Finite Group Databasebeta

← C3×#10592561 #13162900

Subgroups — C3×#10592561 #13162900

Order 1728.

768 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64106C211yesmaximal, characteristic
2C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic
3C33×#106C211yesmaximal, characteristic, complemented
4C3×#63647C211yesmaximal, characteristic
5C3×#63643C211yesmaximal, characteristic, complemented
6C3×#63637C211yesmaximal, characteristic, complemented
7C3×#63635C211yesmaximal, characteristic, complemented
8#10592561C311yesmaximal, characteristic, direct product factor
9C3×#4862—34nomaximal
10C2×C3×C4×C3⋊D3—21no—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C33×C4⋊C4C2211yescharacteristic
16C3×#3752—21no—
17C3×C4⋊(C3⋊Dic3)—21no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×#3750—21no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21#64106C2×C311yescharacteristic
22C4⋊C4×C3⋊D3C2×C311yescharacteristic
23C3×#897—34no—
24C3×D3×C4⋊C4—34no—
25C32×#106—24no—
26C32×#106D314yescomplemented
27C32×#106C2×C311yescharacteristic, complemented
28#63647C2×C311yescharacteristic
29#63643C2×C311yescharacteristic, complemented
30#63637C2×C311yescharacteristic, complemented
31#63635C2×C311yescharacteristic, complemented
32C3×(C2×D3)⋊C8—34no—
33C3×#816—34no—
34C3×#810—34no—
35C3×#808—34no—
36C22×C3×C3⋊D3D411yescharacteristic
37C2×C33×C4—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×C4×C3⋊D3—21no—
44C3×C4×C3⋊D3—21no—
45C3×C4×C3⋊D3—42no—
46C3×#634—21no—
47C3×#634—21no—
48C33×C8—21no—
49C3×C3⋊(C3⋊C8)—21no—
50C3×#483—91no—
51#4862—34no—
52C2×C4×C3⋊D3—21no—
53C2×C4×C3⋊D3C22×C311yescharacteristic
54C2×#634C22×C311yescharacteristic
55C2×C3×C4×D3—34no—
56C2×C3×C4×D3—64no—
57C2×C3×Dic6—34no—
58C2×C32×C8—24no—
59C2×C32×C8C2×D314yes—
60C2×C32×C8C22×C311yescharacteristic
61C32×C4⋊C4—24no—
62C32×C4⋊C4—24no—
63C32×C4⋊C4C2×D314yes—
64C32×C4⋊C4C2×D314yes—
65C32×C4⋊C4C22×C311yescharacteristic
66C32×C4⋊C4C22×C311yescharacteristic
67#3752—21no—
68C4⋊(C3⋊Dic3)—21no—
69C4⋊(C3⋊Dic3)C22×C311yescharacteristic
70#3750—21no—
71C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
72C3×(C2×D3)⋊C4—64no—
73C3×C4⋊Dic3—34no—
74C3×C4⋊Dic3—64no—
75C3×Dic3⋊C4—64no—
76C2×C3×C3⋊C8—34no—
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—21no—
80C2×C3×C3⋊D3SD1612yes—
81C33×C4—41no—
82C33×C4—41no—
83C33×C4#4111yescharacteristic
84C33×C4#4111yescharacteristic
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3—41no—
87C3×C3⋊Dic3—41no—
88C3×C22⋊Q8—91no—
89C2×C3×C4⋊C4—91no—
90#897—34no—
91D3×C4⋊C4—34no—
92C3×#106—24no—
93C3×#106—91no—
94C3×#106C3×D314yescomplemented
95C3×#106C3⋊D311yescharacteristic, complemented
96C3×#103—91no—
97C3×#103—91no—
98C3×C22⋊C8—91no—
99(C2×D3)⋊C8—34no—
100#816—34no—
101#810—34no—
102#808—34no—
