Finite Group Databasebeta

← #27170215

Subgroups — #27170215

Order 12928.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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#20265551C211yesmaximal, characteristic, complemented
2#20265372C212yesmaximal, complemented
3#20264926C211yesmaximal, characteristic, complemented
4D4⋊#11717579C212yesmaximal, complemented
5#555×C101C211yesmaximal, characteristic, complemented
6C2×#13108061—22no—
7C2×#13108059—22no—
8#15692252C2211yescharacteristic
9C4⋊#11717579C2212yes—
10#15692175—22no—
11C4×#11717579C2211yescharacteristic
12C4⋊Q8×C101C2211yescharacteristic
13#124×C101—22no—
14#121×C101—22no—
15C4×D4×C101C2212yes—
16#117×C101—22no—
17C2×D404—22no—
18C2×D404—21no—
19#13108061—44no—
20#13108059—44no—
21C2×#11717579—21no—
22C2×#11717579—22no—
23C2×Q8×C101D412yes—
24C2×D4×C101—22no—
25C2×D4×C101D412yes—
26C22×C4×C101—22no—
27C4⋊C4×C101—22no—
28C4⋊C4×C101D412yes—
29C22⋊C4×C101—24no—
30C22⋊C4×C101—22no—
31C42×C101—22no—
32C42×C101C2311yescharacteristic
33C23×C101—22no—
34C22×D101—41no—
35Q8×C101—42no—
36D4×C101—22no—
37D4×C101—22no—
38D4×C101—82no—
39C2×C4×C101—22no—
40C2×C4×C101—22no—
41C2×C4×C101—42no—
42C2×C4×C101—42no—
43C2×C4×C101C2×D411yescharacteristic
44C2×C4×C101C2×D412yes—
45D404—42no—
46D404—42no—
47D404—42no—
48#11717579—22no—
49#11717579—42no—
50C22×C101—42no—
51C22×C101—42no—
52C22×C101—42no—
53C22×C101#12011yescharacteristic
54C2×D101—82no—
55C2×D101—81no—
56C4×C101—21no—
57C4×C101—42no—
58C4×C101—42no—
59C4×C101#13512yes—
60C4×C101AGL(1,ℤ/8)12yes—
61C2×C101—82no—
62C2×C101#44912yes—
63C2×C101#45311yescharacteristic
64D101—161no—
65#1624—1011nomaximal, Sylow
66C101#162411yescharacteristic, complemented, Sylow
67#555D10111yescharacteristic, complemented
68#492—1011no—
69#459—1012no—
70#333—1011no—
71D4⋊C8—1012no—
72C2×SD16—2022no—
73C2×D8—2022no—
74C4⋊Q8C2×D10111yescharacteristic
75#127—1011no—
76#124—22no—
77#121—22no—
78C4×D4C2×D10112yes—
79#117—22no—
80C4⋊C8—1012no—
81#102—2022no—
82C4×C8—1011no—
83C2×Q8C22⋊D10112yes—
84C2×D4—22no—
85C2×D4—2022no—
86C2×D4—2021no—
87C2×D4C22⋊D10112yes—
88C22×C4—22no—
89SD16—4044no—
90D8—4044no—
91C2×C8—2022no—
92C2×C8—2021no—
93C4⋊C4—22no—
94C4⋊C4C22⋊D10112yes—
95C22⋊C4—24no—
96C22⋊C4—22no—
97C42—22no—
98C42C22×D10111yescharacteristic
99C23—22no—
100C23—4041no—
101Q8—42no—
102D4—22no—
103D4—22no—
104D4—82no—
105D4—4042no—
106D4—4042no—
107D4—4042no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4C2×C22⋊D10111yescharacteristic
113C2×C4C2×C22⋊D10112yes—
114C8—4042no—
115C8—2022no—
116C22—42no—
117C22—42no—
118C22—42no—
119C22—8082no—
120C22—8081no—
121C22#1569222411yescharacteristic
122C4—21no—
123C4—42no—
124C4—42no—
125C4#1569227512yes—
126C4#1569228012yes—
127C2—82no—
128C2—16161no—
129C2#2026532212yes—
130C2#2026534511yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#555×C101C2
socleC22×C101#120
hypercenter#555D101
hypocenterC101#1624
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #20265551C2semidirect
2. #20265372C2semidirect
3. #20264926C2semidirect
4. D4⋊#11717579C2semidirect
5. #555×C101C2semidirect
66. C101#1624semidirect
67. #555D101semidirect

Maximal subgroups

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

subgroupindex
#202655512
#202653722
#202649262
D4⋊#117175792
#555×C1012
#1624101