Finite Group Databasebeta

← #27213312

Subgroups — #27213312

Order 13504.

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#20270023C211yesmaximal, characteristic, complemented
2#20270003C211yesmaximal, characteristic, complemented
3#20269973C211yesmaximal, characteristic, complemented
4#20269904C211yesmaximal, characteristic, complemented
5#20269903C211yesmaximal, characteristic, complemented
6#20269893C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C211C211yesmaximal, characteristic, complemented
8#16809963—21no—
9#16809962—21no—
10#16809962C2211yescharacteristic
11C2×D844—21no—
12C2×D844C2211yescharacteristic
13C2×C22⋊D211—22no—
14D4×D211—22no—
15D4×D211—21no—
16#16809950—21no—
17#16809947—21no—
18#16809946C2211yescharacteristic, complemented
19C4×Dic211C2211yescharacteristic, complemented
20#41×C211C2211yescharacteristic
21C2×D4×C211C2211yescharacteristic, complemented
22SD16×C211—21no—
23D8×C211—21no—
24M16×C211C2211yescharacteristic, complemented
25C22×D211—22no—
26C22×D211D412yes—
27Dic422D411yescharacteristic, complemented
28D844—22no—
29D844—21no—
30D844—21no—
31D844D411yescharacteristic, complemented
32C22⋊D211—22no—
33C22⋊D211—21no—
34C22⋊D211—44no—
35C2×Dic211—21no—
36C4×D211—22no—
37C4×D211—21no—
38C211⋊C8—21no—
39C23×C211—21no—
40Q8×C211D411yescharacteristic, complemented
41D4×C211—21no—
42D4×C211—21no—
43D4×C211—21no—
44D4×C211D411yescharacteristic, complemented
45C2×C4×C211—21no—
46C2×C4×C211C2311yescharacteristic
47C8×C211—21no—
48C22×C211—21no—
49C22×C211—42no—
50C22×C211—41no—
51C22×C211C2×D411yescharacteristic
52C2×D211—22no—
53C2×D211—21no—
54C2×D211—42no—
55C2×D211—82no—
56C4×C211—21no—
57C4×C211C2×D411yescharacteristic
58C4×C211C2×D411yescharacteristic
59Dic211—22no—
60Dic211—21no—
61C2×C211—21no—
62C2×C211—41no—
63C2×C211—81no—
64C2×C211#12011yescharacteristic
65D211—42no—
66D211—41no—
67C211#45311yescharacteristic, complemented, Sylow
68#453—2111nomaximal, Sylow
69H(5,𝔽2)—2111no—
70AGL(1,ℤ/8)—2111no—
71AGL(1,ℤ/8)D21111yescharacteristic, complemented
72#127—2111no—
73#104—2111no—
74#104—2111no—
75#100—2111no—
76#41—4221no—
77#41—4221no—
78#41—2111no—
79#41C2×D21111yescharacteristic
80C2×D4—4221no—
81C2×D4—4222no—
82C2×D4—4222no—
83C2×D4—4221no—
84C2×D4—2111no—
85C2×D4C2×D21111yescharacteristic, complemented
86SD16—21no—
87SD16—4221no—
88D8—21no—
89D8—4221no—
90M16—2111no—
91M16C2×D21111yescharacteristic, complemented
92C42—2111no—
93C23—21no—
94C23—4222no—
95C23—2112no—
96Q8—2111no—
97Q8C22⋊D21111yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—8444no—
102D4—4222no—
103D4—4222no—
104D4—4221no—
105D4—4221no—
106D4—4221no—
107D4—2111no—
108D4C22⋊D21111yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—4222no—
111C2×C4—4221no—
112C2×C4—4221no—
113C2×C4C22×D21111yescharacteristic
114C8—21no—
115C8—4221no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—8442no—
120C22—16882no—
121C22—4222no—
122C22—4221no—
123C22D4×D21111yescharacteristic
124C4—21no—
125C4—4222no—
126C4—4221no—
127C4D4×D21111yescharacteristic
128C4C2×C22⋊D21111yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—8442no—
133C2—8441no—
134C2#2026994511yescharacteristic
subgroupG/H
centerC2#20269945
commutator, cosocleC2×C4×C211C23
FrattiniC2×C4C22×D211
FittingAGL(1,ℤ/8)×C211C2
socleC2×C211#120
hypercenterAGL(1,ℤ/8)D211
hypocenterC211#453
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #20270023C2semidirect
2. #20270003C2semidirect
3. #20269973C2semidirect
4. #20269904C2semidirect
5. #20269903C2semidirect
6. #20269893C2semidirect
7. AGL(1,ℤ/8)×C211C2semidirect
18. #16809946C22semidirect
19. C4×Dic211C22semidirect
21. C2×D4×C211C22semidirect
24. M16×C211C22semidirect
27. Dic422D4semidirect
31. D844D4semidirect
40. Q8×C211D4semidirect
44. D4×C211D4semidirect
67. C211#453semidirect
71. AGL(1,ℤ/8)D211semidirect
85. C2×D4C2×D211semidirect
91. M16C2×D211semidirect
97. Q8C22⋊D211semidirect
108. D4C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
#202700232
#202700032
#202699732
#202699042
#202699032
#202698932
AGL(1,ℤ/8)×C2112
#453211