Finite Group Databasebeta

← C32×#5432 #13172986

Subgroups — C32×#5432 #13172986

Order 1728.

1294 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1294 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C32×A4C211yesmaximal, characteristic
2C2×C32×SL(2,ℤ/4)C212yesmaximal, complemented
3C3×#5432C314yesmaximal, direct product factor
4C32×#379—31nomaximal
5C22×C32×A4—22no—
6C22×C32×A4C412yescomplemented
7C22×C32×A4C2211yescharacteristic
8C32×SL(2,ℤ/4)—22no—
9C32×#217—41nomaximal
10C25×C32D311yescharacteristic
11C23×C3×A4—24no—
12C23×C3×A4C2×C314yes—
13C2×C3×SL(2,ℤ/4)C2×C318yescomplemented
14C2×C32×C22⋊C4—32no—
15C2×C32×C22⋊C4—31no—
16C2×C32×C22⋊C4—31no—
17C2×C32×C22⋊C4—32no—
18C23×C33—41no—
19C2×C32×A4—22no—
20C2×C32×A4C2×C411yescharacteristic
21C2×C32×A4D412yes—
22C2×C32×Dic3—42no—
23#5432C3211yescharacteristic, direct product factor
24C3×#379—34no—
25C24×C32—22no—
26C24×C32—31no—
27C24×C32—31no—
28C24×C32—32no—
29C24×C32—62no—
30C24×C32Dic312yescomplemented
31C24×C32C2×D311yescharacteristic
32C22×C3×A4—216no—
33C22×C3×A4—28no—
34C22×C3×A4—24no—
35C22×C3×A4—28no—
36C22×C3×A4C3×C418yescomplemented
37C22×C3×A4C22×C314yes—
38C22×C32×C4—32no—
39C22×C32×C4—32no—
40C3×SL(2,ℤ/4)—28no—
41C32×C22⋊C4—62no—
42C32×C22⋊C4—64no—
43C32×C22⋊C4—62no—
44C32×C22⋊C4—61no—
45C32×C22⋊C4—62no—
46C32×C22⋊C4—61no—
47C3×#217—44no—
48C22×C33—41no—
49C22×C33—42no—
50C22×C33—82no—
51C32×A4C22⋊C411yescharacteristic, complemented
52C32×Dic3—82no—
53C25×C3C3×D314yes—
54C23×A4—24no—
55C23×A4C2×C3211yescharacteristic
56C2×SL(2,ℤ/4)C2×C3212yescomplemented
57C2×C3×C22⋊C4—38no—
58C2×C3×C22⋊C4—34no—
59C2×C3×C22⋊C4—34no—
60C2×C3×C22⋊C4—38no—
61C23×C32—22no—
62C23×C32—31no—
63C23×C32—32no—
64C23×C32—31no—
65C23×C32—32no—
66C23×C32—34no—
67C23×C32—32no—
68C23×C32—32no—
69C23×C32—31no—
70C23×C32—44no—
71C23×C32—62no—
72C23×C32—62no—
73C23×C32—62no—
74C23×C32—61no—
75C23×C32—64no—
76C23×C32—62no—
77C23×C32—64no—
78C23×C32—84no—
79C23×C32C2×Dic311yescharacteristic
80C23×C32#6712yes—
81C23×C32S411yescharacteristic
82C2×C3×A4—216no—
83C2×C3×A4—28no—
84C2×C3×A4—24no—
85C2×C3×A4—28no—
86C2×C3×A4C2×C3×C414yes—
87C2×C3×A4C3×D418yes—
88C2×C32×C4—62no—
89C2×C32×C4—62no—
90C2×C32×C4—62no—
91C2×C32×C4—62no—
92C2×C32×C4—62no—
93C2×C32×C4—62no—
94C2×C3×Dic3—48no—
95#379—31noSylow
96C2×C33—41no—
97C2×C33—42no—
98C2×C33—82no—
99C24×C3—28no—
100C24×C3—34no—
101C24×C3—34no—
102C24×C3—38no—
103C24×C3—68no—
104C24×C3C3×Dic318yescomplemented
105C24×C3C2×C3×D314yes—
106C22×A4—216no—
107C22×A4—22no—
