Finite Group Databasebeta

← #78564

Subgroups — #78564

Order 384.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C2×#909C211yesmaximal, characteristic, complemented
2C2×#908C211yesmaximal, characteristic, complemented
3C2×#904C211yesmaximal, characteristic, complemented
4C22×#204C211yesmaximal, characteristic, complemented
5#5526C211yesmaximal, characteristic, complemented
6#5525C211yesmaximal, characteristic, complemented
7C3×#405C211yesmaximal, characteristic, complemented
8#4715C212yesmaximal, complemented
9#4712C212yesmaximal, complemented
10#4696C212yesmaximal, complemented
11#4693C212yesmaximal, complemented
12#3154—31nomaximal, Sylow
13C2×#235C2211yescharacteristic
14C2×#235C2211yescharacteristic
15C22×D12C2211yescharacteristic
16C22×Dic6C2211yescharacteristic
17C2×C3×M16C2211yescharacteristic
18C22×C3×C8C2211yescharacteristic
19C3×#117C2211yescharacteristic
20#909—24no—
21#908—24no—
22C2×Dic12C2212yescomplemented
23C2×Dic12C2211yescharacteristic, complemented
24#904—24no—
25C2×D24C2212yes—
26C2×D24C2211yescharacteristic, complemented
27C2×#204—24no—
28C2×#204C2212yes—
29C2×#204C2212yes—
30C2×#204C2212yes—
31C2×#204C2211yescharacteristic, complemented
32C2×#204C2212yescomplemented
33C2×#204C2211yescharacteristic, complemented
34#897—22no—
35#895—22no—
36#886—22no—
37#882—22no—
38#875C2212yes—
39#875C2212yescomplemented
40#874C2212yes—
41#869C2212yescomplemented
42C3×#97C2212yescomplemented
43C3×C4×C8C2212yescomplemented
44C2×#137—31no—
45C2×AGL(1,ℤ/8)—31no—
46C2×#135—31no—
47C22×SD16—31no—
48#536—31no—
49#535—31no—
50#497—32no—
51#496—32no—
52#495—32no—
53#492—32no—
54#405D311yescharacteristic, complemented
55C23×D3—21no—
56C22×C3×C4C2311yescharacteristic
57C2×#67—21no—
58C2×#67—21no—
59C22×Dic3—21no—
60#235—22no—
61#235—22no—
62#235—22no—
63#235—22no—
64C2×D12—22no—
65C2×D12—22no—
66C2×D12—21no—
67C2×D12—24no—
68C2×D12C2311yescharacteristic
69C2×D12C2312yes—
70C2×D12C2311yescharacteristic
71C2×C4×D3—21no—
72C2×C4×D3—21no—
73C2×Dic6—22no—
74C2×Dic6—22no—
75C2×Dic6—21no—
76C2×Dic6C2311yescharacteristic
77C2×Dic6C2312yes—
78C2×Dic6C2311yescharacteristic
79C3×M16D414yescomplemented
80C2×C3×C8D414yescomplemented
81C2×C3×C8C2311yescharacteristic
82C2×C3×C8C2312yes—
83C2×C3×C8C2311yescharacteristic
84C3×C4⋊C4D412yes—
85C3×C22⋊C4D412yes—
86C3×C42C2312yes—
87(C2×D3)⋊C4—24no—
88(C2×D3)⋊C4—24no—
89C4⋊Dic3—24no—
90Dic12—22no—
91Dic12—24no—
92D24—24no—
93D24—22no—
94#204—24no—
95#204—24no—
96#204—24no—
97#204—22no—
98#204—22no—
99#204—24no—
100C2×#41—31no—
101C2×#41—31no—
102C22×Q8—31no—
103C22×D4—31no—
104#137—64no—
105AGL(1,ℤ/8)—64no—
106#135—64no—
107C2×Q16—31no—
108C2×Q16—32no—
109C2×SD16—32no—
110C2×SD16—32no—
111C2×SD16—31no—
112C2×SD16—31no—
113C2×SD16—32no—
114C2×SD16—32no—
115C2×SD16—64no—
116C2×D8—31no—
117C2×D8—32no—
118C2×M16C2×D311yescharacteristic
119C22×C8C2×D311yescharacteristic
120C4⋊Q8—32no—
121#127—32no—
122#124—32no—
123#124—32no—
124#123—62no—
125C22⋊Q8—62no—
126#121—62no—
127#120—62no—
128#117C2×D311yescharacteristic
129#97C2×D312yescomplemented
130C4×C8C2×D312yescomplemented
131C23×C3C2×D411yescharacteristic
132C22×D3—21no—
133C22×D3—22no—
134C22×D3—21no—
135C22×D3—42no—
136C22×D3—82no—
137C2×C3×C4C2×D414yes—
138C2×C3×C4C2×D411yescharacteristic
139C2×C3×C4C2×D412yes—
140C2×C3×C4C2×D412yes—
141C2×C3×C4C2411yescharacteristic
142#67—42no—
143#67—42no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—21no—
147C2×Dic3—42no—
148D12—22no—
149D12—21no—
150D12—22no—
151D12—21no—
