Finite Group Databasebeta

← #13137054

Subgroups — #13137054

Order 1728.

364 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 89 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
1#11721712C211yesmaximal, characteristic
2C9×#870C211yesmaximal, characteristic, complemented
3C3×C4×C9⋊C8C211yesmaximal, characteristic, complemented
4#11720269C212yesmaximal, complemented
5#11720267C212yesmaximal
6#10590293—31nomaximal
7#10588749—31nomaximal
8C2×C4⋊(C3⋊D9)C2211yescharacteristic
9C2×#6226C2211yescharacteristic
10C3×C42×C9C2211yescharacteristic
11#88536—22no—
12C9×C4⋊Dic3C2212yes—
13C9×Dic3⋊C4—22no—
14C2×C3×C9⋊C8C2212yes—
15#64084—31no—
16C3×#870D311yescharacteristic
17C3×C4×C3⋊C8—31no—
18#63567—32no—
19#63564—32no—
20C9×#125—31no—
21#63437—31no—
22#63437—61no—
23#63369—32no—
24#63368—32no—
25C4×C9⋊C8D311yescharacteristic, complemented
26C22×C3⋊D9—21no—
27C2×C3×C4×C9D412yes—
28C2×C3×C4×C9C2311yescharacteristic
29C2×C3⋊Dic9—21no—
30C4⋊(C3⋊D9)—22no—
31#6226—22no—
32C2×C9×Dic3—22no—
33C3×C9⋊C8—22no—
34#5068—91no—
35#4697—91no—
36C2×C4⋊(C3⋊D3)—31no—
37C2×#634—31no—
38C32×C42C2×D311yescharacteristic
39#3752—62no—
40C3×C4⋊Dic3C2×D312yes—
41C3×Dic3⋊C4—22no—
42C2×C3×C3⋊C8—32no—
43C2×D36—31no—
44C2×D36—61no—
45C2×Dic18—31no—
46C2×Dic18—61no—
47C9×C4⋊C4—32no—
48C9×C4⋊C4—62no—
49C42×C9—21no—
50C42×C9C2×D311yescharacteristic
51(C2×D9)⋊C4—64no—
52(C2×D9)⋊C4—62no—
53C2×C9⋊C8C2×D312yes—
54C22×C3×C9C2×D411yescharacteristic
55C2×C3⋊D9—41no—
56C2×C3⋊D9—42no—
57C3×C4×C9—22no—
58C3×C4×C9#4112yes—
59C3⋊Dic9—41no—
60C9×Dic3—42no—
61C3×#125—31no—
62#875—91no—
63#875—91no—
64#875—181no—
65#870D911yescharacteristic, complemented
66C3×C4×C8—91no—
67#821—92no—
68#816—92no—
69#810—92no—
70#809—92no—
71C4×C3⋊C8—31no—
72C22×C3⋊D3—61no—
73C2×C32×C4#6712yes—
74C2×C32×C4C22×D311yescharacteristic
75C2×C3⋊Dic3—61no—
76C4⋊(C3⋊D3)—62no—
77#634—62no—
78C2×C3×Dic3—22no—
79C22×D9—62no—
80C22×D9—61no—
81C3×C3⋊C8—62no—
82C2×C4×C9—22no—
83C2×C4×C9—21no—
84C2×C4×C9—62no—
85C2×C4×C9D1212yes—
86C2×C4×C9C22×D311yescharacteristic
87C2×Dic9—62no—
88C2×Dic9—61no—
89D36—62no—
90D36—122no—
91Dic18—62no—
92Dic18—122no—
93C9⋊C8—22no—
94#488—271noSylow
95C2×C3×C9#12411yescharacteristic
96C2×C3×C9#13512yes—
97C3⋊D9—81no—
98C2×D12—91no—
99C2×D12—91no—
100C2×D12—181no—
101C2×Dic6—91no—
102C2×Dic6—91no—
103C2×Dic6—181no—
104C2×C3×C8—92no—
105C3×C4⋊C4—32no—
106C3×C4⋊C4—62no—
107C3×C42—21no—
108C3×C42C2×D911yescharacteristic
109C3×C42D3211yescharacteristic
110(C2×D3)⋊C4—184no—
111(C2×D3)⋊C4—182no—
112(C2×D3)⋊C4—182no—
113C4⋊Dic3C2×D912yes—
114Dic3⋊C4—22no—
115C2×C3⋊C8—32no—
116C22×C32C2×#6711yescharacteristic
117C2×C3⋊D3—121no—
118C2×C3⋊D3—122no—
119C32×C4—22no—
120C32×C4#23912yes—
121C3⋊Dic3—121no—
122C3×Dic3—42no—
