Finite Group Databasebeta

← C3×C4×#3774 #13166780

Subgroups — C3×C4×#3774 #13166780

Order 1728.

564 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 90 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×C4×#625C212yesmaximal, complemented
3C3×C4×#622C211yesmaximal, characteristic, complemented
4C3×#63743C211yesmaximal, characteristic, complemented
5C3×#63739C212yesmaximal, complemented
6C4×#3774C311yesmaximal, characteristic, direct product factor
7C2×C32×C4×D3—22no—
8C2×C3×#625C2212yes—
9C2×C3×#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11C3×#3774C414yesdirect product factor
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C3×#3724C2212yes—
14C3×(C2×D3)⋊Dic3—22no—
15C2×#3774C2×C311yescharacteristic
16C3×C4×#67—62no—
17C4×#625C2×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—
26C32×C4×D3—44no—
27C3×#625—22no—
28C3×#625C2×C414yescomplemented
29C3×#622—21no—
30C3×#622C2×C412yescomplemented
31C3×C4×D8—91nomaximal
32C22×C32×C4—62no—
33C2×C3×#67—62no—
34C2×C3×C4×D3—22no—
35C2×C3×C4×D3—22no—
36C2×C3×C4×D3—42no—
37C2×#625C22×C312yes—
38C2×#622C22×C311yescharacteristic
39C2×#622C22×C311yescharacteristic
40#3774C3×C414yesdirect product factor
41C4×C3⋊Dic3C22×C311yescharacteristic
42C3×#217—62no—
43C3×(C2×D3)⋊C4—62no—
44C3×Dic3⋊C4—62no—
45C3×C4×Dic3—62no—
46#3724C22×C312yes—
47(C2×D3)⋊Dic3—22no—
48C22×C33C2×D411yescharacteristic
49C2×C32×D3—24no—
50C2×C32×D3—42no—
51C2×C32×D3—42no—
52C33×C4D812yescomplemented
53C3×C3⋊Dic3—21no—
54C3×C3⋊Dic3C22×C411yescharacteristic
55C3×C3⋊Dic3#4111yescharacteristic
56C32×Dic3—42no—
57C2×C3×D8—91no—
58C3×C4×D4—92no—
59C4×#67—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×C4—124no—
71C2×C32×C4C3×D411yescharacteristic
72C2×C3⋊Dic3C3×D411yescharacteristic
73C2×C3⋊Dic3C23×C311yescharacteristic
74C3×#67—64no—
75C3×#67—64no—
76C2×C3×Dic3—22no—
77C2×C3×Dic3—22no—
78C2×C3×Dic3—42no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C3×C4×D3—44no—
82C3×C4×D3—48no—
83C3×C4×D3—44no—
84#625—22no—
85#625C2×C3×C414yescomplemented
86#622—21no—
87#622C2×C3×C412yescomplemented
88C4×D8—91noSylow
89C2×C33C4×D411yescharacteristic
90C2×C33C2×D811yescharacteristic
91C2×C33#13511yescharacteristic
92C32×D3—44no—
93C2×C3×D4—92no—
94C22×C3×C4—62no—
95C22×C3×C4—62no—
96C22×C3×C4—122no—
97C2×#67—62no—
98C2×C4×D3—22no—
99C3×D8—94no—
100C2×C3×C8—91no—
101C2×C3×C8—91no—
102C3×C4⋊C4—92no—
103C3×C22⋊C4—182no—
104C3×C42—91no—
105#217—62no—
106(C2×D3)⋊C4—62no—
107Dic3⋊C4—62no—
108C4×Dic3—62no—
109C22×C32—22no—
110C22×C32—42no—
111C22×C32—64no—
112C22×C32—122no—
113C22×C32—122no—
114C22×C32C2×C3×D411yescharacteristic
115C2×C3×D3—24no—
116C2×C3×D3—24no—
117C2×C3×D3—44no—
118C2×C3×D3—42no—
119C2×C3×D3—42no—
120C2×C3×D3—44no—
121C2×C3×D3—44no—
122C2×C3×D3—42no—
123C2×C3×D3—42no—
124C32×C4—24no—
125C32×C4—44no—
126C32×C4—122no—
127C32×C4C3×D812yescomplemented
128C3⋊Dic3—21no—
129C3⋊Dic3C22×C3×C411yescharacteristic
130C3⋊Dic3C3×#4111yescharacteristic
131C3×Dic3—42no—
132C3×Dic3—44no—
133C3×Dic3—42no—
134C3×Dic3—62no—
135C3×Dic3—62no—
136C3×Dic3—122no—
137C2×D8—91no—
138C4×D4—92no—
139#107—91no—
140#102—92no—
141C4×C8—91no—
