Finite Group Databasebeta

← #79894

Subgroups — #79894

Order 384.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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#5769C212yesmaximal, complemented
2#5752C212yesmaximal, complemented
3C3×#443C211yesmaximal, characteristic, complemented
4#5142C212yesmaximal, complemented
5#4897C212yesmaximal, complemented
6#4822C214yesmaximal, complemented
7#4700C211yesmaximal, characteristic, complemented
8#4695C211yesmaximal, characteristic, complemented
9#3292—31nomaximal, Sylow
10#1009C2214yescomplemented
11C3×#135C2211yescharacteristic
12C3×#131C2212yescomplemented
13#949C2214yescomplemented
14#919—22no—
15#914—24no—
16D3×D8—22no—
17#908—24no—
18#908C2214yescomplemented
19#904—22no—
20#904C2211yescharacteristic
21C2×D24—22no—
22C2×D24C2212yescomplemented
23C2×D24C2212yescomplemented
24#901C2212yescomplemented
25#874C2212yescomplemented
26C3×#108C2211yescharacteristic, complemented
27C3×#104C2212yescomplemented
28C3×C4×C8C2211yescharacteristic, complemented
29#823C2214yescomplemented
30#819C2211yescharacteristic, complemented
31#805C2212yescomplemented
32#576—32no—
33#576—32no—
34#493—31no—
35#472—32no—
36#472—32no—
37#453—34no—
38#443—31no—
39#443D311yescharacteristic, complemented
40C3×#41C2312yes—
41#239—24no—
42D3×D4—24no—
43D3×D4—44no—
44#235—24no—
45#235C2312yes—
46C2×D12—24no—
47C2×D12—42no—
48C2×D12C2314yes—
49C3×Q16D411yescharacteristic, complemented
50C3×SD16D412yescomplemented
51C3×D8D411yescharacteristic, complemented
52C3×M16—22no—
53C3×M16C2312yes—
54C2×C3×C8—22no—
55C2×C3×C8C2311yescharacteristic
56C2×C3×C8C2311yescharacteristic
57C3×C42C2311yescharacteristic
58#215—24no—
59#213—24no—
60#208C2312yes—
61Dic12D411yescharacteristic, complemented
62D24—24no—
63D24—22no—
64D24—22no—
65D24—22no—
66D24—22no—
67D24D411yescharacteristic, complemented
68#204—24no—
69#204D412yescomplemented
70ΓL(1,𝔽25)—22no—
71D3×C8—22no—
72H(5,𝔽2)—34no—
73AGL(1,ℤ/8)—34no—
74AGL(1,ℤ/8)—34no—
75AGL(1,ℤ/8)—64no—
76AGL(1,ℤ/8)—64no—
77#135—31no—
78#135—62no—
79#135—62no—
80#135C2×D311yescharacteristic
81C2×D8—32no—
82C2×D8—32no—
83C2×D8—62no—
84C2×D8—62no—
85#131—32no—
86#131C2×D312yescomplemented
87#127—32no—
88#108—31no—
89#108C2×D311yescharacteristic, complemented
90#104—32no—
91#104C2×D312yescomplemented
92#100—34no—
93C4×C8C2×D311yescharacteristic, complemented
94C22×D3—24no—
95C22×D3—44no—
96C3×Q8C2×D412yes—
97C3×D4—22no—
98C3×D4C2×D412yes—
99C2×C3×C4—21no—
100C2×C3×C4—22no—
101C2×C3×C4C2411yescharacteristic
102#67—22no—
103#67—44no—
104D12—24no—
105D12—24no—
106D12—44no—
107D12—84no—
108D12C2×D412yes—
109C4×D3—22no—
110C4×D3—44no—
111Dic6C2×D412yes—
112C3×C8—22no—
113C3×C8C2×D411yescharacteristic
114C3×C8C2×D411yescharacteristic
115C3×C8C2×D411yescharacteristic
116C3×C8C2×D411yescharacteristic
117C3⋊C8—22no—
118#41—32no—
