Finite Group Databasebeta

← C3×#10593522 #13166888

Subgroups — C3×#10593522 #13166888

Order 1728.

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

normal subgroups only

This group has 556 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, complemented
2C2×C3×#3774C211yesmaximal, characteristic, complemented
3C3×#63980C212yesmaximal, complemented
4C3×#63742C211yesmaximal, characteristic, complemented
5C3×#63739C212yesmaximal, complemented
6#10593522C311yesmaximal, characteristic, direct product factor
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C32×#67—22no—
9C2×C3×#625C2212yes—
10C3×#3792—22no—
11C2×C3×#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C3×#3774—22no—
14C3×#3724C2212yes—
15C3×(C2×D3)⋊Dic3—22no—
16C2×#3792C2×C311yescharacteristic, complemented
17C2×#3774C2×C311yescharacteristic, complemented
18C3×#939—62no—
19#63980C2×C312yescomplemented
20#63742C2×C311yescharacteristic, complemented
21#63739C2×C312yescomplemented
22C23×C33D411yescharacteristic
23C22×C32×D3—22no—
24C2×C3×C3⋊Dic3—21no—
25C2×C3×C3⋊Dic3—21no—
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C32×#67—44no—
29C2×C32×Dic3—22no—
30C3×#625—22no—
31C3×#625—22no—
32C3×#622—21no—
33C3×#622D412yescomplemented
34C3×#469—91nomaximal
35C2×C32×D4—62no—
36C22×C3⋊Dic3C22×C311yescharacteristic
37C2×C3×#67—22no—
38C2×C3×#67—22no—
39C2×C3×#67—42no—
40C2×C3×#67—62no—
41C22×C3×Dic3—62no—
42C2×#625C22×C312yes—
43#3792—22no—
44C2×#622C22×C311yescharacteristic
45C2×#622C22×C311yescharacteristic
46#3774—22no—
47C3×#217—62no—
48C3×(C2×D3)⋊C4—62no—
49C3×Dic3⋊C4—62no—
50#3724C22×C312yes—
51(C2×D3)⋊Dic3—22no—
52C22×C33—21no—
53C22×C33—22no—
54C22×C33C2×D411yescharacteristic
55C2×C32×D3—42no—
56C2×C32×D3—42no—
57C2×C32×D3—42no—
58C3×C3⋊Dic3—21no—
59C3×C3⋊Dic3C2×D411yescharacteristic
60C3×C3⋊Dic3#4111yescharacteristic
61C32×Dic3—42no—
62C2×C3×D8—91no—
63C2×C3×M16—91no—
64C3×#121—92no—
65#939—62no—
66C3×#106—91no—
67C3×#102—92no—
68C23×C32—22no—
69C23×C32—42no—
70C23×C32—62no—
71C23×C32C3×D411yescharacteristic
72C22×C3×D3—22no—
73C22×C3×D3—22no—
74C22×C3×D3—42no—
75C32×D4—124no—
76C2×C32×C4—62no—
77C2×C3⋊Dic3—21no—
78C2×C3⋊Dic3—21no—
79C2×C3⋊Dic3C3×D411yescharacteristic
80C2×C3⋊Dic3C23×C311yescharacteristic
81C3×#67—44no—
82C3×#67—44no—
83C3×#67—48no—
84C3×#67—122no—
85C3×#67—122no—
86C2×C3×Dic3—22no—
87C2×C3×Dic3—22no—
88C2×C3×Dic3—42no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—122no—
92C2×C3×Dic3—122no—
93#625—22no—
94#625—22no—
95#622—21no—
96#622C3×D412yescomplemented
97#469—91noSylow
98C2×C33—41no—
99C2×C33#12111yescharacteristic
100C2×C33AGL(1,ℤ/8)12yes—
101C32×D3—82no—
102C2×C3×D4—62no—
103C2×C3×D4—62no—
104C2×C3×D4—92no—
105C2×C3×D4—122no—
106C22×C3×C4—91no—
107C2×#67—22no—
108C2×#67—62no—
109C22×Dic3—62no—
110C3×D8—182no—
111C3×M16—182no—
112C2×C3×C8—91no—
113C2×C3×C8—91no—
114C3×C4⋊C4—92no—
115C3×C22⋊C4—182no—
116#217—62no—
117(C2×D3)⋊C4—62no—
118Dic3⋊C4—62no—
119C22×C32—24no—
120C22×C32—22no—
121C22×C32—22no—
122C22×C32—21no—
123C22×C32—42no—
124C22×C32—42no—
125C22×C32—42no—
126C22×C32—44no—
127C22×C32—82no—
128C22×C32—82no—
129C22×C32—122no—
130C22×C32—122no—
131C22×C32—122no—
132C22×C32C2×C3×D411yescharacteristic
133C2×C3×D3—42no—
134C2×C3×D3—42no—
135C2×C3×D3—42no—
136C2×C3×D3—44no—
137C2×C3×D3—42no—
138C2×C3×D3—42no—
139C2×C3×D3—42no—
