Finite Group Databasebeta

← #13143799

Subgroups — #13143799

Order 1728.

720 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 191 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×#88525C212yesmaximal, complemented
2C2×#88524C212yesmaximal, complemented
3#11721712C211yesmaximal, characteristic, complemented
4#11721711C211yesmaximal, characteristic, complemented
5C3×C9×#97C211yesmaximal, characteristic, complemented
6#10592411—31nomaximal
7#10588393—33nomaximal
8C2×C4⋊(C3⋊D9)—22no—
9C2×C4⋊(C3⋊D9)C2212yes—
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×C8×C9C2212yes—
13C3×C42×C9C2211yescharacteristic
14#88536—22no—
15#88525—24no—
16#88524—24no—
17C2×#3745—32no—
18C2×#3744—32no—
19#64084—31no—
20#64083—31no—
21C32×#97D311yescharacteristic
22C2×D72—36no—
23C2×#3664—36no—
24#63437—33no—
25#63436—33no—
26C9×#97D313yescomplemented
27C22×C3⋊D9—22no—
28C22×C3⋊D9—21no—
29C2×C3×C4×C9D412yes—
30C2×C3×C4×C9C2311yescharacteristic
31C2×C3⋊Dic9—21no—
32C4⋊(C3⋊D9)—22no—
33C4⋊(C3⋊D9)—22no—
34C4⋊(C3⋊D9)—22no—
35C4⋊(C3⋊D9)—44no—
36#6226—22no—
37C3×C8×C9D414yescomplemented
38#4712—93no—
39#4712—91no—
40C2×C4⋊(C3⋊D3)—32no—
41C2×C4⋊(C3⋊D3)—31no—
42C2×C4⋊(C3⋊D3)—62no—
43C2×#634—31no—
44C2×C32×C8C2×D312yes—
45C32×C42C2×D311yescharacteristic
46#3752—62no—
47#3745—64no—
48#3744—64no—
49C2×D36—36no—
50C2×D36—33no—
51C2×D36—66no—
52C2×Dic18—33no—
53C2×C8×C9C2×D316yes—
54C42×C9C2×D313yes—
55(C2×D9)⋊C4—66no—
56D72—612no—
57#3664—612no—
58C22×C3×C9C2×D411yescharacteristic
59C2×C3⋊D9—42no—
60C2×C3⋊D9—41no—
61C2×C3⋊D9—44no—
62C2×C3⋊D9—42no—
63C3×C4×C9—22no—
64C3×C4×C9C2×D412yes—
65C3⋊Dic9—41no—
66C2×D24—96no—
67C2×D24—92no—
68C2×#204—96no—
69C2×#204—92no—
70#875—93no—
71#875—91no—
72#874—93no—
73#874—91no—
74C3×#97D913yescomplemented
75C3×#97C3⋊D311yescharacteristic
76C22×C3⋊D3—61no—
77C22×C3⋊D3—62no—
78C2×C32×C4D1212yes—
79C2×C32×C4C22×D311yescharacteristic
80C2×C3⋊Dic3—61no—
81C4⋊(C3⋊D3)—62no—
82C4⋊(C3⋊D3)—62no—
83C4⋊(C3⋊D3)—62no—
84C4⋊(C3⋊D3)—124no—
85#634—62no—
86C22×D9—63no—
87C22×D9—66no—
88C32×C8D1214yes—
89C2×C4×C9D1216yes—
90C2×C4×C9C22×D313yes—
91C2×Dic9—63no—
92D36—66no—
93D36—66no—
94D36—66no—
95D36—1212no—
96Dic18—66no—
97C8×C9D12112yescomplemented
98#496—271noSylow
99C2×C3×C9#12711yescharacteristic
100C2×C3×C9AGL(1,ℤ/8)12yes—
101C3⋊D9—82no—
102C3⋊D9—81no—
103C2×D12—96no—
104C2×D12—93no—
105C2×D12—92no—
106C2×D12—91no—
107C2×D12—186no—
108C2×D12—182no—
109C2×Dic6—93no—
110C2×Dic6—91no—
111C2×C3×C8C2×D916yes—
112C2×C3×C8C2×C3⋊D312yes—
113C3×C42C2×D913yes—
114C3×C42C2×C3⋊D311yescharacteristic
115(C2×D3)⋊C4—186no—
116(C2×D3)⋊C4—182no—
117D24—1812no—
118D24—184no—
119#204—1812no—
120#204—184no—
121C22×C32C2×D1211yescharacteristic
122C2×C3⋊D3—122no—
123C2×C3⋊D3—121no—
124C2×C3⋊D3—124no—
125C2×C3⋊D3—122no—
126C32×C4—22no—
127C32×C4C2×D1212yes—
128C3⋊Dic3—121no—
