Finite Group Databasebeta

← #3672⋊A4 #13147159

Subgroups — #3672⋊A4 #13147159

Order 1728.

265 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 41 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
1D4×C9⋊A4C211yesmaximal, characteristic
2Dic18⋊A4C211yesmaximal, characteristic, complemented
3(C9⋊C8)⋊A4C211yesmaximal, characteristic, complemented
4A4×#214—31nomaximal
5C22×#3672C311yesmaximal, characteristic, complemented
6C22×C9⋊A4—21no—
7C4×C9⋊A4C2211yescharacteristic
8Dic9⋊A4—21no—
9#3672⋊C3—41nomaximal
10C3×D4×A4D311yescharacteristic
11A4×Dic6—31no—
12A4×C3⋊C8—31no—
13C22×D4×C9C2×C311yescharacteristic
14C22×Dic18C2×C311yescharacteristic, complemented
15D4×C22⋊C9—21no—
16C2×#3672—31no—
17C2×#3672—31no—
18C2×#3672—31no—
19C2×#3672—31no—
20C22×C9⋊C8C2×C311yescharacteristic, complemented
21C2×C9⋊A4—41no—
22C2×C9⋊A4D411yescharacteristic
23D4×3−1+2—41no—
243−1+2⋊Q8—41no—
253−1+2⋊C8—41no—
26C22×#214—31no—
27A4×SD16—91no—
28C22×C3×A4—21no—
29C3×C4×A4C2×D311yescharacteristic
30Dic3×A4—61no—
31C24×C9—21no—
32C22×C22⋊C9—22no—
33C3×#214—121no—
34C2×D4×C9—31no—
35C2×D4×C9—31no—
36C2×D4×C9—61no—
37C22×C4×C9C22×C311yescharacteristic
38C22×Dic9—21no—
39C2×Dic18—31no—
40C2×Dic18—31no—
41C2×Dic18—61no—
42C4×C22⋊C9—21no—
43#3672—32no—
44#3672—31no—
45#3672—31no—
46#3672—31no—
47#3672A411yescharacteristic, complemented
48C2×C9⋊C8—31no—
49C2×C9⋊C8—31no—
50C22×3−1+2—81no—
51C9⋊A4SD1611yescharacteristic, complemented
52C4×3−1+2—41no—
533−1+2⋊C4—81no—
54C22×C3×D4C3×D311yescharacteristic
55C22×Dic6—31no—
56Q8×A4—91no—
57D4×A4—31no—
58C2×#214—91no—
59C2×#214—91no—
60C2×#214—91no—
61C2×#214—91no—
62C22×C3⋊C8—31no—
63C8×A4—91no—
64C2×C3×A4—41no—
65C2×C3×A4#6711yescharacteristic
66C32×D4—41no—
67C3×Dic6—121no—
68C23×C9—41no—
69C23×C9—61no—
70C23×C9—61no—
71C23×C9—121no—
72C23×C9C3×D411yescharacteristic
73C2×C22⋊C9—21no—
74C2×C22⋊C9—42no—
75C3×C3⋊C8—121no—
76D4×C9—31no—
77D4×C9—61no—
78D4×C9—61no—
79D4×C9—81no—
80D4×C9C2×A411yescharacteristic
81C2×C4×C9—31no—
82C2×C4×C9—31no—
83C2×Dic9—61no—
84C2×Dic9—61no—
85Dic18—31no—
86Dic18—61no—
87Dic18—61no—
88Dic18C2×A411yescharacteristic, complemented
89C9⋊C8—31no—
90C9⋊C8C2×A411yescharacteristic, complemented
91C22×SD16—91noSylow
92C2×3−1+2—41no—
93C2×3−1+2—161no—
94C24×C3—21no—
95C22×A4—61no—
96C2×C3×D4—31no—
97C2×C3×D4—31no—
98C2×C3×D4—61no—
99C22×C3×C4C2×C3×D311yescharacteristic
100C22×Dic3—61no—
101C2×Dic6—91no—
102C2×Dic6—91no—
103C2×Dic6—181no—
104C4×A4—31no—
105C4×A4—181no—
106C3×SD16—361no—
107#214—31no—
108#214—91no—
109#214—91no—
110#214—92no—
111#214—91no—
112C2×C3⋊C8—91no—
113C2×C3⋊C8—91no—
114C22×C32—81no—
115C3×A4#21411yescharacteristic
116C32×C4—41no—
117C3×Dic3—241no—
118C22×C9—21no—
119C22×C9—31no—
120C22×C9—31no—
121C22×C9—61no—
122C22×C9—82no—
123C22×C9—121no—
124C22×C9—121no—
125C22×C9—121no—
126C22×C9—121no—
127C22×C9C3×SD1611yescharacteristic, complemented
128C22⋊C9—21no—
129C4×C9—31no—
130C4×C9—81no—
131C4×C9C22×A411yescharacteristic
132Dic9—21no—
