Finite Group Databasebeta

← C3×#73110 #11920887

Subgroups — C3×#73110 #11920887

Order 1152.

332 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 72 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×C8×A4C211yesmaximal, characteristic
2C3×C4⋊SL(2,ℤ/4)C212yesmaximal
3#73110C311yesmaximal, characteristic, direct product factor
4C3×#1807—31nomaximal
5C2×C3×C4×A4C2211yescharacteristic
6C2×C3×SL(2,ℤ/4)—22no—
7C3×C8×A4C412yescomplemented
8C3×#817—41nomaximal
9C23×C3×C8D311yescharacteristic
10C2×C8×A4—21no—
11C2×C8×A4C2×C311yescharacteristic
12C4⋊SL(2,ℤ/4)C2×C312yes—
13C2×C3×#106—31no—
14C2×C3×#106—31no—
15C3×#380—32no—
16C3×#340—32no—
17C22×C3×A4D411yescharacteristic
18C3×C4×A4C2×C411yescharacteristic
19C3×C4×A4Q811yescharacteristic
20C3×SL(2,ℤ/4)—42no—
21C2×C32×C8—41no—
22C3×C4⋊Dic3—42no—
23#1807—31noSylow
24C23×C3×C4C2×D311yescharacteristic
25C2×C4×A4—21no—
26C2×C4×A4C22×C311yescharacteristic
27C2×SL(2,ℤ/4)—22no—
28C22×C3×C8—31no—
29C22×C3×C8—32no—
30C22×C3×C8—31no—
31C22×C3×C8Dic312yescomplemented
32C2×C3×C4⋊C4—32no—
33C2×C3×C4⋊C4—32no—
34C2×C3×C22⋊C4—62no—
35C8×A4—22no—
36C8×A4C3×C412yescomplemented
37C3×#106—62no—
38C3×#106—61no—
39C3×#106—61no—
40C3×#95—62no—
41#817—41no—
42C2×C3×A4C4⋊C411yescharacteristic
43C2×C3×A4SD1612yes—
44C2×C32×C4—41no—
45C2×C3×Dic3—82no—
46C32×C8—42no—
47C23×C8C3×D311yescharacteristic
48C2×#106—31no—
49C2×#106—31no—
50#380—32no—
51#340—32no—
52C24×C3D1211yescharacteristic
53C22×A4—21no—
54C22×A4C3×D411yescharacteristic
55C22×C3×C4—31no—
56C22×C3×C4—31no—
57C22×C3×C4—31no—
58C22×C3×C4—31no—
59C22×C3×C4—62no—
60C22×C3×C4—62no—
61C22×C3×C4Dic611yescharacteristic
62C22×C3×C4C2×Dic311yescharacteristic
63C4×A4—21no—
64C4×A4—21no—
65C4×A4C2×C3×C411yescharacteristic
66C4×A4C3×Q811yescharacteristic
67SL(2,ℤ/4)—42no—
68C2×C3×C8—31no—
69C2×C3×C8—32no—
70C2×C3×C8—32no—
71C2×C3×C8—32no—
72C2×C3×C8—41no—
73C2×C3×C8—61no—
74C2×C3×C8—81no—
75C2×C3×C8S411yescharacteristic
76C3×C4⋊C4—62no—
77C3×C4⋊C4—62no—
78C3×C4⋊C4—62no—
79C3×C4⋊C4—62no—
80C3×C22⋊C4—122no—
81C3×C22⋊C4—122no—
82C4⋊Dic3—42no—
83C22×C32—41no—
84C3×A4#10611yescharacteristic, complemented
85C32×C4—41no—
86C32×C4—41no—
87C3×Dic3—162no—
88C23×C4C2×C3×D311yescharacteristic
89C22×C8—31no—
90C22×C8—31no—
91C22×C8—32no—
92C22×C8C3×Dic312yescomplemented
93C2×C4⋊C4—32no—
94C2×C4⋊C4—32no—
95C2×C22⋊C4—62no—
96#106—62no—
97#106—61no—
98#106—61no—
99#95—62no—
100C23×C3—31no—
101C23×C3—32no—
102C23×C3—31no—
103C23×C3#20412yes—
104C23×C3C4⋊Dic311yescharacteristic
105C2×A4—22no—
106C2×A4—21no—
107C2×A4C3×C4⋊C411yescharacteristic
108C2×A4C3×SD1612yes—
109C2×C3×C4—31no—
110C2×C3×C4—31no—
111C2×C3×C4—31no—
112C2×C3×C4—31no—
113C2×C3×C4—31no—
114C2×C3×C4—31no—
115C2×C3×C4—31no—
116C2×C3×C4—41no—
117C2×C3×C4—61no—
118C2×C3×C4—81no—
119C2×C3×C4—122no—
120C2×C3×C4—122no—
121C2×C3×C4—122no—
122C2×C3×C4—122no—