108C22×A4—28no—
109C22×A4—24no—
110C22×A4C32×C412yescomplemented
111C22×A4C22×C3211yescharacteristic
112C22×C3×C4—38no—
113C22×C3×C4—38no—
114SL(2,ℤ/4)—22no—
115C3×C22⋊C4—68no—
116C3×C22⋊C4—616no—
117C3×C22⋊C4—68no—
118C3×C22⋊C4—64no—
119C3×C22⋊C4—68no—
120C3×C22⋊C4—64no—
121#217—41no—
122C22×C32—22no—
123C22×C32—31no—
124C22×C32—32no—
125C22×C32—31no—
126C22×C32—31no—
127C22×C32—32no—
128C22×C32—32no—
129C22×C32—34no—
130C22×C32—32no—
131C22×C32—44no—
132C22×C32—48no—
133C22×C32—62no—
134C22×C32—64no—
135C22×C32—62no—
136C22×C32—61no—
137C22×C32—62no—
138C22×C32—64no—
139C22×C32—62no—
140C22×C32—88no—
141C22×C32—816no—
142C22×C32—84no—
143C22×C32—88no—
144C22×C32#21711yescharacteristic, complemented
145C22×C32SL(2,ℤ/4)12yescomplemented
146C22×C32C2×S411yescharacteristic
147C3×A4—24no—
148C3×A4C3×C22⋊C414yescomplemented
149C32×C4—122no—
150C32×C4—122no—
151C3×Dic3—88no—
152C25C32×D311yescharacteristic
153C2×C22⋊C4—32no—
154C2×C22⋊C4—31no—
155C2×C22⋊C4—31no—
156C2×C22⋊C4—32no—
157C33—41noSylow
158C23×C3—28no—
159C23×C3—34no—
160C23×C3—38no—
161C23×C3—38no—
162C23×C3—34no—
163C23×C3—316no—
164C23×C3—34no—
165C23×C3—38no—
166C23×C3—38no—
167C23×C3—41no—
168C23×C3—68no—
169C23×C3—68no—
170C23×C3—68no—
171C23×C3—64no—
172C23×C3—616no—
173C23×C3—68no—
174C23×C3—616no—
175C23×C3—84no—
176C23×C3C2×C3×Dic314yes—
177C23×C3C3×#6718yes—
178C23×C3C3×S414yes—
179C2×A4—216no—
180C2×A4—22no—
181C2×A4—24no—
182C2×A4—28no—
183C2×A4C2×C32×C411yescharacteristic
184C2×A4C32×D412yes—
185C2×C3×C4—68no—
186C2×C3×C4—68no—
187C2×C3×C4—68no—
188C2×C3×C4—68no—
189C2×C3×C4—68no—
190C2×C3×C4—68no—
191C2×Dic3—42no—
192C2×C32—22no—
193C2×C32—32no—
194C2×C32—31no—
195C2×C32—31no—
196C2×C32—44no—
197C2×C32—48no—
198C2×C32—62no—
199C2×C32—88no—
200C2×C32—816no—
201C2×C32—84no—
202C2×C32—88no—
203C2×C32C2×SL(2,ℤ/4)11yescharacteristic
204C2×C32GL(2,ℤ/4)12yes—
205C24—22no—
206C24—31no—
207C24—31no—
208C24—32no—
209C24—62no—
210C24C32×Dic312yescomplemented
211C24C2×C32×D311yescharacteristic
212C22×C4—32no—
213C22×C4—32no—
214C22⋊C4—62no—
215C22⋊C4—64no—
216C22⋊C4—62no—
217C22⋊C4—61no—
218C22⋊C4—62no—
219C22⋊C4—61no—
220C22×C3—28no—
221C22×C3—34no—
222C22×C3—38no—
223C22×C3—34no—
224C22×C3—34no—
225C22×C3—38no—
226C22×C3—38no—
227C22×C3—316no—
228C22×C3—38no—
229C22×C3—41no—
230C22×C3—42no—
231C22×C3—68no—
232C22×C3—616no—
233C22×C3—68no—
234C22×C3—64no—
235C22×C3—68no—
236C22×C3—616no—
237C22×C3—68no—
238C22×C3—82no—
239C22×C3—816no—
240C22×C3—84no—
241C22×C3—88no—
242C22×C3C3×#21714yescomplemented
