Finite Group Databasebeta

← #11724201

Subgroups — #11724201

Order 864.

381 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 46 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⋊(C3⋊S4)C211yesmaximal, characteristic
2C3×Dic3×A4C211yesmaximal, characteristic, complemented
3C3×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4C4×C3⋊S4—31nomaximal
5(C3⋊S4)⋊C4—33nomaximal
6#63952—31nomaximal
7C2×C32×A4C2211yescharacteristic
8C3⋊(C3⋊S4)C412yescomplemented
9#6289—41nomaximal
10C2×C3⋊S4—31no—
11C2×C3⋊S4—33no—
12C2×C3⋊S4—61no—
13C2×C3⋊S4—63no—
14C2×#638—31no—
15C22×C3×Dic3D311yescharacteristic, complemented
16C3×C4×A4—31no—
17C2×#624—31no—
18Dic3×A4D313yescomplemented
19C3⋊SL(2,ℤ/4)D311yescharacteristic, complemented
20C3×SL(2,ℤ/4)—33no—
21C3×#217—31no—
22#3724—31no—
23#3722—31no—
24Dic32—31no—
25C2×C3⋊(C3⋊D3)—41no—
26C32×A4C2×C411yescharacteristic, complemented
27C3×C3⋊Dic3—41no—
28C32×Dic3—41no—
29C4×S4—93no—
30C4×#67—91no—
31#67⋊C4—91no—
32C23×C32C2×D311yescharacteristic
33C22×C3⋊D3—31no—
34C2×C3×A4—21no—
35C2×C3×A4—23no—
36C2×C3×A4C2×D313yes—
37C2×C3×A4C2×D311yescharacteristic
38C3⋊S4—32no—
39C3⋊S4—36no—
40C3⋊S4—62no—
41C3⋊S4—66no—
42#638—32no—
43#638—32no—
44C2×C3⋊Dic3—31no—
45C4×C3⋊D3—121no—
46C2×C3×Dic3—31no—
47C2×C3×Dic3—31no—
48C2×C3×Dic3—31no—
49C2×C3×Dic3—31no—
50#624—61no—
51#624—61no—
52#624—123no—
53C2×C33—41no—
54C3⋊(C3⋊D3)—42no—
55C2×S4—93no—
56C2×S4—183no—
57C22×C3×C4—31no—
58C2×#67—91no—
59C2×#67—91no—
60C2×#67—181no—
61C22×Dic3C3⋊D311yescharacteristic, complemented
62C2×C4×D3—91no—
63C2×C4×D3—91no—
64C4×A4—33no—
65SL(2,ℤ/4)—33no—
66C3×C22⋊C4—91no—
67#217—31no—
68(C2×D3)⋊C4—91no—
69(C2×D3)⋊C4—91no—
70Dic3⋊C4—91no—
71Dic3⋊C4—91no—
72C4×Dic3—91no—
73C4×Dic3—91no—
74C22×C32—31no—
75C22×C32—31no—
76C22×C32C4×D311yescharacteristic, complemented
77C2×C3⋊D3—32no—
78C2×C3⋊D3—61no—
79C2×C3⋊D3—61no—
80C2×C3⋊D3—121no—
81C2×C3⋊D3—123no—
82C2×C3⋊D3—241no—
83C2×C3⋊D3—243no—
84C3×A4—21no—
85C3×A4—23no—
86C3×A4C4×D313yescomplemented
87C3×A4C4×D311yescharacteristic, complemented
88C32×C4—121no—
89C3⋊Dic3—32no—
90C3⋊Dic3—41no—
91C3×Dic3—31no—
92C3×Dic3—43no—
93C3×Dic3—61no—
94C3×Dic3—61no—
95C3×Dic3—123no—
96C3×Dic3S411yescharacteristic, complemented
97C4×D4—271noSylow
98C33—41noSylow
99C23×C3—21no—
100C23×C3D3211yescharacteristic
101C23×C3C2×C3⋊D311yescharacteristic
102C22×D3—91no—
103C22×D3—91no—
104C22×D3—181no—
105C2×A4—23no—
106C2×A4D3213yes—
107S4—96no—
108S4—186no—
109C2×C3×C4—91no—
110C2×C3×C4—91no—
111C2×C3×C4—91no—
112C2×C3×C4—91no—
113#67—92no—
114#67—92no—
115#67—92no—
116#67—92no—
117#67—182no—
118#67—182no—
119C2×Dic3—31no—
120C2×Dic3—31no—
121C2×Dic3—31no—
122C2×Dic3—31no—
123C2×Dic3—91no—
124C2×Dic3—91no—
125C2×Dic3—181no—
126C4×D3—181no—
127C4×D3—181no—
128C4×D3—181no—
129C4×D3—181no—
