Finite Group Databasebeta

← C23×C3×SL(2,ℤ/4) #11923711

Subgroups — C23×C3×SL(2,ℤ/4) #11923711

Order 1152.

3444 conjugacy classes of subgroups; only the 504 normal subgroups are shown.

normal subgroups only

This group has 3444 conjugacy classes of subgroups — too many to draw. Only its 504 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
1C24×C3×A4C211yesmaximal, characteristic
2C22×C3×SL(2,ℤ/4)C2114yesmaximal, direct product factor
3C23×SL(2,ℤ/4)C311yesmaximal, characteristic, direct product factor
4C23×C3×C22⋊C4—31nomaximal
5C23×C3×A4C418yescomplemented
6C23×C3×A4C2217yes—
7C23×C3×Dic3—41nomaximal
8C2×C3×SL(2,ℤ/4)C22128yesdirect product factor
9C26×C3D311yescharacteristic
10C24×A4—21no—
11C24×A4C2×C311yescharacteristic
12C24×C3×C4—31no—
13C24×C3×C4—31no—
14C22×SL(2,ℤ/4)C2×C3114yesdirect product factor
15C22×C3×C22⋊C4—314no—
16C22×C3×C22⋊C4—314no—
17C24×C32—41no—
18C22×C3×A4C2×C4128yescomplemented
19C22×C3×A4C2317yes—
20C22×C3×Dic3—414no—
21C3×SL(2,ℤ/4)C2318yesdirect product factor
22C23×C22⋊C4—31noSylow
23C25×C3—31no—
24C25×C3—38no—
25C25×C3—37no—
26C25×C3Dic318yescomplemented
27C25×C3C2×D317yes—
28C23×A4—28no—
29C23×A4—27no—
30C23×A4C3×C418yescomplemented
31C23×A4C22×C317yes—
32C23×C3×C4—314no—
33C23×C3×C4—314no—
34C23×C3×C4—68no—
35C23×C3×C4—67no—
36C23×C3×C4—61no—
37C23×Dic3—41no—
38C2×SL(2,ℤ/4)C22×C3128yesdirect product factor
39C2×C3×C22⋊C4—384no—
40C2×C3×C22⋊C4—328no—
41C23×C32—48no—
42C23×C32—47no—
43C2×C3×A4C22×C4114yescomplemented
44C2×C3×A4C2411yescharacteristic
45C2×C3×Dic3—428no—
46C26C3×D311yescharacteristic
47C24×C4—31no—
48C24×C4—31no—
49C22×C22⋊C4—314no—
50C22×C22⋊C4—314no—
51C24×C3—38no—
52C24×C3—38no—
53C24×C3—37no—
54C24×C3—37no—
55C24×C3—356no—
56C24×C3—328no—
57C24×C3—321no—
58C24×C3—41no—
59C24×C3—628no—
60C24×C3—67no—
61C24×C3—81no—
62C24×C3C2×Dic3128yescomplemented
63C24×C3S411yescharacteristic
64C24×C3C22×D317yes—
65C22×A4—228no—
66C22×A4—27no—
67C22×A4C2×C3×C4128yescomplemented
68C22×A4C23×C317yes—
69C22×C3×C4—328no—
70C22×C3×C4—328no—
71C22×C3×C4—656no—
72C22×C3×C4—642no—
73C22×C3×C4—614no—
74C22×Dic3—414no—
75SL(2,ℤ/4)C23×C318yesdirect product factor
76C3×C22⋊C4—356no—
77C3×C22⋊C4—38no—
78C22×C32—428no—
79C22×C32—47no—
80C3×A4C23×C411yescharacteristic, complemented
81C3×Dic3—48no—
82C25—31no—
83C25—37no—
84C25—38no—
85C25C3×Dic318yescomplemented
86C25C2×C3×D317yes—
87C23×C4—314no—
88C23×C4—314no—
89C23×C4—68no—
90C23×C4—67no—
91C23×C4—61no—
92C2×C22⋊C4—384no—
93C2×C22⋊C4—328no—
94C23×C3—328no—
95C23×C3—328no—
96C23×C3—356no—
97C23×C3—37no—
98C23×C3—321no—
99C23×C3—37no—
100C23×C3—384no—
101C23×C3—314no—
102C23×C3—47no—
103C23×C3—48no—
104C23×C3—642no—
105C23×C3—656no—
106C23×C3—67no—
107C23×C3—87no—
108C23×C3—88no—
109C23×C3SL(2,ℤ/4)18yesdirect product factor
110C23×C3C22×Dic3114yescomplemented
111C23×C3C2×S417yes—
112C23×C3C23×D311yescharacteristic
113C2×A4—21no—
114C2×A4—214no—
115C2×A4C22×C3×C4114yescomplemented
116C2×A4C24×C311yescharacteristic
117C2×C3×C4—38no—
118C2×C3×C4—38no—
119C2×C3×C4—656no—
120C2×C3×C4—628no—
121C2×C3×C4—628no—
122C2×Dic3—428no—
123C2×C32—414no—
124C2×C32—41no—
125C24—37no—
