Finite Group Databasebeta

← #13157219

Subgroups — #13157219

Order 1728.

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

normal subgroups only

This group has 568 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×#3752C211yesmaximal, characteristic
2#11723112C212yesmaximal, complemented
3#10590204—31nomaximal
4C23×C3×C3⋊D3C411yescharacteristic
5C22×C33×C4C411yescharacteristic
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C2×C3⋊#622—22no—
8C3×#3752—22no—
9C2×#3752D311yescharacteristic
10C2×C3×(C2×D3)⋊C4—62no—
11C22⋊#622—32no—
12C23×C33C2×C411yescharacteristic
13C22×C3×C3⋊D3—21no—
14C22×C3×C3⋊D3—21no—
15C22×C3×C3⋊D3—41no—
16C22×C3×C3⋊D3—41no—
17C22×C3×C3⋊D3C812yescomplemented
18C2×C33×C4—21no—
19C2×C33×C4—41no—
20C2×C3×C3⋊Dic3—41no—
21C2×C3×C3⋊Dic3D412yes—
22C3⋊#622—42no—
23#4525—91nomaximal
24C23×C3⋊D3Dic311yescharacteristic
25C23×C3×D3—62no—
26C22×C32×C4—22no—
27C22×C32×C4—42no—
28C22×C32×C4Dic311yescharacteristic
29C22×C3⋊Dic3C2×D311yescharacteristic
30C22×C3×Dic3—62no—
31C2×#622—62no—
32#3752—22no—
33C3×(C2×D3)⋊C4—68no—
34C22×C33—21no—
35C22×C33—21no—
36C22×C33C22⋊C411yescharacteristic
37C22×C33C2×C811yescharacteristic
38C22×C33M1611yescharacteristic
39C2×C3×C3⋊D3—22no—
40C2×C3×C3⋊D3—42no—
41C2×C3×C3⋊D3—41no—
42C2×C3×C3⋊D3—41no—
43C2×C3×C3⋊D3—42no—
44C33×C4—41no—
45C3×C3⋊Dic3—22no—
46C2×C3×C22⋊C4—91no—
47C2×(C2×D3)⋊C4—62no—
48#830—92no—
49C23×C32—22no—
50C23×C32—42no—
51C23×C32C2×Dic311yescharacteristic
52C22×C3⋊D3—21no—
53C22×C3⋊D3—21no—
54C22×C3⋊D3—41no—
55C22×C3⋊D3—41no—
56C22×C3⋊D3C3⋊C812yescomplemented
57C22×C3×D3—64no—
58C22×C3×D3—64no—
59C22×C3×D3—64no—
60C22×C3×D3—124no—
61C22×C3×D3—124no—
62C2×C32×C4—21no—
63C2×C32×C4—24no—
64C2×C32×C4—44no—
65C2×C32×C4—44no—
66C2×C32×C4—48no—
67C2×C32×C4—41no—
68C2×C3⋊Dic3—41no—
69C2×C3⋊Dic3#6712yes—
70C2×C3×Dic3—64no—
71C2×C3×Dic3—124no—
72#622—122no—
73#323—271noSylow
74C2×C33—21no—
75C2×C33—21no—
76C2×C33C22⋊C811yescharacteristic
77C2×C33#9911yescharacteristic
78C2×C33#10011yescharacteristic
79C3×C3⋊D3—42no—
80C24×C3—91no—
81C23×D3—62no—
82C22×C3×C4—22no—
83C22×C3×C4—42no—
84C22×C3×C4—91no—
85C22×C3×C4#15711yescharacteristic
86C22×Dic3—62no—
87C3×C22⋊C4—182no—
88(C2×D3)⋊C4—68no—
89C2×C3⋊C8—182no—
90C22×C32—21no—
91C22×C32—21no—
92C22×C32—24no—
93C22×C32—24no—
94C22×C32—22no—
95C22×C32—22no—
96C22×C32—22no—
97C22×C32—44no—
98C22×C32—44no—
99C22×C32—42no—
100C22×C32—42no—
101C22×C32—42no—
102C22×C32C2×C3⋊C811yescharacteristic
103C22×C32#20811yescharacteristic
104C22×C32#21711yescharacteristic
105C2×C3⋊D3—22no—
106C2×C3⋊D3—42no—
107C2×C3⋊D3—41no—
108C2×C3⋊D3—41no—
109C2×C3⋊D3—42no—
110C2×C3×D3—68no—
111C2×C3×D3—128no—
112C2×C3×D3—128no—
113C2×C3×D3—124no—
114C2×C3×D3—124no—
115C32×C4—44no—
116C32×C4—48no—
117C32×C4—41no—
118C3⋊Dic3—22no—
119C3×Dic3—124no—
120C2×C22⋊C4—91no—
121C22⋊C8—272no—
122C33#32311yescharacteristic, complemented, Sylow
123C23×C3—22no—
