Finite Group Databasebeta

← C2×#5812⋊C3 #11923364

Subgroups — C2×#5812⋊C3 #11923364

Order 1152.

889 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 889 conjugacy classes of subgroups — too many to draw. Only its 151 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
2#5812⋊C3C214yesmaximal, direct product factor
3C2×Dic3×SL(2,𝔽3)C212yesmaximal, complemented
4C2×#5812C311yesmaximal, characteristic, complemented
5C2×#519⋊C3—31nomaximal
6C22×C3×SL(2,𝔽3)C414yescomplemented
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C22×C3×SL(2,𝔽3)C2212yes—
9C2×C3×#217—41nomaximal
10Dic3×SL(2,𝔽3)C2214yescomplemented
11C23×C3×Q8C2×C311yescharacteristic
12C23×SL(2,𝔽3)—21no—
13C23×SL(2,𝔽3)D311yescharacteristic
14#5812—34no—
15#5812C2×C314yescomplemented
16C2×Q8×Dic3—32no—
17C2×Q8×Dic3C2×C312yescomplemented
18C2×#926—31no—
19C2×#926—31no—
20#519⋊C3—34no—
21C2×C4×SL(2,𝔽3)—32no—
22C24×C32—41no—
23C22×C3×Dic3—42no—
24C2×C3×SL(2,𝔽3)C2×C412yes—
25C2×C3×SL(2,𝔽3)C2×C414yescomplemented
26C2×C3×SL(2,𝔽3)C2311yescharacteristic
27C3×#217—44no—
28C2×#519—31noSylow
29C22×C3×Q8—34no—
30C22×C3×Q8—31no—
31C22×C3×Q8—32no—
32C22×C3×Q8C3×C414yescomplemented
33C22×C3×Q8C22×C311yescharacteristic
34C22×C3×Q8C22×C312yes—
35C23×C3×C4—31no—
36C22×SL(2,𝔽3)—24no—
37C22×SL(2,𝔽3)—22no—
38C22×SL(2,𝔽3)—21no—
39C22×SL(2,𝔽3)Dic314yescomplemented
40C22×SL(2,𝔽3)C2×D311yescharacteristic
41C22×SL(2,𝔽3)C2×D312yes—
42C2×C3×C22⋊C4—121no—
43C2×#217—31no—
44C2×#217A411yescharacteristic
45Q8×Dic3—38no—
46Q8×Dic3—38no—
47Q8×Dic3—34no—
48Q8×Dic3C22×C314yescomplemented
49#926—34no—
50#926—34no—
51#926—34no—
52#926—34no—
53C2×C4⋊Dic3—32no—
54C2×C4⋊Dic3—32no—
55C2×C4×Dic3—32no—
56C2×C4×Dic3—32no—
57C4×SL(2,𝔽3)—34no—
58C23×C32—44no—
59C23×C32—41no—
60C23×C32—42no—
61C23×C32—84no—
62C2×C3×Dic3—44no—
63C2×C3×Dic3—84no—
64C3×SL(2,𝔽3)C22×C411yescharacteristic
65C23×Q8C3×D311yescharacteristic
66#519—34no—
67#519—94no—
68C2×C4×Q8—32no—
69C2×C4×Q8—92no—
70C2×#117—91no—
71C2×#117—91no—
72C24×C3—41no—
73C24×C3—81no—
74C24×C3C2×A411yescharacteristic
75C2×C3×Q8—32no—
76C2×C3×Q8—34no—
77C2×C3×Q8—38no—
78C2×C3×Q8—31no—
79C2×C3×Q8—32no—
80C2×C3×Q8—34no—
81C2×C3×Q8—38no—
82C2×C3×Q8—32no—
83C2×C3×Q8—34no—
84C2×C3×Q8C2×C3×C412yes—
85C2×C3×Q8C2×C3×C414yescomplemented
86C2×C3×Q8C23×C311yescharacteristic
87C22×C3×C4—34no—
