Finite Group Databasebeta

← C3×#10592457 #13162825

Subgroups — C3×#10592457 #13162825

Order 1728.

932 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3×#3754C211yesmaximal, characteristic
3C3×#64096C211yesmaximal, characteristic, complemented
4C33×#102C211yesmaximal, characteristic
5C3×#63643C211yesmaximal, characteristic, complemented
6C3×C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
7C3×#63635C211yesmaximal, characteristic, complemented
8#10592457C311yesmaximal, characteristic, direct product factor
9C3×#4758—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C22×C3×C3⋊Dic3—21no—
12C2×C3×#634C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C3×#3757—21no—
16C3×#3754C414yescomplemented
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×#3750—21no—
19C3×C4×C3⋊Dic3—21no—
20C3×C4×C3⋊Dic3C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22D4×C3⋊Dic3C2×C311yescharacteristic
23C2×#3754C2×C311yescharacteristic
24#64096C2×C311yescharacteristic, complemented
25C3×D4×Dic3—34no—
26C2×C3×#214—34no—
27C3×#887—34no—
28C32×#102—24no—
29C32×#102D314yes—
30C32×#102C2×C311yescharacteristic
31#63643C2×C311yescharacteristic, complemented
32C8×C3⋊Dic3C2×C311yescharacteristic, complemented
33#63635C2×C311yescharacteristic, complemented
34C3×#816—34no—
35C3×C8×Dic3—34no—
36C3×#808—34no—
37C23×C33—21no—
38C33×D4—21no—
39C33×D4C2×C412yes—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3—42no—
45C2×C3×C3⋊Dic3D411yescharacteristic
46C2×C3×C3⋊Dic3D411yescharacteristic
47C3×#634—21no—
48C3×#634C2×C412yescomplemented
49C33×C8—21no—
50C3×C3⋊(C3⋊C8)C2×C412yescomplemented
51C3×C4×SD16—91no—
52#4758—34no—
53C2×C32×D4—24no—
54C2×C32×D4C2×D314yes—
55C2×C32×D4C22×C311yescharacteristic
56C22×C3⋊Dic3—21no—
57C2×#634C22×C311yescharacteristic
58C22×C3×Dic3—64no—
59C2×C3×Dic6—34no—
60C2×C32×C8—24no—
61C2×C32×C8C2×D314yes—
62C2×C32×C8C22×C311yescharacteristic
63C32×C4⋊C4—24no—
64C32×C4⋊C4C2×D314yes—
65C32×C4⋊C4C22×C311yescharacteristic
66#3757—21no—
67#3754C3×C414yescomplemented
68C4⋊(C3⋊Dic3)C22×C311yescharacteristic
69#3750—21no—
70C4×C3⋊Dic3—21no—
71C4×C3⋊Dic3C22×C311yescharacteristic
72C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
73C3×#217—64no—
74C3×#214—316no—
75C3×C4⋊Dic3—34no—
76C3×Dic3⋊C4—64no—
77C3×C4×Dic3—34no—
78C3×C4×Dic3—64no—
79C2×C3×C3⋊C8—34no—
80C22×C33—22no—
81C22×C33—41no—
82C22×C33—41no—
83C22×C33C2×D411yescharacteristic
84C33×C4—41no—
85C33×C4C22×C411yescharacteristic
86C33×C4#4111yescharacteristic
87C3×C3⋊Dic3—22no—
88C3×C3⋊Dic3—21no—
89C3×C3⋊Dic3—41no—
90C3×C3⋊Dic3SD1612yescomplemented
91C2×C3×SD16—91no—
92C3×C4×Q8—91no—
93C3×C4×D4—91no—
94D4×Dic3—34no—
95C2×#214—34no—
96#887—34no—
97C3×#106—91no—
98C3×#103—91no—
99C3×#102—24no—
100C3×#102C3×D314yes—
101C3×#102C3⋊D311yescharacteristic
102C3×C4×C8—91no—
103#816—34no—
104C8×Dic3—34no—
105#808—34no—
106C23×C32—21no—
107C23×C32—24no—
108C23×C32—44no—
109C32×D4—21no—
110C32×D4—28no—
