Finite Group Databasebeta

← C2×C3×#520⋊C3 #11919682

Subgroups — C2×C3×#520⋊C3 #11919682

Order 1152.

936 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 936 conjugacy classes of subgroups — too many to draw. Only its 238 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×#520⋊C3C214yesmaximal, direct product factor
3C2×C3×C4×SL(2,𝔽3)C212yesmaximal, complemented
4C2×C3×#520C311yesmaximal, characteristic, complemented
5C2×#520⋊C3C313yesmaximal, direct product factor
6C2×C3×#41⋊C3C414yescomplemented
7C2×C3×#41⋊C3C2212yes—
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×C32×C4⋊C4—41nomaximal
10C3×C4×SL(2,𝔽3)C2214yescomplemented
11C22×C3×#41C2×C311yescharacteristic
12C22×#41⋊C3C2×C313yes—
13C3×#520—34no—
14C3×#520C2×C314yescomplemented
15C2×C3×C4×Q8C2×C312yes—
16C2×C3×C4×D4—32no—
17C2×C3×#117—31no—
18C22×C3×C4⋊C4—31no—
19#520⋊C3C2×C3112yesdirect product factor
20C2×C4×SL(2,𝔽3)C2×C316yescomplemented
21C22×C32×C4—41no—
22C22×C32×C4—42no—
23C3×#41⋊C3C2×C414yescomplemented
24C2×C3×SL(2,𝔽3)C2×C412yescomplemented
25C2×C3×SL(2,𝔽3)C2311yescharacteristic
26C32×C4⋊C4—44no—
27C2×#520C3211yescharacteristic, 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×C4112yescomplemented
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×C4⋊C4—34no—
47C2×C3×C4⋊C4—31no—
48C2×C3×C4⋊C4—34no—
49C2×C3×C4⋊C4—43no—
50C2×C3×C4⋊C4A411yescharacteristic
51C2×C3×C22⋊C4—32no—
52C2×C3×C42—32no—
53C4×SL(2,𝔽3)C22×C3112yescomplemented
54C23×C32—41no—
55C2×C32×C4—44no—
56C2×C32×C4—42no—
57C2×C32×C4—44no—
58C2×C32×C4—84no—
59C3×SL(2,𝔽3)C22×C411yescharacteristic, complemented
60C22×#41C2×C3211yescharacteristic
61#520—34no—
62#520C2×C3214yescomplemented
63C2×C4×Q8C2×C3212yes—
64C2×C4×D4—32no—
65C2×#117—31no—
66C22×C4⋊C4—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—124no—
90C22×C3×C4C2×A412yes—
91C22×C3×C4C2×A411yescharacteristic
92#41⋊C3C2×C3×C4112yescomplemented
93C2×SL(2,𝔽3)C2×C3×C416yescomplemented
94C2×SL(2,𝔽3)C23×C313yes—
95C3×C4⋊C4—34no—
96C3×C4⋊C4—34no—
97C3×C4⋊C4—34no—
98C3×C4⋊C4—412no—
99C3×C4⋊C4C2×A414yes—
100C3×C22⋊C4—38no—
101C3×C42—34no—
102C3×C42—34no—
103C22×C32—42no—
104C22×C32—42no—
105C22×C32—42no—
106C22×C32—41no—
107C32×C4—44no—
108C32×C4—84no—
109C2×#41—32no—
110C2×#41—34no—
111C2×#41C32×C414yescomplemented
112C2×#41C22×C3212yes—
113C22×Q8C22×C3211yescharacteristic
114C22×D4—31no—
115C23×C4—31no—
116C23×C4—32no—
117C4×Q8—34no—
118C4×Q8C22×C3214yes—
119C4×D4—38no—
120C4×D4—38no—
121#117—34no—
122#117—34no—
123C2×C4⋊C4—32no—
124C2×C4⋊C4—34no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—34no—
127C2×C4⋊C4C3×A411yescharacteristic
128C2×C22⋊C4—32no—
129C2×C42—32no—
130C23×C3—34no—
131C23×C3—32no—
132C23×C3—43no—
133C23×C3—62no—
134C23×C3—62no—
135C23×C3C22×A411yescharacteristic
136C3×Q8—32no—
137C3×Q8—31no—
138C3×Q8—32no—
139C3×Q8C22×C3×C411yescharacteristic
140C3×D4—34no—
141C3×D4—34no—
142C3×D4—38no—
143C2×C3×C4—24no—
144C2×C3×C4—32no—
145C2×C3×C4—32no—
146C2×C3×C4—31no—
147C2×C3×C4—31no—
148C2×C3×C4—34no—
149C2×C3×C4—38no—
150C2×C3×C4—34no—
151C2×C3×C4—34no—
152C2×C3×C4—46no—
153C2×C3×C4—412no—
154C2×C3×C4—412no—
155C2×C3×C4—64no—
156C2×C3×C4—812no—
157C2×C3×C4—128no—
158C2×C3×C4C4×A414yes—
159C2×C3×C4C22×A414yes—
160C2×C3×C4C22×A412yes—
161SL(2,𝔽3)C22×C3×C413yescomplemented
