Finite Group Databasebeta

← #80941

Subgroups — #80941

Order 384.

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

normal subgroups only

This group has 476 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#5774C212yesmaximal, complemented
2#5757C211yesmaximal, characteristic, complemented
3#5752C211yesmaximal, characteristic, complemented
4C3×#472C211yesmaximal, characteristic, complemented
5#5198C212yesmaximal, complemented
6#5151C211yesmaximal, characteristic, complemented
7#5144C211yesmaximal, characteristic, complemented
8#4897C211yesmaximal, characteristic, complemented
9#4895C211yesmaximal, characteristic, complemented
10#4749C212yesmaximal, complemented
11#4747C212yesmaximal, complemented
12#3292—31nomaximal, Sylow
13#1009C2212yescomplemented
14#1004C2212yescomplemented
15C3×AGL(1,ℤ/8)C2212yescomplemented
16C2×C3×D8C2211yescharacteristic, complemented
17C3×#131C2211yescharacteristic, complemented
18#950C2211yescharacteristic
19#940C2212yescomplemented
20#932C2212yescomplemented
21C2×#213C2212yescomplemented
22#916—22no—
23#914—22no—
24#912—21no—
25#911—22no—
26#911—22no—
27D3×D8—22no—
28D3×D8—21no—
29#908—22no—
30#908C2212yescomplemented
31#908C2212yescomplemented
32#907C2212yescomplemented
33#904—21no—
34#904C2211yescharacteristic
35C2×D24—21no—
36C2×D24C2211yescharacteristic, complemented
37#901C2211yescharacteristic, complemented
38C3×#108C2211yescharacteristic, complemented
39C3×#100C2212yescomplemented
40#837C2211yescharacteristic, complemented
41#825C2211yescharacteristic, complemented
42#825C2212yescomplemented
43#823C2212yescomplemented
44#822C2211yescharacteristic, complemented
45C8×Dic3C2211yescharacteristic, complemented
46#576—32no—
47#576—31no—
48#576—31no—
49#493—31no—
50#472—31no—
51#472—32no—
52#472D311yescharacteristic, complemented
53#453—32no—
54#453—32no—
55#443—31no—
56#443—31no—
57C3×#41C2311yescharacteristic
58C2×C3×D4C2312yes—
59C2×#67—42no—
60C2×#67—42no—
61#239—22no—
62#237—22no—
63#237—22no—
64D3×D4—22no—
65D3×D4—22no—
66D3×D4—22no—
67D3×D4—42no—
68#235—22no—
69#235C2311yescharacteristic
70#235C2312yes—
71C2×D12—22no—
72C2×D12C2312yes—
73C3×SD16—22no—
74C3×D8—22no—
75C3×D8—21no—
76C3×D8—21no—
77C3×M16—21no—
78C3×M16C2311yescharacteristic
79C3×M16C2312yes—
80C2×C3×C8—21no—
81C2×C3×C8C2311yescharacteristic
82#216D411yescharacteristic, complemented
83#215D411yescharacteristic, complemented
84#214—22no—
85#214D411yescharacteristic, complemented
86#213—22no—
87#213—22no—
88#213—22no—
89#213D411yescharacteristic, complemented
90C4×Dic3C2311yescharacteristic
91#208C2311yescharacteristic
92C2×C3⋊C8C2311yescharacteristic
93Dic12D411yescharacteristic, complemented
94D24—22no—
95D24—22no—
96D24—21no—
97D24—21no—
98D24D411yescharacteristic, complemented
99#204—22no—
100#204—22no—
101#204D412yescomplemented
102ΓL(1,𝔽25)—21no—
103ΓL(1,𝔽25)—22no—
104D3×C8—21no—
105D3×C8—22no—
106H(5,𝔽2)—32no—
107H(5,𝔽2)—32no—
108AGL(1,ℤ/8)—32no—
109AGL(1,ℤ/8)—32no—
110AGL(1,ℤ/8)—32no—
111AGL(1,ℤ/8)—62no—
112AGL(1,ℤ/8)—62no—
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—61no—
120#135—62no—
121C2×D8—32no—
122C2×D8—31no—
123C2×D8—61no—
124C2×D8—62no—
125C2×D8—61no—
126C2×D8C2×D311yescharacteristic, complemented
127#131—32no—
128#131—31no—
129#131C2×D311yescharacteristic, complemented
130#127—32no—
131#108—31no—
132#108C2×D311yescharacteristic, complemented
133#104—31no—
134#104—32no—
135#104—31no—
136#100—32no—
137#100C2×D312yescomplemented
138C4×C8—31no—
139C23×C3—22no—
140C22×D3—22no—
141C22×D3—42no—
142C22×D3—42no—
143C3×Q8C2×D411yescharacteristic
144C3×D4—21no—
145C3×D4—22no—
146C3×D4—22no—
147C3×D4C2×D411yescharacteristic
148C2×C3×C4—21no—
149C2×C3×C4C2411yescharacteristic
150#67—22no—
151#67—21no—
152#67—42no—
153#67—42no—
154#67—42no—
155#67—84no—
156C2×Dic3—21no—
157C2×Dic3—42no—
158D12—22no—
159D12—22no—
160D12—42no—
161D12C2×D412yes—
