Finite Group Databasebeta

← #29505576

Subgroups — #29505576

Order 15296.

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

normal subgroups only

This group has 268 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#21440006C211yesmaximal, characteristic, complemented
2#21440005C211yesmaximal, characteristic, complemented
3#21439988C211yesmaximal, characteristic, complemented
4#21439987C212yesmaximal, complemented
5#21439984C212yesmaximal, complemented
6#21439981C212yesmaximal, complemented
7#21439980C211yesmaximal, characteristic, complemented
8C2×#16838533C211yesmaximal, characteristic, complemented
9D8×D239C212yesmaximal, complemented
10#21439968C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C239C211yesmaximal, characteristic, complemented
12#16838549C2212yescomplemented
13#16838548—21no—
14#16838548C2211yescharacteristic
15#16838548C2211yescharacteristic, complemented
16#16838547—21no—
17#16838547C2212yescomplemented
18C2×D956—21no—
19C2×D956C2211yescharacteristic, complemented
20C2×C22⋊D239—22no—
21D4×D239—21no—
22D4×D239—22no—
23D4×D239C2212yescomplemented
24D4×D239C2212yescomplemented
25#16838538C2211yescharacteristic, complemented
26#16838537C2212yescomplemented
27#16838536C2211yescharacteristic, complemented
28#16838535C2211yescharacteristic, complemented
29D1912C2212yescomplemented
30#16838533C2211yescharacteristic, complemented
31#16838533C2212yescomplemented
32#16838533C2212yescomplemented
33#16838531C2212yescomplemented
34C2×C239⋊C8C2211yescharacteristic, complemented
35C8×D239C2212yescomplemented
36#41×C239C2211yescharacteristic, complemented
37C2×D4×C239C2211yescharacteristic, complemented
38SD16×C239C2212yescomplemented
39D8×C239C2212yescomplemented
40M16×C239C2211yescharacteristic, complemented
41C22×D239—21no—
42C22×D239—22no—
43C22×D239—22no—
44Dic478D411yescharacteristic
45Dic478C2311yescharacteristic
46D956—22no—
47D956—21no—
48D956D411yescharacteristic
49D956C2312yes—
50D956C2311yescharacteristic
51C22⋊D239—21no—
52C22⋊D239—22no—
53C22⋊D239—22no—
54C22⋊D239—22no—
55C22⋊D239D412yes—
56C2×Dic239—22no—
57C4×D239—21no—
58C4×D239—22no—
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—22no—
77C2×D239—82no—
78C2×D239—82no—
79C2×D239—81no—
80C2×D239C2×D412yes—
81C4×C239—21no—
82C4×C239C2×D411yescharacteristic
83C4×C239C2411yescharacteristic
84Dic239—21no—
85Dic239C2×D412yes—
86C2×C239—21no—
87C2×C239—41no—
88C2×C239—42no—
89C2×C239C22×D411yescharacteristic
90D239—22no—
91D239—41no—
92D239—42no—
93C239#57611yescharacteristic, complemented, Sylow
94#576—2391nomaximal, Sylow
95H(5,𝔽2)—2391no—
96H(5,𝔽2)—2391no—
97AGL(1,ℤ/8)—2392no—
98AGL(1,ℤ/8)—2392no—
99AGL(1,ℤ/8)—2391no—
100AGL(1,ℤ/8)D23911yescharacteristic, complemented
101#135—2391no—
102#135—2392no—
103C2×D8—2391no—
104C2×D8—2392no—
105#131—2391no—
106#41—4781no—
107#41—4781no—
108#41—2392no—
109#41—2392no—
110#41—2391no—
111#41—2391no—
112#41C2×D23911yescharacteristic, complemented
113C2×D4—4781no—
114C2×D4—4782no—
115C2×D4—4781no—
116C2×D4—4782no—
117C2×D4—2392no—
118C2×D4—2392no—
119C2×D4—2391no—
120C2×D4C2×D23911yescharacteristic, complemented
121Q16—2391no—
122SD16—2391no—
123SD16—2391no—
124SD16—2392no—
125SD16C2×D23912yescomplemented
126D8—2392no—
127D8—2391no—
128D8—2392no—
129D8—2392no—
130D8C2×D23912yescomplemented
131M16—2392no—
132M16C2×D23911yescharacteristic, complemented
133C2×C8—2391no—
134C2×C8—2392no—
135C23—21no—
136C23—4782no—
137C23—4782no—
138C23—4781no—
139Q8—2391no—
140Q8—2391no—
141Q8C22×D23911yescharacteristic
142D4—21no—
143D4—21no—
144D4—4782no—
145D4—4782no—
146D4—4781no—
147D4—4782no—
148D4—4782no—
149D4—4781no—
150D4—2392no—
151D4—2391no—
152D4—2392no—
153D4—2391no—
154D4C22×D23912yes—
155D4C22×D23911yescharacteristic
156C2×C4—21no—
157C2×C4—4782no—
158C2×C4—4782no—
159C2×C4—4781no—
160C2×C4—2392no—
161C2×C4C22×D23911yescharacteristic
162C8—2391no—
163C8—2391no—
164C8C22×D23912yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—19121no—
169C22—19122no—
170C22—19122no—
171C22—4781no—
172C22—4782no—
173C22—2392no—
174C22D4×D23911yescharacteristic
175C4—21no—
176C4—4781no—
177C4—2392no—
178C4D4×D23911yescharacteristic
179C4C23×D23911yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—9562no—
184C2—9561no—
185C2—4782no—
186C2C2×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#576
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. #21440006C2semidirect
2. #21440005C2semidirect
3. #21439988C2semidirect
4. #21439987C2semidirect
5. #21439984C2semidirect
6. #21439981C2semidirect
7. #21439980C2semidirect
8. C2×#16838533C2semidirect
9. D8×D239C2semidirect
10. #21439968C2semidirect
11. AGL(1,ℤ/8)×C239C2semidirect
12. #16838549C22semidirect
15. #16838548C22semidirect
17. #16838547C22semidirect
19. C2×D956C22semidirect
23. D4×D239C22semidirect
24. D4×D239C22semidirect
25. #16838538C22semidirect
26. #16838537C22semidirect
27. #16838536C22semidirect
28. #16838535C22semidirect
29. D1912C22semidirect
30. #16838533C22semidirect
31. #16838533C22semidirect
32. #16838533C22semidirect
33. #16838531C22semidirect
34. C2×C239⋊C8C22semidirect
35. C8×D239C22semidirect
36. #41×C239C22semidirect
37. C2×D4×C239C22semidirect
38. SD16×C239C22semidirect
39. D8×C239C22semidirect
40. M16×C239C22semidirect
93. C239#576semidirect
100. AGL(1,ℤ/8)D239semidirect
112. #41C2×D239semidirect
120. C2×D4C2×D239semidirect
125. SD16C2×D239semidirect
130. D8C2×D239semidirect
132. M16C2×D239semidirect

Maximal subgroups

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

subgroupindex
#214400062
#214400052
#214399882
#214399872
#214399842
#214399812
#214399802
C2×#168385332
D8×D2392
#214399682
AGL(1,ℤ/8)×C2392
#576239