Finite Group Databasebeta

← #78926

Subgroups — #78926

Order 384.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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×D3×D4C211yesmaximal, characteristic, complemented
2C2×#932C211yesmaximal, characteristic, complemented
3C2×#908C211yesmaximal, characteristic, complemented
4#5524C211yesmaximal, characteristic, complemented
5C3×#418C211yesmaximal, characteristic, complemented
6#5126C211yesmaximal, characteristic, complemented
7#5001C211yesmaximal, characteristic, complemented
8#4776C214yesmaximal, complemented
9#4773C214yesmaximal, complemented
10#3003—31nomaximal, Sylow
11C24×D3—21no—
12C22×C3×D4C2211yescharacteristic
13C22×#67—21no—
14C2×D3×D4—24no—
15C2×D3×D4—24no—
16C2×D3×D4—22no—
17C2×D3×D4C2214yes—
18C2×#235C2211yescharacteristic, complemented
19C22×D12C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C2×C3×M16C2211yescharacteristic, complemented
22C3×#117C2211yescharacteristic, complemented
23C2×#214C2212yescomplemented
24#932—24no—
25C2×#213C2212yescomplemented
26C2×#208C2211yescharacteristic, complemented
27#908—24no—
28C2×D24C2212yescomplemented
29C2×#204C2212yescomplemented
30#897C2212yescomplemented
31#895C2212yescomplemented
32#886—21no—
33#882—21no—
34C4×D12—22no—
35C3×#102C2214yescomplemented
36(C2×D3)⋊C8C2214yescomplemented
37#809C2214yescomplemented
38C23×D4—31no—
39C2×AGL(1,ℤ/8)—31no—
40C2×AGL(1,ℤ/8)—31no—
41#525—31no—
42#450—34no—
43#447—34no—
44#418—31no—
45#418D311yescharacteristic, complemented
46#407—31no—
47C24×C3—21no—
48C23×D3—28no—
49C23×D3—21no—
50C23×D3—24no—
51C23×D3—48no—
52C23×D3D411yescharacteristic
53C2×C3×D4—21no—
54C2×C3×D4—22no—
55C2×C3×D4D414yescomplemented
56C2×C3×D4C2312yes—
57C22×C3×C4C2311yescharacteristic
58C2×#67—28no—
59C2×#67—21no—
60C2×#67—24no—
61C22×Dic3D411yescharacteristic
62D3×D4—216no—
63D3×D4—28no—
64D3×D4—28no—
65#235—22no—
66#235—22no—
67C2×D12—22no—
68C2×D12—21no—
69C2×D12—22no—
70C2×D12D414yescomplemented
71C2×D12C2312yes—
72C2×D12C2311yescharacteristic, complemented
73C2×C4×D3—22no—
74C2×C4×D3—22no—
75C2×C4×D3—21no—
76C2×C4×D3D412yes—
77C2×C4×D3C2312yes—
78C2×Dic6C2311yescharacteristic, complemented
79C3×M16—22no—
80C2×C3×C8C2312yescomplemented
81C3×C4⋊C4C2312yescomplemented
82C3×C22⋊C4—21no—
83C3×C42—21no—
84#214—24no—
85#213—24no—
86(C2×D3)⋊C4—22no—
87(C2×D3)⋊C4—22no—
88C4⋊Dic3—22no—
89#208—22no—
90C2×C3⋊C8C2312yescomplemented
91D24—24no—
92#204—24no—
93C25—61no—
94C2×#41—31no—
95C22×D4—34no—
96C22×D4—31no—
97C22×D4—62no—
98C22×D4—61no—
99C22×D4—64no—
100C22×D4—64no—
101C22×D4C2×D311yescharacteristic
102C23×C4—31no—
103AGL(1,ℤ/8)—64no—
104AGL(1,ℤ/8)—64no—
105C2×SD16—32no—
106C2×SD16—32no—
107C2×D8—32no—
108C2×D8—32no—
109C2×M16—31no—
110C2×M16C2×D311yescharacteristic, complemented
111#123—61no—
112C22⋊Q8—32no—
113#121—32no—
114#120—61no—
115C4×D4—62no—
116#117C2×D311yescharacteristic, complemented
117#102—34no—
118#102C2×D314yescomplemented
119C22⋊C8—34no—
120C23×C3—24no—
121C23×C3—22no—
122C23×C3—44no—
123C23×C3C2×D411yescharacteristic
124C22×D3—22no—
125C22×D3—24no—
126C22×D3—216no—
127C22×D3—24no—
128C22×D3—22no—
129C22×D3—21no—
130C22×D3—432no—
131C22×D3—48no—
132C22×D3—44no—
133C22×D3—48no—
134C22×D3C2×D412yes—
135C3×D4—24no—
136C3×D4—22no—
137C3×D4C2×D414yes—
138C2×C3×C4—22no—
139C2×C3×C4C2×D412yes—
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2×D412yes—
142C2×C3×C4C2411yescharacteristic
143#67—216no—
144#67—42no—
145C2×Dic3—22no—
146C2×Dic3—21no—
147C2×Dic3C2×D412yes—
148D12—24no—
149D12—22no—
150D12—21no—
151D12—21no—
152D12—44no—
153D12C2×D414yes—
154C4×D3—24no—
155C4×D3—22no—
156C4×D3—22no—
157C4×D3—42no—
158Dic6—21no—
159Dic6—21no—
160C3×C8—22no—
161C3⋊C8—22no—
162C24—21no—
163C24—31no—
