Finite Group Databasebeta

← D3×#237 #64311

Subgroups — D3×#237 #64311

Order 288.

348 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 112 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
1#3830C211yesmaximal, characteristic, complemented
2C3×#237C211yesmaximal, characteristic, complemented
3C3×D3×D4C211yesmaximal, characteristic, complemented
4D3×#67C212yesmaximal, complemented
5#3805C212yesmaximal, complemented
6#3804C212yesmaximal, complemented
7C2×D3×Dic3C212yesmaximal, complemented
8C4×D32C211yesmaximal, characteristic, complemented
9#3796C211yesmaximal, characteristic, complemented
10#3795C211yesmaximal, characteristic, complemented
11D3×Dic6C211yesmaximal, characteristic, complemented
12D3×#41—31nomaximal
13C2×#237—31nomaximal
14C22×C3×D3C2212yes—
15C2×D32C2211yescharacteristic
16C32×D4C2211yescharacteristic, complemented
17#638C2212yescomplemented
18C2×C3⋊Dic3C2212yescomplemented
19C4×C3⋊D3C2211yescharacteristic, complemented
20#634C2211yescharacteristic, complemented
21C3×#67C2212yescomplemented
22C3×#67C2212yescomplemented
23C2×C3×Dic3C2212yescomplemented
24C3×D12C2211yescharacteristic, complemented
25C3×C4×D3C2211yescharacteristic
26C3×C4×D3C2211yescharacteristic, complemented
27C3×Dic6C2211yescharacteristic, complemented
28#627C2212yescomplemented
29#626C2212yescomplemented
30#625C2212yescomplemented
31#624C2211yescharacteristic, complemented
32D3×Dic3C2212yes—
33D3×Dic3C2211yescharacteristic
34D3×Dic3C2212yescomplemented
35D3×Dic3C2212yescomplemented
36D3×Dic3C2211yescharacteristic, complemented
37C3×#41—31no—
38C2×C3×D4—31no—
39C2×#67—32no—
40C22×Dic3—32no—
41#239—31no—
42D3×Q8—31no—
43#237—31no—
44#237—31no—
45#237—32no—
46#237—32no—
47#237—32no—
48#237—31no—
49#237—31no—
50#237—61no—
51#237D311yescharacteristic, direct product factor
52D3×D4—32no—
53D3×D4D311yescharacteristic, complemented
54#235—32no—
55#235—31no—
56C2×C4×D3—32no—
57C2×C4×D3—31no—
58C2×C4×D3—31no—
59C2×Dic6—31no—
60C22×C32C2312yes—
61C2×C3⋊D3C2311yescharacteristic
62C2×C3×D3—24no—
63C2×C3×D3C2312yes—
64C2×C3×D3C2311yescharacteristic
65C2×C3×D3C2311yescharacteristic
66D32—22no—
67C32×C4C2311yescharacteristic
68C3⋊Dic3C2311yescharacteristic
69C3⋊Dic3C2312yes—
70C3×Dic3C2311yescharacteristic
71C3×Dic3C2312yes—
72C3×Dic3C2311yescharacteristic
73C2×#41—91noSylow
74C23×C3—32no—
75C22×D3—31no—
76C22×D3—31no—
77C22×D3C2×D312yes—
78C3×Q8—31no—
79C3×D4—21no—
80C3×D4—32no—
81C3×D4—32no—
82C3×D4—31no—
83C3×D4C2×D311yescharacteristic, complemented
84C3×D4C2×D311yescharacteristic, complemented
85C2×C3×C4—32no—
86C2×C3×C4—31no—
87C2×C3×C4—31no—
88#67—32no—
89#67—32no—
90#67—32no—
91#67—32no—
92#67—32no—
93#67—62no—
94#67C2×D312yescomplemented
95#67C2×D312yescomplemented
96C2×Dic3—32no—
97C2×Dic3—32no—
98C2×Dic3—32no—
99C2×Dic3—31no—
100C2×Dic3—31no—
101C2×Dic3—32no—
102C2×Dic3—32no—
103C2×Dic3—62no—
104C2×Dic3C2×D312yescomplemented
105D12—32no—
106D12C2×D311yescharacteristic, complemented
107C4×D3—31no—
108C4×D3—32no—
109C4×D3—32no—
110C4×D3—31no—
111C4×D3—32no—
112C4×D3—31no—
113C4×D3—31no—
114C4×D3—31no—
115C4×D3—31no—
116C4×D3—61no—
117C4×D3C2×D311yescharacteristic
118C4×D3C2×D311yescharacteristic, complemented
119Dic6—31no—
120Dic6—31no—
121Dic6—32no—
122Dic6—32no—
123Dic6—61no—
124Dic6C2×D311yescharacteristic, complemented
125C2×C32—22no—
126C2×C32C2411yescharacteristic
