Finite Group Databasebeta

← #83387

Subgroups — #83387

Order 384.

442 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 117 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×#535C211yesmaximal, characteristic, complemented
2#5830C211yesmaximal, characteristic, complemented
3#5234C211yesmaximal, characteristic, complemented
4#5217C212yesmaximal, complemented
5#5039C212yesmaximal, complemented
6#5038C212yesmaximal, complemented
7#5036C212yesmaximal, complemented
8#5035C212yesmaximal, complemented
9#5003C212yesmaximal, complemented
10#3199—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12C22×C3×D4C2211yescharacteristic, complemented
13C22×D12C2211yescharacteristic, complemented
14C3×#127—21no—
15C3×#124—21no—
16C3×#121C2212yes—
17C3×#121C2212yescomplemented
18C3×#120—21no—
19C3×#120—21no—
20C3×#117C2211yescharacteristic, complemented
21#947—21no—
22#940—21no—
23#937—21no—
24#937—21no—
25C2×#213C2212yes—
26C2×#213C2212yes—
27#930C2212yes—
28#930C2212yescomplemented
29#926C2211yescharacteristic, complemented
30C2×#208C2211yescharacteristic, complemented
31C22×C3⋊C8C2211yescharacteristic, complemented
32#832C2212yescomplemented
33#832C2212yescomplemented
34#830C2212yescomplemented
35#830C2212yescomplemented
36#809C2212yescomplemented
37#809C2212yescomplemented
38#808C2212yescomplemented
39#807C2212yescomplemented
40#535—31no—
41#535D311yescharacteristic, complemented
42#482—32no—
43#482—32no—
44#469—32no—
45#466—32no—
46#447—32no—
47#447—32no—
48#408—31no—
49C24×C3—21no—
50C23×D3—21no—
51C3×#41—41no—
52C3×#41—41no—
53C2×C3×Q8D411yescharacteristic
54C2×C3×D4—22no—
55C2×C3×D4—22no—
56C2×C3×D4—21no—
57C2×C3×D4—41no—
58C2×C3×D4—41no—
59C2×C3×D4D411yescharacteristic
60C2×C3×D4D411yescharacteristic
61C2×C3×D4D411yescharacteristic
62C2×C3×D4C2312yes—
63C22×C3×C4C2311yescharacteristic
64C22×C3×C4C2311yescharacteristic
65C22×C3×C4C2311yescharacteristic
66C2×#67—22no—
67C2×#67—22no—
68C2×D12—21no—
69C2×D12—41no—
70C2×D12—41no—
71C2×D12C2312yes—
72C3×C4⋊C4C2312yes—
73C3×C22⋊C4—21no—
74C3×C22⋊C4—22no—
75C3×C22⋊C4—22no—
76C3×C42—21no—
77#217—21no—
78#213—42no—
79#213—42no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—22no—
82C4⋊Dic3C2312yes—
83C4×Dic3—21no—
84#208—41no—
85C2×C3⋊C8—41no—
86C2×C3⋊C8C2312yes—
87C2×C3⋊C8C2312yes—
88C2×#41C2×D311yescharacteristic
89C22×D4—31no—
90C22×D4C2×D311yescharacteristic, complemented
91C2×D8—32no—
92C2×D8—32no—
93C2×M16—31no—
94C22×C8—31no—
95#127—21no—
96#127—61no—
97#124—21no—
98#124—61no—
99#121—32no—
100#121—32no—
101#121C2×D312yes—
102#121C2×D312yescomplemented
103#120—21no—
104#120—21no—
105#120—61no—
106#120—61no—
107#117—31no—
108#117C2×D311yescharacteristic, complemented
109#107—32no—
110#106—32no—
111#102—32no—
112#102—32no—
113#102—32no—
114#102—32no—
115C22⋊C8—32no—
116C22⋊C8—32no—
117C23×C3—22no—
118C23×C3—42no—
119C23×C3—82no—
120C23×C3C2×D411yescharacteristic
121C23×C3C2×D411yescharacteristic
122C23×C3C2×D411yescharacteristic
123C22×D3—22no—
124C22×D3—42no—
125C22×D3—82no—
126C3×Q8—41no—
127C3×D4—22no—
128C3×D4—22no—
129C3×D4—41no—
130C3×D4—41no—
131C3×D4—41no—
132C3×D4—42no—
133C3×D4—82no—
134C3×D4—82no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—22no—
142C2×C3×C4C2×D411yescharacteristic
143C2×C3×C4C2×D411yescharacteristic
144C2×C3×C4C2×D411yescharacteristic
145C2×C3×C4C2411yescharacteristic
146#67—82no—
147#67—82no—
148C2×Dic3—22no—
149D12—22no—
150D12—22no—
151D12—42no—
152C3⋊C8—22no—
