Finite Group Databasebeta

← C3×#10590164 #13157088

Subgroups — C3×#10590164 #13157088

Order 1728.

424 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 76 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×C3×C4×#157C211yesmaximal, characteristic
2C2×C3×#3788C211yesmaximal, characteristic
3C3×#64101C211yesmaximal, characteristic
4#10590164C311yesmaximal, characteristic, direct product factor
5C22×C3×#157—21no—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C3×#622—21no—
8C3×C4×#157—21no—
9C3×C4×#157C412yescomplemented
10C3×#3788—21no—
11C3×#3788C412yescomplemented
12C33×C4⋊C4C411yescharacteristic, complemented
13C3×#3752—21no—
14C3×C4⋊(C3⋊Dic3)C411yescharacteristic, complemented
15C3×C4×C3⋊Dic3—21no—
16C2×C4×#157C2×C311yescharacteristic
17C2×#3788C2×C311yescharacteristic
18#64101C2×C311yescharacteristic
19C3×#892—62no—
20C22×C3×C3⋊D3D411yescharacteristic
21C2×C3×#157—22no—
22C2×C3×#157—22no—
23C2×C3×#157—22no—
24C2×C33×C4—21no—
25C2×C33×C4C2×C411yescharacteristic
26C2×C3×C3⋊Dic3—21no—
27C2×C3×C3⋊Dic3D411yescharacteristic
28C3×C4×C3⋊D3C2×C411yescharacteristic
29C3×C4×C3⋊D3C2×C411yescharacteristic
30C3×C4×C3⋊D3D411yescharacteristic
31C3×C4×C3⋊D3Q811yescharacteristic
32C3×#622—22no—
33C3×#339—91nomaximal
34C22×#157—21no—
35C2×C4×C3⋊D3C22×C311yescharacteristic
36C2×C3×C4×D3—62no—
37C2×#622—21no—
38C4×#157—21no—
39C4×#157C3×C412yescomplemented
40#3788—21no—
41#3788C3×C412yescomplemented
42C32×C4⋊C4—22no—
43C32×C4⋊C4—42no—
44C32×C4⋊C4C3×C411yescharacteristic, complemented
45#3752—21no—
46C4⋊(C3⋊Dic3)C3×C411yescharacteristic, complemented
47C4×C3⋊Dic3—21no—
48C3×(C2×D3)⋊C4—64no—
49C3×C4⋊Dic3—62no—
50C3×C4×Dic3—64no—
51C22×C33C22⋊C411yescharacteristic
52C2×C3×C3⋊D3—21no—
53C2×C3×C3⋊D3—21no—
54C2×C3×C3⋊D3C22⋊C411yescharacteristic
55C2×C3×C3⋊D3C4⋊C411yescharacteristic
56C3×#157—24no—
57C33×C4—41no—
58C33×C4C4211yescharacteristic
59C33×C4C4⋊C411yescharacteristic
60C3×C3⋊Dic3—41no—
61C3×C3⋊Dic3C22⋊C411yescharacteristic
62C3×C3⋊Dic3C4⋊C411yescharacteristic
63C2×C3×M16—91no—
64C3×#117—91no—
65C2×C3×C42—91no—
66#892—62no—
67C22×C3⋊D3C3×D411yescharacteristic
68C22×C3×D3—62no—
69C2×#157—22no—
70C2×#157—22no—
71C2×#157—22no—
72C2×C32×C4—21no—
73C2×C32×C4—24no—
74C2×C32×C4—22no—
75C2×C32×C4—44no—
76C2×C32×C4—42no—
77C2×C32×C4C2×C3×C411yescharacteristic
78C2×C3⋊Dic3—21no—
79C2×C3⋊Dic3C3×D411yescharacteristic
80C4×C3⋊D3C2×C3×C411yescharacteristic
81C4×C3⋊D3C2×C3×C411yescharacteristic
82C4×C3⋊D3C3×D411yescharacteristic
83C4×C3⋊D3C3×Q811yescharacteristic
84C2×C3×Dic3—62no—
85C2×C3×Dic3—64no—
86C3×C4×D3—62no—
87C3×C4×D3—62no—
88C3×C4×D3—62no—
89C3×C4×D3—62no—
90#622—22no—
91#339—91noSylow
92C2×C33#9511yescharacteristic
93C2×C33#10411yescharacteristic
94C2×C33#10411yescharacteristic
95C3×C3⋊D3—21no—
96C3×C3⋊D3—21no—
97C22×C3×C4—91no—
98C22×C3×C4—181no—
99C2×C4×D3—62no—
100C3×M16—92no—
101C3×M16—181no—
102C2×C3×C8—181no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4—42no—
105C3×C4⋊C4—91no—
106C3×C4⋊C4#15711yescharacteristic, complemented
107C3×C22⋊C4—181no—
108C3×C42—92no—
109C3×C42—181no—
110C3×C42—181no—
111(C2×D3)⋊C4—64no—
112C4⋊Dic3—62no—
113C4×Dic3—64no—
114C22×C32—22no—
115C22×C32—42no—
116C22×C32C3×C22⋊C411yescharacteristic
117C2×C3⋊D3—21no—
118C2×C3⋊D3—21no—
119C2×C3⋊D3C3×C22⋊C411yescharacteristic
120C2×C3⋊D3C3×C4⋊C411yescharacteristic
121C2×C3×D3—62no—
122C2×C3×D3—62no—
123C2×C3×D3—122no—
124C2×C3×D3—122no—
125#157—24no—
126C32×C4—22no—
127C32×C4—22no—
128C32×C4—42no—
129C32×C4—42no—
