Finite Group Databasebeta

← C32×#5584 #13178961

Subgroups — C32×#5584 #13178961

Order 1728.

1552 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×#946C212yesmaximal, complemented
3C32×#898C214yesmaximal, complemented
4C32×#892C212yesmaximal, complemented
5C32×#875C212yesmaximal
6C32×C4×D12C211yesmaximal, characteristic
7C32×#869C211yesmaximal, characteristic, complemented
8C32×C4×Dic6C212yesmaximal, complemented
9C32×#556—31nomaximal
10C3×#5584C314yesmaximal, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3C2212yes—
14C2×C32×Dic6C2212yes—
15C33×C4⋊C4C2211yescharacteristic
16C33×C4⋊C4C2212yes—
17C33×C42C2212yes—
18C33×C42C2211yescharacteristic
19C32×(C2×D3)⋊C4C2214yes—
20C32×(C2×D3)⋊C4C2212yes—
21C32×(C2×D3)⋊C4C2214yes—
22C32×C4⋊Dic3C2212yes—
23C32×C4⋊Dic3C2211yescharacteristic
24C32×C4⋊Dic3C2212yes—
25C32×Dic3⋊C4C2214yes—
26C32×C4×Dic3C2214yes—
27C32×C4⋊Q8—31no—
28C32×#126—34no—
29C32×#124—32no—
30C32×C22⋊Q8—32no—
31C32×C4×Q8—24no—
32C32×C4×Q8—32no—
33C32×C4×Q8D311yescharacteristic, complemented
34C32×C4×Q8C2×C314yescomplemented
35C32×C4×D4—31no—
36C32×#117—32no—
37C3×#946C2×C318yescomplemented
38C3×#898C2×C3116yescomplemented
39C3×#892C2×C318yescomplemented
40C3×#875C2×C318yes—
41C3×C4×D12C2×C314yes—
42C3×#869C2×C314yescomplemented
43C3×C4×Dic6C2×C318yescomplemented
44C22×C32×D3C2312yes—
45C33×Q8—22no—
46C2×C33×C4C2312yes—
47C2×C33×C4C2311yescharacteristic
48C2×C33×C4C2312yes—
49C2×C33×C4C2311yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C32×Dic3C2314yes—
52C2×C32×Dic3C2312yes—
53C32×D12—22no—
54C32×C4×D3—24no—
55C32×Dic6—24no—
56C3×#556—34no—
57#5584C3211yescharacteristic, direct product factor
58C2×C32×Q8—24no—
59C2×C32×Q8—32no—
60C2×C32×Q8C2×D311yescharacteristic
61C2×C32×Q8C22×C314yes—
62C2×C32×D4—31no—
63C22×C32×C4—32no—
64C2×C3×D12C22×C314yes—
65C2×C3×C4×D3C22×C318yes—
66C2×C3×Dic6C22×C318yes—
67C32×C4⋊C4—24no—
68C32×C4⋊C4—28no—
69C32×C4⋊C4—32no—
70C32×C4⋊C4—31no—
71C32×C4⋊C4—34no—
72C32×C4⋊C4—32no—
73C32×C4⋊C4C2×D311yescharacteristic
74C32×C4⋊C4C2×D312yes—
75C32×C4⋊C4C22×C314yes—
76C32×C4⋊C4C22×C318yes—
77C32×C22⋊C4—34no—
78C32×C22⋊C4—32no—
79C32×C22⋊C4—34no—
80C32×C42—24no—
81C32×C42—28no—
82C32×C42—34no—
83C32×C42C2×D311yescharacteristic
84C32×C42C2×D312yes—
85C32×C42C22×C318yes—
86C32×C42C22×C314yes—
87C3×(C2×D3)⋊C4C22×C3116yes—
88C3×(C2×D3)⋊C4C22×C318yes—
89C3×(C2×D3)⋊C4C22×C3116yes—
90C3×C4⋊Dic3C22×C318yes—
91C3×C4⋊Dic3C22×C318yes—
92C3×C4⋊Dic3C22×C314yes—
93C3×Dic3⋊C4C22×C3116yes—
94C3×C4×Dic3C22×C3116yes—
95C22×C33C2411yescharacteristic
96C2×C32×D3—22no—
97C2×C32×D3—22no—
98C2×C32×D3—22no—
99C33×C4—22no—
100C33×C4—22no—
101C33×C4—21no—
102C33×C4#4112yes—
103C33×C4#4112yes—
104C32×Dic3—24no—
105C32×Dic3—22no—
106C3×C4⋊Q8—34no—
107C3×#126—316no—
108C3×#124—38no—
109C3×C22⋊Q8—38no—
110C3×C4×Q8—24no—
111C3×C4×Q8—38no—
112C3×C4×Q8C3×D314yescomplemented
113C3×C4×Q8C2×C3211yescharacteristic, complemented
114C3×C4×D4—34no—
115C3×#117—38no—
116#946C2×C3212yescomplemented
117#898C2×C3214yescomplemented
118#892C2×C3212yescomplemented
119#875C2×C3212yes—
120C4×D12C2×C3211yescharacteristic
