Finite Group Databasebeta

← C3×#10593449 #13166818

Subgroups — C3×#10593449 #13166818

Order 1728.

420 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 66 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×#3788C211yesmaximal, characteristic
2C2×C3×#3776C211yesmaximal, characteristic, complemented
3C3×#63865C212yesmaximal
4C3×#63742C211yesmaximal, characteristic, complemented
5C3×#63740C212yesmaximal, complemented
6#10593449C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3×#627C2212yes—
9C2×C3×#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11C3×#3788—22no—
12C3×#3776—22no—
13C32×Dic3⋊C4—22no—
14C3×#3724C2212yes—
15C3×#3722—22no—
16C2×#3788C2×C311yescharacteristic
17C2×#3776C2×C311yescharacteristic, complemented
18C3×#896—62no—
19#63865C2×C312yes—
20#63742C2×C311yescharacteristic, complemented
21#63740C2×C312yescomplemented
22C22×C3×C3⋊D3D411yescharacteristic
23C2×C33×C4D411yescharacteristic
24C2×C3×C3⋊Dic3C2311yescharacteristic
25C3×C4×C3⋊D3—21no—
26C3×C4×C3⋊D3—21no—
27C2×C32×Dic3—22no—
28C2×C32×Dic3—22no—
29C3×#627—22no—
30C3×#627—22no—
31C3×#622—21no—
32C3×#622D412yes—
33C3×#470—91nomaximal
34C2×C4×C3⋊D3C22×C311yescharacteristic
35C2×C3×C4×D3—62no—
36C2×C3×Dic6—62no—
37C2×#627C22×C312yes—
38C2×#622C22×C311yescharacteristic
39C2×#622C22×C311yescharacteristic
40#3788—22no—
41#3776—22no—
42C32×C4⋊C4—62no—
43C3×(C2×D3)⋊C4—62no—
44C3×(C2×D3)⋊C4—62no—
45C3×Dic3⋊C4—22no—
46C3×Dic3⋊C4—22no—
47C3×Dic3⋊C4—42no—
48C3×Dic3⋊C4—62no—
49#3724C22×C312yes—
50#3722—22no—
51C22×C33C2×D411yescharacteristic
52C2×C3×C3⋊D3—21no—
53C2×C3×C3⋊D3—22no—
54C33×C4—21no—
55C3×C3⋊Dic3C2×D411yescharacteristic
56C3×C3⋊Dic3#4111yescharacteristic
57C32×Dic3—42no—
58C32×Dic3—42no—
59C2×C3×Q16—91no—
60C2×C3×M16—91no—
61C3×C22⋊Q8—92no—
62#896—62no—
63C3×#106—91no—
64C3×#103—92no—
65C22×C3⋊D3C3×D411yescharacteristic
66C22×C3×D3—62no—
67C2×C32×C4—22no—
68C2×C32×C4—42no—
69C2×C32×C4—62no—
70C2×C32×C4—62no—
71C2×C32×C4C3×D411yescharacteristic
72C2×C3⋊Dic3C23×C311yescharacteristic
73C4×C3⋊D3—21no—
74C4×C3⋊D3—21no—
75C2×C3×Dic3—22no—
76C2×C3×Dic3—22no—
77C2×C3×Dic3—22no—
78C2×C3×Dic3—22no—
79C2×C3×Dic3—42no—
80C2×C3×Dic3—42no—
81C2×C3×Dic3—62no—
82C3×C4×D3—122no—
83C3×C4×D3—122no—
84C3×Dic6—122no—
85C3×Dic6—122no—
86#627—22no—
87#627—22no—
88#622—21no—
89#622C3×D412yes—
90#470—91noSylow
91C2×C33#12111yescharacteristic
92C2×C33#13712yes—
93C3×C3⋊D3—41no—
94C2×C3×Q8—92no—
95C22×C3×C4—91no—
96C2×C4×D3—62no—
97C2×Dic6—62no—
98C3×Q16—182no—
99C3×M16—182no—
100C2×C3×C8—91no—
101C2×C3×C8—91no—
102C3×C4⋊C4—62no—
103C3×C4⋊C4—62no—
104C3×C4⋊C4—92no—
105C3×C4⋊C4—122no—
106C3×C22⋊C4—182no—
107(C2×D3)⋊C4—62no—
108(C2×D3)⋊C4—62no—
109Dic3⋊C4—22no—
110Dic3⋊C4—62no—
111C22×C32—22no—
112C22×C32—42no—
113C22×C32C2×C3×D411yescharacteristic
114C2×C3⋊D3—22no—
115C2×C3⋊D3—21no—
116C2×C3×D3—64no—
