Finite Group Databasebeta

← #13141359

Subgroups — #13141359

Order 1728.

1203 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1203 conjugacy classes of subgroups — too many to draw. Only its 164 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
1C22×C4⋊((C3⋊D3)⋊C3)C213yesmaximal, complemented
2C2×#3752⋊C3C213yesmaximal, complemented
3H(3,𝔽3)×#95C211yesmaximal, characteristic, complemented
4#10592370C311yesmaximal, characteristic, complemented
5C3×#4671—31nomaximal
6C23×(C3⋊D3)⋊C3C2213yes—
7C22×C4×H(3,𝔽3)C2213yes—
8C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
9C2×C4⋊((C3⋊D3)⋊C3)—212no—
10C2×C4⋊((C3⋊D3)⋊C3)—26no—
11#3752⋊C3—26no—
12C22×C4⋊(C3⋊D3)C2×C313yescomplemented
13C22×C3×D12—33no—
14C2×#3752C2×C313yescomplemented
15C2×C3×(C2×D3)⋊C4—33no—
16C32×#95—21no—
17C32×#95D311yescharacteristic, complemented
18C32×#95C2×C311yescharacteristic, complemented
19C23×H(3,𝔽3)C2311yescharacteristic
20C22×(C3⋊D3)⋊C3—26no—
21C22×(C3⋊D3)⋊C3—26no—
22C22×(C3⋊D3)⋊C3—23no—
23C22×(C3⋊D3)⋊C3—23no—
24C22×(C3⋊D3)⋊C3D416yescomplemented
25C2×C4×H(3,𝔽3)—26no—
26C2×C4×H(3,𝔽3)D416yescomplemented
27C2×(C3⋊Dic3)⋊C3—23no—
28C4⋊((C3⋊D3)⋊C3)—224no—
29C3×#392—91no—
30#4671—31no—
31#4671—91no—
32C23×C3⋊D3C22×C313yes—
33C23×C3×D3—33no—
34C22×C32×C4—23no—
35C22×C32×C4C2×D313yes—
36C22×C32×C4C22×C313yes—
37C22×C3⋊Dic3C22×C311yescharacteristic
38C2×C4⋊(C3⋊D3)—212no—
39C2×C4⋊(C3⋊D3)—26no—
40C22×C3×Dic3—31no—
41C2×C3×D12—612no—
42C2×C3×D12—66no—
43#3752—26no—
44C3×(C2×D3)⋊C4—66no—
45C22×H(3,𝔽3)C2×D413yes—
46C22×H(3,𝔽3)C2×D413yes—
47C22×H(3,𝔽3)#4111yescharacteristic
48C2×(C3⋊D3)⋊C3—212no—
49C2×(C3⋊D3)⋊C3—26no—
50C2×(C3⋊D3)⋊C3—43no—
51C2×(C3⋊D3)⋊C3—46no—
52C2×(C3⋊D3)⋊C3—43no—
53C4×H(3,𝔽3)—26no—
54(C3⋊Dic3)⋊C3—41no—
55C22×C3×D4—93no—
56C22×D12—33no—
57C22×D12—93no—
58C2×C3×C22⋊C4—93no—
59C2×(C2×D3)⋊C4—33no—
60C2×(C2×D3)⋊C4—93no—
61C3×#95—31no—
62C3×#95—31no—
63C3×#95—61no—
64C3×#95C3×D311yescharacteristic
65C23×C32—21no—
66C23×C32C22×D311yescharacteristic
67C23×C32C23×C311yescharacteristic
68C22×C3⋊D3—23no—
69C22×C3⋊D3—26no—
70C22×C3⋊D3—26no—
71C22×C3⋊D3—23no—
72C22×C3⋊D3C3×D416yescomplemented
73C22×C3×D3—36no—
74C22×C3×D3—63no—
75C22×C3×D3—66no—
76C22×C3×D3—66no—
77C22×C3×D3—63no—
78C2×C32×C4—26no—
79C2×C32×C4—26no—
80C2×C32×C4—26no—
81C2×C32×C4—46no—
82C2×C32×C4D1216yescomplemented
83C2×C32×C4C3×D416yescomplemented
84C2×C3⋊Dic3—23no—
85C4⋊(C3⋊D3)—224no—
86C2×C3×Dic3—63no—
87C3×D12—624no—
88#392—91noSylow
89C2×H(3,𝔽3)#12013yes—
90C2×H(3,𝔽3)#12113yes—
91C2×H(3,𝔽3)#12711yescharacteristic
92(C3⋊D3)⋊C3—46no—
93C24×C3—93no—
94C23×D3—33no—
95C23×D3—93no—
96C2×C3×D4—186no—
97C2×C3×D4—1812no—
98C22×C3×C4—33no—
99C22×C3×C4—33no—
100C22×C3×C4—63no—
101C22×C3×C4—91no—
102C22×C3×C4C2×C3×D313yes—
103C22×Dic3—31no—
104C22×Dic3—91no—
105C2×D12—612no—
106C2×D12—66no—
107C2×D12—186no—
108C2×D12—1812no—
109C3×C22⋊C4—186no—
110(C2×D3)⋊C4—66no—
111(C2×D3)⋊C4—186no—
112C22×C32—23no—
113C22×C32—23no—
114C22×C32—21no—
115C22×C32C2×D1213yes—
116C22×C32D3×D413yes—
117C22×C32#23911yescharacteristic
118C22×C32C2×C3×D413yes—
119C22×C32C2×C3×D413yes—
120C22×C32C3×#4111yescharacteristic
121C2×C3⋊D3—212no—
122C2×C3⋊D3—26no—
123C2×C3⋊D3—43no—
124C2×C3⋊D3—46no—
125C2×C3⋊D3—43no—
126C2×C3×D3—612no—
127C2×C3×D3—66no—
128C2×C3×D3—123no—
129C2×C3×D3—123no—
130C2×C3×D3—126no—
131C32×C4—26no—
132C32×C4—26no—
133C32×C4—46no—
134C3⋊Dic3—41no—
135C3×Dic3—121no—
