Finite Group Databasebeta

← C3×#5565 #10595662

Subgroups — C3×#5565 #10595662

Order 576.

523 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C4×D4C211yesmaximal, characteristic, complemented
2C3×#939C212yesmaximal, complemented
3C3×#930C211yesmaximal, characteristic, complemented
4C3×#930C211yesmaximal, characteristic, complemented
5C3×#926C211yesmaximal, characteristic, complemented
6C2×C3×C4⋊Dic3C211yesmaximal, characteristic, complemented
7C3×#898C212yesmaximal, complemented
8C3×#883C212yesmaximal, complemented
9C3×#67⋊C4C212yesmaximal, complemented
10C3×C4×D12C211yesmaximal, characteristic, complemented
11C3×#870C211yesmaximal, characteristic, complemented
12C3×#553—31nomaximal
13#5565C311yesmaximal, characteristic, direct product factor
14C2×C32×D4C2211yescharacteristic
15C22×C32×C4C2211yescharacteristic
16C22×C32×C4C2211yescharacteristic
17C2×C3×#67C2212yes—
18C2×C3×#67C2212yes—
19C22×C3×Dic3C2212yes—
20C2×C3×D12C2211yescharacteristic
21C2×C3×C4×D3C2212yes—
22C32×C4⋊C4C2211yescharacteristic
23C32×C22⋊C4C2211yescharacteristic
24C32×C22⋊C4C2211yescharacteristic
25C32×C42C2211yescharacteristic
26C3×#217C2212yes—
27C3×(C2×D3)⋊C4C2212yes—
28C3×(C2×D3)⋊C4C2212yes—
29C3×(C2×D3)⋊C4C2212yes—
30C3×C4⋊Dic3C2212yes—
31C3×C4⋊Dic3C2211yescharacteristic
32C3×C4⋊Dic3C2211yescharacteristic
33C3×C4⋊Dic3C2211yescharacteristic
34C3×Dic3⋊C4C2212yes—
35C3×Dic3⋊C4C2212yes—
36C3×C4×Dic3C2212yes—
37C3×#126—32no—
38C3×#125—31no—
39C3×#123—32no—
40C3×#121—32no—
41C3×#121—31no—
42C3×#121—31no—
43C3×C4×D4—21no—
44C3×C4×D4—32no—
45C3×C4×D4—31no—
46C3×C4×D4D311yescharacteristic, complemented
47C3×C4×D4C2×C311yescharacteristic, complemented
48C3×#117—31no—
49C2×C3×C4⋊C4—31no—
50#939C2×C312yescomplemented
51#930C2×C311yescharacteristic, complemented
52#930C2×C311yescharacteristic, complemented
53#926C2×C311yescharacteristic, complemented
54C2×C4⋊Dic3C2×C311yescharacteristic, complemented
55#898C2×C312yescomplemented
56#883C2×C312yescomplemented
57#67⋊C4C2×C312yescomplemented
58C4×D12C2×C311yescharacteristic, complemented
59#870C2×C311yescharacteristic, complemented
60C23×C32C2311yescharacteristic
61C23×C32C2311yescharacteristic
62C22×C3×D3C2312yes—
63C32×D4—22no—
64C2×C32×C4—22no—
65C2×C32×C4—22no—
66C2×C32×C4C2311yescharacteristic
67C2×C32×C4C2311yescharacteristic
68C2×C32×C4C2311yescharacteristic
69C2×C32×C4C2311yescharacteristic
70C2×C32×C4C2311yescharacteristic
71C3×#67—24no—
72C3×#67—42no—
73C2×C3×Dic3—42no—
74C2×C3×Dic3C2312yes—
75C2×C3×Dic3C2312yes—
76C2×C3×Dic3C2312yes—
77C3×D12—22no—
78C3×C4×D3—42no—
79#553—31noSylow
80C2×C3×D4—21no—
81C2×C3×D4—32no—
82C2×C3×D4—32no—
83C2×C3×D4—31no—
84C2×C3×D4C2×D311yescharacteristic
85C2×C3×D4C22×C311yescharacteristic
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—32no—
89C22×C3×C4—32no—
90C22×C3×C4C2×D311yescharacteristic
91C22×C3×C4C2×D311yescharacteristic
92C22×C3×C4C22×C311yescharacteristic
93C22×C3×C4C22×C311yescharacteristic
94C2×#67C22×C312yes—
95C2×#67C22×C312yes—
96C22×Dic3C22×C312yes—
97C2×D12C22×C311yescharacteristic
98C2×C4×D3C22×C312yes—
