Finite Group Databasebeta

← C32×#5730 #13179107

Subgroups — C32×#5730 #13179107

Order 1728.

1640 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4⋊Q8C211yesmaximal, characteristic, complemented
2C32×#946C214yesmaximal, complemented
3C32×#897C212yesmaximal, complemented
4C32×#893C212yesmaximal
5C32×D3×C4⋊C4C212yesmaximal, complemented
6C32×#890C212yesmaximal, complemented
7C32×C4×D12C211yesmaximal, characteristic
8C32×C4×Dic6C211yesmaximal, characteristic, complemented
9C32×#554—31nomaximal
10C3×#5730C314yesmaximal, direct product factor
11C2×C33×Q8C2212yes—
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3C2212yes—
14C2×C32×C4×D3C2214yes—
15C2×C32×Dic6C2211yescharacteristic
16C33×C4⋊C4C2212yes—
17C33×C4⋊C4C2212yes—
18C33×C42C2211yescharacteristic
19C32×(C2×D3)⋊C4C2212yes—
20C32×(C2×D3)⋊C4C2214yes—
21C32×C4⋊Dic3C2214yes—
22C32×C4⋊Dic3C2211yescharacteristic
23C32×C4⋊Dic3C2211yescharacteristic
24C32×Dic3⋊C4C2212yes—
25C32×Dic3⋊C4C2214yes—
26C32×C4×Dic3C2212yes—
27C32×C4⋊Q8—24no—
28C32×C4⋊Q8D311yescharacteristic, complemented
29C32×C4⋊Q8C2×C314yescomplemented
30C32×#125—32no—
31C32×C22⋊Q8—34no—
32C32×C22⋊Q8—32no—
33C32×C4×Q8—31no—
34C32×C4×D4—32no—
35C32×C4×D4—31no—
36C2×C32×C4⋊C4—32no—
37C3×#946C2×C3116yescomplemented
38C3×#897C2×C318yescomplemented
39C3×#893C2×C318yes—
40C3×D3×C4⋊C4C2×C318yescomplemented
41C3×#890C2×C318yescomplemented
42C3×C4×D12C2×C314yes—
43C3×C4×Dic6C2×C314yescomplemented
44C22×C32×D3C2312yes—
45C33×Q8—42no—
46C2×C33×C4C2311yescharacteristic
47C2×C33×C4C2314yes—
48C2×C33×C4C2311yescharacteristic
49C2×C33×C4C2311yescharacteristic
50C2×C32×Dic3C2312yes—
51C2×C32×Dic3C2312yes—
52C2×C32×Dic3C2312yes—
53C32×D12Q814yescomplemented
54C32×C4×D3—24no—
55C32×C4×D3—44no—
56C32×Dic6—22no—
57C3×#554—34no—
58#5730C3211yescharacteristic, direct product factor
59C2×C32×Q8—28no—
60C2×C32×Q8—31no—
61C2×C32×Q8C2×D312yes—
62C2×C32×Q8C22×C318yes—
63C2×C32×D4—31no—
64C22×C32×C4—32no—
65C22×C32×C4—34no—
66C2×C3×D12C22×C314yes—
67C2×C3×C4×D3C22×C318yes—
68C2×C3×C4×D3C22×C3116yes—
69C2×C3×Dic6C22×C314yes—
70C32×C4⋊C4—28no—
71C32×C4⋊C4—28no—
72C32×C4⋊C4—31no—
73C32×C4⋊C4—34no—
74C32×C4⋊C4—31no—
75C32×C4⋊C4—32no—
76C32×C4⋊C4—34no—
77C32×C4⋊C4C2×D312yes—
78C32×C4⋊C4C2×D312yes—
79C32×C4⋊C4C22×C318yes—
80C32×C4⋊C4C22×C318yes—
81C32×C22⋊C4—34no—
82C32×C22⋊C4—32no—
83C32×C42—24no—
84C32×C42—32no—
85C32×C42C2×D311yescharacteristic
86C32×C42C22×C314yes—
87C3×(C2×D3)⋊C4C22×C3116yes—
