Finite Group Databasebeta

← #13144089

Subgroups — #13144089

Order 1728.

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

normal subgroups only

This group has 480 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
1#11721744C211yesmaximal, characteristic
2#11721743C211yesmaximal, characteristic
3C3×C9×#106C211yesmaximal, characteristic
4#11720539C211yesmaximal, characteristic
5#11720537C211yesmaximal, characteristic
6#11720527C211yesmaximal, characteristic
7#11720526C211yesmaximal, characteristic
8#10592553—31nomaximal
9#10588534—33nomaximal
10C2×#6226—21no—
11C2×#6226C2211yescharacteristic
12C2×C3×C8×C9C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16#88534—21no—
17#88534—21no—
18C4×C3⋊Dic9—21no—
19C4×C3⋊Dic9C2211yescharacteristic
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21#64097—31no—
22#64096—31no—
23C32×#106D311yescharacteristic
24#63643—31no—
25#63641—31no—
26#63637—31no—
27#63635—31no—
28#63450—33no—
29#63449—33no—
30C9×#106D313yes—
31#63376—33no—
32Dic9⋊C8—33no—
33#63368—33no—
34#63367—33no—
35C2×C3×C4×C9—21no—
36C2×C3×C4×C9—21no—
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9—21no—
39C2×C3⋊Dic9—21no—
40C2×C3⋊Dic9D411yescharacteristic
41C2×C3⋊Dic9D411yescharacteristic
42#6226—21no—
43#6226—42no—
44#6226Q812yescomplemented
45C3×C8×C9—21no—
46C3⋊(C9⋊C8)—21no—
47#4854—93no—
48#4854—91no—
49C2×#634—31no—
50C2×#634—61no—
51C2×C32×C8C2×D311yescharacteristic
52C32×C4⋊C4C2×D311yescharacteristic
53C32×C4⋊C4C2×D311yescharacteristic
54C4⋊(C3⋊Dic3)—31no—
55#3750—61no—
56#3750—61no—
57C4×C3⋊Dic3—31no—
58C4×C3⋊Dic3—61no—
59C2×C3⋊(C3⋊C8)—31no—
60C2×Dic18—33no—
61C2×Dic18—63no—
62C2×C8×C9C2×D313yes—
63C9×C4⋊C4C2×D313yes—
64C9×C4⋊C4C2×D313yes—
65C4⋊Dic9—33no—
66Dic9⋊C4—63no—
67Dic9⋊C4—63no—
68C4×Dic9—33no—
69C4×Dic9—63no—
70C2×C9⋊C8—33no—
71C22×C3×C9C2×D411yescharacteristic
72C3×C4×C9—41no—
73C3×C4×C9—41no—
74C3×C4×C9C2×Q811yescharacteristic
75C3×C4×C9#4111yescharacteristic
76C3⋊Dic9—22no—
77C3⋊Dic9—21no—
78C3⋊Dic9—41no—
79C3⋊Dic9SD1612yes—
80#888—93no—
81#888—91no—
82#887—93no—
83#887—91no—
84C3×#106D913yes—
85C3×#106C3⋊D311yescharacteristic
86#816—93no—
87#816—91no—
88Dic3⋊C8—93no—
89Dic3⋊C8—91no—
90#810—93no—
91#810—91no—
92#808—93no—
93#808—91no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4C22×D311yescharacteristic
97C2×C3⋊Dic3—31no—
98C2×C3⋊Dic3—31no—
99C2×C3⋊Dic3—61no—
100C2×C3⋊Dic3—61no—
101#634—32no—
102#634—61no—
103#634—122no—
104C32×C8—21no—
105C3⋊(C3⋊C8)—61no—
106C2×C4×C9—23no—
107C2×C4×C9—23no—
108C2×C4×C9C22×D313yes—
109C2×Dic9—33no—
110C2×Dic9—33no—
111C2×Dic9—63no—
112C2×Dic9—63no—
113Dic18—36no—
114Dic18—63no—
115Dic18—126no—
116C8×C9—23no—
117C9⋊C8—63no—
118#475—271noSylow
119C2×C3×C9C22⋊Q811yescharacteristic
120C2×C3×C9C2×SD1611yescharacteristic
121C2×C3×C9#13711yescharacteristic
122C2×Dic6—93no—
123C2×Dic6—91no—
124C2×Dic6—181no—
125C2×Dic6—183no—
126C2×C3×C8C2×D913yes—
127C2×C3×C8C2×C3⋊D311yescharacteristic
128C3×C4⋊C4C2×D913yes—
129C3×C4⋊C4C2×D913yes—
130C3×C4⋊C4C2×C3⋊D311yescharacteristic
131C3×C4⋊C4C2×C3⋊D311yescharacteristic
132C4⋊Dic3—93no—
133C4⋊Dic3—91no—
134Dic3⋊C4—181no—
