Finite Group Databasebeta

← C2×C3×#5424 #11922799

Subgroups — C2×C3×#5424 #11922799

Order 1152.

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

normal subgroups only

This group has 729 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic
2C3×#5424C214yesmaximal, direct product factor
3C2×C3×#859C212yesmaximal, complemented
4C2×#5424C311yesmaximal, characteristic, direct product factor
5C2×C3×#416—31nomaximal
6C2×C3×#41⋊C3C414yescomplemented
7C2×C3×#41⋊C3C2212yes—
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×C3×C4×Dic3—41nomaximal
10C3×#859C2214yescomplemented
11C22×C3×#41D311yescharacteristic
12C22×#41⋊C3—21no—
13C22×#41⋊C3C2×C311yescharacteristic
14C23×C3×C8—31no—
15C2×C3×#117—31no—
16#5424C2×C314yesdirect product factor
17C2×#859C2×C312yescomplemented
18C3×#416—34no—
19C3×#416—34no—
20C2×C3×#103—32no—
21C2×C3×#102—32no—
22C22×C32×C4—41no—
23C22×C3×Dic3—42no—
24C3×#41⋊C3C2×C414yescomplemented
25C2×C3×SL(2,𝔽3)C2×C412yes—
26C2×C3×SL(2,𝔽3)C2311yescharacteristic
27C3×C4×Dic3—44no—
28C2×#416—31noSylow
29C2×C3×#41—34no—
30C2×C3×#41—32no—
31C2×C3×#41Dic314yescomplemented
32C2×C3×#41C2×D312yes—
33C22×C3×Q8C2×D311yescharacteristic
34C22×C3×D4—31no—
35C23×C3×C4—31no—
36C2×#41⋊C3—24no—
37C2×#41⋊C3—22no—
38C2×#41⋊C3C3×C414yescomplemented
39C2×#41⋊C3C22×C312yes—
40C22×SL(2,𝔽3)—21no—
41C22×SL(2,𝔽3)C22×C311yescharacteristic
42C22×C3×C8—32no—
43C22×C3×C8—34no—
44C22×C3×C8—64no—
45C3×#117—34no—
46C3×#117—62no—
47C2×C3×C4⋊C4—32no—
48C2×C3×C22⋊C4—61no—
49C2×C3×C42—61no—
50C2×C4×Dic3—41no—
51#859C22×C314yescomplemented
52C3×#103—34no—
53C3×#103—34no—
54C3×#102—38no—
55C23×C32—41no—
56C2×C32×C4—44no—
57C2×C32×C4—42no—
58C2×C3×Dic3—44no—
59C2×C3×Dic3—48no—
60C3×SL(2,𝔽3)C22×C411yescharacteristic
61C22×#41C3×D311yescharacteristic
62C23×C8—31no—
63C2×#117—31no—
64#416—34no—
65#416—34no—
66C2×#103—32no—
67C2×#102—32no—
68C24×C3—31no—
69C3×#41—38no—
70C3×#41—34no—
71C3×#41—64no—
72C3×#41C2×Dic314yescomplemented
73C2×C3×Q8—31no—
74C2×C3×Q8—32no—
75C2×C3×Q8C2×Dic312yes—
76C2×C3×Q8C22×D311yescharacteristic
77C2×C3×D4—32no—
78C2×C3×D4—34no—
79C2×C3×D4—62no—
80C2×C3×D4—61no—
81C22×C3×C4—31no—
82C22×C3×C4—34no—
83C22×C3×C4—34no—
84C22×C3×C4—32no—
85C22×C3×C4—32no—
86C22×C3×C4—41no—
87C22×C3×C4—62no—
88C22×C3×C4—81no—
89C22×C3×C4S411yescharacteristic
90C22×Dic3—42no—
91#41⋊C3—24no—
92#41⋊C3C2×C3×C414yescomplemented
93C2×SL(2,𝔽3)—22no—
94C2×SL(2,𝔽3)—21no—
95C2×SL(2,𝔽3)C2×C3×C412yes—
96C2×SL(2,𝔽3)C23×C311yescharacteristic
97C2×C3×C8—34no—
98C2×C3×C8—64no—
99C2×C3×C8—68no—
100C3×C4⋊C4—34no—
101C3×C4⋊C4—62no—
102C3×C22⋊C4—64no—
103C3×C42—64no—
104C4×Dic3—44no—
105C22×C32—44no—
106C22×C32—42no—
107C22×C32—41no—
108C32×C4—44no—
109C3×Dic3—48no—
110C2×#41—32no—
111C2×#41—34no—
112C2×#41C3×Dic314yescomplemented
113C2×#41C2×C3×D312yes—
114C22×Q8C2×C3×D311yescharacteristic
115C22×D4—31no—
116C23×C4—31no—
117C22×C8—32no—
118C22×C8—34no—
119C22×C8—64no—
120#117—34no—
121#117—62no—
122C2×C4⋊C4—32no—
123C2×C22⋊C4—61no—
124C2×C42—61no—
125#103—34no—
126#103—34no—
127#102—38no—
128C23×C3—34no—
129C23×C3—32no—
130C23×C3—41no—
131C23×C3—64no—
132C23×C3—81no—
133C23×C3C2×S411yescharacteristic
134C3×Q8—32no—
135C3×Q8—31no—
136C3×Q8—61no—
