Finite Group Databasebeta

← #13144752

Subgroups — #13144752

Order 1728.

550 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 153 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×C3×C9×Q16C211yesmaximal, characteristic, complemented
2#11721856C212yesmaximal
3C2×#88523C211yesmaximal, characteristic
4#11720582C212yesmaximal, complemented
5#11720540C211yesmaximal, characteristic, complemented
6#10592888—31nomaximal
7#10588869—33nomaximal
8C2×C3×Q8×C9C2212yes—
9C2×C4×C3⋊D9C2211yescharacteristic
10C3×C9×Q16—22no—
11C2×C3×C8×C9C2211yescharacteristic
12#88536—22no—
13C4⋊(C3⋊Dic9)C2212yes—
14#88534—22no—
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16#88523—22no—
17C2×C32×Q16D311yescharacteristic
18#64155—32no—
19C2×#3743—31no—
20#63662—32no—
21#63644—31no—
22C2×C9×Q16D313yescomplemented
23#63508—36no—
24C2×#3663—33no—
25#63395—36no—
26#63377—33no—
27C22×C3⋊D9D411yescharacteristic
28C3×Q8×C9—22no—
29C3×Q8×C9—22no—
30C2×C3×C4×C9—22no—
31C2×C3×C4×C9C2311yescharacteristic
32C2×C3⋊Dic9—22no—
33C2×C3⋊Dic9D411yescharacteristic
34C4×C3⋊D9—21no—
35C4×C3⋊D9—21no—
36C3×C8×C9D412yes—
37C3⋊(C9⋊C8)—21no—
38#5189—93no—
39#5189—91no—
40C2×C32×Q8C2×D312yes—
41C2×C4×C3⋊D3—31no—
42C32×Q16—22no—
43C2×C32×C8C2×D311yescharacteristic
44#3752—62no—
45C4⋊(C3⋊Dic3)—32no—
46#3750—62no—
47C2×C3⋊(C3⋊C8)—31no—
48#3743—62no—
49C2×Q8×C9C2×D316yes—
50C2×C4×D9—33no—
51C9×Q16—26no—
52C2×C8×C9C2×D313yes—
53(C2×D9)⋊C4—66no—
54C4⋊Dic9—36no—
55Dic9⋊C4—66no—
56C2×C9⋊C8—33no—
57#3663—66no—
58C22×C3×C9C2×D411yescharacteristic
59C2×C3⋊D9—21no—
60C2×C3⋊D9—22no—
61C3×C4×C9—42no—
62C3×C4×C9C2×D411yescharacteristic
63C3×C4×C9#4111yescharacteristic
64C3⋊Dic9—21no—
65C3⋊Dic9—42no—
66C2×C3×Q16D913yescomplemented
67C2×C3×Q16C3⋊D311yescharacteristic
68#946—96no—
69#946—92no—
70C2×ΓL(1,𝔽25)—93no—
71C2×ΓL(1,𝔽25)—91no—
72#835—96no—
73#835—92no—
74#817—93no—
75#817—91no—
76C22×C3⋊D3—31no—
77C32×Q8—22no—
78C32×Q8—22no—
79C2×C32×C4—22no—
80C2×C32×C4C22×D311yescharacteristic
81C2×C3⋊Dic3—31no—
82C2×C3⋊Dic3—62no—
83C4×C3⋊D3—61no—
84C4×C3⋊D3—61no—
85C22×D9—33no—
86C32×C8#6712yes—
87C3⋊(C3⋊C8)—61no—
88Q8×C9—26no—
89Q8×C9—26no—
90C2×C4×C9—26no—
91C2×C4×C9C22×D313yes—
92C2×Dic9—33no—
93C2×Dic9—66no—
94C4×D9—63no—
95C4×D9—63no—
96C8×C9#6716yes—
97C9⋊C8—63no—
98#470—271noSylow
99C2×C3×C9#12111yescharacteristic
100C2×C3×C9#13712yes—
101C3⋊D9—41no—
102C2×C3×Q8C2×D916yes—
103C2×C3×Q8C2×C3⋊D312yes—
104C2×C4×D3—93no—
105C2×C4×D3—91no—
106C3×Q16—26no—
107C3×Q16—22no—
108C2×C3×C8C2×D913yes—
109C2×C3×C8C2×C3⋊D311yescharacteristic
110(C2×D3)⋊C4—182no—
111(C2×D3)⋊C4—186no—
112C4⋊Dic3—96no—
113C4⋊Dic3—92no—
114Dic3⋊C4—182no—
115Dic3⋊C4—186no—
116C2×C3⋊C8—93no—
117C2×C3⋊C8—91no—
118ΓL(1,𝔽25)—186no—
119ΓL(1,𝔽25)—182no—