119#41—64no—
120#41—64no—
121#41C22×D312yes—
122C2×D4—34no—
123C2×D4—64no—
124C2×D4—64no—
125C2×D4—122no—
126C2×D4—124no—
127Q16—31no—
128Q16D1211yescharacteristic, complemented
129SD16—32no—
130SD16—64no—
131SD16—64no—
132SD16D1212yescomplemented
133D8—31no—
134D8—64no—
135D8—64no—
136D8—62no—
137D8—62no—
138D8—62no—
139D8—62no—
140D8D1211yescharacteristic, complemented
141M16—22no—
142M16—32no—
143M16—62no—
144M16C22×D312yes—
145C2×C8—22no—
146C2×C8—62no—
147C2×C8C22×D311yescharacteristic
148C2×C8C22×D311yescharacteristic
149C42C22×D311yescharacteristic
150C22×C3—22no—
151C22×C3H(5,𝔽2)11yescharacteristic
152C2×D3—22no—
153C2×D3—44no—
154C2×D3—84no—
155C2×D3—164no—
156C3×C4—22no—
157C3×C4—22no—
158C3×C4C22×D411yescharacteristic
159C3×C4C22×D411yescharacteristic
160Dic3—22no—
161C23—64no—
162C23—124no—
163Q8—32no—
164Q8C2×D1212yes—
165D4—22no—
166D4—32no—
167D4—62no—
168D4—64no—
169D4—64no—
170D4—124no—
171D4—124no—
172D4—244no—
173D4C2×D1212yes—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—62no—
177C2×C4—124no—
178C2×C4C23×D311yescharacteristic
179C8—22no—
180C8—62no—
181C8C2×D1211yescharacteristic
182C8C2×D1211yescharacteristic
183C8D3×D411yescharacteristic
184C8D3×D411yescharacteristic
185C2×C3—21no—
186C2×C3—42no—
187C2×C3D4211yescharacteristic
188D3—42no—
189D3—84no—
190C22—22no—
191C22—62no—
192C22—124no—
193C22—244no—
194C22—484no—
195C22#100911yescharacteristic
196C4—22no—
197C4—22no—
198C4—62no—
199C4C22×D1211yescharacteristic
200C4C2×D3×D411yescharacteristic
201C3#329211yescharacteristic, complemented, Sylow
202C2—21no—
203C2—42no—
204C2—122no—
205C2—244no—
206C2D4×D1211yescharacteristic
subgroupG/H
centerC2D4×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#443C2
socleC2×C3D42
hypercenter#443D3
hypocenterC3#3292
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #5769C2semidirect
2. #5752C2semidirect
3. C3×#443C2semidirect
4. #5142C2semidirect
5. #4897C2semidirect
6. #4822C2semidirect
7. #4700C2semidirect
8. #4695C2semidirect
10. #1009C22semidirect
12. C3×#131C22semidirect
13. #949C22semidirect
18. #908C22semidirect
22. C2×D24C22semidirect
23. C2×D24C22semidirect
24. #901C22semidirect
25. #874C22semidirect
26. C3×#108C22semidirect
27. C3×#104C22semidirect
28. C3×C4×C8C22semidirect
29. #823C22semidirect
30. #819C22semidirect
31. #805C22semidirect
39. #443D3semidirect
49. C3×Q16D4semidirect
50. C3×SD16D4semidirect
51. C3×D8D4semidirect
61. Dic12D4semidirect
67. D24D4semidirect
69. #204D4semidirect
86. #131C2×D3semidirect
89. #108C2×D3semidirect
91. #104C2×D3semidirect
93. C4×C8C2×D3semidirect
128. Q16D12semidirect
132. SD16D12semidirect
140. D8D12semidirect
201. C3#3292semidirect

Maximal subgroups

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

subgroupindex
#57692
#57522
C3×#4432
#51422
#48972
#48222
#47002
#46952
#32923