Finite Group Databasebeta

← #38762688

Subgroups — #38762688

Order 25408.

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#27163142C211yesmaximal, characteristic, complemented
2#27163122C211yesmaximal, characteristic, complemented
3#27163092C211yesmaximal, characteristic, complemented
4#27163023C211yesmaximal, characteristic, complemented
5#27163022C211yesmaximal, characteristic, complemented
6#27163012C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C397C211yesmaximal, characteristic, complemented
8#20263732—21no—
9#20263731—21no—
10#20263731C2211yescharacteristic
11C2×D1588—21no—
12C2×D1588C2211yescharacteristic
13C2×C22⋊D397—22no—
14D4×D397—22no—
15D4×D397—21no—
16#20263719—21no—
17#20263716—21no—
18#20263715C2211yescharacteristic, complemented
19C4×Dic397C2211yescharacteristic, complemented
20#41×C397C2211yescharacteristic
21C2×D4×C397C2211yescharacteristic, complemented
22SD16×C397—21no—
23D8×C397—21no—
24M16×C397C2211yescharacteristic, complemented
25C22×D397—22no—
26C22×D397D412yes—
27Dic794D411yescharacteristic, complemented
28D1588—22no—
29D1588—21no—
30D1588—21no—
31D1588D411yescharacteristic, complemented
32C22⋊D397—22no—
33C22⋊D397—21no—
34C22⋊D397—44no—
35C2×Dic397—21no—
36C4×D397—22no—
37C4×D397—21no—
38#15691787—21no—
39C23×C397—21no—
40Q8×C397D411yescharacteristic, complemented
41D4×C397—21no—
42D4×C397—21no—
43D4×C397—21no—
44D4×C397D411yescharacteristic, complemented
45C2×C4×C397—21no—
46C2×C4×C397C2311yescharacteristic
47C8×C397—21no—
48C22×C397—21no—
49C22×C397—42no—
50C22×C397—41no—
51C22×C397C2×D411yescharacteristic
52C2×D397—22no—
53C2×D397—21no—
54C2×D397—42no—
55C2×D397—82no—
56C4×C397—21no—
57C4×C397C2×D411yescharacteristic
58C4×C397C2×D411yescharacteristic
59Dic397—22no—
60Dic397—21no—
61C2×C397—21no—
62C2×C397—41no—
63C2×C397—81no—
64C2×C397#12011yescharacteristic
65D397—42no—
66D397—41no—
67C397#45311yescharacteristic, complemented, Sylow
68#453—3971nomaximal, Sylow
69H(5,𝔽2)—3971no—
70AGL(1,ℤ/8)—3971no—
71AGL(1,ℤ/8)D39711yescharacteristic, complemented
72#127—3971no—
73#104—3971no—
74#104—3971no—
75#100—3971no—
76#41—7941no—
77#41—7941no—
78#41—3971no—
79#41C2×D39711yescharacteristic
80C2×D4—7941no—
81C2×D4—7942no—
82C2×D4—7941no—
83C2×D4—7942no—
84C2×D4—3971no—
85C2×D4C2×D39711yescharacteristic, complemented
86SD16—21no—
87SD16—7941no—
88D8—21no—
89D8—7941no—
90M16—3971no—
91M16C2×D39711yescharacteristic, complemented
92C42—3971no—
93C23—21no—
94C23—7942no—
95C23—3972no—
96Q8—3971no—
97Q8C22⋊D39711yescharacteristic, complemented
98D4—21no—
99D4—21no—
100D4—21no—
101D4—7942no—
102D4—7942no—
103D4—7941no—
104D4—7941no—
105D4—7941no—
106D4—15884no—
107D4—3971no—
108D4C22⋊D39711yescharacteristic, complemented
109C2×C4—21no—
110C2×C4—7942no—
111C2×C4—7941no—
112C2×C4—7941no—
113C2×C4C22×D39711yescharacteristic
114C8—21no—
115C8—7941no—
116C22—21no—
117C22—42no—
118C22—41no—
119C22—7942no—
120C22—7941no—
121C22—15882no—
122C22—31762no—
123C22D4×D39711yescharacteristic
124C4—21no—
125C4—7942no—
126C4—7941no—
127C4D4×D39711yescharacteristic
128C4C2×C22⋊D39711yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—15882no—
133C2—15881no—
134C2#2716306411yescharacteristic
subgroupG/H
centerC2#27163064
commutator, cosocleC2×C4×C397C23
FrattiniC2×C4C22×D397
FittingAGL(1,ℤ/8)×C397C2
socleC2×C397#120
hypercenterAGL(1,ℤ/8)D397
hypocenterC397#453
LCSG ▷ C2×C4×C397 ▷ C2×C397 ▷ C397
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #27163142C2semidirect
2. #27163122C2semidirect
3. #27163092C2semidirect
4. #27163023C2semidirect
5. #27163022C2semidirect
6. #27163012C2semidirect
7. AGL(1,ℤ/8)×C397C2semidirect
18. #20263715C22semidirect
19. C4×Dic397C22semidirect
21. C2×D4×C397C22semidirect
24. M16×C397C22semidirect
27. Dic794D4semidirect
31. D1588D4semidirect
40. Q8×C397D4semidirect
44. D4×C397D4semidirect
67. C397#453semidirect
71. AGL(1,ℤ/8)D397semidirect
85. C2×D4C2×D397semidirect
91. M16C2×D397semidirect
97. Q8C22⋊D397semidirect
108. D4C22⋊D397semidirect

Maximal subgroups

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

subgroupindex
#271631422
#271631222
#271630922
#271630232
#271630222
#271630122
AGL(1,ℤ/8)×C3972
#453397