164C24—64no—
165C24—68no—
166C24—61no—
167C24—128no—
168#41—62no—
169#41—62no—
170C2×Q8—31no—
171C2×D4—21no—
172C2×D4—22no—
173C2×D4—34no—
174C2×D4—32no—
175C2×D4—31no—
176C2×D4—62no—
177C2×D4—64no—
178C2×D4—68no—
179C2×D4—616no—
180C2×D4—68no—
181C2×D4—68no—
182C2×D4—61no—
183C2×D4—62no—
184C2×D4—61no—
185C2×D4D1214yescomplemented
186C2×D4C22×D312yes—
187C22×C4—31no—
188C22×C4—32no—
189C22×C4—32no—
190C22×C4—62no—
191C22×C4—62no—
192C22×C4—61no—
193C22×C4C22×D311yescharacteristic
194SD16—64no—
195SD16—64no—
196D8—64no—
197D8—64no—
198M16—22no—
199M16—62no—
200C2×C8—32no—
201C2×C8C22×D312yescomplemented
202C4⋊C4—62no—
203C4⋊C4C22×D312yescomplemented
204C22⋊C4—21no—
205C22⋊C4—62no—
206C22⋊C4—62no—
207C42—21no—
208C22×C3—22no—
209C22×C3—24no—
210C22×C3—48no—
211C22×C3—44no—
212C22×C3#12012yes—
213C22×C3C22×D411yescharacteristic
214C2×D3—28no—
215C2×D3—22no—
216C2×D3—22no—
217C2×D3—24no—
218C2×D3—432no—
219C2×D3—48no—
220C2×D3—44no—
221C2×D3—41no—
222C2×D3—42no—
223C2×D3AGL(1,ℤ/8)14yescomplemented
224C3×C4—42no—
225C3×C4#12012yes—
226C3×C4C22×D411yescharacteristic
227C3×C4C22×D411yescharacteristic
228Dic3—22no—
229Dic3—41no—
230C23—24no—
231C23—22no—
232C23—32no—
233C23—44no—
234C23—64no—
235C23—616no—
236C23—62no—
237C23—64no—
238C23—62no—
239C23—61no—
240C23—124no—
241C23—128no—
242C23—1232no—
243C23—128no—
244C23D3×D411yescharacteristic
245Q8—61no—
246Q8—61no—
247D4—24no—
248D4—22no—
249D4—34no—
250D4—616no—
251D4—64no—
252D4—62no—
253D4—61no—
254D4—61no—
255D4—124no—
256D4—122no—
257D4C2×D1214yes—
258C2×C4—22no—
259C2×C4—32no—
260C2×C4—62no—
261C2×C4—64no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—61no—
265C2×C4—122no—
266C2×C4C2×D1212yes—
267C2×C4D3×D411yescharacteristic
268C2×C4D3×D412yes—
269C2×C4C23×D311yescharacteristic
270C8—22no—
271C8—62no—
272C2×C3—22no—
273C2×C3—44no—
274C2×C3C2×#12011yescharacteristic
275C2×C3C2×AGL(1,ℤ/8)12yes—
276D3—24no—
277D3—44no—
278D3—81no—
279C22—22no—
280C22—24no—
281C22—34no—
282C22—48no—
283C22—44no—
284C22—68no—
285C22—62no—
286C22—62no—
287C22—64no—
288C22—128no—
289C22—1232no—
290C22—124no—
291C22—121no—
292C22—122no—
293C22#88212yes—
294C22C2×D3×D411yescharacteristic
295C4—42no—
296C4—62no—
297C4—121no—
298C4#88212yes—
299C4C22×D1211yescharacteristic
300C4C2×D3×D411yescharacteristic
301C3#300311yescharacteristic, complemented, Sylow
302C2—22no—
303C2—44no—
304C2—64no—
305C2—124no—
306C2—241no—
307C2C2×#88211yescharacteristic
308C2D3×AGL(1,ℤ/8)12yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#418C2
socleC22×C3C22×D4
hypercenter#418D3
hypocenterC3#3003
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×D3×D4C2semidirect
2. C2×#932C2semidirect
3. C2×#908C2semidirect
4. #5524C2semidirect
5. C3×#418C2semidirect
6. #5126C2semidirect
7. #5001C2semidirect
8. #4776C2semidirect
9. #4773C2semidirect
18. C2×#235C22semidirect
21. C2×C3×M16C22semidirect
22. C3×#117C22semidirect
23. C2×#214C22semidirect
25. C2×#213C22semidirect
26. C2×#208C22semidirect
28. C2×D24C22semidirect
29. C2×#204C22semidirect
30. #897C22semidirect
31. #895C22semidirect
35. C3×#102C22semidirect
36. (C2×D3)⋊C8C22semidirect
37. #809C22semidirect
45. #418D3semidirect
55. C2×C3×D4D4semidirect
70. C2×D12D4semidirect
72. C2×D12C23semidirect
78. C2×Dic6C23semidirect
80. C2×C3×C8C23semidirect
81. C3×C4⋊C4C23semidirect
90. C2×C3⋊C8C23semidirect
110. C2×M16C2×D3semidirect
116. #117C2×D3semidirect
118. #102C2×D3semidirect
185. C2×D4D12semidirect
201. C2×C8C22×D3semidirect
203. C4⋊C4C22×D3semidirect
223. C2×D3AGL(1,ℤ/8)semidirect
301. C3#3003semidirect

Maximal subgroups

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

subgroupindex
C22×D3×D42
C2×#9322
C2×#9082
#55242
C3×#4182
#51262
#50012
#47762
#47732
#30033