Finite Group Databasebeta

← #81871

Subgroups — #81871

Order 384.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#918C212yesmaximal, complemented
2C2×#914C212yesmaximal, complemented
3#5725C211yesmaximal, characteristic, complemented
4D3×#124C211yesmaximal, characteristic, complemented
5C3×#497C211yesmaximal, characteristic, complemented
6#5191C212yesmaximal, complemented
7#5168C212yesmaximal, complemented
8#5086C211yesmaximal, characteristic, complemented
9#5058C211yesmaximal, characteristic, complemented
10#4712C211yesmaximal, characteristic, complemented
11#4707C211yesmaximal, characteristic, complemented
12#3153—31nomaximal, Sylow
13C2×#239C2212yes—
14C2×D3×Q8C2211yescharacteristic
15C2×D3×D4C2211yescharacteristic
16C2×C3×Q16C2212yescomplemented
17C2×C3×SD16C2212yes—
18C3×C4⋊Q8C2211yescharacteristic, complemented
19C3×#124C2211yescharacteristic
20#947C2211yescharacteristic
21#947C2212yescomplemented
22C2×#216C2212yescomplemented
23C2×#215C2212yes—
24C2×#215C2212yescomplemented
25#936C2211yescharacteristic
26C2×#213C2212yes—
27#918—28no—
28#914—28no—
29C2×D24C2212yes—
30C2×#204C2212yescomplemented
31C2×ΓL(1,𝔽25)C2212yes—
32#895—22no—
33#893—22no—
34#885—22no—
35D3×C22⋊C4—22no—
36#875C2211yescharacteristic
37#874C2211yescharacteristic, complemented
38C42×D3C2211yescharacteristic
39C3×#97C2211yescharacteristic, complemented
40C8×Dic3C2212yescomplemented
41#803C2211yescharacteristic, complemented
42C2×#137—32no—
43C2×AGL(1,ℤ/8)—32no—
44#532—31no—
45C2×#124—31no—
46#497—31no—
47#497D311yescharacteristic, complemented
48#496—31no—
49#496—31no—
50#495—32no—
51#495—32no—
52C4×M16—31no—
53C23×D3—21no—
54C2×C3×Q8C2312yes—
55C2×C3×Q8C2311yescharacteristic
56C2×C3×D4C2311yescharacteristic
57C2×#67—21no—
58#239—24no—
59#239—24no—
60D3×Q8—24no—
61D3×D4—24no—
62C2×D12—22no—
63C2×D12—22no—
64C2×D12C2312yes—
65C2×D12C2311yescharacteristic
66C2×C4×D3—21no—
67C2×C4×D3—22no—
68C2×C4×D3D412yes—
69C2×C4×D3C2311yescharacteristic
70C2×Dic6—21no—
71C2×Dic6C2311yescharacteristic
72C3×Q16—24no—
73C3×SD16—24no—
74C2×C3×C8C2312yes—
75C3×C4⋊C4—22no—
76C3×C22⋊C4—22no—
77C3×C42C2311yescharacteristic
78#217—22no—
79#216—24no—
80#215—24no—
81#215—24no—
82#213—24no—
83(C2×D3)⋊C4—22no—
84(C2×D3)⋊C4—22no—
85(C2×D3)⋊C4—24no—
86C4×Dic3D412yes—
87C4×Dic3C2311yescharacteristic
88C2×C3⋊C8C2312yes—
89D24—24no—
90#204—24no—
91ΓL(1,𝔽25)D418yescomplemented
92C2×#41—32no—
93C22×Q8—31no—
94C22×D4—31no—
95#137—68no—
96AGL(1,ℤ/8)—68no—
97C2×Q16—32no—
98C2×Q16C2×D312yescomplemented
99C2×SD16—32no—
100C2×SD16—32no—
101C2×SD16—32no—
102C2×SD16C2×D312yes—
103C2×D8—32no—
104C2×D8—32no—
105C2×M16—32no—
106C4⋊Q8C2×D311yescharacteristic, complemented
107#127—31no—
108#124—31no—
109#124—32no—
110#124—31no—
111#124—31no—
112#124—62no—
113#124C2×D311yescharacteristic
114#121—62no—
115C4×D4—62no—
116C2×C22⋊C4—62no—
117C2×C42—31no—
118#97—31no—
119#97C2×D311yescharacteristic, complemented
120C4×C8—32no—
121C23×C3—21no—
122C22×D3—21no—
123C22×D3—22no—
124C22×D3—44no—
125C22×D3—42no—
126C22×D3C2×D411yescharacteristic
127C3×Q8—22no—
128C3×Q8—22no—
129C3×Q8—22no—
130C3×D4—22no—
131C2×C3×C4—21no—
132C2×C3×C4—22no—
133C2×C3×C4C2×D412yes—
134C2×C3×C4C2411yescharacteristic
