Finite Group Databasebeta

← C2×C3×#5184 #11860106

Subgroups — C2×C3×#5184 #11860106

Order 1152.

1089 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×Q16C211yesmaximal, characteristic
2C2×C3×Q8×Dic3C212yesmaximal
3C2×C3×#835C212yesmaximal
4C3×#5184C218yesmaximal, direct product factor
5C2×C3×#817C211yesmaximal, characteristic
6C2×C3×#815C211yesmaximal, characteristic
7C2×C3×#441—31nomaximal
8C2×#5184C311yesmaximal, characteristic, direct product factor
9C22×C32×Q8C2212yes—
10C2×C32×Q16C418yescomplemented
11C2×C32×Q16C2214yes—
12C22×C32×C8C2211yescharacteristic
13C3×Q8×Dic3—24no—
14C3×Q8×Dic3C2218yes—
15C2×C3×C4⋊Dic3—22no—
16C2×C3×C4⋊Dic3C2212yes—
17C2×C3×C4×Dic3—22no—
18C2×C3×C4×Dic3C2211yescharacteristic
19C22×C3×C3⋊C8C2211yescharacteristic
20C3×#835C2218yes—
21C3×#817C2214yes—
22C3×#815C2214yes—
23C22×C3×Q16—21no—
24C22×C3×Q16D311yescharacteristic
25C22×C3×Q16C2×C311yescharacteristic
26C2×C3×C4×Q8—32no—
27C2×Q8×Dic3C2×C312yes—
28C3×#441—38no—
29C2×C3×#106—31no—
30C2×C3×#103—32no—
31C2×C3×#97—31no—
32C2×#835C2×C312yes—
33#5184C2×C318yesdirect product factor
34C2×#817C2×C311yescharacteristic
35C2×#815C2×C311yescharacteristic
36C2×C32×Q8—22no—
37C2×C32×Q8—24no—
38C2×C32×Q8C2×C418yes—
39C2×C32×Q8C2314yes—
40C22×C32×C4—22no—
41C22×C32×C4C2311yescharacteristic
42C22×C3×Dic3—22no—
43C22×C3×Dic3D412yes—
44C32×Q16C2×C4116yescomplemented
45C2×C32×C8C2×C414yes—
46C2×C32×C8C2312yes—
47C3×C4⋊Dic3—22no—
48C3×C4⋊Dic3—28no—
49C3×C4⋊Dic3C2314yes—
50C3×C4×Dic3—28no—
51C3×C4×Dic3D412yes—
52C3×C4×Dic3C2312yes—
53C2×C3×C3⋊C8—22no—
54C2×C3×C3⋊C8C2312yes—
55C2×#441—31noSylow
56C22×C3×Q8—22no—
57C22×C3×Q8C2×D312yes—
58C22×C3×Q8C22×C312yes—
59C2×C3×Q16—28no—
60C2×C3×Q16—24no—
61C2×C3×Q16Dic318yescomplemented
62C2×C3×Q16C3×C418yescomplemented
63C2×C3×Q16C2×D314yes—
64C2×C3×Q16C22×C314yes—
65C22×C3×C8—21no—
66C22×C3×C8—31no—
67C22×C3×C8C2×D311yescharacteristic
68C22×C3×C8C22×C311yescharacteristic
69C3×C4×Q8—38no—
70C3×C4×Q8—64no—
71C2×C3×C4⋊C4—32no—
72C2×C3×C4⋊C4—62no—
73C2×C3×C42—31no—
74C2×C3×C42—62no—
75Q8×Dic3—24no—
76Q8×Dic3C22×C318yes—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3C22×C312yes—
79C2×C4×Dic3—22no—
80C2×C4×Dic3C22×C311yescharacteristic
81C22×C3⋊C8C22×C311yescharacteristic
82C3×#106—34no—
83C3×#103—38no—
84C3×#97—34no—
85#835C22×C318yes—
86#817C22×C314yes—
87#815C22×C314yes—
88C23×C32C2×D411yescharacteristic
89C32×Q8—28no—
90C32×Q8—24no—
91C32×Q8C22×C418yes—
92C2×C32×C4—28no—
93C2×C32×C4—24no—
94C2×C32×C4C22×C412yes—
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4#4112yes—
97C2×C32×C4C2411yescharacteristic
98C2×C3×Dic3—24no—
99C2×C3×Dic3—24no—
100C2×C3×Dic3—44no—
101C2×C3×Dic3C2×D414yes—
102C32×C8C22×C414yes—
103C3×C3⋊C8—22no—
104C22×Q16C3×D311yescharacteristic
105C2×C4×Q8—32no—
