Finite Group Databasebeta

← #13154218

Subgroups — #13154218

Order 1728.

1100 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1100 conjugacy classes of subgroups — too many to draw. Only its 251 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
1C3×C9×#125C211yesmaximal, characteristic, complemented
2#11721752C214yesmaximal, complemented
3#11721751C214yesmaximal, complemented
4#11721750C214yesmaximal, complemented
5#11721712C212yesmaximal, complemented
6#10596171—31nomaximal
7#10589802—33nomaximal
8C2×C4⋊(C3⋊D9)C2214yes—
9C2×C4⋊(C3⋊D9)C2212yes—
10C2×C4×C3⋊D9C2214yes—
11C2×#6226C2212yes—
12C3×C9×C4⋊C4C2212yes—
13C3×C9×C4⋊C4C2214yes—
14C3×C42×C9C2211yescharacteristic
15#88536C2214yes—
16#88536C2214yes—
17#88536C2214yes—
18#88534C2214yes—
19C32×#125D311yescharacteristic
20#64105—34no—
21#64104—34no—
22#64103—34no—
23#64084—32no—
24C9×#125D313yescomplemented
25#63458—312no—
26#63457—312no—
27#63456—312no—
28#63437—36no—
29C22×C3⋊D9C2314yes—
30C2×C3×C4×C9C2312yes—
31C2×C3×C4×C9C2314yes—
32C2×C3×C4×C9C2311yescharacteristic
33C2×C3⋊Dic9C2314yes—
34C4⋊(C3⋊D9)—42no—
35C4⋊(C3⋊D9)—44no—
36C4×C3⋊D9—44no—
37#6226—42no—
38#5700—93no—
39#5700—91no—
40C2×C4⋊(C3⋊D3)—32no—
41C2×C4⋊(C3⋊D3)—34no—
42C2×C4×C3⋊D3—34no—
43C2×#634—32no—
44C32×C4⋊C4C2×D312yes—
45C32×C4⋊C4C2×D314yes—
46C32×C42C2×D311yescharacteristic
47#3752—34no—
48#3752—34no—
49#3752—34no—
50#3750—34no—
51C2×D36—36no—
52C2×D36—312no—
53C2×C4×D9—312no—
54C2×Dic18—36no—
55C9×C4⋊C4C2×D316yes—
56C9×C4⋊C4C2×D3112yes—
57C42×C9C2×D313yes—
58(C2×D9)⋊C4—312no—
59(C2×D9)⋊C4—312no—
60(C2×D9)⋊C4—312no—
61Dic9⋊C4—312no—
62C22×C3×C9C2411yescharacteristic
63C2×C3⋊D9—24no—
64C2×C3⋊D9—24no—
65C2×C3⋊D9—24no—
66C3×C4×C9—22no—
67C3×C4×C9—24no—
68C3×C4×C9—21no—
69C3⋊Dic9—24no—
70C3×#125D913yescomplemented
71C3×#125C3⋊D311yescharacteristic
72#896—912no—
73#896—94no—
74#895—912no—
75#895—94no—
76#894—912no—
77#894—94no—
78#875—96no—
79#875—92no—
80C22×C3⋊D3—34no—
81C2×C32×C4C22×D314yes—
82C2×C32×C4C22×D311yescharacteristic
83C2×C32×C4C22×D312yes—
84C2×C3⋊Dic3—34no—
85C4⋊(C3⋊D3)—122no—
86C4⋊(C3⋊D3)—124no—
87C4×C3⋊D3—124no—
88#634—122no—
89C22×D9—312no—
90C2×C4×C9C22×D3112yes—
91C2×C4×C9C22×D313yes—
92C2×C4×C9C22×D316yes—
93C2×Dic9—312no—
94D36—126no—
95D36—1212no—
96C4×D9—1212no—
97Dic18—126no—
98#562—271noSylow
99C2×C3×C9H(5,𝔽2)12yes—
100C2×C3×C92−1+411yescharacteristic
101C3⋊D9—44no—
102C2×D12—96no—
103C2×D12—912no—
104C2×D12—92no—
105C2×D12—94no—
106C2×C4×D3—912no—
107C2×C4×D3—94no—
108C2×Dic6—96no—
109C2×Dic6—92no—
110C3×C4⋊C4C2×D916yes—
111C3×C4⋊C4C2×D9112yes—
112C3×C4⋊C4C2×C3⋊D312yes—
113C3×C4⋊C4C2×C3⋊D314yes—
114C3×C42C2×D913yes—
115C3×C42C2×C3⋊D311yescharacteristic
116(C2×D3)⋊C4—912no—
117(C2×D3)⋊C4—912no—
118(C2×D3)⋊C4—912no—
