Finite Group Databasebeta

← C4×C3⋊((C22⋊C9)⋊C4) #13147037

Subgroups — C4×C3⋊((C22⋊C9)⋊C4) #13147037

Order 1728.

690 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 135 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×C4×C22⋊C9C211yesmaximal, characteristic
2C2×C3⋊((C22⋊C9)⋊C4)C212yesmaximal
3C4×#3757—31nomaximal
4C4×(C22⋊C9)⋊C4—33nomaximal
5C22×C3×C22⋊C9C2211yescharacteristic
6C3×C4×C22⋊C9C412yescomplemented
7C3⋊((C22⋊C9)⋊C4)C414yesdirect product factor
8C4×C3⋊Dic9—41nomaximal
9C23×C32×C4D311yescharacteristic
10C2×#3757—32no—
11C2×C4×C3⋊Dic3—31no—
12C2×C4×C3⋊Dic3—31no—
13C2×C4×C22⋊C9D313yes—
14C2×(C22⋊C9)⋊C4—36no—
15#63658—32no—
16C2×C3×C22⋊C9C2×C412yes—
17C2×C3×C22⋊C9C2×C411yescharacteristic
18C2×C3×C4×C9—41no—
19C2×C3⋊Dic9—42no—
20C4×#217—93no—
21C4×#217—91no—
22C24×C32C2×D311yescharacteristic
23C22×C32×C4—31no—
24C22×C32×C4—32no—
25C22×C32×C4—31no—
26C22×C32×C4Dic312yes—
27C22×C3⋊Dic3—32no—
28C22×C3⋊Dic3—32no—
29C22×C22⋊C9C2×D313yes—
30#3757—34no—
31#3757—34no—
32C4×C3⋊Dic3—62no—
33C4×C3⋊Dic3—61no—
34C4×C3⋊Dic3—61no—
35C4×C22⋊C9Dic316yescomplemented
36(C22⋊C9)⋊C4—312no—
37C4×Dic9—123no—
38C22×C3×C9—41no—
39C3×C22⋊C9C4211yescharacteristic, complemented
40C3×C4×C9—42no—
41C3⋊Dic9—44no—
42C23×C3×C4D913yes—
43C23×C3×C4C3⋊D311yescharacteristic
44C2×#217—96no—
45C2×#217—92no—
46C2×C4×Dic3—93no—
47C2×C4×Dic3—91no—
48C2×C4×Dic3—93no—
49C2×C4×Dic3—91no—
50C3⋊#95—96no—
51C3⋊#95—92no—
52C23×C32—31no—
53C23×C32—32no—
54C23×C32—31no—
55C23×C32C4×D312yes—
56C23×C32C2×Dic311yescharacteristic
57C2×C32×C4—31no—
58C2×C32×C4—32no—
59C2×C32×C4—32no—
60C2×C32×C4—32no—
61C2×C32×C4—61no—
62C2×C32×C4S411yescharacteristic
63C2×C3⋊Dic3—34no—
64C2×C3⋊Dic3—34no—
65C2×C3⋊Dic3—62no—
66C2×C3⋊Dic3—62no—
67C2×C3⋊Dic3—62no—
68C2×C3⋊Dic3—62no—
69C2×C22⋊C9C4×D316yes—
70C2×C22⋊C9C2×Dic313yes—
71C2×C4×C9—43no—
72C2×Dic9—126no—
73C4×C22⋊C4—271noSylow
74C2×C3×C9—42no—
75C2×C3×C9—41no—
76C24×C3C2×D913yes—
77C24×C3C2×C3⋊D311yescharacteristic
78C22×C3×C4—33no—
79C22×C3×C4—31no—
80C22×C3×C4—33no—
81C22×C3×C4—31no—
82C22×C3×C4—36no—
83C22×C3×C4—32no—
84C22×C3×C4Dic916yescomplemented
85C22×C3×C4C3⋊Dic312yes—
86C22×Dic3—92no—
87C22×Dic3—96no—
88C22×Dic3—92no—
89C22×Dic3—96no—
90#217—912no—
91#217—94no—
92#217—912no—
93#217—94no—
94C4×Dic3—186no—
95C4×Dic3—182no—
96C4×Dic3—181no—
97C4×Dic3—183no—
98C4×Dic3—183no—
99C4×Dic3—181no—
100C22×C32—32no—
101C22×C32—31no—
102C22×C32—32no—
103C22×C32—32no—
104C22×C32—31no—
105C22×C32—31no—
106C22×C32—61no—
107C22×C32C4×Dic311yescharacteristic
108C22×C32C2×S411yescharacteristic
109C32×C4—32no—
110C32×C4SL(2,ℤ/4)12yes—
111C3⋊Dic3—64no—
112C3⋊Dic3—64no—
113C22×C9—43no—
114C22⋊C9C4×Dic313yescomplemented
115C4×C9—46no—
116Dic9—1212no—
117C23×C4C3⋊D911yescharacteristic
118C2×C22⋊C4—272no—
119C2×C42—271no—
120C2×C42—271no—
121#95—272no—
122C3×C9—41noSylow
123C23×C3—31no—
124C23×C3—33no—
125C23×C3—36no—
126C23×C3—32no—
127C23×C3—33no—
128C23×C3—31no—
129C23×C3C4×D916yes—
130C23×C3C2×Dic913yes—
131C23×C3C4×C3⋊D312yes—
132C23×C3C2×C3⋊Dic311yescharacteristic
133C2×C3×C4—31no—
134C2×C3×C4—33no—
135C2×C3×C4—36no—
136C2×C3×C4—36no—
