Finite Group Databasebeta

← #38762693

Subgroups — #38762693

Order 25408.

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

normal subgroups only

This group has 146 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#27163143C211yesmaximal, characteristic, complemented
2#27163122C211yesmaximal, characteristic, complemented
3#27163083C211yesmaximal, characteristic, complemented
4#27163023C211yesmaximal, characteristic, complemented
5#27163022C211yesmaximal, characteristic, complemented
6#27163014C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C397C211yesmaximal, characteristic, complemented
8#20263731—21no—
9#20263731C2211yescharacteristic
10#20263730—22no—
11#20263730—21no—
12C2×Dic794—21no—
13C2×Dic794C2211yescharacteristic
14Q8×D397—21no—
15#20263719—21no—
16#20263716—21no—
17#20263715C2211yescharacteristic
18C22⋊Dic397—22no—
19C4×Dic397C2211yescharacteristic
20#41×C397C2211yescharacteristic
21C2×D4×C397C2211yescharacteristic
22SD16×C397—21no—
23D8×C397—21no—
24M16×C397C2211yescharacteristic
25Dic794—22no—
26Dic794—21no—
27Dic794—21no—
28Dic794D411yescharacteristic
29D1588D411yescharacteristic
30C22⋊D397—22no—
31C22⋊D397—21no—
32C2×Dic397—22no—
33C2×Dic397—21no—
34C2×Dic397D412yes—
35C4×D397—22no—
36C4×D397—21no—
37#15691787—21no—
38C23×C397—21no—
39Q8×C397D411yescharacteristic
40D4×C397—21no—
41D4×C397—21no—
42D4×C397—21no—
43D4×C397D411yescharacteristic
44C2×C4×C397—21no—
45C2×C4×C397C2311yescharacteristic
46C8×C397—21no—
47C22×C397—21no—
48C22×C397—42no—
49C22×C397—41no—
50C22×C397C2×D411yescharacteristic
51C2×D397—21no—
52C4×C397—21no—
53C4×C397C2×D411yescharacteristic
54C4×C397C2×D411yescharacteristic
55Dic397—21no—
56Dic397—22no—
57Dic397—41no—
58C2×C397—21no—
59C2×C397—41no—
60C2×C397—81no—
61C2×C397#12011yescharacteristic
62D397—41no—
63C397#45411yescharacteristic, complemented, Sylow
64#454—3971nomaximal, Sylow
652−1+4—3971no—
66AGL(1,ℤ/8)—3971no—
67AGL(1,ℤ/8)D39711yescharacteristic, complemented
68#124—3971no—
69#104—3971no—
70#104—3971no—
71#101—3971no—
72#41—7941no—
73#41—7941no—
74#41—7942no—
75#41—3971no—
76#41C2×D39711yescharacteristic
77C2×Q8—7941no—
78C2×Q8—7941no—
79C2×Q8—3971no—
80C2×D4C2×D39711yescharacteristic
81SD16—21no—
82SD16—7941no—
83D8—21no—
84D8—7941no—
85M16—3971no—
86M16C2×D39711yescharacteristic
87C22⋊C4—7942no—
88C42—3971no—
89C23—21no—
90Q8—7942no—
91Q8—7941no—
92Q8—7941no—
93Q8—3971no—
94Q8C22⋊D39711yescharacteristic
95D4—21no—
96D4—21no—
97D4—21no—
98D4—7942no—
99D4—7941no—
100D4—3971no—
101D4C22⋊D39711yescharacteristic
102C2×C4—21no—
103C2×C4—7942no—
104C2×C4—7942no—
105C2×C4—7941no—
106C2×C4—7941no—
107C2×C4—3972no—
108C2×C4C22×D39711yescharacteristic
109C8—21no—
110C8—7941no—
111C22—21no—
112C22—42no—
113C22—41no—
114C22—7941no—
115C22D4×D39711yescharacteristic
116C4—21no—
117C4—7941no—
118C4—7942no—
119C4—15881no—
120C4D4×D39711yescharacteristic
121C4C2×C22⋊D39711yescharacteristic
122C2—21no—
123C2—41no—
124C2—81no—
125C2—15881no—
126C2#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#454
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. #27163143C2semidirect
2. #27163122C2semidirect
3. #27163083C2semidirect
4. #27163023C2semidirect
5. #27163022C2semidirect
6. #27163014C2semidirect
7. AGL(1,ℤ/8)×C397C2semidirect
63. C397#454semidirect
67. AGL(1,ℤ/8)D397semidirect

Maximal subgroups

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

subgroupindex
#271631432
#271631222
#271630832
#271630232
#271630222
#271630142
AGL(1,ℤ/8)×C3972
#454397