Finite Group Databasebeta

← C3×#10592510 #13162846

Subgroups — C3×#10592510 #13162846

Order 1728.

792 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64155C211yesmaximal, characteristic
2C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic
3C33×#103C211yesmaximal, characteristic, complemented
4C3×#63662C211yesmaximal, characteristic, complemented
5C3×#63647C211yesmaximal, characteristic
6C3×#63645C211yesmaximal, characteristic, complemented
7C3×#63634C211yesmaximal, characteristic, complemented
8#10592510C311yesmaximal, characteristic, direct product factor
9C3×#4811—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×C4×C3⋊D3—21no—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×#3750—21no—
19C3×#3750—21no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21#64155C2×C311yescharacteristic
22C4⋊C4×C3⋊D3C2×C311yescharacteristic
23C3×#946—34no—
24C3×D3×C4⋊C4—34no—
25C32×#103—24no—
26C32×#103D314yescomplemented
27C32×#103C2×C311yescharacteristic, complemented
28#63662C2×C311yescharacteristic, complemented
29#63647C2×C311yescharacteristic
30#63645C2×C311yescharacteristic, complemented
31#63634C2×C311yescharacteristic, complemented
32C3×#835—34no—
33C3×(C2×D3)⋊C8—34no—
34C3×#818—34no—
35C3×#807—34no—
36C22×C3×C3⋊D3D411yescharacteristic
37C33×Q8—21no—
38C33×Q8—21no—
39C2×C33×C4—21no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C3×C4×C3⋊D3—21no—
46C3×C4×C3⋊D3—21no—
47C3×C4×C3⋊D3—42no—
48C33×C8—21no—
49C3×C3⋊(C3⋊C8)—21no—
50C3×#484—91no—
51#4811—34no—
52C2×C32×Q8—24no—
53C2×C32×Q8C2×D314yes—
54C2×C32×Q8C22×C311yescharacteristic
55C2×C4×C3⋊D3—21no—
56C2×C4×C3⋊D3C22×C311yescharacteristic
57C2×C3×C4×D3—34no—
58C2×C3×C4×D3—64no—
59C2×C32×C8—24no—
60C2×C32×C8C2×D314yes—
61C2×C32×C8C22×C311yescharacteristic
62C32×C4⋊C4—24no—
63C32×C4⋊C4C2×D314yes—
64C32×C4⋊C4C22×C311yescharacteristic
65#3752—21no—
66C4⋊(C3⋊Dic3)C22×C311yescharacteristic
67C4⋊(C3⋊Dic3)C22×C311yescharacteristic
68#3750—21no—
69#3750—21no—
70C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
71C3×(C2×D3)⋊C4—64no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3—34no—
74C3×Dic3⋊C4—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⋊D3Q1612yes—
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#946—34no—
91D3×C4⋊C4—34no—
92C3×#107—91no—
93C3×#107—91no—
94C3×#103—24no—
95C3×#103—91no—
96C3×#103C3×D314yescomplemented
97C3×#103C3⋊D311yescharacteristic, complemented
98C3×C22⋊C8—91no—
99#835—34no—
100(C2×D3)⋊C8—34no—
101#818—34no—
102#807—34no—
103C22×C3⋊D3C3×D411yescharacteristic
104C22×C3×D3—34no—
105C32×Q8—28no—
106C32×Q8—24no—
107C32×Q8—21no—
108C32×Q8—21no—
109C32×Q8—24no—
110C32×Q8—44no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—24no—
