Finite Group Databasebeta

← #13143886

Subgroups — #13143886

Order 1728.

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

normal subgroups only

This group has 710 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#11722667C211yesmaximal, characteristic
2#11721854C211yesmaximal, characteristic, complemented
3#11721784C212yesmaximal, complemented
4C3×C9×#101C211yesmaximal, characteristic, complemented
5#11720558C212yesmaximal, complemented
6#10592451—31nomaximal
7#10588432—33nomaximal
8C2×C3×Q8×C9C2211yescharacteristic
9#88746—22no—
10Q8×C3⋊D9—22no—
11#88736—22no—
12#88736C2212yes—
13C2×#6226C2211yescharacteristic
14C3×C9×M16C2212yes—
15#88534—22no—
16C4×C3⋊Dic9C2211yescharacteristic
17#88524—22no—
18#88521—22no—
19#64364—31no—
20#64153—31no—
21#64118—32no—
22#63713—33no—
23C32×#101D311yescharacteristic
24#63652—32no—
25#63506—33no—
26#63471—36no—
27C9×#101D313yescomplemented
28#63384—36no—
29C3×Q8×C9—22no—
30C2×C3×C4×C9D412yes—
31C2×C3×C4×C9C2311yescharacteristic
32#6233—22no—
33C2×C3⋊Dic9—21no—
34C2×C3⋊Dic9—21no—
35C4⋊(C3⋊D9)—22no—
36C4⋊(C3⋊D9)D412yes—
37C4×C3⋊D9—24no—
38C4×C3⋊D9—22no—
39#6226—22no—
40#6226—22no—
41#6226D412yes—
42C3×C8×C9—22no—
43#4752—93no—
44#4752—91no—
45C2×C32×Q8C2×D311yescharacteristic
46#3832—62no—
47Q8×C3⋊D3—62no—
48#3828—32no—
49#3828—62no—
50C2×#634—31no—
51C32×M16C2×D312yes—
52#3750—62no—
53C4×C3⋊Dic3—31no—
54#3746—62no—
55#3744—62no—
56C2×Q8×C9C2×D313yes—
57#3701—66no—
58Q8×D9—66no—
59#3697—36no—
60#3697—66no—
61C2×Dic18—33no—
62C9×M16C2×D316yes—
63Dic9⋊C4—66no—
64C4×Dic9—33no—
65#3664—66no—
66Dic36—66no—
67C22×C3×C9C2×D411yescharacteristic
68C2×C3⋊D9—22no—
69C3×C4×C9—22no—
70C3×C4×C9C2×D412yes—
71C3⋊Dic9—22no—
72C3⋊Dic9—22no—
73C3⋊Dic9—41no—
74#1007—93no—
75#1007—91no—
76#944—93no—
77#944—91no—
78#909—96no—
79#909—92no—
80C3×#101D913yescomplemented
81C3×#101C3⋊D311yescharacteristic
82#825—96no—
83#825—92no—
84C32×Q8—22no—
85C2×C32×C4D1212yes—
86C2×C32×C4C22×D311yescharacteristic
87#638—62no—
88C2×C3⋊Dic3—61no—
89C2×C3⋊Dic3—61no—
90C4⋊(C3⋊D3)—32no—
91C4⋊(C3⋊D3)—62no—
92C4×C3⋊D3—64no—
93C4×C3⋊D3—62no—
94#634—32no—
95#634—62no—
96#634—62no—
97C32×C8—22no—
98Q8×C9—26no—
99C2×C4×C9D1216yes—
100C2×C4×C9C22×D313yes—
101#611—66no—
102C2×Dic9—63no—
103C2×Dic9—63no—
104D36—36no—
105D36—66no—
106C4×D9—612no—
107C4×D9—66no—
108Dic18—36no—
109Dic18—66no—
110Dic18—66no—
111C8×C9—26no—
112#456—271noSylow
113C2×C3×C9—21no—
114C2×C3×C9#12011yescharacteristic
115C3⋊D9—42no—
116C2×C3×Q8C2×D913yes—
117C2×C3×Q8C2×C3⋊D311yescharacteristic
118#239—182no—
119#239—186no—
120D3×Q8—182no—
121D3×Q8—186no—
122#235—96no—
123#235—92no—
124#235—186no—
125#235—182no—
126C2×Dic6—93no—
127C2×Dic6—91no—
128C3×M16C2×D916yes—
129C3×M16C2×C3⋊D312yes—
130Dic3⋊C4—186no—
131Dic3⋊C4—182no—
132C4×Dic3—93no—
133C4×Dic3—91no—
134Dic12—182no—
135Dic12—186no—
