Finite Group Databasebeta

← C3×#10593429 #13166790

Subgroups — C3×#10593429 #13166790

Order 1728.

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

normal subgroups only

This group has 516 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, complemented
2C2×C3×#3774C211yesmaximal, characteristic, complemented
3C3×#63912C212yesmaximal, complemented
4C3×#63742C211yesmaximal, characteristic, complemented
5C3×#63739C212yesmaximal, complemented
6#10593429C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3×#626—22no—
9C2×C3×#625C2212yes—
10C2×C3×#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C3×#3788—22no—
13C3×#3774—22no—
14C32×(C2×D3)⋊C4—22no—
15C3×#3724C2212yes—
16C2×#3788C2×C311yescharacteristic, complemented
17C2×#3774C2×C311yescharacteristic, complemented
18C3×#884—62no—
19#63912C2×C312yescomplemented
20#63742C2×C311yescharacteristic, complemented
21#63739C2×C312yescomplemented
22C22×C3×C3⋊D3D411yescharacteristic
23C22×C32×D3—22no—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C3×C4×C3⋊D3—21no—
27C3×C4×C3⋊D3—21no—
28C2×C32×Dic3—22no—
29C3×#626—44no—
30C3×#625—22no—
31C3×#625—22no—
32C3×#622—21no—
33C3×#622D412yescomplemented
34C3×#469—91nomaximal
35C2×C4×C3⋊D3C22×C311yescharacteristic
36C2×C3×#67—62no—
37C2×C3×#67—62no—
38C2×C3×D12—62no—
39C2×C3×C4×D3—62no—
40C2×#626—22no—
41C2×#625C22×C312yes—
42C2×#622C22×C311yescharacteristic
43C2×#622C22×C311yescharacteristic
44#3788—22no—
45#3774—22no—
46C32×C22⋊C4—62no—
47C3×(C2×D3)⋊C4—22no—
48C3×(C2×D3)⋊C4—22no—
49C3×(C2×D3)⋊C4—42no—
50C3×Dic3⋊C4—62no—
51#3724C22×C312yes—
52C22×C33C2×D411yescharacteristic
53C2×C3×C3⋊D3—21no—
54C2×C3×C3⋊D3—22no—
55C2×C32×D3—42no—
56C2×C32×D3—42no—
57C2×C32×D3—42no—
58C33×C4—21no—
59C3×C3⋊Dic3C2×D411yescharacteristic
60C3×C3⋊Dic3#4111yescharacteristic
61C32×Dic3—42no—
62C2×C3×D8—91no—
63C2×C3×M16—91no—
64C3×#121—92no—
65#884—62no—
66C3×#106—91no—
67C3×#102—92no—
68C23×C32—62no—
69C22×C3⋊D3C3×D411yescharacteristic
70C22×C3×D3—22no—
71C22×C3×D3—22no—
72C22×C3×D3—42no—
73C22×C3×D3—62no—
74C2×C32×C4—22no—
75C2×C32×C4—42no—
76C2×C32×C4—62no—
77C2×C32×C4C3×D411yescharacteristic
78C2×C3⋊Dic3C23×C311yescharacteristic
79C4×C3⋊D3—21no—
80C4×C3⋊D3—21no—
81C3×#67—122no—
82C3×#67—122no—
83C3×#67—124no—
84C2×C3×Dic3—22no—
85C2×C3×Dic3—22no—
86C2×C3×Dic3—42no—
87C2×C3×Dic3—62no—
88C3×D12—124no—
89C3×C4×D3—122no—
90C3×C4×D3—122no—
91#626—44no—
92#625—22no—
93#625—22no—
94#622—21no—
95#622C3×D412yescomplemented
96#469—91noSylow
97C2×C33#12111yescharacteristic
98C2×C33AGL(1,ℤ/8)12yes—
99C3×C3⋊D3—41no—
100C32×D3—82no—
101C2×C3×D4—92no—
102C2×C3×D4—182no—
103C22×C3×C4—91no—
104C2×#67—62no—
105C2×#67—62no—
106C2×D12—62no—
107C2×C4×D3—62no—
108C3×D8—182no—
109C3×M16—182no—
110C2×C3×C8—91no—
111C2×C3×C8—91no—
112C3×C4⋊C4—92no—
113C3×C22⋊C4—62no—
114C3×C22⋊C4—62no—
115C3×C22⋊C4—122no—
116(C2×D3)⋊C4—22no—
117Dic3⋊C4—62no—
118C22×C32—22no—
119C22×C32—42no—
120C22×C32—122no—
121C22×C32—122no—
122C22×C32—122no—
123C22×C32C2×C3×D411yescharacteristic
124C2×C3⋊D3—22no—
125C2×C3⋊D3—21no—
126C2×C3×D3—42no—
127C2×C3×D3—42no—
128C2×C3×D3—42no—
129C2×C3×D3—44no—
130C2×C3×D3—42no—
131C2×C3×D3—42no—
132C2×C3×D3—42no—
133C2×C3×D3—64no—
134C2×C3×D3—82no—
135C2×C3×D3—82no—
