Finite Group Databasebeta

← #13157216

Subgroups — #13157216

Order 1728.

264 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 37 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
1C3×Q16×C3⋊D3C211yesmaximal, characteristic
2#11723095C211yesmaximal, characteristic
3#11723090C211yesmaximal, characteristic
4#10590199—31nomaximal
5C3×Q8×C3⋊D3—21no—
6#88989—21no—
7C33×Q16C411yescharacteristic
8C3×#3756—21no—
9C3×#3746C411yescharacteristic
10C3×C8×C3⋊D3C2211yescharacteristic
11C3⋊#3708—21no—
12Q16×C3⋊D3D311yescharacteristic
13C3×D3×Q16—62no—
14#63770—31no—
15#63765—31no—
16C33×Q8—21no—
17C3×C4×C3⋊D3—41no—
18C3×C4×C3⋊D3D411yescharacteristic
19C3×#634—21no—
20C3×#634—41no—
21C3⋊#622—41no—
22C33×C8C2×C411yescharacteristic
23C3×C3⋊(C3⋊C8)D411yescharacteristic
24#4565—91nomaximal
25Q8×C3⋊D3—21no—
26C3×D3×Q8—64no—
27#3788—61no—
28C32×Q16—22no—
29C32×Q16—42no—
30C32×Q16Dic311yescharacteristic
31#3756—21no—
32#3746Dic311yescharacteristic
33C8×C3⋊D3C2×D311yescharacteristic
34C3×#216—64no—
35C3×Dic12—62no—
36C3×D3×C8—62no—
37#3708—61no—
38C2×C3×C3⋊D3SD1611yescharacteristic
39C33×C4—41no—
40C33×C4C22⋊C411yescharacteristic
41C3×C3⋊Dic3—41no—
42C3×C3⋊Dic3D811yescharacteristic
43C2×C3×Q16—91no—
44D3×Q16—62no—
45#819—91no—
46#812—91no—
47C32×Q8—21no—
48C32×Q8—24no—
49C32×Q8—44no—
50C4×C3⋊D3—41no—
51C4×C3⋊D3#6711yescharacteristic
52#634—21no—
53#634—41no—
54C3×C4×D3—62no—
55C3×C4×D3—124no—
56C3×Dic6—64no—
57C3×Dic6—124no—
58#622—121no—
59C32×C8—22no—
60C32×C8—42no—
61C32×C8C2×Dic311yescharacteristic
62C3⋊(C3⋊C8)#6711yescharacteristic
63C3×C3⋊C8—62no—
64#362—271noSylow
65C2×C33#10211yescharacteristic
66C3×C3⋊D3—21no—
67C2×C3×Q8—181no—
68D3×Q8—64no—
69C3×Q16—22no—
70C3×Q16—42no—
71C3×Q16—91no—
72C3×Q16—181no—
73C3×Q16#15711yescharacteristic
74C2×C3×C8—91no—
75#216—64no—
76#208—181no—
77Dic12—62no—
78D3×C8—62no—
79C3⋊C16—181no—
80C2×C3⋊D3#21411yescharacteristic
81C2×C3×D3—62no—
82C32×C4—22no—
83C32×C4—42no—
84C32×C4—41no—
85C32×C4—44no—
86C32×C4—84no—
87C32×C4#21711yescharacteristic
88C3⋊Dic3—41no—
89C3⋊Dic3#21311yescharacteristic
90C3×Dic3—62no—
91C3×Dic3—124no—
92C2×Q16—91no—
93M32—271no—
94#108—271no—
95C33#36211yescharacteristic, complemented, Sylow
96C3×Q8—24no—
97C3×Q8—21no—
98C3×Q8—44no—
99C3×Q8—181no—
100C3×Q8—361no—
101C2×C3×C4—91no—
102C2×C3×C4—361no—
103C4×D3—62no—
104C4×D3—124no—
105Dic6—64no—
106Dic6—124no—
107C3×C8—22no—
108C3×C8—42no—
109C3×C8—91no—
110C3×C8C2×#15711yescharacteristic
111C3⋊C8—62no—
112C3⋊C8—361no—
113C2×C32—22no—
114C2×C32—42no—
115C2×C32#83211yescharacteristic
116C3⋊D3—21no—
117C3×D3—64no—
118C2×Q8—181no—
119Q16—91no—
120Q16—181no—
121Q16C3⋊#15711yescharacteristic
122M16—541no—
123C2×C8—91no—
124C16—541no—
125C22×C3—91no—
126C2×D3—62no—
127C3×C4—22no—
128C3×C4—44no—
129C3×C4—41no—
130C3×C4—42no—
131C3×C4—84no—
132C3×C4—91no—
133C3×C4—361no—
134C3×C4C22⋊#15711yescharacteristic
135Dic3—62no—
136Dic3—124no—
137C32—22no—
138C32—42no—
139C32#456511yescharacteristic, complemented
140Q8—21no—
141Q8—181no—
142Q8—361no—
143C2×C4—91no—
144C2×C4—361no—
145C8—91no—
146C8—1081no—
147C8C2×C3⋊#15711yescharacteristic
148C2×C3—22no—
149C2×C3—42no—
150C2×C3—181no—
151C2×C3#6378211yescharacteristic
152D3—64no—
153C22—91no—
154C4—41no—
155C4—91no—
156C4—361no—
157C4#8899411yescharacteristic
158C3—22no—
159C3—42no—
160C3#1059019911yescharacteristic, complemented
161C2—181no—
162C2#1172310711yescharacteristic
subgroupG/H
centerC2#11723107
commutatorC33×C8C2×C4
FrattiniC8C2×C3⋊#157
FittingC33×Q16C4
socleC2×C33#102
prime radicalC2×C3×C3⋊D3SD16
cosocleC3×C8×C3⋊D3C22
hypercenterQ16C3⋊#157
hypocenterC33#362
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16

Complements

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

NH ≅ G/Ntype
95. C33#362semidirect
139. C32#4565semidirect
160. C3#10590199semidirect

Maximal subgroups

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

subgroupindex
C3×Q16×C3⋊D32
#117230952
#117230902
#105901993
#45659