Finite Group Databasebeta

← #80287

Subgroups — #80287

Order 384.

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

normal subgroups only

This group has 464 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#5837C212yesmaximal, complemented
2#5774C212yesmaximal, complemented
3C3×#454C211yesmaximal, characteristic, complemented
4#5199C212yesmaximal, complemented
5#5061C211yesmaximal, characteristic, complemented
6#5058C211yesmaximal, characteristic, complemented
7#4823C212yesmaximal, complemented
8#4822C212yesmaximal, complemented
9#4754C212yesmaximal, complemented
10#3288—31nomaximal, Sylow
11C3×2−1+4C2211yescharacteristic, complemented
12#1009C2212yescomplemented
13#1004C2211yescharacteristic
14C3×AGL(1,ℤ/8)C2212yescomplemented
15C3×#124C2211yescharacteristic, complemented
16#950—22no—
17#950C2212yes—
18#949—22no—
19#949C2212yescomplemented
20#948C2212yescomplemented
21#942—22no—
22#942C2212yescomplemented
23C2×#215—22no—
24C2×#215C2212yescomplemented
25C2×#213C2212yescomplemented
26#916—22no—
27#914—22no—
28#911—22no—
29D3×D8—22no—
30#908C2212yes—
31#907C2212yescomplemented
32#874C2211yescharacteristic, complemented
33C3×#104C2212yescomplemented
34C3×#101C2211yescharacteristic, complemented
35#833C2211yescharacteristic, complemented
36#823C2212yescomplemented
37#822C2212yescomplemented
38#805C2212yescomplemented
39#803C2211yescharacteristic, complemented
40#577—32no—
41#576—32no—
42#496—31no—
43#472—32no—
44#471—32no—
45#455—31no—
46#454D311yescharacteristic, complemented
47#453—32no—
48#444—32no—
49C3×#41—22no—
50C3×#41—41no—
51C3×#41C2312yes—
52C2×C3×Q8—22no—
53C2×C3×Q8C2311yescharacteristic
54C2×C3×D4C2311yescharacteristic
55C2×#67—41no—
56#239—22no—
57#237—22no—
58D3×D4—22no—
59D3×D4—22no—
60D3×D4—42no—
61#235—22no—
62#235C2312yes—
63C2×D12—22no—
64C2×D12—41no—
65C2×D12C2312yes—
66C3×SD16—22no—
67C3×D8—22no—
68C3×M16C2312yes—
69C3×C22⋊C4—41no—
70C3×C42C2311yescharacteristic
71#216—22no—
72#216D412yescomplemented
73#215—22no—
74#215—22no—
75#215—22no—
76#215—22no—
77#215D412yescomplemented
78#214D412yescomplemented
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—
912−1+4C2×D311yescharacteristic, complemented
92H(5,𝔽2)—32no—
93H(5,𝔽2)—31no—
94#137—32no—
95#137—62no—
96AGL(1,ℤ/8)—32no—
97AGL(1,ℤ/8)—32no—
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×SD16—32no—
106C2×SD16—62no—
107C2×D8—32no—
108C2×D8—62no—
109#131—32no—
110#131—32no—
111#127—31no—
112#124C2×D311yescharacteristic, complemented
113#108—32no—
114#104—32no—
115#104C2×D312yescomplemented
116#101C2×D311yescharacteristic, complemented
117#100—31no—
118#100—32no—
119#97—31no—
120C23×C3—21no—
121C22×D3—22no—
122C22×D3—41no—
123C22×D3—42no—
124C3×Q8—22no—
125C3×Q8—41no—
126C3×Q8C2×D412yes—
127C3×D4—22no—
128C3×D4—22no—
129C3×D4—41no—
130C3×D4C2×D412yes—
131C2×C3×C4—22no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—42no—
135C2×C3×C4C2411yescharacteristic
136#67—22no—
137#67—42no—
138#67—42no—
139C2×Dic3—41no—
140D12—22no—
141D12—22no—
142D12—42no—
143D12—82no—
144D12C2×D412yes—
145C4×D3—22no—
146C4×D3—42no—
147Dic6C2×D412yes—
