Finite Group Databasebeta

← #83578

Subgroups — #83578

Order 384.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C3×#550C211yesmaximal, characteristic, complemented
2C2×#949C212yesmaximal, complemented
3D4×D12C211yesmaximal, characteristic, complemented
4#5238C212yesmaximal, complemented
5#5073C211yesmaximal, characteristic, complemented
6#5072C211yesmaximal, characteristic, complemented
7#5039C212yesmaximal, complemented
8#5036C212yesmaximal, complemented
9#5027C211yesmaximal, characteristic, complemented
10C4×#215C211yesmaximal, characteristic, complemented
11#5012C211yesmaximal, characteristic, complemented
12#3274—31nomaximal, Sylow
13C2×C3×#41C2212yescomplemented
14C2×D3×D4—21no—
15C22×D12C2212yes—
16C3×#127—21no—
17C3×#127C2211yescharacteristic, complemented
18C3×#121—22no—
19C3×#121C2212yescomplemented
20C3×C4×Q8C2211yescharacteristic, complemented
21C3×C4×D4—21no—
22C3×C4×D4C2211yescharacteristic
23#949—28no—
24C2×#215C2211yescharacteristic
25C2×#215C2212yescomplemented
26C2×#213C2212yes—
27C2×#213C2212yescomplemented
28C2×#213C2212yescomplemented
29#930C2212yes—
30C2×#208C2212yescomplemented
31#895—21no—
32#882—22no—
33#874C2211yescharacteristic
34C4×D12C2211yescharacteristic
35#835C2211yescharacteristic, complemented
36#832C2212yescomplemented
37#830C2212yescomplemented
38#809C2211yescharacteristic
39#809C2212yescomplemented
40#808C2211yescharacteristic, complemented
41C4⋊(C3⋊C8)C2211yescharacteristic, complemented
42C4×C3⋊C8C2211yescharacteristic, complemented
43C2×AGL(1,ℤ/8)—32no—
44#550D311yescharacteristic, complemented
45D42—31no—
46#493—31no—
47#469—32no—
48#460—31no—
49#459—31no—
50#449—32no—
51#447—32no—
52C4×SD16—31no—
53#436—31no—
54C23×D3—22no—
55C3×#41—24no—
56C3×#41—42no—
57C2×C3×Q8C2311yescharacteristic
58C2×C3×D4—22no—
59C2×C3×D4—22no—
60C2×C3×D4—21no—
61C2×C3×D4—21no—
62C2×C3×D4D411yescharacteristic
63C2×C3×D4C2312yes—
64C22×C3×C4—22no—
65C22×C3×C4C2312yes—
66C2×#67—22no—
67D3×D4—44no—
68C2×D12—24no—
69C2×D12—21no—
70C2×D12—22no—
71C2×D12—21no—
72C2×D12—42no—
73C2×D12C2312yes—
74C2×D12C2311yescharacteristic
75C2×C4×D3—21no—
76C3×C4⋊C4—21no—
77C3×C4⋊C4D411yescharacteristic
78C3×C4⋊C4C2311yescharacteristic
79C3×C22⋊C4—22no—
80C3×C22⋊C4D412yes—
81C3×C42—21no—
82C3×C42C2311yescharacteristic
83#215—24no—
84#215D414yescomplemented
85#213—24no—
86#213—24no—
87#213—42no—
88(C2×D3)⋊C4—21no—
89(C2×D3)⋊C4—22no—
90C4⋊Dic3C2311yescharacteristic
91#208—24no—
92C2×C3⋊C8C2311yescharacteristic
93C2×C3⋊C8C2312yes—
94C2×C3⋊C8C2311yescharacteristic
95C2×#41C2×D312yescomplemented
96C22×D4—32no—
97C22×D4—61no—
98AGL(1,ℤ/8)—68no—
99C2×SD16—31no—
100C2×SD16—32no—
101C2×D8—32no—
102C2×D8—32no—
103C2×D8—32no—
104C2×M16—32no—
105#127—21no—
106#127—31no—
107#127C2×D311yescharacteristic, complemented
108#121—22no—
109#121—32no—
110#121—61no—
111#121C2×D312yescomplemented
112#120—62no—
113C4×Q8C2×D311yescharacteristic, complemented
114C4×D4—21no—
115C4×D4—31no—
116C4×D4C2×D311yescharacteristic
117#106—31no—
118C4⋊C8—31no—
119#103—31no—
120#102—32no—
121#102—31no—
122#102—32no—
123C22⋊C8—32no—
124C4×C8—31no—
125C23×C3—22no—
126C23×C3C2×D412yes—
127C22×D3—21no—
128C22×D3—22no—
129C22×D3—44no—
130C22×D3—44no—
131C22×D3—82no—
132C3×Q8—21no—
133C3×Q8C2×D412yescomplemented
134C3×D4—22no—
135C3×D4—22no—
136C3×D4—22no—
137C3×D4—44no—
138C3×D4—42no—
139C3×D4—81no—
140C3×D4—82no—
141C2×C3×C4—21no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—21no—
145C2×C3×C4—44no—
146C2×C3×C4C2×D411yescharacteristic
147C2×C3×C4C2×D412yes—
148C2×C3×C4C2×D411yescharacteristic
149C2×C3×C4C2411yescharacteristic
150#67—44no—
151C2×Dic3—21no—
152D12—22no—
153D12—21no—
154D12—22no—
155D12—44no—
156D12—42no—
157D12—81no—
158D12C2×D412yes—
