Finite Group Databasebeta

← #11724205

Subgroups — #11724205

Order 864.

405 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 44 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×D3×A4C211yesmaximal, characteristic, complemented
2C2×C3⋊(C3⋊S4)C211yesmaximal, characteristic, complemented
3C3×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4#64269—31nomaximal
5#64210—33nomaximal
6#63977—31nomaximal
7C2×C32×A4C2211yescharacteristic
8C3×D3×A4—21no—
9C3⋊(C3⋊S4)—21no—
10#6291—41nomaximal
11C23×C3×D3D311yescharacteristic, complemented
12C22×C3×A4—31no—
13C2×D3×A4D313yescomplemented
14C2×C3⋊S4—31no—
15C2×C3⋊S4—33no—
16C2×C3⋊S4—61no—
17C2×C3⋊S4—63no—
18C2×#638—31no—
19C2×#626—31no—
20C2×#626—31no—
21C3⋊SL(2,ℤ/4)D311yescharacteristic, complemented
22C3×SL(2,ℤ/4)—33no—
23C3×#217—31no—
24(C2×D3)⋊Dic3—31no—
25(C2×D3)⋊Dic3—31no—
26C2×C3⋊(C3⋊D3)—41no—
27C2×C32×D3—41no—
28C32×A4D411yescharacteristic, complemented
29C3×C3⋊Dic3—41no—
30GL(2,ℤ/4)—93no—
31#953—91no—
32#882—91no—
33C23×C32C2×D311yescharacteristic
34C22×C3⋊D3—31no—
35C22×C3×D3—21no—
36C22×C3×D3—31no—
37C22×C3×D3—31no—
38C22×C3×D3—61no—
39C2×C3×A4—21no—
40C2×C3×A4—23no—
41C2×C3×A4—61no—
42C2×C3×A4C2×D313yes—
43C2×C3×A4C2×D311yescharacteristic
44D3×A4—23no—
45C3⋊S4—66no—
46C3⋊S4—62no—
47C3⋊S4—61no—
48C3⋊S4—63no—
49#638—61no—
50#638—61no—
51#638—121no—
52C2×C3⋊Dic3—31no—
53C2×C3×Dic3—31no—
54C2×C3×Dic3—31no—
55#626—62no—
56#626—61no—
57#626—61no—
58#626—123no—
59C2×C33—41no—
60C3⋊(C3⋊D3)—81no—
61C32×D3—81no—
62C24×C3—31no—
63C23×D3C3⋊D311yescharacteristic, complemented
64C22×A4—33no—
65C2×S4—93no—
66C2×S4—183no—
67C2×#67—91no—
68C2×#67—91no—
69C2×#67—91no—
70C2×#67—91no—
71C2×#67—181no—
72C2×D12—91no—
73C2×D12—91no—
74SL(2,ℤ/4)—33no—
75C3×C22⋊C4—91no—
76#217—31no—
77#217—91no—
78#217—91no—
79(C2×D3)⋊C4—91no—
80(C2×D3)⋊C4—91no—
81C22×C32—31no—
82C22×C32—31no—
83C22×C32—121no—
84C22×C32#6711yescharacteristic, complemented
85C2×C3⋊D3—61no—
86C2×C3⋊D3—61no—
87C2×C3⋊D3—61no—
88C2×C3⋊D3—121no—
89C2×C3⋊D3—123no—
90C2×C3⋊D3—241no—
91C2×C3⋊D3—243no—
92C2×C3×D3—32no—
93C2×C3×D3—31no—
94C2×C3×D3—43no—
95C2×C3×D3—61no—
96C2×C3×D3—61no—
97C2×C3×D3—61no—
98C2×C3×D3S411yescharacteristic, complemented
99C3×A4—21no—
100C3×A4—23no—
101C3×A4D1211yescharacteristic, complemented
102C3×A4#6713yescomplemented
103C3⋊Dic3—41no—
104C3⋊Dic3—61no—
105C3×Dic3—61no—
106C3×Dic3—61no—
107C3×Dic3—123no—
108#120—271noSylow
109C33—41noSylow
110C23×C3—21no—
111C23×C3—61no—
112C23×C3—91no—
113C23×C3—91no—
114C23×C3—181no—
115C23×C3D3211yescharacteristic
116C23×C3C2×C3⋊D311yescharacteristic
117C22×D3—21no—
118C22×D3—31no—
119C22×D3—31no—
120C22×D3—61no—
121C22×D3—91no—
122C22×D3—91no—
123C22×D3—181no—
124C2×A4—23no—
125C2×A4—63no—
126C2×A4D3213yes—
127S4—186no—
128S4—183no—
129C2×C3×C4—91no—
130C2×C3×C4—91no—
131#67—181no—
132#67—182no—
133#67—181no—
134#67—182no—
135#67—181no—
136#67—181no—
137#67—182no—
138#67—181no—
139#67—181no—
140#67—363no—
141C2×Dic3—31no—
142C2×Dic3—31no—
143C2×Dic3—91no—
144C2×Dic3—91no—
145C2×Dic3—181no—
146D12—182no—
147D12—181no—
148D12—181no—
