Finite Group Databasebeta

← #80349

Subgroups — #80349

Order 384.

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

normal subgroups only

This group has 556 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#5718C211yesmaximal, characteristic, complemented
3D4×D12C211yesmaximal, characteristic, complemented
4C3×#459C211yesmaximal, characteristic, complemented
5#5156C212yesmaximal, complemented
6#5073C211yesmaximal, characteristic, complemented
7C4×#214C211yesmaximal, characteristic, complemented
8#4848C211yesmaximal, characteristic, complemented
9#4838C211yesmaximal, characteristic, complemented
10#4782C212yesmaximal, complemented
11#4781C212yesmaximal, complemented
12#3274—31nomaximal, Sylow
13C2×#237C2212yescomplemented
14C2×D3×D4C2212yes—
15C22×D12—21no—
16C2×C3×D8C2212yes—
17C3×#127C2211yescharacteristic, complemented
18C3×C4×D4C2211yescharacteristic
19#940—21no—
20#939—22no—
21#938C2212yescomplemented
22D4×Dic3—21no—
23C2×#214C2211yescharacteristic
24C2×#213C2212yescomplemented
25C2×#213C2212yescomplemented
26#930—21no—
27#911—28no—
28C2×#204C2212yescomplemented
29C2×ΓL(1,𝔽25)C2212yescomplemented
30#895C2212yes—
31#882—22no—
32#874C2211yescharacteristic
33C4×D12C2211yescharacteristic
34C4×Dic6C2211yescharacteristic, complemented
35C3×C4⋊C8C2211yescharacteristic, complemented
36C3×#102C2212yescomplemented
37#832C2211yescharacteristic
38#821C2212yescomplemented
39(C2×D3)⋊C8C2212yescomplemented
40#810C2211yescharacteristic, complemented
41#808C2211yescharacteristic, complemented
42C4×C3⋊C8C2211yescharacteristic, complemented
43C2×AGL(1,ℤ/8)—32no—
44#550—31no—
45D42—31no—
46#493—31no—
47#469—32no—
48#460—31no—
49#459D311yescharacteristic, complemented
50#449—32no—
51#447—32no—
52C4×SD16—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—
68C2×D12—22no—
69C2×D12—21no—
70C2×D12—21no—
71C2×D12—44no—
72C2×D12D411yescharacteristic
73C2×D12C2312yes—
74C2×C4×D3C2312yes—
75C2×Dic6C2311yescharacteristic
76C3×D8—24no—
77C2×C3×C8C2312yes—
78C3×C4⋊C4C2311yescharacteristic
79C3×C22⋊C4—21no—
80C3×C42C2311yescharacteristic
81#217—22no—
82#214D414yescomplemented
83#213—24no—
84#213—42no—
85(C2×D3)⋊C4—22no—
86(C2×D3)⋊C4D412yes—
87C4⋊Dic3D411yescharacteristic
88C4⋊Dic3C2311yescharacteristic
89Dic3⋊C4—21no—
90C4×Dic3—21no—
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—31no—
100C2×SD16—32no—
101C2×D8—32no—
102C2×D8—32no—
103C2×D8C2×D312yes—
104C2×M16—32no—
105#127—31no—
106#127—61no—
107#127C2×D311yescharacteristic, complemented
108#121—32no—
109#121—32no—
110#121—62no—
111#121—61no—
112#120—62no—
113C4×Q8—31no—
114C4×D4—31no—
115C4×D4—61no—
116C4×D4C2×D311yescharacteristic
117#106—31no—
118C4⋊C8C2×D311yescharacteristic, complemented
119#103—31no—
120#102—31no—
121#102—32no—
122#102C2×D312yescomplemented
123C22⋊C8—32no—
124C4×C8—31no—
125C23×C3—21no—
126C23×C3—22no—
127C22×D3—22no—
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×D412yes—
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—82no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—44no—
149C2×Dic3C2×D412yes—
150D12—22no—
151D12—22no—
152D12—22no—
153D12—44no—
154D12—42no—
155D12—81no—
156C4×D3—22no—
157C4×D3—22no—
158Dic6—21no—
159Dic6C2×D412yescomplemented
160C3×C8—22no—
161C3⋊C8—21no—
162C3⋊C8C2×D412yescomplemented
163C24—62no—
164#41—64no—
165#41—122no—
