Finite Group Databasebeta

← #11724213

Subgroups — #11724213

Order 864.

283 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 36 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
2A4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C3×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4(C3⋊S4)⋊C4—31nomaximal
5Dic3×S4—31nomaximal
6#67⋊Dic3—31nomaximal
7C2×C32×A4C2211yescharacteristic
8C3×C3⋊S4C412yescomplemented
9#6294—41nomaximal
10C2×C3⋊S4D311yescharacteristic
11C2×C3×S4—31no—
12C2×C3×S4—61no—
13C22×C3⋊Dic3D311yescharacteristic, complemented
14C2×C3×#67—31no—
15C2×D3×Dic3—31no—
16Dic3×A4—31no—
17Dic3×A4—31no—
18Dic3×A4—61no—
19C3⋊SL(2,ℤ/4)D311yescharacteristic, complemented
20C3×SL(2,ℤ/4)—31no—
21C3×SL(2,ℤ/4)—61no—
22C3×#217—31no—
23Dic3⋊Dic3—31no—
24(C2×D3)⋊Dic3—31no—
25Dic32—31no—
26C2×C3×C3⋊D3—41no—
27C32×A4C2×C411yescharacteristic, complemented
28C3×C3⋊Dic3—41no—
29C3×C3⋊Dic3—41no—
30C4×S4—91no—
31D4×Dic3—91no—
32#67⋊C4—91no—
33C23×C32C2×D311yescharacteristic
34C22×C3×D3—31no—
35C2×C3×A4—21no—
36C2×C3×A4—21no—
37C2×C3×A4—21no—
38C2×C3×A4—41no—
39C2×C3×A4C2×D311yescharacteristic
40C2×C3×A4C2×D311yescharacteristic
41C3⋊S4Dic312yescomplemented
42C3×S4—32no—
43C3×S4—62no—
44C2×C3⋊Dic3—31no—
45C2×C3⋊Dic3—31no—
46C3×#67—32no—
47C3×#67—32no—
48C2×C3×Dic3—31no—
49C2×C3×Dic3—31no—
50C2×C3×Dic3—31no—
51#624—121no—
52D3×Dic3—61no—
53D3×Dic3—61no—
54D3×Dic3—121no—
55C2×C33—41no—
56C3×C3⋊D3—42no—
57C2×S4—31no—
58C2×S4—61no—
59C2×C3×D4—91no—
60C2×#67—31no—
61C22×Dic3—31no—
62C22×Dic3—31no—
63C22×Dic3—61no—
64C22×Dic3—91no—
65C2×C4×D3—91no—
66C4×A4—91no—
67SL(2,ℤ/4)—31no—
68SL(2,ℤ/4)—61no—
69C3×C22⋊C4—91no—
70#217—31no—
71#217—91no—
72(C2×D3)⋊C4—91no—
73C4⋊Dic3—91no—
74Dic3⋊C4—91no—
75C4×Dic3—91no—
76C4×Dic3—91no—
77C22×C32—31no—
78C22×C32—31no—
79C22×C32C4×D311yescharacteristic, complemented
80C2×C3⋊D3—41no—
81C2×C3×D3—32no—
82C2×C3×D3—61no—
83C2×C3×D3—61no—
84C2×C3×D3—121no—
85C2×C3×D3—241no—
86C3×A4—21no—
87C3×A4—21no—
88C3×A4—21no—
89C3×A4—41no—
90C3×A4C4×D311yescharacteristic, complemented
91C3×A4C2×Dic311yescharacteristic, complemented
92C3⋊Dic3—31no—
93C3⋊Dic3—41no—
94C3⋊Dic3S411yescharacteristic, complemented
95C3×Dic3—32no—
96C3×Dic3—61no—
97C3×Dic3—61no—
98C3×Dic3—121no—
99C3×Dic3—121no—
100C3×Dic3—121no—
101C3×Dic3—241no—
102C3×Dic3—241no—
103C4×D4—271noSylow
104C33—41noSylow
105C23×C3—21no—
106C23×C3—91no—
107C23×C3D3211yescharacteristic
108C23×C3D3211yescharacteristic
109C22×D3—31no—
110C2×A4—21no—
111C2×A4—21no—
112C2×A4—41no—
113C2×A4D3211yescharacteristic
114S4—32no—
115S4—62no—
116C3×D4—92no—
117C3×D4—92no—
118C2×C3×C4—91no—
119C2×C3×C4—91no—
120C2×C3×C4—91no—
121#67—32no—
122#67—32no—
123C2×Dic3—31no—
124C2×Dic3—31no—
125C2×Dic3—31no—
126C2×Dic3—91no—
127C2×Dic3—91no—
128C2×Dic3—91no—
