Finite Group Databasebeta

← #80946

Subgroups — #80946

Order 384.

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

normal subgroups only

This group has 454 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#5778C212yesmaximal, complemented
2#5766C211yesmaximal, characteristic, complemented
3#5752C211yesmaximal, characteristic, complemented
4C3×#473C211yesmaximal, characteristic, complemented
5#5202C212yesmaximal, complemented
6#5191C211yesmaximal, characteristic, complemented
7#5144C211yesmaximal, characteristic, complemented
8#4897C211yesmaximal, characteristic, complemented
9#4895C211yesmaximal, characteristic, complemented
10#4754C212yesmaximal, complemented
11#4753C212yesmaximal, complemented
12#3290—31nomaximal, Sylow
13#1009C2212yes—
14#1007C2212yescomplemented
15C3×#137C2212yes—
16C2×C3×Q16C2211yescharacteristic, complemented
17C3×#131C2211yescharacteristic
18#950C2211yescharacteristic
19#947C2212yescomplemented
20#942C2212yescomplemented
21C2×#215C2212yescomplemented
22#919—22no—
23#919—21no—
24#918—22no—
25#918—22no—
26D3×Q16—21no—
27#914—22no—
28D3×SD16—22no—
29#908—22no—
30#908C2212yes—
31#908C2212yescomplemented
32#907C2212yescomplemented
33#904—21no—
34#904C2211yescharacteristic
35C2×D24—21no—
36C2×D24C2211yescharacteristic
37#901C2211yescharacteristic, complemented
38C3×#108C2211yescharacteristic, complemented
39C3×#101C2212yescomplemented
40#837C2211yescharacteristic
41#825C2211yescharacteristic, complemented
42#825C2212yescomplemented
43#823C2212yescomplemented
44#822C2211yescharacteristic, complemented
45C8×Dic3C2211yescharacteristic, complemented
46#578—31no—
47#577—32no—
48#576—31no—
49#495—31no—
50#473D311yescharacteristic, complemented
51#472—31no—
52#471—32no—
53#455—32no—
54#454—32no—
55#443—31no—
56#443—31no—
57C3×#41C2311yescharacteristic
58C2×C3×Q8C2312yes—
59#239—22no—
60#239—22no—
61#239—22no—
62D3×Q8—22no—
63D3×Q8—22no—
64D3×D4—22no—
65D3×D4—42no—
66#235—22no—
67#235—42no—
68#235C2311yescharacteristic
69#235C2312yes—
70C2×D12—22no—
71C2×D12C2312yes—
72C3×Q16—22no—
73C3×Q16—21no—
74C3×Q16—21no—
75C3×SD16—22no—
76C3×M16—21no—
77C3×M16C2311yescharacteristic
78C3×M16C2312yes—
79C2×C3×C8—21no—
80C2×C3×C8C2311yescharacteristic
81#216—22no—
82#216D411yescharacteristic
83#215—22no—
84#215—22no—
85#215—22no—
86#215D411yescharacteristic
87#214D411yescharacteristic
88#213D411yescharacteristic
89(C2×D3)⋊C4—42no—
90C4×Dic3C2311yescharacteristic
91#208C2311yescharacteristic
92C2×C3⋊C8C2311yescharacteristic
93Dic12D411yescharacteristic
94D24—22no—
95D24—22no—
96D24—21no—
97D24—21no—
98D24D411yescharacteristic
99#204—22no—
100#204—22no—
101#204D412yes—
102ΓL(1,𝔽25)—21no—
103ΓL(1,𝔽25)—22no—
104D3×C8—21no—
105D3×C8—22no—
1062−1+4—32no—
107H(5,𝔽2)—32no—
108#137—32no—
109#137—62no—
110#137—62no—
111#137C2×D312yes—
112AGL(1,ℤ/8)—32no—
113AGL(1,ℤ/8)—32no—
114AGL(1,ℤ/8)—62no—
115AGL(1,ℤ/8)—62no—
116#135—31no—
117#135—31no—
118#135—61no—
119#135—61no—
120#135—62no—
121C2×Q16—61no—
122C2×Q16C2×D311yescharacteristic, complemented
123C2×SD16—32no—
124C2×SD16—62no—
125C2×D8—31no—
126C2×D8—61no—
127#131—32no—
128#131—31no—
129#131C2×D311yescharacteristic
130#124—32no—
131#108—31no—
132#108C2×D311yescharacteristic, complemented
133#104—31no—
134#104—31no—
135#104—32no—
136#101C2×D312yescomplemented
137#100—32no—
138C4×C8—31no—
139C22×D3—22no—
140C22×D3—42no—
141C3×Q8—22no—
142C3×Q8—22no—
143C3×Q8C2×D411yescharacteristic
144C3×D4—21no—
145C3×D4C2×D411yescharacteristic
146C2×C3×C4—21no—
147C2×C3×C4—22no—
148C2×C3×C4C2411yescharacteristic
149#67—21no—
150#67—22no—