129C22×C9C2×D1213yes—
130C2×D9—126no—
131C2×D9—123no—
132C2×D9—1212no—
133C2×D9—126no—
134C4×C9—26no—
135C4×C9C2×D1216yes—
136Dic9—123no—
137C2×SD16—272no—
138C2×D8—272no—
139#127—271no—
140#124—271no—
141#97C3⋊D911yescharacteristic, complemented
142C3×C9#49611yescharacteristic, complemented, Sylow
143C22×D3—183no—
144C22×D3—186no—
145C22×D3—181no—
146C22×D3—182no—
147C2×C3×C4D3616yes—
148C2×C3×C4C22×D913yes—
149C2×C3×C4C4⋊(C3⋊D3)12yes—
150C2×C3×C4C22×C3⋊D311yescharacteristic
151C2×Dic3—183no—
152C2×Dic3—181no—
153D12—186no—
154D12—186no—
155D12—186no—
156D12—182no—
157D12—182no—
158D12—182no—
159D12—364no—
160D12—3612no—
161Dic6—186no—
162Dic6—182no—
163C3×C8D36112yescomplemented
164C3×C8C4⋊(C3⋊D3)14yes—
165C2×C32#87411yescharacteristic
166C2×C32#90812yes—
167C3⋊D3—242no—
168C3⋊D3—241no—
169C2×C9#87413yes—
170C2×C9#90816yes—
171D9—246no—
172D9—243no—
173C2×Q8—271no—
174C2×D4—271no—
175C2×D4—272no—
176C2×D4—542no—
177SD16—544no—
178D8—544no—
179C2×C8C2×C3⋊D912yes—
180C22⋊C4—542no—
181C42C2×C3⋊D911yescharacteristic
182C22×C3C2×D3613yes—
183C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
184C2×D3—362no—
185C2×D3—361no—
186C2×D3—364no—
187C2×D3—362no—
188C2×D3—366no—
189C2×D3—363no—
190C2×D3—3612no—
191C2×D3—366no—
192C3×C4—22no—
193C3×C4—26no—
194C3×C4C2×D3616yes—
195C3×C4C2×C4⋊(C3⋊D3)12yes—
196Dic3—361no—
197Dic3—363no—
198C32#471211yescharacteristic
199C9#471213yescomplemented
200C23—541no—
201C23—542no—
202Q8—542no—
203D4—542no—
204D4—542no—
205D4—542no—
206D4—1084no—
207C2×C4—541no—
208C2×C4C4⋊(C3⋊D9)12yes—
209C2×C4C22×C3⋊D911yescharacteristic
210C8C4⋊(C3⋊D9)14yescomplemented
211C2×C3#6343613yes—
212C2×C3#6347016yes—
213C2×C3#6408311yescharacteristic
214C2×C3#6411712yes—
215D3—722no—
216D3—721no—
217D3—726no—
218D3—723no—
219C22—1084no—
220C22—1082no—
221C22—1081no—
222C22—1082no—
223C22C2×C4⋊(C3⋊D9)11yescharacteristic
224C4—22no—
225C4—1081no—
226C4C2×C4⋊(C3⋊D9)12yes—
227C3#1058839313yescomplemented
228C3#1059241111yescharacteristic
229C2—2162no—
230C2—2161no—
231C2#1172171111yescharacteristic
232C2#1172178312yes—
subgroupG/H
centerC22C2×C4⋊(C3⋊D9)
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#97C2
socleC22×C32C2×D12
hypercenter#97C3⋊D9
hypocenterC3×C9#496
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
1. C2×#88525C2semidirect
2. C2×#88524C2semidirect
3. #11721712C2semidirect
4. #11721711C2semidirect
5. C3×C9×#97C2semidirect
26. C9×#97D3semidirect
37. C3×C8×C9D4semidirect
74. C3×#97D9semidirect
97. C8×C9D12semidirect
141. #97C3⋊D9semidirect
142. C3×C9#496semidirect
163. C3×C8D36semidirect
199. C9#4712semidirect
210. C8C4⋊(C3⋊D9)semidirect
227. C3#10588393semidirect

Maximal subgroups

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

subgroupindex
C2×#885252
C2×#885242
#117217122
#117217112
C3×C9×#972
#105924113
#105883933