Finite Group Databasebeta

← C2×#11724463 #13181393

Subgroups — C2×#11724463 #13181393

Order 1728.

1598 conjugacy classes of subgroups; only the 283 normal subgroups are shown.

normal subgroups only

This group has 1598 conjugacy classes of subgroups — too many to draw. Only its 283 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×C32×C4×A4C211yesmaximal, characteristic
2C2×C3⋊(C3⋊SL(2,ℤ/4))C212yesmaximal
3#11724463C214yesmaximal, direct product factor
4C2×#64259—312nomaximal
5C2×#64133—31nomaximal
6C22×C32×A4C2211yescharacteristic
7C2×#6308—41nomaximal
8C32×C4×A4C2212yes—
9C3⋊(C3⋊SL(2,ℤ/4))C2214yes—
10C23×C32×C4D311yescharacteristic
11C22×#634—31no—
12C2×C3×C4×A4D3112yes—
13C2×C3⋊SL(2,ℤ/4)—324no—
14#64259—348no—
15C2×#3757—32no—
16C2×C4⋊(C3⋊Dic3)—31no—
17#64133—34no—
18#64133—34no—
19C2×#3750—32no—
20C2×C32×A4Q812yescomplemented
21C2×C32×A4C2311yescharacteristic
22C2×C33×C4—41no—
23C2×C3⋊(C3⋊Dic3)—42no—
24#6308—44no—
25C2×A4⋊Q8—99no—
26C2×#924—94no—
27C24×C32C2×D311yescharacteristic
28C22×C3×A4C2×D3112yes—
29C22×C32×C4—31no—
30C22×C32×C4—31no—
31C22×C32×C4—32no—
32C22×C32×C4C2×D312yes—
33C22×C3⋊Dic3—32no—
34C22×C3⋊Dic3—32no—
35C2×#634—34no—
36C2×#634—61no—
37C2×#634—62no—
38C2×#634—61no—
39C2×#634—1212no—
40C3×C4×A4C2×D3124yes—
41C3⋊SL(2,ℤ/4)—348no—
42#3757—34no—
43#3757—34no—
44C4⋊(C3⋊Dic3)—34no—
45#3750—38no—
46C22×C33—41no—
47C32×A4C2×Q811yescharacteristic, complemented
48C33×C4—42no—
49C3⋊(C3⋊Dic3)—44no—
50C23×C3×C4C3⋊D314yes—
51C22×Dic6—94no—
52C2×C4×A4C3⋊D319yes—
53C2×SL(2,ℤ/4)—918no—
54A4⋊Q8—936no—
55C2×#217—98no—
56C2×C4⋊Dic3—94no—
57#924—916no—
58#924—916no—
59C2×Dic3⋊C4—98no—
60C23×C32—31no—
61C23×C32—32no—
62C23×C32—31no—
63C23×C32Dic612yescomplemented
64C23×C32C22×D311yescharacteristic
65C2×C3×A4Dic6124yescomplemented
66C2×C3×A4C22×D3112yes—
67C2×C32×C4—31no—
68C2×C32×C4—32no—
69C2×C32×C4—32no—
70C2×C32×C4—32no—
71C2×C32×C4—412no—
72C2×C32×C4—61no—
73C2×C32×C4S411yescharacteristic
74C2×C3⋊Dic3—34no—
75C2×C3⋊Dic3—34no—
76C2×C3⋊Dic3—62no—
77C2×C3⋊Dic3—64no—
78C2×C3⋊Dic3—62no—
79C2×C3⋊Dic3—1224no—
80#634—64no—
81#634—64no—
82#634—1248no—
83C2×C22⋊Q8—271noSylow
84C2×C33—42no—
85C2×C33—41no—
86C24×C3C2×C3⋊D314yes—
87C22×A4C2×C3⋊D319yes—
88C22×C3×C4—34no—
89C22×C3×C4—34no—
90C22×C3×C4—38no—
91C22×C3×C4C2×C3⋊D318yes—
92C22×Dic3—98no—
93C22×Dic3—98no—
94C2×Dic6—916no—
95C2×Dic6—184no—
96C2×Dic6—188no—
97C2×Dic6—184no—
98C2×Dic6—369no—
99C4×A4C2×C3⋊D3118yes—
100SL(2,ℤ/4)—936no—
101#217—916no—
102#217—916no—
103C4⋊Dic3—916no—
104Dic3⋊C4—932no—
105C22×C32—32no—
106C22×C32—31no—
107C22×C32—32no—
108C22×C32—32no—
109C22×C32—31no—
110C22×C32—31no—
111C22×C32—412no—
112C22×C32—61no—
113C22×C32C2×Dic611yescharacteristic, complemented
114C22×C32C2×S411yescharacteristic
115C3×A4C2×Dic6112yescomplemented
116C32×C4—32no—
117C32×C4—424no—
118C32×C4C2×S412yes—
119C3⋊Dic3—64no—
120C3⋊Dic3—64no—
121C3⋊Dic3—1248no—
122C22×Q8—271no—
123C23×C4C3⋊(C3⋊D3)11yescharacteristic
124C22⋊Q8—274no—
125C22⋊Q8—274no—
126C2×C4⋊C4—272no—
127C2×C4⋊C4—271no—
128C2×C22⋊C4—272no—
