Finite Group Databasebeta

← #29505582

Subgroups — #29505582

Order 15296.

258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#21440007C211yesmaximal, characteristic, complemented
2#21440005C211yesmaximal, characteristic, complemented
3#21439993C211yesmaximal, characteristic, complemented
4#21439989C212yesmaximal, complemented
5#21439984C212yesmaximal, complemented
6#21439980C211yesmaximal, characteristic, complemented
7#21439979C212yesmaximal, complemented
8C2×#16838536C211yesmaximal, characteristic, complemented
9SD16×D239C212yesmaximal, complemented
10#21439968C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C239C211yesmaximal, characteristic, complemented
12#16838548—21no—
13#16838548C2211yescharacteristic
14#16838548C2211yescharacteristic
15#16838547—21no—
16#16838547—22no—
17#16838547C2212yes—
18#16838547C2212yescomplemented
19C2×Dic478—21no—
20C2×Dic478C2211yescharacteristic, complemented
21Q8×D239C2212yescomplemented
22C2×C22⋊D239—22no—
23D4×D239—21no—
24D4×D239C2212yes—
25Dic956C2212yescomplemented
26#16838538C2211yescharacteristic, complemented
27#16838537C2212yescomplemented
28#16838536C2212yescomplemented
29#16838536C2212yescomplemented
30#16838536C2211yescharacteristic, complemented
31#16838535C2211yescharacteristic, complemented
32#16838533C2211yescharacteristic, complemented
33#16838531C2212yescomplemented
34C2×C239⋊C8C2211yescharacteristic, complemented
35C8×D239C2212yescomplemented
36#41×C239C2211yescharacteristic, complemented
37C2×D4×C239C2211yescharacteristic
38SD16×C239C2212yescomplemented
39D8×C239C2212yescomplemented
40M16×C239C2211yescharacteristic, complemented
41C22×D239—22no—
42Dic478—21no—
43Dic478—22no—
44Dic478D411yescharacteristic
45Dic478C2312yes—
46Dic478C2311yescharacteristic
47D956D411yescharacteristic
48D956C2311yescharacteristic
49C22⋊D239—21no—
50C22⋊D239—22no—
51C22⋊D239—22no—
52C22⋊D239—22no—
53C22⋊D239D412yes—
54C2×Dic239—21no—
55C2×Dic239—22no—
56C2×Dic239—22no—
57C4×D239—22no—
58C4×D239—21no—
59C4×D239C2312yes—
60C239⋊C8C2311yescharacteristic
61C239⋊C8C2311yescharacteristic
62C23×C239—21no—
63Q8×C239C2311yescharacteristic
64D4×C239—21no—
65D4×C239—21no—
66D4×C239C2311yescharacteristic
67D4×C239C2312yes—
68C2×C4×C239—21no—
69C2×C4×C239C2311yescharacteristic
70C8×C239C2312yes—
71C22×C239—21no—
72C22×C239—22no—
73C22×C239—81no—
74C22×C239C2×D411yescharacteristic
75C2×D239—21no—
76C2×D239—82no—
77C2×D239C2×D412yes—
78C4×C239—21no—
79C4×C239C2×D411yescharacteristic
80C4×C239C2411yescharacteristic
81Dic239—21no—
82Dic239—22no—
83Dic239C2×D412yes—
84C2×C239—21no—
85C2×C239—42no—
86C2×C239—41no—
87C2×C239C22×D411yescharacteristic
88D239—22no—
89D239—41no—
90C239#57711yescharacteristic, complemented, Sylow
91#577—2391nomaximal, Sylow
922−1+4—2391no—
93H(5,𝔽2)—2391no—
94#137—2392no—
95#137—2391no—
96AGL(1,ℤ/8)—2392no—
97AGL(1,ℤ/8)D23911yescharacteristic, complemented
98#135—2391no—
