Finite Group Databasebeta

← #78675

Subgroups — #78675

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
1#5830C211yesmaximal, characteristic
2#5825C211yesmaximal, characteristic, complemented
3C3×#408C211yesmaximal, characteristic, complemented
4#5134C212yesmaximal, complemented
5#5114C212yesmaximal, complemented
6#4738C212yesmaximal, complemented
7#4737C212yesmaximal, complemented
8#4731C212yesmaximal, complemented
9#4723C212yesmaximal, complemented
10#3200—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12C22×D12C2211yescharacteristic
13C22×Dic6C2211yescharacteristic, complemented
14C2×C3×M16C2211yescharacteristic, complemented
15C22×C3×C8C2211yescharacteristic, complemented
16#947—21no—
17#944—21no—
18#940—21no—
19#937—21no—
20#937—21no—
21#936—21no—
22#935—21no—
23#935—21no—
24#930C2212yes—
25#930C2212yes—
26#926C2211yescharacteristic
27#926C2211yescharacteristic, complemented
28#924C2212yes—
29#924C2212yescomplemented
30C2×#204C2212yes—
31C2×#204C2212yes—
32C3×C22⋊C8C2212yes—
33C3×C22⋊C8C2212yescomplemented
34#821C2212yes—
35#821C2212yes—
36#818C2212yescomplemented
37#817C2212yescomplemented
38#816C2212yescomplemented
39#816C2212yescomplemented
40#536—31no—
41#535—31no—
42#485—32no—
43#482—32no—
44#468—32no—
45#465—32no—
46#450—32no—
47#448—32no—
48#408D311yescharacteristic, complemented
49C23×D3—21no—
50C3×#41—41no—
51C3×#41—41no—
52C2×C3×Q8D411yescharacteristic
53C2×C3×D4D411yescharacteristic
54C2×C3×D4D411yescharacteristic
55C2×C3×D4D411yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C2×#67—22no—
60C2×#67—22no—
61C22×Dic3—21no—
62C2×D12—21no—
63C2×D12—41no—
64C2×D12—41no—
65C2×D12C2312yes—
66C2×Dic6—21no—
67C2×Dic6—41no—
68C2×Dic6—41no—
69C2×Dic6C2312yes—
70C3×M16—41no—
71C2×C3×C8—41no—
72C2×C3×C8C2312yes—
73C2×C3×C8C2312yes—
74#217—22no—
75#217—22no—
76#217—21no—
77#217—21no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4—22no—
80C4⋊Dic3C2312yes—
81C4⋊Dic3C2312yes—
82Dic3⋊C4—22no—
83Dic3⋊C4—22no—
84C4×Dic3—21no—
85C4×Dic3—21no—
86#204—42no—
87#204—42no—
88C2×#41C2×D311yescharacteristic
89C22×Q8—31no—
90C22×D4—31no—
91C2×SD16—32no—
92C2×SD16—32no—
93C2×M16C2×D311yescharacteristic, complemented
94C22×C8C2×D311yescharacteristic, complemented
95C4⋊Q8—61no—
96#127—61no—
97#124—61no—
98#124—61no—
99#123—61no—
100#123—61no—
101C22⋊Q8—32no—
102C22⋊Q8—32no—
103#121—32no—
104#121—32no—
105#120—61no—
106#120—61no—
107#117—31no—
108#117—31no—
109#107—32no—
110#106—32no—
111#103—32no—
112#103—32no—
113#102—32no—
114#102—32no—
115C22⋊C8C2×D312yes—
116C22⋊C8C2×D312yescomplemented
117C23×C3C2×D411yescharacteristic
118C23×C3C2×D411yescharacteristic
119C23×C3C2×D411yescharacteristic
120C22×D3—22no—
121C22×D3—42no—
122C22×D3—82no—
123C3×Q8—41no—
124C3×D4—41no—
125C3×D4—41no—
126C3×D4—41no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2×D411yescharacteristic
136C2×C3×C4C2411yescharacteristic
137#67—82no—
138#67—82no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—42no—
143D12—22no—
144D12—22no—
145D12—42no—
146Dic6—22no—
147Dic6—22no—
148Dic6—42no—
149C3×C8—22no—
150C3×C8—22no—
151C24—61no—
152#41—41no—
153#41—41no—
154C2×Q8—32no—
155C2×Q8—61no—
156C2×Q8—121no—
157C2×Q8—121no—
158C2×Q8D1211yescharacteristic
159C2×D4—32no—
160C2×D4—62no—
161C2×D4—62no—
162C2×D4—61no—
163C2×D4—121no—
164C2×D4—121no—
165C2×D4D1211yescharacteristic
166C2×D4D1211yescharacteristic
167C2×D4D1211yescharacteristic
168C22×C4—61no—
169C22×C4C22×D311yescharacteristic
170C22×C4C22×D311yescharacteristic
171C22×C4C22×D311yescharacteristic
172SD16—122no—
173SD16—122no—
174M16—41no—
175C2×C8—41no—
176C2×C8C22×D312yes—
177C2×C8C22×D312yes—
178C4⋊C4—32no—
179C4⋊C4—32no—
180C4⋊C4—62no—
181C4⋊C4—62no—
182C22⋊C4—61no—
183C22⋊C4—61no—
184C22⋊C4—62no—
185C22⋊C4—62no—
186C22⋊C4—62no—
187C22⋊C4—62no—
188C42—61no—
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×C4H(5,𝔽2)11yescharacteristic
205C3×C4H(5,𝔽2)11yescharacteristic
206Dic3—42no—
207Dic3—42no—
208Dic3—42no—
209C23—62no—
210C23—122no—
211C23—242no—
212C23C2×D1211yescharacteristic
213C23C2×D1211yescharacteristic
214C23C2×D1211yescharacteristic
215Q8—41no—
216Q8—62no—
217Q8—62no—
218Q8—122no—
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—62no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—122no—
237C2×C4C2×D1211yescharacteristic
238C2×C4C2×D1211yescharacteristic
239C2×C4C2×D1211yescharacteristic
240C2×C4C23×D311yescharacteristic
241C8—22no—
242C8—22no—
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×D1211yescharacteristic
260C4—21no—
261C4—21no—
262C4—21no—
263C4—122no—
264C4—122no—
265C4—122no—
266C4#100411yescharacteristic
267C4#100411yescharacteristic
268C3#320011yescharacteristic, complemented, Sylow
269C2—41no—
270C2—41no—
271C2—41no—
272C2—242no—
273C2#549311yescharacteristic
274C2#575112yes—
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#408C2
socleC22×C3C22×D4
prime radical#5830C2
hypercenter#408D3
hypocenterC3#3200
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #5825C2semidirect
3. C3×#408C2semidirect
4. #5134C2semidirect
5. #5114C2semidirect
6. #4738C2semidirect
7. #4737C2semidirect
8. #4731C2semidirect
9. #4723C2semidirect
13. C22×Dic6C22semidirect
14. C2×C3×M16C22semidirect
15. C22×C3×C8C22semidirect
27. #926C22semidirect
29. #924C22semidirect
33. C3×C22⋊C8C22semidirect
36. #818C22semidirect
37. #817C22semidirect
38. #816C22semidirect
39. #816C22semidirect
48. #408D3semidirect
93. C2×M16C2×D3semidirect
94. C22×C8C2×D3semidirect
116. C22⋊C8C2×D3semidirect
268. C3#3200semidirect

Maximal subgroups

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

subgroupindex
#58302
#58252
C3×#4082
#51342
#51142
#47382
#47372
#47312
#47232
#32003