Finite Group Databasebeta

← #28401059

Subgroups — #28401059

Order 14464.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16813650C211yesmaximal, characteristic, complemented
2C2×#16813642C211yesmaximal, characteristic, complemented
3C22×#14268632C211yesmaximal, characteristic, complemented
4#20299165C211yesmaximal, characteristic, complemented
5#20299162C211yesmaximal, characteristic, complemented
6#20298986C212yesmaximal, complemented
7#20298984C212yesmaximal, complemented
8#20298979C212yesmaximal, complemented
9#20298973C212yesmaximal, complemented
10#20298556C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C113C211yesmaximal, characteristic, complemented
12C2×#14268647C2211yescharacteristic
13C22×D452C2211yescharacteristic
14#16813650—24no—
15#16813642—24no—
16C2×#14268637C2211yescharacteristic, complemented
17C2×#14268636C2212yescomplemented
18C2×#14268635C2211yescharacteristic, complemented
19C2×#14268634C2211yescharacteristic, complemented
20C2×D904C2212yescomplemented
21C2×#14268632—24no—
22C2×#14268632C2212yes—
23C2×#14268632C2212yes—
24C2×#14268632C2211yescharacteristic, complemented
25C22×#11744751C2211yescharacteristic
26#16813618C2212yescomplemented
27#16813618C2212yescomplemented
28#16813611C2212yescomplemented
29#16813609C2212yescomplemented
30#16813596—22no—
31#16813593—22no—
32#16813591—22no—
33#16813590—22no—
34#16813581C2211yescharacteristic
35#16813532C2212yescomplemented
36C8×Dic113C2212yescomplemented
37C2×#41×C113C2211yescharacteristic
38C22×D4×C113C2211yescharacteristic
39AGL(1,ℤ/8)×C113—24no—
40C2×SD16×C113C2212yescomplemented
41C2×D8×C113C2212yescomplemented
42C2×M16×C113C2211yescharacteristic
43C23×D113—21no—
44#14268647—22no—
45#14268647—22no—
46C2×Dic226C2311yescharacteristic
47C2×D452—22no—
48C2×D452—22no—
49C2×D452—21no—
50C2×D452C2312yes—
51C2×D452C2311yescharacteristic
52C2×C22⋊D113—21no—
53C2×C22⋊D113—24no—
54C2×C22⋊D113—24no—
55C2×C4×D113—21no—
56#14268637—22no—
57#14268636—24no—
58#14268635—22no—
59#14268634—22no—
60D904—24no—
61#14268632—24no—
62#14268632—24no—
63#14268632—22no—
64C2×#11744751D414yescomplemented
65C2×#11744751C2311yescharacteristic
66C2×#11744751C2311yescharacteristic
67C4⋊Dic113D412yes—
68C22⋊Dic113—24no—
69C22⋊Dic113D412yes—
70#14268624—24no—
71C4×Dic113C2312yes—
72C24×C113—21no—
73#41×C113—22no—
74#41×C113—22no—
75C2×Q8×C113C2311yescharacteristic
76C2×D4×C113—21no—
77C2×D4×C113—22no—
78C2×D4×C113—22no—
79C2×D4×C113—21no—
80C2×D4×C113C2312yes—
81C2×D4×C113C2311yescharacteristic
82C22×C4×C113—21no—
83C22×C4×C113C2311yescharacteristic
84SD16×C113—24no—
85D8×C113—24no—
86M16×C113D414yescomplemented
87C2×C8×C113C2312yes—
88C23×C113—21no—
89C23×C113—22no—
90C23×C113—42no—
91C23×C113—82no—
92C23×C113C2×D411yescharacteristic
93C22×D113—22no—
94C22×D113—21no—
95C22×D113—42no—
96C22×D113—82no—
97Q8×C113—21no—
98Q8×C113—21no—
99D4×C113—21no—
100D4×C113—22no—
101D4×C113—22no—
102D4×C113—21no—
103D4×C113—42no—
104D4×C113—42no—
105C2×C4×C113—21no—
106C2×C4×C113—42no—
107C2×C4×C113C2×D411yescharacteristic
108C2×C4×C113C2×D412yes—
109C2×C4×C113C2×D412yes—
110C2×C4×C113C2411yescharacteristic
111C22⋊D113—42no—
112C22⋊D113—84no—
113C22⋊D113—84no—
114C2×Dic113—21no—
115C2×Dic113C2×D414yes—
116D452—22no—
117D452—21no—
118D452—21no—
119D452—22no—
120D452—42no—
121C4×D113—42no—
122Dic226—21no—
123Dic226—21no—
124C8×C113C2×D414yes—
125#11744751C2×D412yes—
126#11744751C2×D412yes—
127C22×C113—22no—
128C22×C113—41no—
129C22×C113—42no—
130C22×C113—42no—
131C22×C113—44no—
132C22×C113—84no—
133C22×C113#12712yes—
134C22×C113C22×D411yescharacteristic
135C2×D113—41no—
136C2×D113—42no—
137C2×D113—42no—
138C2×D113—44no—
139C2×D113—84no—
140C4×C113—41no—
141C4×C113#12712yes—
142C4×C113C22×D411yescharacteristic
143C4×C113C22×D411yescharacteristic
144Dic113—24no—
145Dic113—41no—
146C2×C113—22no—
147C2×C113—81no—
148C2×C113—82no—
149C2×C113C2×#12711yescharacteristic
150C2×C113#57612yes—
151D113—81no—
152D113—82no—
153#3155—1131nomaximal, Sylow