137C2×C3×C4—32no—
138C2×C3×C4—32no—
139C2×C3×C4—36no—
140C2×C3×C4—32no—
141C2×C3×C4—61no—
142C2×C3×C4—63no—
143C2×C3×C4(C22⋊C9)⋊C213yes—
144C2×C3×C4C3⋊S411yescharacteristic
145C2×Dic3—94no—
146C2×Dic3—912no—
147C2×Dic3—94no—
148C2×Dic3—912no—
149C2×Dic3—182no—
150C2×Dic3—186no—
151C2×Dic3—182no—
152C2×Dic3—186no—
153C2×Dic3—182no—
154C2×Dic3—186no—
155C2×Dic3—182no—
156C2×Dic3—186no—
157C2×C32—32no—
158C2×C32—31no—
159C2×C32—31no—
160C2×C32C4×S412yes—
161C2×C32C2×SL(2,ℤ/4)11yescharacteristic
162C2×C9—46no—
163C2×C9—43no—
164C24C2×C3⋊D911yescharacteristic
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—32no—
168C22×C4—272no—
169C22×C4—272no—
170C22×C4C3⋊Dic912yescomplemented
171C22⋊C4—274no—
172C22⋊C4—274no—
173C42—541no—
174C42—542no—
175C42—541no—
176C22×C3—36no—
177C22×C3—33no—
178C22×C3—32no—
179C22×C3—31no—
180C22×C3—32no—
181C22×C3—32no—
182C22×C3—31no—
183C22×C3—31no—
184C22×C3—36no—
185C22×C3—36no—
186C22×C3—33no—
187C22×C3—33no—
188C22×C3—63no—
189C22×C3—61no—
190C22×C3C4×Dic913yescomplemented
191C22×C3C4×C3⋊Dic311yescharacteristic
192C22×C3C2×(C22⋊C9)⋊C213yes—
193C22×C3C2×C3⋊S411yescharacteristic
194C3×C4—32no—
195C3×C4—36no—
196C3×C4(C22⋊C9)⋊C416yescomplemented
197C3×C4C3⋊SL(2,ℤ/4)12yes—
198Dic3—1812no—
199Dic3—184no—
200Dic3—184no—
201Dic3—1812no—
202C32C4×SL(2,ℤ/4)11yescharacteristic
203C9—43no—
204C23—31no—
205C23—32no—
206C23—31no—
207C23C4×C3⋊D912yes—
208C23C2×C3⋊Dic911yescharacteristic
209C2×C4—31no—
210C2×C4—32no—
211C2×C4—32no—
212C2×C4—32no—
213C2×C4—61no—
214C2×C4—274no—
215C2×C4—274no—
216C2×C4—542no—
217C2×C4—542no—
218C2×C4—542no—
219C2×C4—542no—
220C2×C4C3⋊((C22⋊C9)⋊C2)11yescharacteristic
221C2×C3—36no—
222C2×C3—33no—
223C2×C3—33no—
224C2×C3—32no—
225C2×C3—31no—
226C2×C3—31no—
227C2×C3C4×(C22⋊C9)⋊C216yes—
228C2×C3C2×(C22⋊C9)⋊C413yes—
229C2×C3C4×C3⋊S412yes—
230C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
231C22—32no—
232C22—31no—
233C22—32no—
234C22—32no—
235C22—31no—
236C22—31no—
237C22—61no—
238C22C4×C3⋊Dic911yescharacteristic, complemented
239C22C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
240C4—32no—
241C4—544no—
242C4—544no—
243C4C3⋊((C22⋊C9)⋊C4)12yesdirect product factor
244C3C4×(C22⋊C9)⋊C413yescomplemented
245C3C4×C3⋊SL(2,ℤ/4)11yescharacteristic
246C2—32no—
247C2—31no—
248C2—31no—
249C2C4×C3⋊((C22⋊C9)⋊C2)12yes—
250C2C2×C3⋊((C22⋊C9)⋊C4)11yescharacteristic
subgroupG/H
center, hypercenterC2×C4C3⋊((C22⋊C9)⋊C2)
commutator, hypocenterC3×C22⋊C9C42
FrattiniC22×C3C2×C3⋊S4
FittingC23×C32×C4D3
socle, prime radicalC24×C32C2×D3
cosocleC22×C3×C22⋊C9C22
LCSG ▷ C3×C22⋊C9
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
6. C3×C4×C22⋊C9C4semidirect
7. C3⋊((C22⋊C9)⋊C4)C4direct (N × H)
35. C4×C22⋊C9Dic3semidirect
39. C3×C22⋊C9C42semidirect
84. C22×C3×C4Dic9semidirect
114. C22⋊C9C4×Dic3semidirect
170. C22×C4C3⋊Dic9semidirect
190. C22×C3C4×Dic9semidirect
196. C3×C4(C22⋊C9)⋊C4semidirect
238. C22C4×C3⋊Dic9semidirect
243. C4C3⋊((C22⋊C9)⋊C4)direct (N × H)
244. C3C4×(C22⋊C9)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C22⋊C92
C2×C3⋊((C22⋊C9)⋊C4)2
C4×#37573
C4×(C22⋊C9)⋊C43
C4×C3⋊Dic94