Finite Group Databasebeta

← C2×C3×#5420 #11922787

Subgroups — C2×C3×#5420 #11922787

Order 1152.

793 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 793 conjugacy classes of subgroups — too many to draw. Only its 140 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×#5420C214yesmaximal, direct product factor
3C2×C3×#859C212yesmaximal, complemented
4C2×#5420C311yesmaximal, characteristic, direct product factor
5C2×C3×#419—31nomaximal
6C22×C3×SL(2,𝔽3)C414yescomplemented
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C22×C3×SL(2,𝔽3)C2212yes—
9C2×C3×#217—41nomaximal
10C3×#859C2214yescomplemented
11C23×C3×Q8D311yescharacteristic
12C23×SL(2,𝔽3)—21no—
13C23×SL(2,𝔽3)C2×C311yescharacteristic
14C22×C3×M16—31no—
15C2×C3×#117—31no—
16#5420C2×C314yesdirect product factor
17C2×#859C2×C312yescomplemented
18C3×#419—34no—
19C3×#419—34no—
20C2×C3×#103—32no—
21C2×C3×#103—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×#419—31noSylow
29C22×C3×Q8—34no—
30C22×C3×Q8—31no—
31C22×C3×Q8—32no—
32C22×C3×Q8Dic314yescomplemented
33C22×C3×Q8C2×D311yescharacteristic
34C22×C3×Q8C2×D312yes—
35C23×C3×C4—31no—
36C22×SL(2,𝔽3)—24no—
37C22×SL(2,𝔽3)—21no—
38C22×SL(2,𝔽3)—22no—
39C22×SL(2,𝔽3)C3×C414yescomplemented
40C22×SL(2,𝔽3)C22×C311yescharacteristic
41C22×SL(2,𝔽3)C22×C312yes—
42C2×C3×M16—34no—
43C2×C3×M16—64no—
44C22×C3×C8—32no—
45C3×#117—34no—
46C3×#117—62no—
47C2×C3×C4⋊C4—32no—
48C2×C3×C22⋊C4—61no—
49C2×C3×C42—61no—
50C2×#217—41no—
51#859C22×C314yescomplemented
52C3×#103—38no—
53C3×#103—34no—
54C3×#103—34no—
55C23×C32—44no—
56C23×C32—42no—
57C23×C32—41no—
58C23×C32—84no—
59C2×C3×Dic3—44no—
60C2×C3×Dic3—84no—
61C3×SL(2,𝔽3)C22×C411yescharacteristic
62C23×Q8C3×D311yescharacteristic
63C22×M16—31no—
64C2×#117—31no—
65#419—34no—
66#419—34no—
67C2×#103—32no—
68C2×#103—32no—
69C24×C3—41no—
70C24×C3—81no—
71C24×C3S411yescharacteristic
72C2×C3×Q8—32no—
73C2×C3×Q8—31no—
74C2×C3×Q8—34no—
75C2×C3×Q8—38no—
76C2×C3×Q8—32no—
77C2×C3×Q8—34no—
78C2×C3×Q8—62no—
79C2×C3×Q8—64no—
80C2×C3×Q8—61no—
81C2×C3×Q8C2×Dic312yes—
82C2×C3×Q8C2×Dic314yescomplemented
83C2×C3×Q8C22×D311yescharacteristic
84C22×C3×C4—34no—
85C22×C3×C4—34no—
86C22×C3×C4—32no—
87C22×C3×C4—32no—
88C22×C3×C4—31no—
89C22×C3×C4—31no—
90C22×C3×C4—62no—
91C22×Dic3—42no—
92C2×SL(2,𝔽3)—22no—
93C2×SL(2,𝔽3)—24no—
94C2×SL(2,𝔽3)—21no—
95C2×SL(2,𝔽3)C2×C3×C412yes—
96C2×SL(2,𝔽3)C2×C3×C414yescomplemented
97C2×SL(2,𝔽3)C23×C311yescharacteristic
98C3×M16—68no—
99C2×C3×C8—34no—
100C2×C3×C8—64no—
101C3×C4⋊C4—34no—
102C3×C4⋊C4—62no—
103C3×C22⋊C4—64no—
104C3×C42—64no—
105#217—44no—
106C22×C32—44no—
107C22×C32—42no—
108C22×C32—44no—
109C22×C32—41no—
110C22×C32—84no—
111C22×C32—88no—
112C3×Dic3—84no—
113C22×Q8—31no—
114C22×Q8—32no—
115C22×Q8—34no—
116C22×Q8C3×Dic314yescomplemented
117C22×Q8C2×C3×D311yescharacteristic
118C22×Q8C2×C3×D312yes—
119C23×C4—31no—
120C2×M16—34no—
121C2×M16—64no—
122C22×C8—32no—
123#117—34no—
124#117—62no—
125C2×C4⋊C4—32no—
126C2×C22⋊C4—61no—
127C2×C42—61no—
128#103—38no—
129#103—34no—
130#103—34no—
131C23×C3—24no—
132C23×C3—41no—
133C23×C3—42no—
134C23×C3—44no—
135C23×C3—84no—
136C23×C3—88no—
137C23×C3—82no—
138C23×C3—81no—
139C23×C3—84no—
140C23×C3SL(2,ℤ/4)14yes—
141C23×C3C2×S412yes—
142C23×C3C2×S411yescharacteristic
143C3×Q8—32no—
144C3×Q8—34no—
145C3×Q8—31no—
