Finite Group Databasebeta

← C32×#5569 #13178946

Subgroups — C32×#5569 #13178946

Order 1728.

1480 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×Q8C211yesmaximal, characteristic
2C32×Q8×Dic3C211yesmaximal, characteristic
3C32×Q8×Dic3C211yesmaximal, characteristic
4C32×#890C213yesmaximal
5C32×#888C213yesmaximal
6C32×#870C213yesmaximal
7C32×C4×Dic6C213yesmaximal
8C32×#557—31nomaximal
9C3×#5569C314yesmaximal, direct product factor
10C2×C33×Q8C2211yescharacteristic
11C2×C32×Dic6C2213yes—
12C33×C4⋊C4C2213yes—
13C33×C42C2213yes—
14C32×C4⋊Dic3C2213yes—
15C32×C4⋊Dic3C2213yes—
16C32×C4⋊Dic3C2213yes—
17C32×Dic3⋊C4C2211yescharacteristic
18C32×Dic3⋊C4C2213yes—
19C32×Dic3⋊C4C2216yes—
20C32×C4×Dic3C2213yes—
21C32×C4×Dic3C2213yes—
22C32×C4⋊Q8—33no—
23C32×#125—33no—
24C32×#125—33no—
25C32×C4×Q8—24no—
26C32×C4×Q8—31no—
27C32×C4×Q8—31no—
28C32×C4×Q8—33no—
29C32×C4×Q8D311yescharacteristic
30C32×C4×Q8C2×C314yes—
31C3×Q8×Dic3C2×C314yes—
32C3×Q8×Dic3C2×C314yes—
33C3×#890C2×C3112yes—
34C3×#888C2×C3112yes—
35C3×#870C2×C3112yes—
36C3×C4×Dic6C2×C3112yes—
37C33×Q8Q814yescomplemented
38C2×C33×C4C2311yescharacteristic
39C2×C33×C4C2313yes—
40C2×C33×C4C2313yes—
41C2×C32×Dic3C2313yes—
42C2×C32×Dic3C2311yescharacteristic
43C2×C32×Dic3C2313yes—
44C2×C32×Dic3C2311yescharacteristic
45C32×Dic6—26no—
46C3×#557—34no—
47#5569C3211yescharacteristic, direct product factor
48C2×C32×Q8—24no—
49C2×C32×Q8—33no—
50C2×C32×Q8C2×D311yescharacteristic
51C2×C32×Q8C22×C314yes—
52C2×C3×Dic6C22×C3112yes—
53C32×C4⋊C4—212no—
54C32×C4⋊C4—31no—
55C32×C4⋊C4—33no—
56C32×C4⋊C4—33no—
57C32×C4⋊C4—33no—
58C32×C4⋊C4—33no—
59C32×C4⋊C4—36no—
60C32×C4⋊C4C2×D313yes—
61C32×C4⋊C4C22×C3112yes—
62C32×C42—212no—
63C32×C42—33no—
64C32×C42—33no—
65C32×C42C2×D313yes—
66C32×C42C22×C3112yes—
67C3×C4⋊Dic3C22×C3112yes—
68C3×C4⋊Dic3C22×C3112yes—
69C3×C4⋊Dic3C22×C3112yes—
70C3×Dic3⋊C4C22×C3124yes—
71C3×Dic3⋊C4C22×C314yes—
72C3×Dic3⋊C4C22×C3112yes—
73C3×C4×Dic3C22×C3112yes—
74C3×C4×Dic3C22×C3112yes—
75C22×C33C2411yescharacteristic
76C33×C4—23no—
77C33×C4—21no—
78C33×C4C2×Q816yes—
79C32×Dic3—23no—
80C32×Dic3—21no—
81C32×Dic3—23no—
82C32×Dic3#4112yes—
83C3×C4⋊Q8—312no—
84C3×#125—312no—
85C3×#125—312no—
86C3×C4×Q8—24no—
87C3×C4×Q8—34no—
88C3×C4×Q8—34no—
89C3×C4×Q8—312no—
90C3×C4×Q8C3×D314yes—
91C3×C4×Q8C2×C3211yescharacteristic
92Q8×Dic3C2×C3211yescharacteristic
93Q8×Dic3C2×C3211yescharacteristic
94#890C2×C3213yes—
95#888C2×C3213yes—
96#870C2×C3213yes—
