Finite Group Databasebeta

← #83407

Subgroups — #83407

Order 384.

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

normal subgroups only

This group has 410 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×#536C211yesmaximal, characteristic, complemented
2#5830C211yesmaximal, characteristic
3#5234C211yesmaximal, characteristic, complemented
4#5227C212yesmaximal, complemented
5#5049C212yesmaximal, complemented
6#5048C212yesmaximal, complemented
7#5046C212yesmaximal, complemented
8#5045C212yesmaximal, complemented
9#5003C212yesmaximal, complemented
10#3200—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12C22×C3×Q8C2211yescharacteristic, complemented
13C22×D12C2211yescharacteristic
14C3×C4⋊Q8—21no—
15C3×#124—21no—
16C3×#123—21no—
17C3×#123—21no—
18C3×C22⋊Q8C2212yes—
19C3×C22⋊Q8C2212yescomplemented
20C3×#117C2211yescharacteristic, complemented
21#947—21no—
22C2×#215C2212yes—
23C2×#215C2212yes—
24#940—21no—
25#937—21no—
26#937—21no—
27#930C2212yes—
28#930C2212yes—
29#926C2211yescharacteristic
30C2×#208C2211yescharacteristic, complemented
31C22×C3⋊C8C2211yescharacteristic, complemented
32#835C2212yescomplemented
33#835C2212yescomplemented
34#830C2212yes—
35#830C2212yescomplemented
36#809C2212yes—
37#809C2212yes—
38#808C2212yescomplemented
39#807C2212yescomplemented
40#536D311yescharacteristic, complemented
41#535—31no—
42#485—32no—
43#482—32no—
44#468—32no—
45#465—32no—
46#450—32no—
47#448—32no—
48#408—31no—
49C23×D3—21no—
50C3×#41—41no—
51C3×#41—41no—
52C2×C3×Q8—21no—
53C2×C3×Q8—41no—
54C2×C3×Q8—41no—
55C2×C3×Q8D411yescharacteristic
56C2×C3×Q8C2312yes—
57C2×C3×D4D411yescharacteristic
58C2×C3×D4D411yescharacteristic
59C2×C3×D4D411yescharacteristic
60C22×C3×C4—21no—
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C22×C3×C4C2311yescharacteristic
64C2×#67—22no—
65C2×#67—22no—
66C2×D12—21no—
67C2×D12—41no—
68C2×D12—41no—
69C2×D12C2312yes—
70C3×C4⋊C4—22no—
71C3×C4⋊C4—22no—
72C3×C4⋊C4C2312yes—
73C3×C22⋊C4—21no—
74C3×C22⋊C4—22no—
75C3×C22⋊C4—22no—
76C3×C42—21no—
77#217—21no—
78#215—42no—
79#215—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×Q8C2×D311yescharacteristic, complemented
90C22×D4—31no—
91C2×SD16—32no—
92C2×SD16—32no—
93C2×M16—31no—
94C22×C8—31no—
95C4⋊Q8—21no—
96#127—61no—
97#124—21no—
98#124—61no—
99#123—21no—
100#123—21no—
101C22⋊Q8C2×D312yes—
102C22⋊Q8C2×D312yescomplemented
103#121—32no—
104#121—32no—
105#120—61no—
106#120—61no—
107#117—31no—
108#117C2×D311yescharacteristic, complemented
109#107—32no—
110#106—32no—
111#103—32no—
112#103—32no—
113#102—32no—
114#102—32no—
115C22⋊C8—32no—
116C22⋊C8—32no—
117C23×C3C2×D411yescharacteristic
118C23×C3C2×D411yescharacteristic
119C23×C3C2×D411yescharacteristic
120C22×D3—22no—
121C22×D3—42no—
122C22×D3—82no—
123C3×Q8—22no—
124C3×Q8—22no—
125C3×Q8—41no—
126C3×Q8—42no—
127C3×D4—41no—
128C3×D4—41no—
129C3×D4—41no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—42no—
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2×D411yescharacteristic
142C2×C3×C4C2411yescharacteristic
143#67—82no—
144#67—82no—
145C2×Dic3—22no—
146D12—22no—