146C3×Q8—61no—
147C3×Q8—62no—
148C3×Q8—64no—
149C3×Q8C22×Dic311yescharacteristic
150C2×C3×C4—32no—
151C2×C3×C4—34no—
152C2×C3×C4—32no—
153C2×C3×C4—34no—
154C2×C3×C4—38no—
155C2×C3×C4—34no—
156C2×C3×C4—32no—
157C2×C3×C4—31no—
158C2×C3×C4—31no—
159C2×C3×C4—64no—
160C2×C3×C4—124no—
161C2×Dic3—44no—
162C2×Dic3—84no—
163SL(2,𝔽3)—21no—
164SL(2,𝔽3)C22×C3×C411yescharacteristic
165C3×C8—64no—
166C2×C32—42no—
167C2×C32—41no—
168C2×C32—44no—
169C2×C32—84no—
170C24C3×S411yescharacteristic
171C2×Q8—32no—
172C2×Q8—31no—
173C2×Q8—34no—
174C2×Q8—38no—
175C2×Q8—32no—
176C2×Q8—34no—
177C2×Q8—62no—
178C2×Q8—64no—
179C2×Q8—61no—
180C2×Q8C2×C3×Dic312yes—
181C2×Q8C2×C3×Dic314yescomplemented
182C2×Q8C22×C3×D311yescharacteristic
183C22×C4—31no—
184C22×C4—31no—
185C22×C4—32no—
186C22×C4—32no—
187C22×C4—34no—
188C22×C4—34no—
189C22×C4—62no—
190M16—68no—
191C2×C8—34no—
192C2×C8—64no—
193C4⋊C4—34no—
194C4⋊C4—62no—
195C22⋊C4—64no—
196C42—64no—
197C22×C3—24no—
198C22×C3—28no—
199C22×C3—42no—
200C22×C3—44no—
201C22×C3—41no—
202C22×C3—44no—
203C22×C3—88no—
204C22×C3—84no—
205C22×C3—816no—
206C22×C3—88no—
207C22×C3—84no—
208C22×C3—82no—
209C22×C3—81no—
210C22×C3—84no—
211C22×C3#98314yes—
212C22×C3C2×SL(2,ℤ/4)14yes—
213C22×C3C2×SL(2,ℤ/4)12yes—
214C22×C3C22×S411yescharacteristic
215C3×C4—31no—
216C3×C4—31no—
217C3×C4—32no—
218C3×C4—34no—
219C3×C4—124no—
220Dic3—84no—
221C32—41noSylow
222C23—24no—
223C23C3×SL(2,ℤ/4)14yes—
224C23C2×C3×S412yes—
225C23C2×C3×S411yescharacteristic
226Q8—32no—
227Q8—34no—
228Q8—31no—
229Q8—61no—
230Q8—62no—
231Q8—64no—
232Q8C22×C3×Dic311yescharacteristic
233C2×C4—32no—
234C2×C4—34no—
235C2×C4—32no—
236C2×C4—34no—
237C2×C4—38no—
238C2×C4—34no—
239C2×C4—32no—
240C2×C4—31no—
241C2×C4—31no—
242C2×C4—64no—
243C2×C4—124no—
244C8—64no—
245C2×C3—24no—
246C2×C3—44no—
247C2×C3—42no—
248C2×C3—41no—
249C2×C3—84no—
250C2×C3—88no—
251C2×C3—84no—
252C2×C3—82no—
253C2×C3—81no—
254C2×C3#542014yesdirect product factor
255C2×C3C2×#98312yes—
256C2×C3C22×SL(2,ℤ/4)11yescharacteristic
257C22—28no—
258C22—24no—
259C22C3×#98314yes—
260C22C2×C3×SL(2,ℤ/4)14yes—
261C22C2×C3×SL(2,ℤ/4)12yes—
262C22C22×C3×S411yescharacteristic
263C4—31no—
264C4—31no—
265C4—32no—
266C4—34no—
267C4—124no—
268C3—41no—
269C3—81no—
270C3C2×#542011yescharacteristic, direct product factor
271C2—24no—
272C2C3×#542014yesdirect product factor
273C2C2×C3×#98312yes—
274C2C22×C3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC23×C3C2×S4
commutator, hypocenterSL(2,𝔽3)C22×C3×C4
FrattiniC22C22×C3×S4
FittingC23×C3×Q8D3
prime radicalC23×C3×SL(2,𝔽3)C2
cosocleC2×SL(2,𝔽3)C23×C3
hypercenterC24×C3S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
2. C3×#5420C2direct (N × H)
3. C2×C3×#859C2semidirect
4. C2×#5420C3direct (N × H)
6. C22×C3×SL(2,𝔽3)C4semidirect
10. C3×#859C22semidirect
16. #5420C2×C3direct (N × H)
17. C2×#859C2×C3semidirect
25. C2×C3×SL(2,𝔽3)C2×C4semidirect
32. C22×C3×Q8Dic3semidirect
39. C22×SL(2,𝔽3)C3×C4semidirect
51. #859C22×C3semidirect
82. C2×C3×Q8C2×Dic3semidirect
96. C2×SL(2,𝔽3)C2×C3×C4semidirect
116. C22×Q8C3×Dic3semidirect
181. C2×Q8C2×C3×Dic3semidirect
254. C2×C3#5420direct (N × H)
270. C3C2×#5420direct (N × H)
272. C2C3×#5420direct (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×#54202
C2×C3×#8592
C2×#54203
C2×C3×#4193
C2×C3×#2174