114C2×C32×C4—24no—
115C2×C32×C4—24no—
116C2×C32×C4—44no—
117C2×C32×C4—44no—
118C2×C32×C4C22×D314yes—
119C2×C32×C4C23×C311yescharacteristic
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3C3×D411yescharacteristic
123C4×C3⋊D3—21no—
124C4×C3⋊D3—21no—
125C4×C3⋊D3—42no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—64no—
128C2×C3×Dic3—64no—
129C3×C4×D3—64no—
130C3×C4×D3—64no—
131C3×C4×D3—128no—
132C32×C8—21no—
133C32×C8—24no—
134C32×C8—44no—
135C3⋊(C3⋊C8)—21no—
136C3×C3⋊C8—64no—
137#484—91noSylow
138C2×C33#12311yescharacteristic
139C2×C33C2×Q1611yescharacteristic
140C2×C33AGL(1,ℤ/8)11yescharacteristic
141C3×C3⋊D3—22no—
142C2×C3×Q8—24no—
143C2×C3×Q8C2×C3×D314yes—
144C2×C3×Q8C2×C3⋊D311yescharacteristic
145C22×C3×C4—91no—
146C22×C3×C4—181no—
147C2×C4×D3—34no—
148C2×C4×D3—64no—
149C2×C3×C8—24no—
150C2×C3×C8—91no—
151C2×C3×C8C2×C3×D314yes—
152C2×C3×C8C2×C3⋊D311yescharacteristic
153C3×C4⋊C4—24no—
154C3×C4⋊C4—91no—
155C3×C4⋊C4—91no—
156C3×C4⋊C4—181no—
157C3×C4⋊C4—181no—
158C3×C4⋊C4C2×C3×D314yes—
159C3×C4⋊C4C2×C3⋊D311yescharacteristic
160C3×C22⋊C4—181no—
161(C2×D3)⋊C4—64no—
162C4⋊Dic3—34no—
163C4⋊Dic3—34no—
164Dic3⋊C4—64no—
165Dic3⋊C4—64no—
166C2×C3⋊C8—34no—
167C22×C32—24no—
168C22×C32D3×D414yes—
169C22×C32C2×C3×D411yescharacteristic
170C2×C3⋊D3—21no—
171C2×C3⋊D3—21no—
172C2×C3⋊D3C3×Q1612yes—
173C2×C3×D3—38no—
174C2×C3×D3—64no—
175C2×C3×D3—64no—
176C32×C4—24no—
177C32×C4—24no—
178C32×C4—48no—
179C32×C4—44no—
180C32×C4—41no—
181C32×C4—41no—
182C32×C4—44no—
183C32×C4—84no—
184C32×C4#23514yes—
185C32×C4#23714yes—
186C32×C4C3×#4111yescharacteristic
187C32×C4C3×#4111yescharacteristic
188C3⋊Dic3—21no—
189C3⋊Dic3—41no—
190C3⋊Dic3—41no—
191C3×Dic3—64no—
192C3×Dic3—124no—
193C3×Dic3—124no—
194C22⋊Q8—91no—
195C2×C4⋊C4—91no—
196#107—91no—
197#107—91no—
198#103—91no—
199#103C3×C3⋊D311yescharacteristic, complemented
200C22⋊C8—91no—
201C33#48411yescharacteristic, complemented, Sylow
202C23×C3—91no—
203C22×D3—34no—
204C3×Q8—28no—
205C3×Q8—24no—
206C3×Q8—24no—
207C3×Q8—21no—
208C3×Q8—21no—
209C3×Q8—44no—
210C2×C3×C4—24no—
211C2×C3×C4—24no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—24no—
215C2×C3×C4—44no—
216C2×C3×C4—44no—
217C2×C3×C4—91no—
218C2×C3×C4—181no—
219C2×C3×C4—181no—
220C2×C3×C4—181no—
221C2×C3×C4—181no—
222C2×C3×C4—362no—
223C2×C3×C4C22×C3×D314yes—
224C2×C3×C4C22×C3⋊D311yescharacteristic
225C2×Dic3—34no—
226C2×Dic3—64no—
227C2×Dic3—64no—
228C4×D3—64no—
229C4×D3—64no—
230C4×D3—128no—
231C3×C8—24no—
232C3×C8—21no—
233C3×C8—44no—
234C3×C8—181no—
235C3⋊C8—64no—
236C2×C32—24no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32#88314yes—
240C2×C32#91414yes—
241C2×C32D3×Q1614yes—
242C2×C32C3×#12311yescharacteristic