121#869C2×C3211yescharacteristic, complemented
122C4×Dic6C2×C3212yescomplemented
123C23×C32—32no—
124C22×C3×D3C23×C318yes—
125C32×Q8—216no—
126C32×Q8—22no—
127C32×Q8—28no—
128C32×Q8—64no—
129C32×D4—62no—
130C2×C32×C4—24no—
131C2×C32×C4—28no—
132C2×C32×C4—24no—
133C2×C32×C4—28no—
134C2×C32×C4—24no—
135C2×C32×C4—32no—
136C2×C32×C4—34no—
137C2×C32×C4—64no—
138C2×C32×C4C22×D312yes—
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D312yes—
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C23×C318yes—
144C2×C32×C4C23×C318yes—
145C2×C32×C4C23×C314yes—
146C2×C32×C4C23×C314yes—
147C2×C32×C4C23×C314yes—
148C2×C3×Dic3C23×C318yes—
149C2×C3×Dic3C23×C3116yes—
150C3×D12—28no—
151C3×C4×D3—216no—
152C3×Dic6—216no—
153#556—31noSylow
154C2×C33C2×#4111yescharacteristic
155C2×C33C2×#4111yescharacteristic
156C2×C332−1+411yescharacteristic
157C32×D3—42no—
158C2×C3×Q8—24no—
159C2×C3×Q8—38no—
160C2×C3×Q8C2×C3×D314yes—
161C2×C3×Q8C22×C3211yescharacteristic
162C2×C3×D4—34no—
163C22×C3×C4—38no—
164C2×D12C22×C3211yescharacteristic
165C2×C4×D3C22×C3212yes—
166C2×Dic6C22×C3212yes—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—28no—
169C3×C4⋊C4—316no—
170C3×C4⋊C4—38no—
171C3×C4⋊C4—34no—
172C3×C4⋊C4—38no—
173C3×C4⋊C4C2×C3×D314yes—
174C3×C4⋊C4C2×C3×D318yes—
175C3×C4⋊C4C22×C3211yescharacteristic
176C3×C4⋊C4C22×C3212yes—
177C3×C22⋊C4—316no—
178C3×C22⋊C4—38no—
179C3×C22⋊C4—316no—
180C3×C42—24no—
181C3×C42—28no—
182C3×C42—316no—
183C3×C42C2×C3×D314yes—
184C3×C42C2×C3×D318yes—
185C3×C42C22×C3212yes—
186C3×C42C22×C3211yescharacteristic
187(C2×D3)⋊C4C22×C3212yes—
188(C2×D3)⋊C4C22×C3214yes—
189(C2×D3)⋊C4C22×C3214yes—
190C4⋊Dic3C22×C3211yescharacteristic
191C4⋊Dic3C22×C3212yes—
192C4⋊Dic3C22×C3212yes—
193Dic3⋊C4C22×C3214yes—
194C4×Dic3C22×C3214yes—
195C22×C32—24no—
196C22×C32—62no—
197C22×C32—62no—
198C22×C32—62no—
199C22×C32C23×D311yescharacteristic
200C22×C32C24×C314yes—
201C2×C3×D3—28no—
202C2×C3×D3—28no—
203C2×C3×D3—28no—
204C32×C4—216no—
205C32×C4—22no—
206C32×C4—22no—
207C32×C4—21no—
208C32×C4—28no—
209C32×C4—28no—
210C32×C4—28no—
211C32×C4—28no—
212C32×C4—24no—
213C32×C4—48no—
214C32×C4—44no—
215C32×C4—62no—
216C32×C4—64no—
217C32×C4#23512yes—
218C32×C4#23912yes—
219C32×C4C3×#4118yes—
220C32×C4C3×#4118yes—
221C3×Dic3—216no—
222C3×Dic3—28no—
223C4⋊Q8—31no—
224#126—34no—
225#124—32no—
226C22⋊Q8—32no—
227C4×Q8—32no—
228C4×Q8C32×D311yescharacteristic, complemented
229C4×D4—31no—
230#117—32no—
231C33#55611yescharacteristic, complemented, Sylow
232C23×C3—38no—
233C22×D3C23×C3212yes—
234C3×Q8—216no—
235C3×Q8—28no—
236C3×Q8—22no—
237C3×Q8—616no—
238C3×D4—68no—
239C2×C3×C4—24no—
240C2×C3×C4—28no—
241C2×C3×C4—24no—
242C2×C3×C4—28no—
243C2×C3×C4—24no—
244C2×C3×C4—38no—
245C2×C3×C4—316no—
246C2×C3×C4—616no—
247C2×C3×C4C22×C3×D318yes—
248C2×C3×C4C22×C3×D314yes—
249C2×C3×C4C22×C3×D318yes—
250C2×C3×C4C22×C3×D314yes—
251C2×C3×C4C22×C3×D314yes—
252C2×C3×C4C23×C3212yes—
253C2×C3×C4C23×C3212yes—
254C2×C3×C4C23×C3211yescharacteristic
255C2×C3×C4C23×C3211yescharacteristic
256C2×C3×C4C23×C3211yescharacteristic
257C2×Dic3C23×C3212yes—
258C2×Dic3C23×C3214yes—
259D12—22no—
