Finite Group Databasebeta

← #83489

Subgroups — #83489

Order 384.

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

normal subgroups only

This group has 546 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×D42C211yesmaximal, characteristic, complemented
2C2×#932C212yesmaximal, complemented
3#5551C211yesmaximal, characteristic, complemented
4#5218C212yesmaximal, complemented
5#5076C211yesmaximal, characteristic, complemented
6#5072C211yesmaximal, characteristic, complemented
7#5042C212yesmaximal, complemented
8#5037C212yesmaximal, complemented
9#5019C211yesmaximal, characteristic, complemented
10C4×#213C211yesmaximal, characteristic, complemented
11#5012C211yesmaximal, characteristic, complemented
12#3272—31nomaximal, Sylow
13C22×C3×D4—21no—
14C22×C3×D4C2212yes—
15C2×#235C2212yescomplemented
16C3×#127C2211yescharacteristic
17C3×#121—21no—
18C3×#121C2212yes—
19C3×#120—22no—
20C3×C4×D4C2211yescharacteristic
21C3×C4×D4C2211yescharacteristic
22#938—21no—
23C2×#214C2212yes—
24C2×#214C2212yescomplemented
25C2×#214C2212yescomplemented
26#932—28no—
27C2×#213C2211yescharacteristic
28C2×#213C2212yescomplemented
29#924C2212yescomplemented
30C2×#208C2212yescomplemented
31D3×C4⋊C4—21no—
32#883—22no—
33C4×D12C2211yescharacteristic, complemented
34#869C2211yescharacteristic, complemented
35#832C2211yescharacteristic
36#832C2212yescomplemented
37#830C2212yescomplemented
38#810C2212yescomplemented
39#809C2211yescharacteristic, complemented
40#807C2211yescharacteristic, complemented
41C4⋊(C3⋊C8)C2211yescharacteristic, complemented
42C4×C3⋊C8C2211yescharacteristic, complemented
43C2×AGL(1,ℤ/8)—32no—
44#547—31no—
45D42D311yescharacteristic, complemented
46#492—31no—
47#468—32no—
48#461—31no—
49#459—31no—
50#450—32no—
51#449—32no—
52C4×D8—31no—
53#436—31no—
54C24×C3—22no—
55C2×C3×D4—24no—
56C2×C3×D4—22no—
57C2×C3×D4—22no—
58C2×C3×D4—21no—
59C2×C3×D4—21no—
60C2×C3×D4—42no—
61C2×C3×D4—44no—
62C2×C3×D4D411yescharacteristic
63C2×C3×D4C2312yes—
64C2×C3×D4C2311yescharacteristic
65C22×C3×C4—21no—
66C22×C3×C4C2312yes—
67C2×#67—22no—
68#235—24no—
69#235—42no—
70C2×D12C2311yescharacteristic
71C2×C4×D3—21no—
72C2×C4×D3—22no—
73C2×Dic6C2312yes—
74C3×C4⋊C4D411yescharacteristic
75C3×C4⋊C4C2311yescharacteristic
76C3×C22⋊C4—21no—
77C3×C22⋊C4—22no—
78C3×C22⋊C4D412yes—
79C3×C42C2311yescharacteristic
80#217—22no—
81#214—24no—
82#214—24no—
83#214—42no—
84#213—24no—
85#213D414yescomplemented
86(C2×D3)⋊C4—21no—
87C4⋊Dic3—21no—
88C4⋊Dic3—21no—
89C4⋊Dic3C2311yescharacteristic
90Dic3⋊C4—22no—
91#208—24no—
92C2×C3⋊C8C2311yescharacteristic
93C2×C3⋊C8C2312yes—
94C2×C3⋊C8C2311yescharacteristic
95C2×#41—32no—
96C22×D4—21no—
97C22×D4C2×D312yes—
98AGL(1,ℤ/8)—68no—
99C2×SD16—32no—
100C2×SD16—32no—
101C2×SD16—32no—
102C2×D8—31no—
103C2×D8—32no—
104C2×M16—32no—
105C4⋊Q8—31no—
106#127C2×D311yescharacteristic
107#123—62no—
108C22⋊Q8—32no—
109#121—21no—
110#121—61no—
111#121C2×D312yes—
112#120—22no—
113C4×D4—31no—
114C4×D4C2×D311yescharacteristic
115C4×D4C2×D311yescharacteristic
116C2×C4⋊C4—61no—
117#107—31no—
118C4⋊C8—31no—
119#103—32no—
120#102—31no—
121#102—32no—
122#102—31no—
123C22⋊C8—32no—
124C4×C8—31no—
125C23×C3—21no—
126C23×C3—22no—
127C23×C3—44no—
128C23×C3—44no—
129C23×C3—82no—
130C23×C3C2×D412yes—
131C22×D3—21no—
132C3×D4—22no—
133C3×D4—22no—
134C3×D4—21no—
135C3×D4—22no—
136C3×D4—44no—
137C3×D4—44no—
138C3×D4—42no—
139C3×D4—81no—
140C3×D4C2×D412yes—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—21no—
144C2×C3×C4—42no—
145C2×C3×C4C2×D412yes—
146C2×C3×C4C2×D411yescharacteristic
147C2×C3×C4C2×D411yescharacteristic
148C2×C3×C4C2411yescharacteristic
149#67—44no—
150C2×Dic3—21no—
151C2×Dic3—22no—
