Finite Group Databasebeta

← Dic3×C3⋊S4 #11724200

Subgroups — Dic3×C3⋊S4 #11724200

Order 864.

343 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 48 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⋊S4C211yesmaximal, characteristic
2C3×Dic3×A4C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
4C4×C3⋊S4—31nomaximal
5Dic3×S4—33nomaximal
6Dic3×#67—31nomaximal
7C2×C32×A4C2211yescharacteristic
8C3×C3⋊S4C412yescomplemented
9Dic3×C3⋊D3—41nomaximal
10C2×C3⋊S4D311yescharacteristic
11C2×C3×S4—33no—
12C2×C3×#67—31no—
13C22×C3×Dic3D311yescharacteristic, complemented
14C3×C4×A4—31no—
15C2×D3×Dic3—31no—
16Dic3×A4D313yescomplemented
17C3⋊SL(2,ℤ/4)—31no—
18C3⋊SL(2,ℤ/4)—33no—
19C3⋊SL(2,ℤ/4)—61no—
20C3⋊SL(2,ℤ/4)—63no—
21#3757—31no—
22Dic3⋊Dic3—31no—
23(C2×D3)⋊Dic3—31no—
24Dic32—31no—
25C2×C3×C3⋊D3—41no—
26C32×A4C2×C411yescharacteristic, complemented
27C3⋊(C3⋊Dic3)—41no—
28C32×Dic3—41no—
29C4×S4—93no—
30D4×Dic3—91no—
31C4×#67—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⋊S4Dic312yesdirect product factor
39C3×S4—36no—
40C2×C3⋊Dic3—31no—
41C2×C3⋊Dic3—31no—
42C4×C3⋊D3—121no—
43C3×#67—32no—
44C3×#67—32no—
45C2×C3×Dic3—31no—
46C2×C3×Dic3—31no—
47C2×C3×Dic3—31no—
48D3×Dic3—61no—
49D3×Dic3—61no—
50D3×Dic3—123no—
51C2×C33—41no—
52C3×C3⋊D3—42no—
53C2×S4—33no—
54C2×C3×D4—91no—
55C22×C3×C4—31no—
56C2×#67—31no—
57C22×Dic3—91no—
58C22×Dic3C3⋊D311yescharacteristic, complemented
59C2×C4×D3—91no—
60C4×A4—33no—
61SL(2,ℤ/4)—93no—
62SL(2,ℤ/4)—183no—
63#217—91no—
64#217—91no—
65#217—91no—
66#217—181no—
67(C2×D3)⋊C4—91no—
68C4⋊Dic3—91no—
69Dic3⋊C4—91no—
70C4×Dic3—91no—
71C4×Dic3—91no—
72C22×C32—31no—
73C22×C32—31no—
74C22×C32C4×D311yescharacteristic, complemented
75C2×C3⋊D3—41no—
76C2×C3×D3—32no—
77C2×C3×D3—61no—
78C2×C3×D3—61no—
79C2×C3×D3—123no—
80C3×A4—21no—
81C3×A4—23no—
82C3×A4C4×D313yescomplemented
83C3×A4C2×Dic311yescharacteristic, complemented
84C32×C4—121no—
85C3⋊Dic3—61no—
86C3⋊Dic3—61no—
87C3⋊Dic3—121no—
88C3⋊Dic3—123no—
89C3⋊Dic3—241no—
90C3⋊Dic3—243no—
91C3×Dic3—32no—
92C3×Dic3—31no—
93C3×Dic3—43no—
94C3×Dic3S411yescharacteristic, complemented
95C4×D4—271noSylow
96C33—41noSylow
97C23×C3—21no—
98C23×C3—91no—
99C23×C3D3211yescharacteristic
100C23×C3C2×C3⋊D311yescharacteristic
101C22×D3—31no—
102C2×A4—23no—
103C2×A4D3213yes—
104S4—36no—
105C3×D4—92no—
106C3×D4—92no—
107C2×C3×C4—91no—
108C2×C3×C4—91no—
109C2×C3×C4—91no—
110#67—32no—
111#67—32no—
112C2×Dic3—31no—
113C2×Dic3—31no—
114C2×Dic3—31no—
115C2×Dic3—91no—
116C2×Dic3—91no—
117C2×Dic3—91no—
118C2×Dic3—91no—
119C2×Dic3—181no—
120C2×Dic3—181no—
121C2×Dic3—181no—
122C2×Dic3—181no—
123C4×D3—181no—
124C4×D3—181no—