106#441—38no—
107C2×#106—31no—
108C2×#103—32no—
109C2×#97—31no—
110C2×C3×Q8—22no—
111C2×C3×Q8—24no—
112C2×C3×Q8—22no—
113C2×C3×Q8—24no—
114C2×C3×Q8—28no—
115C2×C3×Q8—24no—
116C2×C3×Q8—42no—
117C2×C3×Q8—44no—
118C2×C3×Q8C2×Dic318yes—
119C2×C3×Q8C2×C3×C418yes—
120C2×C3×Q8C22×D314yes—
121C2×C3×Q8C23×C314yes—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—32no—
126C22×C3×C4—42no—
127C22×C3×C4—62no—
128C22×C3×C4C22×D311yescharacteristic
129C22×C3×C4C23×C311yescharacteristic
130C22×Dic3—22no—
131C22×Dic3C3×D412yes—
132C3×Q16—216no—
133C3×Q16C2×Dic3116yescomplemented
134C3×Q16C2×C3×C4116yescomplemented
135C2×C3×C8—24no—
136C2×C3×C8—22no—
137C2×C3×C8—32no—
138C2×C3×C8—62no—
139C2×C3×C8C2×Dic314yes—
140C2×C3×C8C2×C3×C414yes—
141C2×C3×C8C22×D312yes—
142C2×C3×C8C23×C312yes—
143C3×C4⋊C4—34no—
144C3×C4⋊C4—68no—
145C3×C4⋊C4—62no—
146C3×C42—32no—
147C3×C42—32no—
148C3×C42—68no—
149C4⋊Dic3—22no—
150C4⋊Dic3—28no—
151C4⋊Dic3C23×C314yes—
152C4×Dic3—28no—
153C4×Dic3C3×D412yes—
154C4×Dic3C23×C312yes—
155C2×C3⋊C8—22no—
156C2×C3⋊C8C23×C312yes—
157C22×C32C4×D412yes—
158C22×C32#13714yes—
159C22×C32C22×D411yescharacteristic
160C32×C4—28no—
161C32×C4C4×D412yes—
162C32×C4C23×C411yescharacteristic
163C32×C4C2×#4111yescharacteristic
164C3×Dic3—24no—
165C3×Dic3—44no—
166C22×Q8C2×C3×D312yes—
167C2×Q16C3×Dic318yescomplemented
168C2×Q16C2×C3×D314yes—
169C22×C8—31no—
170C22×C8C2×C3×D311yescharacteristic
171C4×Q8—38no—
172C4×Q8—64no—
173C2×C4⋊C4—32no—
174C2×C4⋊C4—62no—
175C2×C42—31no—
176C2×C42—62no—
177#106—34no—
178#103—38no—
179#97—34no—
180C23×C3—21no—
181C23×C3D3×D411yescharacteristic
182C23×C3C2×C3×D411yescharacteristic
183C3×Q8—28no—
184C3×Q8—28no—
185C3×Q8—24no—
186C3×Q8—24no—
187C3×Q8—28no—
188C3×Q8—48no—
189C3×Q8—44no—
190C3×Q8C22×Dic318yes—
191C3×Q8C22×C3×C418yes—
192C2×C3×C4—28no—
193C2×C3×C4—28no—
194C2×C3×C4—24no—
195C2×C3×C4—24no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—34no—
201C2×C3×C4—48no—
202C2×C3×C4—44no—
203C2×C3×C4—64no—
204C2×C3×C4—64no—
205C2×C3×C4—124no—
206C2×C3×C4D3×D411yescharacteristic
207C2×C3×C4#23712yes—
208C2×C3×C4C22×Dic312yes—
209C2×C3×C4C22×C3×C412yes—
210C2×C3×C4C2×C3×D411yescharacteristic
211C2×C3×C4C3×#4112yes—
212C2×C3×C4C23×D311yescharacteristic
213C2×C3×C4C24×C311yescharacteristic
214C2×Dic3—24no—
215C2×Dic3—24no—
216C2×Dic3—44no—
217C2×Dic3C2×C3×D414yes—
218C3×C8—24no—
219C3×C8—62no—
220C3×C8C22×Dic314yes—
221C3×C8C22×C3×C414yes—
222C3⋊C8—22no—
223C2×C32#44114yescomplemented
224C2×C32C2×C4×D411yescharacteristic
225C2×C32C2×#13712yes—
226C2×Q8—22no—
227C2×Q8—24no—
228C2×Q8C2×C3×Dic318yes—
229C2×Q8C22×C3×D314yes—
230C22×C4—22no—
231C22×C4—32no—
232C22×C4—62no—
233C22×C4C22×C3×D311yescharacteristic
234Q16C2×C3×Dic3116yescomplemented
235C2×C8—32no—
236C2×C8—62no—
