Finite Group Databasebeta

← #80868

Subgroups — #80868

Order 384.

514 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 113 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
1D3×#121C212yesmaximal, complemented
2C3×#469C211yesmaximal, characteristic, complemented
3#5156C212yesmaximal, complemented
4#5135C211yesmaximal, characteristic, complemented
5#5126C211yesmaximal, characteristic, complemented
6#5039C212yesmaximal, complemented
7#4863C212yesmaximal, complemented
8#4777C212yesmaximal, complemented
9#4773C212yesmaximal, complemented
10#3192—31nomaximal, Sylow
11C2×D3×D4—22no—
12C2×D3×D4C2212yescomplemented
13C22×C4×D3C2211yescharacteristic
14C2×C3×D8C2211yescharacteristic, complemented
15C2×C3×M16C2211yescharacteristic, complemented
16C3×#121C2212yescomplemented
17#939—22no—
18#938C2212yes—
19C2×#213C2212yescomplemented
20#930C2212yescomplemented
21C2×#208C2211yescharacteristic, complemented
22C2×D24C2211yescharacteristic, complemented
23#895C2212yes—
24D3×C4⋊C4C2212yescomplemented
25#884—22no—
26D3×C22⋊C4—22no—
27C3×#106C2211yescharacteristic, complemented
28C3×#102C2212yescomplemented
29#832C2212yescomplemented
30#821C2212yescomplemented
31(C2×D3)⋊C8C2212yescomplemented
32(C2×D3)⋊C8C2212yescomplemented
33#817C2211yescharacteristic, complemented
34#809C2212yescomplemented
35#808C2212yescomplemented
36C2×#121—32no—
37#481—32no—
38#481—32no—
39#469—31no—
40#469—32no—
41#469D311yescharacteristic, complemented
42#447—32no—
43#447—32no—
44#407—31no—
45C23×D3—22no—
46C23×D3D411yescharacteristic
47C2×C3×D4—22no—
48C2×C3×D4C2312yes—
49C22×C3×C4C2311yescharacteristic
50C2×#67—22no—
51C2×#67—22no—
52C2×#67—22no—
53C22×Dic3D411yescharacteristic
54D3×D4—42no—
55D3×D4—42no—
56D3×D4—48no—
57C2×D12—22no—
58C2×D12C2312yes—
59C2×C4×D3—22no—
60C2×C4×D3—41no—
61C2×C4×D3—41no—
62C2×C4×D3D412yes—
63C2×C4×D3C2312yes—
64C3×D8—41no—
65C3×M16—41no—
66C2×C3×C8C2311yescharacteristic
67C2×C3×C8C2311yescharacteristic
68C3×C4⋊C4C2312yes—
69C3×C22⋊C4—22no—
70#217—22no—
71#213—42no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4—22no—
74C4⋊Dic3C2312yes—
75Dic3⋊C4—22no—
76#208—41no—
77C2×C3⋊C8C2312yes—
78D24—41no—
79C22×D4—32no—
80C22×D4—62no—
81C23×C4—31no—
82C2×D8—32no—
83C2×D8—31no—
84C2×D8C2×D311yescharacteristic, complemented
85C2×M16—31no—
86C2×M16C2×D311yescharacteristic, complemented
87#121—32no—
88#121—32no—
89#121—32no—
90#121—62no—
91#121—62no—
92#121C2×D312yescomplemented
93C2×C4⋊C4—32no—
94C2×C22⋊C4—62no—
95#106—31no—
96#106—32no—
97#106C2×D311yescharacteristic, complemented
98#102—32no—
99#102—32no—
100#102—32no—
101#102C2×D312yescomplemented
102C22⋊C8—32no—
103C22⋊C8—32no—
104C23×C3—22no—
105C23×C3C2×D411yescharacteristic
106C22×D3—24no—
107C22×D3—22no—
108C22×D3—41no—
109C22×D3—44no—
110C22×D3—44no—
111C22×D3—44no—
112C22×D3C2×D412yes—
113C3×D4—22no—
114C3×D4—22no—
115C3×D4—44no—
116C2×C3×C4—21no—
117C2×C3×C4—21no—
118C2×C3×C4—22no—
119C2×C3×C4C2×D411yescharacteristic
120C2×C3×C4C2411yescharacteristic
121#67—44no—
122#67—44no—
123#67—44no—
124C2×Dic3—22no—
125C2×Dic3—41no—
126C2×Dic3C2×D412yes—
127D12—22no—
128D12—22no—
129D12—44no—
130C4×D3—22no—
131C4×D3—22no—
132C4×D3—42no—
133C4×D3—44no—
134C3×C8—21no—
135C3×C8—21no—
