Finite Group Databasebeta

← C3×#10593424 #13166793

Subgroups — C3×#10593424 #13166793

Order 1728.

480 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3776C211yesmaximal, characteristic, complemented
2C2×C3×#3775C212yesmaximal, complemented
3C2×C3×#3774C211yesmaximal, characteristic
4C3×#63915C212yesmaximal, complemented
5C3×C4×#622C211yesmaximal, characteristic, complemented
6#10593424C311yesmaximal, characteristic, direct product factor
7C2×C3×#627C2212yes—
8C2×C3×#625C2212yes—
9C2×C3×#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11C3×#3776—22no—
12C3×#3775—24no—
13C3×#3774—22no—
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C32×(C2×D3)⋊C4—22no—
16C3×(C2×D3)⋊Dic3—22no—
17C2×#3776C2×C311yescharacteristic, complemented
18C2×#3775C2×C312yescomplemented
19C2×#3774C2×C311yescharacteristic
20C3×#885—62no—
21#63915C2×C312yescomplemented
22C4×#622C2×C311yescharacteristic, complemented
23C22×C32×D3—22no—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3D411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C2×C32×Dic3—22no—
28C3×#627—22no—
29C3×#627—22no—
30C3×#625—22no—
31C3×#625—22no—
32C3×#622D412yes—
33C3×#622D412yes—
34C3×#495—91nomaximal
35C2×C3×#67—62no—
36C2×C3×Dic6—62no—
37C2×#627C22×C312yes—
38C2×#625C22×C312yes—
39C2×#622C22×C311yescharacteristic
40C2×#622C22×C311yescharacteristic
41#3776—22no—
42#3775—24no—
43#3774—22no—
44C32×C22⋊C4—62no—
45C4×C3⋊Dic3C22×C311yescharacteristic
46C3×#217—62no—
47C3×(C2×D3)⋊C4—22no—
48C3×(C2×D3)⋊C4—22no—
49C3×(C2×D3)⋊C4—42no—
50C3×(C2×D3)⋊C4—62no—
51C3×C4×Dic3—62no—
52(C2×D3)⋊Dic3—22no—
53C22×C33C2×D411yescharacteristic
54C2×C32×D3—42no—
55C2×C32×D3—42no—
56C2×C32×D3—42no—
57C33×C4—21no—
58C3×C3⋊Dic3—21no—
59C3×C3⋊Dic3C2×D411yescharacteristic
60C3×C3⋊Dic3C2×D411yescharacteristic
61C32×Dic3—42no—
62C2×C3×Q16—91no—
63C2×C3×SD16—92no—
64C2×C3×D8—91no—
65C3×#124—92no—
66#885—62no—
67C3×C4×C8—91no—
68C23×C32—62no—
69C22×C3×D3—22no—
70C22×C3×D3—22no—
71C22×C3×D3—42no—
72C2×C32×C4—22no—
73C2×C32×C4—42no—
74C2×C32×C4—62no—
75C2×C32×C4C3×D411yescharacteristic
76C2×C3⋊Dic3C3×D411yescharacteristic
77C2×C3⋊Dic3C23×C311yescharacteristic
78C3×#67—122no—
79C3×#67—122no—
80C2×C3×Dic3—22no—
81C2×C3×Dic3—22no—
82C2×C3×Dic3—42no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—62no—
85C3×Dic6—122no—
86C3×Dic6—122no—
87#627—22no—
88#627—22no—
89#625—22no—
90#625—22no—
91#622C3×D412yes—
92#622C3×D412yes—
93#495—91noSylow
94C2×C33#12711yescharacteristic
95C2×C33#13511yescharacteristic
96C2×C33#13511yescharacteristic
97C32×D3—82no—
98C2×C3×Q8—92no—
99C2×C3×D4—92no—
100C2×#67—62no—
101C2×Dic6—62no—
102C3×Q16—182no—
103C3×SD16—184no—
104C3×D8—182no—
105C2×C3×C8—91no—
106C2×C3×C8—91no—
107C3×C22⋊C4—62no—
108C3×C22⋊C4—62no—
109C3×C22⋊C4—122no—
110C3×C22⋊C4—182no—
111C3×C42—91no—
112#217—62no—
113(C2×D3)⋊C4—22no—
114(C2×D3)⋊C4—62no—
115C4×Dic3—62no—
116C22×C32—22no—
117C22×C32—42no—
118C22×C32—122no—
119C22×C32—122no—
120C22×C32—122no—
121C22×C32C2×C3×D411yescharacteristic
122C2×C3×D3—42no—
123C2×C3×D3—42no—
124C2×C3×D3—42no—
125C2×C3×D3—44no—
126C2×C3×D3—42no—
127C2×C3×D3—42no—
128C2×C3×D3—42no—
129C2×C3×D3—82no—
130C2×C3×D3—82no—
