Finite Group Databasebeta

← #83383

Subgroups — #83383

Order 384.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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×#1009C211yesmaximal, characteristic, complemented
2C3×#535C211yesmaximal, characteristic, complemented
3C2×#932C211yesmaximal, characteristic, complemented
4C22×#213C211yesmaximal, characteristic, complemented
5#5234C211yesmaximal, characteristic, complemented
6#5037C214yesmaximal, complemented
7#5036C214yesmaximal, complemented
8#5002C211yesmaximal, characteristic, complemented
9#5001C211yesmaximal, characteristic, complemented
10#3008—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12C22×C3×D4C2211yescharacteristic, complemented
13#1009—24no—
14#1009—24no—
15C2×#239—21no—
16C2×D3×D4—21no—
17C2×D3×D4—22no—
18C2×#235C2211yescharacteristic
19C2×#235C2212yescomplemented
20C22×D12C2211yescharacteristic
21C22×D12C2212yescomplemented
22C3×#127—21no—
23C3×#124—21no—
24C3×#121C2214yescomplemented
25C3×#120—22no—
26C3×#117C2211yescharacteristic, complemented
27C2×#214C2212yescomplemented
28#932—24no—
29C2×#213—24no—
30C2×#213C2214yescomplemented
31C2×#213C2212yescomplemented
32C2×#208C2211yescharacteristic, complemented
33C22×C3⋊C8C2211yescharacteristic, complemented
34#830C2214yescomplemented
35#810C2212yescomplemented
36#809C2214yescomplemented
37#809C2212yescomplemented
38C2×H(5,𝔽2)—31no—
39C2×AGL(1,ℤ/8)—31no—
40C22×D8—31no—
41#535D311yescharacteristic, complemented
42#449—34no—
43#447—34no—
44#418—31no—
45#416—31no—
46#408—31no—
47C24×C3—21no—
48C23×D3—21no—
49C23×D3—22no—
50C3×#41—22no—
51C3×#41—22no—
52C2×C3×Q8D411yescharacteristic
53C2×C3×D4—24no—
54C2×C3×D4—22no—
55C2×C3×D4—22no—
56C2×C3×D4—21no—
57C2×C3×D4D412yes—
58C2×C3×D4D411yescharacteristic
59C2×C3×D4C2312yes—
60C22×C3×C4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C2×#67—21no—
63C2×#67—22no—
64#239—28no—
65D3×D4—216no—
66D3×D4—28no—
67#235—22no—
68#235—24no—
69#235—24no—
70#235D414yescomplemented
71C2×D12—21no—
72C2×D12—22no—
73C2×D12—22no—
74C2×D12—24no—
75C2×D12—24no—
76C2×D12D414yescomplemented
77C2×D12C2312yes—
78C2×D12C2311yescharacteristic
79C2×C4×D3—21no—
80C2×C4×D3—22no—
81C2×Dic6C2311yescharacteristic
82C3×C4⋊C4C2312yes—
83C3×C22⋊C4—21no—
84C3×C22⋊C4—24no—
85C3×C42—21no—
86#214—24no—
87#213—28no—
88#213—24no—
89#208—22no—
90C2×C3⋊C8—22no—
91C2×C3⋊C8C2312yes—
92C2×C3⋊C8C2312yes—
93H(5,𝔽2)—64no—
94H(5,𝔽2)—64no—
95C2×#41—31no—
96C2×#41—32no—
97C2×#41—61no—
98C2×#41C2×D311yescharacteristic
99C22×D4—31no—
100C22×D4—32no—
101C22×D4—62no—
102C22×D4—61no—
103C22×D4C2×D311yescharacteristic, complemented
104AGL(1,ℤ/8)—64no—
105C2×SD16—32no—
106C2×D8—34no—
107C2×D8—32no—
108C2×D8—64no—
109C2×M16—31no—
110C22×C8—31no—
111#127—21no—
112#124—21no—
113#121C2×D314yescomplemented
114#120—22no—
115#117C2×D311yescharacteristic, complemented
116#103—32no—
117#102—32no—
118#102—34no—
119C22⋊C8—34no—
120C23×C3—22no—
121C23×C3—42no—
122C23×C3—82no—
123C23×C3C2×D412yes—
124C23×C3C2×D411yescharacteristic
125C22×D3—21no—
126C22×D3—22no—
127C22×D3—28no—
128C22×D3—24no—
129C22×D3—82no—
130C22×D3—84no—
131C3×Q8—22no—
132C3×D4—24no—
133C3×D4—22no—
134C3×D4—22no—
135C3×D4—22no—
136C3×D4—42no—
137C3×D4—84no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4C2×D412yes—
142C2×C3×C4C2×D412yes—
143C2×C3×C4C2×D412yes—
144C2×C3×C4C2×D411yescharacteristic
145C2×C3×C4C2411yescharacteristic
146#67—28no—
147#67—24no—
148C2×Dic3—21no—
149D12—28no—
150D12—24no—
151D12—21no—
152D12—22no—
153D12C2×D412yes—
154D12C2×D414yes—