154C113#315511yescharacteristic, complemented, Sylow
155C2×AGL(1,ℤ/8)—1131no—
156C2×AGL(1,ℤ/8)D11311yescharacteristic, complemented
157C2×#135—1131no—
158C22×D8—1131no—
159#535—1131no—
160#535—1131no—
161#496—1132no—
162#496—1132no—
163#495—1132no—
164#493—1132no—
165#405—1131no—
166C2×#41—1131no—
167C2×#41C2×D11311yescharacteristic
168C22×D4—1131no—
169C22×D4C2×D11311yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—2264no—
172#135—2264no—
173C2×Q16—1131no—
174C2×SD16—1131no—
175C2×SD16—1131no—
176C2×SD16—1132no—
177C2×SD16C2×D11312yescomplemented
178C2×D8—1132no—
179C2×D8—1132no—
180C2×D8—1131no—
181C2×D8—1132no—
182C2×D8—2264no—
183C2×D8C2×D11312yescomplemented
184C2×M16C2×D11311yescharacteristic
185C22×C8—1131no—
186#127—1132no—
187#127—1132no—
188#124—1132no—
189#124—1132no—
190#121—2262no—
191#121—2262no—
192#120—2262no—
193#120—2262no—
194#117—1131no—
195#97—1132no—
196C4×C8—1132no—
197C24—21no—
198C24—2261no—
199#41—22no—
200#41—22no—
201#41—2262no—
202#41—2262no—
203C2×Q8—1131no—
204C2×Q8C22×D11311yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—1132no—
210C2×D4—1131no—
211C2×D4—2264no—
212C2×D4—2264no—
213C2×D4—2261no—
214C2×D4—2261no—
215C2×D4—2262no—
216C2×D4—2262no—
217C2×D4C22×D11312yes—
218C2×D4C22×D11311yescharacteristic
219C22×C4—21no—
220C22×C4—2261no—
221C22×C4C22×D11311yescharacteristic
222Q16—2262no—
223SD16—24no—
224SD16—2262no—
225SD16—2262no—
226SD16—2264no—
227D8—24no—
228D8—2264no—
229D8—2264no—
230D8—2262no—
231D8—2264no—
232M16C22⋊D11314yescomplemented
233C2×C8—1134no—
234C2×C8—1131no—
235C2×C8—1131no—
236C2×C8C22×D11312yes—
237C4⋊C4—1132no—
238C22⋊C4—1132no—
239C22⋊C4—2264no—
240C22⋊C4—2264no—
241C42—1132no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—9042no—
247C23—4522no—
248C23—2262no—
249C23—2261no—
250C23D4×D11311yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—2261no—
254Q8—2261no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—9044no—
262D4—9044no—
263D4—4522no—
264D4—4522no—
265D4—2262no—
266D4—2261no—
267D4—2261no—
268D4—2262no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—1134no—
272C2×C4—4522no—
273C2×C4—2261no—
274C2×C4D4×D11311yescharacteristic
275C2×C4D4×D11312yes—
276C2×C4C2×C22⋊D11312yes—
277C2×C4C23×D11311yescharacteristic
278C8—1132no—
279C8—1132no—
280C8C2×C22⋊D11314yes—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—9044no—
288C22—4521no—
289C22—4522no—
290C22—4522no—
291C22—4524no—
292C22#1681361812yes—
293C22C2×D4×D11311yescharacteristic
294C4—41no—
295C4—4521no—
296C4—2264no—
297C4#1681361812yes—
298C4C2×D4×D11311yescharacteristic
299C4C22×C22⋊D11311yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—9042no—
304C2—9041no—
305C2C2×#1681361811yescharacteristic
306C2#2029940212yes—
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC2×AGL(1,ℤ/8)×C113C2
socleC22×C113C22×D4
hypercenterC2×AGL(1,ℤ/8)D113
hypocenterC113#3155
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16813650C2semidirect
2. C2×#16813642C2semidirect
3. C22×#14268632C2semidirect
4. #20299165C2semidirect
5. #20299162C2semidirect
6. #20298986C2semidirect
7. #20298984C2semidirect
8. #20298979C2semidirect
9. #20298973C2semidirect
10. #20298556C2semidirect
11. C2×AGL(1,ℤ/8)×C113C2semidirect
16. C2×#14268637C22semidirect
17. C2×#14268636C22semidirect
18. C2×#14268635C22semidirect
19. C2×#14268634C22semidirect
20. C2×D904C22semidirect
24. C2×#14268632C22semidirect
26. #16813618C22semidirect
27. #16813618C22semidirect
28. #16813611C22semidirect
29. #16813609C22semidirect
35. #16813532C22semidirect
36. C8×Dic113C22semidirect
40. C2×SD16×C113C22semidirect
41. C2×D8×C113C22semidirect
64. C2×#11744751D4semidirect
86. M16×C113D4semidirect
154. C113#3155semidirect
156. C2×AGL(1,ℤ/8)D113semidirect
177. C2×SD16C2×D113semidirect
183. C2×D8C2×D113semidirect
232. M16C22⋊D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136502
C2×#168136422
C22×#142686322
#202991652
#202991622
#202989862
#202989842
#202989792
#202989732
#202985562
C2×AGL(1,ℤ/8)×C1132
#3155113