Finite Group Databasebeta

← #29505568

Subgroups — #29505568

Order 15296.

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

normal subgroups only

This group has 262 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
1C2×#16838547C211yesmaximal, characteristic, complemented
2#41×D239C211yesmaximal, characteristic, complemented
3#21439993C211yesmaximal, characteristic, complemented
4#21439990C211yesmaximal, characteristic, complemented
5#21439989C212yesmaximal, complemented
6#21439986C211yesmaximal, characteristic, complemented
7#21439984C212yesmaximal, complemented
8#21439981C212yesmaximal, complemented
9#21439979C212yesmaximal, complemented
10M16×D239C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C239C211yesmaximal, characteristic, complemented
12#16838549C2211yescharacteristic, complemented
13#16838548—21no—
14#16838548C2211yescharacteristic
15#16838547—21no—
16#16838547—21no—
17#16838547—21no—
18#16838547C2212yes—
19#16838547C2211yescharacteristic, complemented
20#16838547C2212yescomplemented
21C2×Dic478C2211yescharacteristic, complemented
22Q8×D239C2211yescharacteristic, complemented
23C2×C22⋊D239—21no—
24D4×D239—21no—
25D4×D239C2211yescharacteristic, complemented
26C22×Dic239—21no—
27C2×C4×D239—21no—
28C2×C4×D239C2211yescharacteristic
29Dic956C2212yescomplemented
30#16838538C2212yescomplemented
31#16838537C2212yescomplemented
32#16838536C2212yescomplemented
33#16838536C2212yescomplemented
34#16838533C2212yescomplemented
35#16838532C2211yescharacteristic, complemented
36#16838531C2212yescomplemented
37C8×D239C2212yescomplemented
38#41×C239C2211yescharacteristic
39C2×D4×C239C2211yescharacteristic, complemented
40SD16×C239C2212yescomplemented
41D8×C239C2212yescomplemented
42M16×C239C2211yescharacteristic, complemented
43C22×D239—21no—
44C22×D239D411yescharacteristic
45Dic478—21no—
46Dic478—21no—
47Dic478C2311yescharacteristic
48Dic478C2312yes—
49D956—21no—
50D956C2311yescharacteristic
51C22⋊D239—21no—
52C22⋊D239—21no—
53C22⋊D239—22no—
54C22⋊D239—22no—
55C22⋊D239—21no—
56C2×Dic239—21no—
57C2×Dic239—21no—
58C2×Dic239—22no—
59C2×Dic239—22no—
60C2×Dic239D411yescharacteristic
61C4×D239—21no—
62C4×D239—21no—
63C4×D239—21no—
64C4×D239D411yescharacteristic
65C4×D239D411yescharacteristic
66C4×D239C2311yescharacteristic
67C4×D239C2311yescharacteristic
68C239⋊C8C2312yes—
69C23×C239—21no—
70Q8×C239C2311yescharacteristic
71D4×C239—21no—
72D4×C239—21no—
73D4×C239C2311yescharacteristic
74D4×C239C2312yes—
75C2×C4×C239—21no—
76C2×C4×C239C2311yescharacteristic
77C8×C239C2312yes—
78C22×C239—21no—
79C22×C239—22no—
80C22×C239—42no—
81C22×C239C2×D411yescharacteristic
82C2×D239—21no—
83C2×D239—41no—
84C2×D239—42no—
85C2×D239C2×D411yescharacteristic
86C2×D239C2×D411yescharacteristic
87C4×C239—21no—
88C4×C239C2×D411yescharacteristic
89C4×C239C2411yescharacteristic
90Dic239—21no—
91Dic239—22no—
92Dic239C2×D411yescharacteristic
93Dic239C2×D411yescharacteristic
94C2×C239—21no—
95C2×C239—41no—
96C2×C239—42no—
97C2×C239C22×D411yescharacteristic
98D239—21no—
99D239—21no—
100D239—41no—
101C239#57511yescharacteristic, complemented, Sylow
102#575—2391nomaximal, Sylow
