Finite Group Databasebeta

← C2×C3×C4⋊(#41⋊C3) #11919718

Subgroups — C2×C3×C4⋊(#41⋊C3) #11919718

Order 1152.

968 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 190 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
1C22×C3×#41⋊C3C212yesmaximal, complemented
2C3×C4⋊(#41⋊C3)C214yesmaximal, direct product factor
3C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C2×C3×#550C311yesmaximal, characteristic, complemented
5C2×C4⋊(#41⋊C3)C313yesmaximal, direct product factor
6C2×C3×#41⋊C3—24no—
7C2×C3×#41⋊C3C2214yescomplemented
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×C32×C4⋊C4—41nomaximal
10C3×C4×SL(2,𝔽3)C2212yescomplemented
11C22×C3×#41C2×C312yes—
12C22×#41⋊C3C2×C316yescomplemented
13C3×#550—34no—
14C3×#550C2×C314yescomplemented
15C2×C3×#127—31no—
16C2×C3×#121—32no—
17C2×C3×C4×Q8C2×C311yescharacteristic
18C2×C3×C4×D4—31no—
19C4⋊(#41⋊C3)C2×C3112yesdirect product factor
20C2×C4×SL(2,𝔽3)C2×C313yescomplemented
21C22×C32×C4—41no—
22C22×C32×C4—42no—
23C3×#41⋊C3—24no—
24C2×C3×SL(2,𝔽3)D412yescomplemented
25C2×C3×SL(2,𝔽3)C2311yescharacteristic
26C32×C4⋊C4—44no—
27C2×#550C3211yescharacteristic, complemented, Sylow
28C2×C3×#41—24no—
29C2×C3×#41—34no—
30C2×C3×#41—64no—
31C2×C3×#41C22×C314yes—
32C22×C3×Q8C22×C311yescharacteristic
33C22×C3×D4—32no—
34C22×C3×D4—32no—
35C22×C3×D4—31no—
36C23×C3×C4—32no—
37C2×#41⋊C3—212no—
38C2×#41⋊C3C22×C3112yescomplemented
39C22×SL(2,𝔽3)C22×C313yes—
40C3×#127—34no—
41C3×#127—34no—
42C3×#121—38no—
43C3×#121—38no—
44C3×C4×Q8—32no—
45C3×C4×Q8C22×C312yes—
46C3×C4×D4—34no—
47C3×C4×D4—34no—
48C2×C3×C4⋊C4—31no—
49C2×C3×C4⋊C4—43no—
50C2×C3×C4⋊C4A411yescharacteristic
51C2×C3×C22⋊C4—32no—
52C2×C3×C42—31no—
53C4×SL(2,𝔽3)C22×C316yescomplemented
54C23×C32—41no—
55C2×C32×C4—44no—
56C2×C32×C4—42no—
57C2×C32×C4—44no—
58C2×C32×C4—84no—
59C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
60C22×#41C2×C3212yes—
61#550—34no—
62#550C2×C3214yescomplemented
63C2×#127—31no—
64C2×#121—32no—
65C2×C4×Q8C2×C3211yescharacteristic
66C2×C4×D4—31no—
67C24×C3—32no—
68C3×#41—24no—
69C3×#41—68no—
70C3×#41—68no—
71C3×#41—64no—
72C2×C3×Q8—31no—
73C2×C3×Q8—32no—
74C2×C3×Q8C3×D412yes—
75C2×C3×Q8C23×C311yescharacteristic
76C2×C3×D4—34no—
77C2×C3×D4—34no—
78C2×C3×D4—38no—
79C2×C3×D4—34no—
80C2×C3×D4—64no—
81C2×C3×D4—64no—
82C2×C3×D4—64no—
83C2×C3×D4—124no—
84C22×C3×C4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—34no—
