Finite Group Databasebeta

← #84082

Subgroups — #84082

Order 384.

806 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C22×C3×#41C211yesmaximal, characteristic, complemented
2C2×#949C212yesmaximal, complemented
3C2×#930C211yesmaximal, characteristic, complemented
4#5238C214yesmaximal, complemented
5#5232C212yesmaximal, complemented
6#5227C212yesmaximal, complemented
7#5217C212yesmaximal, complemented
8#5207C211yesmaximal, characteristic, complemented
9#3006—31nomaximal, Sylow
10C2×C3×#41—28no—
11C2×C3×#41—21no—
12C2×C3×#41—21no—
13C2×C3×#41C2212yes—
14C2×C3×#41C2212yescomplemented
15C22×C3×Q8C2211yescharacteristic, complemented
16C22×C3×D4—21no—
17C22×C3×D4C2211yescharacteristic, complemented
18C23×C3×C4—21no—
19C23×C3×C4C2211yescharacteristic
20C22×#67—21no—
21C22×D12C2211yescharacteristic, complemented
22#949—28no—
23C2×#215C2214yescomplemented
24C2×#213C2214yescomplemented
25#930—21no—
26#930—21no—
27#930C2212yes—
28#930C2212yescomplemented
29C2×(C2×D3)⋊C4—21no—
30C2×C4⋊Dic3C2211yescharacteristic, complemented
31C2×#208C2212yescomplemented
32#835C2214yescomplemented
33#832C2214yescomplemented
34#830C2212yescomplemented
35#830C2212yescomplemented
36C22×#41D311yescharacteristic, complemented
37C2×AGL(1,ℤ/8)—32no—
38C2×#121—31no—
39#449—34no—
40#448—32no—
41#447—32no—
42#417—32no—
43#407—31no—
44C24×C3—21no—
45C24×C3D411yescharacteristic
46C23×D3—21no—
47C3×#41—28no—
48C3×#41—28no—
49C3×#41—24no—
50C3×#41—24no—
51C3×#41—24no—
52C3×#41D418yescomplemented
53C2×C3×Q8—24no—
54C2×C3×Q8—21no—
55C2×C3×Q8C2312yes—
56C2×C3×D4—28no—
57C2×C3×D4—22no—
58C2×C3×D4—22no—
59C2×C3×D4—24no—
60C2×C3×D4—21no—
61C2×C3×D4C2312yes—
62C22×C3×C4—28no—
63C22×C3×C4—21no—
64C22×C3×C4—22no—
65C22×C3×C4—22no—
66C22×C3×C4—24no—
67C22×C3×C4D411yescharacteristic
68C22×C3×C4D411yescharacteristic
69C22×C3×C4D411yescharacteristic
70C22×C3×C4C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2311yescharacteristic
73C2×#67—22no—
74C2×#67—22no—
75C2×#67—44no—
76C22×Dic3—21no—
77C2×D12—24no—
78C2×D12—21no—
79C2×D12C2312yes—
80#215—28no—
81#213—28no—
82(C2×D3)⋊C4—22no—
83(C2×D3)⋊C4—22no—
84C4⋊Dic3—21no—
85C4⋊Dic3C2312yes—
86#208—24no—
87C2×C3⋊C8C2314yes—
88C2×#41—28no—
89C2×#41—21no—
90C2×#41—21no—
91C2×#41C2×D312yes—
92C2×#41C2×D312yescomplemented
93C22×Q8C2×D311yescharacteristic, complemented
94C22×D4—21no—
95C22×D4—31no—
96C22×D4—61no—
97C22×D4C2×D311yescharacteristic, complemented
98C23×C4—21no—
99C23×C4C2×D311yescharacteristic
100AGL(1,ℤ/8)—68no—
101C2×SD16—34no—
102C2×D8—34no—
103C2×M16—32no—
104#121—32no—
105#121—32no—
106#121—61no—
107#121—61no—
108C2×C4⋊C4—31no—
109C2×C22⋊C4—61no—
110#103—34no—
111#102—34no—
112C22⋊C8—32no—
113C22⋊C8—32no—
114C23×C3—24no—
115C23×C3—22no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—41no—
119C23×C3—42no—
120C23×C3—42no—
121C23×C3C2×D411yescharacteristic
122C23×C3C2×D411yescharacteristic
123C23×C3C2×D411yescharacteristic
124C22×D3—22no—
125C22×D3—42no—
126C22×D3—42no—
127C22×D3—42no—
128C3×Q8—24no—
129C3×Q8—22no—
130C3×Q8C2×D414yescomplemented
131C3×D4—28no—
132C3×D4—24no—
133C3×D4—28no—
134C3×D4—22no—
135C3×D4C2×D414yescomplemented
136C2×C3×C4—24no—
137C2×C3×C4—28no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—28no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4C2×D411yescharacteristic
146C2×C3×C4C2×D411yescharacteristic
147C2×C3×C4C2×D411yescharacteristic
148C2×C3×C4C2×D414yes—
149C2×C3×C4C2411yescharacteristic
150#67—44no—
151#67—44no—
152C2×Dic3—22no—
153C2×Dic3—42no—
154D12—24no—
155D12—42no—
156C3⋊C8—24no—
157C24—21no—
158C24—61no—
159C24#6711yescharacteristic
160#41—28no—
161#41—28no—
162#41—24no—
163#41—24no—
164#41—24no—
165#41#6718yescomplemented
