Finite Group Databasebeta

← C3×#10593420 #13166782

Subgroups — C3×#10593420 #13166782

Order 1728.

524 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 86 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×#3774C211yesmaximal, characteristic
2C3×#63902C212yesmaximal, complemented
3C3×C4×#622C211yesmaximal, characteristic, complemented
4C3×#63743C211yesmaximal, characteristic, complemented
5C3×#63739C212yesmaximal, complemented
6#10593420C311yesmaximal, characteristic, direct product factor
7C2×C3×#625C2212yes—
8C2×C3×D3×Dic3—22no—
9C2×C3×#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11C3×#3774C414yescomplemented
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C32×(C2×D3)⋊C4—22no—
14C3×#3724C2212yes—
15C2×#3774C2×C311yescharacteristic
16C3×#67⋊C4—62no—
17#63902C2×C312yescomplemented
18C4×#622C2×C311yescharacteristic, complemented
19#63743C2×C311yescharacteristic, complemented
20#63739C2×C312yescomplemented
21C22×C32×D3—22no—
22C2×C33×C4D411yescharacteristic
23C2×C3×C3⋊Dic3D411yescharacteristic
24C2×C3×C3⋊Dic3C2311yescharacteristic
25C2×C32×Dic3—22no—
26C3×#625—22no—
27C3×#625C2×C414yescomplemented
28C3×D3×Dic3—44no—
29C3×#622—21no—
30C3×#622C2×C412yescomplemented
31C3×C4×D8—91nomaximal
32C2×C3×#67—62no—
33C22×C3×Dic3—62no—
34C2×C3×C4×D3—62no—
35C2×#625C22×C312yes—
36C2×D3×Dic3—22no—
37C2×#622C22×C311yescharacteristic
38C2×#622C22×C311yescharacteristic
39#3774C3×C414yescomplemented
40C32×C22⋊C4—62no—
41C4×C3⋊Dic3C22×C311yescharacteristic
42C3×(C2×D3)⋊C4—22no—
43C3×(C2×D3)⋊C4—22no—
44C3×(C2×D3)⋊C4—42no—
45C3×Dic3⋊C4—62no—
46C3×C4×Dic3—62no—
47#3724C22×C312yes—
48C22×C33C2×D411yescharacteristic
49C2×C32×D3—24no—
50C2×C32×D3—42no—
51C2×C32×D3—42no—
52C33×C4—21no—
53C3×C3⋊Dic3D812yescomplemented
54C3×C3⋊Dic3C22×C411yescharacteristic
55C3×C3⋊Dic3#4111yescharacteristic
56C32×Dic3—42no—
57C2×C3×D8—91no—
58C3×C4×D4—92no—
59#67⋊C4—62no—
60C3×#107—91no—
61C3×#102—92no—
62C3×C4×C8—91no—
63C23×C32—62no—
64C22×C3×D3—22no—
65C22×C3×D3—22no—
66C22×C3×D3—42no—
67C2×C32×C4—22no—
68C2×C32×C4—42no—
69C2×C32×C4—62no—
70C2×C32×C4C3×D411yescharacteristic
71C2×C3⋊Dic3C3×D411yescharacteristic
72C2×C3⋊Dic3C23×C311yescharacteristic
73C3×#67—64no—
74C3×#67—64no—
75C2×C3×Dic3—22no—
76C2×C3×Dic3—22no—
77C2×C3×Dic3—42no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—124no—
81C3×C4×D3—124no—
82#625—22no—
83#625C2×C3×C414yescomplemented
84D3×Dic3—44no—
85#622—21no—
86#622C2×C3×C412yescomplemented
87C4×D8—91noSylow
88C2×C33C4×D411yescharacteristic
89C2×C33C2×D811yescharacteristic
90C2×C33#13511yescharacteristic
91C32×D3—44no—
92C2×C3×D4—92no—
93C22×C3×C4—182no—
94C2×#67—62no—
95C22×Dic3—62no—
96C2×C4×D3—62no—
97C3×D8—94no—
98C2×C3×C8—91no—
99C2×C3×C8—91no—
100C3×C4⋊C4—92no—
101C3×C22⋊C4—62no—
102C3×C22⋊C4—62no—
103C3×C22⋊C4—122no—
104C3×C42—91no—
105(C2×D3)⋊C4—22no—
106Dic3⋊C4—62no—
107C4×Dic3—62no—
108C22×C32—22no—
109C22×C32—42no—
110C22×C32—64no—
111C22×C32—122no—
112C22×C32—122no—
113C22×C32C2×C3×D411yescharacteristic
114C2×C3×D3—24no—
115C2×C3×D3—24no—
116C2×C3×D3—44no—
117C2×C3×D3—42no—
118C2×C3×D3—42no—
119C2×C3×D3—42no—
120C2×C3×D3—42no—
121C2×C3×D3—82no—
122C2×C3×D3—82no—
123C32×C4—21no—
124C32×C4—42no—
125C32×C4—82no—
126C32×C4—122no—
127C3⋊Dic3C3×D812yescomplemented
