Finite Group Databasebeta

← C3×#5515 #10595612

Subgroups — C3×#5515 #10595612

Order 576.

482 conjugacy classes of subgroups.

The subgroup lattice has 295 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
3C3×#929C212yesmaximal, complemented
4C3×C4×#67C212yesmaximal, complemented
5C3×#924C211yesmaximal, characteristic, complemented
6C3×#898C212yesmaximal, complemented
7C3×#896C212yesmaximal, complemented
8C3×#894C212yesmaximal, complemented
9C3×#889C212yesmaximal, complemented
10C3×#539—31nomaximal
11#5515C311yesmaximal, characteristic, direct product factor
12C22×C32×C4C2211yescharacteristic
13C22×C32×C4C2211yescharacteristic
14C22×C32×C4C2211yescharacteristic
15C2×C3×#67C2212yes—
16C2×C3×D12C2211yescharacteristic
17C2×C3×C4×D3C2212yes—
18C2×C3×Dic6C2211yescharacteristic
19C32×C4⋊C4C2212yes—
20C32×C4⋊C4C2212yes—
21C3×#217C2212yes—
22C3×(C2×D3)⋊C4C2212yes—
23C3×(C2×D3)⋊C4C2212yes—
24C3×(C2×D3)⋊C4C2212yes—
25C3×(C2×D3)⋊C4C2212yes—
26C3×C4⋊Dic3C2211yescharacteristic
27C3×C4⋊Dic3C2211yescharacteristic
28C3×Dic3⋊C4C2212yes—
29C3×Dic3⋊C4C2212yes—
30C3×Dic3⋊C4C2212yes—
31C3×Dic3⋊C4C2212yes—
32C3×C4×Dic3C2212yes—
33C3×#126—32no—
34C3×#125—32no—
35C3×#123—32no—
36C3×#123—32no—
37C3×C22⋊Q8—31no—
38C3×C22⋊Q8—32no—
39C3×#121—31no—
40C3×C4×D4—32no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4D311yescharacteristic, complemented
43C2×C3×C4⋊C4C2×C311yescharacteristic, complemented
44#930C2×C311yescharacteristic
45#929C2×C312yescomplemented
46C4×#67C2×C312yescomplemented
47#924C2×C311yescharacteristic, complemented
48#898C2×C312yescomplemented
49#896C2×C312yescomplemented
50#894C2×C312yescomplemented
51#889C2×C312yescomplemented
52C23×C32C2311yescharacteristic
53C22×C3×D3C2312yes—
54C2×C32×C4—22no—
55C2×C32×C4—41no—
56C2×C32×C4—41no—
57C2×C32×C4C2312yes—
58C2×C32×C4C2311yescharacteristic
59C2×C32×C4C2312yes—
60C2×C32×C4C2311yescharacteristic
61C3×#67—24no—
62C2×C3×Dic3C2312yes—
63C2×C3×Dic3C2312yes—
64C2×C3×Dic3C2312yes—
65C3×D12—41no—
66C3×C4×D3—42no—
67C3×Dic6—41no—
68#539—31noSylow
69C2×C3×Q8—31no—
70C2×C3×D4—32no—
71C2×C3×D4—31no—
72C22×C3×C4—21no—
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—32no—
76C22×C3×C4C2×D311yescharacteristic
77C22×C3×C4C2×D311yescharacteristic
78C22×C3×C4C2×D311yescharacteristic
79C22×C3×C4C22×C311yescharacteristic
80C22×C3×C4C22×C311yescharacteristic
81C22×C3×C4C22×C311yescharacteristic
82C2×#67C22×C312yes—
83C2×D12C22×C311yescharacteristic
84C2×C4×D3C22×C312yes—
85C2×Dic6C22×C311yescharacteristic