152D12—21no—
153D12—21no—
154D12—42no—
155D12—84no—
156C4×D3—42no—
157C4×D3—42no—
158Dic6—22no—
159Dic6—21no—
160Dic6—22no—
161Dic6—21no—
162Dic6—21no—
163Dic6—21no—
164Dic6—42no—
165C3×C8C2×D412yes—
166C3×C8C2×D412yes—
167C3×C8C2×D414yes—
168C24—61no—
169#41—62no—
170#41—62no—
171#41—62no—
172#41—62no—
173C2×Q8—31no—
174C2×Q8—32no—
175C2×Q8—31no—
176C2×Q8—61no—
177C2×Q8—62no—
178C2×Q8—62no—
179C2×D4—31no—
180C2×D4—32no—
181C2×D4—31no—
182C2×D4—61no—
183C2×D4—61no—
184C2×D4—62no—
185C2×D4—62no—
186C2×D4—61no—
187C2×D4—64no—
188C22×C4—61no—
189C22×C4—61no—
190C22×C4—61no—
191C22×C4C22×D311yescharacteristic
192Q16—62no—
193Q16—64no—
194SD16—64no—
195SD16—64no—
196SD16—62no—
197SD16—62no—
198SD16—64no—
199SD16—64no—
200D8—62no—
201D8—64no—
202M16D1214yescomplemented
203C2×C8D1214yescomplemented
204C2×C8C22×D311yescharacteristic
205C2×C8C22×D312yes—
206C2×C8C22×D311yescharacteristic
207C4⋊C4—64no—
208C4⋊C4D1212yes—
209C22⋊C4—64no—
210C22⋊C4—64no—
211C22⋊C4D1212yes—
212C42C22×D312yes—
213C22×C3—22no—
214C22×C3#12712yes—
215C22×C3C22×D411yescharacteristic
216C2×D3—41no—
217C2×D3—41no—
218C2×D3—42no—
219C2×D3—42no—
220C2×D3—44no—
221C2×D3—42no—
222C2×D3—84no—
223C3×C4—24no—
224C3×C4#12712yes—
225C3×C4C22×D411yescharacteristic
226C3×C4C22×D411yescharacteristic
227Dic3—42no—
228Dic3—41no—
229Dic3—41no—
230C23—61no—
231C23—62no—
232C23—61no—
233C23—122no—
234C23—242no—
235C23C2×D1211yescharacteristic
236Q8—62no—
237Q8—61no—
238Q8—62no—
239Q8—61no—
240Q8—61no—
241Q8—61no—
242Q8—122no—
243D4—61no—
244D4—62no—
245D4—61no—
246D4—61no—
247D4—62no—
248D4—61no—
249D4—122no—
250D4—122no—
251D4—122no—
252D4—244no—
253C2×C4—61no—
254C2×C4—62no—
255C2×C4—61no—
256C2×C4—122no—
257C2×C4—122no—
258C2×C4—122no—
259C2×C4C2×D1214yes—
260C2×C4C2×D1211yescharacteristic
261C2×C4C2×D1212yes—
262C2×C4C2×D1212yes—
263C2×C4C23×D311yescharacteristic
264C8C2×D1212yes—
265C8C2×D1212yes—
266C8C2×D1214yes—
267C2×C3—22no—
268C2×C3C2×#12711yescharacteristic
269C2×C3#57712yes—
270D3—81no—
271D3—81no—
272D3—82no—
273C22—22no—
274C22—121no—
275C22—121no—
276C22—122no—
277C22—122no—
278C22—124no—
279C22—122no—
280C22—244no—
281C22#87412yes—
282C22C22×D1211yescharacteristic
283C4—24no—
284C4—121no—
285C4—121no—
286C4—122no—
287C4#87412yes—
288C4C22×D1211yescharacteristic
289C4C22×D1211yescharacteristic
290C3#315411yescharacteristic, complemented, Sylow
291C2—22no—
292C2—242no—
293C2—241no—
294C2—241no—
295C2C2×#87411yescharacteristic
296C2#575112yes—
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#405C2
socleC22×C3C22×D4
hypercenter#405D3
hypocenterC3#3154
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#909C2semidirect
2. C2×#908C2semidirect
3. C2×#904C2semidirect
4. C22×#204C2semidirect
5. #5526C2semidirect
6. #5525C2semidirect
7. C3×#405C2semidirect
8. #4715C2semidirect
9. #4712C2semidirect
10. #4696C2semidirect
11. #4693C2semidirect
22. C2×Dic12C22semidirect
23. C2×Dic12C22semidirect
26. C2×D24C22semidirect
31. C2×#204C22semidirect
32. C2×#204C22semidirect
33. C2×#204C22semidirect
39. #875C22semidirect
41. #869C22semidirect
42. C3×#97C22semidirect
43. C3×C4×C8C22semidirect
54. #405D3semidirect
79. C3×M16D4semidirect
80. C2×C3×C8D4semidirect
129. #97C2×D3semidirect
130. C4×C8C2×D3semidirect
202. M16D12semidirect
203. C2×C8D12semidirect
290. C3#3154semidirect

Maximal subgroups

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

subgroupindex
C2×#9092
C2×#9082
C2×#9042
C22×#2042
#55262
#55252
C3×#4052
#47152
#47122
#46962
#46932
#31543