Finite Group Databasebeta

← #28402144

Subgroups — #28402144

Order 14464.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16813654C211yesmaximal, characteristic, complemented
2Q8×C22⋊D113C211yesmaximal, characteristic, complemented
3#20299259C211yesmaximal, characteristic
4#20298991C211yesmaximal, characteristic, complemented
5#20298853C211yesmaximal, characteristic, complemented
6#20298827C211yesmaximal, characteristic, complemented
7#20298822C211yesmaximal, characteristic, complemented
8#20298806C211yesmaximal, characteristic, complemented
9#20298759C211yesmaximal, characteristic, complemented
10#20298754C211yesmaximal, characteristic, complemented
11#20298752C211yesmaximal, characteristic, complemented
12#20298737C211yesmaximal, characteristic, complemented
13#20298705C211yesmaximal, characteristic, complemented
14#20298669C211yesmaximal, characteristic, complemented
15C22×Q16×C113C211yesmaximal, characteristic, complemented
16C2×#14268648C2211yescharacteristic
17C2×Q8×D113C2211yescharacteristic
18#16813654—24no—
19C2×#14268637C2211yescharacteristic
20C2×#14268637C2211yescharacteristic
21C2×#14268636C2211yescharacteristic
22C2×#14268634C2211yescharacteristic
23C2×#14268634C2211yescharacteristic
24C2×#14268630C2211yescharacteristic
25#16813620—21no—
26#16813620C2211yescharacteristic
27#16813611—21no—
28#16813611C2211yescharacteristic
29#16813601—21no—
30#16813601C2211yescharacteristic
31#16813598—21no—
32#16813598—21no—
33#16813598C2211yescharacteristic
34#16813598C2211yescharacteristic
35#16813596—21no—
36#16813596C2211yescharacteristic
37Q8×Dic113C2211yescharacteristic
38Q8×Dic113C2211yescharacteristic
39C4×C22⋊D113—21no—
40C4×C22⋊D113—21no—
41C4×C22⋊D113C2211yescharacteristic
42#16813554C2211yescharacteristic
43#16813551C2211yescharacteristic
44#16813547C2211yescharacteristic
45#16813546C2211yescharacteristic
46#16813546C2211yescharacteristic
47#16813546C2211yescharacteristic
48#16813546C2211yescharacteristic
49#16813544C2211yescharacteristic
50C22⋊#11744751C2211yescharacteristic
51#16813534C2211yescharacteristic
52#16813532C2211yescharacteristic
53C22×Q8×C113C2211yescharacteristic
54C22×Q8×C113C2211yescharacteristic
55C2×Q16×C113—24no—
56C2×Q16×C113C2211yescharacteristic
57C2×Q16×C113C2211yescharacteristic
58C2×Q16×C113C2211yescharacteristic
59C2×Q16×C113C2211yescharacteristic
60C22×C8×C113C2211yescharacteristic
61#14268648—22no—
62#14268648—41no—
63C2×Dic226—21no—
64C2×Dic226C2311yescharacteristic
65Q8×D113—22no—
66Q8×D113—41no—
67C2×D452—21no—
68C2×D452C2311yescharacteristic
69C2×C22⋊D113—21no—
70C2×C22⋊D113D411yescharacteristic
71C2×C4×D113—21no—
72C2×C4×D113—21no—
73C2×C4×D113C2311yescharacteristic
74#14268637—22no—
75#14268637—41no—
76#14268636—22no—
77#14268634—22no—
78#14268634—41no—
79#14268630—22no—
80C2×#11744751C2311yescharacteristic
81C2×#11744751C2311yescharacteristic
82C4⋊Dic113—21no—
83C4⋊Dic113—21no—
84C4⋊Dic113C2311yescharacteristic
85C4⋊Dic113C2311yescharacteristic
86#14268626—21no—
87#14268626—21no—
88#14268626—21no—
89#14268626—21no—
90#14268626—21no—
91#14268626D411yescharacteristic
92C22⋊Dic113—21no—
93C22⋊Dic113D411yescharacteristic
94#14268624—21no—
95#14268624—21no—
96#14268624—21no—
97#14268624—21no—
98#14268624—21no—
99#14268624D411yescharacteristic
100C4×Dic113—21no—
101C4×Dic113—21no—
