Finite Group Databasebeta

← #79683

Subgroups — #79683

Order 384.

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

normal subgroups only

This group has 448 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#5584C211yesmaximal, characteristic, complemented
2D4×D12C211yesmaximal, characteristic
3C3×C4×SD16C211yesmaximal, characteristic, complemented
4#5169C212yesmaximal, complemented
5#5027C211yesmaximal, characteristic, complemented
6#5007C211yesmaximal, characteristic, complemented
7#4866C212yesmaximal, complemented
8#4812C212yesmaximal, complemented
9#4770C212yesmaximal, complemented
10#4694C211yesmaximal, characteristic, complemented
11#4688C211yesmaximal, characteristic, complemented
12#3324—31nomaximal, Sylow
13C2×D3×D4C2212yes—
14C22×D12—21no—
15C2×C3×SD16C2211yescharacteristic
16C3×C4×Q8C2211yescharacteristic, complemented
17C3×C4×D4C2211yescharacteristic
18#946C2212yescomplemented
19C2×#215C2212yescomplemented
20#930—21no—
21C2×ΓL(1,𝔽25)C2212yescomplemented
22#898—22no—
23#895C2212yes—
24#892C2212yescomplemented
25#882—22no—
26#875—21no—
27#874C2211yescharacteristic
28C4×D12C2211yescharacteristic
29#869C2211yescharacteristic, complemented
30C4×Dic6—21no—
31C3×#106C2211yescharacteristic, complemented
32C3×#103C2211yescharacteristic, complemented
33C3×#102C2211yescharacteristic
34C3×C4×C8C2211yescharacteristic, complemented
35#832C2212yes—
36#821C2212yescomplemented
37#821C2212yescomplemented
38(C2×D3)⋊C8C2212yescomplemented
39#809C2212yescomplemented
40#807C2212yescomplemented
41C4⋊(C3⋊C8)C2211yescharacteristic, 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×SD16D311yescharacteristic, complemented
51#436—31no—
52#418—32no—
53C23×D3—22no—
54C2×C3×Q8C2311yescharacteristic
55C2×C3×D4C2311yescharacteristic
56C22×C3×C4—21no—
57C2×#67—22no—
58D3×D4—24no—
59D3×D4—42no—
60C2×D12—22no—
61C2×D12—21no—
62C2×D12—21no—
63C2×D12—44no—
64C2×D12D411yescharacteristic
65C2×D12C2312yes—
66C2×C4×D3C2312yes—
67C2×Dic6—21no—
68C3×SD16—22no—
69C2×C3×C8C2311yescharacteristic
70C2×C3×C8C2311yescharacteristic
71C3×C4⋊C4—21no—
72C3×C4⋊C4C2311yescharacteristic
73C3×C4⋊C4C2311yescharacteristic
74C3×C22⋊C4—21no—
75C3×C42—21no—
76C3×C42C2311yescharacteristic
77#215—42no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4D412yes—
82C4⋊Dic3—21no—
83C4⋊Dic3D411yescharacteristic
84C4⋊Dic3C2312yes—
85Dic3⋊C4—22no—
86C4×Dic3—22no—
87C2×C3⋊C8C2312yes—
88ΓL(1,𝔽25)—42no—
89C22×D4—32no—
90C22×D4—61no—
91C2×SD16—32no—
92C2×SD16C2×D311yescharacteristic
93C2×M16—32no—
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—61no—
103C4×Q8C2×D311yescharacteristic, complemented
104C4×D4—31no—
105C4×D4C2×D311yescharacteristic
106#117—32no—
107#107—32no—
108#106C2×D311yescharacteristic, complemented
109C4⋊C8—31no—
110#103C2×D311yescharacteristic, complemented
111#102—32no—
112#102—32no—
113#102—32no—
114#102—32no—
115#102C2×D311yescharacteristic
116C22⋊C8—32no—
117C4×C8C2×D311yescharacteristic, complemented
118C23×C3—21no—
119C22×D3—22no—
120C22×D3—44no—
121C22×D3—44no—
122C22×D3—82no—
123C22×D3C2×D412yes—
124C3×Q8—21no—
125C3×Q8—21no—
126C3×D4—21no—
127C3×D4#4112yes—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—42no—
132C2×C3×C4C2×D411yescharacteristic
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2411yescharacteristic
135#67—44no—
136C2×Dic3—22no—
137C2×Dic3C2×D412yes—
138D12—22no—
139D12—22no—
