Finite Group Databasebeta

← C32×#4472 #13145774

Subgroups — C32×#4472 #13145774

Order 1728.

616 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C32×#935C211yesmaximal, characteristic
2C33×#99C211yesmaximal, characteristic
3C32×#833C211yesmaximal, characteristic
4C32×#352—31nomaximal
5C3×#4472C314yesmaximal, direct product factor
6C2×C33×D4C2211yescharacteristic
7C22×C32×Dic3C411yescharacteristic, complemented
8C33×C22⋊C4—21no—
9C32×#217—21no—
10C32×#217C411yescharacteristic, complemented
11C32×Dic3⋊C4—21no—
12C32×#208—21no—
13C32×#123—31no—
14C3×#935C2×C314yes—
15C32×#100—31no—
16C32×#99—24no—
17C32×#99D311yescharacteristic
18C32×#99C2×C314yes—
19C3×#833C2×C314yes—
20C23×C33C2×C411yescharacteristic
21C23×C33D411yescharacteristic
22C33×D4—41no—
23C2×C33×C4—41no—
24C2×C33×C4D411yescharacteristic
25C2×C32×Dic3—21no—
26C2×C32×Dic3—21no—
27C2×C32×Dic3—41no—
28C32×C3⋊C8—41no—
29C3×#352—34no—
30#4472C3211yescharacteristic, direct product factor
31C2×C32×D4—24no—
32C2×C32×D4C2×D311yescharacteristic
33C2×C32×D4C22×C314yes—
34C22×C32×C4—31no—
35C22×C3×Dic3C3×C414yescomplemented
36C32×M16—61no—
37C32×C4⋊C4—61no—
38C32×C22⋊C4—28no—
39C32×C22⋊C4—21no—
40C32×C22⋊C4—24no—
41C32×C22⋊C4—31no—
42C32×C22⋊C4—61no—
43C3×#217—24no—
44C3×#217C3×C414yescomplemented
45C3×Dic3⋊C4—24no—
46C3×#208—24no—
47C22×C33—21no—
48C22×C33—21no—
49C22×C33—41no—
50C22×C33—41no—
51C22×C33C22⋊C411yescharacteristic
52C33×C4—21no—
53C33×C4—81no—
54C32×Dic3—41no—
55C32×Dic3—41no—
56C3×#123—34no—
57#935C2×C3211yescharacteristic
58C3×#100—34no—
59C3×#99—24no—
60C3×#99C3×D314yes—
61C3×#99C2×C3211yescharacteristic
62#833C2×C3211yescharacteristic
63C23×C32—24no—
64C23×C32—24no—
65C23×C32C4×D311yescharacteristic
66C23×C32#6711yescharacteristic
67C23×C32C2×C3×C414yes—
68C23×C32C3×D414yes—
69C32×D4—41no—
70C32×D4—44no—
71C32×D4—48no—
72C2×C32×C4—24no—
73C2×C32×C4—48no—
74C2×C32×C4—41no—
75C2×C32×C4—44no—
76C2×C32×C4—61no—
77C2×C32×C4—61no—
78C2×C32×C4—121no—
79C2×C32×C4D1211yescharacteristic
80C2×C32×C4C3×D414yes—
81C2×C3×Dic3—24no—
82C2×C3×Dic3—24no—
83C2×C3×Dic3—44no—
84C32×C8—121no—
85C3×C3⋊C8—44no—
86#352—31noSylow
87C2×C33—21no—
88C2×C33—41no—
89C2×C33—41no—
90C2×C33#9911yescharacteristic
91C2×C3×D4—24no—
92C2×C3×D4C2×C3×D314yes—
93C2×C3×D4C22×C3211yescharacteristic
94C22×C3×C4—34no—
95C22×Dic3C32×C411yescharacteristic, complemented
96C3×M16—64no—
97C3×C4⋊C4—64no—
98C3×C22⋊C4—28no—
99C3×C22⋊C4—24no—
100C3×C22⋊C4—21no—
101C3×C22⋊C4—34no—
102C3×C22⋊C4—64no—
103#217—21no—
104#217C32×C411yescharacteristic, complemented
105Dic3⋊C4—21no—
106#208—21no—
107C22×C32—28no—
108C22×C32—21no—
109C22×C32—21no—
110C22×C32—24no—
111C22×C32—24no—
112C22×C32—24no—
113C22×C32—44no—
114C22×C32—41no—
115C22×C32—41no—
116C22×C32—44no—
117C22×C32—44no—
118C22×C32—84no—
119C22×C32—84no—
120C22×C32(C2×D3)⋊C411yescharacteristic
121C22×C32C3×C22⋊C414yes—
122C32×C4—21no—
123C32×C4—24no—
124C32×C4—44no—
125C32×C4—88no—
126C32×C4—81no—
127C32×C4—84no—
128C32×C4—121no—
129C32×C4—121no—
130C3×Dic3—44no—
131C3×Dic3—44no—
132#123—31no—
