Finite Group Databasebeta

← #10592761

Subgroups — #10592761

Order 576.

438 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 87 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#64364C211yesmaximal, characteristic, complemented
2C32×#124C211yesmaximal, characteristic, complemented
3#64141C212yesmaximal, complemented
4#63663C211yesmaximal, characteristic, complemented
5#63632C212yesmaximal, complemented
6#5062—34nomaximal
7C2×C32×Q8C2211yescharacteristic
8C2×C32×D4C2211yescharacteristic
9#3832—22no—
10Q8×C3⋊D3—22no—
11#3828—22no—
12#3828C2212yes—
13C32×C22⋊C4—22no—
14C32×C42C2211yescharacteristic
15#3754—22no—
16#3753—22no—
17#3748C2212yes—
18#1007—34no—
19C3×#124D314yescomplemented
20#932—38no—
21#836—34no—
22#805—38no—
23C23×C32—21no—
24C32×Q8—22no—
25C32×D4—22no—
26C2×C32×C4—21no—
27C2×C32×C4D412yes—
28C2×C32×C4C2311yescharacteristic
29#638—22no—
30C4⋊(C3⋊D3)—22no—
31C4⋊(C3⋊D3)D412yes—
32C4×C3⋊D3—24no—
33C4×C3⋊D3—22no—
34#634—22no—
35#634D412yes—
36C3⋊(C3⋊C8)—22no—
37#454—91noSylow
38C2×C3×Q8C2×D314yes—
39C2×C3×D4C2×D314yes—
40#239—68no—
41D3×Q8—68no—
42#235—38no—
43#235—68no—
44C3×C22⋊C4—28no—
45C3×C42C2×D314yes—
46#214—68no—
47#213—68no—
48#208—38no—
49C22×C32—42no—
50C22×C32—41no—
51C22×C32C2×D411yescharacteristic
52C2×C3⋊D3—22no—
53C32×C4—22no—
54C32×C4—41no—
55C32×C4C2×D412yes—
56C3⋊Dic3—22no—
572−1+4—91no—
58AGL(1,ℤ/8)—92no—
59#124C3⋊D311yescharacteristic, complemented
60#104—92no—
61#101—91no—
62C23×C3—24no—
63C3×Q8—28no—
64C3×D4—28no—
65C2×C3×C4—24no—
66C2×C3×C4#6718yes—
67C2×C3×C4C22×D314yes—
68#67—68no—
69D12—38no—
70D12—68no—
71C4×D3—616no—
72C4×D3—68no—
73Dic6—38no—
74Dic6—68no—
75C3⋊C8—68no—
76C2×C32—21no—
77C2×C32—81no—
78C2×C32#12011yescharacteristic
79C3⋊D3—42no—
80#41—92no—
81#41—182no—
82#41—182no—
83C2×Q8—182no—
84C2×Q8C2×C3⋊D311yescharacteristic
85C2×D4C2×C3⋊D311yescharacteristic
86SD16—182no—
87D8—182no—
88M16—92no—
89C22⋊C4—22no—
90C42C2×C3⋊D311yescharacteristic
91C22×C3—48no—
92C22×C3—44no—
93C22×C3C2×#6714yes—
94C2×D3—68no—
95C3×C4—28no—
96C3×C4—44no—
97C3×C4D3×D418yes—
98Dic3—68no—
99C32#45411yescharacteristic, complemented, Sylow
100C23—21no—
101Q8—22no—
102Q8—92no—
103Q8—182no—
104D4—22no—
105D4—92no—
106D4—182no—
107D4—182no—
108C2×C4—21no—
109C2×C4—184no—
110C2×C4—182no—
111C2×C4#63812yes—
112C2×C4C22×C3⋊D311yescharacteristic
113C8—182no—
114C2×C3—24no—
115C2×C3—84no—
116C2×C3#93714yes—
117D3—128no—
118C22—42no—
119C22—41no—
120C22—182no—
121C22C2×#63811yescharacteristic
122C4—22no—
123C4—41no—
124C4—182no—
125C4D4×C3⋊D312yes—
126C3#506214yescomplemented
127C2—21no—
128C2—81no—
129C2—362no—
130C2#6414611yescharacteristic
subgroupG/H
centerC2#64146
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C22×C3⋊D3
FittingC32×#124C2
socleC2×C32#120
hypercenter#124C3⋊D3
hypocenterC32#454
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #64364C2semidirect
2. C32×#124C2semidirect
3. #64141C2semidirect
4. #63663C2semidirect
5. #63632C2semidirect
19. C3×#124D3semidirect
59. #124C3⋊D3semidirect
99. C32#454semidirect
126. C3#5062semidirect

Maximal subgroups

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

subgroupindex
#643642
C32×#1242
#641412
#636632
#636322
#50623