97C4×Dic6C2×C3213yes—
98C32×Q8—216no—
99C32×Q8—66no—
100C32×Q8Dic614yescomplemented
101C32×Q8C3×Q8116yescomplemented
102C2×C32×C4—24no—
103C2×C32×C4—212no—
104C2×C32×C4—212no—
105C2×C32×C4—31no—
106C2×C32×C4—31no—
107C2×C32×C4—33no—
108C2×C32×C4—33no—
109C2×C32×C4C22×D311yescharacteristic
110C2×C32×C4C22×D313yes—
111C2×C32×C4C22×D313yes—
112C2×C32×C4C23×C314yes—
113C2×C32×C4C23×C3112yes—
114C2×C32×C4C23×C3112yes—
115C2×C3×Dic3C23×C314yes—
116C2×C3×Dic3C23×C3112yes—
117C2×C3×Dic3C23×C3112yes—
118C2×C3×Dic3C23×C314yes—
119C3×Dic6—224no—
120#557—31noSylow
121C2×C33C22×Q811yescharacteristic
122C2×C33C2×#4111yescharacteristic
123C2×C332−1+411yescharacteristic
124C2×C3×Q8—24no—
125C2×C3×Q8—312no—
126C2×C3×Q8C2×C3×D314yes—
127C2×C3×Q8C22×C3211yescharacteristic
128C2×Dic6C22×C3213yes—
129C3×C4⋊C4—212no—
130C3×C4⋊C4—34no—
131C3×C4⋊C4—312no—
132C3×C4⋊C4—324no—
133C3×C4⋊C4—312no—
134C3×C4⋊C4—312no—
135C3×C4⋊C4—312no—
136C3×C4⋊C4C2×C3×D3112yes—
137C3×C4⋊C4C22×C3213yes—
138C3×C42—212no—
139C3×C42—312no—
140C3×C42—312no—
141C3×C42C2×C3×D3112yes—
142C3×C42C22×C3213yes—
143C4⋊Dic3C22×C3213yes—
144C4⋊Dic3C22×C3213yes—
145C4⋊Dic3C22×C3213yes—
146Dic3⋊C4C22×C3211yescharacteristic
147Dic3⋊C4C22×C3213yes—
148Dic3⋊C4C22×C3216yes—
149C4×Dic3C22×C3213yes—
150C4×Dic3C22×C3213yes—
151C22×C32—24no—
152C22×C32C23×D311yescharacteristic
153C22×C32C24×C314yes—
154C32×C4—28no—
155C32×C4—21no—
156C32×C4—23no—
157C32×C4—24no—
158C32×C4—224no—
159C32×C4—212no—
160C32×C4—32no—
161C32×C4—412no—
162C32×C4—61no—
163C32×C4—63no—
164C32×C4—63no—
165C32×C4C2×Dic616yes—
166C32×C4C2×C3×Q8124yes—
167C3×Dic3—212no—
168C3×Dic3—24no—
169C3×Dic3—212no—
170C3×Dic3C3×#4118yes—
171C4⋊Q8—33no—
172#125—33no—
173#125—33no—
174C4×Q8—31no—
175C4×Q8—31no—
176C4×Q8—33no—
177C4×Q8C32×D311yescharacteristic
178C33#55711yescharacteristic, complemented, Sylow
179C3×Q8—216no—
180C3×Q8—624no—
181C3×Q8C3×Dic6116yescomplemented
182C3×Q8C32×Q814yescomplemented
183C2×C3×C4—24no—
184C2×C3×C4—212no—
185C2×C3×C4—212no—
186C2×C3×C4—34no—
187C2×C3×C4—34no—
188C2×C3×C4—312no—
189C2×C3×C4—312no—
190C2×C3×C4C22×C3×D314yes—
191C2×C3×C4C22×C3×D3112yes—
192C2×C3×C4C22×C3×D3112yes—
193C2×C3×C4C23×C3211yescharacteristic
194C2×C3×C4C23×C3213yes—
195C2×C3×C4C23×C3213yes—
196C2×Dic3C23×C3211yescharacteristic
197C2×Dic3C23×C3213yes—
198C2×Dic3C23×C3213yes—
199C2×Dic3C23×C3211yescharacteristic
200Dic6—26no—
201C2×C32—24no—
202C2×C32—24no—
203C2×C32—24no—
204C2×C32C22×Dic611yescharacteristic