103C22×C3⋊D3C3×D411yescharacteristic
104C22×C3×D3—34no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4—24no—
110C2×C32×C4—44no—
111C2×C32×C4—44no—
112C2×C32×C4C22×D314yes—
113C2×C32×C4C23×C311yescharacteristic
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3C3×D411yescharacteristic
117C4×C3⋊D3—21no—
118C4×C3⋊D3—21no—
119C4×C3⋊D3—42no—
120#634—21no—
121#634—21no—
122C2×C3×Dic3—34no—
123C2×C3×Dic3—64no—
124C2×C3×Dic3—64no—
125C3×C4×D3—64no—
126C3×C4×D3—64no—
127C3×C4×D3—128no—
128C3×Dic6—64no—
129C3×Dic6—64no—
130C32×C8—28no—
131C32×C8—24no—
132C32×C8—21no—
133C3⋊(C3⋊C8)—21no—
134C3×C3⋊C8—64no—
135#483—91noSylow
136C2×C33#12311yescharacteristic
137C2×C33C2×SD1611yescharacteristic
138C2×C33#13711yescharacteristic
139C3×C3⋊D3—22no—
140C2×C3×Q8—91no—
141C22×C3×C4—91no—
142C22×C3×C4—181no—
143C2×C4×D3—34no—
144C2×C4×D3—64no—
145C2×Dic6—34no—
146C2×C3×C8—24no—
147C2×C3×C8—91no—
148C2×C3×C8C2×C3×D314yes—
149C2×C3×C8C2×C3⋊D311yescharacteristic
150C3×C4⋊C4—24no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—91no—
153C3×C4⋊C4—181no—
154C3×C4⋊C4—181no—
155C3×C4⋊C4C2×C3×D314yes—
156C3×C4⋊C4C2×C3×D314yes—
157C3×C4⋊C4C2×C3⋊D311yescharacteristic
158C3×C4⋊C4C2×C3⋊D311yescharacteristic
159C3×C22⋊C4—181no—
160(C2×D3)⋊C4—64no—
161C4⋊Dic3—34no—
162C4⋊Dic3—64no—
163Dic3⋊C4—64no—
164C2×C3⋊C8—34no—
165C22×C32—24no—
166C22×C32D3×D414yes—
167C22×C32C2×C3×D411yescharacteristic
168C2×C3⋊D3—21no—
169C2×C3⋊D3—21no—
170C2×C3⋊D3C3×SD1612yes—
171C2×C3×D3—38no—
172C2×C3×D3—64no—
173C2×C3×D3—64no—
174C32×C4—24no—
175C32×C4—24no—
176C32×C4—41no—
177C32×C4—41no—
178C32×C4—44no—
179C32×C4—44no—
180C32×C4—84no—
181C32×C4—84no—
182C32×C4#23514yes—
183C32×C4#23914yes—
184C32×C4C3×#4111yescharacteristic
185C32×C4C3×#4111yescharacteristic
186C3⋊Dic3—21no—
187C3⋊Dic3—41no—
188C3⋊Dic3—41no—
189C3×Dic3—64no—
190C3×Dic3—124no—
191C3×Dic3—124no—
192C22⋊Q8—91no—
193C2×C4⋊C4—91no—
194#106—91no—
195#106C3×C3⋊D311yescharacteristic, complemented
196#103—91no—
197#103—91no—
198C22⋊C8—91no—
199C33#48311yescharacteristic, complemented, Sylow
200C23×C3—91no—
201C22×D3—34no—
202C3×Q8—181no—
203C3×Q8—181no—
204C2×C3×C4—24no—
205C2×C3×C4—24no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—24no—
209C2×C3×C4—44no—
210C2×C3×C4—44no—
211C2×C3×C4—91no—
212C2×C3×C4—181no—
213C2×C3×C4—181no—
214C2×C3×C4—181no—
215C2×C3×C4—181no—
216C2×C3×C4—362no—
217C2×C3×C4C22×C3×D314yes—
218C2×C3×C4C22×C3⋊D311yescharacteristic
219C2×Dic3—34no—
220C2×Dic3—64no—
221C2×Dic3—64no—
222C4×D3—64no—
223C4×D3—64no—
224C4×D3—128no—
225Dic6—64no—
226Dic6—64no—
227C3×C8—28no—
228C3×C8—24no—
229C3×C8—21no—
230C3×C8—181no—
231C3⋊C8—64no—
232C2×C32—24no—
233C2×C32—24no—
234C2×C32—24no—
235C2×C32#89414yes—
236C2×C32D3×SD1614yes—
237C2×C32#91514yes—
238C2×C32C3×#12311yescharacteristic
