Finite Group Databasebeta

← #13157217

Subgroups — #13157217

Order 1728.

448 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 53 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
1C2×C3×#3750C211yesmaximal, characteristic
2#11723112C212yesmaximal
3#10590203—31nomaximal
4C22×C33×C4C411yescharacteristic
5C22×C3×C3⋊Dic3C411yescharacteristic
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C2×C3⋊#622—22no—
8C3×#3750—22no—
9C2×#3750D311yescharacteristic
10C2×C3×Dic3⋊C4—62no—
11C22⋊#622—32no—
12C23×C33C2×C411yescharacteristic
13C2×C33×C4—21no—
14C2×C33×C4—41no—
15C2×C3×C3⋊Dic3—21no—
16C2×C3×C3⋊Dic3—21no—
17C2×C3×C3⋊Dic3—41no—
18C2×C3×C3⋊Dic3C812yescomplemented
19C2×C3×C3⋊Dic3D412yes—
20C3⋊#622—42no—
21#4528—91nomaximal
22C22×C32×C4—22no—
23C22×C32×C4—42no—
24C22×C32×C4Dic311yescharacteristic
25C22×C3⋊Dic3Dic311yescharacteristic
26C22×C3⋊Dic3C2×D311yescharacteristic
27C22×C3×Dic3—62no—
28C22×C3×Dic3—62no—
29C2×#622—62no—
30#3750—22no—
31C3×Dic3⋊C4—68no—
32C22×C33—21no—
33C22×C33—21no—
34C22×C33C22⋊C411yescharacteristic
35C22×C33C2×C811yescharacteristic
36C22×C33M1611yescharacteristic
37C33×C4—41no—
38C3×C3⋊Dic3—22no—
39C3×C3⋊Dic3—22no—
40C2×C3×C4⋊C4—91no—
41C2×Dic3⋊C4—62no—
42#830—92no—
43C23×C32—22no—
44C23×C32—42no—
45C23×C32C2×Dic311yescharacteristic
46C2×C32×C4—21no—
47C2×C32×C4—24no—
48C2×C32×C4—44no—
49C2×C32×C4—44no—
50C2×C32×C4—48no—
51C2×C32×C4—41no—
52C2×C3⋊Dic3—21no—
53C2×C3⋊Dic3—21no—
54C2×C3⋊Dic3—41no—
55C2×C3⋊Dic3C3⋊C812yescomplemented
56C2×C3⋊Dic3#6712yes—
57C2×C3×Dic3—64no—
58C2×C3×Dic3—64no—
59C2×C3×Dic3—64no—
60C2×C3×Dic3—64no—
61C2×C3×Dic3—124no—
62#622—122no—
63#324—271noSylow
64C2×C33—21no—
65C2×C33—21no—
66C2×C33C22⋊C811yescharacteristic
67C2×C33#9911yescharacteristic
68C2×C33#10111yescharacteristic
69C22×C3×C4—22no—
70C22×C3×C4—42no—
71C22×C3×C4—91no—
72C22×C3×C4—91no—
73C22×C3×C4#15711yescharacteristic
74C22×Dic3—62no—
75C22×Dic3—62no—
76C3×C4⋊C4—182no—
77Dic3⋊C4—68no—
78C2×C3⋊C8—182no—
79C22×C32—21no—
80C22×C32—21no—
81C22×C32—24no—
82C22×C32—24no—
83C22×C32—22no—
84C22×C32—22no—
85C22×C32—22no—
86C22×C32—44no—
87C22×C32—44no—
88C22×C32—42no—
89C22×C32—42no—
90C22×C32—42no—
91C22×C32C2×C3⋊C811yescharacteristic
92C22×C32#20811yescharacteristic
93C22×C32#21711yescharacteristic
94C32×C4—44no—
95C32×C4—48no—
96C32×C4—41no—
97C3⋊Dic3—22no—
98C3⋊Dic3—22no—
99C3×Dic3—68no—
100C3×Dic3—124no—
101C2×C4⋊C4—91no—
102C22⋊C8—272no—
103C33#32411yescharacteristic, complemented, Sylow
104C23×C3—22no—
105C23×C3—42no—