166C2×Q8—31no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—31no—
170C2×D4—32no—
171C2×D4—62no—
172C2×D4—62no—
173C2×D4—64no—
174C2×D4—62no—
175C2×D4—62no—
176C2×D4—61no—
177C2×D4—61no—
178C2×D4—124no—
179C2×D4—122no—
180C2×D4C22×D312yes—
181C2×D4C22×D311yescharacteristic
182C22×C4—21no—
183C22×C4—32no—
184C22×C4—62no—
185SD16—34no—
186SD16—64no—
187D8—24no—
188D8—64no—
189D8—122no—
190M16—64no—
191C2×C8—31no—
192C2×C8—31no—
193C2×C8C22×D312yes—
194C4⋊C4—31no—
195C4⋊C4—31no—
196C4⋊C4—61no—
197C4⋊C4C22×D311yescharacteristic
198C22⋊C4—21no—
199C22⋊C4—32no—
200C22⋊C4—62no—
201C22⋊C4—62no—
202C42—61no—
203C42C22×D311yescharacteristic
204C22×C3—22no—
205C22×C3—41no—
206C22×C3—41no—
207C22×C3—42no—
208C22×C3—42no—
209C22×C3—42no—
210C22×C3C22×D411yescharacteristic
211C2×D3—22no—
212C2×D3—22no—
213C2×D3—22no—
214C2×D3—42no—
215C2×D3—42no—
216C2×D3—42no—
217C2×D3—88no—
218C3×C4—21no—
219C3×C4—41no—
220C3×C4AGL(1,ℤ/8)12yescomplemented
221C3×C4C22×D411yescharacteristic
222C3×C4H(5,𝔽2)11yescharacteristic
223Dic3—22no—
224Dic3—22no—
225Dic3—41no—
226C23—21no—
227C23—22no—
228C23—32no—
229C23—62no—
230C23—124no—
231C23—124no—
232C23—242no—
233Q8—32no—
234Q8—61no—
235D4—22no—
236D4—21no—
237D4—22no—
238D4—42no—
239D4—62no—
240D4—62no—
241D4—62no—
242D4—81no—
243D4—124no—
244D4—124no—
245D4—124no—
246D4—122no—
247D4—241no—
248D4—242no—
249D4D3×D412yes—
250C2×C4—21no—
251C2×C4—32no—
252C2×C4—42no—
253C2×C4—61no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—61no—
257C2×C4—124no—
258C2×C4D3×D411yescharacteristic
259C2×C4D3×D411yescharacteristic
260C2×C4C23×D311yescharacteristic
261C8—22no—
262C8—32no—
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—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—62no—
278C22—62no—
279C22—62no—
280C22—122no—
281C22—122no—
282C22—122no—
283C22—248no—
284C22C2×D3×D411yescharacteristic
285C4—21no—
286C4—41no—
287C4—62no—
288C4—62no—
289C4—121no—
290C4#91112yescomplemented
291C4C2×D3×D411yescharacteristic
292C4#100911yescharacteristic
293C3#327411yescharacteristic, complemented, Sylow
294C2—42no—
295C2—82no—
296C2—122no—
297C2—242no—
298C2#560911yescharacteristic
299C2C2×#91111yescharacteristic
300C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3274
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. #5718C2semidirect
3. D4×D12C2semidirect
4. C3×#459C2semidirect
5. #5156C2semidirect
6. #5073C2semidirect
7. C4×#214C2semidirect
8. #4848C2semidirect
9. #4838C2semidirect
10. #4782C2semidirect
11. #4781C2semidirect
13. C2×#237C22semidirect
17. C3×#127C22semidirect
21. #938C22semidirect
24. C2×#213C22semidirect
25. C2×#213C22semidirect
28. C2×#204C22semidirect
29. C2×ΓL(1,𝔽25)C22semidirect
34. C4×Dic6C22semidirect
35. C3×C4⋊C8C22semidirect
36. C3×#102C22semidirect
38. #821C22semidirect
39. (C2×D3)⋊C8C22semidirect
40. #810C22semidirect
41. #808C22semidirect
42. C4×C3⋊C8C22semidirect
49. #459D3semidirect
82. #214D4semidirect
107. #127C2×D3semidirect
118. C4⋊C8C2×D3semidirect
122. #102C2×D3semidirect
159. Dic6C2×D4semidirect
162. C3⋊C8C2×D4semidirect
220. C3×C4AGL(1,ℤ/8)semidirect
290. C4#911semidirect
293. C3#3274semidirect

Maximal subgroups

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

subgroupindex
C2×#9112
#57182
D4×D122
C3×#4592
#51562
#50732
C4×#2142
#48482
#48382
#47822
#47812
#32743