Finite Group Databasebeta

← #10592945

Subgroups — #10592945

Order 576.

426 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C32×2−1+4C211yesmaximal, characteristic
2#64160C212yesmaximal, complemented
3#64153C211yesmaximal, characteristic, complemented
4#63664C212yesmaximal, complemented
5#63663C211yesmaximal, characteristic, complemented
6#5246—34nomaximal
7C32×#41—22no—
8C32×#41—22no—
9C32×#41C2212yes—
10C2×C32×Q8—22no—
11C2×C32×Q8C2211yescharacteristic
12C2×#634C2211yescharacteristic
13#3756—22no—
14#3754—22no—
15#3750—22no—
16C4×C3⋊Dic3C2211yescharacteristic
17#3748C2212yes—
18C3×2−1+4D314yes—
19#951—38no—
20#944—34no—
21#837—38no—
22#836—34no—
23C32×Q8—22no—
24C32×Q8—22no—
25C32×Q8D412yes—
26C32×D4—22no—
27C32×D4—22no—
28C32×D4D412yes—
29C2×C32×C4—24no—
30C2×C32×C4—22no—
31C2×C32×C4D412yes—
32C2×C32×C4C2311yescharacteristic
33C2×C3⋊Dic3—21no—
34C2×C3⋊Dic3—21no—
35#634—22no—
36C3⋊(C3⋊C8)—22no—
37#456—91noSylow
38C3×#41—28no—
39C3×#41—28no—
40C3×#41C2×D318yes—
41C2×C3×Q8—28no—
42C2×C3×Q8C2×D314yes—
43C2×Dic6—34no—
44#216—68no—
45#214—68no—
46Dic3⋊C4—68no—
47C4×Dic3—34no—
48#208—38no—
49C22×C32—22no—
50C22×C32C2×D411yescharacteristic
51C32×C4—22no—
52C32×C4—22no—
53C32×C4C2×D412yes—
54C3⋊Dic3—22no—
55C3⋊Dic3—41no—
562−1+4C3⋊D311yescharacteristic
57#137—92no—
58C4⋊Q8—91no—
59#104—92no—
60#101—91no—
61C3×Q8—28no—
62C3×Q8—28no—
63C3×Q8#6718yes—
64C3×D4—28no—
65C3×D4—28no—
66C3×D4#6718yes—
67C2×C3×C4—216no—
68C2×C3×C4—28no—
69C2×C3×C4#6718yes—
70C2×C3×C4C22×D314yes—
71C2×Dic3—64no—
72C2×Dic3—64no—
73Dic6—68no—
74C3⋊C8—68no—
75C2×C32—21no—
76C2×C32—42no—
77C2×C32#12011yescharacteristic
78#41—22no—
79#41—22no—
80#41C2×C3⋊D312yes—
81C2×Q8—22no—
82C2×Q8—91no—
83C2×Q8C2×C3⋊D311yescharacteristic
84Q16—182no—
85SD16—182no—
86M16—92no—
87C4⋊C4—182no—
88C42—91no—
89C22×C3—28no—
90C22×C3C2×#6714yes—
91C3×C4—28no—
92C3×C4—28no—
93C3×C4C2×#6718yes—
94Dic3—68no—
95Dic3—124no—
96C32#45611yescharacteristic, complemented, Sylow
97Q8—22no—
98Q8—22no—
99Q8—182no—
100Q8#63812yes—
101D4—22no—
102D4—22no—
103D4#63812yes—
104C2×C4—24no—
105C2×C4—22no—
106C2×C4—181no—
107C2×C4—181no—
108C2×C4#63812yes—
109C2×C4C22×C3⋊D311yescharacteristic
110C8—182no—
111C2×C3—24no—
112C2×C3—48no—
113C2×C3#95314yes—
114C22—22no—
115C22C2×#63811yescharacteristic
116C4—22no—
117C4—22no—
118C4—182no—
119C4—361no—
120C4C2×#63812yes—
121C3#524614yescomplemented
122C2—21no—
123C2—42no—
124C2#6416211yescharacteristic
subgroupG/H
centerC2#64162
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C22×C3⋊D3
Fitting, prime radicalC32×2−1+4C2
socleC2×C32#120
hypercenter2−1+4C3⋊D3
hypocenterC32#456
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ 2−1+4

Complements

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

NH ≅ G/Ntype
2. #64160C2semidirect
3. #64153C2semidirect
4. #63664C2semidirect
5. #63663C2semidirect
96. C32#456semidirect
121. C3#5246semidirect

Maximal subgroups

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

subgroupindex
C32×2−1+42
#641602
#641532
#636642
#636632
#52463