162D12C2×D411yescharacteristic
163C4×D3—21no—
164C4×D3—22no—
165C4×D3—42no—
166Dic6C2×D411yescharacteristic
167Dic6C2×D412yes—
168C3×C8—22no—
169C3×C8—21no—
170C3×C8C2×D411yescharacteristic
171C3×C8C2×D411yescharacteristic
172C3⋊C8—21no—
173C3⋊C8C2×D411yescharacteristic
174C3⋊C8C2×D411yescharacteristic
175#41—32no—
176#41—31no—
177#41—62no—
178#41—62no—
179#41—62no—
180#41—62no—
181#41C22×D311yescharacteristic
182C2×D4—32no—
183C2×D4—62no—
184C2×D4—62no—
185C2×D4—62no—
186C2×D4—62no—
187C2×D4—122no—
188C2×D4—122no—
189C2×D4—122no—
190C2×D4C22×D312yes—
191Q16—31no—
192Q16—31no—
193SD16—22no—
194SD16—31no—
195SD16—31no—
196SD16—32no—
197SD16—62no—
198SD16—62no—
199SD16—62no—
200D8—22no—
201D8—21no—
202D8—21no—
203D8—31no—
204D8—31no—
205D8—62no—
206D8—62no—
207D8—62no—
208D8—62no—
209D8—62no—
210D8—61no—
211D8—61no—
212M16—21no—
213M16—31no—
214M16—62no—
215M16—61no—
216M16C22×D312yes—
217M16C22×D311yescharacteristic
218C2×C8—21no—
219C2×C8—31no—
220C2×C8—62no—
221C2×C8—61no—
222C2×C8C22×D311yescharacteristic
223C42—31no—
224C22×C3—21no—
225C22×C3—42no—
226C22×C3—82no—
227C22×C3H(5,𝔽2)11yescharacteristic
228C2×D3—22no—
229C2×D3—21no—
230C2×D3—42no—
231C2×D3—82no—
232C2×D3—162no—
233C2×D3—162no—
234C3×C4—21no—
235C3×C4C22×D411yescharacteristic
236C3×C4C22×D411yescharacteristic
237Dic3—22no—
238Dic3—22no—
239Dic3—21no—
240C23—22no—
241C23—62no—
242C23—122no—
243C23—122no—
244Q8—31no—
245Q8—32no—
246Q8D3×D411yescharacteristic
247D4—21no—
248D4—22no—
249D4—22no—
250D4—31no—
251D4—32no—
252D4—62no—
253D4—61no—
254D4—62no—
255D4—62no—
256D4—122no—
257D4—122no—
258D4—122no—
259D4—122no—
260D4—244no—
261D4D3×D411yescharacteristic
262C2×C4—21no—
263C2×C4—61no—
264C2×C4—62no—
265C2×C4—61no—
266C2×C4—122no—
267C2×C4—122no—
268C2×C4C23×D311yescharacteristic
269C8—21no—
270C8—22no—
271C8—31no—
272C8—31no—
273C8—61no—
274C8D3×D411yescharacteristic
275C8D3×D411yescharacteristic
276C2×C3—21no—
277C2×C3—41no—
278C2×C3—82no—
279C2×C3D4211yescharacteristic
280D3—42no—
281D3—41no—
282D3—82no—
283C22—21no—
284C22—42no—
285C22—62no—
286C22—61no—
287C22—82no—
288C22—122no—
289C22—242no—
290C22—482no—
291C22—482no—
292C22#100911yescharacteristic
293C4—21no—
294C4—62no—
295C4—62no—
296C4—61no—
297C4C2×D3×D411yescharacteristic
298C4C2×D3×D411yescharacteristic
299C3#329211yescharacteristic, complemented, Sylow
300C2—21no—
301C2—41no—
302C2—82no—
303C2—122no—
304C2—121no—
305C2—242no—
306C2#560911yescharacteristic
subgroupG/H
centerC2#5609
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#472C2
socleC2×C3D42
hypercenter#472D3
hypocenterC3#3292
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. #5774C2semidirect
2. #5757C2semidirect
3. #5752C2semidirect
4. C3×#472C2semidirect
5. #5198C2semidirect
6. #5151C2semidirect
7. #5144C2semidirect
8. #4897C2semidirect
9. #4895C2semidirect
10. #4749C2semidirect
11. #4747C2semidirect
13. #1009C22semidirect
14. #1004C22semidirect
15. C3×AGL(1,ℤ/8)C22semidirect
16. C2×C3×D8C22semidirect
17. C3×#131C22semidirect
19. #940C22semidirect
20. #932C22semidirect
21. C2×#213C22semidirect
30. #908C22semidirect
31. #908C22semidirect
32. #907C22semidirect
36. C2×D24C22semidirect
37. #901C22semidirect
38. C3×#108C22semidirect
39. C3×#100C22semidirect
40. #837C22semidirect
41. #825C22semidirect
42. #825C22semidirect
43. #823C22semidirect
44. #822C22semidirect
45. C8×Dic3C22semidirect
52. #472D3semidirect
82. #216D4semidirect
83. #215D4semidirect
85. #214D4semidirect
89. #213D4semidirect
93. Dic12D4semidirect
98. D24D4semidirect
101. #204D4semidirect
115. AGL(1,ℤ/8)C2×D3semidirect
126. C2×D8C2×D3semidirect
129. #131C2×D3semidirect
132. #108C2×D3semidirect
137. #100C2×D3semidirect
299. C3#3292semidirect

Maximal subgroups

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

subgroupindex
#57742
#57572
#57522
C3×#4722
#51982
#51512
#51442
#48972
#48952
#47492
#47472
#32923