133Dic9—61no—
134C22×Q8—91no—
135C22×D4AGL(1,ℤ/9)11yescharacteristic
136C2×SD16—271no—
137C2×SD16—271no—
138C2×SD16—271no—
139C2×SD16—271no—
140C22×C8—91no—
1413−1+2—41noSylow
142C23×C3—41no—
143C23×C3—61no—
144C23×C3—61no—
145C23×C3—121no—
146C23×C3C3×#6711yescharacteristic
147C2×A4—31no—
148C2×A4—121no—
149C3×Q8—361no—
150C3×D4—31no—
151C3×D4—61no—
152C3×D4—61no—
153C3×D4—121no—
154C3×D4D3×A411yescharacteristic
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×Dic3—181no—
158C2×Dic3—181no—
159Dic6—31no—
160Dic6—91no—
161Dic6—181no—
162Dic6—181no—
163C3×C8—361no—
164C3⋊C8—31no—
165C3⋊C8—91no—
166C2×C32—41no—
167C2×C32—161no—
168C2×C9—31no—
169C2×C9—31no—
170C2×C9—41no—
171C2×C9—81no—
172C2×C9—121no—
173C2×C9—162no—
174C2×C9D4×A411yescharacteristic
175C24—21no—
176C2×Q8—271no—
177C2×Q8—271no—
178C2×Q8—541no—
179C2×D4—31no—
180C2×D4—31no—
181C2×D4—61no—
182C22×C4—181no—
183C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
184SD16—91no—
185SD16—271no—
186SD16—271no—
187SD16—272no—
188SD16—271no—
189C2×C8—271no—
190C2×C8—271no—
191C22×C3—21no—
192C22×C3—31no—
193C22×C3—31no—
194C22×C3—61no—
195C22×C3—121no—
196C22×C3—121no—
197C22×C3—121no—
198C22×C3—121no—
199C22×C3—241no—
200C22×C3C3×#21411yescharacteristic
201A4—31no—
202C3×C4—31no—
203C3×C4—121no—
204C3×C4—721no—
205C3×C4C2×D3×A411yescharacteristic
206Dic3—61no—
207Dic3—181no—
208C32—41no—
209C9—81no—
210C9A4×SD1611yescharacteristic, complemented
211C23—41no—
212C23—61no—
213C23—61no—
214C23—121no—
215C23#611⋊C311yescharacteristic
216Q8—91no—
217Q8—271no—
218Q8—541no—
219Q8—541no—
220D4—31no—
221D4—61no—
222D4—61no—
223D4D9⋊A411yescharacteristic
224C2×C4—31no—
225C2×C4—31no—
226C2×C4—541no—
227C2×C4—541no—
228C8—91no—
229C8—271no—
230C2×C3—31no—
231C2×C3—31no—
232C2×C3—41no—
233C2×C3—121no—
234C2×C3—121no—
235C2×C3—481no—
236C2×C3A4×#6711yescharacteristic
237C22—21no—
238C22—31no—
239C22—31no—
240C22—61no—
241C22—121no—
242C22—121no—
243C22—121no—
244C22—121no—
245C22#3672⋊C311yescharacteristic, complemented
246C4—31no—
247C4—181no—
248C4—541no—
249C4C2×D9⋊A411yescharacteristic
250C3—121no—
251C3A4×#21411yescharacteristic
252C2—31no—
253C2—31no—
254C2—41no—
255C2—121no—
256C2#611⋊A411yescharacteristic
subgroupG/H
centerC2#611⋊A4
commutator, cosocleC22×C4×C9C22×C3
FrattiniC3×C4C2×D3×A4
FittingC22×D4×C9C2×C3
socleC23×C3C3×#67
prime radicalC3×D4×A4D3
hypercenterD4D9⋊A4
hypocenterC22×C9C3×SD16
LCSG ▷ C22×C4×C9 ▷ C23×C9 ▷ C22×C9
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
2. Dic18⋊A4C2semidirect
3. (C9⋊C8)⋊A4C2semidirect
5. C22×#3672C3semidirect
14. C22×Dic18C2×C3semidirect
20. C22×C9⋊C8C2×C3semidirect
47. #3672A4semidirect
51. C9⋊A4SD16semidirect
88. Dic18C2×A4semidirect
90. C9⋊C8C2×A4semidirect
127. C22×C9C3×SD16semidirect
210. C9A4×SD16semidirect
245. C22#3672⋊C3semidirect

Maximal subgroups

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

subgroupindex
D4×C9⋊A42
Dic18⋊A42
(C9⋊C8)⋊A42
A4×#2143
C22×#36723
#3672⋊C34