Finite Group Databasebeta

← C4⋊(C3⋊#859) #11921016

Subgroups — C4⋊(C3⋊#859) #11921016

Order 1152.

468 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 105 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×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3⋊#859C212yesmaximal
3C4⋊#859—33nomaximal
4#71626—31nomaximal
5C22×C3×SL(2,𝔽3)C2211yescharacteristic
6C2×C4⋊(C3⋊Dic3)—41nomaximal
7C3×C4×SL(2,𝔽3)C412yescomplemented
8C3⋊#859—22no—
9C2×C3×C4×Q8D311yescharacteristic
10C2×C4×SL(2,𝔽3)D313yes—
11C2×#859—36no—
12C2×#835—32no—
13#4935—31no—
14C2×C4⋊(C3⋊C8)—31no—
15#4529—32no—
16C22×C32×C4—41no—
17C22×C3⋊Dic3—42no—
18C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
19C2×C3×SL(2,𝔽3)D411yescharacteristic
20C2×C3×SL(2,𝔽3)Q811yescharacteristic
21C4⋊(C3⋊Dic3)—44no—
22#1803—91noSylow
23C22×C3×Q8C2×D311yescharacteristic
24C22×SL(2,𝔽3)C2×D313yes—
25C3×C4×Q8—32no—
26C3×C4×Q8Dic312yescomplemented
27C2×C3×C4⋊C4—31no—
28C2×C3×C42—31no—
29C2×C4⋊Dic3—32no—
30C2×C4⋊Dic3—61no—
31C2×C4⋊Dic3—61no—
32C2×C4⋊Dic3—123no—
33C22×C3⋊C8—32no—
34C4×SL(2,𝔽3)Dic316yescomplemented
35#859—66no—
36#835—62no—
37#835—62no—
38C4⋊(C3⋊C8)—62no—
39C23×C32—41no—
40C2×C32×C4—42no—
41C2×C32×C4—42no—
42C2×C32×C4—42no—
43C2×C3⋊Dic3—44no—
44C2×C3⋊Dic3—82no—
45C2×C3⋊Dic3—82no—
46C3×SL(2,𝔽3)C4⋊C411yescharacteristic, complemented
47C2×C4×Q8C3⋊D311yescharacteristic
48C2×C4⋊C8—91no—
49C2×#103—92no—
50#384—91no—
51#340—92no—
52C2×C3×Q8—31no—
53C2×C3×Q8—31no—
54C2×C3×Q8—31no—
55C2×C3×Q8Dic611yescharacteristic
56C2×C3×Q8D1211yescharacteristic
57C2×C3×Q8C2×Dic311yescharacteristic
58C22×C3×C4—31no—
59C22×C3×C4—31no—
60C22×C3×C4—43no—
61C22×C3×C4S411yescharacteristic
62C22×Dic3—62no—
63C22×Dic3—126no—
64C2×SL(2,𝔽3)Dic613yes—
65C2×SL(2,𝔽3)D1213yes—
66C2×SL(2,𝔽3)C2×Dic313yes—
67C3×C4⋊C4—32no—
68C3×C4⋊C4—61no—
69C3×C42—32no—
70C3×C42—32no—
71C4⋊Dic3—62no—
72C4⋊Dic3—62no—
73C4⋊Dic3—1212no—
74C4⋊Dic3—122no—
75C4⋊Dic3—122no—
76C2×C3⋊C8—62no—
77C2×C3⋊C8—62no—
78C2×C3⋊C8—62no—
79C22×C32—41no—
80C22×C32—41no—
81C22×C32—41no—
82C22×C32—41no—
83C22×C32—41no—
84C22×C32—41no—
85C22×C32—41no—
86C32×C4—44no—
87C3⋊Dic3—84no—
88C22×Q8C2×C3⋊D311yescharacteristic
89C22×C8—92no—
90C4×Q8—32no—
91C4×Q8C3⋊Dic312yescomplemented
92C2×C4⋊C4—31no—
93C2×C4⋊C4—92no—
94C2×C4⋊C4—181no—
95C2×C4⋊C4—181no—
96C2×C42—31no—
97C4⋊C8—182no—
98#103—182no—
99#103—182no—
100C23×C3—43no—
101C23×C3C2×S411yescharacteristic
102C3×Q8—31no—
103C3×Q8—31no—
104C3×Q8—31no—
105C3×Q8—61no—
106C3×Q8C4⋊Dic311yescharacteristic, complemented
107C2×C3×C4—31no—
108C2×C3×C4—31no—
109C2×C3×C4—31no—
110C2×C3×C4—31no—
111C2×C3×C4—31no—
112C2×C3×C4—31no—
113C2×C3×C4—32no—
114C2×C3×C4—46no—
115C2×C3×C4—46no—
116C2×C3×C4—46no—
117C2×C3×C4—61no—
118C2×C3×C4—61no—
119C2×C3×C4#22612yes—
120C2×C3×C4GL(2,𝔽3)12yes—
121C2×C3×C4SL(2,ℤ/4)12yes—
122C2×Dic3—1212no—
123C2×Dic3—122no—
124C2×Dic3—122no—
125C2×Dic3—122no—
126C2×Dic3—246no—
127C2×Dic3—246no—
128SL(2,𝔽3)C4⋊Dic313yescomplemented
129C3⋊C8—122no—
130C2×C32—41no—
131C2×C32—41no—
132C2×C32—41no—
133C2×C32—41no—
134C2×C32—41no—
135C2×C32—41no—
