Finite Group Databasebeta

← #78637

Subgroups — #78637

Order 384.

994 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×D12C211yesmaximal, characteristic, complemented
2#5791C211yesmaximal, characteristic, complemented
3C2×#908C212yesmaximal, complemented
4C3×#407C211yesmaximal, characteristic, complemented
5#5131C212yesmaximal, complemented
6#4732C214yesmaximal, complemented
7#4731C214yesmaximal, complemented
8#3003—31nomaximal, Sylow
9C24×D3—21no—
10C23×C3×C4C2211yescharacteristic
11C2×#235C2211yescharacteristic, complemented
12C22×D12—22no—
13C22×D12—21no—
14C22×D12—28no—
15C22×D12C2214yes—
16C22×D12C2212yes—
17C2×C3×M16C2212yescomplemented
18#953—21no—
19#930C2212yescomplemented
20#929—21no—
21C4×#67—22no—
22#926C2211yescharacteristic, complemented
23#924C2212yescomplemented
24#908—28no—
25C2×D24C2214yescomplemented
26C2×#204C2214yescomplemented
27C3×C22⋊C8C2214yescomplemented
28#821C2218yescomplemented
29C23×D4—31no—
30C2×AGL(1,ℤ/8)—32no—
31#525—31no—
32#450—34no—
33#447—34no—
34#418—32no—
35#407D311yescharacteristic, complemented
36C24×C3D411yescharacteristic
37C23×D3—28no—
38C23×D3—24no—
39C23×D3—22no—
40C23×D3—48no—
41C22×C3×C4—24no—
42C22×C3×C4D412yes—
43C22×C3×C4D411yescharacteristic
44C22×C3×C4C2312yes—
45C22×C3×C4C2311yescharacteristic
46C2×#67—22no—
47C2×#67—21no—
48#235—24no—
49C2×D12—28no—
50C2×D12—216no—
51C2×D12—24no—
52C2×D12—22no—
53C2×D12—24no—
54C2×D12—216no—
55C2×D12D418yescomplemented
56C2×D12C2314yes—
57C2×D12C2311yescharacteristic, complemented
58C2×C4×D3—21no—
59C2×Dic6C2311yescharacteristic, complemented
60C3×M16—24no—
61C2×C3×C8C2314yescomplemented
62#217—22no—
63#217—21no—
64(C2×D3)⋊C4—22no—
65C4⋊Dic3C2312yescomplemented
66Dic3⋊C4—22no—
67C4×Dic3—21no—
68D24—28no—
69#204—28no—
70C25—61no—
71C2×#41—31no—
72C22×D4—32no—
73C22×D4—34no—
74C22×D4—68no—
75C22×D4—62no—
76C22×D4—61no—
77C23×C4C2×D311yescharacteristic
78AGL(1,ℤ/8)—68no—
79C2×SD16—34no—
80C2×D8—34no—
81C2×M16C2×D312yescomplemented
82#123—61no—
83C22⋊Q8—32no—
84#121—32no—
85#120—61no—
86C4×D4—62no—
87#117—31no—
88#102—38no—
89C22⋊C8C2×D314yescomplemented
90C23×C3—22no—
91C23×C3—24no—
92C23×C3C2×D411yescharacteristic
93C23×C3C2×D412yes—
94C22×D3—216no—
95C22×D3—28no—
96C22×D3—24no—
97C22×D3—21no—
98C22×D3—432no—
99C22×D3—48no—
100C22×D3—48no—
101C22×D3—48no—
102C2×C3×C4—22no—
103C2×C3×C4—24no—
104C2×C3×C4—24no—
105C2×C3×C4C2×D412yes—
106C2×C3×C4C2×D411yescharacteristic
107C2×C3×C4C2×D414yes—
108C2×C3×C4C2411yescharacteristic
109#67—42no—
110#67—44no—
111C2×Dic3—22no—
112C2×Dic3—21no—
113D12—216no—
114D12—28no—
115D12—24no—
116D12—22no—
117D12C2×D418yes—
118C4×D3—42no—
119Dic6—22no—
120C3×C8—24no—
121C24—68no—
122C24—64no—
123C24—62no—
124C24—128no—
125C24D1211yescharacteristic
126#41—64no—
127C2×Q8—31no—
128C2×D4—38no—
129C2×D4—34no—
130C2×D4—31no—