88C3×(C2×D3)⋊C4C22×C318yes—
89C3×C4⋊Dic3C22×C314yes—
90C3×C4⋊Dic3C22×C3116yes—
91C3×C4⋊Dic3C22×C314yes—
92C3×Dic3⋊C4C22×C3116yes—
93C3×Dic3⋊C4C22×C318yes—
94C3×C4×Dic3C22×C318yes—
95C22×C33C2411yescharacteristic
96C2×C32×D3—22no—
97C2×C32×D3—22no—
98C2×C32×D3C2×Q814yes—
99C33×C4—21no—
100C33×C4—24no—
101C33×C4C2×Q812yes—
102C33×C4#4112yes—
103C32×Dic3—22no—
104C32×Dic3—22no—
105C32×Dic3—22no—
106C3×C4⋊Q8—24no—
107C3×C4⋊Q8C3×D314yescomplemented
108C3×C4⋊Q8C2×C3211yescharacteristic, complemented
109C3×#125—38no—
110C3×C22⋊Q8—316no—
111C3×C22⋊Q8—38no—
112C3×C4×Q8—34no—
113C3×C4×D4—38no—
114C3×C4×D4—34no—
115C2×C3×C4⋊C4—38no—
116#946C2×C3214yescomplemented
117#897C2×C3212yescomplemented
118#893C2×C3212yes—
119D3×C4⋊C4C2×C3212yescomplemented
120#890C2×C3212yescomplemented
121C4×D12C2×C3211yescharacteristic
122C4×Dic6C2×C3211yescharacteristic, complemented
123C23×C32—32no—
124C22×C3×D3C23×C318yes—
125C32×Q8—42no—
126C32×Q8—48no—
127C32×Q8—416no—
128C32×Q8—62no—
129C32×D4—34no—
130C2×C32×C4—24no—
131C2×C32×C4—24no—
132C2×C32×C4—24no—
133C2×C32×C4—216no—
134C2×C32×C4—32no—
135C2×C32×C4—32no—
136C2×C32×C4—32no—
137C2×C32×C4—64no—
138C2×C32×C4—124no—
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D314yes—
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C23×C314yes—
144C2×C32×C4C23×C3116yes—
145C2×C32×C4C23×C314yes—
146C2×C32×C4C23×C314yes—
147C2×C3×Dic3C23×C318yes—
148C2×C3×Dic3C23×C318yes—
149C2×C3×Dic3C23×C318yes—
150C3×D12C3×Q8116yescomplemented
151C3×C4×D3—216no—
152C3×C4×D3—416no—
153C3×Dic6—28no—
154#554—31noSylow
155C2×C33C22×Q811yescharacteristic
156C2×C33C2×#4111yescharacteristic
157C2×C33H(5,𝔽2)11yescharacteristic
158C32×D3—24no—
159C2×C3×Q8—28no—
160C2×C3×Q8—34no—
161C2×C3×Q8C2×C3×D318yes—
162C2×C3×Q8C22×C3212yes—
163C2×C3×D4—34no—
164C22×C3×C4—38no—
165C22×C3×C4—316no—
166C2×D12C22×C3211yescharacteristic
167C2×C4×D3C22×C3212yes—
168C2×C4×D3C22×C3214yes—
169C2×Dic6C22×C3211yescharacteristic
170C3×C4⋊C4—28no—
171C3×C4⋊C4—28no—
172C3×C4⋊C4—316no—
173C3×C4⋊C4—38no—
174C3×C4⋊C4—34no—
175C3×C4⋊C4—34no—
176C3×C4⋊C4—316no—
177C3×C4⋊C4C2×C3×D318yes—
178C3×C4⋊C4C2×C3×D318yes—
179C3×C4⋊C4C22×C3212yes—
180C3×C4⋊C4C22×C3212yes—
181C3×C22⋊C4—316no—
182C3×C22⋊C4—38no—
183C3×C42—24no—
184C3×C42—38no—
185C3×C42C2×C3×D314yes—
186C3×C42C22×C3211yescharacteristic
187(C2×D3)⋊C4C22×C3212yes—
188(C2×D3)⋊C4C22×C3214yes—
189C4⋊Dic3C22×C3211yescharacteristic
190C4⋊Dic3C22×C3211yescharacteristic
191C4⋊Dic3C22×C3214yes—