129C33—41noSylow
130C23×C3—34no—
131C23×C3—34no—
132C23×C3—38no—
133C23×C3#63418yescomplemented
134C23×C3C22×C3⋊D314yes—
135C2×A4#634118yescomplemented
136C2×A4C22×C3⋊D319yes—
137C2×C3×C4—34no—
138C2×C3×C4—38no—
139C2×C3×C4—38no—
140C2×C3×C4—38no—
141C2×C3×C4—49no—
142C2×C3×C4—64no—
143C2×C3×C4C3⋊S414yes—
144C2×Dic3—916no—
145C2×Dic3—916no—
146C2×Dic3—188no—
147C2×Dic3—1816no—
148C2×Dic3—188no—
149C2×Dic3—3618no—
150Dic6—1816no—
151Dic6—1816no—
152Dic6—3636no—
153C2×C32—31no—
154C2×C32—31no—
155C2×C32—32no—
156C2×C32—424no—
157C2×C32—412no—
158C2×C32A4⋊Q812yescomplemented
159C2×C32C22×S411yescharacteristic
160C24C2×C3⋊(C3⋊D3)11yescharacteristic
161C2×Q8—274no—
162C2×Q8—541no—
163C2×Q8—542no—
164C2×Q8—541no—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—32no—
168C22×C4—272no—
169C22×C4—272no—
170C22×C4C2×C3⋊(C3⋊D3)12yes—
171C4⋊C4—278no—
172C4⋊C4—274no—
173C22⋊C4—274no—
174C22⋊C4—274no—
175C22×C3—38no—
176C22×C3—34no—
177C22×C3—38no—
178C22×C3—38no—
179C22×C3—34no—
180C22×C3—34no—
181C22×C3—49no—
182C22×C3—64no—
183C22×C3C2×#63414yescomplemented
184C22×C3C2×C3⋊S414yes—
185A4C2×#63419yescomplemented
186C3×C4—38no—
187C3×C4—418no—
188C3×C4C2×C3⋊S418yes—
189Dic3—1816no—
190Dic3—1816no—
191Dic3—3636no—
192C32—412no—
193C32C2×A4⋊Q811yescharacteristic, complemented
194C23—31no—
195C23—31no—
196C23—32no—
197C23#630812yescomplemented
198C23C22×C3⋊(C3⋊D3)11yescharacteristic
199Q8—544no—
200Q8—544no—
201C2×C4—31no—
202C2×C4—32no—
203C2×C4—32no—
204C2×C4—32no—
205C2×C4—61no—
206C2×C4—274no—
207C2×C4—274no—
208C2×C4—542no—
209C2×C4—544no—
210C2×C4—542no—
211C2×C4C3⋊(C3⋊S4)11yescharacteristic
212C2×C3—34no—
213C2×C3—34no—
214C2×C3—38no—
215C2×C3—418no—
216C2×C3—49no—
217C2×C3#6425918yescomplemented
218C2×C3C22×C3⋊S414yes—
219C22—32no—
220C22—31no—
221C22—32no—
222C22—32no—
223C22—31no—
224C22—31no—
225C22—61no—
226C22C2×#630811yescharacteristic, complemented
227C22C2×C3⋊(C3⋊S4)11yescharacteristic
228C4—32no—
229C4—544no—
230C4—544no—
231C4C2×C3⋊(C3⋊S4)12yes—
232C3—49no—
233C3C2×#6425914yescomplemented
234C2—31no—
235C2—31no—
236C2—32no—
237C2#1172446312yesdirect product factor
238C2C22×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
centerC22C2×C3⋊(C3⋊S4)
commutator, cosocleC2×C32×A4C23
FrattiniC2C22×C3⋊(C3⋊S4)
FittingC23×C32×C4D3
socleC24×C32C2×D3
prime radicalC22×C32×A4C22
hypercenterC2×C4C3⋊(C3⋊S4)
hypocenterC32×A4C2×Q8
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11724463C2direct (N × H)
20. C2×C32×A4Q8semidirect
47. C32×A4C2×Q8semidirect
63. C23×C32Dic6semidirect
65. C2×C3×A4Dic6semidirect
113. C22×C32C2×Dic6semidirect
115. C3×A4C2×Dic6semidirect
133. C23×C3#634semidirect
135. C2×A4#634semidirect
158. C2×C32A4⋊Q8semidirect
183. C22×C3C2×#634semidirect
185. A4C2×#634semidirect
193. C32C2×A4⋊Q8semidirect
197. C23#6308semidirect
217. C2×C3#64259semidirect
226. C22C2×#6308semidirect
233. C3C2×#64259semidirect
237. C2#11724463direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C4×A42
C2×C3⋊(C3⋊SL(2,ℤ/4))2
#117244632
C2×#642593
C2×#641333
C2×#63084