136C22×D4—93no—
137C2×C22⋊C4—93no—
138#95(C3⋊D3)⋊C311yescharacteristic, complemented
139H(3,𝔽3)#39211yescharacteristic, complemented, Sylow
140C23×C3—31no—
141C23×C3—31no—
142C23×C3—61no—
143C23×C3—96no—
144C23×C3—183no—
145C23×C3—186no—
146C23×C3—186no—
147C23×C3—183no—
148C23×C3C22×C3×D311yescharacteristic
149C22×D3—36no—
150C22×D3—63no—
151C22×D3—66no—
152C22×D3—66no—
153C22×D3—63no—
154C22×D3—96no—
155C22×D3—183no—
156C22×D3—186no—
157C22×D3—186no—
158C22×D3—183no—
159C3×D4—1824no—
160C2×C3×C4—26no—
161C2×C3×C4—36no—
162C2×C3×C4—36no—
163C2×C3×C4—66no—
164C2×C3×C4—66no—
165C2×C3×C4—66no—
166C2×C3×C4—126no—
167C2×C3×C4—183no—
168C2×C3×C4C3×D1216yes—
169C2×Dic3—63no—
170C2×Dic3—183no—
171D12—624no—
172D12—1824no—
173C2×C32—23no—
174C2×C32—23no—
175C2×C32—21no—
176C2×C32#87411yescharacteristic
177C2×C32#88213yes—
178C2×C32#89513yes—
179C2×C32C3×#12013yes—
180C2×C32C3×#12113yes—
181C2×C32C3×#12711yescharacteristic
182C3⋊D3—46no—
183C3×D3—126no—
184C24—93no—
185C2×D4—1812no—
186C2×D4—186no—
187C22×C4—91no—
188C22×C4C2×(C3⋊D3)⋊C313yes—
189C22⋊C4—186no—
190C22×C3—33no—
191C22×C3—33no—
192C22×C3—33no—
193C22×C3—33no—
194C22×C3—31no—
195C22×C3—31no—
196C22×C3—63no—
197C22×C3—63no—
198C22×C3—61no—
199C22×C3—1812no—
200C22×C3—186no—
201C22×C3—363no—
202C22×C3—363no—
203C22×C3—366no—
204C22×C3C2×C3×D1213yes—
205C22×C3C3×D3×D413yes—
206C22×C3C3×#23911yescharacteristic
207C2×D3—612no—
208C2×D3—66no—
209C2×D3—123no—
210C2×D3—126no—
211C2×D3—123no—
212C2×D3—1812no—
213C2×D3—186no—
214C2×D3—363no—
215C2×D3—363no—
216C2×D3—366no—
217C3×C4—26no—
218C3×C4—66no—
219C3×C4—66no—
220C3×C4—126no—
221C3×C4—361no—
222Dic3—121no—
223Dic3—361no—
224C32—21no—
225C32#467111yescharacteristic, complemented
226C32C3×#39211yescharacteristic, complemented
227C23—96no—
228C23—183no—
229C23—186no—
230C23—186no—
231C23—183no—
232C23C22×(C3⋊D3)⋊C311yescharacteristic
233D4—1824no—
234C2×C4—26no—
235C2×C4—183no—
236C2×C4C4⋊((C3⋊D3)⋊C3)16yescomplemented
237C2×C3—31no—
238C2×C3—31no—
239C2×C3—33no—
240C2×C3—33no—
241C2×C3—33no—
242C2×C3—33no—
243C2×C3—61no—
244C2×C3—63no—
245C2×C3—63no—
246C2×C3—366no—
247C2×C3C3×#87411yescharacteristic
248C2×C3C3×#88213yes—
249C2×C3C3×#89513yes—
250D3—126no—
251D3—366no—
252C22—1812no—
253C22—186no—
254C22—363no—
255C22—363no—
256C22—366no—
257C22C2×C4⋊((C3⋊D3)⋊C3)13yes—
258C22D4×(C3⋊D3)⋊C313yes—
259C22#8872211yescharacteristic
260C4—26no—
261C4—361no—
262C3—31no—
263C3—31no—
264C3—61no—
265C3C3×#467111yescharacteristic
266C2—366no—
267C2#1172138511yescharacteristic
268C2#64091⋊C313yes—
269C2#64104⋊C313yes—
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC23×C32C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingH(3,𝔽3)×#95C2
hypercenter#95(C3⋊D3)⋊C3
hypocenterC32C3×#392
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
2. C2×#3752⋊C3C2semidirect
3. H(3,𝔽3)×#95C2semidirect
4. #10592370C3semidirect
12. C22×C4⋊(C3⋊D3)C2×C3semidirect
14. C2×#3752C2×C3semidirect
17. C32×#95D3semidirect
18. C32×#95C2×C3semidirect
24. C22×(C3⋊D3)⋊C3D4semidirect
26. C2×C4×H(3,𝔽3)D4semidirect
72. C22×C3⋊D3C3×D4semidirect
82. C2×C32×C4D12semidirect
83. C2×C32×C4C3×D4semidirect
138. #95(C3⋊D3)⋊C3semidirect
139. H(3,𝔽3)#392semidirect
225. C32#4671semidirect
226. C32C3×#392semidirect
236. C2×C4C4⋊((C3⋊D3)⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊((C3⋊D3)⋊C3)2
C2×#3752⋊C32
H(3,𝔽3)×#952
#105923703
C3×#46713