Finite Group Databasebeta

← #80984

Subgroups — #80984

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#5732C211yesmaximal, characteristic, complemented
2D4×D12C211yesmaximal, characteristic
3C3×#474C211yesmaximal, characteristic, complemented
4#5088C211yesmaximal, characteristic, complemented
5C4×#214C211yesmaximal, characteristic, complemented
6#4886C212yesmaximal, complemented
7#4885C212yesmaximal, complemented
8#4848C211yesmaximal, characteristic, complemented
9#4838C211yesmaximal, characteristic, complemented
10#4776C212yesmaximal, complemented
11#4770C212yesmaximal, complemented
12#3324—31nomaximal, Sylow
13C2×D3×D4C2212yes—
14C22×D12—21no—
15C3×C4⋊Q8C2211yescharacteristic, complemented
16C3×C4×D4C2211yescharacteristic
17#947—21no—
18Q8×Dic3—21no—
19C2×#214C2211yescharacteristic
20#930—21no—
21C2×#204C2212yescomplemented
22C2×ΓL(1,𝔽25)C2212yescomplemented
23#898—22no—
24#897C2212yescomplemented
25#895C2212yes—
26#892C2212yescomplemented
27#882—22no—
28#874C2211yescharacteristic
29C4×D12C2211yescharacteristic
30C4×Dic6C2211yescharacteristic, complemented
31C3×#107C2212yescomplemented
32C3×C4⋊C8C2211yescharacteristic, complemented
33C3×#102C2212yes—
34#832C2211yescharacteristic
35#821C2212yescomplemented
36(C2×D3)⋊C8C2212yescomplemented
37#810C2211yescharacteristic, complemented
38#809C2212yescomplemented
39#809C2212yescomplemented
40#808C2211yescharacteristic, complemented
41C4×C3⋊C8C2211yescharacteristic, complemented
42#556—31no—
43D42—31no—
44#486—31no—
45#482—32no—
46#474D311yescharacteristic, complemented
47#468—32no—
48#460—31no—
49#450—32no—
50C4×SD16—31no—
51#436—31no—
52#418—32no—
53C23×D3—22no—
54C2×C3×Q8—21no—
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×Dic6C2311yescharacteristic
68C2×C3×C8C2312yes—
69C3×C4⋊C4—21no—
70C3×C4⋊C4C2311yescharacteristic
71C3×C4⋊C4C2312yes—
72C3×C22⋊C4—21no—
73C3×C42C2311yescharacteristic
74#214—22no—
75(C2×D3)⋊C4—22no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4—22no—
78(C2×D3)⋊C4D412yes—
79C4⋊Dic3—22no—
80C4⋊Dic3D411yescharacteristic
81C4⋊Dic3C2311yescharacteristic
82Dic3⋊C4—21no—
83C4×Dic3—21no—
84C4×Dic3—22no—
85C2×C3⋊C8C2311yescharacteristic
86C2×C3⋊C8C2311yescharacteristic
87#204—42no—
88ΓL(1,𝔽25)—42no—
89C22×D4—32no—
90C22×D4—61no—
91C2×SD16—31no—
92C2×SD16—32no—
93C2×M16—32no—
94C4⋊Q8C2×D311yescharacteristic, complemented
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#107C2×D312yescomplemented
108#106—31no—
109C4⋊C8C2×D311yescharacteristic, complemented
110#103—31no—
111#102—31no—
112#102—32no—
113#102—32no—
114#102—32no—
115#102C2×D312yes—
116C22⋊C8—32no—
117C4×C8—31no—
118C23×C3—21no—
119C22×D3—22no—
120C22×D3—44no—
121C22×D3—44no—
122C22×D3—82no—
123C22×D3C2×D412yes—
124C3×Q8—42no—
125C3×D4—21no—
126C3×D4#4112yes—
127C2×C3×C4—22no—
128C2×C3×C4—21no—
129C2×C3×C4—42no—
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4C2×D411yescharacteristic
132C2×C3×C4C2411yescharacteristic
133#67—44no—
134C2×Dic3—21no—
135C2×Dic3—21no—
136C2×Dic3C2×D412yes—
137D12—22no—
138D12—22no—
139D12—22no—
140D12—44no—
141D12—42no—
142D12—81no—
143C4×D3—22no—
144C4×D3—22no—
145Dic6—21no—
146Dic6—21no—
147C3×C8—22no—
148C3⋊C8—21no—
149C3⋊C8—21no—
150C24—62no—
151C2×Q8—21no—
152C2×Q8—31no—
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—62no—
166SD16—122no—
167M16—122no—
168C2×C8—31no—
169C2×C8—31no—
170C2×C8C22×D312yes—
171C4⋊C4—21no—
172C4⋊C4—31no—
173C4⋊C4—31no—
174C4⋊C4—62no—
175C4⋊C4—61no—
176C4⋊C4C22×D312yes—
177C4⋊C4C22×D311yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—32no—
180C22⋊C4—62no—
181C22⋊C4—62no—
182C22⋊C4—62no—
183C42—62no—
184C42—61no—
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—42no—
200C3×C4AGL(1,ℤ/8)12yes—
201C3×C4C2×#4111yescharacteristic
202C3×C4H(5,𝔽2)11yescharacteristic
203Dic3—22no—
204Dic3—41no—
205Dic3—41no—
206C23—21no—
207C23—32no—
208C23—62no—
209C23—124no—
210C23—124no—
211C23—242no—
212Q8—42no—
213Q8—61no—
214Q8—61no—
215D4—21no—
216D4—62no—
217D4—62no—
218D4—62no—
219D4—124no—
220D4—124no—
221D4—122no—
222D4—241no—
223D4#23912yes—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—32no—
227C2×C4—42no—
228C2×C4—62no—
229C2×C4—62no—
230C2×C4—61no—
231C2×C4—61no—
232C2×C4D3×D411yescharacteristic
233C2×C4D3×D411yescharacteristic
234C2×C4C23×D311yescharacteristic
235C8—22no—
236C8—61no—
237C8—61no—
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—42no—
258C4—62no—
259C4—121no—
260C4—121no—
261C4#91112yes—
262C4C2×#23911yescharacteristic
263C4#100911yescharacteristic
264C3#332411yescharacteristic, complemented, Sylow
265C2—42no—
266C2—122no—
267C2—242no—
268C2#563011yescharacteristic
269C2C2×#91111yescharacteristic
270C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#474C2
socleC22×C3C22×D4
prime radicalD4×D12C2
hypercenter#474D3
hypocenterC3#3324
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #5732C2semidirect
3. C3×#474C2semidirect
4. #5088C2semidirect
5. C4×#214C2semidirect
6. #4886C2semidirect
7. #4885C2semidirect
8. #4848C2semidirect
9. #4838C2semidirect
10. #4776C2semidirect
11. #4770C2semidirect
15. C3×C4⋊Q8C22semidirect
21. C2×#204C22semidirect
22. C2×ΓL(1,𝔽25)C22semidirect
24. #897C22semidirect
26. #892C22semidirect
30. C4×Dic6C22semidirect
31. C3×#107C22semidirect
32. C3×C4⋊C8C22semidirect
35. #821C22semidirect
36. (C2×D3)⋊C8C22semidirect
37. #810C22semidirect
38. #809C22semidirect
39. #809C22semidirect
40. #808C22semidirect
41. C4×C3⋊C8C22semidirect
46. #474D3semidirect
94. C4⋊Q8C2×D3semidirect
107. #107C2×D3semidirect
109. 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
#57322
D4×D122
C3×#4742
#50882
C4×#2142
#48862
#48852
#48482
#48382
#47762
#47702
#33243