Finite Group Databasebeta

← #10591189

Subgroups — #10591189

Order 576.

254 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C2×#3748C211yesmaximal, characteristic
2C3×#924C212yesmaximal
3C2×#3718C211yesmaximal, characteristic, complemented
4#63574C211yesmaximal, characteristic, complemented
5#63565C212yesmaximal, complemented
6#5052—32nomaximal
7C22×C32×C4C2211yescharacteristic
8C2×C3×Dic6C2212yes—
9#3748—22no—
10C2×C3⋊(C3⋊C8)C2211yescharacteristic
11C2×C3⋊(C3⋊C8)C2211yescharacteristic
12C3×#217—22no—
13C3×C4⋊Dic3C2212yes—
14C3×Dic3⋊C4—22no—
15#3718—22no—
16C3×C22⋊Q8—32no—
17C2×#216—32no—
18#924D312yes—
19C2×#208—32no—
20C2×#208—61no—
21#835—32no—
22#810—32no—
23#808—32no—
24C23×C32D411yescharacteristic
25C2×C32×C4—21no—
26C2×C32×C4—21no—
27C2×C32×C4D411yescharacteristic
28C2×C32×C4C2311yescharacteristic
29C2×C3×Dic3—22no—
30C2×C3×Dic3—22no—
31C3×Dic6—22no—
32C3×Dic6—22no—
33C3⋊(C3⋊C8)—21no—
34C3⋊(C3⋊C8)D412yes—
35#470—91noSylow
36C2×C3×Q8—32no—
37C22×C3×C4—21no—
38C22×C3×C4C2×D312yes—
39C2×Dic6C2×D312yes—
40C3×C4⋊C4—32no—
41C3×C4⋊C4—62no—
42C3×C22⋊C4—62no—
43#217—22no—
44#216—64no—
45C4⋊Dic3C2×D312yes—
46Dic3⋊C4—22no—
47#208—64no—
48#208—64no—
49C2×C3⋊C8—32no—
50C2×C3⋊C8—32no—
51C2×C3⋊C8—61no—
52C2×C3⋊C8—61no—
53C22×C32—21no—
54C22×C32—22no—
55C22×C32C2×D411yescharacteristic
56C32×C4—21no—
57C32×C4C2×D411yescharacteristic
58C32×C4#4111yescharacteristic
59C3×Dic3—42no—
60C3×Dic3—42no—
61C2×Q16—91no—
62C2×M16—91no—
63C22⋊Q8—32no—
64#106—91no—
65#103—92no—
66C23×C3—21no—
67C23×C3#6712yes—
68C3×Q8—62no—
69C3×Q8—62no—
70C2×C3×C4—22no—
71C2×C3×C4—22no—
72C2×C3×C4—22no—
73C2×C3×C4—22no—
74C2×C3×C4—21no—
75C2×C3×C4—21no—
76C2×C3×C4—62no—
77C2×C3×C4—62no—
78C2×C3×C4#6712yes—
79C2×C3×C4C22×D312yes—
80C2×Dic3—22no—
81C2×Dic3—22no—
82Dic6—22no—
83Dic6—22no—
84C3⋊C8—34no—
85C3⋊C8—62no—
86C3⋊C8—62no—
87C3⋊C8—62no—
88C2×C32—41no—
89C2×C32#12111yescharacteristic
90C2×C32#13712yes—
91C2×Q8—32no—
92C22×C4D3211yescharacteristic
93Q16—182no—
94M16—182no—
95C2×C8—91no—
96C2×C8—91no—
97C4⋊C4—32no—
98C4⋊C4—62no—
99C22⋊C4—62no—
100C22×C3—22no—
101C22×C3—22no—
102C22×C3—22no—
103C22×C3—22no—
104C22×C3—21no—
105C22×C3—42no—
106C22×C3C2×#6712yes—
107C3×C4—22no—
108C3×C4—22no—
109C3×C4—21no—
110C3×C4—21no—
111C3×C4—122no—
112C3×C4—122no—
113C3×C4D3×D412yes—
114C3×C4#23712yes—
115Dic3—42no—
116Dic3—42no—
117C32#47011yescharacteristic, complemented, Sylow
118C23#62511yescharacteristic
119Q8—62no—
120Q8—62no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—62no—
124C2×C4—62no—
125C2×C4#62511yescharacteristic
126C2×C4C2×D3211yescharacteristic
127C8—92no—
128C8—181no—
129C2×C3—22no—
130C2×C3—21no—
131C2×C3—42no—
132C2×C3—42no—
133C2×C3#93912yes—
134C2×C3#94212yes—
135C2×C3#95112yes—
136C22—22no—
137C22—21no—
138C22C2×#62511yescharacteristic
139C4—21no—
140C4—122no—
141C4—122no—
142C4#380511yescharacteristic
143C4#381111yescharacteristic
144C3—21no—
145C3#505212yescomplemented
146C2—41no—
147C2#6382712yes—
148C2#6398011yescharacteristic
subgroupG/H
centerC22C2×#625
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC22×C32×C4C22
socleC22×C32C2×D4
prime radicalC23×C32D4
hypercenterC22×C4D32
hypocenterC32#470
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×#3718C2semidirect
4. #63574C2semidirect
5. #63565C2semidirect
117. C32#470semidirect
145. C3#5052semidirect

Maximal subgroups

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

subgroupindex
C2×#37482
C3×#9242
C2×#37182
#635742
#635652
#50523