Finite Group Databasebeta

← #84038

Subgroups — #84038

Order 384.

994 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 173 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×C3×D4C211yesmaximal, characteristic, complemented
2C2×#932C212yesmaximal, complemented
3#5791C211yesmaximal, characteristic, complemented
4#5218C214yesmaximal, complemented
5#5217C214yesmaximal, complemented
6#5215C212yesmaximal, complemented
7#5207C211yesmaximal, characteristic, complemented
8#3003—31nomaximal, Sylow
9C25×C3—21no—
10C22×C3×D4—28no—
11C22×C3×D4—22no—
12C22×C3×D4—21no—
13C22×C3×D4C2214yes—
14C22×C3×D4C2212yes—
15C23×C3×C4C2211yescharacteristic
16C2×#235C2211yescharacteristic, complemented
17#953—21no—
18C2×#214C2214yescomplemented
19#932—28no—
20C2×#213C2214yescomplemented
21#930C2212yescomplemented
22#929—21no—
23C4×#67—22no—
24#926C2211yescharacteristic, complemented
25#924C2212yescomplemented
26C2×#208C2212yescomplemented
27#832C2218yescomplemented
28#830C2214yescomplemented
29C23×D4D311yescharacteristic, complemented
30C2×AGL(1,ℤ/8)—32no—
31#525—31no—
32#450—34no—
33#447—34no—
34#418—32no—
35#407—31no—
36C24×C3—28no—
37C24×C3—24no—
38C24×C3—22no—
39C24×C3—48no—
40C24×C3D411yescharacteristic
41C2×C3×D4—28no—
42C2×C3×D4—216no—
43C2×C3×D4—216no—
44C2×C3×D4—24no—
45C2×C3×D4—22no—
46C2×C3×D4—24no—
47C2×C3×D4D418yescomplemented
48C2×C3×D4C2314yes—
49C22×C3×C4—24no—
50C22×C3×C4D412yes—
51C22×C3×C4D411yescharacteristic
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C2×#67—22no—
55C2×#67—21no—
56#235—24no—
57C2×D12C2311yescharacteristic, complemented
58C2×C4×D3—21no—
59C2×Dic6C2311yescharacteristic, complemented
60#217—22no—
61#217—21no—
62#214—28no—
63#213—28no—
64(C2×D3)⋊C4—22no—
65C4⋊Dic3C2312yescomplemented
66Dic3⋊C4—22no—
67C4×Dic3—21no—
68#208—24no—
69C2×C3⋊C8C2314yescomplemented
70C25—21no—
71C2×#41—31no—
72C22×D4—28no—
73C22×D4—22no—
74C22×D4—21no—
75C22×D4C2×D314yes—
76C22×D4C2×D312yes—
77C23×C4C2×D311yescharacteristic
78AGL(1,ℤ/8)—68no—
79C2×SD16—34no—
80C2×D8—34no—
81C2×M16—32no—
82#123—61no—
83C22⋊Q8—32no—
84#121—32no—
85#120—61no—
86C4×D4—62no—
87#117—31no—
88#102—38no—
89C22⋊C8—34no—
90C23×C3—22no—
91C23×C3—24no—
92C23×C3—216no—
93C23×C3—28no—
94C23×C3—24no—
95C23×C3—432no—
96C23×C3—48no—
97C23×C3—48no—
98C23×C3—48no—
99C23×C3C2×D411yescharacteristic
100C23×C3C2×D412yes—
101C22×D3—21no—
102C3×D4—216no—
103C3×D4—28no—
104C3×D4—24no—
105C3×D4C2×D418yes—
106C2×C3×C4—22no—
107C2×C3×C4—24no—
108C2×C3×C4—24no—
109C2×C3×C4C2×D412yes—
110C2×C3×C4C2×D411yescharacteristic
111C2×C3×C4C2×D414yes—
112C2×C3×C4C2411yescharacteristic
113#67—42no—
114#67—44no—
115C2×Dic3—22no—
116C2×Dic3—21no—
117D12—22no—
118C4×D3—42no—
119Dic6—22no—
120C3⋊C8—24no—
121C24—28no—
122C24—24no—
123C24—22no—