136C2×C3×D3—122no—
137C2×C3×D3—122no—
138C32×C4—21no—
139C32×C4—42no—
140C32×C4—82no—
141C32×C4—122no—
142C3⋊Dic3C2×C3×D411yescharacteristic
143C3⋊Dic3C3×#4111yescharacteristic
144C3×Dic3—42no—
145C3×Dic3—42no—
146C3×Dic3—62no—
147C3×Dic3—62no—
148C3×Dic3—82no—
149C2×D8—91no—
150C2×M16—91no—
151#121—92no—
152#106—91no—
153#102—92no—
154C33#46911yescharacteristic, complemented, Sylow
155C23×C3—62no—
156C23×C3—62no—
157C23×C3—91no—
158C23×C3—122no—
159C22×D3—22no—
160C22×D3—62no—
161C3×D4—182no—
162C3×D4—182no—
163C3×D4—364no—
164C2×C3×C4—22no—
165C2×C3×C4—42no—
166C2×C3×C4—62no—
167C2×C3×C4—62no—
168C2×C3×C4—91no—
169C2×C3×C4—122no—
170C2×C3×C4—181no—
171C2×C3×C4—181no—
172C2×C3×C4GO+(4,𝔽2)11yescharacteristic
173#67—122no—
174#67—122no—
175#67—124no—
176C2×Dic3—22no—
177C2×Dic3—62no—
178D12—124no—
179C4×D3—122no—
180C4×D3—122no—
181C3×C8—92no—
182C3×C8—181no—
183C2×C32—22no—
184C2×C32—22no—
185C2×C32—22no—
186C2×C32—42no—
187C2×C32—42no—
188C2×C32—42no—
189C2×C32—242no—
190C2×C32C3×#12111yescharacteristic
191C2×C32C3×AGL(1,ℤ/8)12yes—
192C3⋊D3—41no—
193C3×D3—82no—
194C3×D3—84no—
195C3×D3—82no—
196C3×D3—124no—
197C2×D4—92no—
198C2×D4—182no—
199C22×C4—91no—
200D8—182no—
201M16—182no—
202C2×C8—91no—
203C2×C8—91no—
204C4⋊C4—92no—
205C22⋊C4—62no—
206C22×C3—22no—
207C22×C3—42no—
208C22×C3—122no—
209C22×C3—122no—
210C22×C3—122no—
211C22×C3—124no—
212C22×C3—122no—
213C22×C3—122no—
214C22×C3—122no—
215C22×C3—182no—
216C22×C3—181no—
217C22×C3—242no—
218C22×C3—242no—
219C22×C3C2×GO+(4,𝔽2)11yescharacteristic
220C2×D3—42no—
221C2×D3—42no—
222C2×D3—42no—
223C2×D3—64no—
224C2×D3—122no—
225C2×D3—122no—
226C3×C4—21no—
227C3×C4—42no—
228C3×C4—82no—
229C3×C4—91no—
230C3×C4—91no—
231C3×C4—122no—
232C3×C4—122no—
233C3×C4—242no—
234Dic3—42no—
235Dic3—62no—
236Dic3—62no—
237C32—22no—
238C32—42no—
239C32C3×#46911yescharacteristic, complemented
240C23—62no—
241C23—91no—
242D4—182no—
243D4—182no—
244D4—364no—
245C2×C4—62no—
246C2×C4—91no—
247C2×C4—181no—
248C2×C4—181no—
249C2×C4C3×GO+(4,𝔽2)11yescharacteristic
250C8—92no—
251C8—181no—
252C2×C3—22no—
253C2×C3—22no—
254C2×C3—22no—
255C2×C3—42no—
256C2×C3—42no—
257C2×C3—42no—
258C2×C3—242no—
259C2×C3—244no—
260C2×C3—242no—
261C2×C3—361no—
262C2×C3#6422412yes—
263C2×C3#6424211yescharacteristic
264D3—82no—
265D3—124no—
266C22—122no—
267C22—122no—
268C22—122no—
269C22—181no—
270C22—182no—
271C22C2×C3×GO+(4,𝔽2)11yescharacteristic
272C4—21no—
273C4—91no—
274C4—91no—
275C4—122no—
276C3—22no—
277C3—42no—
278C3#1059342911yescharacteristic, direct product factor
279C2—242no—
280C2—361no—
281C2C3×#6422412yes—
282C2C3×#6424211yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#469
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×#3788C2semidirect
2. C2×C3×#3774C2semidirect
3. C3×#63912C2semidirect
4. C3×#63742C2semidirect
5. C3×#63739C2semidirect
6. #10593429C3direct (N × H)
16. C2×#3788C2×C3semidirect
17. C2×#3774C2×C3semidirect
19. #63912C2×C3semidirect
20. #63742C2×C3semidirect
21. #63739C2×C3semidirect
33. C3×#622D4semidirect
95. #622C3×D4semidirect
154. C33#469semidirect
239. C32C3×#469semidirect
278. C3#10593429direct (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×#37742
C3×#639122
C3×#637422
C3×#637392
#105934293
C3×#4699