Finite Group Databasebeta

← C32×#4979 #13162410

Subgroups — C32×#4979 #13162410

Order 1728.

1460 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×Dic6C211yesmaximal, characteristic
2C2×C33×C4⋊C4C211yesmaximal, characteristic
3C2×C32×Dic3⋊C4C212yesmaximal
4C32×C3⋊#95C212yesmaximal
5C32×C3⋊#95C211yesmaximal, characteristic
6C32×#395—31nomaximal
7C3×#4979C314yesmaximal, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C33×C4C2211yescharacteristic
10C22×C33×C4C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2212yes—
13C2×C32×Dic6—22no—
14C2×C32×Dic6—24no—
15C33×C4⋊C4—22no—
16C32×Dic3⋊C4—24no—
17C22×C32×Q8—31no—
18C22×C3×Dic6C2×C314yes—
19C2×C32×C4⋊C4—24no—
20C2×C32×C4⋊C4—32no—
21C2×C32×C4⋊C4D311yescharacteristic
22C2×C32×C4⋊C4C2×C314yes—
23C2×C3×Dic3⋊C4C2×C318yes—
24C32×#95—32no—
25C32×#95—31no—
26C3×C3⋊#95C2×C318yes—
27C3×C3⋊#95C2×C314yes—
28C23×C33C2311yescharacteristic
29C2×C33×C4—21no—
30C2×C33×C4—22no—
31C2×C33×C4—21no—
32C2×C33×C4—21no—
33C2×C33×C4—22no—
34C2×C33×C4D412yes—
35C2×C33×C4Q812yescomplemented
36C2×C32×Dic3—22no—
37C2×C32×Dic3—22no—
38C2×C32×Dic3—22no—
39C2×C32×Dic3—22no—
40C2×C32×Dic3D414yes—
41C2×C32×Dic3Q814yescomplemented
42C32×Dic6—28no—
43C3×#395—34no—
44#4979C3211yescharacteristic, direct product factor
45C2×C32×Q8—64no—
46C2×C32×Q8—62no—
47C22×C32×C4—24no—
48C22×C32×C4—24no—
49C22×C32×C4—24no—
50C22×C32×C4—32no—
51C22×C32×C4—32no—
52C22×C32×C4C2×D311yescharacteristic
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C22×C314yes—
56C22×C32×C4C22×C314yes—
57C22×C32×C4C22×C314yes—
58C22×C3×Dic3C22×C318yes—
59C22×C3×Dic3C22×C318yes—
60C2×C3×Dic6—28no—
61C2×C3×Dic6—216no—
62C32×C4⋊C4—216no—
63C32×C4⋊C4—22no—
64C32×C4⋊C4—28no—
65C32×C4⋊C4—64no—
66C3×Dic3⋊C4—216no—
67C22×C33C2×D412yes—
68C22×C33C2×D411yescharacteristic
69C22×C33C2×Q811yescharacteristic
70C22×C33C2×Q812yes—
71C22×C33#4111yescharacteristic
72C33×C4—22no—
73C33×C4—22no—
74C33×C4—41no—
75C32×Dic3—24no—
76C32×Dic3—24no—
77C22×C3×Q8—34no—
78C22×Dic6C2×C3211yescharacteristic
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4—38no—
81C2×C3×C4⋊C4C3×D314yes—
82C2×C3×C4⋊C4C2×C3211yescharacteristic
83C2×Dic3⋊C4C2×C3212yes—
84C3×#95—34no—
85C3×#95—38no—
86C3⋊#95C2×C3212yes—
87C3⋊#95C2×C3211yescharacteristic
88C23×C32—24no—
89C23×C32C22×D311yescharacteristic
90C23×C32C23×C314yes—
91C32×Q8—68no—
92C2×C32×C4—28no—
93C2×C32×C4—28no—
94C2×C32×C4—21no—
95C2×C32×C4—22no—
96C2×C32×C4—21no—
97C2×C32×C4—24no—
98C2×C32×C4—24no—
99C2×C32×C4—28no—
100C2×C32×C4—22no—
101C2×C32×C4—21no—
102C2×C32×C4—24no—
103C2×C32×C4—28no—
104C2×C32×C4—28no—
105C2×C32×C4—28no—
106C2×C32×C4—28no—
107C2×C32×C4—34no—
108C2×C32×C4—34no—
109C2×C32×C4—48no—
110C2×C32×C4—48no—
111C2×C32×C4—62no—
112C2×C32×C4—62no—