129C2×Dic3—91no—
130C2×Dic3—181no—
131C2×Dic3—181no—
132C2×Dic3—181no—
133C2×Dic3—181no—
134C4×D3—181no—
135C4×D3—181no—
136C4×D3—361no—
137C2×C32—31no—
138C2×C32—31no—
139C2×C32—41no—
140C2×C32—41no—
141C2×C32—81no—
142C2×C32—81no—
143C2×C32—81no—
144C2×C32—161no—
145C2×C32C2×S411yescharacteristic
146C3⋊D3—42no—
147C3×D3—62no—
148C3×D3—122no—
149C3×D3—242no—
150C2×D4—91no—
151C22×C4—91no—
152C22×C4—271no—
153C4⋊C4—271no—
154C22⋊C4—91no—
155C22⋊C4—271no—
156C42—271no—
157C22×C3—21no—
158C22×C3—31no—
159C22×C3—31no—
160C22×C3—31no—
161C22×C3—31no—
162C22×C3—61no—
163C22×C3—61no—
164C22×C3—92no—
165C22×C3—181no—
166C22×C3—181no—
167C22×C3D3×Dic311yescharacteristic, complemented
168C22×C3#62411yescharacteristic, complemented
169C2×D3—32no—
170C2×D3—61no—
171C2×D3—61no—
172C2×D3—121no—
173C2×D3—241no—
174A4—21no—
175A4—21no—
176A4—41no—
177A4D3×Dic311yescharacteristic, complemented
178C3×C4—92no—
179C3×C4—181no—
180C3×C4—181no—
181C3×C4—361no—
182Dic3—32no—
183Dic3—31no—
184Dic3—31no—
185Dic3—61no—
186Dic3—61no—
187Dic3—61no—
188Dic3—91no—
189Dic3—91no—
190Dic3—121no—
191Dic3—181no—
192Dic3—241no—
193C32—41no—
194C32—41no—
195C32—81no—
196C32—81no—
197C32—81no—
198C32—161no—
199C32C4×S411yescharacteristic, complemented
200C23—91no—
201C23#111311yescharacteristic
202D4—92no—
203D4—92no—
204C2×C4—91no—
205C2×C4—91no—
206C2×C4—91no—
207C2×C4—271no—
208C2×C4—271no—
209C2×C4—541no—
210C2×C4—541no—
211C2×C3—21no—
212C2×C3—31no—
213C2×C3—31no—
214C2×C3—31no—
215C2×C3—31no—
216C2×C3—41no—
217C2×C3—61no—
218C2×C3—61no—
219C2×C3—81no—
220C2×C3—81no—
221C2×C3—161no—
222C2×C3—182no—
223C2×C3D3×S411yescharacteristic
224C2×C3D3×S411yescharacteristic
225D3—62no—
226D3—122no—
227D3—242no—
228C22—31no—
229C22—31no—
230C22—92no—
231C22—181no—
232C22—181no—
233C22#629411yescharacteristic, complemented
234C4—92no—
235C4—91no—
236C4—181no—
237C4—181no—
238C4—271no—
239C3—21no—
240C3—41no—
241C3—81no—
242C3—81no—
243C3—161no—
244C3Dic3×S411yescharacteristic, complemented
245C3(C3⋊S4)⋊C411yescharacteristic, complemented
246C2—31no—
247C2—31no—
248C2—182no—
249C2#8910111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2#89101
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. A4×C3⋊Dic3C2semidirect
3. C3×C3⋊SL(2,ℤ/4)C2semidirect
8. C3×C3⋊S4C4semidirect
13. C22×C3⋊Dic3D3semidirect
19. C3⋊SL(2,ℤ/4)D3semidirect
27. C32×A4C2×C4semidirect
41. C3⋊S4Dic3semidirect
79. C22×C32C4×D3semidirect
90. C3×A4C4×D3semidirect
91. C3×A4C2×Dic3semidirect
94. C3⋊Dic3S4semidirect
167. C22×C3D3×Dic3semidirect
168. C22×C3#624semidirect
177. A4D3×Dic3semidirect
199. C32C4×S4semidirect
233. C22#6294semidirect
244. C3Dic3×S4semidirect
245. C3(C3⋊S4)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊S42
A4×C3⋊Dic32
C3×C3⋊SL(2,ℤ/4)2
(C3⋊S4)⋊C43
Dic3×S43
#67⋊Dic33
#62944