Finite Group Databasebeta

← C2×C3×#519⋊C3 #11919680

Subgroups — C2×C3×#519⋊C3 #11919680

Order 1152.

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

normal subgroups only

This group has 992 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
1C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic
2C3×#519⋊C3C214yesmaximal, direct product factor
3C2×C3×C4×SL(2,𝔽3)C212yesmaximal, complemented
4C2×C3×#519C311yesmaximal, characteristic, complemented
5C2×#519⋊C3C313yesmaximal, direct product factor
6C22×C3×SL(2,𝔽3)C414yescomplemented
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C22×C3×SL(2,𝔽3)C2212yes—
9C2×C32×C22⋊C4—41nomaximal
10C3×C4×SL(2,𝔽3)C2214yescomplemented
11C23×C3×Q8C2×C311yescharacteristic
12C23×SL(2,𝔽3)C2×C313yes—
13C3×#519—34no—
14C3×#519C2×C314yescomplemented
15C2×C3×C4×Q8—32no—
16C2×C3×C4×Q8C2×C312yescomplemented
17C2×C3×#117—31no—
18C2×C3×#117—31no—
19#519⋊C3C2×C3112yesdirect product factor
20C2×C4×SL(2,𝔽3)C2×C316yescomplemented
21C24×C32—41no—
22C22×C32×C4—42no—
23C2×C3×SL(2,𝔽3)C2×C412yes—
24C2×C3×SL(2,𝔽3)C2×C414yescomplemented
25C2×C3×SL(2,𝔽3)C2311yescharacteristic
26C32×C22⋊C4—44no—
27C2×#519C3211yescharacteristic, complemented, Sylow
28C22×C3×Q8—34no—
29C22×C3×Q8—31no—
30C22×C3×Q8—32no—
31C22×C3×Q8C3×C414yescomplemented
32C22×C3×Q8C22×C311yescharacteristic
33C22×C3×Q8C22×C312yes—
34C23×C3×C4—31no—
35C22×SL(2,𝔽3)C3×C4112yescomplemented
36C22×SL(2,𝔽3)C22×C313yes—
37C22×SL(2,𝔽3)C22×C316yes—
38C3×C4×Q8—38no—
39C3×C4×Q8—38no—
40C3×C4×Q8—34no—
41C3×C4×Q8C22×C314yescomplemented
42C3×#117—34no—
43C3×#117—34no—
44C3×#117—34no—
45C3×#117—34no—
46C2×C3×C4⋊C4—32no—
47C2×C3×C4⋊C4—32no—
48C2×C3×C22⋊C4—31no—
49C2×C3×C22⋊C4—43no—
50C2×C3×C22⋊C4A411yescharacteristic
51C2×C3×C42—32no—
52C2×C3×C42—32no—
53C4×SL(2,𝔽3)C22×C3112yescomplemented
54C23×C32—44no—
55C23×C32—42no—
56C23×C32—41no—
57C23×C32—84no—
58C2×C32×C4—44no—
59C2×C32×C4—84no—
60C3×SL(2,𝔽3)C22×C411yescharacteristic
61C23×Q8C2×C3211yescharacteristic
62#519—34no—
63#519C2×C3214yescomplemented
64C2×C4×Q8—32no—
65C2×C4×Q8C2×C3212yescomplemented
66C2×#117—31no—
67C2×#117—31no—
68C24×C3—43no—
69C24×C3C2×A411yescharacteristic
70C2×C3×Q8—32no—
71C2×C3×Q8—34no—
72C2×C3×Q8—38no—
73C2×C3×Q8—31no—
74C2×C3×Q8—32no—
75C2×C3×Q8—34no—
76C2×C3×Q8—38no—
77C2×C3×Q8—32no—
78C2×C3×Q8—34no—
79C2×C3×Q8C2×C3×C412yes—
80C2×C3×Q8C2×C3×C414yescomplemented