192Dic3⋊C4C22×C3212yes—
193Dic3⋊C4C22×C3214yes—
194C4×Dic3C22×C3212yes—
195C22×C32—24no—
196C22×C32—34no—
197C22×C32—62no—
198C22×C32—62no—
199C22×C32C23×D311yescharacteristic
200C22×C32C24×C314yes—
201C2×C3×D3—28no—
202C2×C3×D3—28no—
203C2×C3×D3C2×C3×Q8116yes—
204C32×C4—28no—
205C32×C4—24no—
206C32×C4—21no—
207C32×C4—24no—
208C32×C4—28no—
209C32×C4—28no—
210C32×C4—216no—
211C32×C4—416no—
212C32×C4—62no—
213C32×C4—62no—
214C32×C4—62no—
215C32×C4D3×Q812yes—
216C32×C4#23912yes—
217C32×C4C2×C3×Q818yes—
218C32×C4C3×#4118yes—
219C3×Dic3—28no—
220C3×Dic3—28no—
221C3×Dic3—28no—
222C4⋊Q8C32×D311yescharacteristic, complemented
223#125—32no—
224C22⋊Q8—34no—
225C22⋊Q8—32no—
226C4×Q8—31no—
227C4×D4—32no—
228C4×D4—31no—
229C2×C4⋊C4—32no—
230C33#55411yescharacteristic, complemented, Sylow
231C23×C3—38no—
232C22×D3C23×C3212yes—
233C3×Q8—48no—
234C3×Q8—42no—
235C3×Q8—416no—
236C3×Q8—68no—
237C3×D4—316no—
238C2×C3×C4—24no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—216no—
242C2×C3×C4—38no—
243C2×C3×C4—38no—
244C2×C3×C4—38no—
245C2×C3×C4—616no—
246C2×C3×C4—1216no—
247C2×C3×C4C22×C3×D314yes—
248C2×C3×C4C22×C3×D3116yes—
249C2×C3×C4C22×C3×D314yes—
250C2×C3×C4C22×C3×D314yes—
251C2×C3×C4C23×C3211yescharacteristic
252C2×C3×C4C23×C3214yes—
253C2×C3×C4C23×C3211yescharacteristic
254C2×C3×C4C23×C3211yescharacteristic
255C2×Dic3C23×C3212yes—
256C2×Dic3C23×C3212yes—
257C2×Dic3C23×C3212yes—
258D12C32×Q814yescomplemented
259C4×D3—24no—
260C4×D3—44no—
261Dic6—22no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32—24no—
265C2×C32—64no—
266C2×C32#100411yescharacteristic
267C2×C32C2×D3×Q811yescharacteristic
268C2×C32C2×#23911yescharacteristic
269C2×C32C22×C3×Q814yes—
270C2×C32C2×C3×#4114yes—
271C2×C32C3×H(5,𝔽2)14yes—
272C3×D3—216no—
273C2×Q8—31no—
274C2×Q8C2×C32×D312yes—
275C2×D4—31no—
276C22×C4—32no—
277C22×C4—34no—
278C4⋊C4—31no—
279C4⋊C4—34no—
280C4⋊C4—31no—
281C4⋊C4—32no—
282C4⋊C4—34no—
283C4⋊C4C2×C32×D312yes—
284C4⋊C4C2×C32×D312yes—
285C22⋊C4—34no—
286C22⋊C4—32no—
287C42—32no—
288C42C2×C32×D311yescharacteristic
289C22×C3—24no—
290C22×C3—316no—
291C22×C3—68no—
292C22×C3—68no—
293C22×C3C23×C3×D314yes—
294C22×C3C24×C3211yescharacteristic
295C2×D3—22no—
296C2×D3—22no—
297C2×D3C2×C32×Q814yes—
298C3×C4—28no—
299C3×C4—216no—
300C3×C4—24no—
301C3×C4—24no—
302C3×C4—21no—
303C3×C4—28no—
304C3×C4—28no—
305C3×C4—416no—
306C3×C4—68no—
307C3×C4—68no—
308C3×C4—68no—
309C3×C4C3×D3×Q818yes—
310C3×C4C3×#23918yes—
311C3×C4C2×C32×Q812yes—