124C24—48no—
125C24#6711yescharacteristic
126#41—64no—
127C2×Q8—31no—
128C2×D4—28no—
129C2×D4—216no—
130C2×D4—216no—
131C2×D4—24no—
132C2×D4—22no—
133C2×D4—24no—
134C2×D4—31no—
135C2×D4—62no—
136C2×D4—61no—
137C2×D4#6718yescomplemented
138C2×D4C22×D314yes—
139C22×C4—24no—
140C22×C4—61no—
141C22×C4#6712yes—
142C22×C4#6711yescharacteristic
143C22×C4C22×D312yes—
144C22×C4C22×D311yescharacteristic
145SD16—68no—
146D8—68no—
147M16—64no—
148C2×C8—34no—
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—28no—
160C22×C3—416no—
161C22×C3—432no—
162C22×C3—48no—
163C22×C3#12012yes—
164C22×C3AGL(1,ℤ/8)14yescomplemented
165C22×C3C22×D411yescharacteristic
166C2×D3—41no—
167C2×D3—42no—
168C3×C4—22no—
169C3×C4#12012yes—
170C3×C4C22×D412yes—
171Dic3—41no—
172Dic3—42no—
173C23—24no—
174C23—22no—
175C23—216no—
176C23—28no—
177C23—24no—
178C23—432no—
179C23—48no—
180C23—48no—
181C23—48no—
182C23—61no—
183C23C2×#6712yes—
184C23C2×#6711yescharacteristic
185Q8—62no—
186D4—216no—
187D4—28no—
188D4—24no—
189D4—62no—
190D4—124no—
191D4—122no—
192D4C2×#6718yes—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—62no—
197C2×C4—61no—
198C2×C4—122no—
199C2×C4C2×#6712yes—
200C2×C4C2×#6711yescharacteristic
201C2×C4C2×#6714yes—
202C2×C4C23×D311yescharacteristic
203C8—64no—
204C2×C3—22no—
205C2×C3—24no—
206C2×C3—48no—
207C2×C3C2×#12011yescharacteristic
208C2×C3C2×AGL(1,ℤ/8)12yes—
209D3—81no—
210C22—28no—
211C22—22no—
212C22—22no—
213C22—22no—
214C22—28no—
215C22—432no—
216C22—416no—
217C22—48no—
218C22—122no—
219C22—121no—
220C22#93214yescomplemented
221C22#95312yes—
222C22C22×#6711yescharacteristic
223C4—22no—
224C4—122no—
225C4—121no—
226C4#95312yes—
227C4C22×#6712yes—
228C3#300311yescharacteristic, complemented, Sylow
229C2—24no—
230C2—22no—
231C2—48no—
232C2—241no—
233C2C2×#93212yes—
234C2C2×#95311yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC23×C3×D4C2
socleC22×C3C22×D4
hypercenterC23×D4D3
hypocenterC3#3003
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C3×D4C2semidirect
2. C2×#932C2semidirect
3. #5791C2semidirect
4. #5218C2semidirect
5. #5217C2semidirect
6. #5215C2semidirect
7. #5207C2semidirect
16. C2×#235C22semidirect
18. C2×#214C22semidirect
20. C2×#213C22semidirect
21. #930C22semidirect
24. #926C22semidirect
25. #924C22semidirect
26. C2×#208C22semidirect
27. #832C22semidirect
28. #830C22semidirect
29. C23×D4D3semidirect
47. C2×C3×D4D4semidirect
57. C2×D12C23semidirect
59. C2×Dic6C23semidirect
65. C4⋊Dic3C23semidirect
69. C2×C3⋊C8C23semidirect
137. C2×D4#67semidirect
164. C22×C3AGL(1,ℤ/8)semidirect
220. C22#932semidirect
228. C3#3003semidirect

Maximal subgroups

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

subgroupindex
C23×C3×D42
C2×#9322
#57912
#52182
#52172
#52152
#52072
#30033