Finite Group Databasebeta

← C3×#5514 #10595611

Subgroups — C3×#5514 #10595611

Order 576.

522 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4⋊C4C211yesmaximal, characteristic, complemented
2C3×#930C211yesmaximal, characteristic, complemented
3C3×#930C212yesmaximal, complemented
4C3×C4×#67C211yesmaximal, characteristic, complemented
5C3×C4×#67C212yesmaximal, complemented
6C3×#898C212yesmaximal, complemented
7C3×#895C211yesmaximal, characteristic, complemented
8C3×#894C212yesmaximal, complemented
9C3×#893C211yesmaximal, characteristic, complemented
10C3×#892C211yesmaximal, characteristic, complemented
11C3×#890C211yesmaximal, characteristic, complemented
12C3×#553—31nomaximal
13#5514C311yesmaximal, characteristic, direct product factor
14C22×C32×C4C2211yescharacteristic
15C22×C32×C4C2212yes—
16C2×C3×#67C2211yescharacteristic
17C2×C3×#67C2212yes—
18C2×C3×D12C2211yescharacteristic
19C2×C3×D12C2212yes—
20C2×C3×C4×D3C2211yescharacteristic
21C2×C3×C4×D3C2212yes—
22C32×C4⋊C4C2211yescharacteristic
23C32×C4⋊C4C2211yescharacteristic
24C32×C4⋊C4C2212yes—
25C3×#217C2211yescharacteristic
26C3×(C2×D3)⋊C4C2212yes—
27C3×(C2×D3)⋊C4C2212yes—
28C3×(C2×D3)⋊C4C2211yescharacteristic
29C3×(C2×D3)⋊C4C2212yes—
30C3×(C2×D3)⋊C4C2212yes—
31C3×C4⋊Dic3C2211yescharacteristic
32C3×C4⋊Dic3C2212yes—
33C3×Dic3⋊C4C2212yes—
34C3×Dic3⋊C4C2211yescharacteristic
35C3×C4×Dic3C2211yescharacteristic
36C3×C4×Dic3C2212yes—
37C3×#126—32no—
38C3×#125—31no—
39C3×#123—32no—
40C3×#121—32no—
41C3×#121—31no—
42C3×#121—31no—
43C3×C4×D4—32no—
44C3×C4×D4—31no—
45C3×C4×D4—31no—
46C3×#117—31no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D311yescharacteristic, complemented
49C2×C3×C4⋊C4C2×C311yescharacteristic, complemented
50#930C2×C311yescharacteristic, complemented
51#930C2×C312yescomplemented
52C4×#67C2×C311yescharacteristic, complemented
53C4×#67C2×C312yescomplemented
54#898C2×C312yescomplemented
55#895C2×C311yescharacteristic, complemented
56#894C2×C312yescomplemented
57#893C2×C311yescharacteristic, complemented
58#892C2×C311yescharacteristic, complemented
59#890C2×C311yescharacteristic, complemented
60C23×C32C2311yescharacteristic
61C22×C3×D3C2311yescharacteristic
62C22×C3×D3C2312yes—
63C2×C32×C4—22no—
64C2×C32×C4—42no—
65C2×C32×C4C2312yes—
66C2×C32×C4C2312yes—
67C2×C32×C4C2311yescharacteristic
68C2×C32×C4C2311yescharacteristic
69C3×#67—22no—
70C3×#67—24no—
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2312yes—
75C3×D12—22no—
76C3×D12—42no—
77C3×C4×D3—22no—
78C3×C4×D3—42no—
79#553—31noSylow
80C2×C3×D4—32no—
81C2×C3×D4—31no—
82C2×C3×D4—31no—
83C2×C3×D4—32no—
84C22×C3×C4—21no—
85C22×C3×C4—22no—
86C22×C3×C4—31no—
87C22×C3×C4—32no—
88C22×C3×C4C2×D311yescharacteristic
89C22×C3×C4C2×D312yes—
90C22×C3×C4C22×C311yescharacteristic
91C22×C3×C4C22×C312yes—
92C2×#67C22×C311yescharacteristic
93C2×#67C22×C312yes—
94C2×D12C22×C311yescharacteristic
95C2×D12C22×C312yes—
96C2×C4×D3C22×C311yescharacteristic
97C2×C4×D3C22×C312yes—
98C3×C4⋊C4—21no—