152D12—21no—
153D12C2×D412yescomplemented
154C4×D3—44no—
155C4×D3—42no—
156Dic6—22no—
157Dic6—22no—
158C3⋊C8—22no—
159C3⋊C8—21no—
160C3⋊C8C2×D412yescomplemented
161C24—22no—
162#41—64no—
163#41—122no—
164C2×Q8—32no—
165C2×D4—24no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—31no—
171C2×D4—42no—
172C2×D4—44no—
173C2×D4—62no—
174C2×D4#6711yescharacteristic
175C2×D4C22×D312yes—
176C2×D4C22×D311yescharacteristic
177C22×C4—21no—
178C22×C4—61no—
179C22×C4—62no—
180C22×C4C22×D312yes—
181SD16—64no—
182SD16—64no—
183SD16—122no—
184D8—34no—
185D8—64no—
186M16—64no—
187C2×C8—31no—
188C2×C8—31no—
189C2×C8—32no—
190C4⋊C4—31no—
191C4⋊C4—61no—
192C4⋊C4—62no—
193C4⋊C4—61no—
194C4⋊C4#6711yescharacteristic
195C4⋊C4C22×D311yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—22no—
198C22⋊C4—62no—
199C22⋊C4—61no—
200C22⋊C4#6712yes—
201C42C22×D311yescharacteristic
202C22×C3—22no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—41no—
207C22×C3—41no—
208C22×C3—42no—
209C22×C3—42no—
210C22×C3—42no—
211C22×C3—88no—
212C22×C3C22×D411yescharacteristic
213C2×D3—22no—
214C2×D3—41no—
215C2×D3—41no—
216C3×C4—22no—
217C3×C4—21no—
218C3×C4—41no—
219C3×C4AGL(1,ℤ/8)12yescomplemented
220C3×C4C22×D411yescharacteristic
221C3×C4H(5,𝔽2)11yescharacteristic
222Dic3—42no—
223Dic3—41no—
224C23—21no—
225C23—22no—
226C23—44no—
227C23—44no—
228C23—61no—
229C23—82no—
230C23C2×#6712yes—
231Q8—62no—
232Q8—62no—
233D4—22no—
234D4—22no—
235D4—21no—
236D4—22no—
237D4—32no—
238D4—44no—
239D4—44no—
240D4—42no—
241D4—61no—
242D4—81no—
243D4—124no—
244D4D3×D412yes—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—42no—
249C2×C4—61no—
250C2×C4—62no—
251C2×C4—124no—
252C2×C4—122no—
253C2×C4C2×#6711yescharacteristic
254C2×C4C2×#6712yes—
255C2×C4C2×#6711yescharacteristic
256C2×C4C23×D311yescharacteristic
257C8—32no—
258C8—62no—
259C8—61no—
260C2×C3—42no—
261C2×C3—42no—
262C2×C3—82no—
263C2×C3D4211yescharacteristic
264C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
265C2×C3#57711yescharacteristic
266D3—42no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—41no—
272C22—41no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—62no—
277C22—88no—
278C22—121no—
279C22—121no—
280C22C22×#6711yescharacteristic
281C4—22no—
282C4—21no—
283C4—41no—
284C4—121no—
285C4—122no—
286C4#93212yescomplemented
287C4C2×D3×D411yescharacteristic
288C4#100411yescharacteristic
289C3#327211yescharacteristic, complemented, Sylow
290C2—42no—
291C2—42no—
292C2—82no—
293C2—122no—
294C2C2×#93211yescharacteristic
295C2D4×#6711yescharacteristic
296C2#583611yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×D42C2
socleC22×C3C22×D4
hypercenterD42D3
hypocenterC3#3272
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C3×D42C2semidirect
2. C2×#932C2semidirect
3. #5551C2semidirect
4. #5218C2semidirect
5. #5076C2semidirect
6. #5072C2semidirect
7. #5042C2semidirect
8. #5037C2semidirect
9. #5019C2semidirect
10. C4×#213C2semidirect
11. #5012C2semidirect
15. C2×#235C22semidirect
24. C2×#214C22semidirect
25. C2×#214C22semidirect
28. C2×#213C22semidirect
29. #924C22semidirect
30. C2×#208C22semidirect
33. C4×D12C22semidirect
34. #869C22semidirect
36. #832C22semidirect
37. #830C22semidirect
38. #810C22semidirect
39. #809C22semidirect
40. #807C22semidirect
41. C4⋊(C3⋊C8)C22semidirect
42. C4×C3⋊C8C22semidirect
45. D42D3semidirect
85. #213D4semidirect
153. D12C2×D4semidirect
160. C3⋊C8C2×D4semidirect
219. C3×C4AGL(1,ℤ/8)semidirect
286. C4#932semidirect
289. C3#3272semidirect

Maximal subgroups

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

subgroupindex
C3×D422
C2×#9322
#55512
#52182
#50762
#50722
#50422
#50372
#50192
C4×#2132
#50122
#32723