Finite Group Databasebeta

← #31824267

Subgroups — #31824267

Order 17984.

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#22582925C211yesmaximal, characteristic, complemented
2#22582924C211yesmaximal, characteristic, complemented
3#22582907C211yesmaximal, characteristic, complemented
4#22582906C212yesmaximal, complemented
5#22582903C212yesmaximal, complemented
6#22582900C212yesmaximal, complemented
7#22582899C211yesmaximal, characteristic, complemented
8C2×#17985123C211yesmaximal, characteristic, complemented
9D8×D281C212yesmaximal, complemented
10#22582887C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C281C211yesmaximal, characteristic, complemented
12#17985139C2212yescomplemented
13#17985138—21no—
14#17985138C2211yescharacteristic
15#17985138C2211yescharacteristic, complemented
16#17985137—21no—
17#17985137C2212yescomplemented
18C2×D1124—21no—
19C2×D1124C2211yescharacteristic, complemented
20C2×C22⋊D281—22no—
21D4×D281—21no—
22D4×D281—22no—
23D4×D281C2212yescomplemented
24D4×D281C2212yescomplemented
25#17985128C2211yescharacteristic, complemented
26#17985127C2212yescomplemented
27#17985126C2211yescharacteristic, complemented
28#17985125C2211yescharacteristic, complemented
29D2248C2212yescomplemented
30#17985123C2211yescharacteristic, complemented
31#17985123C2212yescomplemented
32#17985123C2212yescomplemented
33#17985121C2212yescomplemented
34C2×#14555886C2211yescharacteristic, complemented
35C8×D281C2212yescomplemented
36#41×C281C2211yescharacteristic, complemented
37C2×D4×C281C2211yescharacteristic, complemented
38SD16×C281C2212yescomplemented
39D8×C281C2212yescomplemented
40M16×C281C2211yescharacteristic, complemented
41C22×D281—21no—
42C22×D281—22no—
43C22×D281—22no—
44Dic562D411yescharacteristic
45Dic562C2311yescharacteristic
46D1124—22no—
47D1124—21no—
48D1124D411yescharacteristic
49D1124C2312yes—
50D1124C2311yescharacteristic
51C22⋊D281—21no—
52C22⋊D281—22no—
53C22⋊D281—22no—
54C22⋊D281—22no—
55C22⋊D281D412yes—
56C2×Dic281—22no—
57C4×D281—21no—
58C4×D281—22no—
59C4×D281C2312yes—
60#14555886C2311yescharacteristic
61#14555886C2311yescharacteristic
62C23×C281—21no—
63Q8×C281C2311yescharacteristic
64D4×C281—21no—
65D4×C281—21no—
66D4×C281C2311yescharacteristic
67D4×C281C2312yes—
68C2×C4×C281—21no—
69C2×C4×C281C2311yescharacteristic
70C8×C281C2312yes—
71C22×C281—21no—
72C22×C281—22no—
73C22×C281—81no—
74C22×C281C2×D411yescharacteristic
75C2×D281—21no—
76C2×D281—22no—
77C2×D281—82no—
78C2×D281—82no—
79C2×D281—81no—
80C2×D281C2×D412yes—
81C4×C281—21no—
82C4×C281C2×D411yescharacteristic
83C4×C281C2411yescharacteristic
84Dic281—21no—
85Dic281C2×D412yes—
86C2×C281—21no—
87C2×C281—41no—
88C2×C281—42no—
89C2×C281C22×D411yescharacteristic
90D281—22no—
91D281—41no—
92D281—42no—
93C281#57611yescharacteristic, complemented, Sylow
94#576—2811nomaximal, Sylow
95H(5,𝔽2)—2811no—
96H(5,𝔽2)—2811no—
97AGL(1,ℤ/8)—2812no—
98AGL(1,ℤ/8)—2812no—
99AGL(1,ℤ/8)—2811no—
100AGL(1,ℤ/8)D28111yescharacteristic, complemented
101#135—2811no—
102#135—2812no—
103C2×D8—2811no—
