Finite Group Databasebeta

← #37588517

Subgroups — #37588517

Order 24512.

168 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 39 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#27133887C211yesmaximal, characteristic, complemented
2#27133867C211yesmaximal, characteristic, complemented
3#27133837C211yesmaximal, characteristic, complemented
4#27133768C211yesmaximal, characteristic, complemented
5#27133767C211yesmaximal, characteristic, complemented
6#27133757C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C383C211yesmaximal, characteristic, complemented
8#20259458—21no—
9#20259457—21no—
10#20259457C2211yescharacteristic
11C2×D1532—21no—
12C2×D1532C2211yescharacteristic
13C2×C22⋊D383—22no—
14D4×D383—22no—
15D4×D383—21no—
16#20259445—21no—
17#20259442—21no—
18#20259441C2211yescharacteristic, complemented
19C4×Dic383C2211yescharacteristic, complemented
20#41×C383C2211yescharacteristic
21C2×D4×C383C2211yescharacteristic, complemented
22SD16×C383—21no—
23D8×C383—21no—
24M16×C383C2211yescharacteristic, complemented
25C22×D383—22no—
26C22×D383D412yes—
27Dic766D411yescharacteristic, complemented
28D1532—22no—
29D1532—21no—
30D1532—21no—
31D1532D411yescharacteristic, complemented
32C22⋊D383—22no—
33C22⋊D383—21no—
34C22⋊D383—44no—
35C2×Dic383—21no—
36C4×D383—22no—
37C4×D383—21no—
38C383⋊C8—21no—
39C23×C383—21no—
40Q8×C383D411yescharacteristic, complemented
41D4×C383—21no—
42D4×C383—21no—
43D4×C383—21no—
44D4×C383D411yescharacteristic, complemented
45C2×C4×C383—21no—
46C2×C4×C383C2311yescharacteristic
47C8×C383—21no—
48C22×C383—21no—
49C22×C383—42no—
50C22×C383—41no—
51C22×C383C2×D411yescharacteristic
52C2×D383—22no—
53C2×D383—21no—
54C2×D383—42no—
55C2×D383—82no—
56C4×C383—21no—
57C4×C383C2×D411yescharacteristic
58C4×C383C2×D411yescharacteristic
59Dic383—22no—
60Dic383—21no—
61C2×C383—21no—
62C2×C383—41no—
63C2×C383—81no—
64C2×C383#12011yescharacteristic
65D383—42no—
66D383—41no—
67C383#45311yescharacteristic, complemented, Sylow
68#453—3831nomaximal, Sylow
69H(5,𝔽2)—3831no—
70AGL(1,ℤ/8)—3831no—
71AGL(1,ℤ/8)D38311yescharacteristic, complemented
72#127—3831no—
73#104—3831no—
74#104—3831no—
75#100—3831no—
76#41—3831no—
77#41—7661no—
78#41—7661no—
79#41C2×D38311yescharacteristic
80C2×D4—3831no—
81C2×D4—7661no—
82C2×D4—7662no—
83C2×D4—7662no—
84C2×D4—7661no—
85C2×D4C2×D38311yescharacteristic, complemented
86SD16—21no—
87SD16—7661no—
88D8—21no—
89D8—7661no—
90M16—3831no—
91M16C2×D38311yescharacteristic, complemented
92C42—3831no—
93C23—21no—
94C23—3832no—
95C23—7662no—
96Q8—3831no—
97Q8C22⋊D38311yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—3831no—
102D4—15324no—
103D4—7662no—
104D4—7662no—
105D4—7661no—
106D4—7661no—
107D4—7661no—
108D4C22⋊D38311yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—7662no—
111C2×C4—7661no—
112C2×C4—7661no—
113C2×C4C22×D38311yescharacteristic
114C8—21no—
115C8—7661no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—30642no—
120C22—15322no—
121C22—7661no—
122C22—7662no—
123C22D4×D38311yescharacteristic
124C4—21no—
125C4—7662no—
126C4—7661no—
127C4D4×D38311yescharacteristic
128C4C2×C22⋊D38311yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—15321no—
133C2—15322no—
134C2#2713380911yescharacteristic
subgroupG/H
centerC2#27133809
commutator, cosocleC2×C4×C383C23
FrattiniC2×C4C22×D383
FittingAGL(1,ℤ/8)×C383C2
socleC2×C383#120
hypercenterAGL(1,ℤ/8)D383
hypocenterC383#453
LCSG ▷ C2×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. #27133887C2semidirect
2. #27133867C2semidirect
3. #27133837C2semidirect
4. #27133768C2semidirect
5. #27133767C2semidirect
6. #27133757C2semidirect
7. AGL(1,ℤ/8)×C383C2semidirect
18. #20259441C22semidirect
19. C4×Dic383C22semidirect
21. C2×D4×C383C22semidirect
24. M16×C383C22semidirect
27. Dic766D4semidirect
31. D1532D4semidirect
40. Q8×C383D4semidirect
44. D4×C383D4semidirect
67. C383#453semidirect
71. AGL(1,ℤ/8)D383semidirect
85. C2×D4C2×D383semidirect
91. M16C2×D383semidirect
97. Q8C22⋊D383semidirect
108. D4C22⋊D383semidirect

Maximal subgroups

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

subgroupindex
#271338872
#271338672
#271338372
#271337682
#271337672
#271337572
AGL(1,ℤ/8)×C3832
#453383