113C2×C32×C4—62no—
114C2×C32×C4—62no—
115C2×C32×C4Dic612yescomplemented
116C2×C32×C4#6712yes—
117C2×C32×C4C3×D418yes—
118C2×C32×C4C3×Q818yescomplemented
119C2×C3×Dic3—28no—
120C2×C3×Dic3—28no—
121C2×C3×Dic3—28no—
122C2×C3×Dic3—28no—
123C2×C3×Dic3C3×D4116yes—
124C2×C3×Dic3C3×Q8116yescomplemented
125C3×Dic6—232no—
126#395—31noSylow
127C2×C33#12011yescharacteristic
128C2×C33C22⋊Q812yes—
129C2×C33C22⋊Q811yescharacteristic
130C2×C33C4⋊Q812yes—
131C2×C33C4⋊Q811yescharacteristic
132C2×C3×Q8—616no—
133C2×C3×Q8—68no—
134C22×C3×C4—24no—
135C22×C3×C4—24no—
136C22×C3×C4—24no—
137C22×C3×C4—38no—
138C22×C3×C4—38no—
139C22×C3×C4C2×C3×D314yes—
140C22×C3×C4C2×C3×D314yes—
141C22×C3×C4C2×C3×D314yes—
142C22×C3×C4C22×C3211yescharacteristic
143C22×C3×C4C22×C3211yescharacteristic
144C22×C3×C4C22×C3211yescharacteristic
145C22×Dic3C22×C3212yes—
146C22×Dic3C22×C3212yes—
147C2×Dic6—22no—
148C2×Dic6—24no—
149C3×C4⋊C4—216no—
150C3×C4⋊C4—28no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—616no—
153Dic3⋊C4—24no—
154C22×C32—24no—
155C22×C32—24no—
156C22×C32—28no—
157C22×C32—28no—
158C22×C32—24no—
159C22×C32C2×Dic611yescharacteristic
160C22×C32#23511yescharacteristic
161C22×C32D3×D412yes—
162C22×C32D3×Q812yes—
163C22×C32C2×#6711yescharacteristic
164C22×C32C2×C3×D418yes—
165C22×C32C2×C3×D414yes—
166C22×C32C2×C3×Q814yes—
167C22×C32C2×C3×Q818yes—
168C22×C32C3×#4114yes—
169C32×C4—216no—
170C32×C4—22no—
171C32×C4—22no—
172C32×C4—28no—
173C32×C4—28no—
174C32×C4—48no—
175C32×C4—48no—
176C32×C4—41no—
177C32×C4—44no—
178C32×C4—64no—
179C32×C4—64no—
180C3×Dic3—216no—
181C3×Dic3—216no—
182C22×Q8—31no—
183C2×C4⋊C4—32no—
184C2×C4⋊C4C32×D311yescharacteristic
185#95—31no—
186#95—32no—
187C33#39511yescharacteristic, complemented, Sylow
188C23×C3—24no—
189C23×C3C22×C3×D314yes—
190C23×C3C23×C3211yescharacteristic
191C3×Q8—632no—
192C2×C3×C4—28no—
193C2×C3×C4—28no—
194C2×C3×C4—24no—
195C2×C3×C4—28no—
196C2×C3×C4—24no—
197C2×C3×C4—21no—
198C2×C3×C4—22no—
199C2×C3×C4—21no—
200C2×C3×C4—28no—
201C2×C3×C4—28no—
202C2×C3×C4—24no—
203C2×C3×C4—22no—
204C2×C3×C4—21no—
205C2×C3×C4—28no—
206C2×C3×C4—28no—
207C2×C3×C4—316no—
208C2×C3×C4—316no—
209C2×C3×C4—48no—
210C2×C3×C4—48no—
211C2×C3×C4—68no—
212C2×C3×C4—68no—
213C2×C3×C4—68no—
214C2×C3×C4—68no—
215C2×C3×C4C3×Dic618yescomplemented
216C2×C3×C4C3×#6718yes—
217C2×C3×C4C32×D412yes—
218C2×C3×C4C32×Q812yescomplemented
219C2×Dic3—22no—
220C2×Dic3—22no—
221C2×Dic3—22no—
222C2×Dic3—22no—
223C2×Dic3C32×D414yes—
224C2×Dic3C32×Q814yescomplemented
225Dic6—28no—
226C2×C32—28no—
227C2×C32—28no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32#88812yes—
232C2×C32#89612yes—
233C2×C32#92411yescharacteristic
234C2×C32#93711yescharacteristic
235C2×C32#94411yescharacteristic