155C4×D3—28no—
156C4×D3—24no—
157Dic6—21no—
158Dic6C2×D412yes—
159C3⋊C8—22no—
160C3⋊C8—22no—
161C24—21no—
162C24—62no—
163C24—61no—
164#41—22no—
165#41—22no—
166#41—34no—
167#41—64no—
168#41—64no—
169#41—68no—
170#41—62no—
171C2×Q8—31no—
172C2×Q8#6711yescharacteristic
173C2×D4—24no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—21no—
177C2×D4—34no—
178C2×D4—32no—
179C2×D4—31no—
180C2×D4—64no—
181C2×D4—64no—
182C2×D4—616no—
183C2×D4—68no—
184C2×D4—62no—
185C2×D4—61no—
186C2×D4—62no—
187C2×D4—62no—
188C2×D4—61no—
189C2×D4#6712yes—
190C2×D4#6711yescharacteristic
191C2×D4C22×D312yes—
192C22×C4—62no—
193C22×C4—61no—
194C22×C4C22×D312yes—
195C22×C4C22×D311yescharacteristic
196SD16—64no—
197D8—68no—
198D8—64no—
199M16—62no—
200C2×C8—32no—
201C2×C8—32no—
202C2×C8—62no—
203C4⋊C4C22×D312yes—
204C22⋊C4—21no—
205C22⋊C4—24no—
206C42—21no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—24no—
210C22×C3—42no—
211C22×C3—44no—
212C22×C3—84no—
213C22×C3#12012yes—
214C22×C3C22×D411yescharacteristic
215C2×D3—24no—
216C2×D3—22no—
217C2×D3—44no—
218C2×D3—42no—
219C2×D3—84no—
220C2×D3—88no—
221C3×C4—22no—
222C3×C4—42no—
223C3×C4#12012yes—
224C3×C4C22×D411yescharacteristic
225C3×C4C22×D411yescharacteristic
226Dic3—22no—
227C23—22no—
228C23—42no—
229C23—68no—
230C23—62no—
231C23—61no—
232C23—64no—
233C23—82no—
234C23—242no—
235C23—244no—
236C23C2×#6712yes—
237C23C2×#6711yescharacteristic
238Q8—22no—
239Q8—32no—
240Q8—61no—
241D4—24no—
242D4—22no—
243D4—22no—
244D4—22no—
245D4—34no—
246D4—32no—
247D4—42no—
248D4—68no—
249D4—68no—
250D4—64no—
251D4—64no—
252D4—61no—
253D4—62no—
254D4—84no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—68no—
259C2×C4—61no—
260C2×C4—64no—
261C2×C4D3×D412yes—
262C2×C4D3×D412yes—
263C2×C4C2×#6712yes—
264C2×C4C2×#6711yescharacteristic
265C2×C4C23×D311yescharacteristic
266C8—62no—
267C8—62no—
268C2×C3—22no—
269C2×C3—42no—
270C2×C3—82no—
271C2×C3C2×#12011yescharacteristic
272C2×C3#57612yes—
273D3—44no—
274D3—42no—
275C22—22no—
276C22—22no—
277C22—24no—
278C22—42no—
279C22—44no—
280C22—64no—
281C22—62no—
282C22—84no—
283C22—124no—
284C22—122no—
285C22—244no—
286C22—248no—
287C22#93712yes—
288C22C22×#6711yescharacteristic
289C4—22no—
290C4—42no—
291C4—62no—
292C4#93712yes—
293C4C2×D3×D411yescharacteristic
294C4C2×D3×D411yescharacteristic
295C3#300811yescharacteristic, complemented, Sylow
296C2—22no—
297C2—42no—
298C2—82no—
299C2—124no—
300C2—122no—
301C2C2×#93711yescharacteristic
302C2#583512yes—
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#535C2
socleC22×C3C22×D4
hypercenter#535D3
hypocenterC3#3008
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#1009C2semidirect
2. C3×#535C2semidirect
3. C2×#932C2semidirect
4. C22×#213C2semidirect
5. #5234C2semidirect
6. #5037C2semidirect
7. #5036C2semidirect
8. #5002C2semidirect
9. #5001C2semidirect
12. C22×C3×D4C22semidirect
19. C2×#235C22semidirect
21. C22×D12C22semidirect
24. C3×#121C22semidirect
26. C3×#117C22semidirect
27. C2×#214C22semidirect
30. C2×#213C22semidirect
31. C2×#213C22semidirect
32. C2×#208C22semidirect
33. C22×C3⋊C8C22semidirect
34. #830C22semidirect
35. #810C22semidirect
36. #809C22semidirect
37. #809C22semidirect
41. #535D3semidirect
70. #235D4semidirect
76. C2×D12D4semidirect
103. C22×D4C2×D3semidirect
113. #121C2×D3semidirect
115. #117C2×D3semidirect
295. C3#3008semidirect

Maximal subgroups

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

subgroupindex
C2×#10092
C3×#5352
C2×#9322
C22×#2132
#52342
#50372
#50362
#50022
#50012
#30083