Finite Group Databasebeta

← #64314

Subgroups — #64314

Order 288.

359 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 108 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
1D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C3×#237C211yesmaximal, characteristic, complemented
3C3×D3×D4C211yesmaximal, characteristic, complemented
4#3811C212yesmaximal, complemented
5D3×#67C212yesmaximal, complemented
6C2×#626C212yesmaximal, complemented
7#3804C212yesmaximal, complemented
8D3×D12C211yesmaximal, characteristic, complemented
9#3799C211yesmaximal, characteristic, complemented
10#3798C211yesmaximal, characteristic, complemented
11#3796C211yesmaximal, characteristic, complemented
12#1009—31nomaximal
13#1004—31nomaximal
14C22×C3⋊D3C2212yescomplemented
15C22×C3×D3C2212yescomplemented
16C2×D32C2212yescomplemented
17C32×D4C2211yescharacteristic, complemented
18#638C2212yescomplemented
19C4⋊(C3⋊D3)C2211yescharacteristic, complemented
20C4×C3⋊D3C2211yescharacteristic, complemented
21C3×#67C2212yescomplemented
22C3×#67C2212yescomplemented
23C2×C3×Dic3C2212yescomplemented
24C3×D12C2211yescharacteristic, complemented
25C3×C4×D3C2211yescharacteristic, complemented
26C3×C4×D3C2211yescharacteristic, complemented
27C3×Dic6C2211yescharacteristic, complemented
28#627C2211yescharacteristic, complemented
29#626C2212yescomplemented
30#626C2211yescharacteristic, complemented
31#626C2211yescharacteristic, complemented
32#626C2212yescomplemented
33#626C2212yescomplemented
34#625C2211yescharacteristic, complemented
35#624C2212yescomplemented
36D3×Dic3C2212yescomplemented
37C3×#41—31no—
38C2×C3×D4—31no—
39C2×#67—32no—
40C2×#67—32no—
41#239—31no—
42#239—31no—
43#237—32no—
44#237—31no—
45#237D311yescharacteristic, complemented
46D3×D4—32no—
47D3×D4—31no—
48D3×D4—31no—
49D3×D4—32no—
50D3×D4—32no—
51D3×D4—31no—
52D3×D4—61no—
53D3×D4D311yescharacteristic, complemented
54#235—32no—
55#235—31no—
56#235—31no—
57#235—31no—
58C2×D12—32no—
59C2×D12—31no—
60C22×C32C2312yes—
61C2×C3⋊D3—42no—
62C2×C3⋊D3C2312yes—
63C2×C3⋊D3C2311yescharacteristic
64C2×C3×D3—42no—
65C2×C3×D3C2312yes—
66C2×C3×D3C2311yescharacteristic
67C2×C3×D3C2311yescharacteristic
68D32—42no—
69C32×C4C2311yescharacteristic
70C3⋊Dic3C2311yescharacteristic
71C3×Dic3C2312yes—
72C3×Dic3C2311yescharacteristic
73C3×Dic3C2311yescharacteristic
74H(5,𝔽2)—91noSylow
75C23×C3—32no—
76C22×D3—32no—
77C22×D3—32no—
78C22×D3—32no—
79C22×D3—32no—
80C22×D3—62no—
81C22×D3C2×D312yescomplemented
82C3×Q8—31no—
83C3×D4—21no—
84C3×D4—32no—
85C3×D4—32no—
86C3×D4—31no—
87C3×D4C2×D311yescharacteristic, complemented
88C3×D4C2×D311yescharacteristic, complemented
89C2×C3×C4—32no—
90C2×C3×C4—31no—
91C2×C3×C4—31no—
92#67—32no—
93#67—32no—
94#67—31no—
95#67—31no—
96#67—31no—
97#67—31no—
98#67—32no—
99#67—32no—
100#67—32no—
101#67—62no—
102#67C2×D312yescomplemented
103#67C2×D312yescomplemented
104C2×Dic3—32no—
105C2×Dic3C2×D312yescomplemented
106D12—32no—
107D12—31no—
108D12—32no—
109D12—32no—
110D12—31no—
111D12—31no—
112D12—31no—
113D12—61no—
114D12C2×D311yescharacteristic, complemented
115C4×D3—32no—
116C4×D3—31no—
117C4×D3—31no—
118C4×D3—32no—
119C4×D3—32no—
120C4×D3—61no—
121C4×D3C2×D311yescharacteristic, complemented
122C4×D3C2×D311yescharacteristic, complemented
123Dic6—31no—
124Dic6—31no—
125Dic6C2×D311yescharacteristic, complemented
126C2×C32—22no—
127C2×C32C2411yescharacteristic
128C3⋊D3—21no—
129C3⋊D3—22no—
130C3×D3—21no—
131C3×D3—22no—
