Finite Group Databasebeta

← C3⋊D3×#123 #10596123

Subgroups — C3⋊D3×#123 #10596123

Order 576.

1026 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 199 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×D4×C3⋊D3C211yesmaximal, characteristic
2C22×C4×C3⋊D3C211yesmaximal, characteristic, complemented
3C32×#123C211yesmaximal, characteristic, complemented
4#64144C211yesmaximal, characteristic, complemented
5#64138C211yesmaximal, characteristic, complemented
6#64103C212yesmaximal, complemented
7C4⋊C4×C3⋊D3C212yesmaximal, complemented
8#64095C211yesmaximal, characteristic, complemented
9#64092C212yesmaximal, complemented
10C22⋊C4×C3⋊D3C212yesmaximal, complemented
11C22⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
12D3×#123—34nomaximal
13C23×C3⋊D3C2211yescharacteristic
14C23×C3⋊D3C2211yescharacteristic
15C2×C32×D4C2211yescharacteristic
16C22×C32×C4C2211yescharacteristic, complemented
17C2×#638C2211yescharacteristic
18C2×#638C2211yescharacteristic
19C22×C3⋊Dic3C2211yescharacteristic, complemented
20D4×C3⋊D3—24no—
21C2×C4⋊(C3⋊D3)C2211yescharacteristic
22C2×C4×C3⋊D3—24no—
23C2×C4×C3⋊D3C2212yes—
24C2×C4×C3⋊D3C2212yes—
25C2×C4×C3⋊D3C2211yescharacteristic
26C2×C4×C3⋊D3C2212yescomplemented
27C32×C4⋊C4C2212yescomplemented
28C32×C22⋊C4C2212yescomplemented
29C32×C22⋊C4C2211yescharacteristic, complemented
30#3757C2212yescomplemented
31#3757C2211yescharacteristic, complemented
32#3752C2212yescomplemented
33#3752C2212yescomplemented
34#3752C2212yescomplemented
35C4⋊(C3⋊Dic3)C2212yescomplemented
36#3750C2212yescomplemented
37#3750C2212yescomplemented
38C2×D3×D4—34no—
39C22×C4×D3—34no—
40C3×#123D314yescomplemented
41#935—34no—
42#929—34no—
43#894—38no—
44D3×C4⋊C4—38no—
45#886—34no—
46#883—38no—
47D3×C22⋊C4—34no—
48D3×C22⋊C4—38no—
49C23×C32C2311yescharacteristic
50C23×C32C2311yescharacteristic
51C22×C3⋊D3—24no—
52C22×C3⋊D3—22no—
53C22×C3⋊D3—24no—
54C22×C3⋊D3D414yes—
55C22×C3⋊D3C2311yescharacteristic
56C22×C3⋊D3C2311yescharacteristic
57C22×C3⋊D3C2311yescharacteristic
58C32×D4—22no—
59C2×C32×C4—22no—
60C2×C32×C4C2311yescharacteristic
61C2×C32×C4C2312yes—
62C2×C32×C4C2312yes—
63#638—22no—
64#638—22no—
65C2×C3⋊Dic3—22no—
66C2×C3⋊Dic3C2311yescharacteristic
67C2×C3⋊Dic3C2312yes—
68C2×C3⋊Dic3C2312yes—
69C4⋊(C3⋊D3)—22no—
70C4×C3⋊D3—24no—
71C4×C3⋊D3—24no—
72C4×C3⋊D3—22no—
73C4×C3⋊D3—24no—
74C2×#123—91noSylow
75C23×D3—34no—
76C23×D3—34no—