86C3×C4⋊C4—22no—
87C3×C4⋊C4—22no—
88C3×C4⋊C4—32no—
89C3×C4⋊C4—32no—
90C3×C4⋊C4—32no—
91C3×C4⋊C4—32no—
92C3×C4⋊C4—31no—
93C3×C4⋊C4—31no—
94C3×C4⋊C4C2×D312yes—
95C3×C4⋊C4C2×D312yes—
96C3×C4⋊C4C22×C312yes—
97C3×C4⋊C4C22×C312yes—
98C3×C22⋊C4—32no—
99C3×C22⋊C4—32no—
100C3×C22⋊C4—32no—
101C3×C22⋊C4—32no—
102C3×C22⋊C4—32no—
103C3×C42—32no—
104#217C22×C312yes—
105(C2×D3)⋊C4C22×C312yes—
106(C2×D3)⋊C4C22×C312yes—
107(C2×D3)⋊C4C22×C312yes—
108(C2×D3)⋊C4C22×C312yes—
109C4⋊Dic3C22×C311yescharacteristic
110C4⋊Dic3C22×C311yescharacteristic
111Dic3⋊C4C22×C312yes—
112Dic3⋊C4C22×C312yes—
113Dic3⋊C4C22×C312yes—
114Dic3⋊C4C22×C312yes—
115C4×Dic3C22×C312yes—
116C22×C32—21no—
117C22×C32—21no—
118C22×C32#4112yes—
119C22×C32C2411yescharacteristic
120C2×C3×D3—22no—
121C2×C3×D3—22no—
122C2×C3×D3—22no—
123C32×C4—22no—
124C32×C4—21no—
125C32×C4—22no—
126C32×C4—21no—
127C3×Dic3—22no—
128C3×Dic3—22no—
129C3×Dic3—22no—
130#126—32no—
131#125—32no—
132#123—32no—
133#123—32no—
134C22⋊Q8—31no—
135C22⋊Q8—32no—
136#121—31no—
137C4×D4—32no—
138C2×C4⋊C4C3×D311yescharacteristic, complemented
139C23×C3—21no—
140C23×C3—32no—
141C23×C3C22×D311yescharacteristic
142C23×C3C23×C311yescharacteristic
143C22×D3C23×C312yes—
144C3×Q8—121no—
145C3×D4—64no—
146C3×D4—121no—
147C2×C3×C4—24no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—32no—
155C2×C3×C4—32no—
156C2×C3×C4—32no—
157C2×C3×C4—41no—
158C2×C3×C4—41no—
159C2×C3×C4—41no—
160C2×C3×C4—41no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4—122no—
164C2×C3×C4C22×D312yes—
165C2×C3×C4C22×D312yes—
166C2×C3×C4C22×D311yescharacteristic
167C2×C3×C4C22×D311yescharacteristic
168C2×C3×C4C23×C312yes—
169C2×C3×C4C23×C311yescharacteristic
170C2×C3×C4C23×C312yes—
171C2×C3×C4C23×C311yescharacteristic
172#67—24no—
173C2×Dic3C23×C312yes—
174C2×Dic3C23×C312yes—
175C2×Dic3C23×C312yes—
176D12—41no—
177C4×D3—42no—
178Dic6—41no—
179C2×C32—22no—
180C2×C32C2×#4111yescharacteristic
181C2×C32H(5,𝔽2)11yescharacteristic
182C2×C322−1+411yescharacteristic
183C3×D3—42no—
184C2×Q8—31no—
185C2×D4—32no—
186C2×D4—31no—
187C22×C4—32no—
188C22×C4C2×C3×D311yescharacteristic
189C22×C4C2×C3×D311yescharacteristic
190C22×C4C2×C3×D311yescharacteristic
191C4⋊C4—32no—
192C4⋊C4—31no—
193C4⋊C4—32no—
194C4⋊C4—32no—
195C4⋊C4—31no—
196C4⋊C4—32no—
197C4⋊C4C2×C3×D312yes—
198C4⋊C4C2×C3×D312yes—
199C22⋊C4—32no—
200C22⋊C4—32no—
201C22⋊C4—32no—
202C22⋊C4—32no—
203C22⋊C4—32no—
204C42—32no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—22no—
213C22×C3—62no—
214C22×C3—62no—
