Finite Group Databasebeta

← #80359

Subgroups — #80359

Order 384.

546 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 125 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×#911C212yesmaximal, complemented
2#5717C211yesmaximal, characteristic, complemented
3#5555C211yesmaximal, characteristic, complemented
4C3×#459C211yesmaximal, characteristic, complemented
5#5158C212yesmaximal, complemented
6#5076C211yesmaximal, characteristic, complemented
7C4×#213C211yesmaximal, characteristic, complemented
8#4841C211yesmaximal, characteristic, complemented
9#4838C211yesmaximal, characteristic, complemented
10#4780C212yesmaximal, complemented
11#4776C212yesmaximal, complemented
12#3272—31nomaximal, Sylow
13C2×#237C2212yescomplemented
14C2×D3×D4—21no—
15C2×D3×D4C2212yes—
16C2×C3×D8C2212yescomplemented
17C3×#127C2211yescharacteristic
18C3×C4×D4C2211yescharacteristic, complemented
19#938—21no—
20#938C2212yes—
21#937—22no—
22C2×#214C2212yescomplemented
23C2×#214C2212yescomplemented
24C2×#213C2211yescharacteristic
25#930—21no—
26C2×C4⋊Dic3—21no—
27#911—28no—
28C2×#204C2212yes—
29C2×ΓL(1,𝔽25)C2212yescomplemented
30#897C2212yescomplemented
31#886—22no—
32C4×D12C2211yescharacteristic
33C4×D12C2211yescharacteristic
34#869C2211yescharacteristic, complemented
35C3×C4⋊C8C2211yescharacteristic, complemented
36C3×#102C2212yescomplemented
37#832C2211yescharacteristic, complemented
38(C2×D3)⋊C8C2212yescomplemented
39#816C2212yescomplemented
40#809C2211yescharacteristic
41#807C2211yescharacteristic, complemented
42C4×C3⋊C8C2211yescharacteristic, complemented
43C2×AGL(1,ℤ/8)—32no—
44#547—31no—
45D42—31no—
46#492—31no—
47#468—32no—
48#461—31no—
49#459D311yescharacteristic, complemented
50#450—32no—
51#449—32no—
52C4×D8—31no—
53#436—31no—
54C23×D3—22no—
55C2×C3×D4—21no—
56C2×C3×D4—21no—
57C2×C3×D4C2312yes—
58C2×C3×D4C2311yescharacteristic
59C22×C3×C4—21no—
60C2×#67—22no—
61C2×#67—22no—
62C2×#67—22no—
63C22×Dic3—22no—
64#237—24no—
65#237—42no—
66D3×D4—24no—
67D3×D4—42no—
68D3×D4—44no—
69C2×D12D411yescharacteristic
70C2×D12C2311yescharacteristic
71C2×C4×D3—21no—
72C2×C4×D3C2312yes—
73C2×Dic6C2312yes—
74C3×D8—24no—
75C2×C3×C8C2312yes—
76C3×C4⋊C4C2311yescharacteristic
77C3×C22⋊C4—21no—
78C3×C42C2311yescharacteristic
79#217—22no—
80#214—24no—
81#214—42no—
82#213D414yescomplemented
83(C2×D3)⋊C4—22no—
84(C2×D3)⋊C4—21no—
85(C2×D3)⋊C4D412yes—
86C4⋊Dic3—21no—
87C4⋊Dic3—21no—
88C4⋊Dic3—22no—
89C4⋊Dic3D411yescharacteristic
90C4⋊Dic3C2311yescharacteristic
91C2×C3⋊C8C2311yescharacteristic
92C2×C3⋊C8C2311yescharacteristic
93#204—24no—
94ΓL(1,𝔽25)—24no—
95C2×#41—32no—
96C22×D4—32no—
97C22×D4—61no—
98AGL(1,ℤ/8)—68no—
99C2×SD16—32no—
100C2×SD16—32no—
101C2×SD16—32no—
102C2×D8—31no—
103C2×D8C2×D312yescomplemented
104C2×M16—32no—
105C4⋊Q8—31no—
106#127C2×D311yescharacteristic
107#123—62no—
108C22⋊Q8—32no—
109#121—32no—
110#121—61no—
111#121—61no—
112#120—62no—
113C4×D4—31no—
114C4×D4—31no—
115C4×D4C2×D311yescharacteristic, complemented
116C2×C4⋊C4—61no—
117#107—31no—
118C4⋊C8C2×D311yescharacteristic, complemented
119#103—32no—
120#102—31no—
121#102—31no—
122#102C2×D312yescomplemented
123C22⋊C8—32no—
124C4×C8—31no—
125C23×C3—21no—
126C23×C3—22no—
127C22×D3—21no—
128C22×D3—44no—
129C22×D3—44no—
130C22×D3—82no—
131C22×D3C2×D412yes—
132C3×D4—22no—
133C3×D4—21no—
134C3×D4—22no—
135C3×D4—42no—
136C3×D4—81no—
137C3×D4C2×D412yescomplemented
138C2×C3×C4—21no—
139C2×C3×C4—42no—
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2×D411yescharacteristic
142C2×C3×C4C2411yescharacteristic
143#67—44no—
144#67—44no—
145#67—44no—
146C2×Dic3—22no—
147C2×Dic3—21no—
148C2×Dic3—44no—
149C2×Dic3C2×D412yes—
150D12—22no—
151D12—21no—
152D12C2×D412yes—
153C4×D3—22no—