127C3⋊D3—21no—
128C3×D3—21no—
129C3×D3—22no—
130C3×D3#4112yescomplemented
131#41—31no—
132#41—92no—
133#41—91no—
134#41—92no—
135#41—91no—
136#41—91no—
137C2×Q8—91no—
138C2×D4—31no—
139C2×D4—92no—
140C22×C4—91no—
141C22×C4—92no—
142C22×C3—22no—
143C22×C3—31no—
144C22×C3—32no—
145C22×C3—31no—
146C22×C3—64no—
147C22×C3C22×D312yes—
148C22×C3C22×D312yes—
149C2×D3—24no—
150C2×D3—31no—
151C2×D3—31no—
152C2×D3—61no—
153C2×D3—62no—
154C2×D3—62no—
155C2×D3C22×D312yes—
156C2×D3C22×D311yescharacteristic
157C2×D3C22×D311yescharacteristic
158C3×C4—21no—
159C3×C4—31no—
160C3×C4—32no—
161C3×C4—31no—
162C3×C4C22×D311yescharacteristic
163C3×C4C22×D311yescharacteristic
164Dic3—32no—
165Dic3—32no—
166Dic3—31no—
167Dic3—31no—
168Dic3—62no—
169Dic3—61no—
170Dic3C22×D311yescharacteristic
171Dic3C22×D312yes—
172Dic3C22×D311yescharacteristic
173C32C2×#4111yescharacteristic, complemented, Sylow
174C23—32no—
175C23—91no—
176Q8—31no—
177Q8—92no—
178Q8—91no—
179D4—32no—
180D4—32no—
181D4—31no—
182D4—92no—
183D4—92no—
184D4—92no—
185D4D3211yescharacteristic, complemented
186C2×C4—32no—
187C2×C4—31no—
188C2×C4—31no—
189C2×C4—91no—
190C2×C4—91no—
191C2×C4—92no—
192C2×C4—92no—
193C2×C4—91no—
194C2×C4—92no—
195C2×C4—92no—
196C2×C4—91no—
197C2×C3—22no—
198C2×C3—22no—
199C2×C3—21no—
200C2×C3—32no—
201C2×C3—42no—
202C2×C3—61no—
203C2×C3—62no—
204C2×C3C23×D311yescharacteristic
205C2×C3C23×D311yescharacteristic
206D3—22no—
207D3—21no—
208D3—61no—
209D3—61no—
210D3—121no—
211D3#23712yesdirect product factor
212C22—31no—
213C22—31no—
214C22—32no—
215C22—64no—
216C22—91no—
217C22—182no—
218C22C2×D3212yes—
219C4—31no—
220C4—31no—
221C4—32no—
222C4—91no—
223C4—92no—
224C4C2×D3211yescharacteristic
225C3—21no—
226C3C2×#23711yescharacteristic, complemented
227C3D3×#4111yescharacteristic, complemented
228C2—22no—
229C2—32no—
230C2—62no—
231C2—61no—
232C2—181no—
233C2C22×D3211yescharacteristic
subgroupG/H
center, FrattiniC2C22×D32
commutator, socle, cosocleC2×C32C24
FittingC32×D4C22
hypercenterD4D32
hypocenterC32C2×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. #3830C2semidirect
2. C3×#237C2semidirect
3. C3×D3×D4C2semidirect
4. D3×#67C2semidirect
5. #3805C2semidirect
6. #3804C2semidirect
7. C2×D3×Dic3C2semidirect
8. C4×D32C2semidirect
9. #3796C2semidirect
10. #3795C2semidirect
11. D3×Dic6C2semidirect
16. C32×D4C22semidirect
17. #638C22semidirect
18. C2×C3⋊Dic3C22semidirect
19. C4×C3⋊D3C22semidirect
20. #634C22semidirect
21. C3×#67C22semidirect
22. C3×#67C22semidirect
23. C2×C3×Dic3C22semidirect
24. C3×D12C22semidirect
26. C3×C4×D3C22semidirect
27. C3×Dic6C22semidirect
28. #627C22semidirect
29. #626C22semidirect
30. #625C22semidirect
31. #624C22semidirect
34. D3×Dic3C22semidirect
35. D3×Dic3C22semidirect
36. D3×Dic3C22semidirect
51. #237D3direct (N × H)
53. D3×D4D3semidirect
83. C3×D4C2×D3semidirect
84. C3×D4C2×D3semidirect
94. #67C2×D3semidirect
95. #67C2×D3semidirect
104. C2×Dic3C2×D3semidirect
106. D12C2×D3semidirect
118. C4×D3C2×D3semidirect
124. Dic6C2×D3semidirect
130. C3×D3#41semidirect
173. C32C2×#41semidirect
185. D4D32semidirect
211. D3#237direct (N × H)
226. C3C2×#237semidirect
227. C3D3×#41semidirect

Maximal subgroups

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

subgroupindex
#38302
C3×#2372
C3×D3×D42
D3×#672
#38052
#38042
C2×D3×Dic32
C4×D322
#37962
#37952
D3×Dic62
D3×#413
C2×#2373