Finite Group Databasebeta

← #44483607

Subgroups — #44483607

Order 30592.

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#29505148C211yesmaximal, characteristic
2#29504972C212yesmaximal, complemented
3#29504516C211yesmaximal, characteristic, complemented
4Q8⋊(C239⋊C8)C212yesmaximal, complemented
5#559×C239C211yesmaximal, characteristic, complemented
6C2×#16838535—22no—
7C2×#16838533—22no—
8#21439957C2211yescharacteristic
9C4⋊(C239⋊C8)C2212yes—
10#21439880—22no—
11C4×C239⋊C8C2211yescharacteristic
12#127×C239C2211yescharacteristic
13#124×C239—22no—
14#123×C239—22no—
15C4×Q8×C239C2212yes—
16C4×D4×C239—22no—
17C2×D956—22no—
18C2×D956—21no—
19#16838535—44no—
20#16838533—44no—
21C2×C239⋊C8—21no—
22C2×C239⋊C8—22no—
23C2×Q8×C239D412yes—
24C2×D4×C239—22no—
25C2×D4×C239D412yes—
26C22×C4×C239—22no—
27C4⋊C4×C239—22no—
28C4⋊C4×C239D412yes—
29C22⋊C4×C239—24no—
30C22⋊C4×C239—22no—
31C42×C239—22no—
32C42×C239C2311yescharacteristic
33C22×D239—41no—
34D956—42no—
35D956—42no—
36D956—42no—
37C239⋊C8—22no—
38C239⋊C8—42no—
39C23×C239—22no—
40Q8×C239—22no—
41Q8×C239—22no—
42D4×C239—42no—
43D4×C239—42no—
44D4×C239—42no—
45C2×C4×C239—22no—
46C2×C4×C239—22no—
47C2×C4×C239—82no—
48C2×C4×C239C2×D411yescharacteristic
49C2×C4×C239C2×D412yes—
50C22×C239—44no—
51C22×C239—42no—
52C22×C239#12011yescharacteristic
53C2×D239—82no—
54C2×D239—81no—
55C4×C239—21no—
56C4×C239—42no—
57C4×C239—42no—
58C4×C239#13512yes—
59C4×C239AGL(1,ℤ/8)12yes—
60C2×C239—82no—
61C2×C239#44912yes—
62C2×C239#45311yescharacteristic
63D239—161no—
64C239#162011yescharacteristic, complemented, Sylow
65#1620—2391nomaximal, Sylow
66#559D23911yescharacteristic, complemented
67#493—2391no—
68#460—2392no—
69#331—2391no—
70Q8⋊C8—2392no—
71C2×SD16—4782no—
72C2×D8—4782no—
73#127—2391no—
74#127C2×D23911yescharacteristic
75#124—22no—
76#123—22no—
77C4×Q8C2×D23912yes—
78C4×D4—22no—
79C4⋊C8—2392no—
80#102—4782no—
81C4×C8—2391no—
82C2×Q8C22⋊D23912yes—
83C2×D4—22no—
84C2×D4—4782no—
85C2×D4—4781no—
86C2×D4C22⋊D23912yes—
87C22×C4—22no—
88SD16—9564no—
89D8—9564no—
90C2×C8—4782no—
91C2×C8—4781no—
92C4⋊C4—22no—
93C4⋊C4C22⋊D23912yes—
94C22⋊C4—24no—
95C22⋊C4—22no—
96C42—22no—
97C42C22×D23911yescharacteristic
98C23—22no—
99C23—9561no—
100Q8—22no—
101Q8—22no—
102D4—42no—
103D4—42no—
104D4—42no—
105D4—9562no—
106D4—9562no—
107D4—9562no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—82no—
111C2×C4C2×C22⋊D23911yescharacteristic
112C2×C4C2×C22⋊D23912yes—
113C8—9562no—
114C8—4782no—
115C22—44no—
116C22—42no—
117C22—19122no—
118C22—19121no—
119C22#2143992911yescharacteristic
120C4—21no—
121C4—42no—
122C4—42no—
123C4#2143998012yes—
124C4#2143998512yes—
125C2—82no—
126C2—38241no—
127C2#2950491712yes—
128C2#2950494011yescharacteristic
subgroupG/H
centerC22#21439929
commutator, cosocleC42×C239C23
FrattiniC42C22×D239
Fitting#559×C239C2
socleC22×C239#120
prime radical#29505148C2
hypercenter#559D239
hypocenterC239#1620
LCSG ▷ C42×C239 ▷ C22×C239 ▷ C239
UCSC1 ◁ C22 ◁ C42 ◁ #559

Complements

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

NH ≅ G/Ntype
2. #29504972C2semidirect
3. #29504516C2semidirect
4. Q8⋊(C239⋊C8)C2semidirect
5. #559×C239C2semidirect
64. C239#1620semidirect
66. #559D239semidirect

Maximal subgroups

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

subgroupindex
#295051482
#295049722
#295045162
Q8⋊(C239⋊C8)2
#559×C2392
#1620239