Finite Group Databasebeta

← C3×#5597 #10595694

Subgroups — C3×#5597 #10595694

Order 576.

635 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#237C212yesmaximal, complemented
2C32×#121C211yesmaximal, characteristic, complemented
3C3×#939C212yesmaximal, complemented
4C3×#939C212yesmaximal, complemented
5C3×#930C211yesmaximal, characteristic, complemented
6C2×C3×Dic3⋊C4C211yesmaximal, characteristic, complemented
7C3×#896C212yesmaximal, complemented
8C3×#884C212yesmaximal, complemented
9C3×#878C212yesmaximal, complemented
10C3×#537—31nomaximal
11#5597C311yesmaximal, characteristic, direct product factor
12C2×C32×D4C2212yes—
13C2×C32×D4C2211yescharacteristic
14C22×C32×C4C2211yescharacteristic
15C2×C3×#67C2212yes—
16C2×C3×#67C2212yes—
17C2×C3×#67C2212yes—
18C22×C3×Dic3C2212yes—
19C22×C3×Dic3C2211yescharacteristic
20C22×C3×Dic3C2211yescharacteristic
21C3×#237—28no—
22C2×C3×D12C2211yescharacteristic
23C2×C3×C4×D3C2212yes—
24C2×C3×Dic6C2212yes—
25C32×C4⋊C4C2211yescharacteristic
26C32×C22⋊C4C2212yes—
27C3×#217C2212yes—
28C3×(C2×D3)⋊C4C2212yes—
29C3×(C2×D3)⋊C4C2212yes—
30C3×C4⋊Dic3C2211yescharacteristic
31C3×Dic3⋊C4C2212yes—
32C3×Dic3⋊C4C2212yes—
33C3×Dic3⋊C4C2212yes—
34C2×C3×#41—32no—
35C2×#237C2×C312yescomplemented
36C3×C22⋊Q8—32no—
37C3×C22⋊Q8—32no—
38C3×#121—21no—
39C3×#121—32no—
40C3×#121—32no—
41C3×#121—31no—
42C3×#121—32no—
43C3×#121D311yescharacteristic, complemented
44C3×#121C2×C311yescharacteristic, complemented
45C2×C3×C4⋊C4—31no—
46#939C2×C312yescomplemented
47#939C2×C312yescomplemented
48#930C2×C311yescharacteristic, complemented
49C2×Dic3⋊C4C2×C311yescharacteristic, complemented
50#896C2×C312yescomplemented
51#884C2×C312yescomplemented
52#878C2×C312yescomplemented
53C23×C32C2312yes—
54C23×C32C2311yescharacteristic
55C22×C3×D3C2312yes—
56C32×D4—24no—
57C32×D4—41no—
58C2×C32×C4—41no—
59C2×C32×C4C2311yescharacteristic
60C2×C32×C4C2311yescharacteristic
61C2×C32×C4C2312yes—
62C3×#67—24no—
63C3×#67—24no—
64C3×#67—42no—
65C2×C3×Dic3—24no—
66C2×C3×Dic3—41no—
67C2×C3×Dic3D414yescomplemented
68C2×C3×Dic3C2312yes—
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2312yes—
71C3×D12—41no—
72C3×C4×D3—24no—
73C3×Dic6—24no—
74#537—31noSylow
75C3×#41—68no—
76C2×C3×Q8—32no—
77C2×C3×D4—22no—
78C2×C3×D4—21no—
79C2×C3×D4—32no—
80C2×C3×D4—32no—
81C2×C3×D4—32no—
82C2×C3×D4—31no—
83C2×C3×D4C2×D312yes—
84C2×C3×D4C2×D311yescharacteristic
85C2×C3×D4C22×C312yes—
86C2×C3×D4C22×C311yescharacteristic
87C22×C3×C4—21no—
88C22×C3×C4—31no—
89C22×C3×C4—32no—
90C22×C3×C4—31no—
91C22×C3×C4—32no—
92C22×C3×C4C2×D311yescharacteristic
93C22×C3×C4C22×C311yescharacteristic
94C2×#67C22×C312yes—
95C2×#67C22×C312yes—
96C2×#67C22×C312yes—
97C22×Dic3C22×C312yes—
98C22×Dic3C22×C311yescharacteristic
99C22×Dic3C22×C311yescharacteristic
100#237—28no—
101C2×D12C22×C311yescharacteristic
