Finite Group Databasebeta

← #79555

Subgroups — #79555

Order 384.

448 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 123 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
1D4×D12C211yesmaximal, characteristic
2#5542C211yesmaximal, characteristic, complemented
3C3×#436C211yesmaximal, characteristic, complemented
4#5134C212yesmaximal, complemented
5#5131C212yesmaximal, complemented
6#4848C211yesmaximal, characteristic, complemented
7#4846C211yesmaximal, characteristic, complemented
8#4737C212yesmaximal, complemented
9#4731C212yesmaximal, complemented
10#4694C211yesmaximal, characteristic, complemented
11C4×#204C211yesmaximal, characteristic, complemented
12#3324—31nomaximal, Sylow
13C2×D3×D4—21no—
14C22×D12C2212yes—
15C2×C3×M16C2212yescomplemented
16C3×C4×D4C2211yescharacteristic
17#936—21no—
18#930C2212yes—
19#926C2212yescomplemented
20#924C2212yescomplemented
21C2×#204C2211yescharacteristic
22C2×#204C2212yescomplemented
23#895—21no—
24#887—21no—
25#882—22no—
26#879—22no—
27#874C2211yescharacteristic
28C4×D12C2211yescharacteristic
29#869C2211yescharacteristic, complemented
30C4×Dic6C2211yescharacteristic, complemented
31C3×C4⋊C8C2211yescharacteristic, complemented
32C3×C22⋊C8C2212yescomplemented
33C3×C4×C8C2211yescharacteristic, complemented
34#821C2211yescharacteristic
35#821C2212yes—
36#821C2212yescomplemented
37#821C2212yescomplemented
38#821C2212yescomplemented
39#818C2212yescomplemented
40#817C2211yescharacteristic, complemented
41#816C2211yescharacteristic, complemented
42#556—31no—
43D42—31no—
44#486—31no—
45#482—32no—
46#474—31no—
47#468—32no—
48#460—31no—
49#450—32no—
50C4×SD16—31no—
51#436D311yescharacteristic, complemented
52#418—32no—
53C23×D3—22no—
54C2×C3×D4D411yescharacteristic
55C22×C3×C4C2312yes—
56C2×#67—22no—
57D3×D4—44no—
58C2×D12—24no—
59C2×D12—21no—
60C2×D12—22no—
61C2×D12—21no—
62C2×D12—42no—
63C2×D12C2312yes—
64C2×D12C2311yescharacteristic
65C2×C4×D3—21no—
66C2×Dic6—21no—
67C2×Dic6C2311yescharacteristic
68C3×M16—42no—
69C2×C3×C8C2311yescharacteristic
70C2×C3×C8C2311yescharacteristic
71C2×C3×C8C2312yes—
72C3×C4⋊C4D411yescharacteristic
73C3×C22⋊C4D412yes—
74C3×C42C2311yescharacteristic
75#217—22no—
76#217—22no—
77(C2×D3)⋊C4—21no—
78(C2×D3)⋊C4—22no—
79C4⋊Dic3—21no—
80C4⋊Dic3C2311yescharacteristic
81C4⋊Dic3C2311yescharacteristic
82C4⋊Dic3C2312yes—
83Dic3⋊C4—21no—
84Dic3⋊C4—22no—
85C4×Dic3—21no—
86C4×Dic3—22no—
87#204—22no—
88#204—42no—
89C22×D4—32no—
90C22×D4—61no—
91C2×SD16—31no—
92C2×SD16—32no—
93C2×M16C2×D312yescomplemented
94C4⋊Q8—31no—
95#127—31no—
96#126—62no—
97#124—61no—
98C22⋊Q8—32no—
99#121—32no—
100#121—61no—
101#120—62no—
102C4×Q8—31no—
103C4×Q8—61no—
104C4×D4—31no—
105C4×D4C2×D311yescharacteristic
106#117—32no—
107#107—32no—
108#106—31no—
109C4⋊C8C2×D311yescharacteristic, complemented
110#103—31no—
111#102—31no—
112#102—32no—
113#102—32no—
114#102—32no—
115#102—32no—
116C22⋊C8C2×D312yescomplemented
117C4×C8C2×D311yescharacteristic, complemented
118C23×C3C2×D412yes—
119C22×D3—21no—
120C22×D3—22no—
121C22×D3—44no—
122C22×D3—44no—
123C22×D3—82no—
124C3×D4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4C2×D412yes—
128C2×C3×C4C2×D411yescharacteristic
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2411yescharacteristic
131#67—44no—
132C2×Dic3—21no—
133C2×Dic3—21no—
134C2×Dic3—22no—
135C2×Dic3—21no—
136D12—22no—
137D12—21no—
138D12—22no—
139D12—44no—
140D12—42no—
141D12—81no—
