Finite Group Databasebeta

← C4⋊(C3⋊(#131⋊C3)) #11921274

Subgroups — C4⋊(C3⋊(#131⋊C3)) #11921274

Order 1152.

291 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C3×C4×#41⋊C3C211yesmaximal, characteristic
2C2×C3⋊(#131⋊C3)C212yesmaximal
3#82512C311yesmaximal, characteristic, complemented
4C4⋊(#131⋊C3)—31nomaximal
5C2×C3×#41⋊C3C2211yescharacteristic
6C3×C4×SL(2,𝔽3)C411yescharacteristic
7C3×C4×SL(2,𝔽3)C411yescharacteristic
8C3⋊(#131⋊C3)—22no—
9C3×C4⋊(C3⋊C8)—41nomaximal
10C3×C4×#41C2×C311yescharacteristic
11C2×#948C2×C312yes—
12C2×#131⋊C3—32no—
13C4×#41⋊C3—21no—
14C4×#41⋊C3D311yescharacteristic
15#4999—32no—
16#4924—31no—
17C2×C4⋊(C3⋊C8)—31no—
18C3×#41⋊C3D411yescharacteristic
19C3×#41⋊C3Q811yescharacteristic
20C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
21C32×C42—41no—
22C2×C3×C3⋊C8—42no—
23#2906—31noSylow
24C2×C3×#41C22×C311yescharacteristic
25C2×#41⋊C3—21no—
26C2×#41⋊C3C2×D311yescharacteristic
27C3×C4×Q8C3×C411yescharacteristic
28C3×C4×Q8C3×C411yescharacteristic
29C3×C4×D4—31no—
30C3×C4×D4—31no—
31C3×#117—31no—
32C2×C3×C42—31no—
33#948—22no—
34#948—62no—
35C2×#208—32no—
36C22×C3⋊C8—32no—
37#131⋊C3—62no—
38C4×SL(2,𝔽3)—21no—
39C4×SL(2,𝔽3)—21no—
40C4×SL(2,𝔽3)Dic311yescharacteristic
41C4×SL(2,𝔽3)Dic311yescharacteristic
42C3×C4⋊C8—121no—
43C4⋊(C3⋊C8)—32no—
44C4⋊(C3⋊C8)—31no—
45C4⋊(C3⋊C8)—32no—
46C4⋊(C3⋊C8)A411yescharacteristic
47C2×C32×C4—41no—
48C2×C32×C4—41no—
49C2×C32×C4—41no—
50C3×SL(2,𝔽3)C4⋊C411yescharacteristic
51C3×C3⋊C8—82no—
52C2×#131—32no—
53C4×#41C3×D311yescharacteristic
54#424—92no—
55#423—91no—
56C2×C4⋊C8—91no—
57C3×#41—31no—
58C3×#41—31no—
59C3×#41C3×D411yescharacteristic
60C3×#41C3×Q811yescharacteristic
61C2×C3×Q8C2×C3×C411yescharacteristic
62C2×C3×D4—31no—
63C22×C3×C4—31no—
64C22×C3×C4—31no—
65C22×C3×C4—31no—
66#41⋊C3—21no—
67#41⋊C3—21no—
68#41⋊C3Dic611yescharacteristic
69#41⋊C3D1211yescharacteristic
70C2×SL(2,𝔽3)—21no—
71C2×SL(2,𝔽3)C2×Dic311yescharacteristic
72C2×C3×C8—122no—
73C3×C4⋊C4—31no—
74C3×C4⋊C4—31no—
75C3×C22⋊C4—32no—
76C3×C42—31no—
77C3×C42—31no—
78C3×C42—31no—
79C3×C42—41no—
80C3×C42—81no—
81C3×C42C2×A411yescharacteristic
82#208—62no—
83#208—62no—
84C2×C3⋊C8—32no—
85C2×C3⋊C8—62no—
86C2×C3⋊C8—62no—
87C2×C3⋊C8C2×A412yes—
88C22×C32—41no—
89C32×C4—42no—
90C32×C4—41no—
91C32×C4—41no—
92C32×C4—81no—
93C2×#41C2×C3×D311yescharacteristic
94#131—62no—
95#131—182no—
96C2×M16—92no—
97C22×C8—92no—
98C4×Q8C3×Dic311yescharacteristic
99C4×Q8C3×Dic311yescharacteristic
100C4×D4—31no—
101C4×D4—31no—
102#117—31no—
103C2×C42—31no—
104C4⋊C8—31no—
105C4⋊C8—92no—
106C4⋊C8—92no—
107C4⋊C8—91no—
108C23×C3—31no—
109C3×Q8—31no—
110C3×Q8C3×C4⋊C411yescharacteristic
111C3×D4—31no—
112C3×D4—31no—
113C3×D4—32no—
114C2×C3×C4—31no—
115C2×C3×C4—31no—
116C2×C3×C4—31no—
117C2×C3×C4—31no—
118C2×C3×C4—31no—
119C2×C3×C4—31no—