102C2×C4×D3C22×C312yes—
103C2×Dic6C22×C312yes—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—32no—
106C3×C4⋊C4—32no—
107C3×C4⋊C4—32no—
108C3×C4⋊C4—31no—
109C3×C4⋊C4C2×D311yescharacteristic
110C3×C4⋊C4C22×C311yescharacteristic
111C3×C22⋊C4—22no—
112C3×C22⋊C4—32no—
113C3×C22⋊C4—32no—
114C3×C22⋊C4—32no—
115C3×C22⋊C4C2×D312yes—
116C3×C22⋊C4C22×C312yes—
117#217C22×C312yes—
118(C2×D3)⋊C4C22×C312yes—
119(C2×D3)⋊C4C22×C312yes—
120C4⋊Dic3C22×C311yescharacteristic
121Dic3⋊C4C22×C312yes—
122Dic3⋊C4C22×C312yes—
123Dic3⋊C4C22×C312yes—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32—22no—
127C22×C32—22no—
128C22×C32—22no—
129C22×C32C2×D412yes—
130C22×C32C2411yescharacteristic
131C2×C3×D3—22no—
132C2×C3×D3—22no—
133C2×C3×D3—22no—
134C32×C4—21no—
135C32×C4—22no—
136C32×C4—21no—
137C3×Dic3—22no—
138C3×Dic3—22no—
139C3×Dic3C2×D414yes—
140C2×#41—32no—
141C22⋊Q8—32no—
142C22⋊Q8—32no—
143#121—32no—
144#121—32no—
145#121—31no—
146#121—32no—
147#121C3×D311yescharacteristic, complemented
148C2×C4⋊C4—31no—
149C23×C3—21no—
150C23×C3—22no—
151C23×C3—32no—
152C23×C3C22×D312yes—
153C23×C3C22×D311yescharacteristic
154C23×C3C23×C312yes—
155C23×C3C23×C311yescharacteristic
156C22×D3C23×C312yes—
157C3×Q8—64no—
158C3×D4—24no—
159C3×D4—24no—
160C3×D4—44no—
161C3×D4—41no—
162C3×D4—41no—
163C3×D4—42no—
164C3×D4—64no—
165C3×D4—64no—
166C3×D4—121no—
167C3×D4—122no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—22no—
171C2×C3×C4—34no—
172C2×C3×C4—32no—
173C2×C3×C4—32no—
174C2×C3×C4—32no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—42no—
178C2×C3×C4—64no—
179C2×C3×C4—64no—
180C2×C3×C4—121no—
181C2×C3×C4C22×D311yescharacteristic
182C2×C3×C4C22×D312yes—
183C2×C3×C4C22×D311yescharacteristic
184C2×C3×C4C23×C311yescharacteristic
185C2×C3×C4C23×C311yescharacteristic
186C2×C3×C4C23×C312yes—
187#67—24no—
188#67—24no—
189#67—42no—
190C2×Dic3—24no—
191C2×Dic3—41no—
192C2×Dic3C3×D414yescomplemented
193C2×Dic3C23×C312yes—
194C2×Dic3C23×C312yes—
195C2×Dic3C23×C312yes—
196D12—41no—
197C4×D3—24no—
198Dic6—24no—
199C2×C32—22no—
200C2×C32—42no—
201C2×C32C22×D411yescharacteristic
202C2×C32H(5,𝔽2)11yescharacteristic
203C2×C322−1+411yescharacteristic
204C3×D3—42no—
205#41—68no—
206C2×Q8—32no—
207C2×D4—32no—
208C2×D4—32no—
209C2×D4—32no—
210C2×D4—31no—
211C2×D4C2×C3×D312yes—
212C2×D4C2×C3×D311yescharacteristic
213C22×C4—31no—
214C22×C4—32no—
215C22×C4—31no—
216C22×C4—32no—
217C22×C4C2×C3×D311yescharacteristic
218C4⋊C4—32no—
219C4⋊C4—32no—
220C4⋊C4—32no—
221C4⋊C4—31no—
222C4⋊C4C2×C3×D311yescharacteristic
223C22⋊C4—32no—
224C22⋊C4—32no—
225C22⋊C4—32no—
226C22⋊C4C2×C3×D312yes—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—22no—
232C22×C3—22no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—22no—
236C22×C3—22no—
237C22×C3—21no—