147D12—22no—
148D12—42no—
149C3⋊C8—22no—
150C3⋊C8—22no—
151C24—61no—
152#41—41no—
153#41—41no—
154C2×Q8—21no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8#6711yescharacteristic
158C2×Q8C22×D312yes—
159C2×D4—32no—
160C2×D4—62no—
161C2×D4—62no—
162C2×D4—61no—
163C2×D4—121no—
164C2×D4—121no—
165C2×D4#6711yescharacteristic
166C2×D4#6711yescharacteristic
167C2×D4#6711yescharacteristic
168C22×C4—21no—
169C22×C4C22×D311yescharacteristic
170C22×C4C22×D311yescharacteristic
171C22×C4C22×D311yescharacteristic
172SD16—122no—
173SD16—122no—
174M16—121no—
175C2×C8—32no—
176C2×C8—32no—
177C2×C8—121no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—32no—
181C4⋊C4C22×D312yes—
182C22⋊C4—21no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—61no—
186C22⋊C4—62no—
187C22⋊C4—62no—
188C42—21no—
189C42—61no—
190C22×C3—21no—
191C22×C3—21no—
192C22×C3—21no—
193C22×C3—22no—
194C22×C3—22no—
195C22×C3—22no—
196C22×C3C22×D411yescharacteristic
197C2×D3—42no—
198C2×D3—42no—
199C2×D3—42no—
200C2×D3—84no—
201C3×C4—21no—
202C3×C4—21no—
203C3×C4—21no—
204C3×C4—42no—
205C3×C4—42no—
206C3×C4H(5,𝔽2)11yescharacteristic
207C3×C4H(5,𝔽2)11yescharacteristic
208Dic3—42no—
209C23—62no—
210C23—122no—
211C23—242no—
212C23C2×#6711yescharacteristic
213C23C2×#6711yescharacteristic
214C23C2×#6711yescharacteristic
215Q8—22no—
216Q8—22no—
217Q8—41no—
218Q8—42no—
219D4—41no—
220D4—41no—
221D4—41no—
222D4—62no—
223D4—62no—
224D4—122no—
225D4—242no—
226D4—242no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—62no—
237C2×C4C2×#6711yescharacteristic
238C2×C4C2×#6711yescharacteristic
239C2×C4C2×#6711yescharacteristic
240C2×C4C23×D311yescharacteristic
241C8—62no—
242C8—62no—
243C2×C3—41no—
244C2×C3—41no—
245C2×C3—41no—
246C2×C3#53411yescharacteristic
247C2×C3#57712yes—
248D3—82no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—122no—
256C22—122no—
257C22—122no—
258C22—244no—
259C22C22×#6711yescharacteristic
260C4—21no—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—122no—
266C4#100411yescharacteristic
267C4#100411yescharacteristic
268C3#320011yescharacteristic, complemented, Sylow
269C2—41no—
270C2—41no—
271C2—41no—
272C2—242no—
273C2#580411yescharacteristic
274C2#583712yes—
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#536C2
socleC22×C3C22×D4
prime radical#5830C2
hypercenter#536D3
hypocenterC3#3200
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C3×#536C2semidirect
3. #5234C2semidirect
4. #5227C2semidirect
5. #5049C2semidirect
6. #5048C2semidirect
7. #5046C2semidirect
8. #5045C2semidirect
9. #5003C2semidirect
12. C22×C3×Q8C22semidirect
19. C3×C22⋊Q8C22semidirect
20. C3×#117C22semidirect
30. C2×#208C22semidirect
31. C22×C3⋊C8C22semidirect
32. #835C22semidirect
33. #835C22semidirect
35. #830C22semidirect
38. #808C22semidirect
39. #807C22semidirect
40. #536D3semidirect
89. C22×Q8C2×D3semidirect
102. C22⋊Q8C2×D3semidirect
108. #117C2×D3semidirect
268. C3#3200semidirect

Maximal subgroups

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

subgroupindex
C3×#5362
#58302
#52342
#52272
#50492
#50482
#50462
#50452
#50032
#32003