99C3×C4⋊C4—21no—
100C3×C4⋊C4—32no—
101C3×C4⋊C4—32no—
102C3×C4⋊C4—31no—
103C3×C4⋊C4—31no—
104C3×C4⋊C4—32no—
105C3×C4⋊C4—31no—
106C3×C4⋊C4C2×D311yescharacteristic
107C3×C4⋊C4C22×C311yescharacteristic
108C3×C22⋊C4—21no—
109C3×C22⋊C4—21no—
110C3×C22⋊C4—32no—
111C3×C22⋊C4—32no—
112C3×C22⋊C4—32no—
113C3×C22⋊C4—32no—
114C3×C22⋊C4C2×D311yescharacteristic
115C3×C22⋊C4C2×D311yescharacteristic
116C3×C22⋊C4C22×C311yescharacteristic
117C3×C22⋊C4C22×C311yescharacteristic
118C3×C42—21no—
119C3×C42—32no—
120C3×C42C2×D311yescharacteristic
121C3×C42C22×C311yescharacteristic
122#217C22×C312yes—
123(C2×D3)⋊C4C22×C312yes—
124(C2×D3)⋊C4C22×C312yes—
125(C2×D3)⋊C4C22×C312yes—
126C4⋊Dic3C22×C311yescharacteristic
127C4⋊Dic3C22×C311yescharacteristic
128C4⋊Dic3C22×C311yescharacteristic
129C4⋊Dic3C22×C312yes—
130Dic3⋊C4C22×C312yes—
131Dic3⋊C4C22×C312yes—
132C4×Dic3C22×C312yes—
133C22×C32—21no—
134C22×C32—21no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32#4112yes—
139C22×C32C2411yescharacteristic
140C2×C3×D3—22no—
141C2×C3×D3—22no—
142C2×C3×D3—22no—
143C32×C4—21no—
144C32×C4—21no—
145C32×C4—21no—
146C32×C4—21no—
147C32×C4#4112yes—
148C3×Dic3—22no—
149C3×Dic3—22no—
150C3×Dic3—22no—
151#126—32no—
152#125—31no—
153#123—32no—
154#121—32no—
155#121—31no—
156#121—31no—
157C4×D4—32no—
158C4×D4—31no—
159C4×D4C3×D311yescharacteristic, complemented
160#117—31no—
161C2×C4⋊C4—31no—
162C23×C3—21no—
163C23×C3—21no—
164C23×C3—32no—
165C23×C3C22×D311yescharacteristic
166C23×C3C22×D311yescharacteristic
167C23×C3C23×C311yescharacteristic
168C23×C3C23×C311yescharacteristic
169C22×D3C23×C312yes—
170C3×D4—24no—
171C3×D4—22no—
172C3×D4—22no—
173C3×D4—64no—
174C3×D4—62no—
175C3×D4—122no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—32no—
186C2×C3×C4—32no—
187C2×C3×C4—32no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—122no—
191C2×C3×C4—122no—
192C2×C3×C4C22×D311yescharacteristic
193C2×C3×C4C22×D311yescharacteristic
194C2×C3×C4C22×D311yescharacteristic
195C2×C3×C4C22×D311yescharacteristic
196C2×C3×C4C22×D311yescharacteristic
197C2×C3×C4C23×C311yescharacteristic
198C2×C3×C4C23×C311yescharacteristic
199C2×C3×C4C23×C311yescharacteristic
200C2×C3×C4C23×C311yescharacteristic
201C2×C3×C4C23×C311yescharacteristic
202#67—24no—
203#67—42no—
204C2×Dic3—42no—
205C2×Dic3C23×C312yes—
206C2×Dic3C23×C312yes—
207C2×Dic3C23×C312yes—
208D12—22no—
209C4×D3—42no—
210C2×C32—22no—
211C2×C32—41no—
212C2×C32C2×#4111yescharacteristic
213C2×C32C2×#4111yescharacteristic
214C2×C32H(5,𝔽2)11yescharacteristic
215C3×D3—42no—
216C2×D4—32no—
217C2×D4—32no—
218C2×D4—31no—
219C2×D4C2×C3×D311yescharacteristic
220C22×C4—32no—
221C22×C4—32no—
222C22×C4C2×C3×D311yescharacteristic
223C22×C4C2×C3×D311yescharacteristic
224C4⋊C4—31no—
225C4⋊C4—31no—
226C4⋊C4—32no—
227C4⋊C4—31no—
228C4⋊C4—32no—
229C4⋊C4—32no—
230C4⋊C4C2×C3×D311yescharacteristic
231C22⋊C4—32no—
232C22⋊C4—32no—
233C22⋊C4—32no—
234C22⋊C4—32no—
235C22⋊C4C2×C3×D311yescharacteristic
236C22⋊C4C2×C3×D311yescharacteristic
237C42—32no—
238C42C2×C3×D311yescharacteristic
