Finite Group Databasebeta

← C3×#10593508 #13166896

Subgroups — C3×#10593508 #13166896

Order 1728.

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

normal subgroups only

This group has 436 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×#3792C211yesmaximal, characteristic
2C2×C3×#3776C211yesmaximal, characteristic, complemented
3C3×#63982C212yesmaximal
4C3×#63742C211yesmaximal, characteristic, complemented
5C3×#63740C212yesmaximal, complemented
6#10593508C311yesmaximal, characteristic, direct product factor
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C3×#627C2212yes—
9C3×#3792—22no—
10C2×C3×#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C3×#3776—22no—
13C32×#217—22no—
14C3×#3724C2212yes—
15C3×Dic3⋊Dic3—22no—
16C2×#3792C2×C311yescharacteristic
17C2×#3776C2×C311yescharacteristic, complemented
18C3×#878—62no—
19#63982C2×C312yes—
20#63742C2×C311yescharacteristic, complemented
21#63740C2×C312yescomplemented
22C23×C33D411yescharacteristic
23C2×C3×C3⋊Dic3—21no—
24C2×C3×C3⋊Dic3—21no—
25C2×C3×C3⋊Dic3D411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C2×C32×Dic3—22no—
28C2×C32×Dic3—22no—
29C3×#627—22no—
30C3×#627—22no—
31C3×#622—21no—
32C3×#622D412yes—
33C3×#470—91nomaximal
34C22×C3⋊Dic3C22×C311yescharacteristic
35C22×C3×Dic3—62no—
36C2×C3×Dic6—62no—
37C2×#627C22×C312yes—
38#3792—22no—
39C2×#622C22×C311yescharacteristic
40C2×#622C22×C311yescharacteristic
41#3776—22no—
42C32×C22⋊C4—62no—
43C3×#217—22no—
44C3×#217—22no—
45C3×#217—42no—
46C3×C4⋊Dic3—62no—
47C3×Dic3⋊C4—62no—
48C3×Dic3⋊C4—62no—
49#3724C22×C312yes—
50Dic3⋊Dic3—22no—
51C22×C33—21no—
52C22×C33—22no—
53C22×C33C2×D411yescharacteristic
54C3×C3⋊Dic3—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#878—62no—
63C3×#106—91no—
64C3×#103—92no—
65C23×C32—22no—
66C23×C32—42no—
67C23×C32C3×D411yescharacteristic
68C2×C32×C4—62no—
69C2×C32×C4—62no—
70C2×C3⋊Dic3—21no—
71C2×C3⋊Dic3—21no—
72C2×C3⋊Dic3C3×D411yescharacteristic
73C2×C3⋊Dic3C23×C311yescharacteristic
74C2×C3×Dic3—22no—
75C2×C3×Dic3—22no—
76C2×C3×Dic3—22no—
77C2×C3×Dic3—22no—
78C2×C3×Dic3—42no—
79C2×C3×Dic3—42no—
80C2×C3×Dic3—62no—
81C2×C3×Dic3—62no—
82C2×C3×Dic3—122no—
83C2×C3×Dic3—122no—
84C3×Dic6—122no—
85C3×Dic6—122no—
86#627—22no—
87#627—22no—
88#622—21no—
89#622C3×D412yes—
90#470—91noSylow
91C2×C33—41no—
92C2×C33#12111yescharacteristic
93C2×C33#13712yes—
94C2×C3×Q8—92no—
95C22×C3×C4—91no—
96C22×Dic3—62no—
97C2×Dic6—62no—
98C3×Q16—182no—
99C3×M16—182no—
100C2×C3×C8—91no—
101C2×C3×C8—91no—
102C3×C4⋊C4—92no—
103C3×C4⋊C4—182no—
104C3×C22⋊C4—62no—
105C3×C22⋊C4—62no—
106C3×C22⋊C4—122no—
107#217—22no—
108C4⋊Dic3—62no—
109Dic3⋊C4—62no—
110Dic3⋊C4—62no—
111C22×C32—24no—
112C22×C32—22no—
113C22×C32—22no—
114C22×C32—21no—
115C22×C32—42no—
116C22×C32—42no—
117C22×C32—42no—
118C22×C32—44no—
119C22×C32—82no—
120C22×C32—82no—
121C22×C32C2×C3×D411yescharacteristic
122C32×C4—122no—
123C32×C4—122no—
