Finite Group Databasebeta

← #83143

Subgroups — #83143

Order 384.

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

normal subgroups only

This group has 450 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
1C3×#529C211yesmaximal, characteristic, complemented
2C2×C4×D12C211yesmaximal, characteristic, complemented
3#5066C212yesmaximal, complemented
4#5057C212yesmaximal, complemented
5#5046C211yesmaximal, characteristic, complemented
6#5043C211yesmaximal, characteristic, complemented
7#5036C211yesmaximal, characteristic
8#5033C211yesmaximal, characteristic, complemented
9#5026C212yesmaximal, complemented
10#5014C212yesmaximal, complemented
11#4926C211yesmaximal, characteristic, complemented
12#3114—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×C4×D3—21no—
15C3×#126—21no—
16C3×#125C2211yescharacteristic, complemented
17C3×#124C2211yescharacteristic, complemented
18C3×#123—21no—
19C3×C22⋊Q8C2211yescharacteristic
20C3×#121C2211yescharacteristic
21C3×C4×Q8C2211yescharacteristic, complemented
22C3×C4×D4—21no—
23C3×C4×D4C2211yescharacteristic, complemented
24C3×#117—21no—
25C2×C3×C42C2211yescharacteristic
26C2×#215C2212yes—
27C2×#213C2212yes—
28C2×(C2×D3)⋊C4—21no—
29C2×C4⋊Dic3C2211yescharacteristic
30C4×D12—21no—
31C4×D12—21no—
32C4×D12C2212yes—
33C4×D12C2212yes—
34#835C2212yescomplemented
35#832C2212yescomplemented
36#830C2211yescharacteristic
37#830C2211yescharacteristic
38#809C2212yes—
39#809C2212yes—
40#808C2212yescomplemented
41#807C2212yescomplemented
42C4⋊(C3⋊C8)C2212yescomplemented
43#803C2212yescomplemented
44#529D311yescharacteristic, complemented
45C2×C4×D4—31no—
46#483—31no—
47#480—31no—
48#478—32no—
49#463—32no—
50#450—31no—
51#447—31no—
52#442—32no—
53#440—32no—
54#432—31no—
55C23×D3—21no—
56C2×C3×Q8C2311yescharacteristic
57C2×C3×D4—21no—
58C2×C3×D4C2311yescharacteristic
59C22×C3×C4—21no—
60C22×C3×C4D411yescharacteristic
61C22×C3×C4D411yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C22×Dic3—21no—
64C2×D12—21no—
65C2×D12—24no—
66C2×D12C2312yes—
67C2×C4×D3—22no—
68C2×C4×D3—22no—
69C2×C4×D3—44no—
70C3×C4⋊C4—21no—
71C3×C4⋊C4—21no—
72C3×C4⋊C4—21no—
73C3×C4⋊C4—21no—
74C3×C4⋊C4C2311yescharacteristic
75C3×C4⋊C4C2311yescharacteristic
76C3×C22⋊C4—21no—
77C3×C22⋊C4—21no—
78C3×C22⋊C4—21no—
79C3×C22⋊C4—21no—
80C3×C22⋊C4—21no—
81C3×C42—21no—
82C3×C42D411yescharacteristic
83C3×C42D411yescharacteristic
84C3×C42C2311yescharacteristic
85C3×C42C2311yescharacteristic
86#215—24no—
87#213—24no—
88(C2×D3)⋊C4—22no—
89(C2×D3)⋊C4—22no—
90C4⋊Dic3—21no—
91C4⋊Dic3C2312yes—
92C2×C3⋊C8C2312yes—
93C2×C3⋊C8C2312yes—
94C22×D4—31no—
95C23×C4—61no—
96C2×SD16—32no—
97C2×D8—32no—
98#126—21no—
99#125C2×D311yescharacteristic, complemented
100#124C2×D311yescharacteristic, complemented
101#123—21no—
102C22⋊Q8C2×D311yescharacteristic
103#121C2×D311yescharacteristic
104C4×Q8C2×D311yescharacteristic, complemented
105C4×D4—21no—
106C4×D4—32no—
107C4×D4—32no—
108C4×D4—61no—
109C4×D4—61no—
110C4×D4C2×D311yescharacteristic, complemented
111#117—21no—
112C2×C4⋊C4—31no—
113C2×C22⋊C4—61no—
114C2×C42C2×D311yescharacteristic
115#107—32no—
116#106—32no—
117C4⋊C8—32no—
118#103—32no—
119#102—32no—
120#102—32no—
121#102—32no—
122C22⋊C8—31no—
123C22⋊C8—31no—
124#97—32no—
125C23×C3—21no—
126C23×C3C2×D411yescharacteristic
127C22×D3—24no—
128C22×D3—22no—
129C22×D3—42no—
130C22×D3—42no—
131C3×Q8—22no—
132C3×D4—22no—
133C3×D4—42no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—41no—
140C2×C3×C4—42no—
141C2×C3×C4C2×D411yescharacteristic
