Finite Group Databasebeta

← C2×#27133868 #37589139

Subgroups — C2×#27133868 #37589139

Order 24512.

298 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 103 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×#20259458C211yesmaximal, characteristic, complemented
2C2×D4×D383C211yesmaximal, characteristic, complemented
3#27133868C218yesmaximal, direct product factor
4C2×#20259444C211yesmaximal, characteristic, complemented
5C2×D3064C211yesmaximal, characteristic, complemented
6C2×#20259442C211yesmaximal, characteristic, complemented
7C2×#20259440C211yesmaximal, characteristic, complemented
8C2×SD16×C383C211yesmaximal, characteristic, complemented
9C23×D383—21no—
10#20259458—22no—
11#20259458C2214yescomplemented
12C2×D1532—21no—
13C2×D1532C2211yescharacteristic
14C2×D1532C2211yescharacteristic, complemented
15C2×C22⋊D383—21no—
16D4×D383—22no—
17D4×D383C2214yescomplemented
18C2×C4×D383—21no—
19C2×C4×D383C2211yescharacteristic
20#20259444C2214yescomplemented
21D3064C2214yescomplemented
22#20259442C2214yescomplemented
23#20259440C2214yescomplemented
24C2×C383⋊C8C2211yescharacteristic, complemented
25C2×Q8×C383C2211yescharacteristic, complemented
26C2×D4×C383C2211yescharacteristic
27SD16×C383C2214yescomplemented
28C2×C8×C383C2211yescharacteristic, complemented
29C22×D383—21no—
30C22×D383—21no—
31C22×D383—24no—
32C22×D383—44no—
33C22×D383D411yescharacteristic
34D1532—24no—
35D1532—21no—
36D1532—21no—
37D1532C2312yes—
38D1532C2312yescomplemented
39C22⋊D383—24no—
40C2×Dic383D411yescharacteristic
41C4×D383—24no—
42C4×D383D412yes—
43C4×D383C2312yes—
44C383⋊C8C2312yescomplemented
45C23×C383—21no—
46Q8×C383—21no—
47Q8×C383C2312yescomplemented
48D4×C383—21no—
49D4×C383C2312yes—
50C2×C4×C383—21no—
51C2×C4×C383C2311yescharacteristic
52C8×C383C2312yescomplemented
53C22×C383—22no—
54C22×C383—42no—
55C22×C383C2×D411yescharacteristic
56C2×D383—22no—
57C2×D383—22no—
58C2×D383—22no—
59C2×D383—48no—
60C2×D383—42no—
61C2×D383—42no—
62C2×D383C2×D412yes—
63C4×C383—22no—
64C4×C383C2×D411yescharacteristic
65C4×C383C2411yescharacteristic
66Dic383C2×D412yes—
67C2×C383—42no—
68C2×C383AGL(1,ℤ/8)12yescomplemented
69C2×C383C22×D411yescharacteristic
70D383—22no—
71D383—42no—
72D383—42no—
73C383C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
74C2×AGL(1,ℤ/8)—3831nomaximal, Sylow
75C2×#41—3831no—
76C22×D4—3831no—
77AGL(1,ℤ/8)—3838no—
78C2×SD16—3831no—
79C2×SD16D38311yescharacteristic, complemented
80C2×D8—3831no—
81C2×D8—3831no—
82C2×M16—3831no—
83C24—7661no—
84#41—3834no—
85#41—7662no—
86C2×Q8C2×D38311yescharacteristic, complemented
87C2×D4—3834no—
88C2×D4—3831no—
89C2×D4—3831no—
90C2×D4—7661no—
91C2×D4—7662no—
92C2×D4—7661no—
93C2×D4C2×D38311yescharacteristic
94C22×C4—3831no—
95C22×C4—7661no—
96SD16—3834no—
97SD16C2×D38314yescomplemented
98D8—3834no—
99D8—3834no—
100M16—3834no—
101C2×C8—3831no—
102C2×C8C2×D38311yescharacteristic, complemented
103C23—21no—
104C23—3831no—
105C23—15324no—
106C23—7664no—
107C23—7661no—
108C23—7661no—
109Q8—21no—
110Q8C22×D38312yescomplemented
111D4—21no—
112D4—3832no—
113D4—3832no—
114D4—7664no—
115D4—7664no—
116D4—7661no—
117D4—7661no—
118D4C22×D38312yes—
119C2×C4—21no—
120C2×C4—3831no—
121C2×C4—3832no—
122C2×C4—3832no—
123C2×C4—7664no—
124C2×C4C22×D38311yescharacteristic
125C8—3832no—
126C8C22×D38312yescomplemented
127C22—22no—
128C22—42no—
129C22—3832no—
130C22—15328no—
131C22—15322no—
132C22—15322no—
133C22—7662no—
134C22—7662no—
135C22—7662no—
136C22D4×D38311yescharacteristic
137C4—22no—
138C4—3832no—
139C4D4×D38311yescharacteristic
140C4C23×D38311yescharacteristic
141C2—42no—
142C2—15322no—
143C2—15322no—
144C2—7662no—
145C2#2713386812yesdirect product factor
146C2C2×D4×D38311yescharacteristic
subgroupG/H
centerC22D4×D383
commutator, cosocleC4×C383C24
FrattiniC4C23×D383
FittingC2×SD16×C383C2
socleC22×C383C2×D4
hypercenterC2×SD16D383
hypocenterC383C2×AGL(1,ℤ/8)
LCSG ▷ C4×C383 ▷ C2×C383 ▷ C383
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#20259458C2semidirect
2. C2×D4×D383C2semidirect
3. #27133868C2direct (N × H)
4. C2×#20259444C2semidirect
5. C2×D3064C2semidirect
6. C2×#20259442C2semidirect
7. C2×#20259440C2semidirect
8. C2×SD16×C383C2semidirect
11. #20259458C22semidirect
14. C2×D1532C22semidirect
17. D4×D383C22semidirect
20. #20259444C22semidirect
21. D3064C22semidirect
22. #20259442C22semidirect
23. #20259440C22semidirect
24. C2×C383⋊C8C22semidirect
25. C2×Q8×C383C22semidirect
27. SD16×C383C22semidirect
28. C2×C8×C383C22semidirect
38. D1532C23semidirect
44. C383⋊C8C23semidirect
47. Q8×C383C23semidirect
52. C8×C383C23semidirect
68. C2×C383AGL(1,ℤ/8)semidirect
73. C383C2×AGL(1,ℤ/8)semidirect
79. C2×SD16D383semidirect
86. C2×Q8C2×D383semidirect
97. SD16C2×D383semidirect
102. C2×C8C2×D383semidirect
110. Q8C22×D383semidirect
126. C8C22×D383semidirect
145. C2#27133868direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#202594582
C2×D4×D3832
#271338682
C2×#202594442
C2×D30642
C2×#202594422
C2×#202594402
C2×SD16×C3832
C2×AGL(1,ℤ/8)383