Finite Group Databasebeta

← C32×#4981 #13162412

Subgroups — C32×#4981 #13162412

Order 1728.

1268 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×C4⋊C4C211yesmaximal, characteristic
2C2×C32×C4⋊Dic3C211yesmaximal, characteristic
3C2×C32×Dic3⋊C4C212yesmaximal
4C32×C3⋊#95C212yesmaximal
5C32×C3⋊#95C211yesmaximal, characteristic
6C32×#398—31nomaximal
7C3×#4981C314yesmaximal, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C33×C4C2211yescharacteristic
10C22×C33×C4C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2212yes—
13C33×C4⋊C4—22no—
14C32×C4⋊Dic3—22no—
15C32×Dic3⋊C4—24no—
16C2×C32×C4⋊C4—24no—
17C2×C32×C4⋊C4—32no—
18C2×C32×C4⋊C4—31no—
19C2×C32×C4⋊C4D311yescharacteristic
20C2×C32×C4⋊C4C2×C314yes—
21C2×C3×C4⋊Dic3C2×C314yes—
22C2×C3×Dic3⋊C4C2×C318yes—
23C32×#95—31no—
24C32×#95—32no—
25C3×C3⋊#95C2×C318yes—
26C3×C3⋊#95C2×C314yes—
27C23×C33C2311yescharacteristic
28C2×C33×C4—21no—
29C2×C33×C4—21no—
30C2×C33×C4—22no—
31C2×C33×C4—21no—
32C2×C33×C4—22no—
33C2×C33×C4D412yes—
34C2×C33×C4D412yes—
35C2×C32×Dic3—22no—
36C2×C32×Dic3—22no—
37C2×C32×Dic3—22no—
38C2×C32×Dic3—22no—
39C2×C32×Dic3—22no—
40C2×C32×Dic3Q814yes—
41C3×#398—34no—
42#4981C3211yescharacteristic, direct product factor
43C22×C32×C4—24no—
44C22×C32×C4—24no—
45C22×C32×C4—24no—
46C22×C32×C4—32no—
47C22×C32×C4—32no—
48C22×C32×C4C2×D311yescharacteristic
49C22×C32×C4C2×D311yescharacteristic
50C22×C32×C4C2×D311yescharacteristic
51C22×C32×C4C22×C314yes—
52C22×C32×C4C22×C314yes—
53C22×C32×C4C22×C314yes—
54C22×C3×Dic3C22×C318yes—
55C22×C3×Dic3C22×C318yes—
56C32×C4⋊C4—216no—
57C32×C4⋊C4—22no—
58C32×C4⋊C4—28no—
59C32×C4⋊C4—64no—
60C32×C4⋊C4—62no—
61C3×C4⋊Dic3—28no—
62C3×Dic3⋊C4—216no—
63C22×C33C2×D411yescharacteristic
64C22×C33C2×D411yescharacteristic
65C22×C33C2×Q812yes—
66C22×C33#4111yescharacteristic
67C22×C33#4112yes—
68C33×C4—22no—
69C33×C4—22no—
70C33×C4—41no—
71C32×Dic3—24no—
72C32×Dic3—42no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—38no—
75C2×C3×C4⋊C4—34no—
76C2×C3×C4⋊C4C3×D314yes—
77C2×C3×C4⋊C4C2×C3211yescharacteristic
78C2×C4⋊Dic3C2×C3211yescharacteristic
79C2×Dic3⋊C4C2×C3212yes—
80C3×#95—34no—
81C3×#95—38no—
82C3⋊#95C2×C3212yes—
83C3⋊#95C2×C3211yescharacteristic
84C23×C32—24no—
85C23×C32C22×D311yescharacteristic
86C23×C32C23×C314yes—
87C2×C32×C4—28no—
88C2×C32×C4—28no—
89C2×C32×C4—21no—
90C2×C32×C4—22no—
91C2×C32×C4—21no—
92C2×C32×C4—24no—
93C2×C32×C4—24no—
94C2×C32×C4—28no—
95C2×C32×C4—22no—
96C2×C32×C4—21no—
97C2×C32×C4—24no—
98C2×C32×C4—28no—
99C2×C32×C4—28no—
100C2×C32×C4—28no—
101C2×C32×C4—28no—
102C2×C32×C4—34no—
103C2×C32×C4—48no—
104C2×C32×C4—48no—
105C2×C32×C4—62no—
106C2×C32×C4—62no—