153C3⋊C8—22no—
154C24—21no—
155C24—61no—
156#41—41no—
157#41—41no—
158C2×Q8#6711yescharacteristic
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—21no—
162C2×D4—32no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—62no—
166C2×D4—62no—
167C2×D4—61no—
168C2×D4—121no—
169C2×D4—121no—
170C2×D4#6711yescharacteristic
171C2×D4#6711yescharacteristic
172C2×D4#6711yescharacteristic
173C2×D4C22×D312yes—
174C22×C4C22×D311yescharacteristic
175C22×C4C22×D311yescharacteristic
176C22×C4C22×D311yescharacteristic
177D8—122no—
178D8—122no—
179M16—121no—
180C2×C8—32no—
181C2×C8—32no—
182C2×C8—121no—
183C4⋊C4—32no—
184C4⋊C4C22×D312yes—
185C22⋊C4—21no—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—61no—
189C22⋊C4—62no—
190C22⋊C4—62no—
191C42—21no—
192C42—61no—
193C22×C3—21no—
194C22×C3—21no—
195C22×C3—21no—
196C22×C3—22no—
197C22×C3—22no—
198C22×C3—22no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3—42no—
202C22×C3—84no—
203C22×C3C22×D411yescharacteristic
204C2×D3—42no—
205C2×D3—42no—
206C2×D3—42no—
207C2×D3—84no—
208C3×C4—21no—
209C3×C4—21no—
210C3×C4—21no—
211C3×C4—42no—
212C3×C4H(5,𝔽2)11yescharacteristic
213C3×C4H(5,𝔽2)11yescharacteristic
214Dic3—42no—
215C23—22no—
216C23—42no—
217C23—62no—
218C23—82no—
219C23—122no—
220C23—242no—
221C23C2×#6711yescharacteristic
222C23C2×#6711yescharacteristic
223C23C2×#6711yescharacteristic
224Q8—41no—
225D4—22no—
226D4—22no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—42no—
231D4—62no—
232D4—62no—
233D4—82no—
234D4—82no—
235D4—122no—
236D4—242no—
237D4—242no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—62no—
246C2×C4C2×#6711yescharacteristic
247C2×C4C2×#6711yescharacteristic
248C2×C4C2×#6711yescharacteristic
249C2×C4C23×D311yescharacteristic
250C8—62no—
251C8—62no—
252C2×C3—41no—
253C2×C3—41no—
254C2×C3—41no—
255C2×C3—82no—
256C2×C3#53411yescharacteristic
257C2×C3#57612yes—
258D3—82no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—22no—
263C22—22no—
264C22—22no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—84no—
269C22—122no—
270C22—122no—
271C22—122no—
272C22—244no—
273C22C22×#6711yescharacteristic
274C4—21no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—122no—
279C4#100411yescharacteristic
280C4#100411yescharacteristic
281C3#319911yescharacteristic, complemented, Sylow
282C2—41no—
283C2—41no—
284C2—41no—
285C2—82no—
286C2—242no—
287C2#580411yescharacteristic
288C2#583512yes—
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#535C2
socleC22×C3C22×D4
hypercenter#535D3
hypocenterC3#3199
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. C3×#535C2semidirect
2. #5830C2semidirect
3. #5234C2semidirect
4. #5217C2semidirect
5. #5039C2semidirect
6. #5038C2semidirect
7. #5036C2semidirect
8. #5035C2semidirect
9. #5003C2semidirect
12. C22×C3×D4C22semidirect
13. C22×D12C22semidirect
17. C3×#121C22semidirect
20. C3×#117C22semidirect
28. #930C22semidirect
29. #926C22semidirect
30. C2×#208C22semidirect
31. C22×C3⋊C8C22semidirect
32. #832C22semidirect
33. #832C22semidirect
34. #830C22semidirect
35. #830C22semidirect
36. #809C22semidirect
37. #809C22semidirect
38. #808C22semidirect
39. #807C22semidirect
41. #535D3semidirect
90. C22×D4C2×D3semidirect
102. #121C2×D3semidirect
108. #117C2×D3semidirect
281. C3#3199semidirect

Maximal subgroups

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

subgroupindex
C3×#5352
#58302
#52342
#52172
#50392
#50382
#50362
#50352
#50032
#31993