88C22×C3×C4—34no—
89C22×C3×C4—31no—
90C22×C3×C4—31no—
91C22×C3×C4—32no—
92C22×C3×C4—32no—
93C22×C3×C4—122no—
94C22×Dic3—32no—
95C22×Dic3C2×A412yes—
96C2×SL(2,𝔽3)—22no—
97C2×SL(2,𝔽3)—24no—
98C2×SL(2,𝔽3)—21no—
99C2×SL(2,𝔽3)C2×Dic312yes—
100C2×SL(2,𝔽3)C2×Dic314yescomplemented
101C2×SL(2,𝔽3)C22×D311yescharacteristic
102C3×C22⋊C4—124no—
103#217—34no—
104#217C2×A414yes—
105C4⋊Dic3—34no—
106C4⋊Dic3—38no—
107C4⋊Dic3—34no—
108C4×Dic3—38no—
109C4×Dic3—34no—
110C4×Dic3—34no—
111C22×C32—42no—
112C22×C32—44no—
113C22×C32—44no—
114C22×C32—41no—
115C22×C32—84no—
116C22×C32—88no—
117C3×Dic3—84no—
118C22×Q8—31no—
119C22×Q8—32no—
120C22×Q8—34no—
121C22×Q8C3×Dic314yescomplemented
122C22×Q8C2×C3×D311yescharacteristic
123C22×Q8C2×C3×D312yes—
124C23×C4—31no—
125C4×Q8—34no—
126C4×Q8—98no—
127C4×Q8—98no—
128C4×Q8—94no—
129#117—94no—
130#117—94no—
131#117—94no—
132#117—94no—
133C2×C4⋊C4—92no—
134C2×C4⋊C4—92no—
135C2×C22⋊C4—31no—
136C2×C22⋊C4—91no—
137C2×C42—92no—
138C2×C42—92no—
139C23×C3—24no—
140C23×C3—42no—
141C23×C3—41no—
142C23×C3—44no—
143C23×C3—84no—
144C23×C3—88no—
145C23×C3—82no—
146C23×C3—81no—
147C23×C3—84no—
148C23×C3C4×A414yes—
149C23×C3C22×A412yes—
150C23×C3C22×A411yescharacteristic
151C3×Q8—32no—
152C3×Q8—34no—
153C3×Q8—31no—
154C3×Q8—38no—
155C3×Q8—32no—
156C3×Q8—34no—
157C3×Q8C22×C3×C411yescharacteristic
158C2×C3×C4—34no—
159C2×C3×C4—38no—
160C2×C3×C4—32no—
161C2×C3×C4—34no—
162C2×C3×C4—32no—
163C2×C3×C4—34no—
164C2×C3×C4—32no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—124no—
168C2×C3×C4—244no—
169C2×Dic3—24no—
170C2×Dic3—34no—
171C2×Dic3—64no—
172C2×Dic3C22×A414yes—
173SL(2,𝔽3)—21no—
174SL(2,𝔽3)C22×Dic311yescharacteristic
175C2×C32—41no—
176C2×C32—42no—
177C2×C32—44no—
178C2×C32—84no—
179C24D3×A411yescharacteristic
180C2×Q8—32no—
181C2×Q8—34no—
182C2×Q8—38no—
183C2×Q8—31no—
184C2×Q8—32no—
185C2×Q8—34no—
186C2×Q8—38no—
187C2×Q8—32no—
188C2×Q8—34no—
189C2×Q8C2×C3×Dic312yes—
190C2×Q8C2×C3×Dic314yescomplemented
191C2×Q8C22×C3×D311yescharacteristic
192C22×C4—32no—
193C22×C4—32no—
194C22×C4—31no—
195C22×C4—31no—
196C22×C4—34no—
197C22×C4—34no—
198C22×C4—32no—
199C22×C4—92no—
200C4⋊C4—98no—
201C4⋊C4—94no—
202C4⋊C4—94no—
203C22⋊C4—34no—
204C22⋊C4—94no—
205C42—94no—
206C42—94no—
207C42—98no—
208C22×C3—24no—
209C22×C3—28no—
210C22×C3—44no—