119(C2×D3)⋊C4—94no—
120(C2×D3)⋊C4—94no—
121(C2×D3)⋊C4—94no—
122Dic3⋊C4—912no—
123Dic3⋊C4—94no—
124C22×C32C23×D311yescharacteristic
125C2×C3⋊D3—64no—
126C2×C3⋊D3—64no—
127C2×C3⋊D3—64no—
128C32×C4—21no—
129C32×C4—22no—
130C32×C4—24no—
131C3⋊Dic3—64no—
132C22×C9C23×D313yes—
133C2×D9—612no—
134C2×D9—612no—
135C2×D9—612no—
136C4×C9—23no—
137C4×C9—26no—
138C4×C9—212no—
139Dic9—612no—
140#125C3⋊D911yescharacteristic, complemented
141#124—272no—
142#123—274no—
143C22⋊Q8—274no—
144#121—274no—
145C3×C9#56211yescharacteristic, complemented, Sylow
146C22×D3—94no—
147C22×D3—912no—
148C2×C3×C4C22×D9112yes—
149C2×C3×C4C22×D913yes—
150C2×C3×C4C22×D916yes—
151C2×C3×C4C22×C3⋊D314yes—
152C2×C3×C4C22×C3⋊D311yescharacteristic
153C2×C3×C4C22×C3⋊D312yes—
154C2×Dic3—94no—
155C2×Dic3—912no—
156D12—362no—
157D12—364no—
158D12—366no—
159D12—3612no—
160C4×D3—364no—
161C4×D3—3612no—
162Dic6—362no—
163Dic6—366no—
164C2×C32#100711yescharacteristic
165C2×C32#100912yes—
166C3⋊D3—124no—
167C2×C9#100713yes—
168C2×C9#100916yes—
169D9—1212no—
170C2×Q8—272no—
171C2×D4—274no—
172C2×D4—272no—
173C22×C4—274no—
174C4⋊C4—274no—
175C4⋊C4C2×C3⋊D914yes—
176C4⋊C4C2×C3⋊D912yes—
177C22⋊C4—274no—
178C22⋊C4—274no—
179C22⋊C4—274no—
180C42C2×C3⋊D911yescharacteristic
181C22×C3C23×D913yes—
182C22×C3C23×C3⋊D311yescharacteristic
183C2×D3—184no—
184C2×D3—184no—
185C2×D3—184no—
186C2×D3—1812no—
187C2×D3—1812no—
188C2×D3—1812no—
189C3×C4—21no—
190C3×C4—22no—
191C3×C4—24no—
192C3×C4—23no—
193C3×C4—26no—
194C3×C4—212no—
195Dic3—184no—
196Dic3—1812no—
197C32#570011yescharacteristic
198C9#570013yescomplemented
199C23—274no—
200Q8—1082no—
201D4—1084no—
202D4—1082no—
203C2×C4—274no—
204C2×C4—1084no—
205C2×C4C22×C3⋊D911yescharacteristic
206C2×C4C22×C3⋊D912yes—
207C2×C4C22×C3⋊D914yes—
208C2×C3#6371313yes—
209C2×C3#6371516yes—
210C2×C3#6436411yescharacteristic
211C2×C3#6436612yes—
212D3—364no—
213D3—3612no—
214C22—544no—
215C22—544no—
216C22—544no—
217C22C23×C3⋊D911yescharacteristic
218C4—21no—
219C4—24no—
220C4—22no—
221C4—544no—
222C3#1058980213yescomplemented
223C3#1059617111yescharacteristic
224C2—1084no—
225C2#1172266711yescharacteristic
226C2#1172267212yes—
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C9×#125C2
socleC22×C32C23×D3
hypercenter#125C3⋊D9
hypocenterC3×C9#562
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. C3×C9×#125C2semidirect
2. #11721752C2semidirect
3. #11721751C2semidirect
4. #11721750C2semidirect
5. #11721712C2semidirect
24. C9×#125D3semidirect
70. C3×#125D9semidirect
140. #125C3⋊D9semidirect
145. C3×C9#562semidirect
198. C9#5700semidirect
222. C3#10589802semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1252
#117217522
#117217512
#117217502
#117217122
#105961713
#105898023