111C32×D4—24no—
112C32×D4—44no—
113C32×D4C4×D318yes—
114C32×D4C2×C3×C412yes—
115C2×C32×C4—21no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—44no—
119C2×C32×C4C22×D314yes—
120C2×C32×C4C23×C311yescharacteristic
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—42no—
124C2×C3⋊Dic3C3×D411yescharacteristic
125C2×C3⋊Dic3C3×D411yescharacteristic
126#634—21no—
127#634C2×C3×C412yescomplemented
128C2×C3×Dic3—34no—
129C2×C3×Dic3—34no—
130C2×C3×Dic3—64no—
131C2×C3×Dic3—64no—
132C2×C3×Dic3—128no—
133C3×Dic6—38no—
134C3×Dic6—64no—
135C32×C8—21no—
136C32×C8—24no—
137C32×C8—44no—
138C3⋊(C3⋊C8)C2×C3×C412yescomplemented
139C3×C3⋊C8—38no—
140C4×SD16—91noSylow
141C2×C33—42no—
142C2×C33C4×D411yescharacteristic
143C2×C33C2×SD1611yescharacteristic
144C2×C33#13511yescharacteristic
145C2×C3×Q8—91no—
146C2×C3×D4—24no—
147C2×C3×D4C2×C3×D314yes—
148C2×C3×D4C2×C3⋊D311yescharacteristic
149C22×C3×C4—181no—
150C22×Dic3—64no—
151C2×Dic6—34no—
152C3×SD16—94no—
153C2×C3×C8—24no—
154C2×C3×C8—91no—
155C2×C3×C8C2×C3×D314yes—
156C2×C3×C8C2×C3⋊D311yescharacteristic
157C3×C4⋊C4—24no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4—181no—
160C3×C4⋊C4C2×C3×D314yes—
161C3×C4⋊C4C2×C3⋊D311yescharacteristic
162C3×C22⋊C4—181no—
163C3×C42—91no—
164C3×C42—181no—
165#217—64no—
166#214—316no—
167C4⋊Dic3—34no—
168Dic3⋊C4—64no—
169C4×Dic3—34no—
170C4×Dic3—64no—
171C2×C3⋊C8—34no—
172C22×C32—22no—
173C22×C32—24no—
174C22×C32—28no—
175C22×C32—48no—
176C22×C32—41no—
177C22×C32—41no—
178C22×C32—44no—
179C22×C32—44no—
180C22×C32—84no—
181C22×C32—84no—
182C22×C32D3×D414yes—
183C22×C32C2×C3×D411yescharacteristic
184C32×C4—24no—
185C32×C4—24no—
186C32×C4—41no—
187C32×C4—44no—
188C32×C4—84no—
189C32×C4C2×C4×D314yes—
190C32×C4#23714yes—
191C32×C4C22×C3×C411yescharacteristic
192C32×C4C3×#4111yescharacteristic
193C3⋊Dic3—22no—
194C3⋊Dic3—21no—
195C3⋊Dic3—41no—
196C3⋊Dic3C3×SD1612yescomplemented
197C3×Dic3—38no—
198C3×Dic3—68no—
199C3×Dic3—64no—
200C3×Dic3—124no—
201C2×SD16—91no—
202C4×Q8—91no—
203C4×D4—91no—
204#106—91no—
205#103—91no—
206#102C3×C3⋊D311yescharacteristic
207C4×C8—91no—
208C33C4×SD1611yescharacteristic, complemented, Sylow
209C23×C3—24no—
210C23×C3—21no—
211C23×C3—44no—
212C3×Q8—92no—
213C3×Q8—181no—
214C3×D4—24no—
215C3×D4—21no—
216C3×D4—28no—
217C3×D4—44no—
218C3×D4C3×C4×D318yes—
219C3×D4C4×C3⋊D312yes—
220C2×C3×C4—24no—
221C2×C3×C4—21no—
222C2×C3×C4—24no—
223C2×C3×C4—44no—
224C2×C3×C4—91no—
225C2×C3×C4—91no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4—362no—
229C2×C3×C4C22×C3×D314yes—
230C2×C3×C4C22×C3⋊D311yescharacteristic
231C2×Dic3—34no—
232C2×Dic3—34no—
233C2×Dic3—64no—
234C2×Dic3—64no—
235C2×Dic3—128no—
236Dic6—38no—
237Dic6—64no—
238C3×C8—24no—
239C3×C8—21no—
240C3×C8—44no—
241C3×C8—92no—
242C3⋊C8—38no—
243C2×C32—24no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32—42no—
247C2×C32—48no—
248C2×C32—88no—
249C2×C32#67⋊C414yes—
250C2×C32#91214yes—
251C2×C32D3×SD1614yes—
