Finite Group Databasebeta

← C3×#10592560 #13162899

Subgroups — C3×#10592560 #13162899

Order 1728.

852 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3743C211yesmaximal, characteristic
2C3×#64101C211yesmaximal, characteristic
3C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic
4C33×#106C211yesmaximal, characteristic, complemented
5C3×#63645C211yesmaximal, characteristic, complemented
6C3×#63635C211yesmaximal, characteristic, complemented
7C3×#63634C211yesmaximal, characteristic, complemented
8#10592560C311yesmaximal, characteristic, direct product factor
9C3×#4861—34nomaximal
10C2×C3×C4×C3⋊D3—21no—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×#3750—21no—
19C3×C4×C3⋊Dic3—21no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C3×#3743C414yescomplemented
22C2×#3743C2×C311yescharacteristic
23#64101C2×C311yescharacteristic
24C4⋊C4×C3⋊D3C2×C311yescharacteristic
25C2×C3×ΓL(1,𝔽25)—34no—
26C3×#892—34no—
27C3×D3×C4⋊C4—34no—
28C32×#106—24no—
29C32×#106D314yescomplemented
30C32×#106C2×C311yescharacteristic, complemented
31#63645C2×C311yescharacteristic, complemented
32#63635C2×C311yescharacteristic, complemented
33#63634C2×C311yescharacteristic, complemented
34C3×#818—34no—
35C3×#808—34no—
36C3×#807—34no—
37C22×C3×C3⋊D3D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×C4×C3⋊D3—42no—
45C3×C4×C3⋊D3C2×C412yes—
46C3×C4×C3⋊D3Q812yes—
47C33×C8C2×C412yescomplemented
48C3×C3⋊(C3⋊C8)C2×C412yescomplemented
49C3×#428—91no—
50#4861—34no—
51C2×C4×C3⋊D3—21no—
52C2×C4×C3⋊D3C22×C311yescharacteristic
53C2×C3×C4×D3—34no—
54C2×C3×C4×D3—64no—
55C2×C32×C8—24no—
56C2×C32×C8C2×D314yes—
57C2×C32×C8C22×C311yescharacteristic
58C32×C4⋊C4—24no—
59C32×C4⋊C4—24no—
60C32×C4⋊C4C2×D314yes—
61C32×C4⋊C4C2×D314yes—
62C32×C4⋊C4C22×C311yescharacteristic
63C32×C4⋊C4C22×C311yescharacteristic
64#3752—21no—
65C4⋊(C3⋊Dic3)C22×C311yescharacteristic
66C4⋊(C3⋊Dic3)C22×C311yescharacteristic
67#3750—21no—
68C4×C3⋊Dic3—21no—
69C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
70#3743C3×C414yescomplemented
71C3×(C2×D3)⋊C4—64no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3—34no—
74C3×Dic3⋊C4—64no—
75C3×C4×Dic3—64no—
76C2×C3×C3⋊C8—34no—
77C3×ΓL(1,𝔽25)—316no—
78C22×C33C2×D411yescharacteristic
79C2×C3×C3⋊D3—21no—
80C2×C3×C3⋊D3—21no—
81C2×C3×C3⋊D3C4⋊C412yes—
82C33×C4—41no—
83C33×C4—41no—
84C33×C4C22×C411yescharacteristic
85C33×C4C2×Q811yescharacteristic
86C3×C3⋊Dic3—41no—
87C3×C3⋊Dic3—41no—
88C3×C3⋊Dic3C4⋊C412yes—
89C2×C3×M16—91no—
90C3×#117—91no—
91C2×C3×C4⋊C4—91no—
92C2×ΓL(1,𝔽25)—34no—
93#892—34no—
94D3×C4⋊C4—34no—
95C3×#107—91no—
96C3×#107—91no—
97C3×#106—24no—
98C3×#106—91no—
99C3×#106C3×D314yescomplemented
100C3×#106C3⋊D311yescharacteristic, complemented
101#818—34no—
102#808—34no—
103#807—34no—
104C22×C3⋊D3C3×D411yescharacteristic
105C22×C3×D3—34no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—24no—
109C2×C32×C4—24no—
