Finite Group Databasebeta

← #13157176

Subgroups — #13157176

Order 1728.

284 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 41 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×#64099C211yesmaximal, characteristic
2#11723102C211yesmaximal, characteristic
3#11723102C211yesmaximal, characteristic
4#10590159—31nomaximal
5C2×C3⋊#622—21no—
6C2×C3⋊#622—21no—
7C33×C4⋊C4C411yescharacteristic
8C3×C4⋊(C3⋊Dic3)—21no—
9C3×C4⋊(C3⋊Dic3)C411yescharacteristic
10C3×#3750—21no—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12#64099D311yescharacteristic
13C3×#890—62no—
14#63777—31no—
15#63777—31no—
16C2×C33×C4—21no—
17C2×C33×C4C2×C411yescharacteristic
18C2×C3×C3⋊Dic3—21no—
19C2×C3×C3⋊Dic3D411yescharacteristic
20C2×C3×C3⋊Dic3D411yescharacteristic
21C3⋊#622—22no—
22C3⋊#622—22no—
23#4543—91nomaximal
24C2×#622—61no—
25C2×#622—61no—
26C32×C4⋊C4—22no—
27C32×C4⋊C4—42no—
28C32×C4⋊C4Dic311yescharacteristic
29C4⋊(C3⋊Dic3)—21no—
30C4⋊(C3⋊Dic3)Dic311yescharacteristic
31#3750—21no—
32C4×C3⋊Dic3C2×D311yescharacteristic
33C3×C4⋊Dic3—62no—
34C3×C4⋊Dic3—64no—
35C3×Dic3⋊C4—64no—
36C3×C4×Dic3—62no—
37C22×C33C22⋊C411yescharacteristic
38C33×C4—21no—
39C33×C4—41no—
40C3×C3⋊Dic3—21no—
41C3×C3⋊Dic3—21no—
42C3×C3⋊Dic3—41no—
43C3×#125—91no—
44#890—62no—
45#803—91no—
46#803—91no—
47C2×C32×C4—21no—
48C2×C32×C4—24no—
49C2×C32×C4—22no—
50C2×C32×C4—44no—
51C2×C32×C4—42no—
52C2×C32×C4C2×Dic311yescharacteristic
53C2×C3⋊Dic3—21no—
54C2×C3⋊Dic3#6711yescharacteristic
55C2×C3⋊Dic3#6711yescharacteristic
56C2×C3×Dic3—62no—
57C2×C3×Dic3—62no—
58C2×C3×Dic3—64no—
59#622—62no—
60#622—62no—
61#330—271noSylow
62C2×C33#10111yescharacteristic
63C2×C33#10411yescharacteristic
64C2×C33#10411yescharacteristic
65C3×C4⋊C4—22no—
66C3×C4⋊C4—42no—
67C3×C4⋊C4—91no—
68C3×C4⋊C4—181no—
69C3×C4⋊C4—181no—
70C3×C4⋊C4#15711yescharacteristic
71C3×C42—91no—
72C4⋊Dic3—62no—
73C4⋊Dic3—64no—
74Dic3⋊C4—64no—
75C4×Dic3—62no—
76C2×C3⋊C8—181no—
77C2×C3⋊C8—181no—
78C22×C32—22no—
79C22×C32—42no—
80C22×C32#21711yescharacteristic
81C32×C4—21no—
82C32×C4—24no—
83C32×C4—44no—
84C32×C4—41no—
85C32×C4—44no—
86C32×C4—84no—
87C3⋊Dic3—21no—
88C3⋊Dic3—21no—
89C3⋊Dic3—41no—
90C3×Dic3—122no—
91C3×Dic3—122no—
92C3×Dic3—124no—
93#125—91no—
94#97—271no—
95#97—271no—
96C33#33011yescharacteristic, complemented, Sylow
97C2×C3×C4—24no—
98C2×C3×C4—21no—
99C2×C3×C4—22no—
100C2×C3×C4—44no—
101C2×C3×C4—42no—
102C2×C3×C4—91no—
103C2×C3×C4—91no—
104C2×C3×C4—181no—
105C2×C3×C4C2×#15711yescharacteristic
106C2×Dic3—62no—
107C2×Dic3—62no—
108C2×Dic3—64no—
109C3⋊C8—182no—
110C3⋊C8—182no—
111C2×C32—22no—
112C2×C32—22no—
113C2×C32—22no—
114C2×C32—42no—
115C2×C32—42no—
116C2×C32—42no—
117C2×C32#83611yescharacteristic
118C2×C32#83711yescharacteristic
119C2×C32#83711yescharacteristic
120C2×C8—541no—
121C2×C8—541no—
122C4⋊C4—91no—
123C4⋊C4—181no—
124C4⋊C4—181no—
125C4⋊C4C3⋊#15711yescharacteristic
126C42—91no—
127C22×C3—22no—
128C22×C3—42no—
129C22×C3C22⋊#15711yescharacteristic
130C3×C4—24no—
131C3×C4—21no—
132C3×C4—44no—
133C3×C4—44no—
134C3×C4—41no—
135C3×C4—84no—
136C3×C4—181no—
137C3×C4—181no—
138C3×C4—361no—
139Dic3—122no—
140Dic3—122no—
141Dic3—124no—
142C32—22no—
143C32—42no—
144C32#454311yescharacteristic, complemented
145C2×C4—21no—
146C2×C4—91no—
147C2×C4—91no—
148C2×C4—181no—
149C2×C4C2×C3⋊#15711yescharacteristic
150C8—542no—
151C8—542no—
152C2×C3—22no—
153C2×C3—22no—
154C2×C3—22no—
155C2×C3—42no—
156C2×C3—42no—
157C2×C3—42no—
158C2×C3#6378011yescharacteristic
159C2×C3#6378311yescharacteristic
160C2×C3#6378511yescharacteristic
161C22#8899411yescharacteristic
162C4—21no—
163C4—41no—
164C4—181no—
165C4—181no—
166C4—361no—
167C3—22no—
168C3—42no—
169C3#1059015911yescharacteristic, complemented
170C2#1172310511yescharacteristic
171C2#1172310811yescharacteristic
172C2#1172311011yescharacteristic
subgroupG/H
centerC22#88994
commutatorC2×C33×C4C2×C4
FrattiniC2×C4C2×C3⋊#157
FittingC33×C4⋊C4C4
socle, prime radicalC22×C33C22⋊C4
cosocleC3×C4×C3⋊Dic3C22
hypercenterC4⋊C4C3⋊#157
hypocenterC33#330
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
96. C33#330semidirect
144. C32#4543semidirect
169. C3#10590159semidirect

Maximal subgroups

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

subgroupindex
C3×#640992
#117231022
#105901593
#45439