Finite Group Databasebeta

← C3×#10593465 #13166834

Subgroups — C3×#10593465 #13166834

Order 1728.

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

normal subgroups only

This group has 416 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
1C3×D4×#157C211yesmaximal, characteristic
2C3×#64222C211yesmaximal, characteristic, complemented
3C3×#63930C212yesmaximal
4C3×#63732C211yesmaximal, characteristic, complemented
5C3×#63730C212yesmaximal, complemented
6#10593465C311yesmaximal, characteristic, direct product factor
7C22×C3×#157—21no—
8C3×D4×C3⋊D3C2211yescharacteristic
9C3×#3797C2212yes—
10C3×C22⋊#157—21no—
11C3×C4⋊#157C2211yescharacteristic
12C3×C4×#157C2211yescharacteristic
13C3×#3773—21no—
14C3×#3773—21no—
15C32×#214—22no—
16C3×#3717—22no—
17C3×#3710C2212yes—
18D4×#157C2×C311yescharacteristic
19#64222C2×C311yescharacteristic, complemented
20C3×D3×SD16—62no—
21#63930C2×C312yes—
22#63732C2×C311yescharacteristic, complemented
23#63730C2×C312yescomplemented
24C22×C3×C3⋊D3—21no—
25C2×C3×#157—21no—
26C2×C3×#157—42no—
27C2×C3×#157D411yescharacteristic
28C2×C3×#157D411yescharacteristic
29C33×D4D411yescharacteristic
30C3×#638—21no—
31C3×C4⋊(C3⋊D3)D411yescharacteristic
32C3×C4×C3⋊D3C2311yescharacteristic
33C32×Dic6—22no—
34C3×#627—22no—
35C3×#624—22no—
36C32×C3⋊C8—22no—
37C3×#461—91nomaximal
38C22×#157—21no—
39D4×C3⋊D3C22×C311yescharacteristic
40C3×D3×Q8—62no—
41C3×D3×D4—62no—
42#3797C22×C312yes—
43C22⋊#157—21no—
44C4⋊#157C22×C311yescharacteristic
45C4×#157C22×C311yescharacteristic
46#3773—21no—
47#3773—21no—
48C32×SD16—62no—
49C3×#215—62no—
50C3×#214—22no—
51C3×#214—22no—
52C3×#214—42no—
53C3×#204—62no—
54C3×D3×C8—62no—
55#3717—22no—
56#3710C22×C312yes—
57C22×C33—21no—
58C2×C3×C3⋊D3—21no—
59C2×C3×C3⋊D3—41no—
60C2×C3×C3⋊D3—41no—
61C2×C3×C3⋊D3C2×D411yescharacteristic
62C3×#157—21no—
63C3×#157—41no—
64C3×#157SD1612yescomplemented
65C33×C4C2×D411yescharacteristic
66C3×C3⋊Dic3#4111yescharacteristic
67C32×Dic3—42no—
68C2×C3×SD16—92no—
69C3×C4⋊Q8—91no—
70C3×C4×D4—91no—
71D3×SD16—62no—
72C3×C4⋊C8—91no—
73C3×#103—92no—
74C22×C3⋊D3—21no—
75C22×C3×D3—122no—
76C2×#157—21no—
77C2×#157—42no—
78C2×#157C3×D411yescharacteristic
79C2×#157C3×D411yescharacteristic
80C32×Q8—62no—
81C32×D4—22no—
82C32×D4—42no—
83C32×D4C3×D411yescharacteristic
84#638—21no—
85C4⋊(C3⋊D3)C3×D411yescharacteristic
86C4×C3⋊D3C23×C311yescharacteristic
87C3×#67—122no—
88C3×D12—62no—
89C3×C4×D3—62no—
90C3×C4×D3—122no—
91C3×Dic6—22no—
92C3×Dic6—22no—
93C3×Dic6—42no—
94C3×Dic6—122no—
95#627—22no—
96#624—22no—
97C32×C8—62no—
98C3×C3⋊C8—22no—
99C3×C3⋊C8—22no—
100C3×C3⋊C8—42no—
101#461—91noSylow
102C2×C33—41no—
103C2×C33#12111yescharacteristic
104C3×C3⋊D3—41no—
105C3×C3⋊D3C2×SD1611yescharacteristic
106C3×C3⋊D3#13711yescharacteristic
107C2×C3×Q8—92no—
108C2×C3×D4—91no—
109C22×C3×C4—181no—
110D3×Q8—62no—
111D3×D4—62no—
112C3×SD16—62no—
113C3×SD16—62no—
114C3×SD16—122no—
115C3×SD16—182no—
116C2×C3×C8—92no—
117C3×C4⋊C4—91no—
118C3×C4⋊C4—181no—
119C3×C4⋊C4—181no—
120C3×C22⋊C4—181no—
121C3×C42—91no—
122#215—62no—
123#214—22no—
124#204—62no—
125D3×C8—62no—
126C22×C32—21no—
127C22×C32—42no—
128C22×C32—82no—
129C2×C3⋊D3—21no—
130C2×C3⋊D3—41no—
131C2×C3⋊D3—41no—
132C2×C3⋊D3C2×C3×D411yescharacteristic