131C32×C4—21no—
132C32×C4—42no—
133C32×C4—82no—
134C32×C4—122no—
135C3⋊Dic3—21no—
136C3⋊Dic3C2×C3×D411yescharacteristic
137C3⋊Dic3C2×C3×D411yescharacteristic
138C3×Dic3—42no—
139C3×Dic3—42no—
140C3×Dic3—62no—
141C3×Dic3—62no—
142C3×Dic3—82no—
143C3×Dic3—122no—
144C2×Q16—91no—
145C2×SD16—92no—
146C2×D8—91no—
147#124—92no—
148C4×C8—91no—
149C33#49511yescharacteristic, complemented, Sylow
150C23×C3—62no—
151C23×C3—62no—
152C23×C3—122no—
153C22×D3—22no—
154C3×Q8—182no—
155C3×Q8—182no—
156C3×D4—182no—
157C3×D4—182no—
158C2×C3×C4—22no—
159C2×C3×C4—42no—
160C2×C3×C4—62no—
161C2×C3×C4—62no—
162C2×C3×C4—91no—
163C2×C3×C4—91no—
164C2×C3×C4—122no—
165C2×C3×C4GO+(4,𝔽2)11yescharacteristic
166#67—122no—
167#67—122no—
168C2×Dic3—22no—
169C2×Dic3—62no—
170C2×Dic3—62no—
171Dic6—122no—
172Dic6—122no—
173C3×C8—92no—
174C3×C8—92no—
175C2×C32—22no—
176C2×C32—22no—
177C2×C32—22no—
178C2×C32—42no—
179C2×C32—42no—
180C2×C32—42no—
181C2×C32—242no—
182C2×C32C3×#12711yescharacteristic
183C2×C32C3×#13511yescharacteristic
184C2×C32C3×#13511yescharacteristic
185C3×D3—82no—
186C3×D3—84no—
187C3×D3—82no—
188C2×Q8—92no—
189C2×D4—92no—
190Q16—182no—
191SD16—184no—
192D8—182no—
193C2×C8—91no—
194C2×C8—91no—
195C22⋊C4—62no—
196C22⋊C4—182no—
197C42—91no—
198C22×C3—22no—
199C22×C3—42no—
200C22×C3—122no—
201C22×C3—122no—
202C22×C3—122no—
203C22×C3—124no—
204C22×C3—122no—
205C22×C3—122no—
206C22×C3—122no—
207C22×C3—242no—
208C22×C3—242no—
209C22×C3C2×GO+(4,𝔽2)11yescharacteristic
210C2×D3—42no—
211C2×D3—42no—
212C2×D3—42no—
213C3×C4—21no—
214C3×C4—42no—
215C3×C4—82no—
216C3×C4—91no—
217C3×C4—91no—
218C3×C4—122no—
219C3×C4—122no—
220C3×C4—181no—
221C3×C4—242no—
222Dic3—42no—
223Dic3—62no—
224Dic3—62no—
225Dic3—122no—
226C32—22no—
227C32—42no—
228C32C3×#49511yescharacteristic, complemented
229C23—62no—
230Q8—182no—
231Q8—182no—
232D4—182no—
233D4—182no—
234C2×C4—62no—
235C2×C4—91no—
236C2×C4—91no—
237C2×C4C3×GO+(4,𝔽2)11yescharacteristic
238C8—92no—
239C8—92no—
240C2×C3—22no—
241C2×C3—22no—
242C2×C3—22no—
243C2×C3—42no—
244C2×C3—42no—
245C2×C3—42no—
246C2×C3—242no—
247C2×C3—244no—
248C2×C3—242no—
249C2×C3#6422311yescharacteristic
250C2×C3#6423111yescharacteristic
251C2×C3#6423711yescharacteristic
252D3—82no—
253C22—122no—
254C22—122no—
255C22—122no—
256C22C2×C3×GO+(4,𝔽2)11yescharacteristic
257C4—21no—
258C4—91no—
259C4—91no—
260C4—122no—
261C4—181no—
262C3—22no—
263C3—42no—
264C3#1059342411yescharacteristic, direct product factor
265C2—242no—
266C2C3×#6422311yescharacteristic
267C2C3×#6423111yescharacteristic
268C2C3×#6423711yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#3774C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#495
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
1. C2×C3×#3776C2semidirect
2. C2×C3×#3775C2semidirect
4. C3×#63915C2semidirect
5. C3×C4×#622C2semidirect
6. #10593424C3direct (N × H)
17. C2×#3776C2×C3semidirect
18. C2×#3775C2×C3semidirect
21. #63915C2×C3semidirect
22. C4×#622C2×C3semidirect
149. C33#495semidirect
228. C32C3×#495semidirect
264. C3#10593424direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37762
C2×C3×#37752
C2×C3×#37742
C3×#639152
C3×C4×#6222
#105934243
C3×#4959