Finite Group Databasebeta

← #13157221

Subgroups — #13157221

Order 1728.

392 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
2C2×C3⋊#3708C212yesmaximal, complemented
3C22⋊#3708—31nomaximal
4C22×C33×C4C411yescharacteristic
5C2×C3×C3⋊(C3⋊C8)—22no—
6C2×C3×C3⋊(C3⋊C8)C411yescharacteristic
7C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
8C3⋊#3708—22no—
9C22×C3⋊(C3⋊C8)D311yescharacteristic
10C22×C3×C3⋊C8—62no—
11C2×#3708—32no—
12C23×C33C811yescharacteristic
13C2×C33×C4—21no—
14C2×C33×C4—21no—
15C2×C33×C4C811yescharacteristic
16C2×C33×C4C2×C411yescharacteristic
17C3×C3⋊(C3⋊C8)—21no—
18C3×C3⋊(C3⋊C8)D412yes—
19#4549—91nomaximal
20C22×C32×C4—22no—
21C22×C32×C4—42no—
22C22×C32×C4Dic311yescharacteristic
23C2×C3⋊(C3⋊C8)—22no—
24C2×C3⋊(C3⋊C8)Dic311yescharacteristic
25C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
26C2×C3×C3⋊C8—62no—
27C2×C3×C3⋊C8—62no—
28C2×C3×C3⋊C8—68no—
29#3708—62no—
30C22×C33—21no—
31C22×C33—21no—
32C22×C33C1612yescomplemented
33C22×C33C2×C811yescharacteristic
34C33×C4—21no—
35C33×C4C22⋊C411yescharacteristic
36C33×C4M1611yescharacteristic
37C22×C3×C8—91no—
38C22×C3⋊C8—62no—
39C2×C3⋊C16—92no—
40C23×C32—22no—
41C23×C32—42no—
42C23×C32C3⋊C811yescharacteristic
43C2×C32×C4—21no—
44C2×C32×C4—21no—
45C2×C32×C4—24no—
46C2×C32×C4—24no—
47C2×C32×C4—22no—
48C2×C32×C4—22no—
49C2×C32×C4—44no—
50C2×C32×C4—44no—
51C2×C32×C4—42no—
52C2×C32×C4—42no—
53C2×C32×C4C3⋊C811yescharacteristic
54C2×C32×C4C2×Dic311yescharacteristic
55C3⋊(C3⋊C8)—21no—
56C3⋊(C3⋊C8)#6712yes—
57C3×C3⋊C8—64no—
58C3×C3⋊C8—64no—
59C22⋊C16—271noSylow
60C2×C33—22no—
61C2×C33C22⋊C811yescharacteristic
62C2×C33C2×C1611yescharacteristic
63C2×C33M3211yescharacteristic
64C22×C3×C4—22no—
65C22×C3×C4—42no—
66C22×C3×C4#15711yescharacteristic
67C2×C3×C8—91no—
68C2×C3×C8—91no—
69C2×C3×C8—182no—
70C2×C3⋊C8—62no—
71C2×C3⋊C8—62no—
72C2×C3⋊C8—68no—
73C3⋊C16—182no—
74C22×C32—21no—
75C22×C32—21no—
76C22×C32—24no—
77C22×C32—24no—
78C22×C32—24no—
79C22×C32—22no—
80C22×C32—44no—
81C22×C32—44no—
82C22×C32—44no—
83C22×C32—42no—
84C22×C32C3⋊C1612yescomplemented
85C22×C32C2×C3⋊C811yescharacteristic
86C32×C4—21no—
87C32×C4—24no—
88C32×C4—22no—
89C32×C4—22no—
90C32×C4—44no—
91C32×C4—42no—
92C32×C4—42no—
93C32×C4#20811yescharacteristic
94C32×C4#21711yescharacteristic
95C22×C8—91no—
96C2×C16—272no—
97C33C22⋊C1611yescharacteristic, complemented, Sylow
98C23×C3—22no—
99C23×C3—42no—
100C23×C3#62211yescharacteristic
101C2×C3×C4—24no—
102C2×C3×C4—24no—
103C2×C3×C4—21no—
104C2×C3×C4—21no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—44no—
108C2×C3×C4—44no—
109C2×C3×C4—42no—
110C2×C3×C4—42no—
111C2×C3×C4#62211yescharacteristic
112C2×C3×C4C2×#15711yescharacteristic
113C3×C8—92no—
114C3×C8—181no—
115C3⋊C8—64no—
116C3⋊C8—64no—
117C2×C32—22no—
118C2×C32—28no—
119C2×C32—22no—
120C2×C32—22no—
121C2×C32—22no—
122C2×C32—48no—
123C2×C32—42no—
124C2×C32—42no—
125C2×C32—42no—
126C2×C32C2×C3⋊C1611yescharacteristic
127C2×C32#81211yescharacteristic
128C2×C32#83011yescharacteristic
129C22×C4C3⋊#15711yescharacteristic
130C2×C8—91no—
131C2×C8—91no—
132C2×C8—182no—
133C16—542no—
134C22×C3—24no—
135C22×C3—24no—
136C22×C3—21no—
137C22×C3—21no—
138C22×C3—22no—
139C22×C3—24no—
140C22×C3—44no—
141C22×C3—44no—
142C22×C3—42no—
143C22×C3—44no—
144C22×C3#370812yescomplemented
145C22×C3C2×#62211yescharacteristic
146C3×C4—24no—
147C3×C4—21no—
148C3×C4—22no—
149C3×C4—22no—
150C3×C4—44no—
151C3×C4—42no—
152C3×C4—42no—
153C3×C4#379211yescharacteristic
154C3×C4C22⋊#15711yescharacteristic
155C32—22no—
156C32—42no—
157C32#454911yescharacteristic, complemented
158C23C3⋊#62211yescharacteristic
159C2×C4—21no—
160C2×C4—21no—
161C2×C4C3⋊#62211yescharacteristic
162C2×C4C2×C3⋊#15711yescharacteristic
163C8—92no—
164C8—181no—
165C2×C3—28no—
166C2×C3—22no—
167C2×C3—22no—
168C2×C3—22no—
169C2×C3—22no—
170C2×C3—48no—
171C2×C3—42no—
172C2×C3—42no—
173C2×C3—42no—
174C2×C3C2×#370811yescharacteristic
175C2×C3#6377311yescharacteristic
176C2×C3C22⋊#62211yescharacteristic
177C22—21no—
178C22—21no—
179C22C3⋊#370812yescomplemented
180C22C2×C3⋊#62211yescharacteristic
181C4—21no—
182C4#8899311yescharacteristic
183C4#8899411yescharacteristic
184C3—22no—
185C3—42no—
186C3C22⋊#370811yescharacteristic, complemented
187C2—22no—
188C2C2×C3⋊#370811yescharacteristic
189C2#1172309811yescharacteristic
190C2#1172311211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×C3⋊#157
commutatorC2×C33C2×C16
FittingC22×C33×C4C4
socleC22×C33C2×C8
prime radicalC23×C33C8
cosocleC2×C3×C3⋊(C3⋊C8)C22
hypercenterC22×C4C3⋊#157
hypocenterC33C22⋊C16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊#3708C2semidirect
32. C22×C33C16semidirect
84. C22×C32C3⋊C16semidirect
97. C33C22⋊C16semidirect
144. C22×C3#3708semidirect
157. C32#4549semidirect
179. C22C3⋊#3708semidirect
186. C3C22⋊#3708semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊(C3⋊C8)2
C2×C3⋊#37082
C22⋊#37083
#45499