133#100—31no—
134#99C32×D311yescharacteristic
135C33#35211yescharacteristic, complemented, Sylow
136C23×C3—24no—
137C23×C3—24no—
138C23×C3C3×C4×D314yes—
139C23×C3C3×#6714yes—
140C23×C3C2×C32×C411yescharacteristic
141C23×C3C32×D411yescharacteristic
142C3×D4—44no—
143C3×D4—41no—
144C3×D4—48no—
145C2×C3×C4—24no—
146C2×C3×C4—48no—
147C2×C3×C4—44no—
148C2×C3×C4—41no—
149C2×C3×C4—64no—
150C2×C3×C4—64no—
151C2×C3×C4—124no—
152C2×C3×C4C3×D1214yes—
153C2×C3×C4C32×D411yescharacteristic
154C2×Dic3—21no—
155C2×Dic3—21no—
156C2×Dic3—41no—
157C3×C8—124no—
158C3⋊C8—41no—
159C2×C32—21no—
160C2×C32—24no—
161C2×C32—24no—
162C2×C32—44no—
163C2×C32—48no—
164C2×C32—41no—
165C2×C32—41no—
166C2×C32—44no—
167C2×C32—44no—
168C2×C32—84no—
169C2×C32#80611yescharacteristic
170C2×C32C3×#9914yes—
171C2×D4C2×C32×D311yescharacteristic
172C22×C4—31no—
173M16—61no—
174C4⋊C4—61no—
175C22⋊C4—21no—
176C22⋊C4—31no—
177C22⋊C4—61no—
178C22×C3—28no—
179C22×C3—24no—
180C22×C3—24no—
181C22×C3—21no—
182C22×C3—21no—
183C22×C3—24no—
184C22×C3—44no—
185C22×C3—44no—
186C22×C3—44no—
187C22×C3—41no—
188C22×C3—41no—
189C22×C3—84no—
190C22×C3—84no—
191C22×C3C3×(C2×D3)⋊C414yes—
192C22×C3C32×C22⋊C411yescharacteristic
193C3×C4—24no—
194C3×C4—21no—
195C3×C4—44no—
196C3×C4—88no—
197C3×C4—81no—
198C3×C4—84no—
199C3×C4—124no—
200C3×C4—124no—
201Dic3—41no—
202Dic3—41no—
203C32—24no—
204C32#447211yescharacteristic, direct product factor
205C32C3×#35214yescomplemented
206C23C32×C4×D311yescharacteristic
207C23C32×#6711yescharacteristic
208D4—41no—
209C2×C4—41no—
210C2×C4—61no—
211C2×C4—61no—
212C2×C4—121no—
213C2×C4C32×D1211yescharacteristic
214C8—121no—
215C2×C3—21no—
216C2×C3—24no—
217C2×C3—24no—
218C2×C3—44no—
219C2×C3—48no—
220C2×C3—41no—
221C2×C3—41no—
222C2×C3—44no—
223C2×C3—44no—
224C2×C3—84no—
225C2×C3C3×#80614yes—
226C2×C3C32×#9911yescharacteristic
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22C32×(C2×D3)⋊C411yescharacteristic
232C4—21no—
233C4—81no—
234C4—121no—
235C4—121no—
236C3—24no—
237C3C3×#447214yesdirect product factor
238C3C32×#35211yescharacteristic, complemented
239C2—21no—
240C2—41no—
241C2—41no—
242C2C32×#80611yescharacteristic
subgroupG/H
centerC2×C32#806
commutatorC23×C3C2×C32×C4
FrattiniC2×D4C2×C32×D3
FittingC33×#99C2
socleC2×C33#99
prime radicalC2×C33×D4C22
cosocleC2×C3×D4C22×C32
hypercenterC32×#99D3
hypocenterC3C32×#352
LCSG ▷ C23×C3 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C32 ◁ C22×C32 ◁ C2×C32×D4 ◁ C32×#99

Complements

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

NH ≅ G/Ntype
5. C3×#4472C3direct (N × H)
7. C22×C32×Dic3C4semidirect
10. C32×#217C4semidirect
30. #4472C32direct (N × H)
35. C22×C3×Dic3C3×C4semidirect
44. C3×#217C3×C4semidirect
95. C22×Dic3C32×C4semidirect
104. #217C32×C4semidirect
135. C33#352semidirect
204. C32#4472direct (N × H)
205. C32C3×#352semidirect
237. C3C3×#4472direct (N × H)
238. C3C32×#352semidirect

Maximal subgroups

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

subgroupindex
C32×#9352
C33×#992
C32×#8332
C32×#3523
C3×#44723