148C3×C8—22no—
149C3⋊C8—22no—
150C3⋊C8C2×D412yes—
151C3⋊C8C2×D412yes—
152#41—22no—
153#41—32no—
154#41—41no—
155#41—62no—
156#41—62no—
157#41—62no—
158#41C22×D312yes—
159C2×Q8—22no—
160C2×Q8C22×D311yescharacteristic
161C2×D4—32no—
162C2×D4—62no—
163C2×D4—62no—
164C2×D4—62no—
165C2×D4—121no—
166C2×D4—122no—
167C2×D4—121no—
168C2×D4C22×D311yescharacteristic
169Q16—32no—
170Q16—62no—
171SD16—22no—
172SD16—32no—
173SD16—32no—
174SD16—62no—
175SD16—62no—
176SD16—62no—
177SD16—62no—
178SD16—62no—
179D8—22no—
180D8—32no—
181D8—62no—
182D8—62no—
183D8—62no—
184D8—62no—
185M16—32no—
186M16—62no—
187M16—62no—
188M16C22×D312yes—
189C2×C8—32no—
190C2×C8—62no—
191C2×C8—62no—
192C22⋊C4—41no—
193C42C22×D311yescharacteristic
194C22×C3—22no—
195C22×C3—41no—
196C22×C3—81no—
197C22×C3H(5,𝔽2)11yescharacteristic
198C2×D3—22no—
199C2×D3—42no—
200C2×D3—82no—
201C2×D3—161no—
202C2×D3—162no—
203C3×C4—22no—
204C3×C4—41no—
205C3×C4—41no—
206C3×C4C22×D412yes—
207Dic3—22no—
208C23—21no—
209C23—62no—
210C23—122no—
211C23—121no—
212Q8—22no—
213Q8—32no—
214Q8—41no—
215Q8D3×D412yes—
216D4—22no—
217D4—22no—
218D4—32no—
219D4—41no—
220D4—62no—
221D4—62no—
222D4—62no—
223D4—122no—
224D4—122no—
225D4—122no—
226D4—242no—
227D4D3×D412yes—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—42no—
232C2×C4—62no—
233C2×C4—122no—
234C2×C4—121no—
235C2×C4C23×D311yescharacteristic
236C8—22no—
237C8—32no—
238C8—32no—
239C8—62no—
240C2×C3—21no—
241C2×C3—42no—
242C2×C3—81no—
243C2×C3D4211yescharacteristic
244D3—42no—
245D3—82no—
246C22—22no—
247C22—41no—
248C22—62no—
249C22—81no—
250C22—122no—
251C22—242no—
252C22—482no—
253C22—481no—
254C22#100411yescharacteristic
255C4—22no—
256C4—41no—
257C4—41no—
258C4—62no—
259C4C2×D3×D412yes—
260C3#328811yescharacteristic, complemented, Sylow
261C2—21no—
262C2—42no—
263C2—81no—
264C2—122no—
265C2—242no—
266C2#559011yescharacteristic
subgroupG/H
centerC2#5590
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#454C2
socleC2×C3D42
hypercenter#454D3
hypocenterC3#3288
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #5837C2semidirect
2. #5774C2semidirect
3. C3×#454C2semidirect
4. #5199C2semidirect
5. #5061C2semidirect
6. #5058C2semidirect
7. #4823C2semidirect
8. #4822C2semidirect
9. #4754C2semidirect
11. C3×2−1+4C22semidirect
12. #1009C22semidirect
14. C3×AGL(1,ℤ/8)C22semidirect
15. C3×#124C22semidirect
19. #949C22semidirect
20. #948C22semidirect
22. #942C22semidirect
24. C2×#215C22semidirect
25. C2×#213C22semidirect
31. #907C22semidirect
32. #874C22semidirect
33. C3×#104C22semidirect
34. C3×#101C22semidirect
35. #833C22semidirect
36. #823C22semidirect
37. #822C22semidirect
38. #805C22semidirect
39. #803C22semidirect
46. #454D3semidirect
72. #216D4semidirect
77. #215D4semidirect
78. #214D4semidirect
82. #213D4semidirect
91. 2−1+4C2×D3semidirect
101. AGL(1,ℤ/8)C2×D3semidirect
112. #124C2×D3semidirect
115. #104C2×D3semidirect
116. #101C2×D3semidirect
260. C3#3288semidirect

Maximal subgroups

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

subgroupindex
#58372
#57742
C3×#4542
#51992
#50612
#50582
#48232
#48222
#47542
#32883