126C24—37no—
127C24—38no—
128C24—38no—
129C24—321no—
130C24—356no—
131C24—328no—
132C24—628no—
133C24—67no—
134C24C2×C3×Dic3128yescomplemented
135C24C3×S411yescharacteristic
136C24C22×C3×D317yes—
137C22×C4—328no—
138C22×C4—328no—
139C22×C4—656no—
140C22×C4—642no—
141C22×C4—614no—
142C22⋊C4—356no—
143C22⋊C4—38no—
144C22×C3—314no—
145C22×C3—31no—
146C22×C3—314no—
147C22×C3—314no—
148C22×C3—384no—
149C22×C3—31no—
150C22×C3—37no—
151C22×C3—428no—
152C22×C3—47no—
153C22×C3—67no—
154C22×C3—628no—
155C22×C3—828no—
156C22×C3—87no—
157C22×C3C2×SL(2,ℤ/4)128yesdirect product factor
158C22×C3C23×Dic311yescharacteristic, complemented
159C22×C3C22×S417yes—
160A4—21no—
161A4C23×C3×C411yescharacteristic, complemented
162C3×C4—68no—
163C3×C4—68no—
164Dic3—48no—
165C32—41noSylow
166C23—328no—
167C23—328no—
168C23—356no—
169C23—37no—
170C23—321no—
171C23—37no—
172C23—384no—
173C23—314no—
174C23—67no—
175C23—642no—
176C23—656no—
177C23C3×SL(2,ℤ/4)18yesdirect product factor
178C23C22×C3×Dic3114yescomplemented
179C23C2×C3×S417yes—
180C23C23×C3×D311yescharacteristic
181C2×C4—38no—
182C2×C4—38no—
183C2×C4—656no—
184C2×C4—628no—
185C2×C4—628no—
186C2×C3—31no—
187C2×C3—31no—
188C2×C3—314no—
189C2×C3—41no—
190C2×C3—414no—
191C2×C3—81no—
192C2×C3—814no—
193C2×C3C22×SL(2,ℤ/4)114yesdirect product factor
194C2×C3C23×S411yescharacteristic
195C22—31no—
196C22—314no—
197C22—31no—
198C22—37no—
199C22—314no—
200C22—314no—
201C22—384no—
202C22—67no—
203C22—628no—
204C22C2×C3×SL(2,ℤ/4)128yesdirect product factor
205C22C23×C3×Dic311yescharacteristic, complemented
206C22C22×C3×S417yes—
207C4—68no—
208C4—68no—
209C3—41no—
210C3—81no—
211C3C23×SL(2,ℤ/4)11yescharacteristic, direct product factor
212C2—31no—
213C2—31no—
214C2—314no—
215C2C22×C3×SL(2,ℤ/4)114yesdirect product factor
216C2C23×C3×S411yescharacteristic
subgroupG/H
center, hypercenterC24×C3S4
commutator, hypocenterA4C23×C3×C4
FrattiniC2C23×C3×S4
Fitting, socleC26×C3D3
prime radicalC24×C3×A4C2
cosocleC2×A4C24×C3
LCSG ▷ A4
UCSC1 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
2. C22×C3×SL(2,ℤ/4)C2direct (N × H)
3. C23×SL(2,ℤ/4)C3direct (N × H)
5. C23×C3×A4C4semidirect
8. C2×C3×SL(2,ℤ/4)C22direct (N × H)
14. C22×SL(2,ℤ/4)C2×C3direct (N × H)
18. C22×C3×A4C2×C4semidirect
21. C3×SL(2,ℤ/4)C23direct (N × H)
26. C25×C3Dic3semidirect
30. C23×A4C3×C4semidirect
38. C2×SL(2,ℤ/4)C22×C3direct (N × H)
43. C2×C3×A4C22×C4semidirect
62. C24×C3C2×Dic3semidirect
67. C22×A4C2×C3×C4semidirect
75. SL(2,ℤ/4)C23×C3direct (N × H)
80. C3×A4C23×C4semidirect
85. C25C3×Dic3semidirect
109. C23×C3SL(2,ℤ/4)direct (N × H)
110. C23×C3C22×Dic3semidirect
115. C2×A4C22×C3×C4semidirect
134. C24C2×C3×Dic3semidirect
157. C22×C3C2×SL(2,ℤ/4)direct (N × H)
158. C22×C3C23×Dic3semidirect
161. A4C23×C3×C4semidirect
177. C23C3×SL(2,ℤ/4)direct (N × H)
178. C23C22×C3×Dic3semidirect
193. C2×C3C22×SL(2,ℤ/4)direct (N × H)
204. C22C2×C3×SL(2,ℤ/4)direct (N × H)
205. C22C23×C3×Dic3semidirect
211. C3C23×SL(2,ℤ/4)direct (N × H)
215. C2C22×C3×SL(2,ℤ/4)direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3×A42
C22×C3×SL(2,ℤ/4)2
C23×SL(2,ℤ/4)3
C23×C3×C22⋊C43
C23×C3×Dic34