Finite Group Databasebeta

← #44397889

Subgroups — #44397889

Order 29056.

106 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 106 conjugacy classes of subgroups — too many to draw. Only its 23 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#28409214C211yesmaximal, characteristic
2#28408831C211yesmaximal, characteristic, complemented
3#333×C227C211yesmaximal, characteristic, complemented
4#20300336C411yescharacteristic
5#20300319—21no—
6#20300216—21no—
7#20300207—21no—
8#20300207C411yescharacteristic
9C4⋊Q8×C227C2211yescharacteristic
10C4⋊C8×C227—21no—
11#16813885—41no—
12#16813884—21no—
13#16813884—41no—
14Q8×D227—41no—
15#16813874—41no—
16#16813871—41no—
17#16813868—21no—
18#16813868—41no—
19C2×Q8×C227C2×C411yescharacteristic
20C2×Q8×C227D411yescharacteristic
21C2×C8×C227—41no—
22C4⋊C4×C227—41no—
23C42×C227D411yescharacteristic
24Dic454—21no—
25Dic454—41no—
26D908—21no—
27D908—41no—
28C22⋊D227—41no—
29C4×D227—41no—
30C4×D227—42no—
31C227⋊C8—41no—
32C227⋊C8—81no—
33Q8×C227—41no—
34Q8×C227—41no—
35C2×C4×C227—21no—
36C2×C4×C227—21no—
37C2×C4×C227—21no—
38C2×C4×C227C22⋊C411yescharacteristic
39C8×C227—81no—
40C22×C227#9911yescharacteristic
41C2×D227—41no—
42C4×C227—21no—
43C4×C227—21no—
44C4×C227—21no—
45C4×C227—41no—
46C4×C227—41no—
47Dic227—41no—
48C2×C227—21no—
49C2×C227AΣL(1,𝔽16)11yescharacteristic
50D227—81no—
51C227#140511yescharacteristic, complemented, Sylow
52#1405—2271nomaximal, Sylow
53#456—2271no—
54#356—2271no—
55#333D22711yescharacteristic, complemented
562−1+4—2271no—
57#137—4541no—
58C4⋊Q8C2×D22711yescharacteristic
59C4⋊C8—21no—
60#104—4541no—
61#101—4541no—
62#101—2271no—
63#41—9081no—
64#41—9081no—
65#41—4541no—
66C2×Q8—9081no—
67C2×Q8C4×D22711yescharacteristic
68C2×Q8D90811yescharacteristic
69Q16—9081no—
70SD16—9081no—
71M16—9081no—
72M16—4541no—
73C2×C8—41no—
74C4⋊C4—41no—
75C42C22⋊D22711yescharacteristic
76Q8—41no—
77Q8—41no—
78Q8—9081no—
79Q8—4541no—
80D4—9081no—
81D4—9081no—
82D4—4541no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—9081no—
87C2×C4—9082no—
88C2×C4(C2×D227)⋊C411yescharacteristic
89C8—81no—
90C8—18161no—
91C8—9081no—
92C22—9081no—
93C22#2030020211yescharacteristic
94C4—21no—
95C4—21no—
96C4—21no—
97C4—41no—
98C4—41no—
99C4—9081no—
100C2—21no—
101C2—18161no—
102C2#2840882011yescharacteristic
subgroupG/H
centerC2#28408820
commutatorC2×Q8×C227C2×C4
FrattiniC4⋊Q8C2×D227
Fitting#333×C227C2
socleC2×C227AΣL(1,𝔽16)
prime radical#20300336C4
cosocleC4⋊Q8×C227C22
hypercenter#333D227
hypocenterC227#1405
LCSG ▷ C2×Q8×C227 ▷ C2×C4×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #28408831C2semidirect
3. #333×C227C2semidirect
51. C227#1405semidirect
55. #333D227semidirect

Maximal subgroups

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

subgroupindex
#284092142
#284088312
#333×C2272
#1405227