Finite Group Databasebeta

← C3×#10590156 #13157081

Subgroups — C3×#10590156 #13157081

Order 1728.

400 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 68 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×#3792C211yesmaximal, characteristic
2C2×C3×#3792C211yesmaximal, characteristic
3C3×#64086C211yesmaximal, characteristic
4#10590156C311yesmaximal, characteristic, direct product factor
5C22×C3×C3⋊Dic3C2211yescharacteristic
6C3×#3792—21no—
7C3×#3792—21no—
8C3×#3792—21no—
9C3×#3792C412yescomplemented
10C2×C3×#622—21no—
11C2×C3×#622C412yescomplemented
12C33×C22⋊C4C411yescharacteristic
13C3×#3757C411yescharacteristic
14C3×#3750—21no—
15C3×C4×C3⋊Dic3—21no—
16C2×#3792C2×C311yescharacteristic
17C2×#3792C2×C311yescharacteristic
18#64086C2×C311yescharacteristic
19C3×#877—62no—
20C23×C33C2×C411yescharacteristic
21C2×C33×C4—21no—
22C2×C3×C3⋊Dic3—21no—
23C2×C3×C3⋊Dic3C2×C411yescharacteristic
24C2×C3×C3⋊Dic3C2×C411yescharacteristic
25C2×C3×C3⋊Dic3D411yescharacteristic
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3D411yescharacteristic
28C2×C3×C3⋊Dic3Q811yescharacteristic
29C3×#622—22no—
30C3×#622—22no—
31C3×#344—91nomaximal
32C22×C3⋊Dic3C22×C311yescharacteristic
33C22×C3×Dic3—62no—
34#3792—21no—
35#3792—21no—
36#3792—21no—
37#3792C3×C412yescomplemented
38C2×#622—21no—
39C2×#622C3×C412yescomplemented
40C32×C22⋊C4—22no—
41C32×C22⋊C4—42no—
42C32×C22⋊C4C3×C411yescharacteristic
43#3757C3×C411yescharacteristic
44#3750—21no—
45C4×C3⋊Dic3—21no—
46C3×#217—62no—
47C3×Dic3⋊C4—64no—
48C3×C4×Dic3—64no—
49C22×C33—41no—
50C22×C33C4211yescharacteristic
51C22×C33C22⋊C411yescharacteristic
52C22×C33C4⋊C411yescharacteristic
53C33×C4—41no—
54C3×C3⋊Dic3—41no—
55C3×C3⋊Dic3C22⋊C411yescharacteristic
56C3×C3⋊Dic3C22⋊C411yescharacteristic
57C3×C3⋊Dic3C4⋊C411yescharacteristic
58C3×C3⋊Dic3C4⋊C411yescharacteristic
59C2×C3×M16—91no—
60C2×C3×M16—91no—
61C3×#117—91no—
62#877—62no—
63C23×C32—22no—
64C23×C32—42no—
65C23×C32C2×C3×C411yescharacteristic
66C2×C32×C4—21no—
67C2×C32×C4—24no—
68C2×C32×C4—44no—
69C2×C3⋊Dic3—21no—
70C2×C3⋊Dic3C2×C3×C411yescharacteristic
71C2×C3⋊Dic3C2×C3×C411yescharacteristic
72C2×C3⋊Dic3C3×D411yescharacteristic
73C2×C3⋊Dic3C3×D411yescharacteristic
74C2×C3⋊Dic3C3×D411yescharacteristic
75C2×C3⋊Dic3C3×Q811yescharacteristic
76C2×C3×Dic3—62no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C2×C3×Dic3—62no—
82C2×C3×Dic3—64no—
83#622—22no—
84#622—22no—
85#344—91noSylow
86C2×C33—21no—
87C2×C33—21no—
88C2×C33—21no—
89C2×C33#9511yescharacteristic
90C22×C3×C4—91no—
91C22×Dic3—62no—
92C3×M16—92no—
93C3×M16—181no—
94C3×M16—181no—
95C3×M16—181no—
96C2×C3×C8—92no—
97C2×C3×C8—181no—
98C3×C4⋊C4—181no—
99C3×C22⋊C4—22no—
100C3×C22⋊C4—42no—
101C3×C22⋊C4—91no—
102C3×C22⋊C4#15711yescharacteristic
103C3×C42—181no—
104#217—62no—
105Dic3⋊C4—64no—
106C4×Dic3—64no—
107C22×C32—22no—
108C22×C32—22no—
109C22×C32—22no—
110C22×C32—42no—
111C22×C32—42no—
112C22×C32—42no—
113C22×C32—41no—
114C22×C32—44no—
115C22×C32—84no—
116C22×C32C3×C4211yescharacteristic
117C22×C32C3×C22⋊C411yescharacteristic
