Finite Group Databasebeta

← #80940

Subgroups — #80940

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#5775C212yesmaximal, complemented
2#5757C211yesmaximal, characteristic, complemented
3#5753C211yesmaximal, characteristic, complemented
4C3×#472C211yesmaximal, characteristic, complemented
5#5201C212yesmaximal, complemented
6#5155C211yesmaximal, characteristic, complemented
7#5144C211yesmaximal, characteristic, complemented
8#4896C211yesmaximal, characteristic, complemented
9#4895C211yesmaximal, characteristic, complemented
10#4748C212yesmaximal, complemented
11#4746C212yesmaximal, complemented
12#3290—31nomaximal, Sylow
13#1010C2212yescomplemented
14#1004C2212yes—
15C3×AGL(1,ℤ/8)C2212yescomplemented
16C2×C3×D8C2211yescharacteristic
17C3×#131C2211yescharacteristic, complemented
18#950C2211yescharacteristic
19#936C2212yescomplemented
20C2×#214C2212yescomplemented
21#932C2212yes—
22#915—22no—
23D3×SD16—22no—
24#912—22no—
25#912—21no—
26#911—22no—
27#911—22no—
28D3×D8—21no—
29#909—22no—
30#909C2212yes—
31#909C2212yescomplemented
32#907C2212yescomplemented
33C2×Dic12—21no—
34C2×Dic12C2211yescharacteristic, complemented
35#904—21no—
36#904C2211yescharacteristic
37#901C2211yescharacteristic
38C3×#108C2211yescharacteristic, complemented
39C3×#100C2212yescomplemented
40#837C2211yescharacteristic, complemented
41#825C2211yescharacteristic
42#825C2212yescomplemented
43#824C2212yescomplemented
44#822C2211yescharacteristic, complemented
45C8×Dic3C2211yescharacteristic, complemented
46#578—31no—
47#577—32no—
48#576—31no—
49#495—31no—
50#473—31no—
51#472D311yescharacteristic, complemented
52#471—32no—
53#455—32no—
54#454—32no—
55#443—31no—
56#443—31no—
57C3×#41C2311yescharacteristic
58C2×C3×D4C2312yes—
59C2×#67—42no—
60D3×Q8—22no—
61#237—22no—
62#237—22no—
63#237—22no—
64#237—42no—
65D3×D4—22no—
66D3×D4—22no—
67#235—22no—
68#235C2311yescharacteristic
69#235C2312yes—
70C2×Dic6—22no—
71C2×Dic6C2312yes—
72C3×SD16—22no—
73C3×D8—22no—
74C3×D8—21no—
75C3×D8—21no—
76C3×M16—21no—
77C3×M16C2311yescharacteristic
78C3×M16C2312yes—
79C2×C3×C8—21no—
80C2×C3×C8C2311yescharacteristic
81#217—42no—
82#216D411yescharacteristic
83#215D411yescharacteristic
84#214—22no—
85#214—22no—
86#214—22no—
87#214D411yescharacteristic
88#213—22no—
89#213D411yescharacteristic
90C4×Dic3C2311yescharacteristic
91#208C2311yescharacteristic
92C2×C3⋊C8C2311yescharacteristic
93Dic12—22no—
94Dic12—22no—
95Dic12—21no—
96Dic12—21no—
97Dic12D411yescharacteristic
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—32no—
110#137—62no—
111#137—62no—
112AGL(1,ℤ/8)—32no—
113AGL(1,ℤ/8)—62no—
114AGL(1,ℤ/8)—62no—
115AGL(1,ℤ/8)C2×D312yescomplemented
116#135—31no—
117#135—31no—
118#135—61no—
119#135—62no—
120#135—61no—
121C2×Q16—31no—
122C2×Q16—61no—
123C2×SD16—32no—
124C2×SD16—62no—
125C2×D8—61no—
126C2×D8C2×D311yescharacteristic
127#131—32no—
128#131—31no—
129#131C2×D311yescharacteristic, complemented
130#124—32no—
131#108—31no—
132#108C2×D311yescharacteristic, complemented
133#104—31no—
134#104—31no—
135#104—32no—
136#101—32no—
137#100C2×D312yescomplemented
138C4×C8—31no—
139C23×C3—22no—
140C22×D3—42no—
141C3×Q8C2×D411yescharacteristic
142C3×D4—21no—
143C3×D4—22no—
144C3×D4—22no—
145C3×D4C2×D411yescharacteristic
146C2×C3×C4—21no—
147C2×C3×C4C2411yescharacteristic
148#67—22no—
149#67—21no—
150#67—42no—
151#67—42no—
152#67—42no—
153C2×Dic3—21no—
154C2×Dic3—22no—