99C3×C4⋊C4—21no—
100C3×C4⋊C4—22no—
101C3×C4⋊C4—32no—
102C3×C4⋊C4—31no—
103C3×C4⋊C4—31no—
104C3×C4⋊C4—32no—
105C3×C4⋊C4C2×D311yescharacteristic
106C3×C4⋊C4C2×D311yescharacteristic
107C3×C4⋊C4C2×D312yes—
108C3×C4⋊C4C22×C311yescharacteristic
109C3×C4⋊C4C22×C311yescharacteristic
110C3×C4⋊C4C22×C312yes—
111C3×C22⋊C4—31no—
112C3×C22⋊C4—32no—
113C3×C22⋊C4—32no—
114C3×C22⋊C4—32no—
115C3×C22⋊C4—31no—
116C3×C22⋊C4—32no—
117C3×C42—32no—
118C3×C42—31no—
119#217C22×C311yescharacteristic
120(C2×D3)⋊C4C22×C312yes—
121(C2×D3)⋊C4C22×C312yes—
122(C2×D3)⋊C4C22×C311yescharacteristic
123(C2×D3)⋊C4C22×C312yes—
124(C2×D3)⋊C4C22×C312yes—
125C4⋊Dic3C22×C311yescharacteristic
126C4⋊Dic3C22×C312yes—
127Dic3⋊C4C22×C312yes—
128Dic3⋊C4C22×C311yescharacteristic
129C4×Dic3C22×C311yescharacteristic
130C4×Dic3C22×C312yes—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32#4112yes—
134C22×C32C2411yescharacteristic
135C2×C3×D3—21no—
136C2×C3×D3—22no—
137C2×C3×D3—21no—
138C2×C3×D3—22no—
139C2×C3×D3—22no—
140C2×C3×D3—21no—
141C32×C4—21no—
142C32×C4—22no—
143C32×C4—22no—
144C32×C4#4112yes—
145C3×Dic3—21no—
146C3×Dic3—21no—
147C3×Dic3—22no—
148C3×Dic3—21no—
149#126—32no—
150#125—31no—
151#123—32no—
152#121—32no—
153#121—31no—
154#121—31no—
155C4×D4—32no—
156C4×D4—31no—
157C4×D4—31no—
158#117—31no—
159C2×C4⋊C4C3×D311yescharacteristic, complemented
160C23×C3—21no—
161C23×C3—31no—
162C23×C3—32no—
163C23×C3C22×D311yescharacteristic
164C23×C3C23×C311yescharacteristic
165C22×D3C23×C311yescharacteristic
166C22×D3C23×C312yes—
167C3×D4—64no—
168C3×D4—62no—
169C3×D4—62no—
170C3×D4—122no—
171C2×C3×C4—24no—
172C2×C3×C4—22no—
173C2×C3×C4—22no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—31no—
179C2×C3×C4—32no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—42no—
183C2×C3×C4—42no—
184C2×C3×C4—44no—
185C2×C3×C4—62no—
186C2×C3×C4—122no—
187C2×C3×C4C22×D312yes—
188C2×C3×C4C22×D311yescharacteristic
189C2×C3×C4C22×D312yes—
190C2×C3×C4C22×D311yescharacteristic
191C2×C3×C4C23×C312yes—
192C2×C3×C4C23×C312yes—
193C2×C3×C4C23×C311yescharacteristic
194C2×C3×C4C23×C311yescharacteristic
195#67—22no—
196#67—24no—
197C2×Dic3C23×C311yescharacteristic
198C2×Dic3C23×C311yescharacteristic
199C2×Dic3C23×C311yescharacteristic
200C2×Dic3C23×C312yes—
201D12—22no—
202D12—42no—
203C4×D3—22no—
204C4×D3—42no—
205C2×C32—22no—
206C2×C32C2×#4111yescharacteristic
207C2×C32C2×#4111yescharacteristic
208C2×C32H(5,𝔽2)11yescharacteristic
209C3×D3—41no—
210C3×D3—42no—
211C2×D4—32no—
212C2×D4—31no—
213C2×D4—31no—
214C2×D4—32no—
215C22×C4—31no—
216C22×C4—32no—
217C22×C4C2×C3×D311yescharacteristic
218C22×C4C2×C3×D312yes—
219C4⋊C4—31no—
220C4⋊C4—32no—
221C4⋊C4—31no—
222C4⋊C4—32no—
223C4⋊C4C2×C3×D311yescharacteristic
224C4⋊C4C2×C3×D311yescharacteristic
225C4⋊C4C2×C3×D312yes—
226C22⋊C4—31no—
227C22⋊C4—32no—
228C22⋊C4—32no—
229C22⋊C4—32no—
230C22⋊C4—31no—
231C22⋊C4—32no—
232C42—32no—
233C42—31no—
234C22×C3—22no—
235C22×C3—22no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—22no—
