Finite Group Databasebeta

← #27169287

Subgroups — #27169287

Order 12928.

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#20265359C211yesmaximal, characteristic
2#20264976C211yesmaximal, characteristic, complemented
3#333×C101C211yesmaximal, characteristic, complemented
4#15692303C411yescharacteristic
5#15692286—21no—
6#15692183—21no—
7#15692174—21no—
8#15692174C411yescharacteristic
9C4⋊Q8×C101C2211yescharacteristic
10C4⋊C8×C101—21no—
11#13108075—41no—
12#13108074—21no—
13#13108074—41no—
14Q8×D101—41no—
15#13108064—41no—
16#13108061—41no—
17#13108058—21no—
18#13108058—41no—
19C2×Q8×C101C2×C411yescharacteristic
20C2×Q8×C101D411yescharacteristic
21C2×C8×C101—41no—
22C4⋊C4×C101—41no—
23C42×C101D411yescharacteristic
24Q8×C101—41no—
25Q8×C101—41no—
26C2×C4×C101—21no—
27C2×C4×C101—21no—
28C2×C4×C101—21no—
29C2×C4×C101C22⋊C411yescharacteristic
30C22⋊D101—41no—
31D404—21no—
32D404—41no—
33C4×D101—41no—
34C4×D101—42no—
35Dic202—21no—
36Dic202—41no—
37C8×C101—81no—
38#11717579—41no—
39#11717579—81no—
40C22×C101#9911yescharacteristic
41C2×D101—41no—
42C4×C101—21no—
43C4×C101—21no—
44C4×C101—21no—
45C4×C101—41no—
46C4×C101—41no—
47Dic101—41no—
48C2×C101—21no—
49C2×C101AΣL(1,𝔽16)11yescharacteristic
50D101—81no—
51#1405—1011nomaximal, Sylow
52C101#140511yescharacteristic, complemented, Sylow
53#456—1011no—
54#356—1011no—
55#333D10111yescharacteristic, complemented
562−1+4—1011no—
57#137—2021no—
58C4⋊Q8C2×D10111yescharacteristic
59C4⋊C8—21no—
60#104—2021no—
61#101—1011no—
62#101—2021no—
63#41—4041no—
64#41—4041no—
65#41—2021no—
66C2×Q8—4041no—
67C2×Q8C4×D10111yescharacteristic
68C2×Q8D40411yescharacteristic
69Q16—4041no—
70SD16—4041no—
71M16—4041no—
72M16—2021no—
73C2×C8—41no—
74C4⋊C4—41no—
75C42C22⋊D10111yescharacteristic
76Q8—41no—
77Q8—41no—
78Q8—4041no—
79Q8—2021no—
80D4—4041no—
81D4—4041no—
82D4—2021no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—4041no—
87C2×C4—4042no—
88C2×C4#1310805111yescharacteristic
89C8—81no—
90C8—8081no—
91C8—4041no—
92C22—4041no—
93C22#1569216911yescharacteristic
94C4—21no—
95C4—21no—
96C4—21no—
97C4—41no—
98C4—41no—
99C4—4041no—
100C2—21no—
101C2—8081no—
102C2#2026496511yescharacteristic
subgroupG/H
centerC2#20264965
commutatorC2×Q8×C101C2×C4
FrattiniC4⋊Q8C2×D101
Fitting#333×C101C2
socleC2×C101AΣL(1,𝔽16)
prime radical#15692303C4
cosocleC4⋊Q8×C101C22
hypercenter#333D101
hypocenterC101#1405
LCSG ▷ C2×Q8×C101 ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #20264976C2semidirect
3. #333×C101C2semidirect
52. C101#1405semidirect
55. #333D101semidirect

Maximal subgroups

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

subgroupindex
#202653592
#202649762
#333×C1012
#1405101