312C3×C4C32×#4112yes—
313Dic3—22no—
314Dic3—22no—
315Dic3—22no—
316C32—24no—
317C32#573011yescharacteristic, direct product factor
318C32C3×#55414yescomplemented
319C23—32no—
320Q8—42no—
321Q8—62no—
322D4—34no—
323C2×C4—32no—
324C2×C4—32no—
325C2×C4—32no—
326C2×C4—64no—
327C2×C4—124no—
328C2×C4C22×C32×D311yescharacteristic
329C2×C4C22×C32×D314yes—
330C2×C4C22×C32×D311yescharacteristic
331C2×C4C22×C32×D311yescharacteristic
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—24no—
335C2×C3—616no—
336C2×C3C3×#100414yes—
337C2×C3C2×C3×D3×Q814yes—
338C2×C3C2×C3×#23914yes—
339C2×C3C22×C32×Q811yescharacteristic
340C2×C3C2×C32×#4111yescharacteristic
341C2×C3C32×H(5,𝔽2)11yescharacteristic
342D3—24no—
343C22—34no—
344C22—62no—
345C22—62no—
346C22C23×C32×D311yescharacteristic
347C4—24no—
348C4—21no—
349C4—62no—
350C4—62no—
351C4—62no—
352C4C32×D3×Q812yes—
353C4C32×#23912yes—
354C3—24no—
355C3C3×#573014yesdirect product factor
356C3C32×#55411yescharacteristic, complemented
357C2—64no—
358C2C32×#100411yescharacteristic
359C2C2×C32×D3×Q811yescharacteristic
360C2C2×C32×#23911yescharacteristic
subgroupG/H
centerC22×C32C23×D3
commutator, cosocleC22×C3C24×C32
FrattiniC22C23×C32×D3
FittingC33×C4⋊Q8C2
socleC22×C33C24
prime radicalC2×C32×D12C22
hypercenterC32×C4⋊Q8D3
hypocenterC3C32×#554
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C32×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C33×C4⋊Q8C2semidirect
2. C32×#946C2semidirect
3. C32×#897C2semidirect
5. C32×D3×C4⋊C4C2semidirect
6. C32×#890C2semidirect
8. C32×C4×Dic6C2semidirect
10. C3×#5730C3direct (N × H)
28. C32×C4⋊Q8D3semidirect
29. C32×C4⋊Q8C2×C3semidirect
37. C3×#946C2×C3semidirect
38. C3×#897C2×C3semidirect
40. C3×D3×C4⋊C4C2×C3semidirect
41. C3×#890C2×C3semidirect
43. C3×C4×Dic6C2×C3semidirect
53. C32×D12Q8semidirect
58. #5730C32direct (N × H)
107. C3×C4⋊Q8C3×D3semidirect
108. C3×C4⋊Q8C2×C32semidirect
116. #946C2×C32semidirect
117. #897C2×C32semidirect
119. D3×C4⋊C4C2×C32semidirect
120. #890C2×C32semidirect
122. C4×Dic6C2×C32semidirect
150. C3×D12C3×Q8semidirect
222. C4⋊Q8C32×D3semidirect
230. C33#554semidirect
258. D12C32×Q8semidirect
317. C32#5730direct (N × H)
318. C32C3×#554semidirect
355. C3C3×#5730direct (N × H)
356. C3C32×#554semidirect

Maximal subgroups

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

subgroupindex
C33×C4⋊Q82
C32×#9462
C32×#8972
C32×#8932
C32×D3×C4⋊C42
C32×#8902
C32×C4×D122
C32×C4×Dic62
C32×#5543
C3×#57303