118C22×C32C3×C4⋊C411yescharacteristic
119C32×C4—41no—
120C32×C4—44no—
121C32×C4—84no—
122C3⋊Dic3—41no—
123C3⋊Dic3C3×C22⋊C411yescharacteristic
124C3⋊Dic3C3×C22⋊C411yescharacteristic
125C3⋊Dic3C3×C4⋊C411yescharacteristic
126C3⋊Dic3C3×C4⋊C411yescharacteristic
127C3×Dic3—62no—
128C3×Dic3—62no—
129C3×Dic3—62no—
130C3×Dic3—62no—
131C3×Dic3—124no—
132C2×M16—91no—
133C2×M16—91no—
134#117—91no—
135C33#34411yescharacteristic, complemented, Sylow
136C23×C3—22no—
137C23×C3—42no—
138C23×C3C2×#15711yescharacteristic
139C2×C3×C4—24no—
140C2×C3×C4—21no—
141C2×C3×C4—44no—
142C2×C3×C4—91no—
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×C3×C4—91no—
147C2×C3×C4—91no—
148C2×C3×C4—181no—
149C2×Dic3—62no—
150C2×Dic3—62no—
151C2×Dic3—62no—
152C2×Dic3—62no—
153C2×Dic3—62no—
154C2×Dic3—62no—
155C2×Dic3—64no—
156C3×C8—182no—
157C3×C8—182no—
158C2×C32—21no—
159C2×C32—24no—
160C2×C32—22no—
161C2×C32—21no—
162C2×C32—21no—
163C2×C32—44no—
164C2×C32—42no—
165C2×C32—42no—
166C2×C32—42no—
167C2×C32—82no—
168C2×C32—82no—
169C2×C32C3×#9511yescharacteristic
170C22×C4—91no—
171M16—92no—
172M16—181no—
173M16—181no—
174M16—181no—
175C2×C8—92no—
176C2×C8—181no—
177C4⋊C4—181no—
178C22⋊C4—91no—
179C22⋊C4C3×#15711yescharacteristic
180C42—181no—
181C22×C3—22no—
182C22×C3—22no—
183C22×C3—22no—
184C22×C3—41no—
185C22×C3—44no—
186C22×C3—42no—
187C22×C3—42no—
188C22×C3—42no—
189C22×C3—84no—
190C22×C3C4×#15711yescharacteristic
191C22×C3C4⋊#15711yescharacteristic
192C22×C3C22⋊#15711yescharacteristic
193C3×C4—41no—
194C3×C4—44no—
195C3×C4—84no—
196C3×C4—91no—
197C3×C4—91no—
198C3×C4—91no—
199C3×C4—91no—
200C3×C4—361no—
201Dic3—62no—
202Dic3—62no—
203Dic3—62no—
204Dic3—62no—
205Dic3—124no—
206C32—22no—
207C32—42no—
208C32C3×#34411yescharacteristic, complemented
209C23C2×C3×#15711yescharacteristic
210C2×C4—21no—
211C2×C4—91no—
212C2×C4—91no—
213C2×C4—91no—
214C2×C4—91no—
215C2×C4—91no—
216C2×C4—91no—
217C2×C4—181no—
218C8—182no—
219C8—182no—
220C2×C3—24no—
221C2×C3—21no—
222C2×C3—21no—
223C2×C3—21no—
224C2×C3—22no—
225C2×C3—42no—
226C2×C3—42no—
227C2×C3—44no—
228C2×C3—42no—
229C2×C3—82no—
230C2×C3—82no—
231C2×C3#6378111yescharacteristic
232C22—41no—
233C22C3×C4×#15711yescharacteristic
234C22C3×C4⋊#15711yescharacteristic
235C22C3×C22⋊#15711yescharacteristic
236C4—41no—
237C4—91no—
238C4—91no—
239C4—91no—
240C4—91no—
241C4—361no—
242C3—22no—
243C3—42no—
244C3#1059015611yescharacteristic, direct product factor
245C2—21no—
246C2—21no—
247C2—21no—
248C2C3×#6378111yescharacteristic
subgroupG/H
centerC2×C3#63781
commutatorC22×C32C3×C42
FrattiniC23C2×C3×#157
FittingC33×C22⋊C4C4
socleC2×C33#95
prime radicalC23×C33C2×C4
cosocleC22×C3⋊Dic3C22×C3
hypercenterC3×C22⋊C4#157
hypocenterC32C3×#344
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #10590156C3direct (N × H)
9. C3×#3792C4semidirect
11. C2×C3×#622C4semidirect
37. #3792C3×C4semidirect
39. C2×#622C3×C4semidirect
135. C33#344semidirect
208. C32C3×#344semidirect
244. C3#10590156direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37922
C3×#640862
#105901563
C3×#3449