136#204—182no—
137#204—186no—
138C22×C32C2×D1211yescharacteristic
139C2×C3⋊D3—62no—
140C32×C4—22no—
141C32×C4D3×D412yes—
142C3⋊Dic3—62no—
143C3⋊Dic3—62no—
144C3⋊Dic3—121no—
145C22×C9C2×D1213yes—
146C2×D9—66no—
147C4×C9—26no—
148C4×C9D3×D416yes—
149Dic9—66no—
150Dic9—66no—
151Dic9—123no—
1522−1+4—271no—
153#137—272no—
154C4⋊Q8—271no—
155#104—272no—
156#101C3⋊D911yescharacteristic, complemented
157C3×C9#45611yescharacteristic, complemented, Sylow
158C3×Q8—26no—
159C3×Q8—22no—
160C2×C3×C4D3616yes—
161C2×C3×C4C22×D913yes—
162C2×C3×C4C4⋊(C3⋊D3)12yes—
163C2×C3×C4C22×C3⋊D311yescharacteristic
164#67—186no—
165#67—182no—
166C2×Dic3—183no—
167C2×Dic3—181no—
168C2×Dic3—183no—
169C2×Dic3—181no—
170D12—96no—
171D12—92no—
172D12—182no—
173D12—186no—
174C4×D3—184no—
175C4×D3—1812no—
176C4×D3—186no—
177C4×D3—182no—
178Dic6—96no—
179Dic6—92no—
180Dic6—182no—
181Dic6—186no—
182Dic6—186no—
183Dic6—182no—
184C3×C8—26no—
185C3×C8—22no—
186C2×C32—21no—
187C2×C32#88211yescharacteristic
188C3⋊D3—122no—
189C2×C9—23no—
190C2×C9#88213yes—
191D9—126no—
192#41—272no—
193#41—542no—
194#41—542no—
195C2×Q8—271no—
196C2×Q8—542no—
197C2×Q8C2×C3⋊D911yescharacteristic
198Q16—542no—
199SD16—542no—
200M16C2×C3⋊D912yes—
201C4⋊C4—542no—
202C42—271no—
203C22×C3C2×D3613yes—
204C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
205C2×D3—182no—
206C2×D3—186no—
207C3×C4—22no—
208C3×C4—26no—
209C3×C4D4×D916yes—
210C3×C4D4×C3⋊D312yes—
211Dic3—182no—
212Dic3—186no—
213Dic3—182no—
214Dic3—186no—
215Dic3—361no—
216Dic3—363no—
217C32#475211yescharacteristic
218C9#475213yescomplemented
219Q8—22no—
220Q8—272no—
221Q8—542no—
222Q8—542no—
223D4—272no—
224D4—542no—
225D4—542no—
226C2×C4—544no—
227C2×C4—541no—
228C2×C4—542no—
229C2×C4—541no—
230C2×C4C4⋊(C3⋊D9)12yes—
231C2×C4C22×C3⋊D911yescharacteristic
232C8—22no—
233C2×C3—23no—
234C2×C3—21no—
235C2×C3#6344413yes—
236C2×C3#6409111yescharacteristic
237D3—366no—
238D3—362no—
239C22—542no—
240C22C2×C4⋊(C3⋊D9)11yescharacteristic
241C4—22no—
242C4—542no—
243C4—542no—
244C4—1081no—
245C4D4×C3⋊D912yes—
246C3#1058843213yescomplemented
247C3#1059245111yescharacteristic
248C2—21no—
249C2—1082no—
250C2#1172172811yescharacteristic
subgroupG/H
centerC2#11721728
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#101C2
socleC2×C32#882
prime radical#11722667C2
hypercenter#101C3⋊D9
hypocenterC3×C9#456
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
2. #11721854C2semidirect
3. #11721784C2semidirect
4. C3×C9×#101C2semidirect
5. #11720558C2semidirect
27. C9×#101D3semidirect
80. C3×#101D9semidirect
156. #101C3⋊D9semidirect
157. C3×C9#456semidirect
218. C9#4752semidirect
246. C3#10588432semidirect

Maximal subgroups

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

subgroupindex
#117226672
#117218542
#117217842
C3×C9×#1012
#117205582
#105924513
#105884323