125C4×D3—363no—
126C2×C32—31no—
127C2×C32—31no—
128C2×C32—41no—
129C2×C32—43no—
130C2×C32—83no—
131C2×C32—81no—
132C2×C32C2×S411yescharacteristic
133C3⋊D3—42no—
134C3×D3—62no—
135C3×D3—126no—
136C2×D4—91no—
137C22×C4—31no—
138C22×C4—271no—
139C4⋊C4—271no—
140C22⋊C4—271no—
141C22⋊C4—271no—
142C42—271no—
143C22×C3—21no—
144C22×C3—31no—
145C22×C3—31no—
146C22×C3—31no—
147C22×C3—31no—
148C22×C3—61no—
149C22×C3—61no—
150C22×C3—92no—
151C22×C3—181no—
152C22×C3—181no—
153C22×C3D3×Dic311yescharacteristic, complemented
154C22×C3C4×C3⋊D311yescharacteristic, complemented
155C2×D3—32no—
156C2×D3—61no—
157C2×D3—61no—
158C2×D3—123no—
159A4—23no—
160A4D3×Dic313yescomplemented
161C3×C4—31no—
162C3×C4—92no—
163C3×C4—91no—
164C3×C4—123no—
165Dic3—32no—
166Dic3—31no—
167Dic3—181no—
168Dic3—181no—
169Dic3—181no—
170Dic3—181no—
171Dic3—361no—
172Dic3—363no—
173Dic3—361no—
174Dic3—723no—
175Dic3C3⋊S411yescharacteristic, direct product factor
176C32—41no—
177C32—43no—
178C32—83no—
179C32—81no—
180C32C4×S411yescharacteristic, complemented
181C23—91no—
182C23D3×C3⋊D311yescharacteristic
183D4—92no—
184D4—92no—
185C2×C4—91no—
186C2×C4—91no—
187C2×C4—91no—
188C2×C4—271no—
189C2×C4—271no—
190C2×C4—541no—
191C2×C4—541no—
192C2×C3—21no—
193C2×C3—31no—
194C2×C3—31no—
195C2×C3—31no—
196C2×C3—31no—
197C2×C3—43no—
198C2×C3—61no—
199C2×C3—61no—
200C2×C3—83no—
201C2×C3—182no—
202C2×C3D3×S411yescharacteristic
203C2×C3C2×C3⋊S411yescharacteristic
204D3—62no—
205D3—126no—
206C22—31no—
207C22—31no—
208C22—92no—
209C22—181no—
210C22—181no—
211C22Dic3×C3⋊D311yescharacteristic, complemented
212C4—31no—
213C4—92no—
214C4—91no—
215C4—541no—
216C4—541no—
217C3—21no—
218C3—43no—
219C3—83no—
220C3Dic3×S411yescharacteristic, complemented
221C3C4×C3⋊S411yescharacteristic, complemented
222C2—31no—
223C2—31no—
224C2—182no—
225C2D3×C3⋊S411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×C3⋊S4
commutator, hypocenterC32×A4C2×C4
Fitting, socleC23×C32C2×D3
prime radicalC2×C3×C3⋊S4C2
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⋊S4C4semidirect
13. C22×C3×Dic3D3semidirect
16. Dic3×A4D3semidirect
26. C32×A4C2×C4semidirect
38. C3⋊S4Dic3direct (N × H)
58. C22×Dic3C3⋊D3semidirect
74. C22×C32C4×D3semidirect
82. C3×A4C4×D3semidirect
83. C3×A4C2×Dic3semidirect
94. C3×Dic3S4semidirect
153. C22×C3D3×Dic3semidirect
154. C22×C3C4×C3⋊D3semidirect
160. A4D3×Dic3semidirect
175. Dic3C3⋊S4direct (N × H)
180. C32C4×S4semidirect
211. C22Dic3×C3⋊D3semidirect
220. C3Dic3×S4semidirect
221. 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⋊S42
C3×Dic3×A42
C3⋊(C3⋊SL(2,ℤ/4))2
C4×C3⋊S43
Dic3×S43
Dic3×#673
Dic3×C3⋊D34