107C2×C32×C4—62no—
108C2×C32×C4—62no—
109C2×C32×C4—62no—
110C2×C32×C4D1212yes—
111C2×C32×C4#6712yes—
112C2×C32×C4C3×D418yes—
113C2×C32×C4C3×D418yes—
114C2×C3×Dic3—28no—
115C2×C3×Dic3—28no—
116C2×C3×Dic3—28no—
117C2×C3×Dic3—28no—
118C2×C3×Dic3—28no—
119C2×C3×Dic3C3×Q8116yes—
120#398—31noSylow
121C2×C33#12111yescharacteristic
122C2×C33C22⋊Q812yes—
123C2×C33#12311yescharacteristic
124C2×C33#12512yes—
125C2×C33C4⋊Q811yescharacteristic
126C22×C3×C4—24no—
127C22×C3×C4—24no—
128C22×C3×C4—24no—
129C22×C3×C4—38no—
130C22×C3×C4—38no—
131C22×C3×C4C2×C3×D314yes—
132C22×C3×C4C2×C3×D314yes—
133C22×C3×C4C2×C3×D314yes—
134C22×C3×C4C22×C3211yescharacteristic
135C22×C3×C4C22×C3211yescharacteristic
136C22×C3×C4C22×C3211yescharacteristic
137C22×Dic3C22×C3212yes—
138C22×Dic3C22×C3212yes—
139C3×C4⋊C4—216no—
140C3×C4⋊C4—28no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—616no—
143C3×C4⋊C4—68no—
144C4⋊Dic3—22no—
145Dic3⋊C4—24no—
146C22×C32—24no—
147C22×C32—24no—
148C22×C32—28no—
149C22×C32—28no—
150C22×C32—24no—
151C22×C32C2×D1211yescharacteristic
152C22×C32#23511yescharacteristic
153C22×C32#23712yes—
154C22×C32D3×Q812yes—
155C22×C32C2×#6711yescharacteristic
156C22×C32C2×C3×D414yes—
157C22×C32C2×C3×D414yes—
158C22×C32C2×C3×Q818yes—
159C22×C32C3×#4114yes—
160C22×C32C3×#4118yes—
161C32×C4—216no—
162C32×C4—22no—
163C32×C4—22no—
164C32×C4—28no—
165C32×C4—28no—
166C32×C4—48no—
167C32×C4—48no—
168C32×C4—41no—
169C32×C4—44no—
170C32×C4—64no—
171C32×C4—122no—
172C3×Dic3—216no—
173C3×Dic3—48no—
174C2×C4⋊C4—32no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4C32×D311yescharacteristic
177#95—31no—
178#95—32no—
179C33#39811yescharacteristic, complemented, Sylow
180C23×C3—24no—
181C23×C3C22×C3×D314yes—
182C23×C3C23×C3211yescharacteristic
183C2×C3×C4—28no—
184C2×C3×C4—28no—
185C2×C3×C4—24no—
186C2×C3×C4—28no—
187C2×C3×C4—24no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—22no—
191C2×C3×C4—28no—
192C2×C3×C4—28no—
193C2×C3×C4—24no—
194C2×C3×C4—21no—
195C2×C3×C4—22no—
196C2×C3×C4—28no—
197C2×C3×C4—28no—
198C2×C3×C4—316no—
199C2×C3×C4—48no—
200C2×C3×C4—48no—
201C2×C3×C4—68no—
202C2×C3×C4—68no—
203C2×C3×C4—68no—
204C2×C3×C4—68no—
205C2×C3×C4—68no—
206C2×C3×C4C3×D1218yes—
207C2×C3×C4C3×#6718yes—
208C2×C3×C4C32×D412yes—
209C2×C3×C4C32×D412yes—
210C2×Dic3—22no—
211C2×Dic3—22no—
212C2×Dic3—22no—
213C2×Dic3—22no—
214C2×Dic3—22no—
215C2×Dic3C32×Q814yes—
216C2×C32—28no—
217C2×C32—28no—
218C2×C32—24no—
219C2×C32—24no—
220C2×C32—24no—
221C2×C32#88912yes—
222C2×C32#89712yes—
223C2×C32#93011yescharacteristic
224C2×C32#93511yescharacteristic
225C2×C32#94411yescharacteristic
226C2×C32C3×#12114yes—
227C2×C32C3×C22⋊Q818yes—