104C2×D8—2812no—
105#131—2811no—
106#41—2811no—
107#41—2812no—
108#41—2812no—
109#41—2811no—
110#41—5621no—
111#41—5621no—
112#41C2×D28111yescharacteristic, complemented
113C2×D4—2812no—
114C2×D4—2812no—
115C2×D4—2811no—
116C2×D4—5621no—
117C2×D4—5622no—
118C2×D4—5621no—
119C2×D4—5622no—
120C2×D4C2×D28111yescharacteristic, complemented
121Q16—2811no—
122SD16—2811no—
123SD16—2811no—
124SD16—2812no—
125SD16C2×D28112yescomplemented
126D8—2812no—
127D8—2811no—
128D8—2812no—
129D8—2812no—
130D8C2×D28112yescomplemented
131M16—2812no—
132M16C2×D28111yescharacteristic, complemented
133C2×C8—2811no—
134C2×C8—2812no—
135C23—21no—
136C23—5622no—
137C23—5622no—
138C23—5621no—
139Q8—2811no—
140Q8—2811no—
141Q8C22×D28111yescharacteristic
142D4—21no—
143D4—21no—
144D4—2812no—
145D4—2811no—
146D4—2812no—
147D4—2811no—
148D4—5622no—
149D4—5622no—
150D4—5622no—
151D4—5622no—
152D4—5621no—
153D4—5621no—
154D4C22×D28112yes—
155D4C22×D28111yescharacteristic
156C2×C4—21no—
157C2×C4—2812no—
158C2×C4—5622no—
159C2×C4—5622no—
160C2×C4—5621no—
161C2×C4C22×D28111yescharacteristic
162C8—2811no—
163C8—2811no—
164C8C22×D28112yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—2812no—
169C22—5622no—
170C22—5621no—
171C22—22481no—
172C22—22482no—
173C22—22482no—
174C22D4×D28111yescharacteristic
175C4—21no—
176C4—2812no—
177C4—5621no—
178C4D4×D28111yescharacteristic
179C4C23×D28111yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—5622no—
184C2—11242no—
185C2—11241no—
186C2C2×D4×D28111yescharacteristic
subgroupG/H
centerC2C2×D4×D281
commutator, cosocleC4×C281C24
FrattiniC4C23×D281
FittingAGL(1,ℤ/8)×C281C2
socleC2×C281C22×D4
hypercenterAGL(1,ℤ/8)D281
hypocenterC281#576
LCSG ▷ C4×C281 ▷ C2×C281 ▷ C281
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #22582925C2semidirect
2. #22582924C2semidirect
3. #22582907C2semidirect
4. #22582906C2semidirect
5. #22582903C2semidirect
6. #22582900C2semidirect
7. #22582899C2semidirect
8. C2×#17985123C2semidirect
9. D8×D281C2semidirect
10. #22582887C2semidirect
11. AGL(1,ℤ/8)×C281C2semidirect
12. #17985139C22semidirect
15. #17985138C22semidirect
17. #17985137C22semidirect
19. C2×D1124C22semidirect
23. D4×D281C22semidirect
24. D4×D281C22semidirect
25. #17985128C22semidirect
26. #17985127C22semidirect
27. #17985126C22semidirect
28. #17985125C22semidirect
29. D2248C22semidirect
30. #17985123C22semidirect
31. #17985123C22semidirect
32. #17985123C22semidirect
33. #17985121C22semidirect
34. C2×#14555886C22semidirect
35. C8×D281C22semidirect
36. #41×C281C22semidirect
37. C2×D4×C281C22semidirect
38. SD16×C281C22semidirect
39. D8×C281C22semidirect
40. M16×C281C22semidirect
93. C281#576semidirect
100. AGL(1,ℤ/8)D281semidirect
112. #41C2×D281semidirect
120. C2×D4C2×D281semidirect
125. SD16C2×D281semidirect
130. D8C2×D281semidirect
132. M16C2×D281semidirect

Maximal subgroups

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

subgroupindex
#225829252
#225829242
#225829072
#225829062
#225829032
#225829002
#225828992
C2×#179851232
D8×D2812
#225828872
AGL(1,ℤ/8)×C2812
#576281