135#67—42no—
136C2×Dic3—21no—
137C2×Dic3C2×D412yes—
138C2×Dic3C2×D411yescharacteristic
139D12—22no—
140D12—22no—
141D12—22no—
142D12—44no—
143D12—44no—
144C4×D3—42no—
145C4×D3—42no—
146C4×D3—44no—
147C4×D3C2×D414yes—
148Dic6—22no—
149Dic6—42no—
150C3×C8C2×D414yes—
151C3⋊C8C2×D414yes—
152C24—61no—
153#41—64no—
154#41—64no—
155C2×Q8—31no—
156C2×Q8—61no—
157C2×Q8—64no—
158C2×Q8C22×D311yescharacteristic
159C2×Q8C22×D312yes—
160C2×D4—31no—
161C2×D4—32no—
162C2×D4—62no—
163C2×D4—64no—
164C2×D4—61no—
165C2×D4—62no—
166C2×D4C22×D311yescharacteristic
167C22×C4—31no—
168C22×C4—32no—
169C22×C4—62no—
170C22×C4—61no—
171Q16—24no—
172Q16—64no—
173SD16—24no—
174SD16—64no—
175SD16—64no—
176SD16—64no—
177D8—64no—
178D8—64no—
179M16—38no—
180C2×C8—32no—
181C2×C8C22×D312yes—
182C4⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—62no—
185C22⋊C4—64no—
186C22⋊C4—62no—
187C22⋊C4—62no—
188C42—32no—
189C42—31no—
190C42C22×D311yescharacteristic
191C22×C3—41no—
192C22×C3—42no—
193C22×C3C22×D411yescharacteristic
194C2×D3—22no—
195C2×D3—42no—
196C2×D3—41no—
197C2×D3—44no—
198C2×D3—42no—
199C2×D3—84no—
200C2×D3#12712yes—
201C3×C4—22no—
202C3×C4—41no—
203C3×C4—42no—
204C3×C4C22×D412yes—
205Dic3—22no—
206Dic3—41no—
207Dic3#12712yes—
208C23—21no—
209C23—31no—
210C23—61no—
211C23—62no—
212C23—122no—
213C23—124no—
214Q8—22no—
215Q8—22no—
216Q8—22no—
217Q8—62no—
218Q8—122no—
219D4—22no—
220D4—62no—
221D4—62no—
222D4—62no—
223D4—124no—
224D4—122no—
225D4—124no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—31no—
229C2×C4—34no—
230C2×C4—32no—
231C2×C4—61no—
232C2×C4—122no—
233C2×C4—124no—
234C2×C4—122no—
235C2×C4D3×D412yes—
236C2×C4C23×D311yescharacteristic
237C8—34no—
238C8D3×D414yes—
239C2×C3—81no—
240C2×C3C2×#12711yescharacteristic
241C2×C3#57512yes—
242D3—22no—
243D3—82no—
244D3—81no—
245C22—32no—
246C22—41no—
247C22—42no—
248C22—62no—
249C22—122no—
250C22—121no—
251C22—124no—
252C22—122no—
253C22—244no—
254C22C2×D3×D411yescharacteristic
255C4—22no—
256C4—32no—
257C4—42no—
258C4—41no—
259C4—62no—
260C4—121no—
261C4C2×D3×D412yes—
262C3#315311yescharacteristic, complemented, Sylow
263C2—62no—
264C2—81no—
265C2—241no—
266C2—242no—
267C2D3×#12711yescharacteristic
268C2#577712yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#497C2
socleC22×C3C22×D4
hypercenter#497D3
hypocenterC3#3153
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#918C2semidirect
2. C2×#914C2semidirect
3. #5725C2semidirect
4. D3×#124C2semidirect
5. C3×#497C2semidirect
6. #5191C2semidirect
7. #5168C2semidirect
8. #5086C2semidirect
9. #5058C2semidirect
10. #4712C2semidirect
11. #4707C2semidirect
16. C2×C3×Q16C22semidirect
18. C3×C4⋊Q8C22semidirect
21. #947C22semidirect
22. C2×#216C22semidirect
24. C2×#215C22semidirect
30. C2×#204C22semidirect
37. #874C22semidirect
39. C3×#97C22semidirect
40. C8×Dic3C22semidirect
41. #803C22semidirect
47. #497D3semidirect
91. ΓL(1,𝔽25)D4semidirect
98. C2×Q16C2×D3semidirect
106. C4⋊Q8C2×D3semidirect
119. #97C2×D3semidirect
262. C3#3153semidirect

Maximal subgroups

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

subgroupindex
C2×#9182
C2×#9142
#57252
D3×#1242
C3×#4972
#51912
#51682
#50862
#50582
#47122
#47072
#31533