Finite Group Databasebeta

← #10591493

Subgroups — #10591493

Order 576.

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

normal subgroups only

This group has 286 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
1#64145C211yesmaximal, characteristic
2C2×C3×#214C212yesmaximal
3#63873C211yesmaximal, characteristic, complemented
4C2×#3719C212yesmaximal, complemented
5#63555C211yesmaximal, characteristic, complemented
6#5167—32nomaximal
7C2×C32×D4C2211yescharacteristic
8C2×#634C2211yescharacteristic
9C2×C3×Dic6C2212yes—
10#3757—22no—
11C4×C3⋊Dic3C2211yescharacteristic
12C3×#214—24no—
13C2×C3×C3⋊C8C2212yes—
14#3724—22no—
15#3719—24no—
16C2×C3×SD16—32no—
17#944—32no—
18C2×#216—32no—
19#936—32no—
20#936—61no—
21C2×#214D312yes—
22C2×Dic12—32no—
23#815—32no—
24C23×C32—21no—
25C32×D4—22no—
26C2×C32×C4C2311yescharacteristic
27C2×C3⋊Dic3—21no—
28C2×C3⋊Dic3D412yes—
29#634—22no—
30C2×C3×Dic3—22no—
31C3×Dic6—22no—
32C3×Dic6—22no—
33C3×C3⋊C8D414yes—
34#497—91noSylow
35C2×C3×Q8—32no—
36C2×C3×D4—21no—
37C2×C3×D4C2×D312yes—
38C2×Dic6—32no—
39C2×Dic6—61no—
40C2×Dic6C2×D312yes—
41C3×SD16—64no—
42C2×C3×C8—32no—
43#217—64no—
44#217—64no—
45#216—64no—
46#214—24no—
47Dic3⋊C4—64no—
48C4×Dic3—32no—
49C4×Dic3—61no—
50C2×C3⋊C8C2×D312yes—
51Dic12—64no—
52C22×C32—41no—
53C22×C32—42no—
54C22×C32C2×D411yescharacteristic
55C32×C4C2×D412yes—
56C3⋊Dic3—22no—
57C3⋊Dic3—41no—
58C3×Dic3—42no—
59C2×Q16—92no—
60C2×SD16—32no—
61C4⋊Q8—91no—
62#124—91no—
63#97—91no—
64C23×C3—22no—
65C23×C3—22no—
66C3×Q8—62no—
67C3×Q8—62no—
68C3×D4—22no—
69C3×D4—22no—
70C3×D4—42no—
71C2×C3×C4—21no—
72C2×C3×C4—62no—
73C2×C3×C4C22×D312yes—
74C2×Dic3—22no—
75C2×Dic3—34no—
76C2×Dic3—62no—
77C2×Dic3—62no—
78C2×Dic3—62no—
79Dic6—22no—
80Dic6—22no—
81Dic6—62no—
82Dic6—62no—
83Dic6—122no—
84C3×C8—34no—
85C3⋊C8#6714yes—
86C2×C32—81no—
87C2×C32#12711yescharacteristic
88C2×C32#13712yes—
89C2×Q8—32no—
90C2×Q8—91no—
91C2×D4D3211yescharacteristic
92Q16—184no—
93SD16—64no—
94C2×C8—32no—
95C4⋊C4—182no—
96C22⋊C4—182no—
97C42—91no—
98C22×C3—21no—
99C22×C3—42no—
100C22×C3—44no—
101C22×C3—44no—
102C22×C3—42no—
103C22×C3D3×D412yes—
104C3×C4—22no—
105C3×C4—122no—
106C3×C4D3×D412yes—
107C3×C4C2×#6712yes—
108Dic3—42no—
109Dic3—64no—
110Dic3—122no—
111Dic3—122no—
112Dic3—122no—
113C32#49711yescharacteristic, complemented, Sylow
114C23—21no—
115Q8—62no—
116Q8—62no—
117Q8—182no—
118D4—22no—
119C2×C4—62no—
120C2×C4—92no—
121C2×C4—181no—
122C2×C4C2×D3211yescharacteristic
123C8—34no—
124C2×C3—22no—
125C2×C3—21no—
126C2×C3—82no—
127C2×C3—82no—
128C2×C3#91514yes—
129C2×C3#94012yes—
130C22—42no—
131C22—41no—
132C22#381111yescharacteristic
133C4—122no—
134C4—182no—
135C4—361no—
136C4D3×#6712yes—
137C3—21no—
138C3#516712yescomplemented
139C2—81no—
140C2#6393112yes—
141C2#6397911yescharacteristic
subgroupG/H
centerC22#3811
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
Fitting, prime radicalC2×C32×D4C22
socleC22×C32C2×D4
hypercenterC2×D4D32
hypocenterC32#497
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. #63873C2semidirect
4. C2×#3719C2semidirect
5. #63555C2semidirect
113. C32#497semidirect
138. C3#5167semidirect

Maximal subgroups

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

subgroupindex
#641452
C2×C3×#2142
#638732
C2×#37192
#635552
#51673