106C23×C3C2×#15711yescharacteristic
107C2×C3×C4—24no—
108C2×C3×C4—21no—
109C2×C3×C4—44no—
110C2×C3×C4—44no—
111C2×C3×C4—48no—
112C2×C3×C4—41no—
113C2×C3×C4—92no—
114C2×C3×C4—92no—
115C2×C3×C4—181no—
116C2×C3×C4—181no—
117C2×C3×C4—361no—
118C2×Dic3—64no—
119C2×Dic3—64no—
120C2×Dic3—64no—
121C2×Dic3—64no—
122C2×Dic3—124no—
123C3⋊C8—362no—
124C2×C32—21no—
125C2×C32—21no—
126C2×C32—24no—
127C2×C32—24no—
128C2×C32—22no—
129C2×C32—22no—
130C2×C32—22no—
131C2×C32—44no—
132C2×C32—44no—
133C2×C32—42no—
134C2×C32—42no—
135C2×C32—42no—
136C2×C32#83011yescharacteristic
137C2×C32#83411yescharacteristic
138C2×C32#83611yescharacteristic
139C22×C4—91no—
140C22×C4—91no—
141C22×C4C3⋊#15711yescharacteristic
142C2×C8—542no—
143C4⋊C4—182no—
144C22×C3—24no—
145C22×C3—24no—
146C22×C3—21no—
147C22×C3—21no—
148C22×C3—22no—
149C22×C3—22no—
150C22×C3—22no—
151C22×C3—44no—
152C22×C3—44no—
153C22×C3—42no—
154C22×C3—42no—
155C22×C3—42no—
156C22×C3#378711yescharacteristic
157C22×C3#378811yescharacteristic
158C22×C3C22⋊#15711yescharacteristic
159C3×C4—44no—
160C3×C4—48no—
161C3×C4—41no—
162C3×C4—182no—
163C3×C4—182no—
164Dic3—68no—
165Dic3—124no—
166C32—22no—
167C32—42no—
168C32#452811yescharacteristic, complemented
169C23C2×C3⋊#15711yescharacteristic
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—92no—
173C2×C4—92no—
174C2×C4—181no—
175C2×C4—181no—
176C2×C4—361no—
177C8—1082no—
178C2×C3—24no—
179C2×C3—24no—
180C2×C3—21no—
181C2×C3—21no—
182C2×C3—22no—
183C2×C3—22no—
184C2×C3—22no—
185C2×C3—44no—
186C2×C3—44no—
187C2×C3—42no—
188C2×C3—42no—
189C2×C3—42no—
190C2×C3#6377811yescharacteristic
191C2×C3#6378011yescharacteristic
192C2×C3#6378611yescharacteristic
193C22—21no—
194C22—21no—
195C22#8898811yescharacteristic
196C22#8898911yescharacteristic
197C22#8899411yescharacteristic
198C4—41no—
199C4—182no—
200C4—182no—
201C3—22no—
202C3—42no—
203C3#1059020311yescharacteristic, complemented
204C2—21no—
205C2—21no—
206C2#1172310311yescharacteristic
207C2#1172310511yescharacteristic
208C2#1172311111yescharacteristic
subgroupG/H
centerC22#88994
commutatorC22×C33C2×C8
FrattiniC23C2×C3⋊#157
FittingC22×C33×C4C4
socleC22×C33C22⋊C4
prime radicalC23×C33C2×C4
cosocleC22×C3×C3⋊Dic3C22
hypercenterC22×C4C3⋊#157
hypocenterC33#324
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
18. C2×C3×C3⋊Dic3C8semidirect
55. C2×C3⋊Dic3C3⋊C8semidirect
103. C33#324semidirect
168. C32#4528semidirect
203. C3#10590203semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37502
#117231122
#105902033
#45289