Finite Group Databasebeta

← #37589161

Subgroups — #37589161

Order 24512.

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#27133888C211yesmaximal, characteristic, complemented
2#27133886C211yesmaximal, characteristic, complemented
3#27133874C211yesmaximal, characteristic, complemented
4#27133870C212yesmaximal, complemented
5#27133865C212yesmaximal, complemented
6#27133861C211yesmaximal, characteristic, complemented
7#27133860C212yesmaximal, complemented
8C2×#20259445C211yesmaximal, characteristic, complemented
9SD16×D383C212yesmaximal, complemented
10#27133849C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C383C211yesmaximal, characteristic, complemented
12#20259457—21no—
13#20259457C2211yescharacteristic
14#20259457C2211yescharacteristic
15#20259456—21no—
16#20259456—22no—
17#20259456C2212yes—
18#20259456C2212yescomplemented
19C2×Dic766—21no—
20C2×Dic766C2211yescharacteristic, complemented
21Q8×D383C2212yescomplemented
22C2×C22⋊D383—22no—
23D4×D383—21no—
24D4×D383C2212yes—
25Dic1532C2212yescomplemented
26#20259447C2211yescharacteristic, complemented
27#20259446C2212yescomplemented
28#20259445C2212yescomplemented
29#20259445C2212yescomplemented
30#20259445C2211yescharacteristic, complemented
31#20259444C2211yescharacteristic, complemented
32#20259442C2211yescharacteristic, complemented
33#20259440C2212yescomplemented
34C2×C383⋊C8C2211yescharacteristic, complemented
35C8×D383C2212yescomplemented
36#41×C383C2211yescharacteristic, complemented
37C2×D4×C383C2211yescharacteristic
38SD16×C383C2212yescomplemented
39D8×C383C2212yescomplemented
40M16×C383C2211yescharacteristic, complemented
41C22×D383—22no—
42Dic766—21no—
43Dic766—22no—
44Dic766D411yescharacteristic
45Dic766C2312yes—
46Dic766C2311yescharacteristic
47D1532D411yescharacteristic
48D1532C2311yescharacteristic
49C22⋊D383—21no—
50C22⋊D383—22no—
51C22⋊D383—22no—
52C22⋊D383—22no—
53C22⋊D383D412yes—
54C2×Dic383—21no—
55C2×Dic383—22no—
56C2×Dic383—22no—
57C4×D383—22no—
58C4×D383—21no—
59C4×D383C2312yes—
60C383⋊C8C2311yescharacteristic
61C383⋊C8C2311yescharacteristic
62C23×C383—21no—
63Q8×C383C2311yescharacteristic
64D4×C383—21no—
65D4×C383—21no—
66D4×C383C2311yescharacteristic
67D4×C383C2312yes—
68C2×C4×C383—21no—
69C2×C4×C383C2311yescharacteristic
70C8×C383C2312yes—
71C22×C383—21no—
72C22×C383—22no—
73C22×C383—81no—
74C22×C383C2×D411yescharacteristic
75C2×D383—21no—
76C2×D383—82no—
77C2×D383C2×D412yes—
78C4×C383—21no—
79C4×C383C2×D411yescharacteristic
80C4×C383C2411yescharacteristic
81Dic383—21no—
82Dic383—22no—
83Dic383C2×D412yes—
84C2×C383—21no—
85C2×C383—42no—
86C2×C383—41no—
87C2×C383C22×D411yescharacteristic
88D383—22no—
89D383—41no—
90C383#57711yescharacteristic, complemented, Sylow
91#577—3831nomaximal, Sylow
922−1+4—3831no—
93H(5,𝔽2)—3831no—
94#137—3832no—
95#137—3831no—
96AGL(1,ℤ/8)—3832no—
97AGL(1,ℤ/8)D38311yescharacteristic, complemented
98#135—3831no—
