Finite Group Databasebeta

← #39909413

Subgroups — #39909413

Order 26176.

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#27189763C211yesmaximal, characteristic, complemented
2#27189743C211yesmaximal, characteristic, complemented
3#27189713C211yesmaximal, characteristic, complemented
4#27189644C211yesmaximal, characteristic, complemented
5#27189643C211yesmaximal, characteristic, complemented
6#27189633C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C409C211yesmaximal, characteristic, complemented
8#20266989—21no—
9#20266988—21no—
10#20266988C2211yescharacteristic
11C2×D1636—21no—
12C2×D1636C2211yescharacteristic
13C2×C22⋊D409—22no—
14D4×D409—21no—
15D4×D409—22no—
16#20266976—21no—
17#20266973—21no—
18#20266972C2211yescharacteristic, complemented
19C4×Dic409C2211yescharacteristic, complemented
20#41×C409C2211yescharacteristic
21C2×D4×C409C2211yescharacteristic, complemented
22SD16×C409—21no—
23D8×C409—21no—
24M16×C409C2211yescharacteristic, complemented
25C22×D409—22no—
26C22×D409D412yes—
27Dic818D411yescharacteristic, complemented
28D1636—22no—
29D1636—21no—
30D1636—21no—
31D1636D411yescharacteristic, complemented
32C22⋊D409—22no—
33C22⋊D409—21no—
34C22⋊D409—44no—
35C2×Dic409—21no—
36C4×D409—22no—
37C4×D409—21no—
38#15692543—21no—
39C23×C409—21no—
40Q8×C409D411yescharacteristic, complemented
41D4×C409—21no—
42D4×C409—21no—
43D4×C409—21no—
44D4×C409D411yescharacteristic, complemented
45C2×C4×C409—21no—
46C2×C4×C409C2311yescharacteristic
47C8×C409—21no—
48C22×C409—21no—
49C22×C409—42no—
50C22×C409—41no—
51C22×C409C2×D411yescharacteristic
52C2×D409—22no—
53C2×D409—21no—
54C2×D409—42no—
55C2×D409—82no—
56C4×C409—21no—
57C4×C409C2×D411yescharacteristic
58C4×C409C2×D411yescharacteristic
59Dic409—22no—
60Dic409—21no—
61C2×C409—21no—
62C2×C409—41no—
63C2×C409—81no—
64C2×C409#12011yescharacteristic
65D409—42no—
66D409—41no—
67C409#45311yescharacteristic, complemented, Sylow
68#453—4091nomaximal, Sylow
69H(5,𝔽2)—4091no—
70AGL(1,ℤ/8)—4091no—
71AGL(1,ℤ/8)D40911yescharacteristic, complemented
72#127—4091no—
73#104—4091no—
74#104—4091no—
75#100—4091no—
76#41—8181no—
77#41—8181no—
78#41—4091no—
79#41C2×D40911yescharacteristic
80C2×D4—8181no—
81C2×D4—8182no—
82C2×D4—8181no—
83C2×D4—8182no—
84C2×D4—4091no—
85C2×D4C2×D40911yescharacteristic, complemented
86SD16—21no—
87SD16—8181no—
88D8—21no—
89D8—8181no—
90M16—4091no—
91M16C2×D40911yescharacteristic, complemented
92C42—4091no—
93C23—21no—
94C23—8182no—
95C23—4092no—
96Q8—4091no—
97Q8C22⋊D40911yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—8182no—
102D4—8182no—
103D4—8181no—
104D4—8181no—
105D4—8181no—
106D4—16364no—
107D4—4091no—
108D4C22⋊D40911yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—8182no—
111C2×C4—8181no—
112C2×C4—8181no—
113C2×C4C22×D40911yescharacteristic
114C8—21no—
115C8—8181no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—8182no—
120C22—8181no—
121C22—16362no—
122C22—32722no—
123C22D4×D40911yescharacteristic
124C4—21no—
125C4—8182no—
126C4—8181no—
127C4D4×D40911yescharacteristic
128C4C2×C22⋊D40911yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—16362no—
133C2—16361no—
134C2#2718968511yescharacteristic
subgroupG/H
centerC2#27189685
commutator, cosocleC2×C4×C409C23
FrattiniC2×C4C22×D409
FittingAGL(1,ℤ/8)×C409C2
socleC2×C409#120
hypercenterAGL(1,ℤ/8)D409
hypocenterC409#453
LCSG ▷ C2×C4×C409 ▷ C2×C409 ▷ C409
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #27189763C2semidirect
2. #27189743C2semidirect
3. #27189713C2semidirect
4. #27189644C2semidirect
5. #27189643C2semidirect
6. #27189633C2semidirect
7. AGL(1,ℤ/8)×C409C2semidirect
18. #20266972C22semidirect
19. C4×Dic409C22semidirect
21. C2×D4×C409C22semidirect
24. M16×C409C22semidirect
27. Dic818D4semidirect
31. D1636D4semidirect
40. Q8×C409D4semidirect
44. D4×C409D4semidirect
67. C409#453semidirect
71. AGL(1,ℤ/8)D409semidirect
85. C2×D4C2×D409semidirect
91. M16C2×D409semidirect
97. Q8C22⋊D409semidirect
108. D4C22⋊D409semidirect

Maximal subgroups

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

subgroupindex
#271897632
#271897432
#271897132
#271896442
#271896432
#271896332
AGL(1,ℤ/8)×C4092
#453409