151#67—42no—
152C2×Dic3—21no—
153D12—22no—
154D12—22no—
155D12—42no—
156D12—42no—
157D12C2×D412yes—
158D12C2×D411yescharacteristic
159C4×D3—22no—
160C4×D3—21no—
161C4×D3—42no—
162C4×D3—42no—
163C4×D3—42no—
164Dic6—42no—
165Dic6C2×D411yescharacteristic
166Dic6C2×D412yes—
167C3×C8—22no—
168C3×C8—21no—
169C3×C8C2×D411yescharacteristic
170C3×C8C2×D411yescharacteristic
171C3⋊C8—21no—
172C3⋊C8C2×D411yescharacteristic
173C3⋊C8C2×D411yescharacteristic
174#41—31no—
175#41—32no—
176#41—62no—
177#41—62no—
178#41—62no—
179#41—62no—
180#41—122no—
181#41C22×D311yescharacteristic
182C2×Q8—62no—
183C2×Q8—62no—
184C2×Q8C22×D312yes—
185C2×D4—32no—
186C2×D4—62no—
187C2×D4—62no—
188C2×D4—122no—
189Q16—22no—
190Q16—21no—
191Q16—21no—
192Q16—31no—
193Q16—31no—
194Q16—62no—
195SD16—22no—
196SD16—31no—
197SD16—31no—
198SD16—32no—
199SD16—62no—
200SD16—62no—
201SD16—62no—
202SD16—62no—
203SD16—62no—
204D8—31no—
205D8—31no—
206D8—62no—
207D8—62no—
208D8—61no—
209D8—61no—
210M16—21no—
211M16—31no—
212M16—62no—
213M16—61no—
214M16C22×D312yes—
215M16C22×D311yescharacteristic
216C2×C8—21no—
217C2×C8—31no—
218C2×C8—62no—
219C2×C8—61no—
220C2×C8C22×D311yescharacteristic
221C22⋊C4—122no—
222C42—31no—
223C22×C3—21no—
224C22×C3H(5,𝔽2)11yescharacteristic
225C2×D3—21no—
226C2×D3—22no—
227C2×D3—42no—
228C2×D3—82no—
229C2×D3—162no—
230C3×C4—21no—
231C3×C4—42no—
232C3×C4C22×D411yescharacteristic
233C3×C4C22×D411yescharacteristic
234Dic3—21no—
235Dic3—22no—
236Dic3—41no—
237C23—62no—
238C23—122no—
239Q8—22no—
240Q8—22no—
241Q8—31no—
242Q8—32no—
243Q8—122no—
244Q8D3×D411yescharacteristic
245D4—21no—
246D4—31no—
247D4—32no—
248D4—62no—
249D4—61no—
250D4—62no—
251D4—62no—
252D4—122no—
253D4—122no—
254D4—122no—
255D4D3×D411yescharacteristic
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—62no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4C23×D311yescharacteristic
265C8—21no—
266C8—22no—
267C8—31no—
268C8—31no—
269C8—61no—
270C8D3×D411yescharacteristic
271C8D3×D411yescharacteristic
272C2×C3—21no—
273C2×C3—41no—
274C2×C3D4211yescharacteristic
275D3—41no—
276D3—42no—
277D3—82no—
278C22—21no—
279C22—61no—
280C22—62no—
281C22—122no—
282C22—242no—
283C22—482no—
284C22#100911yescharacteristic
285C4—21no—
286C4—42no—
287C4—62no—
288C4—61no—
289C4—121no—
290C4C2×D3×D411yescharacteristic
291C4C2×D3×D411yescharacteristic
292C3#329011yescharacteristic, complemented, Sylow
293C2—21no—
294C2—41no—
295C2—122no—
296C2—121no—
297C2—242no—
298C2#560911yescharacteristic
subgroupG/H
centerC2#5609
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#473C2
socleC2×C3D42
hypercenter#473D3
hypocenterC3#3290
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #5778C2semidirect
2. #5766C2semidirect
3. #5752C2semidirect
4. C3×#473C2semidirect
5. #5202C2semidirect
6. #5191C2semidirect
7. #5144C2semidirect
8. #4897C2semidirect
9. #4895C2semidirect
10. #4754C2semidirect
11. #4753C2semidirect
14. #1007C22semidirect
16. C2×C3×Q16C22semidirect
19. #947C22semidirect
20. #942C22semidirect
21. C2×#215C22semidirect
31. #908C22semidirect
32. #907C22semidirect
37. #901C22semidirect
38. C3×#108C22semidirect
39. C3×#101C22semidirect
41. #825C22semidirect
42. #825C22semidirect
43. #823C22semidirect
44. #822C22semidirect
45. C8×Dic3C22semidirect
50. #473D3semidirect
122. C2×Q16C2×D3semidirect
132. #108C2×D3semidirect
136. #101C2×D3semidirect
292. C3#3290semidirect

Maximal subgroups

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

subgroupindex
#57782
#57662
#57522
C3×#4732
#52022
#51912
#51442
#48972
#48952
#47542
#47532
#32903