Finite Group Databasebeta

← C4×((C3⋊D3)⋊C3)⋊C8 #13137374

Subgroups — C4×((C3⋊D3)⋊C3)⋊C8 #13137374

Order 1728.

611 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 187 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
1C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×((C3⋊D3)⋊C3)⋊C8C212yesmaximal
3C4×#6180C211yesmaximal, characteristic, complemented
4C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
5#11720310C212yesmaximal, complemented
6C4×#3710—31nomaximal
7C4×D3×C3⋊C8—31nomaximal
8C2×C4×(C3⋊D3)⋊C3C412yes—
9C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
10C42×H(3,𝔽3)C2211yescharacteristic
11C2×#6180C2212yes—
12C4×(C3⋊Dic3)⋊C3C412yes—
13C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15((C3⋊D3)⋊C3)⋊C8C418yesdirect product factor
16C42×C3⋊D3D311yescharacteristic
17C3×C42×D3—31no—
18C2×#3710—32no—
19C2×D3×C3⋊C8—32no—
20C4×C3⋊(C3⋊C8)D311yescharacteristic, complemented
21C3×C4×C3⋊C8—31no—
22C3×C4×C3⋊C8—31no—
23C3×C4×C3⋊C8—31no—
24C3×C4×C3⋊C8—61no—
25#63553—32no—
26Dic3×C3⋊C8—32no—
27C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
28C2×C4×H(3,𝔽3)C2×C412yes—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
31C2×(C3⋊Dic3)⋊C3C2×C412yes—
32C4×(C3⋊D3)⋊C3C818yescomplemented
33C4×(C3⋊D3)⋊C3C2×C414yes—
34#6180C2×C414yescomplemented
35(C3⋊(C3⋊C8))⋊C3C2×C414yescomplemented
36C2×C4×C3⋊C8—91no—
37C4×D3×C8—91no—
38C4×D3×C8—91no—
39C2×C4×C3⋊D3Dic312yes—
40C2×C4×C3⋊D3C2×D311yescharacteristic
41C2×C3×C4×D3—32no—
42C2×C3×C4×D3—31no—
43C32×C42—21no—
44C32×C42C2×D311yescharacteristic
45C32×C42C2×D311yescharacteristic
46C4×C3⋊Dic3Dic312yes—
47C4×C3⋊Dic3C2×D311yescharacteristic
48C2×C3⋊(C3⋊C8)C2×D312yes—
49C3×C4×Dic3—32no—
50C3×C4×Dic3—31no—
51C2×C3×C3⋊C8—32no—
52C2×C3×C3⋊C8—32no—
53C2×C3×C3⋊C8—32no—
54C2×C3×C3⋊C8—62no—
55#3710—38no—
56D3×C3⋊C8—38no—
57C22×H(3,𝔽3)C22×C411yescharacteristic
58C2×(C3⋊D3)⋊C3C4212yes—
59C2×(C3⋊D3)⋊C3C2×C814yes—
60C4×H(3,𝔽3)C2×C814yescomplemented
61C4×H(3,𝔽3)C22×C412yes—
62(C3⋊Dic3)⋊C3C4212yes—
63(C3⋊Dic3)⋊C3C2×C814yescomplemented
64C2×C3×C42—91no—
65C22×C3⋊C8—92no—
66C2×D3×C8—92no—
67C2×D3×C8—92no—
68C42×D3—31no—
69C42×D3—91no—
70C3×C4×C8—91no—
71C3×C4×C8—91no—
72C8×Dic3—92no—
73C8×Dic3—92no—
74C4×C3⋊C8—31no—
75C4×C3⋊C8—31no—
76C4×C3⋊C8—31no—
77C4×C3⋊C8—61no—
78C4×C3⋊C8—91no—
79C4×C3⋊C8—92no—
80C22×C3⋊D3C2×Dic311yescharacteristic
81C22×C3×D3—31no—
82C2×C32×C4—22no—
83C2×C32×C4—21no—
84C2×C32×C4C4×D312yes—
85C2×C32×C4C2×Dic312yes—
86C2×C32×C4C22×D311yescharacteristic
87C2×C32×C4C22×D311yescharacteristic
88C2×C3⋊Dic3C2×Dic312yes—
89C2×C3⋊Dic3C2×Dic311yescharacteristic
90C4×C3⋊D3C3⋊C818yescomplemented
91C4×C3⋊D3C4×D314yes—
92C2×C3×Dic3—31no—
93C2×C3×Dic3—32no—
94C3×C4×D3—38no—
95C3×C4×D3—34no—
96C3⋊(C3⋊C8)C4×D314yescomplemented
97C3×C3⋊C8—34no—
98C3×C3⋊C8—34no—
99C3×C3⋊C8—34no—
100C3×C3⋊C8—64no—
101C2×C4×C8—271noSylow
102C2×H(3,𝔽3)C2×C4211yescharacteristic
103C2×H(3,𝔽3)C22×C812yes—
104(C3⋊D3)⋊C3C4×C814yescomplemented
105C22×C3×C4—91no—
106C22×C3×C4—92no—
107C2×C4×D3—32no—
108C2×C4×D3—31no—
109C2×C4×D3—91no—
110C2×C4×D3—92no—
111C2×C3×C8—92no—
112C2×C3×C8—92no—
113C3×C42—31no—
114C3×C42—31no—
115C3×C42—61no—
116C3×C42—91no—
117C3×C42—92no—
118C3×C42D3211yescharacteristic
