Finite Group Databasebeta

← #11724204

Subgroups — #11724204

Order 864.

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

normal subgroups only

This group has 367 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⋊S4C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
4#64269—31nomaximal
5#64209—33nomaximal
6#63976—31nomaximal
7C2×C32×A4C2211yescharacteristic
8C3×D3×A4—21no—
9C3×C3⋊S4—21no—
10#6290—41nomaximal
11C23×C3×D3D311yescharacteristic, complemented
12C22×C3×A4—31no—
13C2×D3×A4D313yescomplemented
14C2×C3⋊S4D311yescharacteristic, complemented
15C2×C3×S4—33no—
16C2×C3×#67—31no—
17C2×#625—31no—
18C2×#625—31no—
19C3⋊SL(2,ℤ/4)—31no—
20C3⋊SL(2,ℤ/4)—33no—
21C3⋊SL(2,ℤ/4)—61no—
22C3⋊SL(2,ℤ/4)—63no—
23#3757—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#937—91no—
33C23×C32C2×D311yescharacteristic
34C22×C3×D3—21no—
35C22×C3×D3—31no—
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—21no—
46C3×S4—63no—
47#638—121no—
48C2×C3⋊Dic3—31no—
49C2×C3⋊Dic3—31no—
50C3×#67—61no—
51C3×#67—61no—
52C2×C3×Dic3—31no—
53#625—62no—
54#625—61no—
55#625—61no—
56#625—123no—
57C2×C33—41no—
58C3×C3⋊D3—81no—
59C32×D3—81no—
60C24×C3—31no—
61C23×D3C3⋊D311yescharacteristic, complemented
62C22×A4—33no—
63C2×S4—33no—
64C2×C3×D4—91no—
65C2×#67—31no—
66C2×#67—91no—
67C2×#67—91no—
68C2×#67—91no—
69C2×#67—91no—
70SL(2,ℤ/4)—93no—
71SL(2,ℤ/4)—183no—
72#217—91no—
73#217—91no—
74#217—91no—
75#217—91no—
76#217—181no—
77(C2×D3)⋊C4—91no—
78(C2×D3)⋊C4—91no—
79C22×C32—31no—
80C22×C32—31no—
81C22×C32—121no—
82C22×C32#6711yescharacteristic, complemented
83C2×C3⋊D3—41no—
84C2×C3×D3—32no—
85C2×C3×D3—31no—
86C2×C3×D3—43no—
87C2×C3×D3—61no—
88C2×C3×D3—61no—
89C2×C3×D3—61no—
90C2×C3×D3—61no—
91C2×C3×D3—61no—
92C2×C3×D3—61no—
93C2×C3×D3—123no—
94C2×C3×D3S411yescharacteristic, complemented
95C3×A4—21no—
96C3×A4—23no—
97C3×A4#6713yescomplemented
98C3×A4#6711yescharacteristic, complemented
99C3⋊Dic3—61no—
100C3⋊Dic3—61no—
101C3⋊Dic3—121no—
102C3⋊Dic3—123no—
103C3⋊Dic3—241no—
104C3⋊Dic3—243no—
105C3×Dic3—61no—
106#120—271noSylow
107C33—41noSylow
108C23×C3—21no—
109C23×C3—61no—
110C23×C3—91no—
111C23×C3—91no—
112C23×C3—91no—
113C23×C3—181no—
114C23×C3D3211yescharacteristic
115C23×C3C2×C3⋊D311yescharacteristic
116C22×D3—21no—
117C22×D3—31no—
118C22×D3—31no—
119C22×D3—31no—
120C22×D3—61no—
121C2×A4—23no—
122C2×A4—63no—
123C2×A4D3213yes—
124S4—63no—
125C3×D4—181no—
126C3×D4—181no—
127C2×C3×C4—91no—
128#67—61no—
129#67—61no—
130#67—182no—
131#67—181no—
132#67—181no—
133#67—182no—
134#67—181no—
135#67—181no—
136#67—363no—
137C2×Dic3—31no—
138C2×Dic3—91no—
139C2×Dic3—91no—
140C2×Dic3—91no—
141C2×Dic3—91no—
142C2×Dic3—181no—
143C2×Dic3—181no—
144C2×C32—31no—
145C2×C32—31no—
146C2×C32—41no—
147C2×C32—43no—