124C23×C3—42no—
125C23×C3—92no—
126C23×C3—181no—
127C23×C3—181no—
128C23×C3—361no—
129C23×C3—361no—
130C23×C3C2×#15711yescharacteristic
131C22×D3—64no—
132C22×D3—64no—
133C22×D3—64no—
134C22×D3—124no—
135C22×D3—124no—
136C2×C3×C4—24no—
137C2×C3×C4—21no—
138C2×C3×C4—44no—
139C2×C3×C4—44no—
140C2×C3×C4—48no—
141C2×C3×C4—41no—
142C2×C3×C4—92no—
143C2×C3×C4—361no—
144C2×Dic3—64no—
145C2×Dic3—124no—
146C3⋊C8—362no—
147C2×C32—21no—
148C2×C32—21no—
149C2×C32—24no—
150C2×C32—24no—
151C2×C32—22no—
152C2×C32—22no—
153C2×C32—22no—
154C2×C32—44no—
155C2×C32—44no—
156C2×C32—42no—
157C2×C32—42no—
158C2×C32—42no—
159C2×C32#83011yescharacteristic
160C2×C32#83311yescharacteristic
161C2×C32#83411yescharacteristic
162C3⋊D3—42no—
163C3×D3—128no—
164C24—91no—
165C22×C4—91no—
166C22×C4C3⋊#15711yescharacteristic
167C2×C8—542no—
168C22⋊C4—182no—
169C22×C3—24no—
170C22×C3—24no—
171C22×C3—21no—
172C22×C3—21no—
173C22×C3—22no—
174C22×C3—22no—
175C22×C3—22no—
176C22×C3—44no—
177C22×C3—44no—
178C22×C3—42no—
179C22×C3—42no—
180C22×C3—42no—
181C22×C3—182no—
182C22×C3—362no—
183C22×C3—362no—
184C22×C3—361no—
185C22×C3—361no—
186C22×C3#378711yescharacteristic
187C22×C3#378811yescharacteristic
188C22×C3C22⋊#15711yescharacteristic
189C2×D3—68no—
190C2×D3—128no—
191C2×D3—128no—
192C2×D3—124no—
193C2×D3—124no—
194C3×C4—44no—
195C3×C4—48no—
196C3×C4—41no—
197C3×C4—182no—
198Dic3—124no—
199C32—22no—
200C32—42no—
201C32#452511yescharacteristic, complemented
202C23—92no—
203C23—181no—
204C23—181no—
205C23—361no—
206C23—361no—
207C23C2×C3⋊#15711yescharacteristic
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—92no—
211C2×C4—361no—
212C8—1082no—
213C2×C3—24no—
214C2×C3—24no—
215C2×C3—21no—
216C2×C3—21no—
217C2×C3—22no—
218C2×C3—22no—
219C2×C3—22no—
220C2×C3—44no—
221C2×C3—44no—
222C2×C3—42no—
223C2×C3—42no—
224C2×C3—42no—
225C2×C3—362no—
226C2×C3#6377411yescharacteristic
227C2×C3#6377811yescharacteristic
228C2×C3#6378811yescharacteristic
229D3—128no—
230C22—21no—
231C22—21no—
232C22—182no—
233C22—362no—
234C22—362no—
235C22—361no—
236C22—361no—
237C22#8898811yescharacteristic
238C22#8898911yescharacteristic
239C22#8899411yescharacteristic
240C4—41no—
241C4—182no—
242C3—22no—
243C3—42no—
244C3#1059020411yescharacteristic, complemented
245C2—21no—
246C2—21no—
247C2—362no—
248C2#1172309911yescharacteristic
249C2#1172310311yescharacteristic
250C2#1172311311yescharacteristic
subgroupG/H
centerC22#88994
commutatorC22×C33C2×C8
FrattiniC23C2×C3⋊#157
FittingC22×C33×C4C4
socleC22×C33C22⋊C4
prime radicalC23×C3×C3⋊D3C4
cosocleC22×C3×C3⋊Dic3C22
hypercenterC22×C4C3⋊#157
hypocenterC33#323
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
2. #11723112C2semidirect
17. C22×C3×C3⋊D3C8semidirect
56. C22×C3⋊D3C3⋊C8semidirect
122. C33#323semidirect
201. C32#4525semidirect
244. C3#10590204semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37522
#117231122
#105902043
#45259