Finite Group Databasebeta

← #13143793

Subgroups — #13143793

Order 1728.

550 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 163 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
1C42×C3⋊D9C211yesmaximal, characteristic
2C3×C9×#97C211yesmaximal, characteristic, complemented
3#11720542C212yesmaximal
4#11720537C212yesmaximal, complemented
5#11720516C211yesmaximal, characteristic, complemented
6#10592404—31nomaximal
7#10588386—33nomaximal
8C2×C4×C3⋊D9C411yescharacteristic
9C2×C4×C3⋊D9C411yescharacteristic
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×C3×C8×C9C2212yes—
12C3×C42×C9C2211yescharacteristic
13C4×C3⋊Dic9C411yescharacteristic
14C4×C3⋊Dic9C411yescharacteristic
15C4×C3⋊Dic9C2211yescharacteristic
16C2×C3⋊(C9⋊C8)C2212yes—
17C42×C3⋊D3—31no—
18C32×#97D311yescharacteristic
19#63647—32no—
20#63641—32no—
21#63630—31no—
22C42×D9—33no—
23C9×#97D313yescomplemented
24(C2×D9)⋊C8—36no—
25Dic9⋊C8—36no—
26#63362—33no—
27C22×C3⋊D9C2×C411yescharacteristic
28C2×C3×C4×C9C2×C411yescharacteristic
29C2×C3×C4×C9C2×C411yescharacteristic
30C2×C3×C4×C9C2311yescharacteristic
31C2×C3⋊Dic9C2×C411yescharacteristic
32C2×C3⋊Dic9C2×C411yescharacteristic
33C2×C3⋊Dic9C2×C411yescharacteristic
34C4×C3⋊D9—22no—
35C4×C3⋊D9—22no—
36C4×C3⋊D9—22no—
37C3×C8×C9—22no—
38C3⋊(C9⋊C8)—22no—
39#4705—93no—
40#4705—91no—
41C2×C4×C3⋊D3—31no—
42C2×C4×C3⋊D3—31no—
43C2×C4×C3⋊D3—31no—
44C2×C32×C8C2×D312yes—
45C32×C42C2×D311yescharacteristic
46C4×C3⋊Dic3—31no—
47C4×C3⋊Dic3—31no—
48C4×C3⋊Dic3—31no—
49C2×C3⋊(C3⋊C8)—32no—
50C2×C4×D9—33no—
51C2×C4×D9—33no—
52C2×C4×D9—33no—
53C2×C8×C9C2×D316yes—
54C42×C9C2×D313yes—
55C4×Dic9—33no—
56C4×Dic9—33no—
57C4×Dic9—33no—
58C2×C9⋊C8—36no—
59C22×C3×C9C22×C411yescharacteristic
60C2×C3⋊D9—21no—
61C2×C3⋊D9—21no—
62C2×C3⋊D9M1612yes—
63C3×C4×C9—21no—
64C3×C4×C9—21no—
65C3×C4×C9#4112yes—
66C3⋊Dic9—21no—
67C3⋊Dic9—21no—
68C3⋊Dic9M1612yes—
69C42×D3—93no—
70C42×D3—91no—
71C3×#97D913yescomplemented
72C3×#97C3⋊D311yescharacteristic
73(C2×D3)⋊C8—96no—
74(C2×D3)⋊C8—92no—
75Dic3⋊C8—96no—
76Dic3⋊C8—92no—
77#803—93no—
78#803—91no—
79C22×C3⋊D3—31no—
80C2×C32×C4C4×D311yescharacteristic
81C2×C32×C4C4×D311yescharacteristic
82C2×C32×C4C22×D311yescharacteristic
83C2×C3⋊Dic3—31no—
84C2×C3⋊Dic3—31no—
85C2×C3⋊Dic3—31no—
86C4×C3⋊D3—62no—
87C4×C3⋊D3—62no—
88C4×C3⋊D3—62no—
89C22×D9—33no—
90C32×C8—22no—
91C3⋊(C3⋊C8)—62no—
92C2×C4×C9C4×D313yes—
93C2×C4×C9C4×D313yes—
94C2×C4×C9C22×D313yes—
95C2×Dic9—33no—
96C2×Dic9—33no—
97C2×Dic9—33no—
98C4×D9—66no—
99C4×D9—66no—
100C4×D9—66no—
101C8×C9—26no—
102C9⋊C8—66no—
103#432—271noSylow
104C2×C3×C9#11711yescharacteristic
105C2×C3×C9C2×M1611yescharacteristic
106C2×C3×C9C2×M1611yescharacteristic
107C3⋊D9—22no—
108C2×C4×D3—93no—
109C2×C4×D3—93no—
110C2×C4×D3—93no—
111C2×C4×D3—91no—
112C2×C4×D3—91no—
113C2×C4×D3—91no—
114C2×C3×C8C2×D916yes—
115C2×C3×C8C2×C3⋊D312yes—
116C3×C42C2×D913yes—
117C3×C42C2×C3⋊D311yescharacteristic
118C4×Dic3—93no—
119C4×Dic3—93no—
120C4×Dic3—93no—
121C4×Dic3—91no—
122C4×Dic3—91no—
123C4×Dic3—91no—
124C2×C3⋊C8—96no—
125C2×C3⋊C8—92no—
126C22×C32C2×C4×D311yescharacteristic
127C2×C3⋊D3—32no—
128C2×C3⋊D3—61no—
129C2×C3⋊D3—61no—
130C32×C4—21no—
131C32×C4—21no—