132C3×D3—21no—
133#41—31no—
134#41—91no—
135#41—92no—
136#41—91no—
137#41—91no—
138C2×D4—31no—
139C2×D4—91no—
140C2×D4—92no—
141C2×D4—92no—
142C2×D4—92no—
143C2×D4—91no—
144C22×C3—22no—
145C22×C3—32no—
146C22×C3—31no—
147C22×C3—31no—
148C22×C3—122no—
149C22×C3C22×D312yes—
150C22×C3C22×D312yes—
151C2×D3—31no—
152C2×D3—32no—
153C2×D3—31no—
154C2×D3—32no—
155C2×D3—42no—
156C2×D3—61no—
157C2×D3—62no—
158C2×D3—122no—
159C2×D3—122no—
160C2×D3—122no—
161C2×D3—122no—
162C2×D3—124no—
163C2×D3C22×D312yes—
164C2×D3C22×D311yescharacteristic
165C2×D3C22×D311yescharacteristic
166C3×C4—21no—
167C3×C4—31no—
168C3×C4—31no—
169C3×C4—32no—
170C3×C4C22×D311yescharacteristic
171C3×C4C22×D311yescharacteristic
172Dic3—31no—
173Dic3—31no—
174Dic3—61no—
175Dic3C22×D311yescharacteristic
176Dic3C22×D311yescharacteristic
177Dic3C22×D312yes—
178C32H(5,𝔽2)11yescharacteristic, complemented, Sylow
179C23—32no—
180C23—92no—
181C23—92no—
182Q8—31no—
183Q8—91no—
184D4—32no—
185D4—32no—
186D4—31no—
187D4—91no—
188D4—92no—
189D4—91no—
190D4—92no—
191D4—92no—
192D4—91no—
193D4—92no—
194D4—91no—
195D4D3211yescharacteristic, complemented
196C2×C4—32no—
197C2×C4—31no—
198C2×C4—31no—
199C2×C4—92no—
200C2×C4—92no—
201C2×C4—91no—
202C2×C3—22no—
203C2×C3—22no—
204C2×C3—21no—
205C2×C3—42no—
206C2×C3—61no—
207C2×C3—62no—
208C2×C3—61no—
209C2×C3C23×D311yescharacteristic
210C2×C3C23×D311yescharacteristic
211D3—21no—
212D3—22no—
213D3—21no—
214D3—62no—
215D3—61no—
216D3—62no—
217D3—61no—
218D3—62no—
219D3—122no—
220C22—31no—
221C22—31no—
222C22—32no—
223C22—91no—
224C22—92no—
225C22—122no—
226C22—362no—
227C22—362no—
228C22C2×D3212yes—
229C4—31no—
230C4—31no—
231C4—32no—
232C4—91no—
233C4C2×D3211yescharacteristic
234C3—21no—
235C3#100411yescharacteristic, complemented
236C3#100911yescharacteristic, complemented
237C2—22no—
238C2—61no—
239C2—62no—
240C2—61no—
241C2—181no—
242C2—182no—
243C2C22×D3211yescharacteristic
subgroupG/H
center, FrattiniC2C22×D32
commutator, socle, cosocleC2×C32C24
FittingC32×D4C22
hypercenterD4D32
hypocenterC32H(5,𝔽2)
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. D4×C3⋊D3C2semidirect
2. C3×#237C2semidirect
3. C3×D3×D4C2semidirect
4. #3811C2semidirect
5. D3×#67C2semidirect
6. C2×#626C2semidirect
7. #3804C2semidirect
8. D3×D12C2semidirect
9. #3799C2semidirect
10. #3798C2semidirect
11. #3796C2semidirect
14. C22×C3⋊D3C22semidirect
15. C22×C3×D3C22semidirect
16. C2×D32C22semidirect
17. C32×D4C22semidirect
18. #638C22semidirect
19. C4⋊(C3⋊D3)C22semidirect
20. C4×C3⋊D3C22semidirect
21. C3×#67C22semidirect
22. C3×#67C22semidirect
23. C2×C3×Dic3C22semidirect
24. C3×D12C22semidirect
25. C3×C4×D3C22semidirect
26. C3×C4×D3C22semidirect
27. C3×Dic6C22semidirect
28. #627C22semidirect
29. #626C22semidirect
30. #626C22semidirect
31. #626C22semidirect
32. #626C22semidirect
33. #626C22semidirect
34. #625C22semidirect
35. #624C22semidirect
36. D3×Dic3C22semidirect
45. #237D3semidirect
53. D3×D4D3semidirect
81. C22×D3C2×D3semidirect
87. C3×D4C2×D3semidirect
88. C3×D4C2×D3semidirect
102. #67C2×D3semidirect
103. #67C2×D3semidirect
105. C2×Dic3C2×D3semidirect
114. D12C2×D3semidirect
121. C4×D3C2×D3semidirect
122. C4×D3C2×D3semidirect
125. Dic6C2×D3semidirect
178. C32H(5,𝔽2)semidirect
195. D4D32semidirect
235. C3#1004semidirect
236. C3#1009semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊D32
C3×#2372
C3×D3×D42
#38112
D3×#672
C2×#6262
#38042
D3×D122
#37992
#37982
#37962
#10093
#10043