110C2×C32×C4—24no—
111C2×C32×C4—44no—
112C2×C32×C4—44no—
113C2×C32×C4C22×D314yes—
114C2×C32×C4C23×C311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3C3×D411yescharacteristic
118C4×C3⋊D3—42no—
119C4×C3⋊D3C2×C3×C412yes—
120C4×C3⋊D3C3×Q812yes—
121C2×C3×Dic3—34no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3—64no—
124C3×C4×D3—38no—
125C3×C4×D3—38no—
126C3×C4×D3—128no—
127C32×C8—28no—
128C32×C8C4×D318yescomplemented
129C32×C8C2×C3×C412yescomplemented
130C3⋊(C3⋊C8)C2×C3×C412yescomplemented
131C3×C3⋊C8—38no—
132#428—91noSylow
133C2×C33C2×C4⋊C411yescharacteristic
134C2×C33AGL(1,ℤ/8)11yescharacteristic
135C2×C33#13711yescharacteristic
136C3×C3⋊D3—22no—
137C22×C3×C4—91no—
138C22×C3×C4—181no—
139C2×C4×D3—34no—
140C2×C4×D3—64no—
141C3×M16—94no—
142C2×C3×C8—24no—
143C2×C3×C8—91no—
144C2×C3×C8C2×C3×D314yes—
145C2×C3×C8C2×C3⋊D311yescharacteristic
146C3×C4⋊C4—24no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4—91no—
149C3×C4⋊C4—91no—
150C3×C4⋊C4—181no—
151C3×C4⋊C4C2×C3×D314yes—
152C3×C4⋊C4C2×C3×D314yes—
153C3×C4⋊C4C2×C3⋊D311yescharacteristic
154C3×C4⋊C4C2×C3⋊D311yescharacteristic
155C3×C22⋊C4—181no—
156C3×C42—181no—
157(C2×D3)⋊C4—64no—
158C4⋊Dic3—34no—
159C4⋊Dic3—34no—
160Dic3⋊C4—64no—
161C4×Dic3—64no—
162C2×C3⋊C8—34no—
163ΓL(1,𝔽25)—316no—
164C22×C32—24no—
165C22×C32D3×D414yes—
166C22×C32C2×C3×D411yescharacteristic
167C2×C3⋊D3—21no—
168C2×C3⋊D3—21no—
169C2×C3⋊D3C3×C4⋊C412yes—
170C2×C3×D3—38no—
171C2×C3×D3—64no—
172C2×C3×D3—64no—
173C32×C4—24no—
174C32×C4—24no—
175C32×C4—41no—
176C32×C4—41no—
177C32×C4—44no—
178C32×C4—44no—
179C32×C4—84no—
180C32×C4—84no—
181C32×C4C2×C4×D314yes—
182C32×C4D3×Q814yes—
183C32×C4C22×C3×C411yescharacteristic
184C32×C4C2×C3×Q811yescharacteristic
185C3⋊Dic3—41no—
186C3⋊Dic3—41no—
187C3⋊Dic3C3×C4⋊C412yes—
188C3×Dic3—38no—
189C3×Dic3—124no—
190C3×Dic3—124no—
191C2×M16—91no—
192#117—91no—
193C2×C4⋊C4—91no—
194#107—91no—
195#107—91no—
196#106—91no—
197#106C3×C3⋊D311yescharacteristic, complemented
198C33#42811yescharacteristic, complemented, Sylow
199C23×C3—91no—
200C22×D3—34no—
201C2×C3×C4—24no—
202C2×C3×C4—24no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—24no—
206C2×C3×C4—44no—
207C2×C3×C4—44no—
208C2×C3×C4—91no—
209C2×C3×C4—92no—
210C2×C3×C4—92no—
211C2×C3×C4—181no—
212C2×C3×C4—181no—
213C2×C3×C4—362no—
214C2×C3×C4C22×C3×D314yes—
215C2×C3×C4C22×C3⋊D311yescharacteristic
216C2×Dic3—34no—
217C2×Dic3—64no—
218C2×Dic3—64no—
219C4×D3—38no—
220C4×D3—38no—
221C4×D3—128no—
222C3×C8—28no—
223C3×C8—92no—
224C3×C8C3×C4×D318yescomplemented
225C3×C8C4×C3⋊D312yescomplemented
226C3⋊C8—38no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32D3×C4⋊C414yes—
231C2×C32#91414yes—
232C2×C32#91514yes—
233C2×C32C2×C3×C4⋊C411yescharacteristic
234C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
235C2×C32C3×#13711yescharacteristic
236C3⋊D3—22no—
237C3×D3—68no—
238C22×C4—91no—
239C22×C4—181no—
240M16—94no—