148C2×C32—83no—
149C2×C32—81no—
150C2×C32—241no—
151C2×C32C2×S411yescharacteristic
152C3⋊D3—81no—
153C3×D3—21no—
154C3×D3—61no—
155C3×D3—83no—
156C3×D3—121no—
157C3×D3—243no—
158C24—31no—
159C2×D4—91no—
160C2×D4—271no—
161C2×D4—271no—
162C22⋊C4—271no—
163C22⋊C4—271no—
164C22⋊C4—271no—
165C22×C3—21no—
166C22×C3—31no—
167C22×C3—31no—
168C22×C3—31no—
169C22×C3—31no—
170C22×C3—31no—
171C22×C3—61no—
172C22×C3—61no—
173C22×C3—92no—
174C22×C3—91no—
175C22×C3—123no—
176C22×C3—181no—
177C22×C3—181no—
178C22×C3—181no—
179C22×C3—181no—
180C22×C3—181no—
181C22×C3—181no—
182C22×C3#62511yescharacteristic, complemented
183C22×C3#63811yescharacteristic, complemented
184C2×D3—32no—
185C2×D3—31no—
186C2×D3—61no—
187C2×D3—61no—
188C2×D3—61no—
189C2×D3—61no—
190C2×D3—61no—
191C2×D3—61no—
192C2×D3—123no—
193C2×D3C3⋊S411yescharacteristic, complemented
194A4—23no—
195A4#62513yescomplemented
196C3×C4—181no—
197Dic3—61no—
198Dic3—181no—
199Dic3—181no—
200Dic3—181no—
201Dic3—181no—
202Dic3—361no—
203Dic3—363no—
204Dic3—361no—
205Dic3—723no—
206C32—41no—
207C32—43no—
208C32—83no—
209C32—81no—
210C32GL(2,ℤ/4)11yescharacteristic, complemented
211C23—61no—
212C23—91no—
213C23—91no—
214C23—91no—
215C23—181no—
216C23D3×C3⋊D311yescharacteristic
217D4—181no—
218D4—181no—
219D4—541no—
220D4—542no—
221D4—541no—
222C2×C4—91no—
223C2×C4—271no—
224C2×C4—271no—
225C2×C3—21no—
226C2×C3—31no—
227C2×C3—31no—
228C2×C3—31no—
229C2×C3—31no—
230C2×C3—43no—
231C2×C3—61no—
232C2×C3—61no—
233C2×C3—61no—
234C2×C3—83no—
235C2×C3—181no—
236C2×C3—243no—
237C2×C3—361no—
238C2×C3D3×S411yescharacteristic
239C2×C3C2×C3⋊S411yescharacteristic
240D3—21no—
241D3—61no—
242D3—121no—
243D3—243no—
244C22—31no—
245C22—31no—
246C22—31no—
247C22—92no—
248C22—91no—
249C22—181no—
250C22—181no—
251C22—181no—
252C22—181no—
253C22—181no—
254C22—181no—
255C22#629011yescharacteristic, complemented
256C4—181no—
257C4—541no—
258C4—541no—
259C3—21no—
260C3—43no—
261C3—83no—
262C3#6420911yescharacteristic, complemented
263C3#6426911yescharacteristic, complemented
264C2—31no—
265C2—31no—
266C2—61no—
267C2—181no—
268C2—361no—
269C2D3×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⋊S4C2semidirect
3. C3⋊(C3⋊SL(2,ℤ/4))C2semidirect
11. C23×C3×D3D3semidirect
13. C2×D3×A4D3semidirect
14. C2×C3⋊S4D3semidirect
28. C32×A4D4semidirect
61. C23×D3C3⋊D3semidirect
82. C22×C32#67semidirect
94. C2×C3×D3S4semidirect
97. C3×A4#67semidirect
98. C3×A4#67semidirect
182. C22×C3#625semidirect
183. C22×C3#638semidirect
193. C2×D3C3⋊S4semidirect
195. A4#625semidirect
210. C32GL(2,ℤ/4)semidirect
255. C22#6290semidirect
262. C3#64209semidirect
263. 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⋊S42
C3⋊(C3⋊SL(2,ℤ/4))2
#642693
#642093
#639763
#62904