215C22×C3—62no—
216C22×C3#23512yes—
217C22×C3C3×#4112yes—
218C22×C3C23×D311yescharacteristic
219C22×C3C24×C311yescharacteristic
220C2×D3—22no—
221C2×D3—22no—
222C2×D3—22no—
223C3×C4—24no—
224C3×C4—22no—
225C3×C4—22no—
226C3×C4—21no—
227C3×C4—21no—
228C3×C4—22no—
229C3×C4—21no—
230C3×C4—22no—
231C3×C4—21no—
232C3×C4—42no—
233C3×C4—41no—
234C3×C4—41no—
235C3×C4—62no—
236C3×C4—62no—
237C3×C4—62no—
238Dic3—22no—
239Dic3—22no—
240Dic3—22no—
241C32#53911yescharacteristic, complemented, Sylow
242C23—32no—
243C23C22×C3×D311yescharacteristic
244Q8—121no—
245D4—64no—
246D4—121no—
247C2×C4—22no—
248C2×C4—32no—
249C2×C4—32no—
250C2×C4—32no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—122no—
254C2×C4C22×C3×D312yes—
255C2×C4C22×C3×D312yes—
256C2×C4C22×C3×D311yescharacteristic
257C2×C4C22×C3×D311yescharacteristic
258C2×C3—24no—
259C2×C3—22no—
260C2×C3—22no—
261C2×C3—21no—
262C2×C3—21no—
263C2×C3—21no—
264C2×C3—122no—
265C2×C3C2×#23511yescharacteristic
266C2×C3#100411yescharacteristic
267C2×C3#100711yescharacteristic
268C2×C3C2×C3×#4111yescharacteristic
269C2×C3C3×H(5,𝔽2)11yescharacteristic
270C2×C3C3×2−1+411yescharacteristic
271D3—42no—
272C22—21no—
273C22—21no—
274C22—62no—
275C22—62no—
276C22—62no—
277C22C3×#23512yes—
278C22C23×C3×D311yescharacteristic
279C4—22no—
280C4—22no—
281C4—21no—
282C4—21no—
283C4—62no—
284C4—62no—
285C4—62no—
286C3—21no—
287C3#551511yescharacteristic, direct product factor
288C3C3×#53911yescharacteristic, complemented
289C2—22no—
290C2—122no—
291C2C2×C3×#23511yescharacteristic
292C2C3×#100411yescharacteristic
293C2C3×#100711yescharacteristic
subgroupG/H
centerC22×C3C23×D3
commutator, cosocleC22×C3C24×C3
FrattiniC22C23×C3×D3
FittingC2×C32×C4⋊C4C2
socleC22×C32C24
prime radicalC3×#930C2
hypercenterC2×C3×C4⋊C4D3
hypocenterC3C3×#539
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
3. C3×#929C2semidirect
4. C3×C4×#67C2semidirect
5. C3×#924C2semidirect
6. C3×#898C2semidirect
7. C3×#896C2semidirect
8. C3×#894C2semidirect
9. C3×#889C2semidirect
11. #5515C3direct (N × H)
42. C2×C3×C4⋊C4D3semidirect
43. C2×C3×C4⋊C4C2×C3semidirect
45. #929C2×C3semidirect
46. C4×#67C2×C3semidirect
47. #924C2×C3semidirect
48. #898C2×C3semidirect
49. #896C2×C3semidirect
50. #894C2×C3semidirect
51. #889C2×C3semidirect
138. C2×C4⋊C4C3×D3semidirect
241. C32#539semidirect
287. C3#5515direct (N × H)
288. C3C3×#539semidirect

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×#9292
C3×C4×#672
C3×#9242
C3×#8982
C3×#8962
C3×#8942
C3×#8892
C3×#5393
#55153