Finite Group Databasebeta

← C2×C3×C4×#41⋊C3 #11919678

Subgroups — C2×C3×C4×#41⋊C3 #11919678

Order 1152.

1008 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 270 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⋊C3C211yesmaximal, characteristic
2C3×C4×#41⋊C3C214yesmaximal, direct product factor
3C2×C3×C4×SL(2,𝔽3)C212yesmaximal, complemented
4C2×C3×C4×#41C311yesmaximal, characteristic, complemented
5C2×C4×#41⋊C3C313yesmaximal, direct product factor
6C2×C3×#41⋊C3C414yesdirect product factor
7C2×C3×#41⋊C3C2212yes—
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×C32×C42—41nomaximal
10C3×C4×SL(2,𝔽3)C2214yescomplemented
11C22×C3×#41C2×C311yescharacteristic
12C22×#41⋊C3C2×C313yes—
13C3×C4×#41—34no—
14C3×C4×#41C2×C314yescomplemented
15C2×C3×C4×Q8C2×C312yes—
16C2×C3×C4×D4—32no—
17C2×C3×#117—31no—
18C22×C3×C42—31no—
19C4×#41⋊C3C2×C3112yesdirect product factor
20C2×C4×SL(2,𝔽3)C2×C316yescomplemented
21C22×C32×C4—41no—
22C22×C32×C4—42no—
23C3×#41⋊C3C2×C414yesdirect product factor
24C2×C3×SL(2,𝔽3)C2×C412yescomplemented
25C2×C3×SL(2,𝔽3)C2311yescharacteristic
26C32×C42—44no—
27C2×C4×#41C3211yescharacteristic, complemented, Sylow
28C2×C3×#41—34no—
29C2×C3×#41—32no—
30C2×C3×#41C3×C414yescomplemented
31C2×C3×#41C22×C312yes—
32C22×C3×Q8C22×C311yescharacteristic
33C22×C3×D4—31no—
34C23×C3×C4—31no—
35C23×C3×C4—32no—
36C2×#41⋊C3C3×C4112yesdirect product factor
37C2×#41⋊C3C22×C316yes—
38C22×SL(2,𝔽3)C22×C313yes—
39C3×C4×Q8—34no—
40C3×C4×Q8C22×C314yes—
41C3×C4×D4—38no—
42C3×C4×D4—38no—
43C3×#117—34no—
44C3×#117—34no—
45C2×C3×C4⋊C4—32no—
46C2×C3×C22⋊C4—32no—
47C2×C3×C42—31no—
48C2×C3×C42—32no—
49C2×C3×C42—34no—
50C2×C3×C42—34no—
51C2×C3×C42—43no—
52C2×C3×C42A411yescharacteristic
53C4×SL(2,𝔽3)C22×C3112yescomplemented
54C23×C32—41no—
55C2×C32×C4—44no—
56C2×C32×C4—48no—
57C2×C32×C4—42no—
58C2×C32×C4—44no—
59C3×SL(2,𝔽3)C22×C411yescharacteristic, complemented
60C22×#41C2×C3211yescharacteristic
61C4×#41—34no—
62C4×#41C2×C3214yescomplemented
63C2×C4×Q8C2×C3212yes—
64C2×C4×D4—32no—
65C2×#117—31no—
66C22×C42—31no—
67C24×C3—31no—
68C3×#41—38no—
69C3×#41—38no—
70C3×#41—34no—
71C3×#41C2×C3×C414yescomplemented
72C2×C3×Q8—31no—
73C2×C3×Q8—32no—
74C2×C3×Q8C2×C3×C412yes—
75C2×C3×Q8C23×C311yescharacteristic
76C2×C3×D4—32no—
77C2×C3×D4—34no—
78C2×C3×D4—32no—
79C2×C3×D4—34no—
80C22×C3×C4—31no—
81C22×C3×C4—34no—
82C22×C3×C4—34no—
83C22×C3×C4—32no—
84C22×C3×C4—32no—
85C22×C3×C4—32no—
86C22×C3×C4—38no—
87C22×C3×C4—43no—
88C22×C3×C4—46no—
89C22×C3×C4—68no—
90C22×C3×C4C2×A411yescharacteristic
91C22×C3×C4C2×A412yes—
92#41⋊C3C2×C3×C4112yesdirect product factor
93C2×SL(2,𝔽3)C2×C3×C416yescomplemented
94C2×SL(2,𝔽3)C23×C313yes—
95C3×C4⋊C4—34no—
96C3×C4⋊C4—34no—
97C3×C22⋊C4—38no—
98C3×C42—34no—
99C3×C42—34no—
100C3×C42—34no—
101C3×C42—412no—
102C3×C42C2×A414yes—
103C22×C32—42no—
104C22×C32—44no—
105C22×C32—41no—
106C32×C4—44no—
107C32×C4—48no—
108C2×#41—32no—
109C2×#41—34no—
110C2×#41C32×C414yescomplemented
111C2×#41C22×C3212yes—
112C22×Q8C22×C3211yescharacteristic
113C22×D4—31no—
114C23×C4—31no—
115C23×C4—32no—
116C4×Q8—34no—
