C2semidirect
14. #20298596C2semidirect
15. #465×C113C2semidirect
20. C2×Dic452C22semidirect
21. C2×#14268637C22semidirect
22. C2×#14268635C22semidirect
45. #16813556C22semidirect
46. #16813555C22semidirect
47. #16813553C22semidirect
48. #16813547C22semidirect
49. #16813547C22semidirect
50. #16813546C22semidirect
51. #16813545C22semidirect
52. #16813545C22semidirect
53. #16813543C22semidirect
54. C2×SD16×C113C22semidirect
58. #106×C113C22semidirect
59. #103×C113C22semidirect
60. #102×C113C22semidirect
87. #14268630D4semidirect
171. C22×C113#137semidirect
200. C113#3056semidirect
212. #465D113semidirect
223. C2×SD16C2×D113semidirect
252. #106C2×D113semidirect
258. #103C2×D113semidirect
260. #102C2×D113semidirect
378. C22#16813651semidirect

Maximal subgroups

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

subgroupindex
C2×#168136512
C22×#142686302
#202991082
C22⋊Q8×D1132
#202988572
#202988552
#202988432
#202988412
#202988392
#202988372
#202988252
#202986372
#202986052
#202985962
#465×C1132
#3056113