Finite Group Databasebeta

← #13138434

Subgroups — #13138434

Order 1728.

378 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 137 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×D3×C8×C9C211yesmaximal, characteristic
2#11720906C212yesmaximal
3#11720540C211yesmaximal, characteristic, complemented
4C3×#63377C211yesmaximal, characteristic, complemented
5#11720279C212yesmaximal, complemented
6#10590706—31nomaximal
7#10588787—31nomaximal
8C2×C4×D3×C9C2211yescharacteristic
9C2×C3×C8×C9C2211yescharacteristic
10C4⋊(C3⋊Dic9)C2212yes—
11C3×C4⋊Dic9C2212yes—
12C2×C9×C3⋊C8C2211yescharacteristic
13D3×C8×C9C414yescomplemented
14(C2×D3)⋊Dic9—22no—
15Dic3×Dic9—22no—
16C2×C3×D3×C8D311yescharacteristic
17#63841—32no—
18#63644—31no—
19C3×#817—31no—
20#63575—32no—
21C22×C8×C9—31no—
22#63488—32no—
23#63378—32no—
24#63377—31no—
25#63377—61no—
26#63377D311yescharacteristic, complemented
27C22×D3×C9D411yescharacteristic
28C2×C3×C4×C9C2311yescharacteristic
29C2×C3⋊Dic9—22no—
30C2×C9×Dic3D411yescharacteristic
31C2×C3×Dic9—22no—
32C4×D3×C9C2×C412yes—
33C4×D3×C9Q812yes—
34C3×C8×C9C2×C412yes—
35C9×C3⋊C8C2×C412yes—
36#5106—91no—
37#4860—91no—
38C2×C3×C4×D3C2×D311yescharacteristic
39C2×C32×C8C2×D311yescharacteristic
40C4⋊(C3⋊Dic3)—32no—
41C3×C4⋊Dic3—32no—
42C2×C3×C3⋊C8C2×D311yescharacteristic
43C3×D3×C8Dic314yes—
44(C2×D3)⋊Dic3—62no—
45Dic32—62no—
46C22×C4×C9—31no—
47C2×C8×C9—21no—
48C2×C8×C9—31no—
49C2×C8×C9—34no—
50C2×C8×C9C2×D311yescharacteristic
51#3676—62no—
52C4⋊Dic9—32no—
53C4⋊Dic9—62no—
54C4⋊Dic9C2×D312yes—
55C4×Dic9—62no—
56C22×C3×C9C2×D411yescharacteristic
57C2×D3×C9—22no—
58C2×D3×C9C4⋊C412yes—
59C3×C4×C9C22×C411yescharacteristic
60C3×C4×C9C2×Q811yescharacteristic
61C3⋊Dic9—42no—
62C9×Dic3C4⋊C412yes—
63C3×Dic9—42no—
64C22×C3×C8—31no—
65#926—92no—
66C2×D3×C8D911yescharacteristic
67#892—92no—
68C3×#106—91no—
69#818—92no—
70#817—31no—
71#817—91no—
72#817—91no—
73#817—181no—
74#807—92no—
75C22×C3×D3D1211yescharacteristic
76C2×C32×C4C22×D311yescharacteristic
77C2×C3⋊Dic3—62no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3D1211yescharacteristic
80C3×C4×D3Dic612yes—
81C3×C4×D3C2×Dic312yes—
82C23×C9—31no—
83C32×C8C2×Dic312yes—
84C3×C3⋊C8C2×Dic312yes—
85C2×C4×C9—21no—
86C2×C4×C9—31no—
87C2×C4×C9—32no—
88C2×C4×C9—32no—
89C2×C4×C9C22×D311yescharacteristic
90C2×Dic9—22no—
91C2×Dic9—62no—
92C2×Dic9—122no—
93C8×C9—22no—
94C8×C9—32no—
95C8×C9C4×D312yes—
96#427—271noSylow
97C2×C3×C9C2×C4⋊C411yescharacteristic
98C2×C3×C9#13512yes—
99D3×C9—22no—
100C22×C3×C4—31no—
101C2×C4×D3C2×D911yescharacteristic
102C2×C3×C8—21no—
103C2×C3×C8—31no—
104C2×C3×C8—34no—
105C2×C3×C8C2×D911yescharacteristic
106C2×C3×C8D3211yescharacteristic
107C3×C4⋊C4—92no—
108#217—182no—
109(C2×D3)⋊C4—182no—
110C4⋊Dic3—32no—
111C4⋊Dic3—92no—
112C4⋊Dic3—92no—
113C4⋊Dic3—182no—
114C4×Dic3—182no—
115C4×Dic3—182no—
116C2×C3⋊C8C2×D911yescharacteristic
117D3×C8Dic914yescomplemented
118C22×C32C2×D1211yescharacteristic
119C2×C3×D3—22no—
120C2×C3×D3C4⋊Dic312yes—
121C32×C4C2×Dic611yescharacteristic