87C22×C3×C4—34no—
88C22×C3×C4—43no—
89C22×C3×C4—46no—
90C22×C3×C4—64no—
91C22×C3×C4—64no—
92C22×C3×C4C2×A412yes—
93C22×C3×C4C2×A411yescharacteristic
94#41⋊C3—212no—
95C2×SL(2,𝔽3)C3×D416yescomplemented
96C2×SL(2,𝔽3)C23×C313yes—
97C3×C4⋊C4—32no—
98C3×C4⋊C4—32no—
99C3×C4⋊C4—412no—
100C3×C4⋊C4C2×A414yes—
101C3×C22⋊C4—38no—
102C3×C42—32no—
103C3×C42—32no—
104C22×C32—42no—
105C22×C32—42no—
106C22×C32—42no—
107C22×C32—41no—
108C32×C4—44no—
109C32×C4—84no—
110C2×#41—24no—
111C2×#41—34no—
112C2×#41—64no—
113C2×#41C22×C3214yes—
114C22×Q8C22×C3211yescharacteristic
115C22×D4—32no—
116C22×D4—32no—
117C22×D4—31no—
118C23×C4—32no—
119#127—34no—
120#127—34no—
121#121—38no—
122#121—38no—
123C4×Q8—32no—
124C4×Q8C22×C3212yes—
125C4×D4—34no—
126C4×D4—34no—
127C2×C4⋊C4—31no—
128C2×C4⋊C4C3×A411yescharacteristic
129C2×C22⋊C4—32no—
130C2×C42—31no—
131C23×C3—34no—
132C23×C3—43no—
133C23×C3—64no—
134C23×C3—64no—
135C23×C3—64no—
136C23×C3C22×A411yescharacteristic
137C3×Q8—32no—
138C3×Q8—31no—
139C3×Q8—32no—
140C3×Q8C2×C3×D411yescharacteristic
141C3×D4—64no—
142C3×D4—64no—
143C3×D4—68no—
144C3×D4—68no—
145C3×D4—128no—
146C2×C3×C4—24no—
147C2×C3×C4—32no—
148C2×C3×C4—32no—
149C2×C3×C4—31no—
150C2×C3×C4—31no—
151C2×C3×C4—32no—
152C2×C3×C4—46no—
153C2×C3×C4—412no—
154C2×C3×C4—412no—
155C2×C3×C4—64no—
156C2×C3×C4—68no—
157C2×C3×C4—64no—
158C2×C3×C4—62no—
159C2×C3×C4—812no—
160C2×C3×C4#41⋊C314yescomplemented
161C2×C3×C4C22×A414yes—
162C2×C3×C4C22×A412yes—
163SL(2,𝔽3)C2×C3×D413yescomplemented
164C2×C32—41no—
165C2×C32—41no—
166C2×C32—41no—
167C2×C32—44no—
168C24—32no—
169#41—24no—
170#41—68no—
171#41—68no—
172#41—64no—
173C2×Q8—31no—
174C2×Q8—32no—
175C2×Q8C32×D412yes—
176C2×Q8C23×C3211yescharacteristic
177C2×D4—34no—
178C2×D4—34no—
179C2×D4—38no—
180C2×D4—34no—
181C2×D4—64no—
182C2×D4—64no—
183C2×D4—64no—
184C2×D4—124no—
185C22×C4—31no—
186C22×C4—31no—
187C22×C4—34no—
188C22×C4—34no—
189C22×C4—64no—
190C22×C4—64no—
191C22×C4C2×C3×A412yes—
192C22×C4C2×C3×A411yescharacteristic
193C4⋊C4—32no—
194C4⋊C4—32no—
195C4⋊C4C2×C3×A414yes—
196C22⋊C4—38no—
197C42—32no—
198C42—32no—
199C22×C3—46no—
200C22×C3—46no—
201C22×C3—46no—
202C22×C3—43no—
203C22×C3—64no—
204C22×C3—64no—
205C22×C3—64no—
206C22×C3—128no—
207C22×C3D4×A412yes—
208C22×C3C2×#41⋊C312yes—
209C22×C3#99412yes—
210C22×C3C23×A411yescharacteristic
211C3×C4—24no—
212C3×C4—31no—
213C3×C4—32no—
214C3×C4—31no—
215C3×C4—412no—
216C3×C4—62no—