252C2×C32C3×C4×D411yescharacteristic
253C2×C32C2×C3×SD1611yescharacteristic
254C2×C32C3×#13511yescharacteristic
255C2×Q8—91no—
256C2×D4C2×C3×C3⋊D311yescharacteristic
257C22×C4—181no—
258SD16—94no—
259C2×C8—91no—
260C2×C8C2×C3×C3⋊D311yescharacteristic
261C4⋊C4—91no—
262C4⋊C4—181no—
263C4⋊C4C2×C3×C3⋊D311yescharacteristic
264C22⋊C4—181no—
265C42—91no—
266C42—181no—
267C22×C3—28no—
268C22×C3—22no—
269C22×C3—24no—
270C22×C3—48no—
271C22×C3—44no—
272C22×C3—44no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—84no—
276C22×C3—84no—
277C22×C3C3×D3×D414yes—
278C22×C3D4×C3⋊D311yescharacteristic
279C3×C4—24no—
280C3×C4—24no—
281C3×C4—44no—
282C3×C4—41no—
283C3×C4—84no—
284C3×C4—92no—
285C3×C4—182no—
286C3×C4—181no—
287C3×C4—361no—
288C3×C4C2×C3×C4×D314yes—
289C3×C4C3×#23714yes—
290C3×C4C2×C4×C3⋊D311yescharacteristic
291C3×C4#383011yescharacteristic
292Dic3—38no—
293Dic3—68no—
294Dic3—64no—
295Dic3—124no—
296C32—24no—
297C32#475814yescomplemented
298C32C3×C4×SD1611yescharacteristic, complemented
299C23—21no—
300Q8—92no—
301Q8—181no—
302D4—21no—
303D4C3×C4×C3⋊D312yes—
304C2×C4—21no—
305C2×C4—91no—
306C2×C4—91no—
307C2×C4—181no—
308C2×C4—181no—
309C2×C4—362no—
310C2×C4C22×C3×C3⋊D311yescharacteristic
311C8—21no—
312C8—92no—
313C2×C3—24no—
314C2×C3—24no—
315C2×C3—24no—
316C2×C3—48no—
317C2×C3—42no—
318C2×C3—88no—
319C2×C3C3×#67⋊C414yes—
320C2×C3C3×#91214yes—
321C2×C3C3×D3×SD1614yes—
322C2×C3#6409011yescharacteristic
323C2×C3#6412111yescharacteristic
324C2×C3SD16×C3⋊D311yescharacteristic
325C22—22no—
326C22—41no—
327C22—41no—
328C22C3×D4×C3⋊D311yescharacteristic
329C4—41no—
330C4—92no—
331C4—182no—
332C4—181no—
333C4—361no—
334C4C2×C3×C4×C3⋊D311yescharacteristic
335C4C3×#383011yescharacteristic
336C3—24no—
337C3C3×#475814yescomplemented
338C3#1059245711yescharacteristic, direct product factor
339C2—42no—
340C2C3×#6409011yescharacteristic
341C2C3×#6412111yescharacteristic
342C2C3×SD16×C3⋊D311yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#102C2
socleC22×C33C2×D4
prime radicalC2×C33×D4C22
cosocleC2×C32×C4C23×C3
hypercenterC3×#102C3⋊D3
hypocenterC32C3×C4×SD16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
3. C3×#64096C2semidirect
5. C3×#63643C2semidirect
6. C3×C8×C3⋊Dic3C2semidirect
7. C3×#63635C2semidirect
8. #10592457C3direct (N × H)
16. C3×#3754C4semidirect
24. #64096C2×C3semidirect
31. #63643C2×C3semidirect
32. C8×C3⋊Dic3C2×C3semidirect
33. #63635C2×C3semidirect
48. C3×#634C2×C4semidirect
50. C3×C3⋊(C3⋊C8)C2×C4semidirect
67. #3754C3×C4semidirect
90. C3×C3⋊Dic3SD16semidirect
127. #634C2×C3×C4semidirect
138. C3⋊(C3⋊C8)C2×C3×C4semidirect
196. C3⋊Dic3C3×SD16semidirect
208. C33C4×SD16semidirect
297. C32#4758semidirect
298. C32C3×C4×SD16semidirect
337. C3C3×#4758semidirect
338. C3#10592457direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D4×C3⋊Dic32
C2×C3×#37542
C3×#640962
C33×#1022
C3×#636432
C3×C8×C3⋊Dic32
C3×#636352
#105924573
C3×#47583