237C2×C8C2×C3×Dic314yes—
238C2×C8C22×C3×D312yes—
239C4⋊C4—34no—
240C4⋊C4—68no—
241C4⋊C4—62no—
242C42—32no—
243C42—32no—
244C42—68no—
245C22×C3—24no—
246C22×C3—22no—
247C22×C3—21no—
248C22×C3#91814yes—
249C22×C3D4×Dic312yes—
250C22×C3C3×C4×D412yes—
251C22×C3C3×#13714yes—
252C22×C3C2×D3×D411yescharacteristic
253C22×C3C22×C3×D411yescharacteristic
254C3×C4—28no—
255C3×C4—28no—
256C3×C4—22no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—48no—
260C3×C4—64no—
261C3×C4—124no—
262C3×C4D4×Dic312yes—
263C3×C4C3×C4×D412yes—
264C3×C4C2×#23711yescharacteristic
265C3×C4C23×Dic311yescharacteristic
266C3×C4C23×C3×C411yescharacteristic
267C3×C4C2×C3×#4111yescharacteristic
268Dic3—24no—
269Dic3—44no—
270C32C2×#44111yescharacteristic, complemented, Sylow
271C23C3×D3×D411yescharacteristic
272Q8—28no—
273Q8—24no—
274Q8C22×C3×Dic318yes—
275C2×C4—28no—
276C2×C4—24no—
277C2×C4—34no—
278C2×C4—64no—
279C2×C4—64no—
280C2×C4—124no—
281C2×C4C3×D3×D411yescharacteristic
282C2×C4C3×#23712yes—
283C2×C4C22×C3×Dic312yes—
284C2×C4C23×C3×D311yescharacteristic
285C8—62no—
286C8C22×C3×Dic314yes—
287C2×C3—22no—
288C2×C3—24no—
289C2×C3—21no—
290C2×C3#518414yesdirect product factor
291C2×C3C3×#44114yescomplemented
292C2×C3C2×#91812yes—
293C2×C3C2×D4×Dic311yescharacteristic
294C2×C3C2×C3×C4×D411yescharacteristic
295C2×C3C2×C3×#13712yes—
296C22C3×#91814yes—
297C22C3×D4×Dic312yes—
298C22C2×C3×D3×D411yescharacteristic
299C4—28no—
300C4—64no—
301C4—124no—
302C4C3×D4×Dic312yes—
303C4C2×C3×#23711yescharacteristic
304C4C23×C3×Dic311yescharacteristic
305C3—21no—
306C3C2×#518411yescharacteristic, direct product factor
307C3C2×C3×#44111yescharacteristic, complemented
308C2C3×#518414yesdirect product factor
309C2C2×C3×#91812yes—
310C2C2×C3×D4×Dic311yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutatorC3×C4C23×C3×C4
FrattiniC2×C4C23×C3×D3
FittingC22×C32×Q16C2
socle, prime radicalC23×C32C2×D4
cosocleC2×C3×C4C24×C3
hypercenterC22×C3×Q16D3
hypocenterC3C2×C3×#441
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×Q16

Complements

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

NH ≅ G/Ntype
4. C3×#5184C2direct (N × H)
8. C2×#5184C3direct (N × H)
10. C2×C32×Q16C4semidirect
33. #5184C2×C3direct (N × H)
44. C32×Q16C2×C4semidirect
61. C2×C3×Q16Dic3semidirect
62. C2×C3×Q16C3×C4semidirect
133. C3×Q16C2×Dic3semidirect
134. C3×Q16C2×C3×C4semidirect
167. C2×Q16C3×Dic3semidirect
223. C2×C32#441semidirect
234. Q16C2×C3×Dic3semidirect
270. C32C2×#441semidirect
290. C2×C3#5184direct (N × H)
291. C2×C3C3×#441semidirect
306. C3C2×#5184direct (N × H)
307. C3C2×C3×#441semidirect
308. C2C3×#5184direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×Q162
C2×C3×Q8×Dic32
C2×C3×#8352
C3×#51842
C2×C3×#8172
C2×C3×#8152
C2×C3×#4413
C2×#51843