228C2×C32C3×#12314yes—
229C2×C32C3×#12518yes—
230C2×C32C3×C4⋊Q814yes—
231C22×C4—32no—
232C22×C4—32no—
233C22×C4C2×C32×D311yescharacteristic
234C22×C4C2×C32×D311yescharacteristic
235C22×C4C2×C32×D311yescharacteristic
236C4⋊C4—22no—
237C4⋊C4—64no—
238C4⋊C4—62no—
239C22×C3—24no—
240C22×C3—24no—
241C22×C3—28no—
242C22×C3—28no—
243C22×C3—24no—
244C22×C3C2×C3×D1214yes—
245C22×C3C3×#23514yes—
246C22×C3C3×#23718yes—
247C22×C3C3×D3×Q818yes—
248C22×C3C2×C3×#6714yes—
249C22×C3C2×C32×D411yescharacteristic
250C22×C3C2×C32×D411yescharacteristic
251C22×C3C2×C32×Q812yes—
252C22×C3C32×#4111yescharacteristic
253C22×C3C32×#4112yes—
254C3×C4—216no—
255C3×C4—28no—
256C3×C4—28no—
257C3×C4—22no—
258C3×C4—22no—
259C3×C4—48no—
260C3×C4—48no—
261C3×C4—44no—
262C3×C4—41no—
263C3×C4—616no—
264C3×C4—128no—
265Dic3—24no—
266Dic3—42no—
267C32—24no—
268C32#498111yescharacteristic, direct product factor
269C32C3×#39814yescomplemented
270C23C22×C32×D311yescharacteristic
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—34no—
277C2×C4—62no—
278C2×C4—62no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4C32×D1212yes—
283C2×C4C32×#6712yes—
284C2×C3—24no—
285C2×C3—24no—
286C2×C3—24no—
287C2×C3—28no—
288C2×C3—28no—
289C2×C3C3×#88918yes—
290C2×C3C3×#89718yes—
291C2×C3C3×#93014yes—
292C2×C3C3×#93514yes—
293C2×C3C3×#94414yes—
294C2×C3C32×#12111yescharacteristic
295C2×C3C32×C22⋊Q812yes—
296C2×C3C32×#12311yescharacteristic
297C2×C3C32×#12512yes—
298C2×C3C32×C4⋊Q811yescharacteristic
299C22C2×C32×D1211yescharacteristic
300C22C32×#23511yescharacteristic
301C22C32×#23712yes—
302C22C32×D3×Q812yes—
303C22C2×C32×#6711yescharacteristic
304C4—22no—
305C4—22no—
306C4—41no—
307C4—64no—
308C4—122no—
309C3—24no—
310C3C3×#498114yesdirect product factor
311C3C32×#39811yescharacteristic, complemented
312C2C32×#88912yes—
313C2C32×#89712yes—
314C2C32×#93011yescharacteristic
315C2C32×#93511yescharacteristic
316C2C32×#94411yescharacteristic
subgroupG/H
centerC23×C32C22×D3
commutator, cosocleC23×C3C23×C32
FrattiniC23C22×C32×D3
FittingC2×C33×C4⋊C4C2
socle, prime radicalC23×C33C23
hypercenterC2×C32×C4⋊C4D3
hypocenterC3C32×#398
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23×C32 ◁ C2×C32×C4⋊C4

Complements

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

NH ≅ G/Ntype
7. C3×#4981C3direct (N × H)
42. #4981C32direct (N × H)
179. C33#398semidirect
268. C32#4981direct (N × H)
269. C32C3×#398semidirect
310. C3C3×#4981direct (N × H)
311. C3C32×#398semidirect

Maximal subgroups

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

subgroupindex
C2×C33×C4⋊C42
C2×C32×C4⋊Dic32
C2×C32×Dic3⋊C42
C32×C3⋊#952
C32×#3983
C3×#49813