130C32×C4—41no—
131C32×C4—44no—
132C32×C4—84no—
133C32×C4C3×C4211yescharacteristic
134C32×C4C3×C4⋊C411yescharacteristic
135C3⋊Dic3—41no—
136C3⋊Dic3C3×C22⋊C411yescharacteristic
137C3⋊Dic3C3×C4⋊C411yescharacteristic
138C3×Dic3—62no—
139C3×Dic3—62no—
140C3×Dic3—124no—
141C2×M16—91no—
142#117—91no—
143C2×C42—91no—
144C33#33911yescharacteristic, complemented, Sylow
145C23×C3—91no—
146C22×D3—62no—
147C2×C3×C4—24no—
148C2×C3×C4—21no—
149C2×C3×C4—22no—
150C2×C3×C4—44no—
151C2×C3×C4—42no—
152C2×C3×C4—91no—
153C2×C3×C4—91no—
154C2×C3×C4—91no—
155C2×C3×C4—91no—
156C2×C3×C4—91no—
157C2×C3×C4—182no—
158C2×C3×C4—182no—
159C2×C3×C4—182no—
160C2×C3×C4—181no—
161C2×C3×C4C2×#15711yescharacteristic
162C2×Dic3—62no—
163C2×Dic3—64no—
164C4×D3—62no—
165C4×D3—62no—
166C4×D3—62no—
167C4×D3—62no—
168C3×C8—182no—
169C2×C32—22no—
170C2×C32—22no—
171C2×C32—22no—
172C2×C32—42no—
173C2×C32—42no—
174C2×C32—42no—
175C2×C32C3×#9511yescharacteristic
176C2×C32C3×#10411yescharacteristic
177C2×C32C3×#10411yescharacteristic
178C3⋊D3—21no—
179C3⋊D3—21no—
180C3×D3—122no—
181C3×D3—122no—
182C22×C4—91no—
183C22×C4—181no—
184M16—92no—
185M16—181no—
186C2×C8—181no—
187C4⋊C4—91no—
188C4⋊C4C3×#15711yescharacteristic, complemented
189C22⋊C4—181no—
190C42—92no—
191C42—181no—
192C42—181no—
193C22×C3—22no—
194C22×C3—42no—
195C22×C3—91no—
196C22×C3—91no—
197C22×C3—181no—
198C22×C3—181no—
199C22×C3C22⋊#15711yescharacteristic
200C2×D3—62no—
201C2×D3—62no—
202C2×D3—122no—
203C2×D3—122no—
204C3×C4—22no—
205C3×C4—22no—
206C3×C4—44no—
207C3×C4—41no—
208C3×C4—42no—
209C3×C4—42no—
210C3×C4—84no—
211C3×C4—91no—
212C3×C4—91no—
213C3×C4—184no—
214C3×C4—361no—
215C3×C4C4×#15711yescharacteristic
216C3×C4C4⋊#15711yescharacteristic
217Dic3—62no—
218Dic3—62no—
219Dic3—124no—
220C32—22no—
221C32—42no—
222C32C3×#33911yescharacteristic, complemented
223C23—91no—
224C2×C4—21no—
225C2×C4—91no—
226C2×C4—91no—
227C2×C4—91no—
228C2×C4—91no—
229C2×C4—91no—
230C2×C4—182no—
231C2×C4—182no—
232C2×C4—182no—
233C2×C4—181no—
234C2×C4C2×C3×#15711yescharacteristic
235C8—182no—
236C2×C3—22no—
237C2×C3—22no—
238C2×C3—22no—
239C2×C3—42no—
240C2×C3—42no—
241C2×C3—42no—
242C2×C3—181no—
243C2×C3—181no—
244C2×C3#6378111yescharacteristic
245C2×C3#6378311yescharacteristic
246C2×C3#6378511yescharacteristic
247D3—122no—
248D3—122no—
249C22—91no—
250C22—91no—
251C22—181no—
252C22—181no—
253C22C3×C22⋊#15711yescharacteristic
254C4—41no—
255C4—91no—
256C4—91no—
257C4—184no—
258C4—361no—
259C4C3×C4×#15711yescharacteristic
260C4C3×C4⋊#15711yescharacteristic
261C3—22no—
262C3—42no—
263C3#1059016411yescharacteristic, direct product factor
264C2—181no—
265C2—181no—
266C2C3×#6378111yescharacteristic
267C2C3×#6378311yescharacteristic
268C2C3×#6378511yescharacteristic
subgroupG/H
centerC22×C3C22⋊#157
commutatorC32×C4C3×C42
FrattiniC2×C4C2×C3×#157
FittingC33×C4⋊C4C4
socleC22×C33C22⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C4×C3⋊D3C22×C3
hypercenterC3×C4⋊C4#157
hypocenterC32C3×#339
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #10590164C3direct (N × H)
9. C3×C4×#157C4semidirect
11. C3×#3788C4semidirect
12. C33×C4⋊C4C4semidirect
14. C3×C4⋊(C3⋊Dic3)C4semidirect
39. C4×#157C3×C4semidirect
41. #3788C3×C4semidirect
44. C32×C4⋊C4C3×C4semidirect
46. C4⋊(C3⋊Dic3)C3×C4semidirect
106. C3×C4⋊C4#157semidirect
144. C33#339semidirect
188. C4⋊C4C3×#157semidirect
222. C32C3×#339semidirect
263. C3#10590164direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×#1572
C2×C3×#37882
C3×#641012
#105901643
C3×#3399