Finite Group Databasebeta

← C9×C4⋊SL(2,ℤ/4) #13146776

Subgroups — C9×C4⋊SL(2,ℤ/4) #13146776

Order 1728.

359 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C2×C4×C9×A4C211yesmaximal, characteristic
2C2×C9×SL(2,ℤ/4)C212yesmaximal
3C3×C4⋊SL(2,ℤ/4)C311yesmaximal, characteristic
4C9×#380—31nomaximal
5C22×C9×A4C2211yescharacteristic
6C4×C9×A4C412yescomplemented
7C9×SL(2,ℤ/4)—22no—
8C9×C4⋊Dic3—41nomaximal
9C2×C3×C4×A4C2×C311yescharacteristic
10C2×C3×SL(2,ℤ/4)C2×C312yes—
11C23×C4×C9D311yescharacteristic
12C2×C4×C22⋊C9—21no—
13C2×C9×C4⋊C4—31no—
14C2×C9×C4⋊C4—31no—
15C2×C9×C22⋊C4—32no—
16C9×#95—32no—
17C2×C9×A4C2×C411yescharacteristic
18C2×C9×A4D411yescharacteristic
19C2×C9×A4Q811yescharacteristic
20C2×C3×C4×C9—41no—
21C2×C9×Dic3—42no—
22C4⋊SL(2,ℤ/4)C911yescharacteristic, direct product factor
23C3×#380—31no—
24C22×C3×A4C22×C311yescharacteristic
25C3×C4×A4C3×C412yes—
26C3×SL(2,ℤ/4)—22no—
27C24×C9C2×D311yescharacteristic
28C22×C22⋊C9—21no—
29C3×C4⋊Dic3—41no—
30C22×C4×C9—31no—
31C22×C4×C9—31no—
32C22×C4×C9—32no—
33C22×C4×C9—32no—
34C22×C4×C9—32no—
35C22×C4×C9Dic312yescomplemented
36C4×C22⋊C9—22no—
37C9×C4⋊C4—61no—
38C9×C4⋊C4—62no—
39C9×C4⋊C4—61no—
40C9×C22⋊C4—62no—
41C9×C22⋊C4—62no—
42C22×C3×C9—41no—
43C9×A4C4⋊C411yescharacteristic, complemented
44C3×C4×C9—42no—
45C9×Dic3—82no—
46C23×C3×C4C3×D311yescharacteristic
47C2×C4×A4—21no—
48C2×C4×A4C2×C911yescharacteristic
49C2×SL(2,ℤ/4)C2×C912yes—
50C2×C3×C4⋊C4—31no—
51C2×C3×C4⋊C4—31no—
52C2×C3×C22⋊C4—32no—
53C3×#95—32no—
54C2×C3×A4C2×C3×C411yescharacteristic
55C2×C3×A4C3×D411yescharacteristic
56C2×C3×A4C3×Q811yescharacteristic
57C2×C32×C4—41no—
58C2×C3×Dic3—42no—
59C23×C9—31no—
60C23×C9—31no—
61C23×C9—31no—
62C23×C9—31no—
63C23×C9Dic611yescharacteristic
64C23×C9D1211yescharacteristic
65C23×C9C2×Dic311yescharacteristic
66C2×C22⋊C9—21no—
67C2×C22⋊C9—21no—
68C2×C22⋊C9—21no—
69C2×C4×C9—31no—
70C2×C4×C9—32no—
71C2×C4×C9—32no—
72C2×C4×C9—32no—
73C2×C4×C9—61no—
74C2×C4×C9—62no—
75C2×C4×C9—62no—
76C2×C4×C9—62no—
77C2×C4×C9—62no—
78C2×C4×C9—62no—
79C2×C4×C9—62no—
80C2×C4×C9—81no—
81C2×C4×C9S411yescharacteristic
82#380—31noSylow
83C2×C3×C9—41no—
84C2×C3×C9—41no—
85C2×C3×C9—41no—
86C24×C3C2×C3×D311yescharacteristic
87C22×A4—21no—
88C22×A4C22×C911yescharacteristic
89C22×C3×C4—31no—
90C22×C3×C4—31no—
91C22×C3×C4—32no—
92C22×C3×C4—32no—
93C22×C3×C4—32no—
94C22×C3×C4C3×Dic312yes—
95C4×A4—22no—
96C4×A4C4×C912yescomplemented
97SL(2,ℤ/4)—22no—
98C3×C4⋊C4—61no—
99C3×C4⋊C4—62no—
100C3×C4⋊C4—61no—
101C3×C22⋊C4—62no—
102C3×C22⋊C4—62no—
103C4⋊Dic3—41no—
104C22×C32—41no—
105C3×A4C3×C4⋊C411yescharacteristic
106C32×C4—42no—
107C3×Dic3—82no—
108C22×C9—31no—
109C22×C9—31no—
110C22×C9—31no—
111C22×C9—31no—
112C22×C9—31no—
113C22×C9—31no—
114C22×C9—31no—
115C22×C9—31no—
116C22×C9—31no—
117C22×C9—61no—
118C22×C9—81no—
119C22×C9C4⋊Dic311yescharacteristic, complemented
120C22×C9C2×S411yescharacteristic
121C22⋊C9—21no—
122C4×C9—32no—
123C4×C9—82no—
124C4×C9—122no—
125C4×C9—122no—
126C4×C9SL(2,ℤ/4)12yescomplemented
127C23×C4D3×C911yescharacteristic
128C2×C4⋊C4—31no—
129C2×C4⋊C4—31no—
130C2×C22⋊C4—32no—
131#95—32no—
132C3×C9—41noSylow
133C23×C3—31no—
134C23×C3—31no—
135C23×C3—31no—
136C23×C3—31no—
137C23×C3C3×Dic611yescharacteristic
138C23×C3C3×D1211yescharacteristic
139C23×C3C2×C3×Dic311yescharacteristic
140C2×A4—21no—
141C2×A4—21no—
142C2×A4—21no—