131C2×D4—616no—
132C2×D4—68no—
133C2×D4—616no—
134C2×D4—62no—
135C2×D4—61no—
136C2×D4—64no—
137C2×D4—62no—
138C2×D4—64no—
139C22×C4—24no—
140C22×C4—61no—
141C22×C4D1212yes—
142C22×C4D1211yescharacteristic
143C22×C4C22×D312yes—
144C22×C4C22×D311yescharacteristic
145SD16—68no—
146D8—68no—
147M16—24no—
148C2×C8C22×D314yescomplemented
149C4⋊C4—32no—
150C4⋊C4—62no—
151C22⋊C4—61no—
152C22⋊C4—62no—
153C22⋊C4—62no—
154C42—61no—
155C22×C3—28no—
156C22×C3—22no—
157C22×C3—22no—
158C22×C3—22no—
159C22×C3#12012yes—
160C22×C3AGL(1,ℤ/8)14yescomplemented
161C22×C3C22×D411yescharacteristic
162C2×D3—28no—
163C2×D3—416no—
164C2×D3—432no—
165C2×D3—48no—
166C2×D3—41no—
167C2×D3—42no—
168C3×C4—22no—
169C3×C4#12012yes—
170C3×C4C22×D412yes—
171Dic3—41no—
172Dic3—42no—
173C23—22no—
174C23—24no—
175C23—616no—
176C23—68no—
177C23—64no—
178C23—61no—
179C23—1232no—
180C23—128no—
181C23—128no—
182C23—128no—
183C23C2×D1211yescharacteristic
184C23C2×D1212yes—
185Q8—62no—
186D4—38no—
187D4—616no—
188D4—68no—
189D4—64no—
190D4—62no—
191D4—122no—
192D4—124no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—62no—
197C2×C4—61no—
198C2×C4—122no—
199C2×C4C2×D1212yes—
200C2×C4C2×D1211yescharacteristic
201C2×C4D3×D414yes—
202C2×C4C23×D311yescharacteristic
203C8—24no—
204C2×C3—22no—
205C2×C3—24no—
206C2×C3C2×#12011yescharacteristic
207C2×C3C2×AGL(1,ℤ/8)12yes—
208D3—48no—
209D3—81no—
210C22—28no—
211C22—22no—
212C22—22no—
213C22—22no—
214C22—68no—
215C22—1216no—
216C22—1232no—
217C22—128no—
218C22—121no—
219C22—122no—
220C22#88212yes—
221C22#90814yescomplemented
222C22C22×D1211yescharacteristic
223C4—22no—
224C4—121no—
225C4—122no—
226C4#88212yes—
227C4C2×D3×D412yes—
228C3#300311yescharacteristic, complemented, Sylow
229C2—22no—
230C2—24no—
231C2—128no—
232C2—241no—
233C2C2×#88211yescharacteristic
234C2C2×#90812yes—
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#407C2
socleC22×C3C22×D4
hypercenter#407D3
hypocenterC3#3003
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C23×D12C2semidirect
2. #5791C2semidirect
3. C2×#908C2semidirect
4. C3×#407C2semidirect
5. #5131C2semidirect
6. #4732C2semidirect
7. #4731C2semidirect
11. C2×#235C22semidirect
17. C2×C3×M16C22semidirect
19. #930C22semidirect
22. #926C22semidirect
23. #924C22semidirect
25. C2×D24C22semidirect
26. C2×#204C22semidirect
27. C3×C22⋊C8C22semidirect
28. #821C22semidirect
35. #407D3semidirect
55. C2×D12D4semidirect
57. C2×D12C23semidirect
59. C2×Dic6C23semidirect
61. C2×C3×C8C23semidirect
65. C4⋊Dic3C23semidirect
81. C2×M16C2×D3semidirect
89. C22⋊C8C2×D3semidirect
148. C2×C8C22×D3semidirect
160. C22×C3AGL(1,ℤ/8)semidirect
221. C22#908semidirect
228. C3#3003semidirect

Maximal subgroups

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

subgroupindex
C23×D122
#57912
C2×#9082
C3×#4072
#51312
#47322
#47312
#30033