77C2×C3×D4C2×D314yes—
78C22×C3×C4C2×D314yescomplemented
79C2×#67—34no—
80C2×#67—34no—
81C22×Dic3—34no—
82D3×D4—616no—
83C2×D12—34no—
84C2×C4×D3—38no—
85C2×C4×D3—38no—
86C2×C4×D3—38no—
87C2×C4×D3—34no—
88C2×C4×D3—616no—
89C3×C4⋊C4C2×D318yescomplemented
90C3×C22⋊C4C2×D318yescomplemented
91C3×C22⋊C4C2×D314yescomplemented
92#217—38no—
93#217—34no—
94(C2×D3)⋊C4—38no—
95(C2×D3)⋊C4—38no—
96(C2×D3)⋊C4—38no—
97C4⋊Dic3—38no—
98Dic3⋊C4—38no—
99Dic3⋊C4—38no—
100C22×C32—22no—
101C22×C32—21no—
102C22×C32—22no—
103C22×C32C2×D412yes—
104C22×C32C2411yescharacteristic
105C2×C3⋊D3—22no—
106C2×C3⋊D3—28no—
107C2×C3⋊D3—21no—
108C2×C3⋊D3—22no—
109C2×C3⋊D3—44no—
110C2×C3⋊D3C2×D412yes—
111C2×C3⋊D3C2×D412yes—
112C2×C3⋊D3#4114yes—
113C32×C4—22no—
114C32×C4—21no—
115C32×C4—22no—
116C3⋊Dic3—22no—
117C3⋊Dic3—21no—
118C3⋊Dic3—22no—
119C22×D4—91no—
120C23×C4—91no—
121#123—91no—
122#123—91no—
123#123—92no—
124#123—92no—
125#123—91no—
126#123C3⋊D311yescharacteristic, direct product factor
127C2×C4⋊C4—92no—
128C2×C22⋊C4—91no—
129C2×C22⋊C4—92no—
130C23×C3C22×D314yes—
131C23×C3C22×D314yes—
132C22×D3—34no—
133C22×D3—316no—
134C22×D3—34no—
135C22×D3—34no—
136C22×D3—616no—
137C22×D3—68no—
138C22×D3—616no—
139C3×D4—28no—
140C2×C3×C4—28no—
141C2×C3×C4C22×D314yes—
142C2×C3×C4C22×D318yes—
143C2×C3×C4C22×D318yes—
144#67—68no—
145#67—68no—
146C2×Dic3—34no—
147C2×Dic3—38no—
148C2×Dic3—38no—
149C2×Dic3—68no—
150D12—68no—
151C4×D3—616no—
152C4×D3—616no—
153C4×D3—616no—
154C4×D3—68no—
155C2×C32—22no—
156C2×C32—41no—
157C2×C32C22×D411yescharacteristic
158C2×C32C2×#4112yes—
159C3⋊D3—22no—
160C3⋊D3—41no—
161C3⋊D3#12314yesdirect product factor
162C24—91no—
163C24—91no—
164C2×D4—91no—
165C2×D4—91no—
166C2×D4—91no—
167C2×D4—184no—
168C2×D4C2×C3⋊D311yescharacteristic
169C22×C4—91no—
170C22×C4—92no—
171C22×C4—92no—
172C22×C4—91no—
173C22×C4—92no—
174C22×C4—184no—
175C22×C4C2×C3⋊D311yescharacteristic, complemented
176C4⋊C4—92no—
177C4⋊C4—92no—
178C4⋊C4—92no—
179C4⋊C4C2×C3⋊D312yescomplemented
180C22⋊C4—91no—
181C22⋊C4—92no—
182C22⋊C4—92no—
183C22⋊C4—92no—
184C22⋊C4—92no—
185C22⋊C4C2×C3⋊D311yescharacteristic, complemented
186C22⋊C4C2×C3⋊D312yescomplemented
187C22×C3—28no—
188C22×C3—24no—
189C22×C3—28no—
190C22×C3D3×D418yes—
191C22×C3C23×D314yes—
192C2×D3—38no—
193C2×D3—316no—
194C2×D3—38no—
195C2×D3—632no—
196C2×D3—68no—
197C2×D3—64no—
198C2×D3—68no—
199C2×D3—1216no—
200C3×C4—28no—
201C3×C4—24no—
202C3×C4—28no—