117C4×Q8C22×C3214yes—
118C4×D4—38no—
119C4×D4—38no—
120#117—34no—
121#117—34no—
122C2×C4⋊C4—32no—
123C2×C22⋊C4—32no—
124C2×C42—31no—
125C2×C42—32no—
126C2×C42—34no—
127C2×C42—34no—
128C2×C42C3×A411yescharacteristic
129C23×C3—34no—
130C23×C3—32no—
131C23×C3—43no—
132C23×C3—64no—
133C23×C3C22×A411yescharacteristic
134C3×Q8—32no—
135C3×Q8—31no—
136C3×Q8—32no—
137C3×Q8C22×C3×C411yescharacteristic
138C3×D4—34no—
139C3×D4—34no—
140C3×D4—38no—
141C2×C3×C4—32no—
142C2×C3×C4—32no—
143C2×C3×C4—31no—
144C2×C3×C4—31no—
145C2×C3×C4—34no—
146C2×C3×C4—38no—
147C2×C3×C4—34no—
148C2×C3×C4—34no—
149C2×C3×C4—412no—
150C2×C3×C4—46no—
151C2×C3×C4—412no—
152C2×C3×C4—424no—
153C2×C3×C4—64no—
154C2×C3×C4—616no—
155C2×C3×C4C4×A414yes—
156C2×C3×C4#41⋊C318yesdirect product factor
157C2×C3×C4C22×A412yes—
158C2×C3×C4C22×A414yes—
159SL(2,𝔽3)C22×C3×C413yescomplemented
160C2×C32—41no—
161C2×C32—42no—
162C2×C32—44no—
163C24—31no—
164#41—38no—
165#41—38no—
166#41—34no—
167#41C2×C32×C414yescomplemented
168C2×Q8—31no—
169C2×Q8—32no—
170C2×Q8C2×C32×C412yes—
171C2×Q8C23×C3211yescharacteristic
172C2×D4—32no—
173C2×D4—34no—
174C2×D4—32no—
175C2×D4—34no—
176C22×C4—31no—
177C22×C4—32no—
178C22×C4—32no—
179C22×C4—34no—
180C22×C4—34no—
181C22×C4—32no—
182C22×C4—38no—
183C22×C4—68no—
184C22×C4C2×C3×A411yescharacteristic
185C22×C4C2×C3×A412yes—
186C4⋊C4—34no—
187C4⋊C4—34no—
188C22⋊C4—38no—
189C42—34no—
190C42—34no—
191C42—34no—
192C42C2×C3×A414yes—
193C22×C3—34no—
194C22×C3—46no—
195C22×C3—412no—
196C22×C3—43no—
197C22×C3—64no—
198C22×C3—68no—
199C22×C3C2×C4×A412yes—
200C22×C3C2×#41⋊C314yes—
201C22×C3C23×A411yescharacteristic
202C3×C4—31no—
203C3×C4—32no—
204C3×C4—31no—
205C3×C4—412no—
206C3×C4—424no—
207C3×C4—64no—
208C3×C4C2×C4×A414yes—
209C3×C4C2×#41⋊C318yesdirect product factor
210C32—41noSylow
211C23—34no—
212C23—32no—
213C23—64no—
214C23C22×C3×A411yescharacteristic
215Q8—32no—
216Q8—31no—
217Q8—32no—
218Q8C22×C32×C411yescharacteristic
219D4—34no—
220D4—34no—
221D4—38no—
222C2×C4—32no—
223C2×C4—32no—
224C2×C4—31no—
225C2×C4—31no—
226C2×C4—34no—
227C2×C4—34no—
228C2×C4—38no—
229C2×C4—34no—
230C2×C4—64no—
231C2×C4—616no—
232C2×C4C3×C4×A414yes—
233C2×C4C3×#41⋊C318yesdirect product factor
234C2×C4C22×C3×A412yes—
235C2×C4C22×C3×A414yes—
236C2×C3—43no—
237C2×C3—46no—
238C2×C3—412no—
239C2×C3—64no—
240C2×C3C4×#41⋊C314yesdirect product factor
241C2×C3C22×C4×A411yescharacteristic
242C2×C3C22×#41⋊C312yes—
243C22—34no—
244C22—64no—
245C22—68no—
246C22C2×C3×C4×A412yes—
247C22C2×C3×#41⋊C314yes—
248C22C23×C3×A411yescharacteristic
249C4—31no—
250C4—32no—
251C4—31no—
252C4—64no—
253C4C2×C3×C4×A414yes—
254C4C2×C3×#41⋊C318yesdirect product factor
255C3—43no—
256C3C2×C4×#41⋊C311yescharacteristic, direct product factor
257C2—64no—
258C2C3×C4×#41⋊C314yesdirect product factor
259C2C22×C3×C4×A411yescharacteristic
260C2C22×C3×#41⋊C312yes—
subgroupG/H
center, hypercenterC2×C3×C42A4
commutator, hypocenterQ8C22×C32×C4
FrattiniC22C23×C3×A4
FittingC2×C3×C4×#41C3
socleC23×C3C22×A4
prime radicalC22×C3×#41⋊C3C2
cosocleC2×Q8C23×C32
LCSG ▷ Q8
UCSC1 ◁ C2×C3×C42