123C22×C9—21no—
124C22×C9C2×D1211yescharacteristic
125C2×D9—122no—
126C2×D9—124no—
127C2×D9—121no—
128C2×D9—122no—
129C4×C9—22no—
130C4×C9—22no—
131C4×C9—42no—
132C4×C9—122no—
133C4×C9#23512yes—
134Dic9—122no—
135Dic9—121no—
136#125—31no—
137#124—271no—
138#103—272no—
139#102—272no—
140C4×C8—91no—
141C3×C9#48811yescharacteristic, complemented, Sylow
142C22×D3—182no—
143C22×D3—181no—
144C22×D3—181no—
145C2×C3×C4—22no—
146C2×C3×C4—21no—
147C2×C3×C4—62no—
148C2×C3×C4#61112yes—
149C2×C3×C4C22×D911yescharacteristic
150C2×C3×C4#62612yes—
151C2×C3×C4C2×D3211yescharacteristic
152C2×Dic3—22no—
153C2×Dic3—182no—
154C2×Dic3—181no—
155C2×Dic3—181no—
156D12—182no—
157D12—182no—
158D12—362no—
159Dic6—182no—
160Dic6—182no—
161Dic6—362no—
162C3×C8—182no—
163C3⋊C8—62no—
164C2×C32#94711yescharacteristic
165C2×C32#95012yes—
166C3⋊D3—241no—
167C2×C9—22no—
168C2×C9—21no—
169C2×C9#87511yescharacteristic
170C2×C9#90412yes—
171D9—242no—
172D9—241no—
173C2×Q8—271no—
174C2×D4—271no—
175C2×C8—92no—
176C4⋊C4—32no—
177C4⋊C4—62no—
178C22⋊C4—542no—
179C42D3×D911yescharacteristic
180C22×C3—21no—
181C22×C3C2×#61111yescharacteristic
182C22×C3C2×#62611yescharacteristic
183C2×D3—362no—
184C2×D3—364no—
185C2×D3—361no—
186C2×D3—362no—
187C2×D3—362no—
188C2×D3—361no—
189C3×C4—22no—
190C3×C4—22no—
191C3×C4—22no—
192C3×C4—42no—
193C3×C4—122no—
194C3×C4#370112yes—
195C3×C4#379912yes—
196Dic3—42no—
197Dic3—362no—
198Dic3—361no—
199Dic3—361no—
200C32#506811yescharacteristic
201C9—21no—
202C9#469711yescharacteristic, complemented
203C23—541no—
204Q8—542no—
205D4—542no—
206C2×C4—62no—
207C2×C4—541no—
208C2×C4#619512yes—
209C2×C4C2×D3×D911yescharacteristic
210C8—182no—
211C2×C3—22no—
212C2×C3—21no—
213C2×C3#6350911yescharacteristic
214C2×C3#6351312yes—
215C2×C3#6382912yes—
216C2×C3#6386411yescharacteristic
217D3—722no—
218D3—721no—
219D3—721no—
220C22—1082no—
221C22—1081no—
222C22C2×#619511yescharacteristic
223C4—22no—
224C4—122no—
225C4—1081no—
226C4#8863512yes—
227C3—21no—
228C3#1058874911yescharacteristic, complemented
229C3#1059029311yescharacteristic
230C2—2161no—
231C2#1172088912yes—
232C2#1172095311yescharacteristic
subgroupG/H
centerC22C2×#6195
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C42×C9C22
socleC22×C32C2×#67
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC42D3×D9
hypocenterC3×C9#488
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C9×#870C2semidirect
3. C3×C4×C9⋊C8C2semidirect
4. #11720269C2semidirect
25. C4×C9⋊C8D3semidirect
65. #870D9semidirect
141. C3×C9#488semidirect
202. C9#4697semidirect
228. C3#10588749semidirect

Maximal subgroups

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

subgroupindex
#117217122
C9×#8702
C3×C4×C9⋊C82
#117202692
#117202672
#105902933
#105887493