142D12#4112yes—
143C4×D3—42no—
144Dic6—21no—
145Dic6—21no—
146Dic6—42no—
147C3×C8—21no—
148C3×C8—22no—
149C3×C8—21no—
150C24—62no—
151C2×Q8—31no—
152C2×Q8—61no—
153C2×D4—31no—
154C2×D4—32no—
155C2×D4—64no—
156C2×D4—62no—
157C2×D4—62no—
158C2×D4—61no—
159C2×D4—61no—
160C2×D4—122no—
161C2×D4—124no—
162C2×D4D1211yescharacteristic
163C22×C4—61no—
164C22×C4C22×D312yes—
165SD16—62no—
166SD16—122no—
167M16—42no—
168C2×C8C22×D311yescharacteristic
169C2×C8C22×D311yescharacteristic
170C2×C8C22×D312yes—
171C4⋊C4—31no—
172C4⋊C4—32no—
173C4⋊C4—31no—
174C4⋊C4—61no—
175C4⋊C4—62no—
176C4⋊C4—61no—
177C4⋊C4D1211yescharacteristic
178C22⋊C4—62no—
179C22⋊C4—62no—
180C22⋊C4—61no—
181C22⋊C4—62no—
182C22⋊C4D1212yes—
183C42—62no—
184C42—61no—
185C42C22×D311yescharacteristic
186C22×C3—22no—
187C22×C3—22no—
188C22×C3—22no—
189C22×C3C22×D411yescharacteristic
190C2×D3—22no—
191C2×D3—41no—
192C2×D3—41no—
193C2×D3—42no—
194C2×D3—42no—
195C2×D3—42no—
196C2×D3—88no—
197C3×C4—22no—
198C3×C4—21no—
199C3×C4AGL(1,ℤ/8)12yes—
200C3×C4C2×#4111yescharacteristic
201C3×C4H(5,𝔽2)11yescharacteristic
202Dic3—41no—
203Dic3—42no—
204Dic3—41no—
205Dic3—41no—
206C23—61no—
207C23—62no—
208C23—124no—
209C23—124no—
210C23—242no—
211C23C2×D1212yes—
212Q8—61no—
213Q8—61no—
214Q8—122no—
215D4—22no—
216D4—32no—
217D4—61no—
218D4—62no—
219D4—62no—
220D4—124no—
221D4—124no—
222D4—122no—
223D4—241no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—61no—
227C2×C4—62no—
228C2×C4—61no—
229C2×C4—61no—
230C2×C4—122no—
231C2×C4C2×D1212yes—
232C2×C4C2×D1211yescharacteristic
233C2×C4C2×D1211yescharacteristic
234C2×C4C23×D311yescharacteristic
235C8—21no—
236C8—22no—
237C8—21no—
238C2×C3—42no—
239C2×C3#54611yescharacteristic
240C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
241C2×C3#57711yescharacteristic
242D3—42no—
243D3—82no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—62no—
248C22—121no—
249C22—121no—
250C22—122no—
251C22—122no—
252C22—122no—
253C22—248no—
254C22C22×D1211yescharacteristic
255C4—21no—
256C4—22no—
257C4—121no—
258C4—122no—
259C4—121no—
260C4—121no—
261C4#90812yes—
262C4#100411yescharacteristic
263C4D3×#4111yescharacteristic
264C3#332411yescharacteristic, complemented, Sylow
265C2—42no—
266C2—122no—
267C2—242no—
268C2#555411yescharacteristic
269C2C2×#90811yescharacteristic
270C2#575111yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#436C2
socleC22×C3C22×D4
prime radicalD4×D12C2
hypercenter#436D3
hypocenterC3#3324
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
2. #5542C2semidirect
3. C3×#436C2semidirect
4. #5134C2semidirect
5. #5131C2semidirect
6. #4848C2semidirect
7. #4846C2semidirect
8. #4737C2semidirect
9. #4731C2semidirect
10. #4694C2semidirect
11. C4×#204C2semidirect
15. C2×C3×M16C22semidirect
19. #926C22semidirect
20. #924C22semidirect
22. C2×#204C22semidirect
29. #869C22semidirect
30. C4×Dic6C22semidirect
31. C3×C4⋊C8C22semidirect
32. C3×C22⋊C8C22semidirect
33. C3×C4×C8C22semidirect
36. #821C22semidirect
37. #821C22semidirect
38. #821C22semidirect
39. #818C22semidirect
40. #817C22semidirect
41. #816C22semidirect
51. #436D3semidirect
93. C2×M16C2×D3semidirect
109. C4⋊C8C2×D3semidirect
116. C22⋊C8C2×D3semidirect
117. C4×C8C2×D3semidirect
264. C3#3324semidirect

Maximal subgroups

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

subgroupindex
D4×D122
#55422
C3×#4362
#51342
#51312
#48482
#48462
#47372
#47312
#46942
C4×#2042
#33243