Finite Group Databasebeta

← #49120344

Subgroups — #49120344

Order 36224.

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#31826839C211yesmaximal, characteristic, complemented
2#31826660C212yesmaximal, complemented
3#31826214C211yesmaximal, characteristic, complemented
4D4⋊(C283⋊C8)C212yesmaximal, complemented
5#555×C283C211yesmaximal, characteristic, complemented
6C2×#17985379—22no—
7C2×#17985377—22no—
8#22583607C2211yescharacteristic
9C4⋊(C283⋊C8)C2212yes—
10#22583530—22no—
11C4×C283⋊C8C2211yescharacteristic
12C4⋊Q8×C283C2211yescharacteristic
13#124×C283—22no—
14#121×C283—22no—
15C4×D4×C283C2212yes—
16#117×C283—22no—
17C2×D1132—22no—
18C2×D1132—21no—
19#17985379—44no—
20#17985377—44no—
21C2×C283⋊C8—21no—
22C2×C283⋊C8—22no—
23C2×Q8×C283D412yes—
24C2×D4×C283—22no—
25C2×D4×C283D412yes—
26C22×C4×C283—22no—
27C4⋊C4×C283—22no—
28C4⋊C4×C283D412yes—
29C22⋊C4×C283—24no—
30C22⋊C4×C283—22no—
31C42×C283—22no—
32C42×C283C2311yescharacteristic
33C22×D283—41no—
34D1132—42no—
35D1132—42no—
36D1132—42no—
37C283⋊C8—22no—
38C283⋊C8—42no—
39C23×C283—22no—
40Q8×C283—42no—
41D4×C283—22no—
42D4×C283—22no—
43D4×C283—82no—
44C2×C4×C283—22no—
45C2×C4×C283—22no—
46C2×C4×C283—42no—
47C2×C4×C283—42no—
48C2×C4×C283C2×D411yescharacteristic
49C2×C4×C283C2×D412yes—
50C22×C283—42no—
51C22×C283—42no—
52C22×C283—42no—
53C22×C283#12011yescharacteristic
54C2×D283—82no—
55C2×D283—81no—
56C4×C283—21no—
57C4×C283—42no—
58C4×C283—42no—
59C4×C283#13512yes—
60C4×C283AGL(1,ℤ/8)12yes—
61C2×C283—82no—
62C2×C283#44912yes—
63C2×C283#45311yescharacteristic
64D283—161no—
65C283#162411yescharacteristic, complemented, Sylow
66#1624—2831nomaximal, Sylow
67#555D28311yescharacteristic, complemented
68#492—2831no—
69#459—2832no—
70#333—2831no—
71D4⋊C8—2832no—
72C2×SD16—5662no—
73C2×D8—5662no—
74C4⋊Q8C2×D28311yescharacteristic
75#127—2831no—
76#124—22no—
77#121—22no—
78C4×D4C2×D28312yes—
79#117—22no—
80C4⋊C8—2832no—
81#102—5662no—
82C4×C8—2831no—
83C2×Q8C22⋊D28312yes—
84C2×D4—22no—
85C2×D4—5662no—
86C2×D4—5661no—
87C2×D4C22⋊D28312yes—
88C22×C4—22no—
89SD16—11324no—
90D8—11324no—
91C2×C8—5661no—
92C2×C8—5662no—
93C4⋊C4—22no—
94C4⋊C4C22⋊D28312yes—
95C22⋊C4—24no—
96C22⋊C4—22no—
97C42—22no—
98C42C22×D28311yescharacteristic
99C23—22no—
100C23—11321no—
101Q8—42no—
102D4—22no—
103D4—22no—
104D4—82no—
105D4—11322no—
106D4—11322no—
107D4—11322no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4C2×C22⋊D28311yescharacteristic
113C2×C4C2×C22⋊D28312yes—
114C8—5662no—
115C8—11322no—
116C22—42no—
117C22—42no—
118C22—42no—
119C22—22642no—
120C22—22641no—
121C22#2258357911yescharacteristic
122C4—21no—
123C4—42no—
124C4—42no—
125C4#2258363012yes—
126C4#2258363512yes—
127C2—82no—
128C2—45281no—
129C2#3182661012yes—
130C2#3182663311yescharacteristic
subgroupG/H
centerC22#22583579
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
Fitting#555×C283C2
socleC22×C283#120
hypercenter#555D283
hypocenterC283#1624
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #31826839C2semidirect
2. #31826660C2semidirect
3. #31826214C2semidirect
4. D4⋊(C283⋊C8)C2semidirect
5. #555×C283C2semidirect
65. C283#1624semidirect
67. #555D283semidirect

Maximal subgroups

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

subgroupindex
#318268392
#318266602
#318262142
D4⋊(C283⋊C8)2
#555×C2832
#1624283