Complements

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

NH ≅ G/Ntype
2. C3×C4×#41⋊C3C2direct (N × H)
3. C2×C3×C4×SL(2,𝔽3)C2semidirect
4. C2×C3×C4×#41C3semidirect
5. C2×C4×#41⋊C3C3direct (N × H)
6. C2×C3×#41⋊C3C4direct (N × H)
10. C3×C4×SL(2,𝔽3)C22semidirect
14. C3×C4×#41C2×C3semidirect
19. C4×#41⋊C3C2×C3direct (N × H)
20. C2×C4×SL(2,𝔽3)C2×C3semidirect
23. C3×#41⋊C3C2×C4direct (N × H)
24. C2×C3×SL(2,𝔽3)C2×C4semidirect
27. C2×C4×#41C32semidirect
30. C2×C3×#41C3×C4semidirect
36. C2×#41⋊C3C3×C4direct (N × H)
53. C4×SL(2,𝔽3)C22×C3semidirect
59. C3×SL(2,𝔽3)C22×C4semidirect
62. C4×#41C2×C32semidirect
71. C3×#41C2×C3×C4semidirect
92. #41⋊C3C2×C3×C4direct (N × H)
93. C2×SL(2,𝔽3)C2×C3×C4semidirect
110. C2×#41C32×C4semidirect
156. C2×C3×C4#41⋊C3direct (N × H)
159. SL(2,𝔽3)C22×C3×C4semidirect
167. #41C2×C32×C4semidirect
209. C3×C4C2×#41⋊C3direct (N × H)
233. C2×C4C3×#41⋊C3direct (N × H)
240. C2×C3C4×#41⋊C3direct (N × H)
254. C4C2×C3×#41⋊C3direct (N × H)
256. C3C2×C4×#41⋊C3direct (N × H)
258. C2C3×C4×#41⋊C3direct (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⋊C32
C2×C3×C4×SL(2,𝔽3)2
C2×C3×C4×#413
C2×C4×#41⋊C33
C2×C32×C424