211C22×C3—42no—
212C22×C3—44no—
213C22×C3—41no—
214C22×C3—84no—
215C22×C3—88no—
216C22×C3—88no—
217C22×C3—816no—
218C22×C3—84no—
219C22×C3—84no—
220C22×C3—82no—
221C22×C3—81no—
222C22×C3C2×C4×A414yes—
223C22×C3C2×C4×A412yes—
224C22×C32−1+4⋊C314yes—
225C22×C3C23×A411yescharacteristic
226C3×C4—31no—
227C3×C4—32no—
228C3×C4—34no—
229C3×C4—31no—
230C3×C4—244no—
231Dic3—24no—
232Dic3—64no—
233C32—41noSylow
234C23—24no—
235C23Dic3×A414yes—
236C23C2×D3×A412yes—
237C23C2×D3×A411yescharacteristic
238Q8—32no—
239Q8—34no—
240Q8—31no—
241Q8—38no—
242Q8—32no—
243Q8—34no—
244Q8C22×C3×Dic311yescharacteristic
245C2×C4—32no—
246C2×C4—34no—
247C2×C4—34no—
248C2×C4—38no—
249C2×C4—32no—
250C2×C4—34no—
251C2×C4—31no—
252C2×C4—31no—
253C2×C4—32no—
254C2×C4—34no—
255C2×C4—64no—
256C2×C4—94no—
257C2×C4—184no—
258C2×C3—24no—
259C2×C3—42no—
260C2×C3—41no—
261C2×C3—44no—
262C2×C3—84no—
263C2×C3—88no—
264C2×C3—82no—
265C2×C3—81no—
266C2×C3—84no—
267C2×C3#519⋊C314yescomplemented
268C2×C3C22×C4×A411yescharacteristic
269C2×C3C2×2−1+4⋊C312yes—
270C22—24no—
271C22—28no—
272C22#1007⋊C314yes—
273C22C2×Dic3×A414yes—
274C22C2×Dic3×A412yes—
275C22C22×D3×A411yescharacteristic
276C4—31no—
277C4—32no—
278C4—34no—
279C4—31no—
280C4—64no—
281C4—184no—
282C3—41no—
283C3—81no—
284C3C2×#519⋊C311yescharacteristic, complemented
285C2—24no—
286C2#5812⋊C314yesdirect product factor
287C2C2×#1007⋊C312yes—
288C2C22×Dic3×A411yescharacteristic
subgroupG/H
centerC23C2×D3×A4
commutator, hypocenterC3×Q8C22×C3×C4
FrattiniC22C22×D3×A4
FittingC23×C3×Q8C2×C3
socleC23×C3C22×A4
prime radicalC23×C3×SL(2,𝔽3)C2
cosocleC2×C3×Q8C23×C3
hypercenterC24D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. #5812⋊C3C2direct (N × H)
3. C2×Dic3×SL(2,𝔽3)C2semidirect
4. C2×#5812C3semidirect
6. C22×C3×SL(2,𝔽3)C4semidirect
10. Dic3×SL(2,𝔽3)C22semidirect
15. #5812C2×C3semidirect
17. C2×Q8×Dic3C2×C3semidirect
25. C2×C3×SL(2,𝔽3)C2×C4semidirect
32. C22×C3×Q8C3×C4semidirect
39. C22×SL(2,𝔽3)Dic3semidirect
48. Q8×Dic3C22×C3semidirect
85. C2×C3×Q8C2×C3×C4semidirect
100. C2×SL(2,𝔽3)C2×Dic3semidirect
121. C22×Q8C3×Dic3semidirect
190. C2×Q8C2×C3×Dic3semidirect
267. C2×C3#519⋊C3semidirect
284. C3C2×#519⋊C3semidirect
286. C2#5812⋊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
#5812⋊C32
C2×Dic3×SL(2,𝔽3)2
C2×#58123
C2×#519⋊C33
C2×C3×#2174