162C2×C32—41no—
163C2×C32—41no—
164C2×C32—41no—
165C2×C32—44no—
166C24—31no—
167#41—38no—
168#41—38no—
169#41—34no—
170#41C2×C32×C414yescomplemented
171C2×Q8—31no—
172C2×Q8—32no—
173C2×Q8C2×C32×C412yes—
174C2×Q8C23×C3211yescharacteristic
175C2×D4—32no—
176C2×D4—34no—
177C2×D4—32no—
178C2×D4—34no—
179C22×C4—31no—
180C22×C4—32no—
181C22×C4—32no—
182C22×C4—34no—
183C22×C4—34no—
184C22×C4—32no—
185C22×C4—38no—
186C22×C4—124no—
187C22×C4C2×C3×A412yes—
188C22×C4C2×C3×A411yescharacteristic
189C4⋊C4—34no—
190C4⋊C4—34no—
191C4⋊C4—34no—
192C4⋊C4C2×C3×A414yes—
193C22⋊C4—38no—
194C42—34no—
195C42—34no—
196C22×C3—34no—
197C22×C3—46no—
198C22×C3—46no—
199C22×C3—46no—
200C22×C3—43no—
201C22×C3—62no—
202C22×C3—62no—
203C22×C3—68no—
204C22×C3C2×C4×A412yes—
205C22×C3#99412yes—
206C22×C32−1+4⋊C312yes—
207C22×C3C23×A411yescharacteristic
208C3×C4—24no—
209C3×C4—31no—
210C3×C4—32no—
211C3×C4—31no—
212C3×C4—412no—
213C3×C4—64no—
214C3×C4—812no—
215C3×C4C2×C4×A414yes—
216C32—41noSylow
217C23—34no—
218C23—32no—
219C23—62no—
220C23—62no—
221C23C22×C3×A411yescharacteristic
222Q8—32no—
223Q8—31no—
224Q8—32no—
225Q8C22×C32×C411yescharacteristic
226D4—34no—
227D4—34no—
228D4—38no—
229C2×C4—24no—
230C2×C4—32no—
231C2×C4—32no—
232C2×C4—31no—
233C2×C4—31no—
234C2×C4—34no—
235C2×C4—34no—
236C2×C4—38no—
237C2×C4—34no—
238C2×C4—64no—
239C2×C4—128no—
240C2×C4C3×C4×A414yes—
241C2×C4C22×C3×A414yes—
242C2×C4C22×C3×A412yes—
243C2×C3—43no—
244C2×C3—43no—
245C2×C3—43no—
246C2×C3—412no—
247C2×C3—64no—
248C2×C3#520⋊C314yesdirect product factor
249C2×C3C22×C4×A411yescharacteristic
250C2×C3C2×#99411yescharacteristic
251C2×C3C2×2−1+4⋊C311yescharacteristic
252C22—34no—
253C22—62no—
254C22—62no—
255C22—68no—
256C22C2×C3×C4×A412yes—
257C22C3×#99412yes—
258C22C3×2−1+4⋊C312yes—
259C22C23×C3×A411yescharacteristic
260C4—24no—
261C4—31no—
262C4—32no—
263C4—31no—
264C4—64no—
265C4C2×C3×C4×A414yes—
266C3—43no—
267C3C2×#520⋊C311yescharacteristic, direct product factor
268C2—64no—
269C2C3×#520⋊C314yesdirect product factor
270C2C22×C3×C4×A411yescharacteristic
271C2C2×C3×#99411yescharacteristic
272C2C2×C3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, socleC23×C3C22×A4
commutator, hypocenterQ8C22×C32×C4
FrattiniC22C23×C3×A4
FittingC2×C3×#520C3
prime radicalC22×C3×#41⋊C3C2
cosocleC2×Q8C23×C32
hypercenterC2×C3×C4⋊C4A4
LCSG ▷ Q8
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#520⋊C3C2direct (N × H)
3. C2×C3×C4×SL(2,𝔽3)C2semidirect
4. C2×C3×#520C3semidirect
5. C2×#520⋊C3C3direct (N × H)
6. C2×C3×#41⋊C3C4semidirect
10. C3×C4×SL(2,𝔽3)C22semidirect
14. C3×#520C2×C3semidirect
19. #520⋊C3C2×C3direct (N × H)
20. C2×C4×SL(2,𝔽3)C2×C3semidirect
23. C3×#41⋊C3C2×C4semidirect
24. C2×C3×SL(2,𝔽3)C2×C4semidirect
27. C2×#520C32semidirect
30. C2×C3×#41C3×C4semidirect
36. C2×#41⋊C3C3×C4semidirect
53. C4×SL(2,𝔽3)C22×C3semidirect
59. C3×SL(2,𝔽3)C22×C4semidirect
62. #520C2×C32semidirect
71. C3×#41C2×C3×C4semidirect
92. #41⋊C3C2×C3×C4semidirect
93. C2×SL(2,𝔽3)C2×C3×C4semidirect
111. C2×#41C32×C4semidirect
161. SL(2,𝔽3)C22×C3×C4semidirect
170. #41C2×C32×C4semidirect
248. C2×C3#520⋊C3direct (N × H)
267. C3C2×#520⋊C3direct (N × H)
269. C2C3×#520⋊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×#520⋊C32
C2×C3×C4×SL(2,𝔽3)2
C2×C3×#5203
C2×#520⋊C33
C2×C32×C4⋊C44