Finite Group Databasebeta

← #80282

Subgroups — #80282

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#5835C212yesmaximal, complemented
2#5774C212yesmaximal, complemented
3C3×#453C211yesmaximal, characteristic, complemented
4#5199C212yesmaximal, complemented
5#5077C211yesmaximal, characteristic, complemented
6#5073C211yesmaximal, characteristic, complemented
7#4824C212yesmaximal, complemented
8#4822C212yesmaximal, complemented
9#4749C212yesmaximal, complemented
10#3292—31nomaximal, Sylow
11C3×H(5,𝔽2)C2211yescharacteristic, complemented
12#1009C2212yescomplemented
13#1004C2211yescharacteristic, complemented
14C3×AGL(1,ℤ/8)C2212yescomplemented
15C3×#127C2211yescharacteristic, complemented
16#950—22no—
17#950C2212yes—
18#949—22no—
19#949C2212yescomplemented
20#948C2212yescomplemented
21#932—22no—
22#932C2212yescomplemented
23C2×#213—22no—
24C2×#213C2212yescomplemented
25C2×#213C2212yescomplemented
26#916—22no—
27#914—22no—
28#911—22no—
29D3×D8—22no—
30#908C2212yescomplemented
31#907C2212yescomplemented
32#874C2211yescharacteristic, complemented
33C3×#104C2212yescomplemented
34C3×#100C2211yescharacteristic, complemented
35#833C2211yescharacteristic, complemented
36#823C2212yescomplemented
37#822C2212yescomplemented
38#805C2212yescomplemented
39C4×C3⋊C8C2211yescharacteristic, complemented
40#576—32no—
41#576—32no—
42#493—31no—
43#472—32no—
44#472—32no—
45#453—31no—
46#453—32no—
47#453D311yescharacteristic, complemented
48#443—32no—
49C3×#41—22no—
50C3×#41C2312yes—
51C2×C3×D4—22no—
52C2×C3×D4—41no—
53C2×C3×D4—41no—
54C2×C3×D4C2311yescharacteristic
55C2×C3×D4C2311yescharacteristic
56C2×#67—41no—
57#239—22no—
58#237—22no—
59D3×D4—22no—
60D3×D4—22no—
61D3×D4—42no—
62#235—22no—
63#235C2312yes—
64C2×D12—22no—
65C2×D12—41no—
66C2×D12C2312yes—
67C3×SD16—22no—
68C3×D8—22no—
69C3×M16C2312yes—
70C3×C42C2311yescharacteristic
71#216D412yescomplemented
72#215—22no—
73#215D412yescomplemented
74#214—22no—
75#214D412yescomplemented
76#213—22no—
77#213—22no—
78#213—22no—
79#213—22no—
80#213—22no—
81#213—22no—
82#213D412yescomplemented
83#208—22no—
84#208C2312yes—
85C2×C3⋊C8—22no—
86C2×C3⋊C8C2312yes—
87D24—22no—
88#204—22no—
89ΓL(1,𝔽25)—22no—
90D3×C8—22no—
91H(5,𝔽2)—32no—
92H(5,𝔽2)—31no—
93H(5,𝔽2)C2×D311yescharacteristic, complemented
94AGL(1,ℤ/8)—32no—
95AGL(1,ℤ/8)—32no—
96AGL(1,ℤ/8)—32no—
97AGL(1,ℤ/8)—62no—
98AGL(1,ℤ/8)—62no—
99AGL(1,ℤ/8)—62no—
100AGL(1,ℤ/8)—62no—
101AGL(1,ℤ/8)C2×D312yescomplemented
102#135—32no—
103#135—62no—
104#135—62no—
105C2×D8—32no—
106C2×D8—32no—
107C2×D8—62no—
108C2×D8—62no—
109#131—32no—
110#131—32no—
111#127—31no—
112#127C2×D311yescharacteristic, complemented
113#108—32no—
114#104—32no—
115#104C2×D312yescomplemented
116#100—31no—
117#100—32no—
118#100C2×D311yescharacteristic, complemented
119C4×C8—31no—
120C23×C3—21no—
121C23×C3—21no—
122C23×C3—41no—
123C22×D3—22no—
124C22×D3—41no—
125C22×D3—42no—
126C3×Q8C2×D412yes—
127C3×D4—22no—
128C3×D4—22no—
129C3×D4—22no—
130C3×D4—42no—
131C3×D4—82no—
132C3×D4C2×D412yes—
133C2×C3×C4—21no—
134C2×C3×C4—22no—
135C2×C3×C4—41no—
136C2×C3×C4C2411yescharacteristic
137#67—22no—
138#67—42no—
139#67—42no—
140C2×Dic3—41no—
141D12—22no—
142D12—22no—
143D12—42no—
144D12—82no—
145D12C2×D412yes—
146C4×D3—22no—
147C4×D3—42no—
148Dic6C2×D412yes—