132C32×C4#23512yes—
133C3⋊Dic3—32no—
134C3⋊Dic3—61no—
135C3⋊Dic3—61no—
136C22×C9C2×C4×D313yes—
137C2×D9—36no—
138C2×D9—63no—
139C2×D9—63no—
140C4×C9—23no—
141C4×C9—23no—
142C4×C9#23516yes—
143Dic9—36no—
144Dic9—63no—
145Dic9—63no—
146C2×C42—271no—
147C4⋊C8—272no—
148C22⋊C8—272no—
149#97—271no—
150#97C3⋊D911yescharacteristic, complemented
151C3×C9#43211yescharacteristic, complemented, Sylow
152C22×D3—93no—
153C22×D3—91no—
154C2×C3×C4C4×D913yes—
155C2×C3×C4C4×D913yes—
156C2×C3×C4C22×D913yes—
157C2×C3×C4C4×C3⋊D311yescharacteristic
158C2×C3×C4C4×C3⋊D311yescharacteristic
159C2×C3×C4C22×C3⋊D311yescharacteristic
160C2×Dic3—93no—
161C2×Dic3—91no—
162C2×Dic3—93no—
163C2×Dic3—93no—
164C2×Dic3—91no—
165C2×Dic3—91no—
166C4×D3—182no—
167C4×D3—186no—
168C4×D3—182no—
169C4×D3—186no—
170C4×D3—186no—
171C4×D3—182no—
172C3×C8—26no—
173C3×C8—22no—
174C3⋊C8—186no—
175C3⋊C8—182no—
176C2×C32#87211yescharacteristic
177C2×C32D3×M1611yescharacteristic
178C2×C32D3×M1611yescharacteristic
179C3⋊D3—62no—
180C2×C9#87213yes—
181C2×C9D3×M1613yes—
182C2×C9D3×M1613yes—
183D9—66no—
184C22×C4—271no—
185C22×C4—271no—
186C22×C4—271no—
187C2×C8—272no—
188C2×C8C2×C3⋊D912yes—
189C42—271no—
190C42—271no—
191C42—271no—
192C42C2×C3⋊D911yescharacteristic
193C22×C3C2×C4×D913yes—
194C22×C3C2×C4×C3⋊D311yescharacteristic
195C2×D3—92no—
196C2×D3—96no—
197C2×D3—181no—
198C2×D3—181no—
199C2×D3—183no—
200C2×D3—183no—
201C3×C4—21no—
202C3×C4—21no—
203C3×C4—23no—
204C3×C4—23no—
205C3×C4#369716yes—
206C3×C4#382812yes—
207Dic3—92no—
208Dic3—96no—
209Dic3—181no—
210Dic3—181no—
211Dic3—183no—
212Dic3—183no—
213C32#470511yescharacteristic
214C9#470513yescomplemented
215C23—271no—
216C2×C4—271no—
217C2×C4—271no—
218C2×C4—271no—
219C2×C4—542no—
220C2×C4—542no—
221C2×C4—542no—
222C2×C4C4×C3⋊D911yescharacteristic
223C2×C4C4×C3⋊D911yescharacteristic
224C2×C4C22×C3⋊D911yescharacteristic
225C8—22no—
226C8—542no—
227C2×C3#6343413yes—
228C2×C3M16×D913yes—
229C2×C3M16×D913yes—
230C2×C3#6408111yescharacteristic
231C2×C3M16×C3⋊D311yescharacteristic
232C2×C3M16×C3⋊D311yescharacteristic
233D3—182no—
234D3—186no—
235C22—272no—
236C22—541no—
237C22—541no—
238C22C2×C4×C3⋊D911yescharacteristic
239C4—21no—
240C4—21no—
241C4—272no—
242C4—541no—
243C4—541no—
244C4#8873612yes—
245C3#1058838613yescomplemented
246C3#1059240411yescharacteristic
247C2—542no—
248C2#1172170911yescharacteristic
249C2M16×C3⋊D911yescharacteristic
250C2M16×C3⋊D911yescharacteristic
subgroupG/H
centerC2×C4C22×C3⋊D9
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#97C2
socleC22×C32C2×C4×D3
prime radicalC22×C3⋊D9C2×C4
cosocleC2×C3×C4×C9C23
hypercenter#97C3⋊D9
hypocenterC3×C9#432
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ #97

Complements

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

NH ≅ G/Ntype
2. C3×C9×#97C2semidirect
4. #11720537C2semidirect
5. #11720516C2semidirect
23. C9×#97D3semidirect
71. C3×#97D9semidirect
150. #97C3⋊D9semidirect
151. C3×C9#432semidirect
214. C9#4705semidirect
245. C3#10588386semidirect

Maximal subgroups

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

subgroupindex
C42×C3⋊D92
C3×C9×#972
#117205422
#117205372
#117205162
#105924043
#105883863