Finite Group Databasebeta

← #13143874

Subgroups — #13143874

Order 1728.

760 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 135 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#11722661C211yesmaximal, characteristic
2#11721836C211yesmaximal, characteristic, complemented
3#11721784C212yesmaximal, complemented
4C3×C9×#100C211yesmaximal, characteristic, complemented
5#11720558C212yesmaximal, complemented
6#10592445—31nomaximal
7#10588426—33nomaximal
8C2×C3×D4×C9C2211yescharacteristic
9C2×#6233—22no—
10#88742—22no—
11D4×C3⋊D9—22no—
12#88736C2212yes—
13C2×#6226C2211yescharacteristic, complemented
14C3×C9×M16C2212yescomplemented
15#88551—22no—
16C4×C3⋊Dic9C2211yescharacteristic, complemented
17#88524—22no—
18#88521—22no—
19#64361—31no—
20#64145—31no—
21#64118—32no—
22#63710—33no—
23C32×#100D311yescharacteristic
24#63652—32no—
25#63498—33no—
26#63471—36no—
27C9×#100D313yescomplemented
28#63384—36no—
29C23×C3×C9D412yes—
30C22×C3⋊D9—22no—
31C3×D4×C9—22no—
32C2×C3×C4×C9C2311yescharacteristic
33#6233—24no—
34#6233—22no—
35C2×C3⋊Dic9—22no—
36C2×C3⋊Dic9—21no—
37C2×C3⋊Dic9—21no—
38C4⋊(C3⋊D9)D412yes—
39C4×C3⋊D9—22no—
40#6226—22no—
41#6226D412yes—
42C3×C8×C9—22no—
43#4746—93no—
44#4746—91no—
45C2×C32×D4C2×D311yescharacteristic
46C2×#638—62no—
47#3830—62no—
48D4×C3⋊D3—62no—
49#3828—32no—
50C2×#634—31no—
51C32×M16C2×D312yes—
52#3757—62no—
53C4×C3⋊Dic3—31no—
54#3746—62no—
55#3744—62no—
56C2×D4×C9C2×D313yes—
57C2×#611—66no—
58#3699—66no—
59D4×D9—66no—
60#3697—36no—
61C2×Dic18—33no—
62C9×M16C2×D316yescomplemented
63#3676—66no—
64C4×Dic9—33no—
65#3664—66no—
66Dic36—66no—
67C22×C3×C9—22no—
68C22×C3×C9—42no—
69C22×C3×C9C2×D411yescharacteristic
70C2×C3⋊D9—22no—
71C2×C3⋊D9—82no—
72C3×C4×C9C2×D412yes—
73C3⋊Dic9—22no—
74C3⋊Dic9—41no—
75C3⋊Dic9—41no—
76#1004—93no—
77#1004—91no—
78#936—93no—
79#936—91no—
80#909—96no—
81#909—92no—
82C3×#100D913yescomplemented
83C3×#100C3⋊D311yescharacteristic
84#825—96no—
85#825—92no—
86C23×C32D1212yes—
87C22×C3⋊D3—62no—
88C32×D4—22no—
89C2×C32×C4C22×D311yescharacteristic
90#638—64no—
91#638—62no—
92C2×C3⋊Dic3—62no—
93C2×C3⋊Dic3—61no—
94C2×C3⋊Dic3—61no—
95C4⋊(C3⋊D3)—32no—
96C4×C3⋊D3—62no—
97#634—32no—
98#634—62no—
99C23×C9D1216yes—
100C22×D9—66no—
101C32×C8—22no—
102D4×C9—26no—
103C2×C4×C9C22×D313yes—
104#611—612no—
105#611—66no—
106C2×Dic9—66no—
107C2×Dic9—63no—
108C2×Dic9—63no—
109D36—36no—
110C4×D9—66no—
111Dic18—36no—
112Dic18—66no—
113C8×C9—26no—
114#455—271noSylow
115C2×C3×C9—21no—
116C2×C3×C9—42no—
117C2×C3×C9#12011yescharacteristic
118C3⋊D9—42no—
119C2×C3×D4C2×D913yes—
120C2×C3×D4C2×C3⋊D311yescharacteristic
121C2×#67—186no—
122C2×#67—182no—
123#237—182no—
124#237—186no—
125D3×D4—182no—
126D3×D4—186no—
127#235—96no—
128#235—92no—
129C2×Dic6—93no—
130C2×Dic6—91no—
131C3×M16C2×D916yescomplemented
132C3×M16C2×C3⋊D312yes—
133#217—186no—
134#217—182no—
135C4×Dic3—93no—
136C4×Dic3—91no—
137Dic12—182no—
138Dic12—186no—
139#204—182no—
140#204—186no—
141C22×C32—22no—
142C22×C32—42no—
143C22×C32C2×D1211yescharacteristic
144C2×C3⋊D3—62no—
145C2×C3⋊D3—242no—
146C32×C4D3×D412yes—
147C3⋊Dic3—62no—
148C3⋊Dic3—121no—
149C3⋊Dic3—121no—