130C4×D3—363no—
131C2×C32—31no—
132C2×C32—31no—
133C2×C32—41no—
134C2×C32—43no—
135C2×C32—83no—
136C2×C32—81no—
137C2×C32C2×S411yescharacteristic
138C3⋊D3—62no—
139C3⋊D3—122no—
140C3⋊D3—126no—
141C3⋊D3—242no—
142C3⋊D3—246no—
143C2×D4—271no—
144C22×C4—31no—
145C22×C4—271no—
146C4⋊C4—271no—
147C22⋊C4—91no—
148C22⋊C4—271no—
149C42—271no—
150C22×C3—21no—
151C22×C3—31no—
152C22×C3—31no—
153C22×C3—31no—
154C22×C3—31no—
155C22×C3—61no—
156C22×C3—61no—
157C22×C3#62411yescharacteristic, complemented
158C22×C3C4×C3⋊D311yescharacteristic, complemented
159C2×D3—92no—
160C2×D3—92no—
161C2×D3—182no—
162C2×D3—181no—
163C2×D3—181no—
164C2×D3—181no—
165C2×D3—181no—
166C2×D3—361no—
167C2×D3—363no—
168C2×D3—361no—
169C2×D3—723no—
170A4—23no—
171A4#62413yescomplemented
172C3×C4—31no—
173C3×C4—91no—
174C3×C4—123no—
175C3×C4—181no—
176C3×C4—181no—
177Dic3—31no—
178Dic3—61no—
179Dic3—61no—
180Dic3—92no—
181Dic3—92no—
182Dic3—123no—
183Dic3—182no—
184Dic3C3⋊S411yescharacteristic, complemented
185C32—41no—
186C32—43no—
187C32—83no—
188C32—81no—
189C32C4×S411yescharacteristic, complemented
190C23—271no—
191C23D3×C3⋊D311yescharacteristic
192D4—272no—
193D4—272no—
194C2×C4—91no—
195C2×C4—91no—
196C2×C4—91no—
197C2×C4—91no—
198C2×C4—271no—
199C2×C4—541no—
200C2×C4—541no—
201C2×C3—21no—
202C2×C3—31no—
203C2×C3—31no—
204C2×C3—31no—
205C2×C3—31no—
206C2×C3—43no—
207C2×C3—61no—
208C2×C3—61no—
209C2×C3—83no—
210C2×C3D3×S411yescharacteristic
211C2×C3C2×C3⋊S411yescharacteristic
212D3—182no—
213D3—182no—
214D3—362no—
215D3—366no—
216D3—726no—
217C22—31no—
218C22—31no—
219C22—272no—
220C22—541no—
221C22—541no—
222C22#628911yescharacteristic, complemented
223C4—31no—
224C4—91no—
225C4—181no—
226C4—181no—
227C4—272no—
228C3—21no—
229C3—43no—
230C3—83no—
231C3(C3⋊S4)⋊C411yescharacteristic, complemented
232C3C4×C3⋊S411yescharacteristic, complemented
233C2—31no—
234C2—31no—
235C2—542no—
236C2D3×C3⋊S411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×C3⋊S4
commutator, hypocenterC32×A4C2×C4
Fitting, socleC23×C32C2×D3
prime radicalC2×C3⋊(C3⋊S4)C2
cosocleC2×C32×A4C22
LCSG ▷ C32×A4
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. C3×Dic3×A4C2semidirect
3. C3×C3⋊SL(2,ℤ/4)C2semidirect
8. C3⋊(C3⋊S4)C4semidirect
15. C22×C3×Dic3D3semidirect
18. Dic3×A4D3semidirect
19. C3⋊SL(2,ℤ/4)D3semidirect
26. C32×A4C2×C4semidirect
61. C22×Dic3C3⋊D3semidirect
76. C22×C32C4×D3semidirect
86. C3×A4C4×D3semidirect
87. C3×A4C4×D3semidirect
96. C3×Dic3S4semidirect
157. C22×C3#624semidirect
158. C22×C3C4×C3⋊D3semidirect
171. A4#624semidirect
184. Dic3C3⋊S4semidirect
189. C32C4×S4semidirect
222. C22#6289semidirect
231. C3(C3⋊S4)⋊C4semidirect
232. C3C4×C3⋊S4semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(C3⋊S4)2
C3×Dic3×A42
C3×C3⋊SL(2,ℤ/4)2
C4×C3⋊S43
(C3⋊S4)⋊C43
#639523
#62894