119C4×Dic3—32no—
120C4×Dic3—31no—
121C4×Dic3—91no—
122C4×Dic3—92no—
123C2×C3⋊C8—32no—
124C2×C3⋊C8—32no—
125C2×C3⋊C8—32no—
126C2×C3⋊C8—62no—
127C2×C3⋊C8—92no—
128C2×C3⋊C8—98no—
129D3×C8—98no—
130D3×C8—98no—
131C22×C32—21no—
132C22×C32C2×C4×D311yescharacteristic
133C22×C32C22×Dic311yescharacteristic
134C2×C3⋊D3C2×C3⋊C814yes—
135C2×C3⋊D3C4×Dic312yes—
136C2×C3×D3—32no—
137C2×C3×D3—34no—
138C32×C4—24no—
139C32×C4—22no—
140C32×C4D3×C814yescomplemented
141C32×C4C2×C3⋊C814yescomplemented
142C32×C4C2×C4×D312yes—
143C32×C4C2×C4×D312yes—
144C3⋊Dic3C2×C3⋊C814yescomplemented
145C3⋊Dic3C4×Dic312yes—
146C3×Dic3—32no—
147C3×Dic3—34no—
148C22×C8—272no—
149C2×C42—91no—
150C4×C8—91no—
151C4×C8—91no—
152C4×C8—272no—
153H(3,𝔽3)C2×C4×C811yescharacteristic, complemented, Sylow
154C23×C3—91no—
155C22×D3—31no—
156C22×D3—91no—
157C2×C3×C4—32no—
158C2×C3×C4—32no—
159C2×C3×C4—31no—
160C2×C3×C4—31no—
161C2×C3×C4—62no—
162C2×C3×C4—61no—
163C2×C3×C4—94no—
164C2×C3×C4—98no—
165C2×C3×C4—92no—
166C2×C3×C4—91no—
167C2×C3×C4D3×Dic312yes—
168C2×C3×C4C2×D3211yescharacteristic
169C2×Dic3—32no—
170C2×Dic3—31no—
171C2×Dic3—92no—
172C2×Dic3—91no—
173C4×D3—38no—
174C4×D3—34no—
175C4×D3—98no—
176C4×D3—94no—
177C3×C8—94no—
178C3×C8—94no—
179C3⋊C8—34no—
180C3⋊C8—34no—
181C3⋊C8—34no—
182C3⋊C8—64no—
183C3⋊C8—94no—
184C2×C32—22no—
185C2×C32—21no—
186C2×C32C42×D311yescharacteristic
187C2×C32C2×D3×C812yes—
188C2×C32C22×C3⋊C812yes—
189C2×C32C2×C4×Dic311yescharacteristic
190C3⋊D3C4×C3⋊C814yescomplemented
191C3×D3—34no—
192C22×C4—92no—
193C22×C4—91no—
194C2×C8—92no—
195C2×C8—92no—
196C2×C8—278no—
197C42—92no—
198C42—91no—
199C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
200C22×C3—31no—
201C22×C3—31no—
202C22×C3—61no—
203C22×C3—94no—
204C22×C3—92no—
205C22×C3C2×D3×Dic311yescharacteristic
206C2×D3—34no—
207C2×D3—32no—
208C2×D3—94no—
209C2×D3—92no—
210C3×C4—32no—
211C3×C4—32no—
212C3×C4—34no—
213C3×C4—34no—
214C3×C4—62no—
215C3×C4—64no—
216C3×C4—94no—
217C3×C4—92no—
218C3×C4D3×C3⋊C814yes—
219C3×C4C4×D3212yes—
220Dic3—34no—
221Dic3—32no—
222Dic3—94no—
223Dic3—92no—
224C32—21no—
225C32C4×D3×C811yescharacteristic, complemented
226C32C2×C4×C3⋊C811yescharacteristic, complemented
227C23—91no—
228C2×C4—92no—
229C2×C4—91no—
230C2×C4—94no—
231C2×C4—98no—
232C2×C4((C3⋊D3)⋊C3)⋊C412yes—
233C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
234C8—94no—
235C8—94no—
236C2×C3—32no—
237C2×C3—32no—
238C2×C3—31no—
239C2×C3—31no—
240C2×C3—62no—
241C2×C3—61no—
242C2×C3—94no—
243C2×C3C2×D3×C3⋊C812yes—
244C2×C3C4×D3×Dic311yescharacteristic
245D3—34no—
246D3—94no—
247C22—94no—
248C22—92no—
249C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
250C4—94no—
251C4—92no—
252C4((C3⋊D3)⋊C3)⋊C814yesdirect product factor
253C4C4×((C3⋊D3)⋊C3)⋊C212yes—
254C3—31no—
255C3—31no—
256C3—61no—
257C3C4×D3×C3⋊C811yescharacteristic
258C2—94no—
259C2C2×((C3⋊D3)⋊C3)⋊C812yes—
260C2C4×((C3⋊D3)⋊C3)⋊C411yescharacteristic
subgroupG/H
center, hypercenterC42((C3⋊D3)⋊C3)⋊C2
commutator, hypocenterH(3,𝔽3)C2×C4×C8
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×D3×Dic3
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C4×H(3,𝔽3)C23
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C42