140C2×C3×D3—82no—
141C2×C3×D3—82no—
142C32×C4—122no—
143C3⋊Dic3—21no—
144C3⋊Dic3C2×C3×D411yescharacteristic
145C3⋊Dic3C3×#4111yescharacteristic
146C3×Dic3—42no—
147C3×Dic3—44no—
148C3×Dic3—42no—
149C3×Dic3—62no—
150C3×Dic3—62no—
151C3×Dic3—122no—
152C2×D8—91no—
153C2×M16—91no—
154#121—92no—
155#106—91no—
156#102—92no—
157C33#46911yescharacteristic, complemented, Sylow
158C23×C3—22no—
159C23×C3—42no—
160C23×C3—62no—
161C23×C3—62no—
162C23×C3—122no—
163C23×C3GO+(4,𝔽2)11yescharacteristic
164C22×D3—22no—
165C3×D4—124no—
166C3×D4—124no—
167C3×D4—128no—
168C3×D4—182no—
169C3×D4—182no—
170C2×C3×C4—62no—
171C2×C3×C4—62no—
172C2×C3×C4—91no—
173C2×C3×C4—91no—
174C2×C3×C4—122no—
175C2×C3×C4—181no—
176C2×C3×C4—181no—
177#67—44no—
178#67—122no—
179#67—122no—
180C2×Dic3—22no—
181C2×Dic3—62no—
182C2×Dic3—62no—
183C2×Dic3—122no—
184C2×Dic3—122no—
185C3×C8—92no—
186C3×C8—181no—
187C2×C32—22no—
188C2×C32—22no—
189C2×C32—22no—
190C2×C32—42no—
191C2×C32—42no—
192C2×C32—42no—
193C2×C32—41no—
194C2×C32—44no—
195C2×C32—84no—
196C2×C32—242no—
197C2×C32C3×#12111yescharacteristic
198C2×C32C3×AGL(1,ℤ/8)12yes—
199C3×D3—82no—
200C3×D3—84no—
201C3×D3—82no—
202C2×D4—62no—
203C2×D4—92no—
204C22×C4—91no—
205D8—182no—
206M16—182no—
207C2×C8—91no—
208C2×C8—91no—
209C4⋊C4—92no—
210C22⋊C4—182no—
211C22×C3—24no—
212C22×C3—22no—
213C22×C3—21no—
214C22×C3—22no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3—44no—
218C22×C3—42no—
219C22×C3—82no—
220C22×C3—82no—
221C22×C3—122no—
222C22×C3—122no—
223C22×C3—122no—
224C22×C3—124no—
225C22×C3—122no—
226C22×C3—122no—
227C22×C3—122no—
228C22×C3—242no—
229C22×C3—242no—
230C22×C3C2×GO+(4,𝔽2)11yescharacteristic
231C2×D3—42no—
232C2×D3—42no—
233C2×D3—42no—
234C3×C4—91no—
235C3×C4—91no—
236C3×C4—122no—
237C3×C4—124no—
238C3×C4—122no—
239C3×C4—181no—
240Dic3—42no—
241Dic3—62no—
242Dic3—62no—
243Dic3—122no—
244C32—22no—
245C32—42no—
246C32C3×#46911yescharacteristic, complemented
247C23—62no—
248C23C3×GO+(4,𝔽2)11yescharacteristic
249D4—124no—
250D4—182no—
251D4—182no—
252C2×C4—62no—
253C2×C4—91no—
254C2×C4—91no—
255C2×C4—181no—
256C2×C4—181no—
257C8—92no—
258C8—181no—
259C2×C3—22no—
260C2×C3—22no—
261C2×C3—22no—
262C2×C3—44no—
263C2×C3—41no—
264C2×C3—42no—
265C2×C3—42no—
266C2×C3—42no—
267C2×C3—84no—
268C2×C3—242no—
269C2×C3—244no—
270C2×C3—242no—
271C2×C3#6423612yes—
272C2×C3#6424211yescharacteristic
273D3—82no—
274C22—22no—
275C22—21no—
276C22—122no—
277C22—122no—
278C22—122no—
279C22C2×C3×GO+(4,𝔽2)11yescharacteristic
280C4—91no—
281C4—91no—
282C4—122no—
283C4—181no—
284C3—22no—
285C3—42no—
286C3#1059352211yescharacteristic, direct product factor
287C2—41no—
288C2—242no—
289C2C3×#6423612yes—
290C2C3×#6424211yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#469
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
1. C2×C3×#3792C2semidirect
2. C2×C3×#3774C2semidirect
3. C3×#63980C2semidirect
4. C3×#63742C2semidirect
5. C3×#63739C2semidirect
6. #10593522C3direct (N × H)
16. C2×#3792C2×C3semidirect
17. C2×#3774C2×C3semidirect
19. #63980C2×C3semidirect
20. #63742C2×C3semidirect
21. #63739C2×C3semidirect
33. C3×#622D4semidirect
96. #622C3×D4semidirect
157. C33#469semidirect
246. C32C3×#469semidirect
286. C3#10593522direct (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×#37742
C3×#639802
C3×#637422
C3×#637392
#105935223
C3×#4699