81C2×C3×Q8C23×C311yescharacteristic
82C22×C3×C4—34no—
83C22×C3×C4—34no—
84C22×C3×C4—32no—
85C22×C3×C4—32no—
86C22×C3×C4—31no—
87C22×C3×C4—31no—
88C22×C3×C4—32no—
89C22×C3×C4—46no—
90C22×C3×C4C2×A412yes—
91C2×SL(2,𝔽3)C2×C3×C416yes—
92C2×SL(2,𝔽3)C2×C3×C4112yescomplemented
93C2×SL(2,𝔽3)C23×C313yes—
94C3×C4⋊C4—34no—
95C3×C4⋊C4—34no—
96C3×C4⋊C4—38no—
97C3×C22⋊C4—34no—
98C3×C22⋊C4—412no—
99C3×C22⋊C4C2×A414yes—
100C3×C42—38no—
101C3×C42—34no—
102C3×C42—34no—
103C22×C32—42no—
104C22×C32—44no—
105C22×C32—44no—
106C22×C32—41no—
107C22×C32—84no—
108C22×C32—88no—
109C32×C4—84no—
110C22×Q8—31no—
111C22×Q8—32no—
112C22×Q8—34no—
113C22×Q8C32×C414yescomplemented
114C22×Q8C22×C3211yescharacteristic
115C22×Q8C22×C3212yes—
116C23×C4—31no—
117C4×Q8—38no—
118C4×Q8—38no—
119C4×Q8—34no—
120C4×Q8C22×C3214yescomplemented
121#117—34no—
122#117—34no—
123#117—34no—
124#117—34no—
125C2×C4⋊C4—32no—
126C2×C4⋊C4—32no—
127C2×C22⋊C4—31no—
128C2×C22⋊C4C3×A411yescharacteristic
129C2×C42—32no—
130C2×C42—32no—
131C23×C3—24no—
132C23×C3—43no—
133C23×C3—46no—
134C23×C3—412no—
135C23×C3—812no—
136C23×C3C4×A414yes—
137C23×C3C22×A412yes—
138C23×C3C22×A411yescharacteristic
139C3×Q8—32no—
140C3×Q8—34no—
141C3×Q8—31no—
142C3×Q8—38no—
143C3×Q8—32no—
144C3×Q8—34no—
145C3×Q8C22×C3×C411yescharacteristic
146C2×C3×C4—24no—
147C2×C3×C4—34no—
148C2×C3×C4—38no—
149C2×C3×C4—32no—
150C2×C3×C4—34no—
151C2×C3×C4—32no—
152C2×C3×C4—34no—
153C2×C3×C4—32no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—34no—
157C2×C3×C4—412no—
158C2×C3×C4—64no—
159C2×C3×C4—812no—
160C2×C3×C4C22×A414yes—
161SL(2,𝔽3)C22×C3×C413yes—
162C2×C32—41no—
163C2×C32—42no—
164C2×C32—44no—
165C2×C32—84no—
166C24C2×C3×A411yescharacteristic
167C2×Q8—32no—
168C2×Q8—34no—
169C2×Q8—38no—
170C2×Q8—31no—
171C2×Q8—32no—
172C2×Q8—34no—
173C2×Q8—38no—
174C2×Q8—32no—
175C2×Q8—34no—
176C2×Q8C2×C32×C412yes—
177C2×Q8C2×C32×C414yescomplemented
178C2×Q8C23×C3211yescharacteristic
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—32no—
182C22×C4—32no—
183C22×C4—34no—
184C22×C4—34no—
185C22×C4—32no—
186C22×C4C2×C3×A412yes—
187C4⋊C4—34no—
188C4⋊C4—34no—
189C4⋊C4—38no—
190C22⋊C4—34no—
191C22⋊C4C2×C3×A414yes—
192C42—38no—
193C42—34no—
194C42—34no—
195C22×C3—24no—
196C22×C3—28no—
197C22×C3—46no—
198C22×C3—412no—
199C22×C3—412no—
200C22×C3—43no—
201C22×C3—812no—
202C22×C3—824no—
203C22×C3C2×C4×A414yes—