242C22×C3—61no—
243C22×C3—62no—
244C22×C3—62no—
245C22×C3—61no—
246C22×C3—61no—
247C22×C3—62no—
248C22×C3#23912yes—
249C22×C3C3×#4112yes—
250C22×C3C23×D311yescharacteristic
251C22×C3C24×C311yescharacteristic
252C2×D3—21no—
253C2×D3—22no—
254C2×D3—21no—
255C2×D3—22no—
256C2×D3—22no—
257C2×D3—21no—
258C3×C4—22no—
259C3×C4—21no—
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—21no—
263C3×C4—22no—
264C3×C4—22no—
265C3×C4—22no—
266C3×C4—42no—
267C3×C4—42no—
268C3×C4—61no—
269C3×C4—61no—
270C3×C4—62no—
271C3×C4—61no—
272C3×C4#23512yes—
273C3×C4C3×#4112yes—
274Dic3—21no—
275Dic3—21no—
276Dic3—22no—
277Dic3—21no—
278C32#55311yescharacteristic, complemented, Sylow
279C23—31no—
280C23—32no—
281C23C22×C3×D311yescharacteristic
282D4—64no—
283D4—62no—
284D4—62no—
285D4—122no—
286C2×C4—22no—
287C2×C4—31no—
288C2×C4—32no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—42no—
292C2×C4—62no—
293C2×C4—122no—
294C2×C4C22×C3×D312yes—
295C2×C4C22×C3×D311yescharacteristic
296C2×C4C22×C3×D312yes—
297C2×C4C22×C3×D311yescharacteristic
298C2×C3—24no—
299C2×C3—22no—
300C2×C3—22no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—121no—
305C2×C3—122no—
306C2×C3C2×#23511yescharacteristic
307C2×C3#100411yescharacteristic
308C2×C3C2×#23911yescharacteristic
309C2×C3C2×C3×#4111yescharacteristic
310C2×C3C2×C3×#4111yescharacteristic
311C2×C3C3×H(5,𝔽2)11yescharacteristic
312D3—41no—
313D3—42no—
314C22—21no—
315C22—21no—
316C22—61no—
317C22—62no—
318C22—62no—
319C22—61no—
320C22—61no—
321C22—62no—
322C22C3×#23912yes—
323C22C23×C3×D311yescharacteristic
324C4—21no—
325C4—22no—
326C4—22no—
327C4—61no—
328C4—61no—
329C4—62no—
330C4—61no—
331C4C3×#23512yes—
332C3—21no—
333C3#551411yescharacteristic, direct product factor
334C3C3×#55311yescharacteristic, complemented
335C2—22no—
336C2—121no—
337C2—122no—
338C2C2×C3×#23511yescharacteristic
339C2C3×#100411yescharacteristic
340C2C2×C3×#23911yescharacteristic
subgroupG/H
centerC22×C3C23×D3
commutator, cosocleC22×C3C24×C3
FrattiniC22C23×C3×D3
FittingC2×C32×C4⋊C4C2
socleC22×C32C24
hypercenterC2×C3×C4⋊C4D3
hypocenterC3C3×#553
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4⋊C4C2semidirect
2. C3×#930C2semidirect
3. C3×#930C2semidirect
4. C3×C4×#67C2semidirect
5. C3×C4×#67C2semidirect
6. C3×#898C2semidirect
7. C3×#895C2semidirect
8. C3×#894C2semidirect
9. C3×#893C2semidirect
10. C3×#892C2semidirect
11. C3×#890C2semidirect
13. #5514C3direct (N × H)
48. C2×C3×C4⋊C4D3semidirect
49. C2×C3×C4⋊C4C2×C3semidirect
50. #930C2×C3semidirect
51. #930C2×C3semidirect
52. C4×#67C2×C3semidirect
53. C4×#67C2×C3semidirect
54. #898C2×C3semidirect
55. #895C2×C3semidirect
56. #894C2×C3semidirect
57. #893C2×C3semidirect
58. #892C2×C3semidirect
59. #890C2×C3semidirect
159. C2×C4⋊C4C3×D3semidirect
278. C32#553semidirect
333. C3#5514direct (N × H)
334. C3C3×#553semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4⋊C42
C3×#9302
C3×C4×#672
C3×#8982
C3×#8952
C3×#8942
C3×#8932
C3×#8922
C3×#8902
C3×#5533
#55143