Complements

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

NH ≅ G/Ntype
3. C4×#6180C2semidirect
4. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
5. #11720310C2semidirect
15. ((C3⋊D3)⋊C3)⋊C8C4direct (N × H)
20. C4×C3⋊(C3⋊C8)D3semidirect
32. C4×(C3⋊D3)⋊C3C8semidirect
34. #6180C2×C4semidirect
35. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
60. C4×H(3,𝔽3)C2×C8semidirect
63. (C3⋊Dic3)⋊C3C2×C8semidirect
90. C4×C3⋊D3C3⋊C8semidirect
96. C3⋊(C3⋊C8)C4×D3semidirect
104. (C3⋊D3)⋊C3C4×C8semidirect
140. C32×C4D3×C8semidirect
141. C32×C4C2×C3⋊C8semidirect
144. C3⋊Dic3C2×C3⋊C8semidirect
153. H(3,𝔽3)C2×C4×C8semidirect
190. C3⋊D3C4×C3⋊C8semidirect
225. C32C4×D3×C8semidirect
226. C32C2×C4×C3⋊C8semidirect
252. C4((C3⋊D3)⋊C3)⋊C8direct (N × H)

Maximal subgroups

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

subgroupindex
C42×(C3⋊D3)⋊C32
C2×((C3⋊D3)⋊C3)⋊C82
C4×#61802
C4×(C3⋊(C3⋊C8))⋊C32
#117203102
C4×#37103
C4×D3×C3⋊C83