142C2×C3×C4C2×D411yescharacteristic
143C2×C3×C4C2×D411yescharacteristic
144C2×C3×C4C2×D411yescharacteristic
145C2×C3×C4C2×D411yescharacteristic
146C2×C3×C4C2411yescharacteristic
147C2×Dic3—22no—
148C2×Dic3—42no—
149D12—24no—
150D12—22no—
151D12#4114yes—
152C4×D3—44no—
153C4×D3—44no—
154C3⋊C8—22no—
155C3⋊C8—22no—
156C24—61no—
157C2×Q8C22×D311yescharacteristic
158C2×D4—21no—
159C2×D4—32no—
160C2×D4—64no—
161C2×D4—61no—
162C2×D4C22×D311yescharacteristic
163C22×C4—21no—
164C22×C4—61no—
165C22×C4—62no—
166C22×C4—62no—
167C22×C4—124no—
168C22×C4#6711yescharacteristic
169C22×C4#6711yescharacteristic
170C22×C4C22×D311yescharacteristic
171SD16—64no—
172D8—64no—
173C2×C8—32no—
174C2×C8—32no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—32no—
180C4⋊C4—61no—
181C4⋊C4C22×D311yescharacteristic
182C4⋊C4C22×D311yescharacteristic
183C22⋊C4—21no—
184C22⋊C4—21no—
185C22⋊C4—21no—
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—62no—
189C22⋊C4—62no—
190C42—21no—
191C42#6711yescharacteristic
192C42#6711yescharacteristic
193C42C22×D311yescharacteristic
194C42C22×D311yescharacteristic
195C22×C3—21no—
196C22×C3—21no—
197C22×C3—41no—
198C22×C3—41no—
199C22×C3—41no—
200C22×C3AGL(1,ℤ/8)12yes—
201C22×C3C22×D411yescharacteristic
202C2×D3—24no—
203C2×D3—48no—
204C2×D3—42no—
205C2×D3—42no—
206C3×C4—21no—
207C3×C4—21no—
208C3×C4—21no—
209C3×C4—21no—
210C3×C4—21no—
211C3×C4—41no—
212C3×C4—41no—
213C3×C4—41no—
214C3×C4C2×#4112yes—
215Dic3—42no—
216C23—21no—
217C23—64no—
218C23—62no—
219C23—122no—
220C23—122no—
221C23C2×#6711yescharacteristic
222Q8—22no—
223D4—22no—
224D4—34no—
225D4—42no—
226D4—64no—
227D4—62no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—62no—
236C2×C4—124no—
237C2×C4—124no—
238C2×C4—122no—
239C2×C4C2×#6711yescharacteristic
240C2×C4C2×#6711yescharacteristic
241C2×C4C2×#6711yescharacteristic
242C2×C4C2×#6711yescharacteristic
243C2×C4C2×#6711yescharacteristic
244C2×C4C23×D311yescharacteristic
245C8—62no—
246C8—62no—
247C2×C3—22no—
248C2×C3—81no—
249C2×C3#52511yescharacteristic
250C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
251C2×C3#57511yescharacteristic
252D3—44no—
253C22—21no—
254C22—21no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—64no—
259C22—128no—
260C22—122no—
261C22—122no—
262C22#94912yes—
263C22C22×#6711yescharacteristic
264C4—21no—
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—122no—
273C4D3×#4112yes—
274C3#311411yescharacteristic, complemented, Sylow
275C2—22no—
276C2—81no—
277C2—124no—
278C2C2×#94911yescharacteristic
279C2#582211yescharacteristic
280C2#582811yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#529C2
socleC22×C3C22×D4
prime radical#5036C2
hypercenter#529D3
hypocenterC3#3114
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C3×#529C2semidirect
2. C2×C4×D12C2semidirect
3. #5066C2semidirect
4. #5057C2semidirect
5. #5046C2semidirect
6. #5043C2semidirect
8. #5033C2semidirect
9. #5026C2semidirect
10. #5014C2semidirect
11. #4926C2semidirect
16. C3×#125C22semidirect
17. C3×#124C22semidirect
21. C3×C4×Q8C22semidirect
23. C3×C4×D4C22semidirect
34. #835C22semidirect
35. #832C22semidirect
40. #808C22semidirect
41. #807C22semidirect
42. C4⋊(C3⋊C8)C22semidirect
43. #803C22semidirect
44. #529D3semidirect
99. #125C2×D3semidirect
100. #124C2×D3semidirect
104. C4×Q8C2×D3semidirect
110. C4×D4C2×D3semidirect
274. C3#3114semidirect

Maximal subgroups

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

subgroupindex
C3×#5292
C2×C4×D122
#50662
#50572
#50462
#50432
#50362
#50332
#50262
#50142
#49262
#31143