Finite Group Databasebeta

← C32×#4983 #13162414

Subgroups — C32×#4983 #13162414

Order 1728.

1268 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×C4⋊C4C211yesmaximal, characteristic
2C2×C32×Dic3⋊C4C211yesmaximal, characteristic
3C2×C32×Dic3⋊C4C212yesmaximal
4C32×C3⋊#95C212yesmaximal
5C32×C3⋊#95C211yesmaximal, characteristic
6C32×#398—31nomaximal
7C3×#4983C314yesmaximal, direct product factor
8C22×C33×C4C2212yes—
9C22×C33×C4C2211yescharacteristic
10C22×C32×Dic3C2211yescharacteristic
11C22×C32×Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C33×C4⋊C4—22no—
14C32×Dic3⋊C4—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×Dic3⋊C4C2×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—22no—
29C2×C33×C4—22no—
30C2×C33×C4—22no—
31C2×C33×C4—22no—
32C2×C33×C4D412yes—
33C2×C32×Dic3—21no—
34C2×C32×Dic3—22no—
35C2×C32×Dic3—21no—
36C2×C32×Dic3—22no—
37C2×C32×Dic3—21no—
38C2×C32×Dic3—22no—
39C2×C32×Dic3D412yes—
40C2×C32×Dic3Q814yes—
41C3×#398—34no—
42#4983C3211yescharacteristic, direct product factor
43C22×C32×C4—24no—
44C22×C32×C4—28no—
45C22×C32×C4—31no—
46C22×C32×C4—32no—
47C22×C32×C4—31no—
48C22×C32×C4C2×D312yes—
49C22×C32×C4C2×D311yescharacteristic
50C22×C32×C4C22×C318yes—
51C22×C32×C4C22×C314yes—
52C22×C3×Dic3C22×C314yes—
53C22×C3×Dic3C22×C314yes—
54C22×C3×Dic3C22×C318yes—
55C32×C4⋊C4—216no—
56C32×C4⋊C4—22no—
57C32×C4⋊C4—28no—
58C32×C4⋊C4—64no—
59C32×C4⋊C4—62no—
60C3×Dic3⋊C4—28no—
61C3×Dic3⋊C4—216no—
62C22×C33C2×D411yescharacteristic
63C22×C33C2×D411yescharacteristic
64C22×C33C2×Q812yes—
65C22×C33#4111yescharacteristic
66C22×C33#4112yes—
67C33×C4—22no—
68C33×C4—42no—
69C32×Dic3—22no—
70C32×Dic3—24no—
71C32×Dic3—41no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4—38no—
74C2×C3×C4⋊C4—34no—
75C2×C3×C4⋊C4C3×D314yes—
76C2×C3×C4⋊C4C2×C3211yescharacteristic
77C2×Dic3⋊C4C2×C3211yescharacteristic
78C2×Dic3⋊C4C2×C3212yes—
79C3×#95—34no—
80C3×#95—38no—
81C3⋊#95C2×C3212yes—
82C3⋊#95C2×C3211yescharacteristic
83C23×C32—24no—
84C23×C32C22×D311yescharacteristic
85C23×C32C23×C314yes—
86C2×C32×C4—216no—
87C2×C32×C4—22no—
88C2×C32×C4—28no—
89C2×C32×C4—216no—
90C2×C32×C4—22no—
91C2×C32×C4—22no—
92C2×C32×C4—22no—
93C2×C32×C4—28no—
94C2×C32×C4—28no—
95C2×C32×C4—28no—
96C2×C32×C4—28no—
97C2×C32×C4—32no—
98C2×C32×C4—34no—
99C2×C32×C4—48no—
100C2×C32×C4—48no—
101C2×C32×C4—62no—
102C2×C32×C4—62no—
103C2×C32×C4—61no—
104C2×C32×C4—61no—
105C2×C32×C4—62no—
106C2×C32×C4—61no—
107C2×C32×C4#6712yes—
108C2×C32×C4C3×D418yes—
109C2×C3×Dic3—28no—
110C2×C3×Dic3—24no—
111C2×C3×Dic3—28no—
112C2×C3×Dic3—24no—
113C2×C3×Dic3—24no—
114C2×C3×Dic3—28no—
115C2×C3×Dic3C3×D418yes—
116C2×C3×Dic3C3×Q8116yes—
117#398—31noSylow
118C2×C33#12111yescharacteristic
119C2×C33C22⋊Q812yes—
120C2×C33#12311yescharacteristic
121C2×C33#12512yes—
122C2×C33C4⋊Q811yescharacteristic
123C22×C3×C4—24no—
124C22×C3×C4—28no—
125C22×C3×C4—38no—
126C22×C3×C4—34no—
127C22×C3×C4—34no—
128C22×C3×C4C2×C3×D318yes—
129C22×C3×C4C2×C3×D314yes—
130C22×C3×C4C22×C3212yes—
131C22×C3×C4C22×C3211yescharacteristic
132C22×Dic3C22×C3211yescharacteristic
133C22×Dic3C22×C3211yescharacteristic
134C22×Dic3C22×C3212yes—
135C3×C4⋊C4—216no—
136C3×C4⋊C4—28no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—616no—
139C3×C4⋊C4—68no—
140Dic3⋊C4—22no—
141Dic3⋊C4—24no—
142C22×C32—28no—
143C22×C32—24no—
144C22×C32—24no—
145C22×C32—28no—
146C22×C32—24no—
147C22×C32#23512yes—
148C22×C32D3×D411yescharacteristic
149C22×C32#23711yescharacteristic
150C22×C32D3×Q812yes—
151C22×C32C2×#6711yescharacteristic
152C22×C32C2×C3×D414yes—
153C22×C32C2×C3×D414yes—