243C22×C3C3×SL(2,ℤ/4)18yescomplemented
244C22×C3C2×C3×S414yes—
245A4—24no—
246A4C32×C22⋊C411yescharacteristic, complemented
247C3×C4—128no—
248C3×C4—128no—
249Dic3—82no—
250C32—44no—
251C32—84no—
252C32#543211yescharacteristic, direct product factor
253C23—22no—
254C23—31no—
255C23—32no—
256C23—32no—
257C23—31no—
258C23—34no—
259C23—31no—
260C23—32no—
261C23—32no—
262C23—62no—
263C23—62no—
264C23—62no—
265C23—61no—
266C23—64no—
267C23—62no—
268C23—64no—
269C23C2×C32×Dic311yescharacteristic
270C23C32×#6712yes—
271C23C32×S411yescharacteristic
272C2×C4—62no—
273C2×C4—62no—
274C2×C4—62no—
275C2×C4—62no—
276C2×C4—62no—
277C2×C4—62no—
278C2×C3—28no—
279C2×C3—38no—
280C2×C3—34no—
281C2×C3—34no—
282C2×C3—41no—
283C2×C3—42no—
284C2×C3—68no—
285C2×C3—82no—
286C2×C3—816no—
287C2×C3—84no—
288C2×C3—88no—
289C2×C3C2×C3×SL(2,ℤ/4)14yes—
290C2×C3C3×GL(2,ℤ/4)18yes—
291C22—22no—
292C22—31no—
293C22—32no—
294C22—31no—
295C22—31no—
296C22—32no—
297C22—32no—
298C22—34no—
299C22—32no—
300C22—62no—
301C22—64no—
302C22—62no—
303C22—61no—
304C22—62no—
305C22—64no—
306C22—62no—
307C22C32×#21711yescharacteristic, complemented
308C22C32×SL(2,ℤ/4)12yescomplemented
309C22C2×C32×S411yescharacteristic
310C4—122no—
311C4—122no—
312C3—41no—
313C3—84no—
314C3C3×#543214yesdirect product factor
315C2—22no—
316C2—32no—
317C2—31no—
318C2—31no—
319C2—62no—
320C2C2×C32×SL(2,ℤ/4)11yescharacteristic
321C2C32×GL(2,ℤ/4)12yes—
subgroupG/H
centerC22×C32C2×S4
commutatorC2×A4C2×C32×C4
FrattiniC22C2×C32×S4
FittingC25×C32D3
socleC24×C32C2×D3
prime radicalC23×C32×A4C2
cosocleC22×A4C22×C32
hypercenterC23×C32S4
hypocenterA4C32×C22⋊C4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C32 ◁ C23×C32

Complements

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

NH ≅ G/Ntype
2. C2×C32×SL(2,ℤ/4)C2semidirect
3. C3×#5432C3direct (N × H)
6. C22×C32×A4C4semidirect
13. C2×C3×SL(2,ℤ/4)C2×C3semidirect
23. #5432C32direct (N × H)
30. C24×C32Dic3semidirect
36. C22×C3×A4C3×C4semidirect
51. C32×A4C22⋊C4semidirect
56. C2×SL(2,ℤ/4)C2×C32semidirect
104. C24×C3C3×Dic3semidirect
110. C22×A4C32×C4semidirect
144. C22×C32#217semidirect
145. C22×C32SL(2,ℤ/4)semidirect
148. C3×A4C3×C22⋊C4semidirect
210. C24C32×Dic3semidirect
242. C22×C3C3×#217semidirect
243. C22×C3C3×SL(2,ℤ/4)semidirect
246. A4C32×C22⋊C4semidirect
252. C32#5432direct (N × H)
307. C22C32×#217semidirect
308. C22C32×SL(2,ℤ/4)semidirect
314. C3C3×#5432direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×A42
C2×C32×SL(2,ℤ/4)2
C3×#54323
C32×#3793
C32×#2174