136C2×C32—41no—
137C2×Q8—31no—
138C2×Q8—31no—
139C2×Q8—31no—
140C2×Q8#63411yescharacteristic
141C2×Q8C4⋊(C3⋊D3)11yescharacteristic
142C2×Q8C2×C3⋊Dic311yescharacteristic
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—182no—
146C22×C4C3⋊S411yescharacteristic
147C2×C8—182no—
148C2×C8—182no—
149C2×C8—182no—
150C4⋊C4—32no—
151C4⋊C4—61no—
152C4⋊C4—182no—
153C4⋊C4—182no—
154C4⋊C4—362no—
155C4⋊C4—362no—
156C42—32no—
157C42—32no—
158C22×C3—43no—
159C22×C3—43no—
160C22×C3—43no—
161C22×C3—43no—
162C22×C3—43no—
163C22×C3—43no—
164C22×C3—43no—
165C22×C3A4⋊Q811yescharacteristic
166C22×C3C4⋊S411yescharacteristic
167C22×C3C2×#22611yescharacteristic
168C22×C3C2×GL(2,𝔽3)11yescharacteristic
169C22×C3#98411yescharacteristic
170C22×C3#98611yescharacteristic
171C22×C3C2×SL(2,ℤ/4)11yescharacteristic
172C3×C4—31no—
173C3×C4—31no—
174C3×C4—31no—
175C3×C4—31no—
176C3×C4—412no—
177C3×C4—62no—
178C3×C4#85914yescomplemented
179Dic3—2412no—
180Dic3—242no—
181C32—41noSylow
182C23C2×C3⋊S411yescharacteristic
183Q8—31no—
184Q8—31no—
185Q8—31no—
186Q8—61no—
187Q8C4⋊(C3⋊Dic3)11yescharacteristic, complemented
188C2×C4—31no—
189C2×C4—31no—
190C2×C4—31no—
191C2×C4—31no—
192C2×C4—31no—
193C2×C4—31no—
194C2×C4—32no—
195C2×C4—61no—
196C2×C4—61no—
197C2×C4—362no—
198C2×C4—362no—
199C2×C4—362no—
200C2×C4C3⋊#22612yes—
201C2×C4C3⋊GL(2,𝔽3)12yes—
202C2×C4C3⋊SL(2,ℤ/4)12yes—
203C8—362no—
204C2×C3—43no—
205C2×C3—43no—
206C2×C3—43no—
207C2×C3—43no—
208C2×C3—43no—
209C2×C3—43no—
210C2×C3—43no—
211C2×C3#538611yescharacteristic
212C2×C3C4⋊#22611yescharacteristic
213C2×C3#539111yescharacteristic
214C2×C3C4⋊GL(2,𝔽3)11yescharacteristic
215C2×C3C4⋊SL(2,ℤ/4)11yescharacteristic
216C2×C3C2×#85911yescharacteristic
217C2×C3#542611yescharacteristic
218C22#6425911yescharacteristic
219C22C4⋊(C3⋊S4)11yescharacteristic
220C22C2×C3⋊#22611yescharacteristic
221C22C2×C3⋊GL(2,𝔽3)11yescharacteristic
222C22#6426511yescharacteristic
223C22#6426711yescharacteristic
224C22C2×C3⋊SL(2,ℤ/4)11yescharacteristic
225C4—31no—
226C4—31no—
227C4—31no—
228C4—31no—
229C4—62no—
230C4—722no—
231C4C3⋊#85914yescomplemented
232C3—43no—
233C3C4⋊#85911yescharacteristic, complemented
234C2#1059362011yescharacteristic
235C2C4⋊(C3⋊#226)11yescharacteristic
236C2#1059362511yescharacteristic
237C2C4⋊(C3⋊GL(2,𝔽3))11yescharacteristic
238C2C4⋊(C3⋊SL(2,ℤ/4))11yescharacteristic
239C2C2×C3⋊#85911yescharacteristic
240C2#1059366011yescharacteristic
subgroupG/H
center, FrattiniC23C2×C3⋊S4
commutatorC2×C3×SL(2,𝔽3)C2×C4
FittingC2×C3×C4×Q8D3
socleC23×C3C2×S4
prime radical, cosocleC22×C3×SL(2,𝔽3)C22
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C4⋊C4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
7. C3×C4×SL(2,𝔽3)C4semidirect
26. C3×C4×Q8Dic3semidirect
34. C4×SL(2,𝔽3)Dic3semidirect
46. C3×SL(2,𝔽3)C4⋊C4semidirect
91. C4×Q8C3⋊Dic3semidirect
106. C3×Q8C4⋊Dic3semidirect
128. SL(2,𝔽3)C4⋊Dic3semidirect
178. C3×C4#859semidirect
187. Q8C4⋊(C3⋊Dic3)semidirect
231. C4C3⋊#859semidirect
233. C3C4⋊#859semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
C2×C3⋊#8592
C4⋊#8593
#716263
C2×C4⋊(C3⋊Dic3)4