120C2×C3×C4—31no—
121C2×C3×C4—41no—
122C2×C3×C4—41no—
123C2×C3×C4—41no—
124C2×C3×C4—62no—
125C2×C3×C4—81no—
126C2×C3×C4—81no—
127C2×C3×C4—81no—
128C2×C3×C4—121no—
129C2×C3×C4C4×A411yescharacteristic
130C2×C3×C4C4×A411yescharacteristic
131C2×C3×C4C22×A411yescharacteristic
132SL(2,𝔽3)—21no—
133SL(2,𝔽3)C4⋊Dic311yescharacteristic
134C3×C8—242no—
135C3⋊C8—22no—
136C3⋊C8—62no—
137C2×C32—41no—
138C2×C32—41no—
139C2×C32—41no—
140#41—31no—
141#41—31no—
142#41C3×Dic611yescharacteristic
143#41C3×D1211yescharacteristic
144C2×Q8C2×C3×Dic311yescharacteristic
145C2×D4—31no—
146C22×C4—31no—
147C22×C4—31no—
148C22×C4—31no—
149M16—182no—
150M16—182no—
151C2×C8—32no—
152C2×C8—92no—
153C2×C8—182no—
154C2×C8—182no—
155C4⋊C4—31no—
156C4⋊C4—31no—
157C22⋊C4—32no—
158C42—31no—
159C42—31no—
160C42—31no—
161C42D3×A411yescharacteristic
162C22×C3—31no—
163C22×C3—31no—
164C22×C3—41no—
165C22×C3—61no—
166C22×C3—61no—
167C22×C3—81no—
168C22×C3C2×C4×A411yescharacteristic
169C3×C4—21no—
170C3×C4—31no—
171C3×C4—31no—
172C3×C4—41no—
173C3×C4—41no—
174C3×C4—42no—
175C3×C4—61no—
176C3×C4—61no—
177C3×C4—81no—
178C3×C4—82no—
179C3×C4—81no—
180C3×C4—81no—
181C3×C4—82no—
182C3×C4#131⋊C312yescomplemented
183C3×C4D4×A411yescharacteristic
184C3×C4Q8×A411yescharacteristic
185C32—41noSylow
186C23—31no—
187Q8—31no—
188Q8C3×C4⋊Dic311yescharacteristic
189D4—31no—
190D4—31no—
191D4—32no—
192C2×C4—31no—
193C2×C4—31no—
194C2×C4—31no—
195C2×C4—31no—
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—31no—
199C2×C4—62no—
200C2×C4—121no—
201C2×C4Dic3×A411yescharacteristic
202C2×C4Dic3×A411yescharacteristic
203C2×C4C2×D3×A411yescharacteristic
204C8—62no—
205C8—182no—
206C2×C3—41no—
207C2×C3—41no—
208C2×C3—41no—
209C2×C3—61no—
210C2×C3—61no—
211C2×C3—81no—
212C2×C3—81no—
213C2×C3—81no—
214C2×C3A4×C4⋊C411yescharacteristic
215C2×C3C2×#131⋊C311yescharacteristic
216C2×C3#568⋊C311yescharacteristic
217C22—31no—
218C22—31no—
219C22—61no—
220C22—61no—
221C22C2×Dic3×A411yescharacteristic
222C4—21no—
223C4—31no—
224C4—31no—
225C4—61no—
226C4—61no—
227C4C3⋊(#131⋊C3)12yescomplemented
228C4A4×Dic611yescharacteristic
229C4A4×D1211yescharacteristic
230C3—41no—
231C3—81no—
232C3C4⋊(#131⋊C3)11yescharacteristic, complemented
233C2—61no—
234C2—61no—
235C2A4×C4⋊Dic311yescharacteristic
236C2C2×C3⋊(#131⋊C3)11yescharacteristic
237C2#5819⋊C311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×A4
commutatorC2×C3×Q8C2×C3×C4
FittingC3×C4×#41C2×C3
socleC22×C3C2×C4×A4
prime radicalC2×C3×#41⋊C3C22
cosocleC2×C3×#41C22×C3
hypercenterC42D3×A4
hypocenterC3×Q8C3×C4⋊C4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
3. #82512C3semidirect
182. C3×C4#131⋊C3semidirect
227. C4C3⋊(#131⋊C3)semidirect
232. C3C4⋊(#131⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#41⋊C32
C2×C3⋊(#131⋊C3)2
#825123
C4⋊(#131⋊C3)3
C3×C4⋊(C3⋊C8)4