236C2×C32C3×#12014yes—
237C2×C32C3×C22⋊Q818yes—
238C2×C32C3×C22⋊Q814yes—
239C2×C32C3×C4⋊Q818yes—
240C2×C32C3×C4⋊Q814yes—
241C2×Q8—64no—
242C2×Q8—62no—
243C22×C4—32no—
244C22×C4—32no—
245C22×C4C2×C32×D311yescharacteristic
246C22×C4C2×C32×D311yescharacteristic
247C22×C4C2×C32×D311yescharacteristic
248C4⋊C4—22no—
249C4⋊C4—64no—
250C22×C3—24no—
251C22×C3—24no—
252C22×C3—28no—
253C22×C3—28no—
254C22×C3—24no—
255C22×C3C2×C3×Dic614yes—
256C22×C3C3×#23514yes—
257C22×C3C3×D3×D418yes—
258C22×C3C3×D3×Q818yes—
259C22×C3C2×C3×#6714yes—
260C22×C3C2×C32×D412yes—
261C22×C3C2×C32×D411yescharacteristic
262C22×C3C2×C32×Q811yescharacteristic
263C22×C3C2×C32×Q812yes—
264C22×C3C32×#4111yescharacteristic
265C3×C4—216no—
266C3×C4—28no—
267C3×C4—28no—
268C3×C4—22no—
269C3×C4—22no—
270C3×C4—48no—
271C3×C4—48no—
272C3×C4—44no—
273C3×C4—41no—
274C3×C4—616no—
275C3×C4—616no—
276Dic3—24no—
277Dic3—24no—
278C32—24no—
279C32#497911yescharacteristic, direct product factor
280C32C3×#39514yescomplemented
281C23C22×C32×D311yescharacteristic
282Q8—68no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—34no—
289C2×C4—34no—
290C2×C4—62no—
291C2×C4—62no—
292C2×C4—62no—
293C2×C4—62no—
294C2×C4C32×Dic612yescomplemented
295C2×C4C32×#6712yes—
296C2×C3—24no—
297C2×C3—24no—
298C2×C3—24no—
299C2×C3—28no—
300C2×C3—28no—
301C2×C3C3×#88818yes—
302C2×C3C3×#89618yes—
303C2×C3C3×#92414yes—
304C2×C3C3×#93714yes—
305C2×C3C3×#94414yes—
306C2×C3C32×#12011yescharacteristic
307C2×C3C32×C22⋊Q811yescharacteristic
308C2×C3C32×C22⋊Q812yes—
309C2×C3C32×C4⋊Q811yescharacteristic
310C2×C3C32×C4⋊Q812yes—
311C22C2×C32×Dic611yescharacteristic
312C22C32×#23511yescharacteristic
313C22C32×D3×D412yes—
314C22C32×D3×Q812yes—
315C22C2×C32×#6711yescharacteristic
316C4—22no—
317C4—22no—
318C4—41no—
319C4—64no—
320C4—64no—
321C3—24no—
322C3C3×#497914yesdirect product factor
323C3C32×#39511yescharacteristic, complemented
324C2C32×#88812yes—
325C2C32×#89612yes—
326C2C32×#92411yescharacteristic
327C2C32×#93711yescharacteristic
328C2C32×#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×#395
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×#4979C3direct (N × H)
35. C2×C33×C4Q8semidirect
41. C2×C32×Dic3Q8semidirect
44. #4979C32direct (N × H)
115. C2×C32×C4Dic6semidirect
118. C2×C32×C4C3×Q8semidirect
124. C2×C3×Dic3C3×Q8semidirect
187. C33#395semidirect
215. C2×C3×C4C3×Dic6semidirect
218. C2×C3×C4C32×Q8semidirect
224. C2×Dic3C32×Q8semidirect
279. C32#4979direct (N × H)
280. C32C3×#395semidirect
294. C2×C4C32×Dic6semidirect
322. C3C3×#4979direct (N × H)
323. C3C32×#395semidirect

Maximal subgroups

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

subgroupindex
C22×C32×Dic62
C2×C33×C4⋊C42
C2×C32×Dic3⋊C42
C32×C3⋊#952
C32×#3953
C3×#49793