102C4×Dic113C2311yescharacteristic
103C2×Q8×C113—21no—
104C2×Q8×C113—21no—
105C2×Q8×C113—22no—
106C2×Q8×C113—22no—
107C2×Q8×C113—41no—
108C2×Q8×C113—41no—
109C2×Q8×C113C2311yescharacteristic
110C2×Q8×C113C2311yescharacteristic
111C2×Q8×C113C2311yescharacteristic
112C2×Q8×C113C2311yescharacteristic
113C22×C4×C113—21no—
114C22×C4×C113—21no—
115C22×C4×C113C2311yescharacteristic
116Q16×C113—22no—
117Q16×C113—22no—
118Q16×C113—22no—
119Q16×C113D414yescomplemented
120C2×C8×C113—22no—
121C2×C8×C113C2311yescharacteristic
122C2×C8×C113C2311yescharacteristic
123C23×C113C2×D411yescharacteristic
124C22×D113—21no—
125C22×D113C2×D411yescharacteristic
126Q8×C113—21no—
127Q8×C113—21no—
128Q8×C113—21no—
129Q8×C113—21no—
130Q8×C113—21no—
131Q8×C113—21no—
132Q8×C113—42no—
133Q8×C113—42no—
134Q8×C113C2×D412yes—
135Q8×C113C2×D412yes—
136C2×C4×C113—21no—
137C2×C4×C113—21no—
138C2×C4×C113—21no—
139C2×C4×C113—21no—
140C2×C4×C113—21no—
141C2×C4×C113—21no—
142C2×C4×C113—42no—
143C2×C4×C113—42no—
144C2×C4×C113C2×D411yescharacteristic
145C2×C4×C113C2411yescharacteristic
146C22⋊D113—22no—
147C22⋊D113—42no—
148C2×Dic113—21no—
149C2×Dic113—21no—
150C2×Dic113—21no—
151C2×Dic113C2×D411yescharacteristic
152C2×Dic113C2×D411yescharacteristic
153C2×Dic113C2×D411yescharacteristic
154D452—21no—
155D452—21no—
156D452—42no—
157C4×D113—21no—
158C4×D113—21no—
159C4×D113—42no—
160C4×D113—42no—
161Dic226—21no—
162Dic226—21no—
163Dic226—42no—
164C8×C113—21no—
165C8×C113C2×D412yes—
166#11744751—21no—
167#11744751—21no—
168C22×C113—21no—
169C22×C113—21no—
170C22×C113#13712yescomplemented
171C22×C113C22×D411yescharacteristic
172C2×D113—21no—
173C2×D113—21no—
174C2×D113—21no—
175C2×D113—41no—
176C2×D113—41no—
177C2×D113—41no—
178C4×C113—22no—
179C4×C113—22no—
180C4×C113—21no—
181C4×C113—41no—
182C4×C113—41no—
183C4×C113C22×D411yescharacteristic
184C4×C113H(5,𝔽2)11yescharacteristic
185Dic113—21no—
186Dic113—21no—
187Dic113—21no—
188Dic113—41no—
189Dic113—41no—
190Dic113—41no—
191C2×C113—22no—
192C2×C113D4211yescharacteristic
193C2×C113C2×#13711yescharacteristic
194C2×C113#57811yescharacteristic
195D113—41no—
196D113—81no—
197#3270—1131nomaximal, Sylow
198C113#327011yescharacteristic, complemented, Sylow
199C2×#137—1131no—
200C22×Q16D11311yescharacteristic, complemented
201D4×Q8—1131no—
202#548—1131no—
203#497—1131no—
204#470—1131no—
205#465—1131no—
206#464—1131no—
207#462—1131no—
208#452—1131no—
209#451—1131no—
210#451—1131no—
211#448—1131no—
212#441—1131no—
213#435—1131no—
214C2×#41—1131no—
215C22×Q8—1131no—
216C22×Q8C2×D11311yescharacteristic
217C22×Q8C2×D11311yescharacteristic
218#137—2264no—
219C2×Q16—24no—
220C2×Q16—1131no—
221C2×Q16—1131no—
222C2×Q16C2×D11311yescharacteristic
223C2×Q16C2×D11311yescharacteristic
224C2×Q16C2×D11311yescharacteristic
225C2×Q16C2×D11311yescharacteristic
226C2×SD16—1131no—
227C2×SD16—1131no—
228C2×SD16—1131no—
229C2×M16—1131no—
230C22×C8C2×D11311yescharacteristic
231C4⋊Q8—1131no—
232C4⋊Q8—2261no—
233#124—1131no—
234#124—2261no—
235C22⋊Q8—1131no—
236C22⋊Q8—1131no—
237C22⋊Q8—1131no—
238C22⋊Q8—2261no—
239C22⋊Q8—2261no—
240C22⋊Q8—2261no—
241#121—1131no—
242#121—2261no—
243C4×Q8—1131no—
244C4×Q8—1131no—
245C4×D4—1131no—
246C4×D4—2261no—
247C4×D4—2261no—
248#106—1131no—
249C4⋊C8—1131no—
250#103—1131no—
251#103—1131no—
252#103—1131no—
253#103—1131no—
254#103—1131no—
255#102—1131no—