154C4×D3—22no—
155C4×D3—42no—
156Dic6—22no—
157Dic6—22no—
158C3×C8—22no—
159C3⋊C8—21no—
160C3⋊C8C2×D412yescomplemented
161C24—62no—
162#41—64no—
163#41—122no—
164C2×Q8—32no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—31no—
168C2×D4—31no—
169C2×D4—62no—
170C2×D4—64no—
171C2×D4—62no—
172C2×D4—62no—
173C2×D4—122no—
174C2×D4—124no—
175C2×D4C22×D312yes—
176C2×D4C22×D311yescharacteristic
177C22×C4—21no—
178C22×C4—32no—
179C22×C4—62no—
180C22×C4—61no—
181SD16—64no—
182SD16—64no—
183SD16—122no—
184D8—24no—
185D8—34no—
186M16—64no—
187C2×C8—31no—
188C2×C8—31no—
189C2×C8C22×D312yes—
190C4⋊C4—31no—
191C4⋊C4—31no—
192C4⋊C4—62no—
193C4⋊C4—61no—
194C4⋊C4—61no—
195C4⋊C4C22×D311yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—32no—
198C22⋊C4—62no—
199C22⋊C4—62no—
200C22⋊C4—61no—
201C42C22×D311yescharacteristic
202C22×C3—22no—
203C22×C3—41no—
204C22×C3—41no—
205C22×C3—42no—
206C22×C3—42no—
207C22×C3—42no—
208C22×C3C22×D411yescharacteristic
209C2×D3—22no—
210C2×D3—22no—
211C2×D3—22no—
212C2×D3—22no—
213C2×D3—41no—
214C2×D3—41no—
215C2×D3—88no—
216C3×C4—21no—
217C3×C4—41no—
218C3×C4AGL(1,ℤ/8)12yescomplemented
219C3×C4C22×D411yescharacteristic
220C3×C4H(5,𝔽2)11yescharacteristic
221Dic3—22no—
222Dic3—42no—
223Dic3—41no—
224C23—21no—
225C23—22no—
226C23—32no—
227C23—61no—
228C23—124no—
229C23—124no—
230C23—242no—
231Q8—62no—
232Q8—62no—
233D4—22no—
234D4—21no—
235D4—22no—
236D4—32no—
237D4—42no—
238D4—62no—
239D4—61no—
240D4—81no—
241D4—124no—
242D4—124no—
243D4—124no—
244D4D3×D412yescomplemented
245C2×C4—21no—
246C2×C4—32no—
247C2×C4—42no—
248C2×C4—62no—
249C2×C4—62no—
250C2×C4—61no—
251C2×C4—62no—
252C2×C4—124no—
253C2×C4—122no—
254C2×C4D3×D411yescharacteristic
255C2×C4D3×D411yescharacteristic
256C2×C4C23×D311yescharacteristic
257C8—22no—
258C8—32no—
259C8—61no—
260C2×C3—42no—
261C2×C3—82no—
262C2×C3D4211yescharacteristic
263C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
264C2×C3#57711yescharacteristic
265D3—42no—
266D3—42no—
267C22—22no—
268C22—41no—
269C22—41no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—62no—
274C22—62no—
275C22—62no—
276C22—62no—
277C22—121no—
278C22—121no—
279C22—248no—
280C22C2×D3×D411yescharacteristic
281C4—21no—
282C4—41no—
283C4—62no—
284C4—121no—
285C4—122no—
286C4#91112yescomplemented
287C4C2×D3×D411yescharacteristic
288C4#100911yescharacteristic
289C3#327211yescharacteristic, complemented, Sylow
290C2—42no—
291C2—82no—
292C2—122no—
293C2—122no—
294C2#560911yescharacteristic
295C2C2×#91111yescharacteristic
296C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3272
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#911C2semidirect
2. #5717C2semidirect
3. #5555C2semidirect
4. C3×#459C2semidirect
5. #5158C2semidirect
6. #5076C2semidirect
7. C4×#213C2semidirect
8. #4841C2semidirect
9. #4838C2semidirect
10. #4780C2semidirect
11. #4776C2semidirect
13. C2×#237C22semidirect
16. C2×C3×D8C22semidirect
18. C3×C4×D4C22semidirect
22. C2×#214C22semidirect
23. C2×#214C22semidirect
29. C2×ΓL(1,𝔽25)C22semidirect
30. #897C22semidirect
34. #869C22semidirect
35. C3×C4⋊C8C22semidirect
36. C3×#102C22semidirect
37. #832C22semidirect
38. (C2×D3)⋊C8C22semidirect
39. #816C22semidirect
41. #807C22semidirect
42. C4×C3⋊C8C22semidirect
49. #459D3semidirect
82. #213D4semidirect
103. C2×D8C2×D3semidirect
115. C4×D4C2×D3semidirect
118. C4⋊C8C2×D3semidirect
122. #102C2×D3semidirect
137. C3×D4C2×D4semidirect
160. C3⋊C8C2×D4semidirect
218. C3×C4AGL(1,ℤ/8)semidirect
244. D4D3×D4semidirect
286. C4#911semidirect
289. C3#3272semidirect

Maximal subgroups

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

subgroupindex
C2×#9112
#57172
#55552
C3×#4592
#51582
#50762
C4×#2132
#48412
#48382
#47802
#47762
#32723