103C2×#41—2391no—
104C2×#41—2391no—
105#137—2391no—
106#137—2392no—
107#137—2391no—
108AGL(1,ℤ/8)—2391no—
109AGL(1,ℤ/8)—2392no—
110AGL(1,ℤ/8)D23911yescharacteristic, complemented
111#135—2392no—
112#135—2392no—
113C2×M16—2391no—
114#41—4781no—
115#41—4781no—
116#41—4781no—
117#41—4781no—
118#41—2391no—
119#41—2392no—
120#41—2391no—
121#41—2392no—
122#41—2391no—
123#41C2×D23911yescharacteristic
124C2×Q8—2391no—
125C2×Q8—2391no—
126C2×D4—4781no—
127C2×D4—4781no—
128C2×D4—2391no—
129C2×D4C2×D23911yescharacteristic, complemented
130C22×C4—4781no—
131C22×C4—4781no—
132C22×C4—2391no—
133Q16—2392no—
134Q16—2392no—
135SD16—2392no—
136SD16—2392no—
137SD16—2392no—
138SD16C2×D23912yescomplemented
139D8—2392no—
140D8C2×D23912yescomplemented
141M16—2391no—
142M16—2392no—
143M16C2×D23911yescharacteristic, complemented
144C2×C8—2392no—
145C23—21no—
146C23—4781no—
147C23—2391no—
148Q8—4781no—
149Q8—4781no—
150Q8—2391no—
151Q8—2392no—
152Q8C22×D23911yescharacteristic
153D4—21no—
154D4—21no—
155D4—4781no—
156D4—4781no—
157D4—4782no—
158D4—4782no—
159D4—4781no—
160D4—4781no—
161D4—2391no—
162D4C22×D23912yes—
163D4C22×D23911yescharacteristic
164C2×C4—21no—
165C2×C4—4781no—
166C2×C4—4781no—
167C2×C4—4781no—
168C2×C4—4781no—
169C2×C4—4782no—
170C2×C4—4782no—
171C2×C4—4781no—
172C2×C4—2391no—
173C2×C4—2391no—
174C2×C4—2391no—
175C2×C4—2391no—
176C2×C4—2391no—
177C2×C4C22×D23911yescharacteristic
178C8—2392no—
179C8C22×D23912yes—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—9561no—
184C22—9562no—
185C22—4781no—
186C22—2391no—
187C22—2391no—
188C22D4×D23911yescharacteristic
189C4—21no—
190C4—4781no—
191C4—4782no—
192C4—2391no—
193C4—2391no—
194C4D4×D23911yescharacteristic
195C4C23×D23911yescharacteristic
196C2—21no—
197C2—42no—
198C2—41no—
199C2—9561no—
200C2—4781no—
201C2—4781no—
202C2C2×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#575
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. C2×#16838547C2semidirect
2. #41×D239C2semidirect
3. #21439993C2semidirect
4. #21439990C2semidirect
5. #21439989C2semidirect
6. #21439986C2semidirect
7. #21439984C2semidirect
8. #21439981C2semidirect
9. #21439979C2semidirect
10. M16×D239C2semidirect
11. AGL(1,ℤ/8)×C239C2semidirect
12. #16838549C22semidirect
19. #16838547C22semidirect
20. #16838547C22semidirect
21. C2×Dic478C22semidirect
22. Q8×D239C22semidirect
25. D4×D239C22semidirect
29. Dic956C22semidirect
30. #16838538C22semidirect
31. #16838537C22semidirect
32. #16838536C22semidirect
33. #16838536C22semidirect
34. #16838533C22semidirect
35. #16838532C22semidirect
36. #16838531C22semidirect
37. C8×D239C22semidirect
39. C2×D4×C239C22semidirect
40. SD16×C239C22semidirect
41. D8×C239C22semidirect
42. M16×C239C22semidirect
101. C239#575semidirect
110. AGL(1,ℤ/8)D239semidirect
129. C2×D4C2×D239semidirect
138. SD16C2×D239semidirect
140. D8C2×D239semidirect
143. M16C2×D239semidirect

Maximal subgroups

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

subgroupindex
C2×#168385472
#41×D2392
#214399932
#214399902
#214399892
#214399862
#214399842
#214399812
#214399792
M16×D2392
AGL(1,ℤ/8)×C2392
#575239