204C22×C3C2×C4×A412yes—
205C22×C32−1+4⋊C314yes—
206C22×C3C23×A411yescharacteristic
207C3×C4—24no—
208C3×C4—31no—
209C3×C4—32no—
210C3×C4—34no—
211C3×C4—31no—
212C3×C4—64no—
213C3×C4—812no—
214C32—41noSylow
215C23—24no—
216C23C3×C4×A414yes—
217C23C22×C3×A412yes—
218C23C22×C3×A411yescharacteristic
219Q8—32no—
220Q8—34no—
221Q8—31no—
222Q8—38no—
223Q8—32no—
224Q8—34no—
225Q8C22×C32×C411yescharacteristic
226C2×C4—24no—
227C2×C4—32no—
228C2×C4—34no—
229C2×C4—34no—
230C2×C4—38no—
231C2×C4—32no—
232C2×C4—34no—
233C2×C4—31no—
234C2×C4—32no—
235C2×C4—31no—
236C2×C4—34no—
237C2×C4—64no—
238C2×C4C22×C3×A414yes—
239C2×C3—24no—
240C2×C3—43no—
241C2×C3—46no—
242C2×C3—412no—
243C2×C3—812no—
244C2×C3#519⋊C314yesdirect product factor
245C2×C3C22×C4×A411yescharacteristic
246C2×C3C2×2−1+4⋊C312yes—
247C22—24no—
248C22—28no—
249C22C2×C3×C4×A414yes—
250C22C2×C3×C4×A412yes—
251C22C3×2−1+4⋊C314yes—
252C22C23×C3×A411yescharacteristic
253C4—24no—
254C4—31no—
255C4—32no—
256C4—34no—
257C4—31no—
258C4—64no—
259C3—43no—
260C3C2×#519⋊C311yescharacteristic, direct product factor
261C2—24no—
262C2C3×#519⋊C314yesdirect product factor
263C2C22×C3×C4×A411yescharacteristic
264C2C2×C3×2−1+4⋊C312yes—
subgroupG/H
center, socleC23×C3C22×A4
commutator, hypocenterQ8C22×C32×C4
FrattiniC22C23×C3×A4
FittingC2×C3×#519C3
prime radicalC23×C3×SL(2,𝔽3)C2
cosocleC2×Q8C23×C32
hypercenterC2×C3×C22⋊C4A4
LCSG ▷ Q8
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#519⋊C3C2direct (N × H)
3. C2×C3×C4×SL(2,𝔽3)C2semidirect
4. C2×C3×#519C3semidirect
5. C2×#519⋊C3C3direct (N × H)
6. C22×C3×SL(2,𝔽3)C4semidirect
10. C3×C4×SL(2,𝔽3)C22semidirect
14. C3×#519C2×C3semidirect
16. C2×C3×C4×Q8C2×C3semidirect
19. #519⋊C3C2×C3direct (N × H)
20. C2×C4×SL(2,𝔽3)C2×C3semidirect
24. C2×C3×SL(2,𝔽3)C2×C4semidirect
27. C2×#519C32semidirect
31. C22×C3×Q8C3×C4semidirect
35. C22×SL(2,𝔽3)C3×C4semidirect
41. C3×C4×Q8C22×C3semidirect
53. C4×SL(2,𝔽3)C22×C3semidirect
63. #519C2×C32semidirect
65. C2×C4×Q8C2×C32semidirect
80. C2×C3×Q8C2×C3×C4semidirect
92. C2×SL(2,𝔽3)C2×C3×C4semidirect
113. C22×Q8C32×C4semidirect
120. C4×Q8C22×C32semidirect
177. C2×Q8C2×C32×C4semidirect
244. C2×C3#519⋊C3direct (N × H)
260. C3C2×#519⋊C3direct (N × H)
262. C2C3×#519⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×SL(2,𝔽3)2
C3×#519⋊C32
C2×C3×C4×SL(2,𝔽3)2
C2×C3×#5193
C2×#519⋊C33
C2×C32×C22⋊C44