99#135—3832no—
100C2×SD16—3831no—
101C2×SD16—3832no—
102#131—3831no—
103#41—3831no—
104#41—3832no—
105#41—3832no—
106#41—3831no—
107#41—7661no—
108#41—7662no—
109#41—7661no—
110#41C2×D38311yescharacteristic, complemented
111C2×Q8—3832no—
112C2×Q8—3831no—
113C2×Q8—7661no—
114C2×D4—3832no—
115C2×D4—7661no—
116C2×D4—7662no—
117C2×D4C2×D38311yescharacteristic
118Q16—3831no—
119Q16—3832no—
120SD16—3831no—
121SD16—3832no—
122SD16—3832no—
123SD16—3831no—
124SD16—3832no—
125SD16C2×D38312yescomplemented
126D8—3831no—
127D8C2×D38312yescomplemented
128M16—3832no—
129M16C2×D38311yescharacteristic, complemented
130C2×C8—3831no—
131C2×C8—3832no—
132C23—21no—
133C23—7662no—
134Q8—3831no—
135Q8—3832no—
136Q8—3831no—
137Q8—7662no—
138Q8—7661no—
139Q8C22×D38311yescharacteristic
140D4—21no—
141D4—21no—
142D4—3832no—
143D4—3831no—
144D4—3831no—
145D4—7662no—
146D4—7662no—
147D4—7662no—
148D4—7661no—
149D4C22×D38311yescharacteristic
150D4C22×D38312yes—
151C2×C4—21no—
152C2×C4—3832no—
153C2×C4—7662no—
154C2×C4—7662no—
155C2×C4—7662no—
156C2×C4—7661no—
157C2×C4—7661no—
158C2×C4C22×D38311yescharacteristic
159C8—3831no—
160C8—3831no—
161C8C22×D38312yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—3832no—
166C22—30642no—
167C22—7661no—
168C22D4×D38311yescharacteristic
169C4—21no—
170C4—3832no—
171C4—7661no—
172C4—7662no—
173C4D4×D38311yescharacteristic
174C4C23×D38311yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—15321no—
179C2—7662no—
180C2C2×D4×D38311yescharacteristic
subgroupG/H
centerC2C2×D4×D383
commutator, cosocleC4×C383C24
FrattiniC4C23×D383
FittingAGL(1,ℤ/8)×C383C2
socleC2×C383C22×D4
hypercenterAGL(1,ℤ/8)D383
hypocenterC383#577
LCSG ▷ C4×C383 ▷ C2×C383 ▷ C383
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #27133888C2semidirect
2. #27133886C2semidirect
3. #27133874C2semidirect
4. #27133870C2semidirect
5. #27133865C2semidirect
6. #27133861C2semidirect
7. #27133860C2semidirect
8. C2×#20259445C2semidirect
9. SD16×D383C2semidirect
10. #27133849C2semidirect
11. AGL(1,ℤ/8)×C383C2semidirect
18. #20259456C22semidirect
20. C2×Dic766C22semidirect
21. Q8×D383C22semidirect
25. Dic1532C22semidirect
26. #20259447C22semidirect
27. #20259446C22semidirect
28. #20259445C22semidirect
29. #20259445C22semidirect
30. #20259445C22semidirect
31. #20259444C22semidirect
32. #20259442C22semidirect
33. #20259440C22semidirect
34. C2×C383⋊C8C22semidirect
35. C8×D383C22semidirect
36. #41×C383C22semidirect
38. SD16×C383C22semidirect
39. D8×C383C22semidirect
40. M16×C383C22semidirect
90. C383#577semidirect
97. AGL(1,ℤ/8)D383semidirect
110. #41C2×D383semidirect
125. SD16C2×D383semidirect
127. D8C2×D383semidirect
129. M16C2×D383semidirect

Maximal subgroups

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

subgroupindex
#271338882
#271338862
#271338742
#271338702
#271338652
#271338612
#271338602
C2×#202594452
SD16×D3832
#271338492
AGL(1,ℤ/8)×C3832
#577383