117C2×C3×D3—122no—
118C2×C3×D3—122no—
119C32×C4—21no—
120C32×C4—42no—
121C32×C4—82no—
122C32×C4—122no—
123C32×C4—122no—
124C3⋊Dic3C2×C3×D411yescharacteristic
125C3⋊Dic3C3×#4111yescharacteristic
126C3×Dic3—42no—
127C3×Dic3—42no—
128C3×Dic3—44no—
129C3×Dic3—42no—
130C3×Dic3—42no—
131C3×Dic3—62no—
132C3×Dic3—62no—
133C3×Dic3—82no—
134C2×Q16—91no—
135C2×M16—91no—
136C22⋊Q8—92no—
137#106—91no—
138#103—92no—
139C33#47011yescharacteristic, complemented, Sylow
140C23×C3—91no—
141C22×D3—62no—
142C3×Q8—182no—
143C3×Q8—182no—
144C2×C3×C4—22no—
145C2×C3×C4—42no—
146C2×C3×C4—62no—
147C2×C3×C4—62no—
148C2×C3×C4—62no—
149C2×C3×C4—62no—
150C2×C3×C4—91no—
151C2×C3×C4—122no—
152C2×C3×C4—122no—
153C2×C3×C4—181no—
154C2×C3×C4—181no—
155C2×C3×C4GO+(4,𝔽2)11yescharacteristic
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3—62no—
159C4×D3—122no—
160C4×D3—122no—
161Dic6—122no—
162Dic6—122no—
163C3×C8—92no—
164C3×C8—181no—
165C2×C32—22no—
166C2×C32—22no—
167C2×C32—22no—
168C2×C32—42no—
169C2×C32—42no—
170C2×C32—42no—
171C2×C32C3×#12111yescharacteristic
172C2×C32C3×#13712yes—
173C3⋊D3—41no—
174C3×D3—124no—
175C2×Q8—92no—
176C22×C4—91no—
177Q16—182no—
178M16—182no—
179C2×C8—91no—
180C2×C8—91no—
181C4⋊C4—62no—
182C4⋊C4—92no—
183C22⋊C4—182no—
184C22×C3—22no—
185C22×C3—42no—
186C22×C3—182no—
187C22×C3—181no—
188C22×C3C2×GO+(4,𝔽2)11yescharacteristic
189C2×D3—64no—
190C2×D3—122no—
191C2×D3—122no—
192C3×C4—21no—
193C3×C4—42no—
194C3×C4—82no—
195C3×C4—91no—
196C3×C4—91no—
197C3×C4—122no—
198C3×C4—122no—
199C3×C4—124no—
200C3×C4—122no—
201C3×C4—122no—
202C3×C4—242no—
203Dic3—42no—
204Dic3—42no—
205Dic3—62no—
206Dic3—62no—
207C32—22no—
208C32—42no—
209C32C3×#47011yescharacteristic, complemented
210C23—91no—
211Q8—182no—
212Q8—182no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—91no—
216C2×C4—181no—
217C2×C4—181no—
218C2×C4C3×GO+(4,𝔽2)11yescharacteristic
219C8—92no—
220C8—181no—
221C2×C3—22no—
222C2×C3—22no—
223C2×C3—22no—
224C2×C3—42no—
225C2×C3—42no—
226C2×C3—42no—
227C2×C3—361no—
228C2×C3#6422612yes—
229C2×C3#6424211yescharacteristic
230D3—124no—
231C22—181no—
232C22—182no—
233C22C2×C3×GO+(4,𝔽2)11yescharacteristic
234C4—21no—
235C4—91no—
236C4—91no—
237C4—122no—
238C4—122no—
239C3—22no—
240C3—42no—
241C3#1059344911yescharacteristic, direct product factor
242C2—361no—
243C2C3×#6422612yes—
244C2C3×#6424211yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#470
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3776C2semidirect
4. C3×#63742C2semidirect
5. C3×#63740C2semidirect
6. #10593449C3direct (N × H)
17. C2×#3776C2×C3semidirect
20. #63742C2×C3semidirect
21. #63740C2×C3semidirect
139. C33#470semidirect
209. C32C3×#470semidirect
241. C3#10593449direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37882
C2×C3×#37762
C3×#638652
C3×#637422
C3×#637402
#105934493
C3×#4709