136C3⋊C8—22no—
137C24—31no—
138C24—62no—
139C2×D4—22no—
140C2×D4—32no—
141C2×D4—62no—
142C2×D4—62no—
143C2×D4—62no—
144C2×D4—62no—
145C2×D4—122no—
146C2×D4—122no—
147C2×D4—128no—
148C2×D4C22×D312yes—
149C22×C4—31no—
150C22×C4—32no—
151C22×C4—32no—
152C22×C4—62no—
153C22×C4—121no—
154C22×C4—121no—
155C22×C4C22×D311yescharacteristic
156D8—41no—
157D8—121no—
158D8—122no—
159M16—41no—
160M16—121no—
161C2×C8—32no—
162C2×C8C22×D311yescharacteristic
163C2×C8C22×D311yescharacteristic
164C4⋊C4—32no—
165C4⋊C4—62no—
166C4⋊C4C22×D312yes—
167C22⋊C4—22no—
168C22⋊C4—62no—
169C22⋊C4—62no—
170C22⋊C4—62no—
171C22×C3—21no—
172C22×C3—22no—
173C22×C3—42no—
174C22×C3—42no—
175C22×C3—42no—
176C22×C3C22×D411yescharacteristic
177C2×D3—22no—
178C2×D3—22no—
179C2×D3—44no—
180C2×D3—42no—
181C2×D3—42no—
182C2×D3—42no—
183C2×D3—88no—
184C2×D3AGL(1,ℤ/8)14yes—
185C3×C4—21no—
186C3×C4—42no—
187C3×C4H(5,𝔽2)11yescharacteristic
188C3×C4H(5,𝔽2)11yescharacteristic
189Dic3—22no—
190Dic3—42no—
191C23—22no—
192C23—32no—
193C23—64no—
194C23—62no—
195C23—121no—
196C23—124no—
197C23—124no—
198C23—124no—
199C23D3×D411yescharacteristic
200D4—22no—
201D4—22no—
202D4—44no—
203D4—62no—
204D4—62no—
205D4—124no—
206D4—124no—
207D4—124no—
208D4—124no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—32no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—122no—
217C2×C4—124no—
218C2×C4—121no—
219C2×C4D3×D411yescharacteristic
220C2×C4C23×D311yescharacteristic
221C8—21no—
222C8—21no—
223C8—62no—
224C2×C3—41no—
225C2×C3—82no—
226C2×C3#53411yescharacteristic
227C2×C3C2×AGL(1,ℤ/8)12yes—
228D3—24no—
229D3—82no—
230C22—22no—
231C22—21no—
232C22—34no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—62no—
237C22—62no—
238C22—124no—
239C22—122no—
240C22—122no—
241C22—122no—
242C22—248no—
243C22C2×D3×D411yescharacteristic
244C4—21no—
245C4—42no—
246C4—62no—
247C4—122no—
248C4#100411yescharacteristic
249C4#100911yescharacteristic
250C3#319211yescharacteristic, complemented, Sylow
251C2—41no—
252C2—64no—
253C2—82no—
254C2—242no—
255C2#560511yescharacteristic
256C2D3×AGL(1,ℤ/8)12yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#469C2
socleC22×C3C22×D4
hypercenter#469D3
hypocenterC3#3192
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. D3×#121C2semidirect
2. C3×#469C2semidirect
3. #5156C2semidirect
4. #5135C2semidirect
5. #5126C2semidirect
6. #5039C2semidirect
7. #4863C2semidirect
8. #4777C2semidirect
9. #4773C2semidirect
12. C2×D3×D4C22semidirect
14. C2×C3×D8C22semidirect
15. C2×C3×M16C22semidirect
16. C3×#121C22semidirect
19. C2×#213C22semidirect
20. #930C22semidirect
21. C2×#208C22semidirect
22. C2×D24C22semidirect
24. D3×C4⋊C4C22semidirect
27. C3×#106C22semidirect
28. C3×#102C22semidirect
29. #832C22semidirect
30. #821C22semidirect
31. (C2×D3)⋊C8C22semidirect
32. (C2×D3)⋊C8C22semidirect
33. #817C22semidirect
34. #809C22semidirect
35. #808C22semidirect
41. #469D3semidirect
84. C2×D8C2×D3semidirect
86. C2×M16C2×D3semidirect
92. #121C2×D3semidirect
97. #106C2×D3semidirect
101. #102C2×D3semidirect
250. C3#3192semidirect

Maximal subgroups

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

subgroupindex
D3×#1212
C3×#4692
#51562
#51352
#51262
#50392
#48632
#47772
#47732
#31923