166C2×Q8—24no—
167C2×Q8—21no—
168C2×Q8C22×D312yes—
169C2×D4—28no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—24no—
173C2×D4—21no—
174C2×D4—32no—
175C2×D4—62no—
176C2×D4—62no—
177C2×D4—64no—
178C2×D4—61no—
179C2×D4—124no—
180C2×D4C22×D312yes—
181C22×C4—28no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—61no—
187C22×C4#6711yescharacteristic
188C22×C4#6711yescharacteristic
189C22×C4#6711yescharacteristic
190C22×C4C22×D311yescharacteristic
191C22×C4C22×D311yescharacteristic
192C22×C4C22×D311yescharacteristic
193SD16—68no—
194D8—68no—
195M16—64no—
196C2×C8—34no—
197C4⋊C4—32no—
198C4⋊C4—61no—
199C22⋊C4—62no—
200C22⋊C4—62no—
201C22×C3—21no—
202C22×C3—21no—
203C22×C3—21no—
204C22×C3—21no—
205C22×C3—21no—
206C22×C3—24no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—44no—
210C22×C3—48no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3#12012yes—
214C22×C3AGL(1,ℤ/8)12yescomplemented
215C22×C3C22×D411yescharacteristic
216C2×D3—42no—
217C2×D3—42no—
218C2×D3—42no—
219C2×D3—84no—
220C3×C4—21no—
221C3×C4—24no—
222C3×C4—21no—
223C3×C4#12012yes—
224C3×C4C22×D412yes—
225Dic3—42no—
226C23—24no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23—62no—
234C23—122no—
235C23—122no—
236C23—122no—
237C23C2×#6711yescharacteristic
238C23C2×#6711yescharacteristic
239C23C2×#6711yescharacteristic
240Q8—24no—
241Q8—22no—
242Q8C2×#6714yescomplemented
243D4—28no—
244D4—24no—
245D4—28no—
246D4—22no—
247D4—64no—
248D4—124no—
249D4—124no—
250D4—122no—
251D4C2×#6714yescomplemented
252C2×C4—24no—
253C2×C4—28no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—28no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—62no—
262C2×C4—122no—
263C2×C4C2×#6711yescharacteristic
264C2×C4C2×#6711yescharacteristic
265C2×C4C2×#6711yescharacteristic
266C2×C4C2×#6714yes—
267C2×C4C23×D311yescharacteristic
268C8—64no—
269C2×C3—22no—
270C2×C3—22no—
271C2×C3—41no—
272C2×C3—44no—
273C2×C3C2×#12011yescharacteristic
274C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
275C2×C3#57511yescharacteristic
276D3—82no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—24no—
285C22—44no—
286C22—48no—
287C22—42no—
288C22—42no—
289C22—122no—
290C22—122no—
291C22—122no—
292C22—244no—
293C22#94912yescomplemented
294C22#95312yes—
295C22C22×#6711yescharacteristic
296C4—21no—
297C4—24no—
298C4—21no—
299C4—122no—
300C4#95312yes—
301C4C22×#6712yes—
302C3#300611yescharacteristic, complemented, Sylow
303C2—22no—
304C2—22no—
305C2—41no—
306C2—44no—
307C2—242no—
308C2C2×#94911yescharacteristic
309C2#582211yescharacteristic
310C2C2×#95311yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC22×C3×#41C2
socleC22×C3C22×D4
hypercenterC22×#41D3
hypocenterC3#3006
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41C2semidirect
2. C2×#949C2semidirect
3. C2×#930C2semidirect
4. #5238C2semidirect
5. #5232C2semidirect
6. #5227C2semidirect
7. #5217C2semidirect
8. #5207C2semidirect
14. C2×C3×#41C22semidirect
15. C22×C3×Q8C22semidirect
17. C22×C3×D4C22semidirect
21. C22×D12C22semidirect
23. C2×#215C22semidirect
24. C2×#213C22semidirect
28. #930C22semidirect
30. C2×C4⋊Dic3C22semidirect
31. C2×#208C22semidirect
32. #835C22semidirect
33. #832C22semidirect
34. #830C22semidirect
35. #830C22semidirect
36. C22×#41D3semidirect
52. C3×#41D4semidirect
92. C2×#41C2×D3semidirect
93. C22×Q8C2×D3semidirect
97. C22×D4C2×D3semidirect
130. C3×Q8C2×D4semidirect
135. C3×D4C2×D4semidirect
165. #41#67semidirect
214. C22×C3AGL(1,ℤ/8)semidirect
242. Q8C2×#67semidirect
251. D4C2×#67semidirect
293. C22#949semidirect
302. C3#3006semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#412
C2×#9492
C2×#9302
#52382
#52322
#52272
#52172
#52072
#30063