120C22×C32D3×D411yescharacteristic
121C2×C3⋊D3—62no—
122C2×C3⋊D3—61no—
123C32×C4—42no—
124C32×C4#23711yescharacteristic
125C32×C4C2×#6711yescharacteristic
126C3⋊Dic3—61no—
127C3⋊Dic3—122no—
128C22×C9D3×D413yes—
129C2×D9—66no—
130C2×D9—63no—
131C4×C9—46no—
132C4×C9#23713yes—
133C4×C9C2×#6713yes—
134Dic9—63no—
135Dic9—126no—
136C2×Q16C3⋊D911yescharacteristic, complemented
137C2×M16—271no—
138C22⋊Q8—272no—
139#106—271no—
140#103—272no—
141C3×C9#47011yescharacteristic, complemented, Sylow
142C22×D3—93no—
143C22×D3—91no—
144C3×Q8—26no—
145C3×Q8—26no—
146C3×Q8—22no—
147C3×Q8—22no—
148C2×C3×C4—26no—
149C2×C3×C4—22no—
150C2×C3×C4C22×D913yes—
151C2×C3×C4C22×C3⋊D311yescharacteristic
152C2×Dic3—93no—
153C2×Dic3—91no—
154C2×Dic3—186no—
155C2×Dic3—182no—
156C4×D3—183no—
157C4×D3—183no—
158C4×D3—181no—
159C4×D3—181no—
160C3×C8#61116yes—
161C3×C8#63812yes—
162C3⋊C8—183no—
163C3⋊C8—181no—
164C2×C32#91812yes—
165C2×C32#93811yescharacteristic
166C3⋊D3—121no—
167C2×C9#91816yes—
168C2×C9#93813yes—
169D9—123no—
170C2×Q8C2×C3⋊D912yes—
171C22×C4—271no—
172Q16—22no—
173M16—542no—
174C2×C8—271no—
175C2×C8C2×C3⋊D911yescharacteristic
176C4⋊C4—272no—
177C4⋊C4—542no—
178C22⋊C4—542no—
179C22×C3D4×D913yes—
180C22×C3D4×C3⋊D311yescharacteristic
181C2×D3—183no—
182C2×D3—182no—
183C2×D3—181no—
184C2×D3—186no—
185C3×C4—46no—
186C3×C4—42no—
187C3×C4#369913yes—
188C3×C4C2×#61113yes—
189C3×C4#383011yescharacteristic
190C3×C4C2×#63811yescharacteristic
191Dic3—183no—
192Dic3—181no—
193Dic3—362no—
194Dic3—366no—
195C32#518911yescharacteristic
196C9#518913yescomplemented
197C23—271no—
198Q8—22no—
199Q8—22no—
200C2×C4—22no—
201C2×C4—271no—
202C2×C4—541no—
203C2×C4—541no—
204C2×C4—542no—
205C2×C4C22×C3⋊D911yescharacteristic
206C8—541no—
207C8#623312yes—
208C2×C3#6348016yes—
209C2×C3#6350013yes—
210C2×C3#6412712yes—
211C2×C3#6414711yescharacteristic
212D3—361no—
213D3—363no—
214C22—542no—
215C22—541no—
216C22D4×C3⋊D911yescharacteristic
217C4—42no—
218C4—541no—
219C4—1082no—
220C4#8874211yescharacteristic
221C4C2×#623311yescharacteristic
222C3#1058886913yescomplemented
223C3#1059288811yescharacteristic
224C2—1081no—
225C2#1172180412yes—
226C2#1172183811yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×C3×C9×Q16C2
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
hypercenterC2×Q16C3⋊D9
hypocenterC3×C9#470
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×Q16C2semidirect
4. #11720582C2semidirect
5. #11720540C2semidirect
22. C2×C9×Q16D3semidirect
66. C2×C3×Q16D9semidirect
136. C2×Q16C3⋊D9semidirect
141. C3×C9#470semidirect
196. C9#5189semidirect
222. C3#10588869semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C9×Q162
#117218562
C2×#885232
#117205822
#117205402
#105928883
#105888693