241C2×C8—91no—
242C2×C8C2×C3×C3⋊D311yescharacteristic
243C4⋊C4—91no—
244C4⋊C4—91no—
245C4⋊C4—181no—
246C4⋊C4C2×C3×C3⋊D311yescharacteristic
247C4⋊C4C2×C3×C3⋊D311yescharacteristic
248C22⋊C4—181no—
249C42—181no—
250C22×C3—24no—
251C22×C3—92no—
252C22×C3—181no—
253C22×C3—181no—
254C22×C3C3×D3×D414yes—
255C22×C3D4×C3⋊D311yescharacteristic
256C2×D3—38no—
257C2×D3—64no—
258C2×D3—64no—
259C3×C4—24no—
260C3×C4—24no—
261C3×C4—44no—
262C3×C4—44no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—84no—
266C3×C4—84no—
267C3×C4—92no—
268C3×C4—361no—
269C3×C4—361no—
270C3×C4C2×C3×C4×D314yes—
271C3×C4C3×D3×Q814yes—
272C3×C4C2×C4×C3⋊D311yescharacteristic
273C3×C4Q8×C3⋊D311yescharacteristic
274Dic3—38no—
275Dic3—124no—
276Dic3—124no—
277C32—24no—
278C32#486114yescomplemented
279C32C3×#42811yescharacteristic, complemented
280C23—91no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—91no—
284C2×C4—92no—
285C2×C4—92no—
286C2×C4—181no—
287C2×C4—181no—
288C2×C4—362no—
289C2×C4C22×C3×C3⋊D311yescharacteristic
290C8—92no—
291C8C3×C4×C3⋊D312yescomplemented
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—24no—
295C2×C3—182no—
296C2×C3C3×D3×C4⋊C414yes—
297C2×C3C3×#91414yes—
298C2×C3C3×#91514yes—
299C2×C3C4⋊C4×C3⋊D311yescharacteristic
300C2×C3#6412311yescharacteristic
301C2×C3#6412411yescharacteristic
302D3—68no—
303C22—92no—
304C22—181no—
305C22—181no—
306C22C3×D4×C3⋊D311yescharacteristic
307C4—41no—
308C4—41no—
309C4—92no—
310C4—361no—
311C4—361no—
312C4C2×C3×C4×C3⋊D311yescharacteristic
313C4C3×Q8×C3⋊D311yescharacteristic
314C3—24no—
315C3C3×#486114yescomplemented
316C3#1059256011yescharacteristic, direct product factor
317C2—182no—
318C2C3×C4⋊C4×C3⋊D311yescharacteristic
319C2C3×#6412311yescharacteristic
320C2C3×#6412411yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#106C2
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C32×C4C23×C3
hypercenterC3×#106C3⋊D3
hypocenterC32C3×#428
LCSG ▷ 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
4. C33×#106C2semidirect
5. C3×#63645C2semidirect
6. C3×#63635C2semidirect
7. C3×#63634C2semidirect
8. #10592560C3direct (N × H)
21. C3×#3743C4semidirect
29. C32×#106D3semidirect
30. C32×#106C2×C3semidirect
31. #63645C2×C3semidirect
32. #63635C2×C3semidirect
33. #63634C2×C3semidirect
47. C33×C8C2×C4semidirect
48. C3×C3⋊(C3⋊C8)C2×C4semidirect
70. #3743C3×C4semidirect
99. C3×#106C3×D3semidirect
100. C3×#106C3⋊D3semidirect
128. C32×C8C4×D3semidirect
129. C32×C8C2×C3×C4semidirect
130. C3⋊(C3⋊C8)C2×C3×C4semidirect
197. #106C3×C3⋊D3semidirect
198. C33#428semidirect
224. C3×C8C3×C4×D3semidirect
225. C3×C8C4×C3⋊D3semidirect
278. C32#4861semidirect
279. C32C3×#428semidirect
291. C8C3×C4×C3⋊D3semidirect
315. C3C3×#4861semidirect
316. C3#10592560direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37432
C3×#641012
C3×C4⋊C4×C3⋊D32
C33×#1062
C3×#636452
C3×#636352
C3×#636342
#105925603
C3×#48613