133C2×C3×D3—62no—
134C2×C3×D3—122no—
135C2×C3×D3—242no—
136C2×C3×D3—242no—
137#157—21no—
138#157—41no—
139#157C3×SD1612yescomplemented
140C32×C4—22no—
141C32×C4—42no—
142C32×C4—122no—
143C32×C4C2×C3×D411yescharacteristic
144C3⋊Dic3C3×#4111yescharacteristic
145C3×Dic3—42no—
146C3×Dic3—42no—
147C3×Dic3—62no—
148C3×Dic3—82no—
149C2×SD16—92no—
150C4⋊Q8—91no—
151C4×D4—91no—
152C4⋊C8—91no—
153#103—92no—
154C33#46111yescharacteristic, complemented, Sylow
155C23×C3—181no—
156C22×D3—122no—
157C3×Q8—62no—
158C3×Q8—62no—
159C3×Q8—122no—
160C3×Q8—182no—
161C3×D4—22no—
162C3×D4—42no—
163C3×D4—91no—
164C3×D4—181no—
165C3×D4GO+(4,𝔽2)11yescharacteristic
166C2×C3×C4—91no—
167C2×C3×C4—91no—
168C2×C3×C4—91no—
169C2×C3×C4—182no—
170C2×C3×C4—181no—
171C2×C3×C4—362no—
172#67—122no—
173D12—62no—
174C4×D3—62no—
175C4×D3—122no—
176Dic6—22no—
177Dic6—122no—
178C3×C8—62no—
179C3×C8—62no—
180C3×C8—122no—
181C3⋊C8—22no—
182C2×C32—22no—
183C2×C32—42no—
184C2×C32—41no—
185C2×C32—82no—
186C2×C32—162no—
187C2×C32C3×#12111yescharacteristic
188C3⋊D3—41no—
189C3⋊D3C2×C3×SD1611yescharacteristic
190C3⋊D3C3×#13711yescharacteristic
191C3×D3—62no—
192C3×D3—62no—
193C3×D3—242no—
194C2×Q8—92no—
195C2×D4—91no—
196C22×C4—181no—
197SD16—62no—
198SD16—182no—
199C2×C8—92no—
200C4⋊C4—91no—
201C4⋊C4—181no—
202C4⋊C4—181no—
203C22⋊C4—181no—
204C42—91no—
205C22×C3—21no—
206C22×C3—42no—
207C22×C3—82no—
208C22×C3—91no—
209C22×C3—181no—
210C22×C3—361no—
211C22×C3—361no—
212C2×D3—62no—
213C2×D3—122no—
214C2×D3—242no—
215C2×D3—242no—
216C3×C4—22no—
217C3×C4—42no—
218C3×C4—92no—
219C3×C4—91no—
220C3×C4—122no—
221C3×C4—122no—
222C3×C4—181no—
223C3×C4—242no—
224C3×C4—361no—
225C3×C4C2×GO+(4,𝔽2)11yescharacteristic
226Dic3—42no—
227Dic3—62no—
228C32—22no—
229C32—42no—
230C32C3×#46111yescharacteristic, complemented
231C23—181no—
232Q8—62no—
233Q8—182no—
234D4—91no—
235D4—181no—
236D4C3×GO+(4,𝔽2)11yescharacteristic
237C2×C4—91no—
238C2×C4—91no—
239C2×C4—91no—
240C2×C4—182no—
241C2×C4—181no—
242C2×C4—362no—
243C8—62no—
244C2×C3—22no—
245C2×C3—41no—
246C2×C3—42no—
247C2×C3—82no—
248C2×C3—91no—
249C2×C3—91no—
250C2×C3—162no—
251C2×C3—361no—
252C2×C3#6424211yescharacteristic
253D3—62no—
254D3—62no—
255D3—242no—
256C22—21no—
257C22—91no—
258C22—181no—
259C22—361no—
260C22—361no—
261C4—92no—
262C4—91no—
263C4—122no—
264C4—181no—
265C4—361no—
266C4C2×C3×GO+(4,𝔽2)11yescharacteristic
267C3—22no—
268C3—42no—
269C3#1059346511yescharacteristic, direct product factor
270C2—41no—
271C2—91no—
272C2—91no—
273C2—361no—
274C2C3×#6424211yescharacteristic
subgroupG/H
centerC2×C3#64242
commutator, cosocleC4×C3⋊D3C23×C3
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×D4D4
socleC2×C33#121
prime radicalC3×D4×C3⋊D3C22
hypercenterC3×D4GO+(4,𝔽2)
hypocenterC32C3×#461
LCSG ▷ C4×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×#64222C2semidirect
4. C3×#63732C2semidirect
5. C3×#63730C2semidirect
6. #10593465C3direct (N × H)
19. #64222C2×C3semidirect
22. #63732C2×C3semidirect
23. #63730C2×C3semidirect
64. C3×#157SD16semidirect
139. #157C3×SD16semidirect
154. C33#461semidirect
230. C32C3×#461semidirect
269. C3#10593465direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D4×#1572
C3×#642222
C3×#639302
C3×#637322
C3×#637302
#105934653
C3×#4619