140D12—22no—
141D12—44no—
142D12—42no—
143D12—81no—
144C4×D3—22no—
145C4×D3—22no—
146Dic6—42no—
147C3×C8—21no—
148C3×C8—21no—
149C3⋊C8—22no—
150C24—62no—
151C2×Q8—61no—
152C2×Q8C22×D311yescharacteristic
153C2×D4—31no—
154C2×D4—32no—
155C2×D4—62no—
156C2×D4—62no—
157C2×D4—64no—
158C2×D4—61no—
159C2×D4—61no—
160C2×D4—124no—
161C2×D4—122no—
162C2×D4C22×D311yescharacteristic
163C22×C4—21no—
164C22×C4—32no—
165SD16—22no—
166SD16—122no—
167M16—122no—
168C2×C8—32no—
169C2×C8C22×D311yescharacteristic
170C2×C8C22×D311yescharacteristic
171C4⋊C4—21no—
172C4⋊C4—31no—
173C4⋊C4—32no—
174C4⋊C4—62no—
175C4⋊C4—61no—
176C4⋊C4C22×D311yescharacteristic
177C4⋊C4C22×D311yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—32no—
180C22⋊C4—62no—
181C22⋊C4—62no—
182C22⋊C4—62no—
183C42—21no—
184C42—62no—
185C42C22×D311yescharacteristic
186C22×C3—22no—
187C22×C3—41no—
188C22×C3—41no—
189C22×C3C22×D411yescharacteristic
190C2×D3—22no—
191C2×D3—22no—
192C2×D3—22no—
193C2×D3—42no—
194C2×D3—42no—
195C2×D3—42no—
196C2×D3—88no—
197C3×C4—21no—
198C3×C4—41no—
199C3×C4—41no—
200C3×C4—41no—
201C3×C4AGL(1,ℤ/8)12yes—
202C3×C4C2×#4111yescharacteristic
203C3×C4H(5,𝔽2)11yescharacteristic
204Dic3—22no—
205Dic3—42no—
206C23—21no—
207C23—32no—
208C23—62no—
209C23—124no—
210C23—124no—
211C23—242no—
212Q8—21no—
213Q8—21no—
214Q8—122no—
215D4—21no—
216D4—62no—
217D4—62no—
218D4—62no—
219D4—124no—
220D4—124no—
221D4—122no—
222D4—241no—
223D4#23512yes—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—32no—
228C2×C4—42no—
229C2×C4—62no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4D3×D411yescharacteristic
233C2×C4D3×D411yescharacteristic
234C2×C4C23×D311yescharacteristic
235C8—21no—
236C8—21no—
237C8—62no—
238C2×C3—42no—
239C2×C3#54611yescharacteristic
240C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
241C2×C3#57711yescharacteristic
242D3—42no—
243D3—82no—
244C22—22no—
245C22—41no—
246C22—41no—
247C22—62no—
248C22—62no—
249C22—62no—
250C22—122no—
251C22—122no—
252C22—122no—
253C22—248no—
254C22C2×D3×D411yescharacteristic
255C4—21no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—62no—
260C4—122no—
261C4#91412yes—
262C4C2×#23511yescharacteristic
263C4#100911yescharacteristic
264C3#332411yescharacteristic, complemented, Sylow
265C2—42no—
266C2—122no—
267C2—242no—
268C2#555011yescharacteristic
269C2C2×#91411yescharacteristic
270C2#577011yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×SD16C2
socleC22×C3C22×D4
prime radicalD4×D12C2
hypercenterC4×SD16D3
hypocenterC3#3324
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #5584C2semidirect
3. C3×C4×SD16C2semidirect
4. #5169C2semidirect
5. #5027C2semidirect
6. #5007C2semidirect
7. #4866C2semidirect
8. #4812C2semidirect
9. #4770C2semidirect
10. #4694C2semidirect
11. #4688C2semidirect
16. C3×C4×Q8C22semidirect
18. #946C22semidirect
19. C2×#215C22semidirect
21. C2×ΓL(1,𝔽25)C22semidirect
24. #892C22semidirect
29. #869C22semidirect
31. C3×#106C22semidirect
32. C3×#103C22semidirect
34. C3×C4×C8C22semidirect
36. #821C22semidirect
37. #821C22semidirect
38. (C2×D3)⋊C8C22semidirect
39. #809C22semidirect
40. #807C22semidirect
41. C4⋊(C3⋊C8)C22semidirect
50. C4×SD16D3semidirect
103. C4×Q8C2×D3semidirect
108. #106C2×D3semidirect
110. #103C2×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
#55842
D4×D122
C3×C4×SD162
#51692
#50272
#50072
#48662
#48122
#47702
#46942
#46882
#33243