149C3×C8—22no—
150C3⋊C8—22no—
151C3⋊C8C2×D412yes—
152C3⋊C8C2×D412yes—
153#41—22no—
154#41—32no—
155#41—62no—
156#41—62no—
157#41—62no—
158#41C22×D312yes—
159C2×D4—22no—
160C2×D4—32no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—62no—
164C2×D4—62no—
165C2×D4—62no—
166C2×D4—121no—
167C2×D4—122no—
168C2×D4—121no—
169C2×D4C22×D311yescharacteristic
170C2×D4C22×D311yescharacteristic
171Q16—32no—
172SD16—22no—
173SD16—32no—
174SD16—32no—
175SD16—62no—
176SD16—62no—
177SD16—62no—
178D8—22no—
179D8—32no—
180D8—62no—
181D8—62no—
182D8—62no—
183D8—62no—
184D8—62no—
185D8—62no—
186D8—62no—
187M16—32no—
188M16—62no—
189M16—62no—
190M16C22×D312yes—
191C2×C8—32no—
192C2×C8—62no—
193C2×C8—62no—
194C42C22×D311yescharacteristic
195C22×C3—22no—
196C22×C3—41no—
197C22×C3—41no—
198C22×C3—81no—
199C22×C3—81no—
200C22×C3—161no—
201C22×C3H(5,𝔽2)11yescharacteristic
202C2×D3—22no—
203C2×D3—42no—
204C2×D3—82no—
205C2×D3—161no—
206C2×D3—162no—
207C3×C4—22no—
208C3×C4—22no—
209C3×C4C22×D412yes—
210Dic3—22no—
211C23—21no—
212C23—21no—
213C23—41no—
214C23—62no—
215C23—122no—
216C23—121no—
217Q8—32no—
218Q8D3×D412yes—
219D4—22no—
220D4—22no—
221D4—22no—
222D4—32no—
223D4—42no—
224D4—62no—
225D4—62no—
226D4—62no—
227D4—82no—
228D4—122no—
229D4—122no—
230D4—122no—
231D4—242no—
232D4D3×D412yes—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—62no—
237C2×C4—122no—
238C2×C4—121no—
239C2×C4C23×D311yescharacteristic
240C8—22no—
241C8—32no—
242C8—32no—
243C8—62no—
244C2×C3—21no—
245C2×C3—42no—
246C2×C3—81no—
247C2×C3—81no—
248C2×C3D4211yescharacteristic
249D3—42no—
250D3—82no—
251C22—22no—
252C22—41no—
253C22—41no—
254C22—62no—
255C22—81no—
256C22—81no—
257C22—122no—
258C22—161no—
259C22—242no—
260C22—482no—
261C22—481no—
262C22#100411yescharacteristic
263C4—22no—
264C4—22no—
265C4—62no—
266C4C2×D3×D412yes—
267C3#329211yescharacteristic, complemented, Sylow
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—122no—
273C2—242no—
274C2#559011yescharacteristic
subgroupG/H
centerC2#5590
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#453C2
socleC2×C3D42
hypercenter#453D3
hypocenterC3#3292
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #5835C2semidirect
2. #5774C2semidirect
3. C3×#453C2semidirect
4. #5199C2semidirect
5. #5077C2semidirect
6. #5073C2semidirect
7. #4824C2semidirect
8. #4822C2semidirect
9. #4749C2semidirect
11. C3×H(5,𝔽2)C22semidirect
12. #1009C22semidirect
13. #1004C22semidirect
14. C3×AGL(1,ℤ/8)C22semidirect
15. C3×#127C22semidirect
19. #949C22semidirect
20. #948C22semidirect
22. #932C22semidirect
24. C2×#213C22semidirect
25. C2×#213C22semidirect
30. #908C22semidirect
31. #907C22semidirect
32. #874C22semidirect
33. C3×#104C22semidirect
34. C3×#100C22semidirect
35. #833C22semidirect
36. #823C22semidirect
37. #822C22semidirect
38. #805C22semidirect
39. C4×C3⋊C8C22semidirect
47. #453D3semidirect
71. #216D4semidirect
73. #215D4semidirect
75. #214D4semidirect
82. #213D4semidirect
93. H(5,𝔽2)C2×D3semidirect
101. AGL(1,ℤ/8)C2×D3semidirect
112. #127C2×D3semidirect
115. #104C2×D3semidirect
118. #100C2×D3semidirect
267. C3#3292semidirect

Maximal subgroups

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

subgroupindex
#58352
#57742
C3×#4532
#51992
#50772
#50732
#48242
#48222
#47492
#32923