205C2×C32#100711yescharacteristic
206C2×C32D3×#4111yescharacteristic
207C2×C32C22×C3×Q814yes—
208C2×C32C2×C3×#4114yes—
209C2×C32C3×2−1+414yes—
210C2×Q8—33no—
211C2×Q8C2×C32×D311yescharacteristic
212C4⋊C4—31no—
213C4⋊C4—33no—
214C4⋊C4—33no—
215C4⋊C4—33no—
216C4⋊C4—33no—
217C4⋊C4—36no—
218C4⋊C4C2×C32×D313yes—
219C42—33no—
220C42—33no—
221C42C2×C32×D313yes—
222C22×C3—24no—
223C22×C3C23×C3×D314yes—
224C22×C3C24×C3211yescharacteristic
225C3×C4—28no—
226C3×C4—24no—
227C3×C4—212no—
228C3×C4—23no—
229C3×C4—21no—
230C3×C4—224no—
231C3×C4—38no—
232C3×C4—412no—
233C3×C4—64no—
234C3×C4—612no—
235C3×C4—612no—
236C3×C4C2×C3×Dic6124yes—
237C3×C4C2×C32×Q816yes—
238Dic3—23no—
239Dic3—21no—
240Dic3—23no—
241Dic3C32×#4112yes—
242C32—24no—
243C32#556911yescharacteristic, direct product factor
244C32C3×#55714yescomplemented
245Q8—66no—
246Q8C32×Dic614yescomplemented
247C2×C4—31no—
248C2×C4—31no—
249C2×C4—33no—
250C2×C4—33no—
251C2×C4C22×C32×D311yescharacteristic
252C2×C4C22×C32×D313yes—
253C2×C4C22×C32×D313yes—
254C2×C3—24no—
255C2×C3—24no—
256C2×C3—24no—
257C2×C3C22×C3×Dic614yes—
258C2×C3C3×#100714yes—
259C2×C3C3×D3×#4114yes—
260C2×C3C22×C32×Q811yescharacteristic
261C2×C3C2×C32×#4111yescharacteristic
262C2×C3C32×2−1+411yescharacteristic
263C22C23×C32×D311yescharacteristic
264C4—21no—
265C4—23no—
266C4—32no—
267C4—61no—
268C4—63no—
269C4—63no—
270C4C2×C32×Dic616yes—
271C3—24no—
272C3C3×#556914yesdirect product factor
273C3C32×#55711yescharacteristic, complemented
274C2C22×C32×Dic611yescharacteristic
275C2C32×#100711yescharacteristic
276C2C32×D3×#4111yescharacteristic
subgroupG/H
centerC22×C32C23×D3
commutator, cosocleC22×C3C24×C32
FrattiniC22C23×C32×D3
FittingC33×C4×Q8C2
socle, prime radicalC22×C33C24
hypercenterC32×C4×Q8D3
hypocenterC3C32×#557
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C32×C4×Q8

Complements

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

NH ≅ G/Ntype
9. C3×#5569C3direct (N × H)
37. C33×Q8Q8semidirect
47. #5569C32direct (N × H)
100. C32×Q8Dic6semidirect
101. C32×Q8C3×Q8semidirect
178. C33#557semidirect
181. C3×Q8C3×Dic6semidirect
182. C3×Q8C32×Q8semidirect
243. C32#5569direct (N × H)
244. C32C3×#557semidirect
246. Q8C32×Dic6semidirect
272. C3C3×#5569direct (N × H)
273. C3C32×#557semidirect

Maximal subgroups

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

subgroupindex
C33×C4×Q82
C32×Q8×Dic32
C32×#8902
C32×#8882
C32×#8702
C32×C4×Dic62
C32×#5573
C3×#55693