142C33C4×D811yescharacteristic, complemented, Sylow
143C23×C3—62no—
144C23×C3—62no—
145C23×C3—122no—
146C22×D3—22no—
147C3×D4—94no—
148C3×D4—182no—
149C2×C3×C4—22no—
150C2×C3×C4—42no—
151C2×C3×C4—62no—
152C2×C3×C4—62no—
153C2×C3×C4—91no—
154C2×C3×C4—91no—
155C2×C3×C4—122no—
156C2×C3×C4—124no—
157C2×C3×C4—128no—
158C2×C3×C4—124no—
159C2×C3×C4GO+(4,𝔽2)11yescharacteristic
160#67—64no—
161#67—64no—
162C2×Dic3—22no—
163C2×Dic3—62no—
164C2×Dic3—62no—
165C4×D3—44no—
166C3×C8—92no—
167C3×C8—181no—
168C2×C32—22no—
169C2×C32—22no—
170C2×C32—22no—
171C2×C32—42no—
172C2×C32—42no—
173C2×C32—42no—
174C2×C32—124no—
175C2×C32C3×C4×D411yescharacteristic
176C2×C32C2×C3×D811yescharacteristic
177C2×C32C3×#13511yescharacteristic
178C3×D3—44no—
179C3×D3—48no—
180C3×D3—44no—
181C2×D4—92no—
182C22×C4—62no—
183D8—94no—
184C2×C8—91no—
185C2×C8—91no—
186C4⋊C4—92no—
187C22⋊C4—182no—
188C42—91no—
189C22×C3—22no—
190C22×C3—42no—
191C22×C3—64no—
192C22×C3—64no—
193C22×C3—124no—
194C22×C3—122no—
195C22×C3—122no—
196C22×C3—124no—
197C22×C3—124no—
198C22×C3—122no—
199C22×C3—122no—
200C22×C3C2×GO+(4,𝔽2)11yescharacteristic
201C2×D3—24no—
202C2×D3—42no—
203C2×D3—42no—
204C3×C4—24no—
205C3×C4—44no—
206C3×C4—91no—
207C3×C4—91no—
208C3×C4—122no—
209C3×C4—124no—
210C3×C4—122no—
211C3×C4—181no—
212C3×C4#377412yesdirect product factor
213Dic3—42no—
214Dic3—62no—
215Dic3—62no—
216Dic3—122no—
217C32—22no—
218C32—42no—
219C32C3×C4×D811yescharacteristic, complemented
220C23—62no—
221D4—94no—
222D4—182no—
223C2×C4—62no—
224C2×C4—91no—
225C2×C4—91no—
226C2×C4—124no—
227C2×C4C3×GO+(4,𝔽2)11yescharacteristic
228C8—92no—
229C8—181no—
230C2×C3—22no—
231C2×C3—22no—
232C2×C3—22no—
233C2×C3—42no—
234C2×C3—42no—
235C2×C3—42no—
236C2×C3—124no—
237C2×C3—128no—
238C2×C3—124no—
239C2×C3#6422311yescharacteristic
240C2×C3C4×GO+(4,𝔽2)11yescharacteristic
241C2×C3C2×#377411yescharacteristic
242D3—44no—
243C22—64no—
244C22—122no—
245C22—122no—
246C22C2×C3×GO+(4,𝔽2)11yescharacteristic
247C4—91no—
248C4—91no—
249C4—122no—
250C4—181no—
251C4C3×#377412yesdirect product factor
252C3—22no—
253C3—42no—
254C3C4×#377411yescharacteristic, direct product factor
255C2—124no—
256C2C3×#6422311yescharacteristic
257C2C3×C4×GO+(4,𝔽2)11yescharacteristic
258C2C2×C3×#377411yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4GO+(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
hypocenterC32C3×C4×D8
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×#625C2semidirect
3. C3×C4×#622C2semidirect
4. C3×#63743C2semidirect
5. C3×#63739C2semidirect
6. C4×#3774C3direct (N × H)
11. C3×#3774C4direct (N × H)
17. C4×#625C2×C3semidirect
18. C4×#622C2×C3semidirect
19. #63743C2×C3semidirect
20. #63739C2×C3semidirect
28. C3×#625C2×C4semidirect
30. C3×#622C2×C4semidirect
40. #3774C3×C4direct (N × H)
52. C33×C4D8semidirect
85. #625C2×C3×C4semidirect
87. #622C2×C3×C4semidirect
127. C32×C4C3×D8semidirect
142. C33C4×D8semidirect
212. C3×C4#3774direct (N × H)
219. C32C3×C4×D8semidirect
251. C4C3×#3774direct (N × H)
254. C3C4×#3774direct (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×C4×#6252
C3×C4×#6222
C3×#637432
C3×#637392
C4×#37743
C3×C4×D89