99#135—2392no—
100C2×SD16—2391no—
101C2×SD16—2392no—
102#131—2391no—
103#41—4781no—
104#41—4781no—
105#41—4782no—
106#41—2392no—
107#41—2392no—
108#41—2391no—
109#41—2391no—
110#41C2×D23911yescharacteristic, complemented
111C2×Q8—4781no—
112C2×Q8—2392no—
113C2×Q8—2391no—
114C2×D4—4781no—
115C2×D4—4782no—
116C2×D4—2392no—
117C2×D4C2×D23911yescharacteristic
118Q16—2391no—
119Q16—2392no—
120SD16—2391no—
121SD16—2392no—
122SD16—2392no—
123SD16—2391no—
124SD16—2392no—
125SD16C2×D23912yescomplemented
126D8—2391no—
127D8C2×D23912yescomplemented
128M16—2392no—
129M16C2×D23911yescharacteristic, complemented
130C2×C8—2391no—
131C2×C8—2392no—
132C23—21no—
133C23—4782no—
134Q8—4782no—
135Q8—4781no—
136Q8—2392no—
137Q8—2391no—
138Q8—2391no—
139Q8C22×D23911yescharacteristic
140D4—21no—
141D4—21no—
142D4—4782no—
143D4—4782no—
144D4—4781no—
145D4—4782no—
146D4—2392no—
147D4—2391no—
148D4—2391no—
149D4C22×D23911yescharacteristic
150D4C22×D23912yes—
151C2×C4—21no—
152C2×C4—4782no—
153C2×C4—4781no—
154C2×C4—4782no—
155C2×C4—4782no—
156C2×C4—4781no—
157C2×C4—2392no—
158C2×C4C22×D23911yescharacteristic
159C8—2391no—
160C8—2391no—
161C8C22×D23912yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—19122no—
166C22—4781no—
167C22—2392no—
168C22D4×D23911yescharacteristic
169C4—21no—
170C4—4781no—
171C4—4782no—
172C4—2392no—
173C4D4×D23911yescharacteristic
174C4C23×D23911yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—9561no—
179C2—4782no—
180C2C2×D4×D23911yescharacteristic
subgroupG/H
centerC2C2×D4×D239
commutator, cosocleC4×C239C24
FrattiniC4C23×D239
FittingAGL(1,ℤ/8)×C239C2
socleC2×C239C22×D4
hypercenterAGL(1,ℤ/8)D239
hypocenterC239#577
LCSG ▷ C4×C239 ▷ C2×C239 ▷ C239
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #21440007C2semidirect
2. #21440005C2semidirect
3. #21439993C2semidirect
4. #21439989C2semidirect
5. #21439984C2semidirect
6. #21439980C2semidirect
7. #21439979C2semidirect
8. C2×#16838536C2semidirect
9. SD16×D239C2semidirect
10. #21439968C2semidirect
11. AGL(1,ℤ/8)×C239C2semidirect
18. #16838547C22semidirect
20. C2×Dic478C22semidirect
21. Q8×D239C22semidirect
25. Dic956C22semidirect
26. #16838538C22semidirect
27. #16838537C22semidirect
28. #16838536C22semidirect
29. #16838536C22semidirect
30. #16838536C22semidirect
31. #16838535C22semidirect
32. #16838533C22semidirect
33. #16838531C22semidirect
34. C2×C239⋊C8C22semidirect
35. C8×D239C22semidirect
36. #41×C239C22semidirect
38. SD16×C239C22semidirect
39. D8×C239C22semidirect
40. M16×C239C22semidirect
90. C239#577semidirect
97. AGL(1,ℤ/8)D239semidirect
110. #41C2×D239semidirect
125. SD16C2×D239semidirect
127. D8C2×D239semidirect
129. M16C2×D239semidirect

Maximal subgroups

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

subgroupindex
#214400072
#214400052
#214399932
#214399892
#214399842
#214399802
#214399792
C2×#168385362
SD16×D2392
#214399682
AGL(1,ℤ/8)×C2392
#577239