122C32×C4C22×Dic311yescharacteristic
123C3⋊Dic3—122no—
124C3×Dic3—122no—
125C3×Dic3C4⋊Dic312yes—
126C22×C9—21no—
127C22×C9—32no—
128C22×C9—62no—
129C22×C9D3×D411yescharacteristic
130C4×C9—21no—
131C4×C9—21no—
132C4×C9—32no—
133C4×C9C2×C4×D311yescharacteristic
134C4×C9D3×Q811yescharacteristic
135Dic9—42no—
136Dic9—122no—
137Dic9—242no—
138C22×C8—31no—
139#117—272no—
140#107—272no—
141#106—91no—
142#106—271no—
143C3×C9#42711yescharacteristic, complemented, Sylow
144C23×C3—31no—
145C22×D3D3611yescharacteristic
146C2×C3×C4—21no—
147C2×C3×C4—31no—
148C2×C3×C4—32no—
149C2×C3×C4—32no—
150C2×C3×C4—182no—
151C2×C3×C4C22×D911yescharacteristic
152C2×C3×C4C2×D3211yescharacteristic
153C2×Dic3—62no—
154C2×Dic3—182no—
155C2×Dic3—182no—
156C2×Dic3—362no—
157C2×Dic3D3611yescharacteristic
158C4×D3Dic1812yes—
159C4×D3C2×Dic912yes—
160C3×C8—22no—
161C3×C8—32no—
162C3×C8C2×Dic912yes—
163C3×C8D3×Dic312yes—
164C3⋊C8C2×Dic912yes—
165C2×C32#90412yes—
166C2×C32C2×C4⋊Dic311yescharacteristic
167C3×D3—22no—
168C2×C9—22no—
169C2×C9—21no—
170C2×C9—62no—
171C2×C9D3×C4⋊C411yescharacteristic
172C2×C9#91612yes—
173C22×C4—31no—
174C2×C8—31no—
175C2×C8—34no—
176C2×C8D3×D911yescharacteristic
177C4⋊C4—92no—
178C4⋊C4—272no—
179C22⋊C4—542no—
180C42—542no—
181C22×C3—21no—
182C22×C3—32no—
183C22×C3—62no—
184C22×C3C2×D3611yescharacteristic
185C22×C3D3×D1211yescharacteristic
186C2×D3—22no—
187C2×D3C4⋊Dic912yes—
188C3×C4—21no—
189C3×C4—21no—
190C3×C4—32no—
191C3×C4—362no—
192C3×C4C2×Dic1811yescharacteristic
193C3×C4C22×Dic911yescharacteristic
194C3×C4D3×Dic611yescharacteristic
195C3×C4C2×D3×Dic311yescharacteristic
196Dic3—122no—
197Dic3—362no—
198Dic3—362no—
199Dic3—722no—
200Dic3C4⋊Dic912yes—
201C32#510611yescharacteristic
202C9—21no—
203C9#486011yescharacteristic, complemented
204C23—31no—
205C2×C4—31no—
206C2×C4—32no—
207C2×C4—32no—
208C2×C4—182no—
209C2×C4—542no—
210C2×C4C2×D3×D911yescharacteristic
211C8—32no—
212C8D3×Dic912yes—
213C2×C3—21no—
214C2×C3—22no—
215C2×C3—62no—
216C2×C3#6346712yes—
217C2×C3C2×C4⋊Dic911yescharacteristic
218C2×C3#6380612yes—
219C2×C3D3×C4⋊Dic311yescharacteristic
220D3—22no—
221C22—32no—
222C22—62no—
223C22D3×D3611yescharacteristic
224C4—32no—
225C4—362no—
226C4—1082no—
227C4D3×Dic1811yescharacteristic
228C4C2×D3×Dic911yescharacteristic
229C3—21no—
230C3#1058878711yescharacteristic, complemented
231C3#1059070611yescharacteristic
232C2—62no—
233C2#1172081012yes—
234C2D3×C4⋊Dic911yescharacteristic
subgroupG/H
centerC22D3×D36
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
prime radicalC22×C3×D3D12
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9#427
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. #11720540C2semidirect
4. C3×#63377C2semidirect
5. #11720279C2semidirect
13. D3×C8×C9C4semidirect
26. #63377D3semidirect
117. D3×C8Dic9semidirect
143. C3×C9#427semidirect
203. C9#4860semidirect
230. C3#10588787semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C8×C92
#117209062
#117205402
C3×#633772
#117202792
#105907063
#105887873