124C3⋊Dic3—21no—
125C3⋊Dic3C2×C3×D411yescharacteristic
126C3⋊Dic3C3×#4111yescharacteristic
127C3×Dic3—42no—
128C3×Dic3—42no—
129C3×Dic3—42no—
130C3×Dic3—42no—
131C3×Dic3—62no—
132C3×Dic3—62no—
133C3×Dic3—82no—
134C3×Dic3—82no—
135C3×Dic3—122no—
136C2×Q16—91no—
137C2×M16—91no—
138C22⋊Q8—92no—
139#106—91no—
140#103—92no—
141C33#47011yescharacteristic, complemented, Sylow
142C23×C3—22no—
143C23×C3—42no—
144C23×C3GO+(4,𝔽2)11yescharacteristic
145C3×Q8—182no—
146C3×Q8—182no—
147C2×C3×C4—62no—
148C2×C3×C4—62no—
149C2×C3×C4—62no—
150C2×C3×C4—62no—
151C2×C3×C4—91no—
152C2×C3×C4—91no—
153C2×C3×C4—122no—
154C2×C3×C4—122no—
155C2×C3×C4—181no—
156C2×C3×C4—181no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3—62no—
160C2×Dic3—62no—
161C2×Dic3—122no—
162C2×Dic3—122no—
163Dic6—122no—
164Dic6—122no—
165C3×C8—92no—
166C3×C8—181no—
167C2×C32—22no—
168C2×C32—22no—
169C2×C32—22no—
170C2×C32—42no—
171C2×C32—42no—
172C2×C32—42no—
173C2×C32—41no—
174C2×C32—44no—
175C2×C32—84no—
176C2×C32C3×#12111yescharacteristic
177C2×C32C3×#13712yes—
178C2×Q8—92no—
179C22×C4—91no—
180Q16—182no—
181M16—182no—
182C2×C8—91no—
183C2×C8—91no—
184C4⋊C4—92no—
185C4⋊C4—182no—
186C22⋊C4—62no—
187C22×C3—24no—
188C22×C3—22no—
189C22×C3—21no—
190C22×C3—22no—
191C22×C3—42no—
192C22×C3—42no—
193C22×C3—44no—
194C22×C3—42no—
195C22×C3—82no—
196C22×C3—82no—
197C22×C3C2×GO+(4,𝔽2)11yescharacteristic
198C3×C4—91no—
199C3×C4—91no—
200C3×C4—122no—
201C3×C4—122no—
202C3×C4—122no—
203C3×C4—122no—
204C3×C4—181no—
205C3×C4—242no—
206C3×C4—242no—
207Dic3—42no—
208Dic3—42no—
209Dic3—62no—
210Dic3—62no—
211Dic3—122no—
212C32—22no—
213C32—42no—
214C32C3×#47011yescharacteristic, complemented
215C23C3×GO+(4,𝔽2)11yescharacteristic
216Q8—182no—
217Q8—182no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—91no—
221C2×C4—91no—
222C2×C4—181no—
223C2×C4—181no—
224C8—92no—
225C8—181no—
226C2×C3—22no—
227C2×C3—22no—
228C2×C3—22no—
229C2×C3—44no—
230C2×C3—41no—
231C2×C3—42no—
232C2×C3—42no—
233C2×C3—42no—
234C2×C3—84no—
235C2×C3#6423912yes—
236C2×C3#6424211yescharacteristic
237C22—22no—
238C22—21no—
239C22C2×C3×GO+(4,𝔽2)11yescharacteristic
240C4—91no—
241C4—91no—
242C4—122no—
243C4—122no—
244C4—181no—
245C3—22no—
246C3—42no—
247C3#1059350811yescharacteristic, direct product factor
248C2—41no—
249C2C3×#6423912yes—
250C2C3×#6424211yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
Fitting, prime radicalC23×C33D4
socleC22×C33C2×D4
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#470
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

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. #10593508C3direct (N × H)
17. C2×#3776C2×C3semidirect
20. #63742C2×C3semidirect
21. #63740C2×C3semidirect
141. C33#470semidirect
214. C32C3×#470semidirect
247. C3#10593508direct (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
C2×C3×#37762
C3×#639822
C3×#637422
C3×#637402
#105935083
C3×#4709