137C3×Q8C22×Dic311yescharacteristic
138C3×D4—34no—
139C3×D4—62no—
140C3×D4—64no—
141C2×C3×C4—32no—
142C2×C3×C4—32no—
143C2×C3×C4—31no—
144C2×C3×C4—31no—
145C2×C3×C4—34no—
146C2×C3×C4—34no—
147C2×C3×C4—38no—
148C2×C3×C4—42no—
149C2×C3×C4—44no—
150C2×C3×C4—64no—
151C2×C3×C4—82no—
152C2×C3×C4—84no—
153C2×C3×C4—124no—
154C2×C3×C4SL(2,ℤ/4)14yes—
155C2×C3×C4C2×S412yes—
156C2×Dic3—44no—
157C2×Dic3—48no—
158SL(2,𝔽3)—21no—
159SL(2,𝔽3)C22×C3×C411yescharacteristic
160C3×C8—64no—
161C2×C32—42no—
162C2×C32—41no—
163C2×C32—44no—
164C24—31no—
165#41—38no—
166#41—34no—
167#41—64no—
168#41C2×C3×Dic314yescomplemented
169C2×Q8—31no—
170C2×Q8—32no—
171C2×Q8C2×C3×Dic312yes—
172C2×Q8C22×C3×D311yescharacteristic
173C2×D4—32no—
174C2×D4—34no—
175C2×D4—62no—
176C2×D4—61no—
177C22×C4—31no—
178C22×C4—32no—
179C22×C4—32no—
180C22×C4—34no—
181C22×C4—34no—
182C22×C4—62no—
183C22×C4C3×S411yescharacteristic
184C2×C8—34no—
185C2×C8—64no—
186C2×C8—68no—
187C4⋊C4—34no—
188C4⋊C4—62no—
189C22⋊C4—64no—
190C42—64no—
191C22×C3—34no—
192C22×C3—42no—
193C22×C3—41no—
194C22×C3—44no—
195C22×C3—64no—
196C22×C3—68no—
197C22×C3—82no—
198C22×C3—81no—
199C22×C3—84no—
200C22×C3#98514yes—
201C22×C3C2×SL(2,ℤ/4)12yes—
202C22×C3C22×S411yescharacteristic
203C3×C4—31no—
204C3×C4—31no—
205C3×C4—32no—
206C3×C4—44no—
207C3×C4—84no—
208C3×C4—124no—
209C3×C4C2×SL(2,ℤ/4)14yes—
210Dic3—48no—
211C32—41noSylow
212C23—34no—
213C23—32no—
214C23—64no—
215C23C2×C3×S411yescharacteristic
216Q8—32no—
217Q8—31no—
218Q8—61no—
219Q8C22×C3×Dic311yescharacteristic
220D4—34no—
221D4—62no—
222D4—64no—
223C2×C4—32no—
224C2×C4—32no—
225C2×C4—31no—
226C2×C4—31no—
227C2×C4—34no—
228C2×C4—34no—
229C2×C4—38no—
230C2×C4—64no—
231C2×C4—124no—
232C2×C4C3×SL(2,ℤ/4)14yes—
233C2×C4C2×C3×S412yes—
234C8—64no—
235C2×C3—44no—
236C2×C3—42no—
237C2×C3—41no—
238C2×C3—64no—
239C2×C3—84no—
240C2×C3—82no—
241C2×C3—81no—
242C2×C3#542414yesdirect product factor
243C2×C3C2×#98512yes—
244C2×C3C22×SL(2,ℤ/4)11yescharacteristic
245C22—34no—
246C22—68no—
247C22—64no—
248C22C3×#98514yes—
249C22C2×C3×SL(2,ℤ/4)12yes—
250C22C22×C3×S411yescharacteristic
251C4—31no—
252C4—31no—
253C4—32no—
254C4—124no—
255C4C2×C3×SL(2,ℤ/4)14yes—
256C3—41no—
257C3—81no—
258C3C2×#542411yescharacteristic, direct product factor
259C2—64no—
260C2C3×#542414yesdirect product factor
261C2C2×C3×#98512yes—
262C2C22×C3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4S4
commutator, hypocenterSL(2,𝔽3)C22×C3×C4
FrattiniC22C22×C3×S4
FittingC22×C3×#41D3
socleC23×C3C2×S4
prime radicalC22×C3×#41⋊C3C2
cosocleC2×SL(2,𝔽3)C23×C3
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#5424C2direct (N × H)
3. C2×C3×#859C2semidirect
4. C2×#5424C3direct (N × H)
6. C2×C3×#41⋊C3C4semidirect
10. C3×#859C22semidirect
16. #5424C2×C3direct (N × H)
17. C2×#859C2×C3semidirect
24. C3×#41⋊C3C2×C4semidirect
31. C2×C3×#41Dic3semidirect
38. C2×#41⋊C3C3×C4semidirect
51. #859C22×C3semidirect
72. C3×#41C2×Dic3semidirect
92. #41⋊C3C2×C3×C4semidirect
112. C2×#41C3×Dic3semidirect
168. #41C2×C3×Dic3semidirect
242. C2×C3#5424direct (N × H)
258. C3C2×#5424direct (N × H)
260. C2C3×#5424direct (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×#54242
C2×C3×#8592
C2×#54243
C2×C3×#4163
C2×C3×C4×Dic34