Finite Group Databasebeta

← C3×#10590712 #13157712

Subgroups — C3×#10590712 #13157712

Order 1728.

626 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 160 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×C8×C3⋊D3C211yesmaximal, characteristic
2C3×#63849C212yesmaximal
3C32×#817C212yesmaximal, complemented
4C3×#63576C212yesmaximal, complemented
5C3×#4860—32nomaximal
6#10590712C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C33×C8C2211yescharacteristic
9C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
10C3×C8×C3⋊D3C414yescomplemented
11C32×C4⋊Dic3C2214yes—
12C3×#3722—22no—
13C3×Dic32—22no—
14C2×C8×C3⋊D3C2×C311yescharacteristic
15C2×C3×D3×C8—32no—
16C2×C3×D3×C8—61no—
17C3×#892—34no—
18#63849C2×C312yes—
19C32×#106—32no—
20C3×#817—22no—
21C3×#817D312yescomplemented
22C3×#817C2×C312yescomplemented
23C3×#807—34no—
24#63576C2×C312yescomplemented
25C22×C3×C3⋊D3D411yescharacteristic
26C2×C33×C4C2311yescharacteristic
27C2×C3×C3⋊Dic3D411yescharacteristic
28C3×C4×C3⋊D3C2×C412yes—
29C3×C4×C3⋊D3Q812yes—
30C2×C32×Dic3—24no—
31C33×C8C2×C412yes—
32C3×C3⋊(C3⋊C8)C2×C412yes—
33C3×#427—91no—
34#4860—32no—
35C2×C4×C3⋊D3C22×C311yescharacteristic
36C2×C3×C4×D3—32no—
37C2×C3×C4×D3—61no—
38C2×C32×C8—21no—
39C2×C32×C8—22no—
40C2×C32×C8—41no—
41C2×C32×C8C2×D312yes—
42C2×C32×C8C22×C311yescharacteristic
43C32×C4⋊C4—34no—
44C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
45C8×C3⋊D3C3×C414yescomplemented
46C3×(C2×D3)⋊C4—64no—
47C3×C4⋊Dic3—24no—
48C3×C4⋊Dic3C2×D314yes—
49C3×C4⋊Dic3C22×C314yes—
50C3×C4×Dic3—64no—
51C2×C3×C3⋊C8—32no—
52C2×C3×C3⋊C8—61no—
53C3×D3×C8—38no—
54C3×D3×C8—64no—
55#3722—22no—
56Dic32—22no—
57C22×C33C2×D411yescharacteristic
58C2×C3×C3⋊D3—22no—
59C2×C3×C3⋊D3C4⋊C412yes—
60C33×C4C22×C411yescharacteristic
61C33×C4C2×Q811yescharacteristic
62C3×C3⋊Dic3C4⋊C412yes—
63C32×Dic3—44no—
64C22×C3×C8—91no—
65C3×#117—92no—
66C2×D3×C8—32no—
67C2×D3×C8—61no—
68#892—34no—
69C3×#107—92no—
70C3×#106—32no—
71C3×#106—32no—
72C3×#106—62no—
73#817C3×D312yescomplemented
74#807—34no—
75C22×C3⋊D3C3×D411yescharacteristic
76C22×C3×D3—32no—
77C22×C3×D3—61no—
78C2×C32×C4—21no—
79C2×C32×C4—22no—
80C2×C32×C4—41no—
81C2×C32×C4—64no—
82C2×C32×C4C22×D312yes—
83C2×C32×C4C23×C311yescharacteristic
84C2×C3⋊Dic3C3×D411yescharacteristic
85C4×C3⋊D3C2×C3×C412yes—
86C4×C3⋊D3C3×Q812yes—
87C2×C3×Dic3—24no—
88C2×C3×Dic3—24no—
89C2×C3×Dic3—32no—
90C2×C3×Dic3—44no—
91C2×C3×Dic3—61no—
92C3×C4×D3—34no—
93C3×C4×D3—34no—
94C3×C4×D3—62no—
95C3×C4×D3—62no—
96C32×C8—22no—
97C32×C8—24no—
98C32×C8—42no—
99C32×C8C4×D314yes—
100C32×C8C2×C3×C412yes—
101C3⋊(C3⋊C8)C2×C3×C412yes—
102C3×C3⋊C8—34no—
103C3×C3⋊C8—62no—
104#427—91noSylow
105C2×C33C2×C4⋊C411yescharacteristic
106C2×C33#13512yes—
107C3×C3⋊D3—22no—
108C22×C3×C4—91no—
109C2×C4×D3—32no—
110C2×C4×D3—61no—
111C2×C3×C8—22no—
112C2×C3×C8—21no—
113C2×C3×C8—41no—
114C2×C3×C8—91no—
115C2×C3×C8—94no—
116C2×C3×C8D3211yescharacteristic
117C2×C3×C8C2×C3×D312yes—
118C3×C4⋊C4—34no—
119C3×C4⋊C4—34no—
120C3×C4⋊C4—64no—
121C3×C22⋊C4—182no—
122C3×C42—182no—
123(C2×D3)⋊C4—64no—
124C4⋊Dic3C2×C3×D314yes—
125C4×Dic3—64no—
126C2×C3⋊C8—32no—
127C2×C3⋊C8—61no—
128D3×C8—38no—
129D3×C8—64no—
130C22×C32—21no—
131C22×C32—22no—
132C22×C32—41no—
133C22×C32D3×D412yes—
134C22×C32C2×C3×D411yescharacteristic
135C2×C3⋊D3—22no—
136C2×C3⋊D3C3×C4⋊C412yes—
137C2×C3×D3—34no—
138C2×C3×D3—62no—
139C2×C3×D3—64no—