217C3×C4—812no—
218C3×C4C2×#41⋊C314yescomplemented
219C32—41noSylow
220C23—34no—
221C23—64no—
222C23—64no—
223C23—64no—
224C23C22×C3×A411yescharacteristic
225Q8—32no—
226Q8—31no—
227Q8—32no—
228Q8C2×C32×D411yescharacteristic
229D4—64no—
230D4—64no—
231D4—68no—
232D4—68no—
233D4—128no—
234C2×C4—24no—
235C2×C4—32no—
236C2×C4—32no—
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—32no—
240C2×C4—64no—
241C2×C4—64no—
242C2×C4—68no—
243C2×C4—62no—
244C2×C4C3×#41⋊C314yescomplemented
245C2×C4C22×C3×A414yes—
246C2×C4C22×C3×A412yes—
247C2×C3—43no—
248C2×C3—43no—
249C2×C3—43no—
250C2×C3—412no—
251C2×C3—124no—
252C2×C3C4⋊(#41⋊C3)14yesdirect product factor
253C2×C3C2×D4×A411yescharacteristic
254C2×C3C22×#41⋊C311yescharacteristic
255C2×C3C2×#99411yescharacteristic
256C22—64no—
257C22—64no—
258C22—64no—
259C22—128no—
260C22C3×D4×A412yes—
261C22C2×C3×#41⋊C312yes—
262C22C3×#99412yes—
263C22C23×C3×A411yescharacteristic
264C4—24no—
265C4—31no—
266C4—32no—
267C4—31no—
268C4—62no—
269C4C2×C3×#41⋊C314yescomplemented
270C3—43no—
271C3C2×C4⋊(#41⋊C3)11yescharacteristic, direct product factor
272C2—124no—
273C2C3×C4⋊(#41⋊C3)14yesdirect product factor
274C2C2×C3×D4×A411yescharacteristic
275C2C22×C3×#41⋊C311yescharacteristic
276C2C2×C3×#99411yescharacteristic
subgroupG/H
center, socleC23×C3C22×A4
commutator, cosocleC2×Q8C23×C32
FrattiniC22C23×C3×A4
FittingC2×C3×#550C3
hypercenterC2×C3×C4⋊C4A4
hypocenterQ8C2×C32×D4
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41⋊C3C2semidirect
2. C3×C4⋊(#41⋊C3)C2direct (N × H)
3. C2×C3×C4×SL(2,𝔽3)C2semidirect
4. C2×C3×#550C3semidirect
5. C2×C4⋊(#41⋊C3)C3direct (N × H)
7. C2×C3×#41⋊C3C22semidirect
10. C3×C4×SL(2,𝔽3)C22semidirect
12. C22×#41⋊C3C2×C3semidirect
14. C3×#550C2×C3semidirect
19. C4⋊(#41⋊C3)C2×C3direct (N × H)
20. C2×C4×SL(2,𝔽3)C2×C3semidirect
24. C2×C3×SL(2,𝔽3)D4semidirect
27. C2×#550C32semidirect
38. C2×#41⋊C3C22×C3semidirect
53. C4×SL(2,𝔽3)C22×C3semidirect
59. C3×SL(2,𝔽3)C2×D4semidirect
62. #550C2×C32semidirect
95. C2×SL(2,𝔽3)C3×D4semidirect
160. C2×C3×C4#41⋊C3semidirect
163. SL(2,𝔽3)C2×C3×D4semidirect
218. C3×C4C2×#41⋊C3semidirect
244. C2×C4C3×#41⋊C3semidirect
252. C2×C3C4⋊(#41⋊C3)direct (N × H)
269. C4C2×C3×#41⋊C3semidirect
271. C3C2×C4⋊(#41⋊C3)direct (N × H)
273. C2C3×C4⋊(#41⋊C3)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C3×C4⋊(#41⋊C3)2
C2×C3×C4×SL(2,𝔽3)2
C2×C3×#5503
C2×C4⋊(#41⋊C3)3
C2×C32×C4⋊C44