260C4×D3—24no—
261Dic6—24no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32—24no—
265C2×C32—122no—
266C2×C32C2×#23511yescharacteristic
267C2×C32C2×#23911yescharacteristic
268C2×C32#101011yescharacteristic
269C2×C32C2×C3×#4114yes—
270C2×C32C2×C3×#4114yes—
271C2×C32C3×2−1+414yes—
272C3×D3—48no—
273C2×Q8—32no—
274C2×Q8C2×C32×D311yescharacteristic
275C2×D4—31no—
276C22×C4—32no—
277C4⋊C4—32no—
278C4⋊C4—31no—
279C4⋊C4—34no—
280C4⋊C4—32no—
281C4⋊C4C2×C32×D311yescharacteristic
282C4⋊C4C2×C32×D312yes—
283C22⋊C4—34no—
284C22⋊C4—32no—
285C22⋊C4—34no—
286C42—34no—
287C42C2×C32×D311yescharacteristic
288C42C2×C32×D312yes—
289C22×C3—24no—
290C22×C3—68no—
291C22×C3—68no—
292C22×C3—68no—
293C22×C3C23×C3×D314yes—
294C22×C3C24×C3211yescharacteristic
295C2×D3—22no—
296C2×D3—22no—
297C2×D3—22no—
298C3×C4—216no—
299C3×C4—28no—
300C3×C4—28no—
301C3×C4—24no—
302C3×C4—22no—
303C3×C4—22no—
304C3×C4—21no—
305C3×C4—28no—
306C3×C4—28no—
307C3×C4—48no—
308C3×C4—44no—
309C3×C4—68no—
310C3×C4—616no—
311C3×C4C3×#23518yes—
312C3×C4C3×#23918yes—
313C3×C4C32×#4112yes—
314C3×C4C32×#4112yes—
315Dic3—24no—
316Dic3—22no—
317C32—24no—
318C32#558411yescharacteristic, direct product factor
319C32C3×#55614yescomplemented
320C23—32no—
321Q8—22no—
322Q8—64no—
323D4—62no—
324C2×C4—32no—
325C2×C4—34no—
326C2×C4—64no—
327C2×C4C22×C32×D312yes—
328C2×C4C22×C32×D311yescharacteristic
329C2×C4C22×C32×D312yes—
330C2×C4C22×C32×D311yescharacteristic
331C2×C4C22×C32×D311yescharacteristic
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—24no—
335C2×C3—128no—
336C2×C3C2×C3×#23514yes—
337C2×C3C2×C3×#23914yes—
338C2×C3C3×#101014yes—
339C2×C3C2×C32×#4111yescharacteristic
340C2×C3C2×C32×#4111yescharacteristic
341C2×C3C32×2−1+411yescharacteristic
342D3—42no—
343C22—62no—
344C22—62no—
345C22—62no—
346C22C23×C32×D311yescharacteristic
347C4—22no—
348C4—22no—
349C4—21no—
350C4—62no—
351C4—64no—
352C4C32×#23512yes—
353C4C32×#23912yes—
354C3—24no—
355C3C3×#558414yesdirect product factor
356C3C32×#55611yescharacteristic, complemented
357C2—122no—
358C2C2×C32×#23511yescharacteristic
359C2C2×C32×#23911yescharacteristic
360C2C32×#101011yescharacteristic
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×#556
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×#898C2semidirect
4. C32×#892C2semidirect
7. C32×#869C2semidirect
8. C32×C4×Dic6C2semidirect
10. C3×#5584C3direct (N × H)
33. C32×C4×Q8D3semidirect
34. C32×C4×Q8C2×C3semidirect
37. C3×#946C2×C3semidirect
38. C3×#898C2×C3semidirect
39. C3×#892C2×C3semidirect
42. C3×#869C2×C3semidirect
43. C3×C4×Dic6C2×C3semidirect
57. #5584C32direct (N × H)
112. C3×C4×Q8C3×D3semidirect
113. C3×C4×Q8C2×C32semidirect
116. #946C2×C32semidirect
117. #898C2×C32semidirect
118. #892C2×C32semidirect
121. #869C2×C32semidirect
122. C4×Dic6C2×C32semidirect
228. C4×Q8C32×D3semidirect
231. C33#556semidirect
318. C32#5584direct (N × H)
319. C32C3×#556semidirect
355. C3C3×#5584direct (N × H)
356. C3C32×#556semidirect

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×#8982
C32×#8922
C32×#8752
C32×C4×D122
C32×#8692
C32×C4×Dic62
C32×#5563
C3×#55843