159C4×D3—42no—
160C3⋊C8—22no—
161C3⋊C8—21no—
162C3⋊C8C2×D412yescomplemented
163C24—62no—
164#41—24no—
165#41—42no—
166C2×Q8C22×D311yescharacteristic
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—31no—
172C2×D4—32no—
173C2×D4—64no—
174C2×D4—62no—
175C2×D4—62no—
176C2×D4—61no—
177C2×D4—61no—
178C2×D4—122no—
179C2×D4—124no—
180C2×D4#6711yescharacteristic
181C2×D4C22×D312yes—
182C22×C4—22no—
183C22×C4—61no—
184C22×C4C22×D312yes—
185SD16—34no—
186SD16—64no—
187D8—64no—
188D8—64no—
189D8—122no—
190M16—64no—
191C2×C8—31no—
192C2×C8—31no—
193C2×C8—32no—
194C4⋊C4—21no—
195C4⋊C4—31no—
196C4⋊C4#6711yescharacteristic
197C4⋊C4C22×D311yescharacteristic
198C22⋊C4—22no—
199C22⋊C4—61no—
200C22⋊C4—62no—
201C22⋊C4#6712yes—
202C42—21no—
203C42C22×D311yescharacteristic
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—42no—
208C22×C3—42no—
209C22×C3—42no—
210C22×C3C22×D411yescharacteristic
211C2×D3—22no—
212C2×D3—41no—
213C2×D3—41no—
214C2×D3—42no—
215C2×D3—42no—
216C2×D3—42no—
217C2×D3—88no—
218C3×C4—22no—
219C3×C4—22no—
220C3×C4—21no—
221C3×C4—41no—
222C3×C4AGL(1,ℤ/8)12yescomplemented
223C3×C4C22×D411yescharacteristic
224C3×C4H(5,𝔽2)11yescharacteristic
225Dic3—41no—
226C23—22no—
227C23—61no—
228C23—62no—
229C23—124no—
230C23—124no—
231C23—242no—
232C23C2×#6712yes—
233Q8—21no—
234Q8D3×D412yescomplemented
235D4—22no—
236D4—22no—
237D4—22no—
238D4—32no—
239D4—44no—
240D4—42no—
241D4—62no—
242D4—61no—
243D4—62no—
244D4—81no—
245D4—82no—
246D4—124no—
247D4—124no—
248D4—122no—
249D4—241no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—44no—
255C2×C4—61no—
256C2×C4—122no—
257C2×C4C2×#6711yescharacteristic
258C2×C4C2×#6712yes—
259C2×C4C2×#6711yescharacteristic
260C2×C4C23×D311yescharacteristic
261C8—32no—
262C8—62no—
263C8—61no—
264C2×C3—42no—
265C2×C3—82no—
266C2×C3D4211yescharacteristic
267C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
268C2×C3#57611yescharacteristic
269D3—42no—
270D3—82no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—62no—
278C22—121no—
279C22—121no—
280C22—122no—
281C22—122no—
282C22—122no—
283C22—248no—
284C22C22×#6711yescharacteristic
285C4—22no—
286C4—22no—
287C4—21no—
288C4—41no—
289C4—121no—
290C4#94912yescomplemented
291C4C2×D3×D411yescharacteristic
292C4#100411yescharacteristic
293C3#327411yescharacteristic, complemented, Sylow
294C2—42no—
295C2—82no—
296C2—122no—
297C2—242no—
298C2D4×#6711yescharacteristic
299C2C2×#94911yescharacteristic
300C2#583511yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#550C2
socleC22×C3C22×D4
hypercenter#550D3
hypocenterC3#3274
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C3×#550C2semidirect
2. C2×#949C2semidirect
3. D4×D12C2semidirect
4. #5238C2semidirect
5. #5073C2semidirect
6. #5072C2semidirect
7. #5039C2semidirect
8. #5036C2semidirect
9. #5027C2semidirect
10. C4×#215C2semidirect
11. #5012C2semidirect
13. C2×C3×#41C22semidirect
17. C3×#127C22semidirect
19. C3×#121C22semidirect
20. C3×C4×Q8C22semidirect
25. C2×#215C22semidirect
27. C2×#213C22semidirect
28. C2×#213C22semidirect
30. C2×#208C22semidirect
35. #835C22semidirect
36. #832C22semidirect
37. #830C22semidirect
39. #809C22semidirect
40. #808C22semidirect
41. C4⋊(C3⋊C8)C22semidirect
42. C4×C3⋊C8C22semidirect
44. #550D3semidirect
84. #215D4semidirect
95. C2×#41C2×D3semidirect
107. #127C2×D3semidirect
111. #121C2×D3semidirect
113. C4×Q8C2×D3semidirect
133. C3×Q8C2×D4semidirect
162. C3⋊C8C2×D4semidirect
222. C3×C4AGL(1,ℤ/8)semidirect
234. Q8D3×D4semidirect
290. C4#949semidirect
293. C3#3274semidirect

Maximal subgroups

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

subgroupindex
C3×#5502
C2×#9492
D4×D122
#52382
#50732
#50722
#50392
#50362
#50272
C4×#2152
#50122
#32743