203Dic3—68no—
204Dic3—64no—
205Dic3—68no—
206C32C2×#12311yescharacteristic, complemented, Sylow
207C23—91no—
208C23—91no—
209C23—94no—
210C23—91no—
211C23—184no—
212C23—182no—
213C23—184no—
214C23C22×C3⋊D311yescharacteristic
215C23C22×C3⋊D311yescharacteristic
216D4—22no—
217D4—182no—
218D4—182no—
219D4—182no—
220C2×C4—22no—
221C2×C4—92no—
222C2×C4—92no—
223C2×C4—91no—
224C2×C4—184no—
225C2×C4—184no—
226C2×C4—182no—
227C2×C4—184no—
228C2×C4—182no—
229C2×C4C22×C3⋊D312yes—
230C2×C4C22×C3⋊D312yes—
231C2×C4C22×C3⋊D311yescharacteristic
232C2×C3—28no—
233C2×C3—44no—
234C2×C3C2×D3×D414yes—
235C2×C3D3×#4118yes—
236D3—316no—
237D3—68no—
238D3—124no—
239C22—22no—
240C22—21no—
241C22—22no—
242C22—92no—
243C22—94no—
244C22—92no—
245C22—188no—
246C22—182no—
247C22—181no—
248C22—182no—
249C22—364no—
250C22D4×C3⋊D312yes—
251C22C23×C3⋊D311yescharacteristic
252C4—22no—
253C4—21no—
254C4—22no—
255C4—182no—
256C4—181no—
257C4—182no—
258C3D3×#12314yescomplemented
259C2—22no—
260C2—41no—
261C2—94no—
262C2—182no—
263C2—361no—
264C2C2×D4×C3⋊D311yescharacteristic
265C2#41×C3⋊D312yes—
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#123C2
prime radicalC2×D4×C3⋊D3C2
hypercenter#123C3⋊D3
hypocenterC32C2×#123
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×C3⋊D3C2semidirect
3. C32×#123C2semidirect
4. #64144C2semidirect
5. #64138C2semidirect
6. #64103C2semidirect
7. C4⋊C4×C3⋊D3C2semidirect
8. #64095C2semidirect
9. #64092C2semidirect
10. C22⋊C4×C3⋊D3C2semidirect
11. C22⋊C4×C3⋊D3C2semidirect
16. C22×C32×C4C22semidirect
19. C22×C3⋊Dic3C22semidirect
26. C2×C4×C3⋊D3C22semidirect
27. C32×C4⋊C4C22semidirect
28. C32×C22⋊C4C22semidirect
29. C32×C22⋊C4C22semidirect
30. #3757C22semidirect
31. #3757C22semidirect
32. #3752C22semidirect
33. #3752C22semidirect
34. #3752C22semidirect
35. C4⋊(C3⋊Dic3)C22semidirect
36. #3750C22semidirect
37. #3750C22semidirect
40. C3×#123D3semidirect
78. C22×C3×C4C2×D3semidirect
89. C3×C4⋊C4C2×D3semidirect
90. C3×C22⋊C4C2×D3semidirect
91. C3×C22⋊C4C2×D3semidirect
126. #123C3⋊D3direct (N × H)
161. C3⋊D3#123direct (N × H)
175. C22×C4C2×C3⋊D3semidirect
179. C4⋊C4C2×C3⋊D3semidirect
185. C22⋊C4C2×C3⋊D3semidirect
186. C22⋊C4C2×C3⋊D3semidirect
206. C32C2×#123semidirect
258. C3D3×#123semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊D32
C22×C4×C3⋊D32
C32×#1232
#641442
#641382
#641032
C4⋊C4×C3⋊D32
#640952
#640922
C22⋊C4×C3⋊D32
D3×#1233