123C2×C3×C4—122no—
124C2×C3×C4—122no—
125C2×C3×C4C2×S411yescharacteristic
126C2×Dic3—82no—
127C3×C8—32no—
128C3×C8—42no—
129C3×C8—82no—
130C3×C8SL(2,ℤ/4)12yescomplemented
131C2×C32—42no—
132C2×C32—41no—
133C24C3×D1211yescharacteristic
134C22×C4—31no—
135C22×C4—31no—
136C22×C4—31no—
137C22×C4—31no—
138C22×C4—62no—
139C22×C4—62no—
140C22×C4C3×Dic611yescharacteristic
141C22×C4C2×C3×Dic311yescharacteristic
142C2×C8—31no—
143C2×C8—32no—
144C2×C8—32no—
145C2×C8—32no—
146C2×C8—61no—
147C2×C8C3×S411yescharacteristic
148C4⋊C4—62no—
149C4⋊C4—62no—
150C4⋊C4—62no—
151C4⋊C4—62no—
152C22⋊C4—122no—
153C22⋊C4—122no—
154C22×C3—32no—
155C22×C3—31no—
156C22×C3—32no—
157C22×C3—32no—
158C22×C3—31no—
159C22×C3—31no—
160C22×C3—41no—
161C22×C3—61no—
162C22×C3—81no—
163C22×C3#81711yescharacteristic, complemented
164C22×C3C4⋊S411yescharacteristic
165A4—21no—
166A4C3×#10611yescharacteristic, complemented
167C3×C4—31no—
168C3×C4—31no—
169C3×C4—41no—
170C3×C4—41no—
171C3×C4—81no—
172C3×C4—81no—
173C3×C4—242no—
174C3×C4—242no—
175C3×C4A4⋊Q811yescharacteristic
176C3×C4C2×SL(2,ℤ/4)11yescharacteristic
177Dic3—162no—
178C32—41noSylow
179C23—31no—
180C23—32no—
181C23—31no—
182C23C3×#20412yes—
183C23C3×C4⋊Dic311yescharacteristic
184C2×C4—31no—
185C2×C4—31no—
186C2×C4—31no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—31no—
190C2×C4—31no—
191C2×C4—61no—
192C2×C4—122no—
193C2×C4—122no—
194C2×C4—122no—
195C2×C4—122no—
196C2×C4—122no—
197C2×C4—122no—
198C2×C4C2×C3×S411yescharacteristic
199C8—32no—
200C8C3×SL(2,ℤ/4)12yescomplemented
201C2×C3—32no—
202C2×C3—31no—
203C2×C3—31no—
204C2×C3—42no—
205C2×C3—41no—
206C2×C3—82no—
207C2×C3—81no—
208C2×C3#540112yes—
209C2×C3C4⋊SL(2,ℤ/4)11yescharacteristic
210C22—32no—
211C22—31no—
212C22—32no—
213C22—32no—
214C22—31no—
215C22—31no—
216C22—61no—
217C22C3×#81711yescharacteristic, complemented
218C22C3×C4⋊S411yescharacteristic
219C4—31no—
220C4—31no—
221C4—242no—
222C4—242no—
223C4C3×A4⋊Q811yescharacteristic
224C4C2×C3×SL(2,ℤ/4)11yescharacteristic
225C3—41no—
226C3—81no—
227C3#7311011yescharacteristic, direct product factor
228C2—32no—
229C2—31no—
230C2—31no—
231C2C3×#540112yes—
232C2C3×C4⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C3C4⋊S4
commutatorC4×A4C2×C3×C4
FrattiniC2×C4C2×C3×S4
FittingC23×C3×C8D3
socleC24×C3D12
prime radicalC22×C3×A4D4
cosocleC2×C4×A4C22×C3
hypercenterC2×C3×C8S4
hypocenterA4C3×#106
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
3. #73110C3direct (N × H)
7. C3×C8×A4C4semidirect
31. C22×C3×C8Dic3semidirect
36. C8×A4C3×C4semidirect
84. C3×A4#106semidirect
92. C22×C8C3×Dic3semidirect
130. C3×C8SL(2,ℤ/4)semidirect
163. C22×C3#817semidirect
166. A4C3×#106semidirect
200. C8C3×SL(2,ℤ/4)semidirect
217. C22C3×#817semidirect
227. C3#73110direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C8×A42
C3×C4⋊SL(2,ℤ/4)2
#731103
C3×#18073
C3×#8174