239C2×C32C2×C3×SD1611yescharacteristic
240C2×C32C3×#13711yescharacteristic
241C3⋊D3—22no—
242C3×D3—68no—
243C2×Q8—91no—
244C22×C4—91no—
245C22×C4—181no—
246C2×C8—91no—
247C2×C8C2×C3×C3⋊D311yescharacteristic
248C4⋊C4—91no—
249C4⋊C4—181no—
250C4⋊C4—181no—
251C4⋊C4C2×C3×C3⋊D311yescharacteristic
252C4⋊C4C2×C3×C3⋊D311yescharacteristic
253C22⋊C4—181no—
254C22×C3—24no—
255C22×C3—92no—
256C22×C3—181no—
257C22×C3—181no—
258C22×C3C3×D3×D414yes—
259C22×C3D4×C3⋊D311yescharacteristic
260C2×D3—38no—
261C2×D3—64no—
262C2×D3—64no—
263C3×C4—24no—
264C3×C4—24no—
265C3×C4—44no—
266C3×C4—44no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4—84no—
270C3×C4—84no—
271C3×C4—181no—
272C3×C4—361no—
273C3×C4—361no—
274C3×C4C3×#23514yes—
275C3×C4C3×#23914yes—
276C3×C4#382811yescharacteristic
277C3×C4#383211yescharacteristic
278Dic3—64no—
279Dic3—124no—
280Dic3—124no—
281C32—24no—
282C32#486214yescomplemented
283C32C3×#48311yescharacteristic, complemented
284C23—91no—
285Q8—181no—
286Q8—181no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—91no—
290C2×C4—181no—
291C2×C4—181no—
292C2×C4—181no—
293C2×C4—181no—
294C2×C4—362no—
295C2×C4C22×C3×C3⋊D311yescharacteristic
296C8—21no—
297C8—181no—
298C2×C3—24no—
299C2×C3—24no—
300C2×C3—24no—
301C2×C3—182no—
302C2×C3C3×#89414yes—
303C2×C3C3×D3×SD1614yes—
304C2×C3C3×#91514yes—
305C2×C3#6410311yescharacteristic
306C2×C3SD16×C3⋊D311yescharacteristic
307C2×C3#6412411yescharacteristic
308D3—68no—
309C22—92no—
310C22—181no—
311C22—181no—
312C22C3×D4×C3⋊D311yescharacteristic
313C4—41no—
314C4—41no—
315C4—181no—
316C4—361no—
317C4—361no—
318C4C3×#382811yescharacteristic
319C4C3×#383211yescharacteristic
320C3—24no—
321C3C3×#486214yescomplemented
322C3#1059256111yescharacteristic, direct product factor
323C2—182no—
324C2C3×#6410311yescharacteristic
325C2C3×SD16×C3⋊D311yescharacteristic
326C2C3×#6412411yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#106C2
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC3×#106C3⋊D3
hypocenterC32C3×#483
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#106

Complements

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

NH ≅ G/Ntype
3. C33×#106C2semidirect
5. C3×#63643C2semidirect
6. C3×#63637C2semidirect
7. C3×#63635C2semidirect
8. #10592561C3direct (N × H)
26. C32×#106D3semidirect
27. C32×#106C2×C3semidirect
29. #63643C2×C3semidirect
30. #63637C2×C3semidirect
31. #63635C2×C3semidirect
94. C3×#106C3×D3semidirect
95. C3×#106C3⋊D3semidirect
195. #106C3×C3⋊D3semidirect
199. C33#483semidirect
282. C32#4862semidirect
283. C32C3×#483semidirect
321. C3C3×#4862semidirect
322. C3#10592561direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#641062
C3×C4⋊C4×C3⋊D32
C33×#1062
C3×#636472
C3×#636432
C3×#636372
C3×#636352
#105925613
C3×#48623