135Dic3⋊C4—181no—
136Dic3⋊C4—183no—
137Dic3⋊C4—183no—
138C4×Dic3—93no—
139C4×Dic3—91no—
140C4×Dic3—181no—
141C4×Dic3—183no—
142C2×C3⋊C8—93no—
143C2×C3⋊C8—91no—
144C22×C32D3×D411yescharacteristic
145C32×C4—41no—
146C32×C4—41no—
147C32×C4#23511yescharacteristic
148C32×C4D3×Q811yescharacteristic
149C3⋊Dic3—32no—
150C3⋊Dic3—62no—
151C3⋊Dic3—61no—
152C3⋊Dic3—121no—
153C22×C9D3×D413yes—
154C4×C9—43no—
155C4×C9—43no—
156C4×C9#23513yes—
157C4×C9D3×Q813yes—
158Dic9—36no—
159Dic9—66no—
160Dic9—63no—
161Dic9—123no—
162C4⋊Q8—271no—
163C4×Q8—271no—
164#106—271no—
165#106C3⋊D911yescharacteristic
166C4⋊C8—271no—
167#103—271no—
168#103—271no—
169C3×C9#47511yescharacteristic, complemented, Sylow
170C2×C3×C4—23no—
171C2×C3×C4—23no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4C22×D913yes—
175C2×C3×C4C22×C3⋊D311yescharacteristic
176C2×Dic3—93no—
177C2×Dic3—91no—
178C2×Dic3—93no—
179C2×Dic3—91no—
180C2×Dic3—183no—
181C2×Dic3—181no—
182C2×Dic3—183no—
183C2×Dic3—181no—
184Dic6—96no—
185Dic6—92no—
186Dic6—183no—
187Dic6—181no—
188Dic6—362no—
189Dic6—366no—
190C3×C8—23no—
191C3×C8—21no—
192C3⋊C8—183no—
193C3⋊C8—181no—
194C2×C32#89611yescharacteristic
195C2×C32D3×SD1611yescharacteristic
196C2×C32#91511yescharacteristic
197C2×C9#89613yes—
198C2×C9D3×SD1613yes—
199C2×C9#91513yes—
200C2×Q8—271no—
201C2×Q8—541no—
202C2×C8—271no—
203C2×C8C2×C3⋊D911yescharacteristic
204C4⋊C4—271no—
205C4⋊C4—541no—
206C4⋊C4—541no—
207C4⋊C4C2×C3⋊D911yescharacteristic
208C4⋊C4C2×C3⋊D911yescharacteristic
209C42—271no—
210C42—541no—
211C22×C3D4×D913yes—
212C22×C3D4×C3⋊D311yescharacteristic
213C3×C4—41no—
214C3×C4—41no—
215C3×C4—43no—
216C3×C4—43no—
217C3×C4#369713yes—
218C3×C4Q8×D913yes—
219C3×C4#382811yescharacteristic
220C3×C4Q8×C3⋊D311yescharacteristic
221Dic3—92no—
222Dic3—96no—
223Dic3—182no—
224Dic3—181no—
225Dic3—186no—
226Dic3—183no—
227Dic3—361no—
228Dic3—363no—
229C32#485411yescharacteristic
230C9#485413yescomplemented
231Q8—272no—
232Q8—541no—
233Q8—1082no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—271no—
237C2×C4—271no—
238C2×C4—541no—
239C2×C4—541no—
240C2×C4C22×C3⋊D911yescharacteristic
241C8—21no—
242C8—541no—
243C2×C3#6345813yes—
244C2×C3SD16×D913yes—
245C2×C3#6347713yes—
246C2×C3#6410511yescharacteristic
247C2×C3SD16×C3⋊D311yescharacteristic
248C2×C3#6412411yescharacteristic
249C22D4×C3⋊D911yescharacteristic
250C4—41no—
251C4—41no—
252C4—272no—
253C4—542no—
254C4—541no—
255C4—1081no—
256C4#8873611yescharacteristic
257C4Q8×C3⋊D911yescharacteristic
258C3#1058853413yescomplemented
259C3#1059255311yescharacteristic
260C2#1172175211yescharacteristic
261C2SD16×C3⋊D911yescharacteristic
262C2#1172179711yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#106C2
socle, prime radicalC22×C32D3×D4
hypercenter#106C3⋊D9
hypocenterC3×C9#475
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
44. #6226Q8semidirect
169. C3×C9#475semidirect
230. C9#4854semidirect
258. C3#10588534semidirect

Maximal subgroups

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

subgroupindex
#117217442
#117217432
C3×C9×#1062
#117205392
#117205372
#117205272
#117205262
#105925533
#105885343