Finite Group Databasebeta

← #13157237

Subgroups — #13157237

Order 1728.

340 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 49 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×#64153C211yesmaximal, characteristic
2#11723104C212yesmaximal
3#10590220—31nomaximal
4C2×C33×Q8C411yescharacteristic
5C2×C3×#634C411yescharacteristic
6C2×C3⋊#622—22no—
7C3×#3750—22no—
8C3×C4×C3⋊Dic3C2211yescharacteristic
9#64153D311yescharacteristic
10C3×#944—62no—
11#63779—32no—
12C33×Q8—41no—
13C2×C33×C4—21no—
14C2×C33×C4C2×C411yescharacteristic
15C2×C3×C3⋊Dic3—21no—
16C2×C3×C3⋊Dic3D412yes—
17C3×#634—41no—
18C3⋊#622—42no—
19#4546—91nomaximal
20C2×C32×Q8—22no—
21C2×C32×Q8—42no—
22C2×C32×Q8Dic311yescharacteristic
23C2×#634Dic311yescharacteristic
24C2×C3×Dic6—62no—
25C2×#622—62no—
26#3750—22no—
27C4×C3⋊Dic3C2×D311yescharacteristic
28C3×Dic3⋊C4—68no—
29C3×C4×Dic3—62no—
30C22×C33C22⋊C411yescharacteristic
31C33×C4—21no—
32C33×C4—41no—
33C3×C3⋊Dic3—41no—
34C3×C3⋊Dic3SD1612yes—
35C3×C3⋊Dic3Q1612yes—
36C3×C4⋊Q8—91no—
37#944—62no—
38C4⋊(C3⋊C8)—92no—
39C32×Q8—41no—
40C32×Q8—44no—
41C32×Q8—48no—
42C2×C32×C4—21no—
43C2×C32×C4—24no—
44C2×C32×C4—22no—
45C2×C32×C4—44no—
46C2×C32×C4—42no—
47C2×C32×C4C2×Dic311yescharacteristic
48C2×C3⋊Dic3—21no—
49C2×C3⋊Dic3#6712yes—
50#634—41no—
51C2×C3×Dic3—64no—
52C2×C3×Dic3—64no—
53C3×Dic6—124no—
54#622—122no—
55#333—271noSylow
56C2×C33#10011yescharacteristic
57C2×C33#10312yes—
58C2×C3×Q8—22no—
59C2×C3×Q8—42no—
60C2×C3×Q8—91no—
61C2×C3×Q8#15711yescharacteristic
62C2×Dic6—62no—
63C3×C4⋊C4—182no—
64C3×C42—91no—
65Dic3⋊C4—68no—
66C4×Dic3—62no—
67C2×C3⋊C8—182no—
68C22×C32—22no—
69C22×C32—42no—
70C22×C32#21711yescharacteristic
71C32×C4—21no—
72C32×C4—24no—
73C32×C4—44no—
74C32×C4—44no—
75C32×C4—48no—
76C32×C4—41no—
77C3⋊Dic3—41no—
78C3⋊Dic3#21512yes—
79C3⋊Dic3#21612yes—
80C3×Dic3—64no—
81C3×Dic3—64no—
82C3×Dic3—124no—
83C4⋊Q8—91no—
84C4⋊C8—272no—
85C33#33311yescharacteristic, complemented, Sylow
86C3×Q8—41no—
87C3×Q8—44no—
88C3×Q8—48no—
89C3×Q8—361no—
90C2×C3×C4—24no—
91C2×C3×C4—21no—
92C2×C3×C4—22no—
93C2×C3×C4—44no—
94C2×C3×C4—42no—
95C2×C3×C4—92no—
96C2×C3×C4—181no—
97C2×C3×C4C2×#15711yescharacteristic
98C2×Dic3—64no—
99C2×Dic3—64no—
100Dic6—124no—
101C3⋊C8—362no—
102C2×C32—24no—
103C2×C32—22no—
104C2×C32—42no—
105C2×C32—44no—
106C2×C32#83311yescharacteristic
107C2×C32#83512yes—
108C2×Q8—91no—
109C2×Q8C3⋊#15711yescharacteristic
110C2×C8—542no—
111C4⋊C4—182no—
112C42—91no—
113C22×C3—22no—
114C22×C3—42no—
115C22×C3C22⋊#15711yescharacteristic
116C3×C4—24no—
117C3×C4—21no—
118C3×C4—44no—
119C3×C4—44no—
120C3×C4—48no—
121C3×C4—41no—
122C3×C4—92no—
123C3×C4—92no—
124C3×C4—361no—
125Dic3—64no—
126Dic3—64no—
127Dic3—124no—
128C32—22no—
129C32—42no—
130C32#454611yescharacteristic, complemented
131Q8—41no—
132Q8—361no—
133C2×C4—21no—
134C2×C4—92no—
135C2×C4—181no—
136C2×C4C2×C3⋊#15711yescharacteristic
137C8—1082no—
138C2×C3—24no—
139C2×C3—22no—
140C2×C3—44no—
141C2×C3—42no—
142C2×C3#6378412yes—
143C2×C3#6378811yescharacteristic
144C22#8899411yescharacteristic
145C4—21no—
146C4—41no—
147C4—92no—
148C4—92no—
149C4—361no—
150C3—22no—
151C3—42no—
152C3#1059022011yescharacteristic, complemented
153C2#1172310912yes—
154C2#1172311311yescharacteristic
subgroupG/H
centerC22#88994
commutatorC2×C33×C4C2×C4
FrattiniC2×C4C2×C3⋊#157
FittingC2×C33×Q8C4
socle, prime radicalC22×C33C22⋊C4
cosocleC3×C4×C3⋊Dic3C22
hypercenterC2×Q8C3⋊#157
hypocenterC33#333
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
85. C33#333semidirect
130. C32#4546semidirect
152. C3#10590220semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
#117231042
#105902203
#45469