143C2×A4C2×C4×C911yescharacteristic
144C2×A4D4×C911yescharacteristic
145C2×A4Q8×C911yescharacteristic
146C2×C3×C4—31no—
147C2×C3×C4—32no—
148C2×C3×C4—32no—
149C2×C3×C4—32no—
150C2×C3×C4—41no—
151C2×C3×C4—61no—
152C2×C3×C4—62no—
153C2×C3×C4—62no—
154C2×C3×C4—62no—
155C2×C3×C4—62no—
156C2×C3×C4—62no—
157C2×C3×C4—62no—
158C2×C3×C4—81no—
159C2×C3×C4C3×S411yescharacteristic
160C2×Dic3—42no—
161C2×C32—41no—
162C2×C32—41no—
163C2×C32—41no—
164C2×C9—31no—
165C2×C9—31no—
166C2×C9—31no—
167C2×C9—31no—
168C2×C9—81no—
169C2×C9—81no—
170C2×C9—81no—
171C2×C9A4⋊Q811yescharacteristic
172C2×C9C4⋊S411yescharacteristic
173C2×C9C2×SL(2,ℤ/4)11yescharacteristic
174C24C2×D3×C911yescharacteristic
175C22×C4—31no—
176C22×C4—31no—
177C22×C4—32no—
178C22×C4—32no—
179C22×C4—32no—
180C22×C4C9×Dic312yescomplemented
181C4⋊C4—61no—
182C4⋊C4—62no—
183C4⋊C4—61no—
184C22⋊C4—62no—
185C22⋊C4—62no—
186C22×C3—31no—
187C22×C3—31no—
188C22×C3—31no—
189C22×C3—31no—
190C22×C3—31no—
191C22×C3—31no—
192C22×C3—31no—
193C22×C3—31no—
194C22×C3—31no—
195C22×C3—41no—
196C22×C3—61no—
197C22×C3—81no—
198C22×C3C3×C4⋊Dic311yescharacteristic
199C22×C3C2×C3×S411yescharacteristic
200A4—21no—
201A4C9×C4⋊C411yescharacteristic, complemented
202C3×C4—32no—
203C3×C4—42no—
204C3×C4—82no—
205C3×C4—122no—
206C3×C4—122no—
207C3×C4C3×SL(2,ℤ/4)12yes—
208Dic3—82no—
209C32—41no—
210C9—81no—
211C9C4⋊SL(2,ℤ/4)11yescharacteristic, direct product factor
212C23—31no—
213C23—31no—
214C23—31no—
215C23—31no—
216C23C9×Dic611yescharacteristic
217C23C9×D1211yescharacteristic
218C23C2×C9×Dic311yescharacteristic
219C2×C4—31no—
220C2×C4—32no—
221C2×C4—32no—
222C2×C4—32no—
223C2×C4—61no—
224C2×C4—62no—
225C2×C4—62no—
226C2×C4—62no—
227C2×C4—62no—
228C2×C4—62no—
229C2×C4—62no—
230C2×C4C9×S411yescharacteristic
231C2×C3—31no—
232C2×C3—31no—
233C2×C3—31no—
234C2×C3—31no—
235C2×C3—41no—
236C2×C3—41no—
237C2×C3—41no—
238C2×C3—81no—
239C2×C3—81no—
240C2×C3—81no—
241C2×C3C3×A4⋊Q811yescharacteristic
242C2×C3C3×C4⋊S411yescharacteristic
243C2×C3C2×C3×SL(2,ℤ/4)11yescharacteristic
244C22—31no—
245C22—31no—
246C22—31no—
247C22—31no—
248C22—31no—
249C22—31no—
250C22—31no—
251C22—31no—
252C22—31no—
253C22—61no—
254C22C9×C4⋊Dic311yescharacteristic, complemented
255C22C2×C9×S411yescharacteristic
256C4—32no—
257C4—122no—
258C4—122no—
259C4C9×SL(2,ℤ/4)12yescomplemented
260C3—41no—
261C3—81no—
262C3C3×C4⋊SL(2,ℤ/4)11yescharacteristic
263C2—31no—
264C2—31no—
265C2—31no—
266C2—31no—
267C2C9×A4⋊Q811yescharacteristic
268C2C9×C4⋊S411yescharacteristic
269C2C2×C9×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C9C2×S4
commutatorC2×A4C2×C4×C9
FrattiniC22×C3C2×C3×S4
FittingC23×C4×C9D3
socleC24×C3C2×C3×D3
prime radical, cosocleC22×C3×A4C22×C3
hypercenterC2×C4×C9S4
hypocenterA4C9×C4⋊C4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C9 ◁ C2×C4×C9

Complements

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

NH ≅ G/Ntype
6. C4×C9×A4C4semidirect
22. C4⋊SL(2,ℤ/4)C9direct (N × H)
35. C22×C4×C9Dic3semidirect
43. C9×A4C4⋊C4semidirect
96. C4×A4C4×C9semidirect
119. C22×C9C4⋊Dic3semidirect
126. C4×C9SL(2,ℤ/4)semidirect
180. C22×C4C9×Dic3semidirect
201. A4C9×C4⋊C4semidirect
211. C9C4⋊SL(2,ℤ/4)direct (N × H)
254. C22C9×C4⋊Dic3semidirect
259. C4C9×SL(2,ℤ/4)semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C9×A42
C2×C9×SL(2,ℤ/4)2
C3×C4⋊SL(2,ℤ/4)3
C9×#3803
C9×C4⋊Dic34