155C2×Dic3—42no—
156C2×Dic3—42no—
157D12C2×D411yescharacteristic
158D12C2×D412yes—
159C4×D3—21no—
160C4×D3—22no—
161C4×D3—42no—
162Dic6—22no—
163Dic6—22no—
164Dic6—42no—
165Dic6C2×D412yes—
166Dic6C2×D411yescharacteristic
167C3×C8—22no—
168C3×C8—21no—
169C3×C8C2×D411yescharacteristic
170C3×C8C2×D411yescharacteristic
171C3⋊C8—21no—
172C3⋊C8C2×D411yescharacteristic
173C3⋊C8C2×D411yescharacteristic
174#41—32no—
175#41—31no—
176#41—62no—
177#41—62no—
178#41—62no—
179#41—62no—
180#41—122no—
181#41C22×D311yescharacteristic
182C2×Q8—32no—
183C2×Q8—62no—
184C2×Q8—62no—
185C2×D4—62no—
186C2×D4—62no—
187C2×D4—122no—
188C2×D4C22×D312yes—
189Q16—31no—
190Q16—31no—
191Q16—62no—
192Q16—62no—
193Q16—61no—
194Q16—61no—
195SD16—22no—
196SD16—31no—
197SD16—31no—
198SD16—32no—
199SD16—62no—
200SD16—62no—
201SD16—62no—
202SD16—62no—
203SD16—62no—
204D8—22no—
205D8—21no—
206D8—21no—
207D8—31no—
208D8—31no—
209D8—62no—
210M16—21no—
211M16—31no—
212M16—62no—
213M16—61no—
214M16C22×D311yescharacteristic
215M16C22×D312yes—
216C2×C8—21no—
217C2×C8—31no—
218C2×C8—62no—
219C2×C8—61no—
220C2×C8C22×D311yescharacteristic
221C22⋊C4—122no—
222C42—31no—
223C22×C3—21no—
224C22×C3—42no—
225C22×C3—82no—
226C22×C3H(5,𝔽2)11yescharacteristic
227C2×D3—22no—
228C2×D3—21no—
229C2×D3—162no—
230C3×C4—21no—
231C3×C4C22×D411yescharacteristic
232C3×C4C22×D411yescharacteristic
233Dic3—22no—
234Dic3—21no—
235Dic3—41no—
236Dic3—42no—
237C23—22no—
238C23—122no—
239Q8—32no—
240Q8—31no—
241Q8—62no—
242Q8—62no—
243Q8—122no—
244Q8D3×D411yescharacteristic
245D4—21no—
246D4—22no—
247D4—22no—
248D4—32no—
249D4—31no—
250D4—61no—
251D4—62no—
252D4—122no—
253D4—122no—
254D4—122no—
255D4D3×D411yescharacteristic
256C2×C4—21no—
257C2×C4—61no—
258C2×C4—62no—
259C2×C4—61no—
260C2×C4—62no—
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×C3—82no—
275C2×C3D4211yescharacteristic
276D3—42no—
277D3—41no—
278C22—21no—
279C22—42no—
280C22—62no—
281C22—61no—
282C22—82no—
283C22—482no—
284C22#100911yescharacteristic
285C4—21no—
286C4—61no—
287C4—62no—
288C4—121no—
289C4—122no—
290C4C2×D3×D411yescharacteristic
291C4C2×D3×D411yescharacteristic
292C3#329011yescharacteristic, complemented, Sylow
293C2—21no—
294C2—41no—
295C2—82no—
296C2—121no—
297C2—122no—
298C2#560911yescharacteristic
subgroupG/H
centerC2#5609
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#472C2
socleC2×C3D42
hypercenter#472D3
hypocenterC3#3290
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #5775C2semidirect
2. #5757C2semidirect
3. #5753C2semidirect
4. C3×#472C2semidirect
5. #5201C2semidirect
6. #5155C2semidirect
7. #5144C2semidirect
8. #4896C2semidirect
9. #4895C2semidirect
10. #4748C2semidirect
11. #4746C2semidirect
13. #1010C22semidirect
15. C3×AGL(1,ℤ/8)C22semidirect
17. C3×#131C22semidirect
19. #936C22semidirect
20. C2×#214C22semidirect
31. #909C22semidirect
32. #907C22semidirect
34. C2×Dic12C22semidirect
38. C3×#108C22semidirect
39. C3×#100C22semidirect
40. #837C22semidirect
42. #825C22semidirect
43. #824C22semidirect
44. #822C22semidirect
45. C8×Dic3C22semidirect
51. #472D3semidirect
115. AGL(1,ℤ/8)C2×D3semidirect
129. #131C2×D3semidirect
132. #108C2×D3semidirect
137. #100C2×D3semidirect
292. C3#3290semidirect

Maximal subgroups

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

subgroupindex
#57752
#57572
#57532
C3×#4722
#52012
#51552
#51442
#48962
#48952
#47482
#47462
#32903