243C2×C32C2×C3×Q1611yescharacteristic
244C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
245C3⋊D3—22no—
246C3×D3—68no—
247C2×Q8C2×C3×C3⋊D311yescharacteristic
248C22×C4—91no—
249C22×C4—181no—
250C2×C8—91no—
251C2×C8C2×C3×C3⋊D311yescharacteristic
252C4⋊C4—91no—
253C4⋊C4—91no—
254C4⋊C4—181no—
255C4⋊C4—181no—
256C4⋊C4C2×C3×C3⋊D311yescharacteristic
257C22⋊C4—181no—
258C22×C3—24no—
259C22×C3—92no—
260C22×C3—181no—
261C22×C3—181no—
262C22×C3C3×D3×D414yes—
263C22×C3D4×C3⋊D311yescharacteristic
264C2×D3—38no—
265C2×D3—64no—
266C2×D3—64no—
267C3×C4—24no—
268C3×C4—24no—
269C3×C4—48no—
270C3×C4—44no—
271C3×C4—44no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—84no—
275C3×C4—181no—
276C3×C4—361no—
277C3×C4—361no—
278C3×C4C3×#23514yes—
279C3×C4C3×#23714yes—
280C3×C4#382811yescharacteristic
281C3×C4#383011yescharacteristic
282Dic3—64no—
283Dic3—124no—
284Dic3—124no—
285C32—24no—
286C32#481114yescomplemented
287C32C3×#48411yescharacteristic, complemented
288C23—91no—
289Q8—21no—
290Q8—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—91no—
294C2×C4—181no—
295C2×C4—181no—
296C2×C4—181no—
297C2×C4—181no—
298C2×C4—362no—
299C2×C4C22×C3×C3⋊D311yescharacteristic
300C8—21no—
301C8—181no—
302C2×C3—24no—
303C2×C3—24no—
304C2×C3—24no—
305C2×C3—182no—
306C2×C3C3×#88314yes—
307C2×C3C3×#91414yes—
308C2×C3C3×D3×Q1614yes—
309C2×C3#6409211yescharacteristic
310C2×C3#6412311yescharacteristic
311C2×C3Q16×C3⋊D311yescharacteristic
312D3—68no—
313C22—92no—
314C22—181no—
315C22—181no—
316C22C3×D4×C3⋊D311yescharacteristic
317C4—41no—
318C4—41no—
319C4—181no—
320C4—361no—
321C4—361no—
322C4C3×#382811yescharacteristic
323C4C3×#383011yescharacteristic
324C3—24no—
325C3C3×#481114yescomplemented
326C3#1059251011yescharacteristic, direct product factor
327C2—182no—
328C2C3×#6409211yescharacteristic
329C2C3×#6412311yescharacteristic
330C2C3×Q16×C3⋊D311yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#103C2
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC3×#103C3⋊D3
hypocenterC32C3×#484
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
3. C33×#103C2semidirect
4. C3×#63662C2semidirect
6. C3×#63645C2semidirect
7. C3×#63634C2semidirect
8. #10592510C3direct (N × H)
26. C32×#103D3semidirect
27. C32×#103C2×C3semidirect
28. #63662C2×C3semidirect
30. #63645C2×C3semidirect
31. #63634C2×C3semidirect
96. C3×#103C3×D3semidirect
97. C3×#103C3⋊D3semidirect
199. #103C3×C3⋊D3semidirect
201. C33#484semidirect
286. C32#4811semidirect
287. C32C3×#484semidirect
325. C3C3×#4811semidirect
326. C3#10592510direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#641552
C3×C4⋊C4×C3⋊D32
C33×#1032
C3×#636622
C3×#636472
C3×#636452
C3×#636342
#105925103
C3×#48113