238C22×C3—22no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3—42no—
242C22×C3—42no—
243C22×C3—42no—
244C22×C3—62no—
245C22×C3—62no—
246C22×C3—62no—
247C22×C3D3×D412yes—
248C22×C3C2×C3×D412yes—
249C22×C3C23×D311yescharacteristic
250C22×C3C24×C311yescharacteristic
251C2×D3—22no—
252C2×D3—22no—
253C2×D3—22no—
254C3×C4—22no—
255C3×C4—22no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—22no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—34no—
262C3×C4—41no—
263C3×C4—42no—
264C3×C4—62no—
265C3×C4—62no—
266Dic3—22no—
267Dic3—22no—
268Dic3C2×C3×D414yes—
269C32#53711yescharacteristic, complemented, Sylow
270C23—32no—
271C23C22×C3×D312yes—
272C23C22×C3×D311yescharacteristic
273Q8—64no—
274D4—24no—
275D4—41no—
276D4—64no—
277D4—64no—
278D4—121no—
279D4—122no—
280C2×C4—32no—
281C2×C4—34no—
282C2×C4—32no—
283C2×C4—32no—
284C2×C4—41no—
285C2×C4—64no—
286C2×C4—64no—
287C2×C4—121no—
288C2×C4C22×C3×D311yescharacteristic
289C2×C4C22×C3×D312yes—
290C2×C4C22×C3×D311yescharacteristic
291C2×C3—22no—
292C2×C3—22no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—42no—
297C2×C3—42no—
298C2×C3—42no—
299C2×C3—82no—
300C2×C3—122no—
301C2×C3C2×D3×D411yescharacteristic
302C2×C3#100411yescharacteristic
303C2×C3#101011yescharacteristic
304C2×C3C22×C3×D411yescharacteristic
305C2×C3C3×H(5,𝔽2)11yescharacteristic
306C2×C3C3×2−1+411yescharacteristic
307D3—42no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—62no—
314C22—62no—
315C22—62no—
316C22C3×D3×D412yes—
317C22C23×C3×D311yescharacteristic
318C4—22no—
319C4—21no—
320C4—21no—
321C4—34no—
322C4—62no—
323C4—62no—
324C3—21no—
325C3#559711yescharacteristic, direct product factor
326C3C3×#53711yescharacteristic, complemented
327C2—22no—
328C2—42no—
329C2—122no—
330C2C2×C3×D3×D411yescharacteristic
331C2C3×#100411yescharacteristic
332C2C3×#101011yescharacteristic
subgroupG/H
centerC22×C3C23×D3
commutator, cosocleC22×C3C24×C3
FrattiniC22C23×C3×D3
FittingC32×#121C2
socleC22×C32C24
hypercenterC3×#121D3
hypocenterC3C3×#537
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×#237C2semidirect
2. C32×#121C2semidirect
3. C3×#939C2semidirect
4. C3×#939C2semidirect
5. C3×#930C2semidirect
6. C2×C3×Dic3⋊C4C2semidirect
7. C3×#896C2semidirect
8. C3×#884C2semidirect
9. C3×#878C2semidirect
11. #5597C3direct (N × H)
35. C2×#237C2×C3semidirect
43. C3×#121D3semidirect
44. C3×#121C2×C3semidirect
46. #939C2×C3semidirect
47. #939C2×C3semidirect
48. #930C2×C3semidirect
49. C2×Dic3⋊C4C2×C3semidirect
50. #896C2×C3semidirect
51. #884C2×C3semidirect
52. #878C2×C3semidirect
67. C2×C3×Dic3D4semidirect
147. #121C3×D3semidirect
192. C2×Dic3C3×D4semidirect
269. C32#537semidirect
325. C3#5597direct (N × H)
326. C3C3×#537semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#2372
C32×#1212
C3×#9392
C3×#9302
C2×C3×Dic3⋊C42
C3×#8962
C3×#8842
C3×#8782
C3×#5373
#55973