Finite Group Databasebeta

← #13157235

Subgroups — #13157235

Order 1728.

332 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 33 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#11723105C212yesmaximal
3#10590218—31nomaximal
4C2×C33×Q8C411yescharacteristic
5C2×C3×#634C2211yescharacteristic
6#88993—22no—
7C3×#3750—22no—
8C3×C4×C3⋊Dic3C411yescharacteristic
9#64153D311yescharacteristic
10C3×#944—62no—
11#63780—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#4548—91nomaximal
20C2×C32×Q8—22no—
21C2×C32×Q8—42no—
22C2×C32×Q8Dic311yescharacteristic
23C2×#634C2×D311yescharacteristic
24C2×C3×Dic6—62no—
25#3792—62no—
26#3750—22no—
27C4×C3⋊Dic3Dic311yescharacteristic
28C3×Dic3⋊C4—68no—
29C3×C4×Dic3—62no—
30C22×C33C22⋊C411yescharacteristic
31C33×C4—21no—
32C33×C4—41no—
33C3×C3⋊Dic3—22no—
34C3×C3⋊Dic3—22no—
35C3×C4⋊Q8—91no—
36#944—62no—
37#836—92no—
38C32×Q8—41no—
39C32×Q8—44no—
40C32×Q8—48no—
41C2×C32×C4—21no—
42C2×C32×C4—24no—
43C2×C32×C4—22no—
44C2×C32×C4—44no—
45C2×C32×C4—42no—
46C2×C32×C4C2×Dic311yescharacteristic
47C2×C3⋊Dic3—21no—
48C2×C3⋊Dic3#6712yes—
49#634—41no—
50C2×C3×Dic3—64no—
51C2×C3×Dic3—64no—
52C3×Dic6—124no—
53#622—122no—
54#356—271noSylow
55C2×C33—21no—
56C2×C33#9911yescharacteristic
57C2×C3×Q8—22no—
58C2×C3×Q8—42no—
59C2×C3×Q8—91no—
60C2×C3×Q8#15711yescharacteristic
61C2×Dic6—62no—
62C3×C4⋊C4—182no—
63C3×C42—91no—
64Dic3⋊C4—68no—
65C4×Dic3—62no—
66#208—182no—
67C22×C32—22no—
68C22×C32—42no—
69C22×C32#21711yescharacteristic
70C32×C4—21no—
71C32×C4—24no—
72C32×C4—44no—
73C32×C4—44no—
74C32×C4—48no—
75C32×C4—41no—
76C3⋊Dic3—22no—
77C3⋊Dic3—22no—
78C3×Dic3—68no—
79C3×Dic3—124no—
80C4⋊Q8—91no—
81#101—272no—
82C33#35611yescharacteristic, complemented, Sylow
83C3×Q8—41no—
84C3×Q8—44no—
85C3×Q8—48no—
86C3×Q8—361no—
87C2×C3×C4—24no—
88C2×C3×C4—21no—
89C2×C3×C4—22no—
90C2×C3×C4—44no—
91C2×C3×C4—42no—
92C2×C3×C4—92no—
93C2×C3×C4—181no—
94C2×C3×C4C2×#15711yescharacteristic
95C2×Dic3—64no—
96C2×Dic3—64no—
97Dic6—124no—
98C3⋊C8—362no—
99C2×C32—21no—
100C2×C32—24no—
101C2×C32—22no—
102C2×C32—44no—
103C2×C32—42no—
104C2×C32#83411yescharacteristic
105C2×Q8—91no—
106C2×Q8C3⋊#15711yescharacteristic
107M16—542no—
108C4⋊C4—182no—
109C42—91no—
110C22×C3—22no—
111C22×C3—42no—
112C22×C3C22⋊#15711yescharacteristic
113C3×C4—24no—
114C3×C4—21no—
115C3×C4—44no—
116C3×C4—44no—
117C3×C4—48no—
118C3×C4—41no—
119C3×C4—182no—
120C3×C4—182no—
121Dic3—68no—
122Dic3—124no—
123C32—22no—
124C32—42no—
125C32#454811yescharacteristic, complemented
126Q8—41no—
127Q8—361no—
128C2×C4—21no—
129C2×C4—92no—
130C2×C4—181no—
131C2×C4C2×C3⋊#15711yescharacteristic
132C8—1082no—
133C2×C3—24no—
134C2×C3—21no—
135C2×C3—22no—
136C2×C3—44no—
137C2×C3—42no—
138C2×C3#6378611yescharacteristic
139C22#8899411yescharacteristic
140C4—21no—
141C4—41no—
142C4—182no—
143C4—182no—
144C3—22no—
145C3—42no—
146C3#1059021811yescharacteristic, complemented
147C2—21no—
148C2#1172311111yescharacteristic
subgroupG/H
centerC2#11723111
commutatorC2×C33×C4C2×C4
FrattiniC2×C4C2×C3⋊#157
FittingC2×C33×Q8C4
socleC2×C33#99
prime radicalC22×C33C22⋊C4
cosocleC2×C3×#634C22
hypercenterC2×Q8C3⋊#157
hypocenterC33#356
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
82. C33#356semidirect
125. C32#4548semidirect
146. C3#10590218semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
#117231052
#105902183
#45489