150C22×C9—26no—
151C22×C9—46no—
152C22×C9C2×D1213yes—
153C2×D9—66no—
154C2×D9—246no—
155C4×C9D3×D416yes—
156Dic9—66no—
157Dic9—123no—
158Dic9—123no—
159H(5,𝔽2)—271no—
160#137—272no—
161#124—271no—
162#104—272no—
163#100C3⋊D911yescharacteristic, complemented
164C3×C9#45511yescharacteristic, complemented, Sylow
165C23×C3D3616yes—
166C23×C3C4⋊(C3⋊D3)12yes—
167C22×D3—182no—
168C22×D3—186no—
169C3×D4—26no—
170C3×D4—22no—
171C2×C3×C4C22×D913yes—
172C2×C3×C4C22×C3⋊D311yescharacteristic
173#67—184no—
174#67—1812no—
175#67—186no—
176#67—182no—
177C2×Dic3—182no—
178C2×Dic3—186no—
179C2×Dic3—183no—
180C2×Dic3—183no—
181C2×Dic3—181no—
182C2×Dic3—181no—
183D12—96no—
184D12—92no—
185C4×D3—186no—
186C4×D3—182no—
187Dic6—96no—
188Dic6—92no—
189Dic6—186no—
190Dic6—182no—
191C3×C8—26no—
192C3×C8—22no—
193C2×C32—21no—
194C2×C32—42no—
195C2×C32#88211yescharacteristic
196C3⋊D3—122no—
197C2×C9—23no—
198C2×C9—46no—
199C2×C9#88213yes—
200D9—126no—
201#41—272no—
202#41—542no—
203C2×Q8—271no—
204C2×D4—542no—
205C2×D4—542no—
206C2×D4C2×C3⋊D911yescharacteristic
207Q16—542no—
208SD16—542no—
209M16C2×C3⋊D912yescomplemented
210C22⋊C4—542no—
211C42—271no—
212C22×C3—22no—
213C22×C3—26no—
214C22×C3—46no—
215C22×C3—42no—
216C22×C3C2×D3613yes—
217C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
218C2×D3—182no—
219C2×D3—186no—
220C2×D3—722no—
221C2×D3—726no—
222C3×C4D4×D916yes—
223C3×C4D4×C3⋊D312yes—
224Dic3—182no—
225Dic3—186no—
226Dic3—361no—
227Dic3—363no—
228Dic3—361no—
229Dic3—363no—
230C32#474611yescharacteristic
231C9#474613yescomplemented
232C23—542no—
233C23C4⋊(C3⋊D9)12yes—
234Q8—272no—
235Q8—542no—
236D4—22no—
237D4—272no—
238D4—544no—
239D4—542no—
240C2×C4—542no—
241C2×C4—542no—
242C2×C4—541no—
243C2×C4—541no—
244C2×C4C22×C3⋊D911yescharacteristic
245C8—22no—
246C2×C3—23no—
247C2×C3—21no—
248C2×C3—46no—
249C2×C3—42no—
250C2×C3#6344413yes—
251C2×C3#6409111yescharacteristic
252D3—366no—
253D3—362no—
254C22—22no—
255C22—42no—
256C22—542no—
257C22—2162no—
258C22C2×C4⋊(C3⋊D9)11yescharacteristic
259C4—542no—
260C4—1081no—
261C4—1081no—
262C4D4×C3⋊D912yes—
263C3#1058842613yescomplemented
264C3#1059244511yescharacteristic
265C2—21no—
266C2—42no—
267C2—1082no—
268C2#1172172811yescharacteristic
subgroupG/H
centerC2#11721728
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#100C2
socleC2×C32#882
prime radical#11722661C2
hypercenter#100C3⋊D9
hypocenterC3×C9#455
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #100

Complements

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

NH ≅ G/Ntype
2. #11721836C2semidirect
3. #11721784C2semidirect
4. C3×C9×#100C2semidirect
5. #11720558C2semidirect
13. C2×#6226C22semidirect
14. C3×C9×M16C22semidirect
16. C4×C3⋊Dic9C22semidirect
27. C9×#100D3semidirect
62. C9×M16C2×D3semidirect
82. C3×#100D9semidirect
131. C3×M16C2×D9semidirect
163. #100C3⋊D9semidirect
164. C3×C9#455semidirect
209. M16C2×C3⋊D9semidirect
231. C9#4746semidirect
263. C3#10588426semidirect

Maximal subgroups

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

subgroupindex
#117226612
#117218362
#117217842
C3×C9×#1002
#117205582
#105924453
#105884263