149C2×C32—31no—
150C2×C32—31no—
151C2×C32—41no—
152C2×C32—43no—
153C2×C32—83no—
154C2×C32—81no—
155C2×C32—241no—
156C2×C32C2×S411yescharacteristic
157C3⋊D3—121no—
158C3⋊D3—243no—
159C3⋊D3—241no—
160C3⋊D3—246no—
161C3⋊D3—242no—
162C3×D3—21no—
163C3×D3—61no—
164C3×D3—83no—
165C24—31no—
166C2×D4—271no—
167C2×D4—271no—
168C2×D4—271no—
169C22⋊C4—91no—
170C22⋊C4—271no—
171C22⋊C4—271no—
172C22×C3—21no—
173C22×C3—31no—
174C22×C3—31no—
175C22×C3—31no—
176C22×C3—31no—
177C22×C3—31no—
178C22×C3—61no—
179C22×C3—61no—
180C22×C3—92no—
181C22×C3—91no—
182C22×C3—123no—
183C22×C3—181no—
184C22×C3—181no—
185C22×C3—181no—
186C22×C3#62611yescharacteristic, complemented
187C22×C3#63811yescharacteristic, complemented
188C2×D3—32no—
189C2×D3—31no—
190C2×D3—61no—
191C2×D3—61no—
192C2×D3—61no—
193C2×D3—181no—
194C2×D3—181no—
195C2×D3—181no—
196C2×D3—181no—
197C2×D3—182no—
198C2×D3—181no—
199C2×D3—181no—
200C2×D3—361no—
201C2×D3—361no—
202C2×D3—363no—
203C2×D3—723no—
204C2×D3C3⋊S411yescharacteristic, complemented
205A4—23no—
206A4#62613yescomplemented
207C3×C4—181no—
208C3×C4—181no—
209Dic3—61no—
210Dic3—61no—
211Dic3—123no—
212Dic3—181no—
213Dic3—181no—
214Dic3—182no—
215C32—41no—
216C32—43no—
217C32—83no—
218C32—81no—
219C32GL(2,ℤ/4)11yescharacteristic, complemented
220C23—61no—
221C23—91no—
222C23—91no—
223C23—181no—
224C23—271no—
225C23D3×C3⋊D311yescharacteristic
226D4—541no—
227D4—542no—
228D4—541no—
229D4—541no—
230D4—541no—
231C2×C4—91no—
232C2×C4—91no—
233C2×C4—271no—
234C2×C3—21no—
235C2×C3—31no—
236C2×C3—31no—
237C2×C3—31no—
238C2×C3—31no—
239C2×C3—43no—
240C2×C3—61no—
241C2×C3—61no—
242C2×C3—61no—
243C2×C3—83no—
244C2×C3—181no—
245C2×C3—243no—
246C2×C3D3×S411yescharacteristic
247C2×C3C2×C3⋊S411yescharacteristic
248D3—21no—
249D3—61no—
250D3—362no—
251D3—361no—
252D3—361no—
253D3—723no—
254D3—726no—
255C22—31no—
256C22—31no—
257C22—31no—
258C22—92no—
259C22—91no—
260C22—181no—
261C22—181no—
262C22—181no—
263C22—541no—
264C22—541no—
265C22—541no—
266C22#629111yescharacteristic, complemented
267C4—181no—
268C4—181no—
269C4—541no—
270C3—21no—
271C3—43no—
272C3—83no—
273C3#6421011yescharacteristic, complemented
274C3#6426911yescharacteristic, complemented
275C2—31no—
276C2—31no—
277C2—61no—
278C2—181no—
279C2—1081no—
280C2D3×C3⋊S411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×C3⋊S4
commutator, cosocleC2×C32×A4C22
Fitting, socleC23×C32C2×D3
hypocenterC32×A4D4
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C3×D3×A4C2semidirect
2. C2×C3⋊(C3⋊S4)C2semidirect
3. C3×C3⋊SL(2,ℤ/4)C2semidirect
11. C23×C3×D3D3semidirect
13. C2×D3×A4D3semidirect
21. C3⋊SL(2,ℤ/4)D3semidirect
28. C32×A4D4semidirect
63. C23×D3C3⋊D3semidirect
84. C22×C32#67semidirect
98. C2×C3×D3S4semidirect
101. C3×A4D12semidirect
102. C3×A4#67semidirect
186. C22×C3#626semidirect
187. C22×C3#638semidirect
204. C2×D3C3⋊S4semidirect
206. A4#626semidirect
219. C32GL(2,ℤ/4)semidirect
266. C22#6291semidirect
273. C3#64210semidirect
274. C3#64269semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×A42
C2×C3⋊(C3⋊S4)2
C3×C3⋊SL(2,ℤ/4)2
#642693
#642103
#639773
#62914