256C22⋊C8—1131no—
257C22⋊C8—1131no—
258#97—1131no—
259#41—4521no—
260#41—2262no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—1131no—
268C2×Q8—4521no—
269C2×Q8—2261no—
270C2×Q8—2262no—
271C2×Q8C22×D11311yescharacteristic
272C2×Q8C22×D11311yescharacteristic
273C2×Q8C22×D11311yescharacteristic
274C2×Q8C22×D11311yescharacteristic
275C2×D4—1131no—
276C2×D4—1131no—
277C2×D4—2261no—
278C2×D4—2261no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—1131no—
282C22×C4—2261no—
283C22×C4—2261no—
284C22×C4C22×D11311yescharacteristic
285Q16—22no—
286Q16—22no—
287Q16—22no—
288Q16—4521no—
289Q16—2262no—
290Q16C22⋊D11314yescomplemented
291SD16—4521no—
292SD16—2262no—
293SD16—2262no—
294M16—2262no—
295C2×C8—22no—
296C2×C8—1131no—
297C2×C8—1131no—
298C2×C8C22×D11311yescharacteristic
299C2×C8C22×D11311yescharacteristic
300C4⋊C4—1131no—
301C4⋊C4—1131no—
302C4⋊C4—1131no—
303C4⋊C4—2261no—
304C4⋊C4—2261no—
305C4⋊C4—2261no—
306C4⋊C4—2261no—
307C4⋊C4—2261no—
308C4⋊C4—2261no—
309C4⋊C4—2261no—
310C22⋊C4—1131no—
311C22⋊C4—1131no—
312C22⋊C4—2261no—
313C22⋊C4—2261no—
314C22⋊C4—2261no—
315C22⋊C4—2261no—
316C22⋊C4—2261no—
317C22⋊C4—2261no—
318C42—1131no—
319C42—2261no—
320C42—2261no—
321C23—1131no—
322C23—2261no—
323C23D4×D11311yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—42no—
331Q8—42no—
332Q8—4522no—
333Q8—2261no—
334Q8—2261no—
335Q8C2×C22⋊D11312yes—
336Q8C2×C22⋊D11312yes—
337D4—4522no—
338D4—4522no—
339D4—2262no—
340D4—2261no—
341D4—2261no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—1131no—
351C2×C4—1131no—
352C2×C4—1131no—
353C2×C4—4522no—
354C2×C4—4522no—
355C2×C4—2261no—
356C2×C4—2261no—
357C2×C4—2261no—
358C2×C4—2261no—
359C2×C4—2261no—
360C2×C4D4×D11311yescharacteristic
361C2×C4C23×D11311yescharacteristic
362C8—21no—
363C8—2261no—
364C8—2261no—
365C8C2×C22⋊D11312yes—
366C22—21no—
367C22—21no—
368C22—4521no—
369C22—4521no—
370C22—4521no—
371C22—2261no—
372C22—2261no—
373C22—2261no—
374C22#1681365412yescomplemented
375C22C2×D4×D11311yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—4521no—
382C4—4521no—
383C4—4521no—
384C4—2261no—
385C4—2261no—
386C4—2261no—
387C4C22×C22⋊D11311yescharacteristic
388C4#1681366711yescharacteristic
389C2—22no—
390C2—9041no—
391C2—4521no—
392C2D4×C22⋊D11311yescharacteristic
393C2C2×#1681365411yescharacteristic
394C2#2029941111yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC22×Q16×C113C2
socleC22×C113C22×D4
prime radical#20299259C2
hypercenterC22×Q16D113
hypocenterC113#3270
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#16813654C2semidirect
2. Q8×C22⋊D113C2semidirect
4. #20298991C2semidirect
5. #20298853C2semidirect
6. #20298827C2semidirect
7. #20298822C2semidirect
8. #20298806C2semidirect
9. #20298759C2semidirect
10. #20298754C2semidirect
11. #20298752C2semidirect
12. #20298737C2semidirect
13. #20298705C2semidirect
14. #20298669C2semidirect
15. C22×Q16×C113C2semidirect
119. Q16×C113D4semidirect
170. C22×C113#137semidirect
198. C113#3270semidirect
200. C22×Q16D113semidirect
290. Q16C22⋊D113semidirect
374. C22#16813654semidirect

Maximal subgroups

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

subgroupindex
C2×#168136542
Q8×C22⋊D1132
#202992592
#202989912
#202988532
#202988272
#202988222
#202988062
#202987592
#202987542
#202987522
#202987372
#202987052
#202986692
C22×Q16×C1132
#3270113