128C3⋊Dic3C22×C3×C411yescharacteristic
129C3⋊Dic3C3×#4111yescharacteristic
130C3×Dic3—42no—
131C3×Dic3—42no—
132C3×Dic3—64no—
133C3×Dic3—62no—
134C3×Dic3—62no—
135C3×Dic3—82no—
136C2×D8—91no—
137C4×D4—92no—
138#107—91no—
139#102—92no—
140C4×C8—91no—
141C33C4×D811yescharacteristic, complemented, Sylow
142C23×C3—62no—
143C23×C3—62no—
144C23×C3—122no—
145C22×D3—22no—
146C3×D4—94no—
147C3×D4—182no—
148C2×C3×C4—22no—
149C2×C3×C4—42no—
150C2×C3×C4—62no—
151C2×C3×C4—62no—
152C2×C3×C4—91no—
153C2×C3×C4—91no—
154C2×C3×C4—122no—
155C2×C3×C4—364no—
156C2×C3×C4GO+(4,𝔽2)11yescharacteristic
157#67—64no—
158#67—64no—
159C2×Dic3—22no—
160C2×Dic3—62no—
161C2×Dic3—62no—
162C2×Dic3—124no—
163C4×D3—124no—
164C3×C8—92no—
165C3×C8—181no—
166C2×C32—22no—
167C2×C32—22no—
168C2×C32—22no—
169C2×C32—42no—
170C2×C32—42no—
171C2×C32—42no—
172C2×C32—124no—
173C2×C32C3×C4×D411yescharacteristic
174C2×C32C2×C3×D811yescharacteristic
175C2×C32C3×#13511yescharacteristic
176C3×D3—44no—
177C3×D3—44no—
178C3×D3—84no—
179C2×D4—92no—
180C22×C4—182no—
181D8—94no—
182C2×C8—91no—
183C2×C8—91no—
184C4⋊C4—92no—
185C22⋊C4—62no—
186C42—91no—
187C22×C3—22no—
188C22×C3—42no—
189C22×C3—64no—
190C22×C3—64no—
191C22×C3—124no—
192C22×C3—122no—
193C22×C3—122no—
194C22×C3—122no—
195C22×C3—122no—
196C22×C3—242no—
197C22×C3—242no—
198C22×C3C2×GO+(4,𝔽2)11yescharacteristic
199C2×D3—24no—
200C2×D3—42no—
201C2×D3—42no—
202C3×C4—21no—
203C3×C4—42no—
204C3×C4—82no—
205C3×C4—92no—
206C3×C4—91no—
207C3×C4—91no—
208C3×C4—122no—
209C3×C4—122no—
210C3×C4—242no—
211Dic3—42no—
212Dic3—64no—
213Dic3—62no—
214Dic3—62no—
215C32—22no—
216C32—42no—
217C32C3×C4×D811yescharacteristic, complemented
218C23—62no—
219D4—94no—
220D4—182no—
221C2×C4—62no—
222C2×C4—91no—
223C2×C4—91no—
224C2×C4—364no—
225C2×C4C3×GO+(4,𝔽2)11yescharacteristic
226C8—92no—
227C8—181no—
228C2×C3—22no—
229C2×C3—22no—
230C2×C3—22no—
231C2×C3—42no—
232C2×C3—42no—
233C2×C3—42no—
234C2×C3—124no—
235C2×C3—124no—
236C2×C3—244no—
237C2×C3#6422511yescharacteristic
238C2×C3C4×GO+(4,𝔽2)11yescharacteristic
239C2×C3#6423711yescharacteristic
240D3—44no—
241C22—64no—
242C22—122no—
243C22—122no—
244C22C2×C3×GO+(4,𝔽2)11yescharacteristic
245C4—21no—
246C4—92no—
247C4—91no—
248C4—91no—
249C4—122no—
250C3—22no—
251C3—42no—
252C3#1059342011yescharacteristic, direct product factor
253C2—124no—
254C2C3×#6422511yescharacteristic
255C2C3×C4×GO+(4,𝔽2)11yescharacteristic
256C2C3×#6423711yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#3774C2
cosocleC2×C3⋊Dic3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×C4×D8
LCSG ▷ 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
2. C3×#63902C2semidirect
3. C3×C4×#622C2semidirect
4. C3×#63743C2semidirect
5. C3×#63739C2semidirect
6. #10593420C3direct (N × H)
11. C3×#3774C4semidirect
17. #63902C2×C3semidirect
18. C4×#622C2×C3semidirect
19. #63743C2×C3semidirect
20. #63739C2×C3semidirect
27. C3×#625C2×C4semidirect
30. C3×#622C2×C4semidirect
39. #3774C3×C4semidirect
53. C3×C3⋊Dic3D8semidirect
83. #625C2×C3×C4semidirect
86. #622C2×C3×C4semidirect
127. C3⋊Dic3C3×D8semidirect
141. C33C4×D8semidirect
217. C32C3×C4×D8semidirect
252. C3#10593420direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37742
C3×#639022
C3×C4×#6222
C3×#637432
C3×#637392
#105934203
C3×C4×D89