154C22×C32C2×C3×Q818yes—
155C22×C32C3×#4118yes—
156C22×C32C3×#4114yes—
157C32×C4—216no—
158C32×C4—22no—
159C32×C4—28no—
160C32×C4—416no—
161C32×C4—42no—
162C32×C4—48no—
163C32×C4—64no—
164C32×C4—62no—
165C32×C4—121no—
166C3×Dic3—28no—
167C3×Dic3—216no—
168C3×Dic3—44no—
169C2×C4⋊C4—32no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4C32×D311yescharacteristic
172#95—31no—
173#95—32no—
174C33#39811yescharacteristic, complemented, Sylow
175C23×C3—24no—
176C23×C3C22×C3×D314yes—
177C23×C3C23×C3211yescharacteristic
178C2×C3×C4—216no—
179C2×C3×C4—28no—
180C2×C3×C4—22no—
181C2×C3×C4—216no—
182C2×C3×C4—28no—
183C2×C3×C4—28no—
184C2×C3×C4—28no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—28no—
189C2×C3×C4—38no—
190C2×C3×C4—316no—
191C2×C3×C4—48no—
192C2×C3×C4—48no—
193C2×C3×C4—68no—
194C2×C3×C4—68no—
195C2×C3×C4—64no—
196C2×C3×C4—64no—
197C2×C3×C4—68no—
198C2×C3×C4—64no—
199C2×C3×C4C3×#6718yes—
200C2×C3×C4C32×D412yes—
201C2×Dic3—22no—
202C2×Dic3—21no—
203C2×Dic3—22no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—22no—
207C2×Dic3C32×D412yes—
208C2×Dic3C32×Q814yes—
209C2×C32—28no—
210C2×C32—28no—
211C2×C32—24no—
212C2×C32—24no—
213C2×C32—24no—
214C2×C32#88912yes—
215C2×C32#89612yes—
216C2×C32#92911yescharacteristic
217C2×C32#93811yescharacteristic
218C2×C32#94411yescharacteristic
219C2×C32C3×#12114yes—
220C2×C32C3×C22⋊Q818yes—
221C2×C32C3×#12314yes—
222C2×C32C3×#12518yes—
223C2×C32C3×C4⋊Q814yes—
224C22×C4—32no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4C2×C32×D312yes—
228C22×C4C2×C32×D311yescharacteristic
229C4⋊C4—22no—
230C4⋊C4—64no—
231C4⋊C4—62no—
232C22×C3—28no—
233C22×C3—24no—
234C22×C3—24no—
235C22×C3—28no—
236C22×C3—24no—
237C22×C3C3×#23518yes—
238C22×C3C3×D3×D414yes—
239C22×C3C3×#23714yes—
240C22×C3C3×D3×Q818yes—
241C22×C3C2×C3×#6714yes—
242C22×C3C2×C32×D411yescharacteristic
243C22×C3C2×C32×D411yescharacteristic
244C22×C3C2×C32×Q812yes—
245C22×C3C32×#4112yes—
246C22×C3C32×#4111yescharacteristic
247C3×C4—216no—
248C3×C4—28no—
249C3×C4—22no—
250C3×C4—416no—
251C3×C4—48no—
252C3×C4—42no—
253C3×C4—616no—
254C3×C4—68no—
255C3×C4—124no—
256Dic3—22no—
257Dic3—24no—
258Dic3—41no—
259C32—24no—
260C32#498311yescharacteristic, direct product factor
261C32C3×#39814yescomplemented
262C23C22×C32×D311yescharacteristic
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—32no—
268C2×C4—34no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—62no—
274C2×C4—61no—
275C2×C4C32×#6712yes—
276C2×C3—24no—
277C2×C3—24no—
278C2×C3—24no—
279C2×C3—28no—
280C2×C3—28no—
281C2×C3C3×#88918yes—
282C2×C3C3×#89618yes—
283C2×C3C3×#92914yes—
284C2×C3C3×#93814yes—
285C2×C3C3×#94414yes—
286C2×C3C32×#12111yescharacteristic
287C2×C3C32×C22⋊Q812yes—
288C2×C3C32×#12311yescharacteristic
289C2×C3C32×#12512yes—
290C2×C3C32×C4⋊Q811yescharacteristic
291C22C32×#23512yes—
292C22C32×D3×D411yescharacteristic
293C22C32×#23711yescharacteristic
294C22C32×D3×Q812yes—
295C22C2×C32×#6711yescharacteristic
296C4—22no—
297C4—42no—
298C4—62no—
299C4—64no—
300C4—121no—
301C3—24no—
302C3C3×#498314yesdirect product factor
303C3C32×#39811yescharacteristic, complemented
304C2C32×#88912yes—
305C2C32×#89612yes—
306C2C32×#92911yescharacteristic
307C2C32×#93811yescharacteristic
308C2C32×#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×#4983C3direct (N × H)
42. #4983C32direct (N × H)
174. C33#398semidirect
260. C32#4983direct (N × H)
261. C32C3×#398semidirect
302. C3C3×#4983direct (N × H)
303. 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×Dic3⋊C42
C32×C3⋊#952
C32×#3983
C3×#49833