140C2×C3×D3—64no—
141C32×C4—21no—
142C32×C4—21no—
143C32×C4—22no—
144C32×C4—22no—
145C32×C4—41no—
146C32×C4—41no—
147C32×C4—124no—
148C32×C4C2×C4×D312yes—
149C32×C4D3×Q812yes—
150C32×C4C22×C3×C411yescharacteristic
151C32×C4C2×C3×Q811yescharacteristic
152C3⋊Dic3C3×C4⋊C412yes—
153C3×Dic3—34no—
154C3×Dic3—44no—
155C3×Dic3—44no—
156C3×Dic3—62no—
157C3×Dic3—84no—
158C22×C8—91no—
159#117—92no—
160#107—92no—
161#106—32no—
162C33#42711yescharacteristic, complemented, Sylow
163C23×C3—91no—
164C22×D3—32no—
165C22×D3—61no—
166C2×C3×C4—22no—
167C2×C3×C4—21no—
168C2×C3×C4—41no—
169C2×C3×C4—64no—
170C2×C3×C4—64no—
171C2×C3×C4—91no—
172C2×C3×C4—92no—
173C2×C3×C4—92no—
174C2×C3×C4—124no—
175C2×C3×C4C2×D3211yescharacteristic
176C2×C3×C4C22×C3×D312yes—
177C2×Dic3—24no—
178C2×Dic3—32no—
179C2×Dic3—61no—
180C4×D3—34no—
181C4×D3—34no—
182C4×D3—62no—
183C4×D3—62no—
184C3×C8—24no—
185C3×C8—22no—
186C3×C8—42no—
187C3×C8—92no—
188C3×C8#62412yes—
189C3×C8C3×C4×D314yes—
190C3⋊C8—34no—
191C3⋊C8—62no—
192C2×C32—22no—
193C2×C32—24no—
194C2×C32—22no—
195C2×C32—21no—
196C2×C32—41no—
197C2×C32—42no—
198C2×C32D3×C4⋊C412yes—
199C2×C32#91614yes—
200C2×C32C2×C3×C4⋊C411yescharacteristic
201C2×C32C3×#13512yes—
202C3⋊D3—22no—
203C3×D3—64no—
204C3×D3—64no—
205C22×C4—91no—
206C2×C8—91no—
207C2×C8—94no—
208C2×C8C3×D3211yescharacteristic
209C4⋊C4—34no—
210C22⋊C4—182no—
211C42—182no—
212C22×C3—21no—
213C22×C3—22no—
214C22×C3—41no—
215C22×C3—92no—
216C22×C3—182no—
217C22×C3#380211yescharacteristic
218C22×C3C3×D3×D412yes—
219C2×D3—34no—
220C2×D3—62no—
221C2×D3—64no—
222C2×D3—64no—
223C3×C4—21no—
224C3×C4—22no—
225C3×C4—21no—
226C3×C4—22no—
227C3×C4—41no—
228C3×C4—41no—
229C3×C4—92no—
230C3×C4—124no—
231C3×C4—124no—
232C3×C4—244no—
233C3×C4#379711yescharacteristic
234C3×C4C2×#62411yescharacteristic
235C3×C4C2×C3×C4×D312yes—
236C3×C4C3×D3×Q812yes—
237Dic3—34no—
238Dic3—44no—
239Dic3—62no—
240C32—22no—
241C32—21no—
242C32—41no—
243C32#486012yescomplemented
244C32C3×#42711yescharacteristic, complemented
245C23—91no—
246C2×C4—64no—
247C2×C4—91no—
248C2×C4—92no—
249C2×C4—92no—
250C2×C4C2×C3×D3211yescharacteristic
251C8—92no—
252C8C3×#62412yes—
253C2×C3—24no—
254C2×C3—22no—
255C2×C3—21no—
256C2×C3—22no—
257C2×C3—42no—
258C2×C3—41no—
259C2×C3—182no—
260C2×C3#6380912yes—
261C2×C3#6390011yescharacteristic
262C2×C3C3×D3×C4⋊C412yes—
263C2×C3C3×#91614yes—
264D3—64no—
265D3—64no—
266C22—92no—
267C22—182no—
268C22C3×#380211yescharacteristic
269C4—92no—
270C4—124no—
271C4C3×#379711yescharacteristic
272C4C2×C3×#62411yescharacteristic
273C3—21no—
274C3—22no—
275C3—41no—
276C3#1059071211yescharacteristic, direct product factor
277C3C3×#486012yescomplemented
278C2—182no—
279C2C3×#6380912yes—
280C2C3×#6390011yescharacteristic
subgroupG/H
centerC22×C3#3802
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×C8D32
hypocenterC32C3×#427
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
3. C32×#817C2semidirect
4. C3×#63576C2semidirect
6. #10590712C3direct (N × H)
10. C3×C8×C3⋊D3C4semidirect
21. C3×#817D3semidirect
22. C3×#817C2×C3semidirect
24. #63576C2×C3semidirect
45. C8×C3⋊D3C3×C4semidirect
73. #817C3×D3semidirect
162. C33#427semidirect
243. C32#4860semidirect
244. C32C3×#427semidirect
276. C3#10590712direct (N × H)
277. C3C3×#4860semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C8×C3⋊D32
C3×#638492
C32×#8172
C3×#635762
C3×#48603
#105907123