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—22no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—22no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3—62no—
257C22×C3—62no—
258C22×C3—62no—
259C22×C3#23512yes—
260C22×C3C3×#4112yes—
261C22×C3C23×D311yescharacteristic
262C22×C3C24×C311yescharacteristic
263C2×D3—22no—
264C2×D3—22no—
265C2×D3—22no—
266C3×C4—22no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—22no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4—41no—
279C3×C4—62no—
280C3×C4—62no—
281C3×C4—62no—
282C3×C4#23712yes—
283C3×C4C3×#4112yes—
284Dic3—22no—
285Dic3—22no—
286Dic3—22no—
287C32#55311yescharacteristic, complemented, Sylow
288C23—32no—
289C23C22×C3×D311yescharacteristic
290C23C22×C3×D311yescharacteristic
291D4—22no—
292D4—64no—
293D4—62no—
294D4—122no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—32no—
298C2×C4—32no—
299C2×C4—32no—
300C2×C4—122no—
301C2×C4—122no—
302C2×C4C22×C3×D311yescharacteristic
303C2×C4C22×C3×D311yescharacteristic
304C2×C4C22×C3×D311yescharacteristic
305C2×C4C22×C3×D311yescharacteristic
306C2×C4C22×C3×D311yescharacteristic
307C2×C3—22no—
308C2×C3—22no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—42no—
313C2×C3—42no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3—122no—
317C2×C3C2×#23511yescharacteristic
318C2×C3C2×#23711yescharacteristic
319C2×C3#100911yescharacteristic
320C2×C3C2×C3×#4111yescharacteristic
321C2×C3C2×C3×#4111yescharacteristic
322C2×C3C3×H(5,𝔽2)11yescharacteristic
323D3—42no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—62no—
330C22—62no—
331C22—62no—
332C22C3×#23512yes—
333C22C23×C3×D311yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—62no—
339C4—62no—
340C4—62no—
341C4C3×#23712yes—
342C3—21no—
343C3#556511yescharacteristic, direct product factor
344C3C3×#55311yescharacteristic, complemented
345C2—22no—
346C2—41no—
347C2—122no—
348C2C2×C3×#23511yescharacteristic
349C2C2×C3×#23711yescharacteristic
350C2C3×#100911yescharacteristic
subgroupG/H
centerC22×C3C23×D3
commutator, cosocleC22×C3C24×C3
FrattiniC22C23×C3×D3
FittingC32×C4×D4C2
socleC22×C32C24
hypercenterC3×C4×D4D3
hypocenterC3C3×#553
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
1. C32×C4×D4C2semidirect
2. C3×#939C2semidirect
3. C3×#930C2semidirect
4. C3×#930C2semidirect
5. C3×#926C2semidirect
6. C2×C3×C4⋊Dic3C2semidirect
7. C3×#898C2semidirect
8. C3×#883C2semidirect
9. C3×#67⋊C4C2semidirect
10. C3×C4×D12C2semidirect
11. C3×#870C2semidirect
13. #5565C3direct (N × H)
46. C3×C4×D4D3semidirect
47. C3×C4×D4C2×C3semidirect
50. #939C2×C3semidirect
51. #930C2×C3semidirect
52. #930C2×C3semidirect
53. #926C2×C3semidirect
54. C2×C4⋊Dic3C2×C3semidirect
55. #898C2×C3semidirect
56. #883C2×C3semidirect
57. #67⋊C4C2×C3semidirect
58. C4×D12C2×C3semidirect
59. #870C2×C3semidirect
159. C4×D4C3×D3semidirect
287. C32#553semidirect
343. C3#5565direct (N × H)
344. C3C3×#553semidirect

Maximal subgroups

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

subgroupindex
C32×C4×D42
C3×#9392
C3×#9302
C3×#9262
C2×C3×C4⋊Dic32
C3×#8982
C3×#8832
C3×#67⋊C42
C3×C4×D122
C3×#8702
C3×#5533
#55653