Finite Group Databasebeta

← #87167

Subgroups — #87167

Order 384.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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#5964C211yesmaximal, characteristic, complemented
2C2×#1009C211yesmaximal, characteristic, complemented
3C3×#568C211yesmaximal, characteristic, complemented
4#5752C218yesmaximal, complemented
5#5751C218yesmaximal, complemented
6#5748C216yesmaximal, complemented
7C2×#908C216yesmaximal, complemented
8#3585—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10#1010C2214yescomplemented
11#1009—24no—
12#1009C2218yescomplemented
13#1009C2214yescomplemented
14D3×#41—24no—
15#1007—21no—
16C2×#239—21no—
17#1004—23no—
18C2×D3×D4—23no—
19C2×#235C2213yescomplemented
20C2×#235C2216yescomplemented
21C22×D12C2213yescomplemented
22C3×#131C2218yescomplemented
23C2×C3×M16C2216yescomplemented
24#909C2214yescomplemented
25#909C22112yescomplemented
26#908C22112yescomplemented
27#908C22124yescomplemented
28#908C22112yescomplemented
29#904C22124yescomplemented
30C2×D24C22112yescomplemented
31C2×#204C22112yescomplemented
32#585—31no—
33C2×H(5,𝔽2)—31no—
34#577—38no—
35#576—38no—
36#575—36no—
37C2×AGL(1,ℤ/8)—36no—
38#568D311yescharacteristic, complemented
39C23×D3—23no—
40C3×#41D414yes—
41C3×#41C2314yes—
42C2×C3×Q8D411yescharacteristic
43C2×C3×D4D413yes—
44C22×C3×C4C2313yes—
45C2×#67—23no—
46C2×#67—26no—
47#239—28no—
48#239—24no—
49D3×Q8—24no—
50#237—212no—
51D3×D4—224no—
52D3×D4—212no—
53#235—26no—
54#235—26no—
55#235—26no—
56#235C23112yes—
57#235C2314yes—
58#235C23112yescomplemented
59C2×D12—23no—
60C2×D12—26no—
61C2×D12C23112yes—
62C2×D12C2313yes—
63C2×D12C2316yescomplemented
64C2×C4×D3—23no—
65C2×C4×D3—26no—
66C2×Dic6C2311yescharacteristic
67C2×Dic6C2316yescomplemented
68C3×M16C2314yes—
69C3×M16C23112yescomplemented
70C2×C3×C8C23112yescomplemented
71Dic12C2318yescomplemented
72D24C23124yescomplemented
73#204C23124yescomplemented
74#204C2318yescomplemented
752−1+4—34no—
762−1+4—61no—
77H(5,𝔽2)—34no—
78H(5,𝔽2)—38no—
79H(5,𝔽2)—64no—
80H(5,𝔽2)—63no—
81C2×#41—36no—
82C2×#41—33no—
83C2×#41—64no—
84C2×#41—61no—
85C2×#41C2×D311yescharacteristic
86C22×D4—33no—
87C22×D4—63no—
88#137—34no—
89#137—312no—
90AGL(1,ℤ/8)—312no—
91AGL(1,ℤ/8)—324no—
92AGL(1,ℤ/8)—312no—
93#135—324no—
94C2×SD16—312no—
95C2×D8—312no—
96#131C2×D318yescomplemented
97C2×M16C2×D316yescomplemented
98C23×C3C2×D413yes—
99C22×D3—23no—
100C22×D3—26no—
101C22×D3—212no—
102C22×D3—86no—
103C3×Q8C2×D414yes—
104C3×D4C2×D4112yes—
105C2×C3×C4C2×D413yes—
106C2×C3×C4C2×D416yes—
107C2×C3×C4C2416yes—
108C2×C3×C4C2411yescharacteristic
109#67—24no—
110#67—212no—
111#67—212no—
112C2×Dic3—21no—
113C2×Dic3—26no—
114D12—23no—
115D12—212no—
116D12—26no—
117D12C2416yes—
118D12C2414yes—
119C4×D3—24no—
120C4×D3—212no—
121C4×D3—212no—
122Dic6—21no—
123Dic6—26no—
124Dic6C2414yes—
125Dic6C2412yes—
126C3×C8C2418yes—
127C24—63no—
128#41—312no—
129#41—312no—
130#41—34no—
131#41—68no—
132#41—64no—
133#41—612no—
134#41—66no—
135#41—66no—
136#41—66no—
137#41D1214yes—
138#41C22×D314yes—
139C2×Q8—36no—
140C2×Q8—31no—
141C2×Q8—64no—
142C2×Q8D1211yescharacteristic
143C2×D4—36no—
144C2×D4—312no—
145C2×D4—33no—
146C2×D4—63no—
147C2×D4—66no—
148C2×D4—624no—
149C2×D4—612no—
150C2×D4—63no—
151C2×D4—66no—
152C2×D4D1213yes—
153C22×C4—63no—
154C22×C4—66no—
155C22×C4C22×D313yes—
156Q16—38no—
157SD16—324no—
158SD16—38no—
159D8—324no—
160M16C22×D314yes—
161M16C22×D3112yescomplemented
162C2×C8C22×D3112yescomplemented
163C22×C3—43no—
164C22×C3C22×D411yescharacteristic
165C22×C3C22×D416yes—
166C2×D3—24no—
167C2×D3—26no—
168C2×D3—46no—
169C2×D3—86no—
170C2×D3—812no—
171C3×C4C22×D411yescharacteristic
172C3×C4C22×D416yes—
173C3×C4C2511yescharacteristic
174Dic3—24no—
175Dic3—22no—
176C23—63no—
177C23—66no—
178C23—612no—
179C23—246no—
180C23C2×D1213yes—
181Q8—34no—
182Q8—32no—
183Q8—66no—
184Q8—61no—
185Q8C2×D1214yes—
186D4—34no—
187D4—36no—
188D4—612no—
189D4—64no—
190D4—612no—
191D4—63no—
192D4—612no—
193D4—66no—
194D4C2×D12112yes—
195C2×C4—61no—
196C2×C4—66no—
197C2×C4—64no—
198C2×C4—612no—
199C2×C4—612no—
200C2×C4C2×D1213yes—
201C2×C4C2×D1216yes—
202C2×C4C23×D316yes—
203C2×C4C23×D311yescharacteristic
204C8C23×D318yes—
205C2×C3—21no—
206C2×C3—26no—
207C2×C3C23×D411yescharacteristic
208D3—46no—
209D3—44no—
210C22—43no—
211C22—64no—
212C22—66no—
213C22—126no—
214C22—2412no—
215C22—246no—
216C22C22×D1211yescharacteristic
217C22C22×D1216yes—
218C4—62no—
219C4—64no—
220C4C22×D1211yescharacteristic
221C4C22×D1216yes—
222C4C24×D311yescharacteristic
223C3#358511yescharacteristic, complemented, Sylow
224C2—21no—
225C2—26no—
226C2—126no—
227C2—124no—
228C2C23×D1211yescharacteristic
subgroupG/H
centerC2C23×D12
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC3×#568C2
socleC2×C3C23×D4
hypercenter#568D3
hypocenterC3#3585
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #5964C2semidirect
2. C2×#1009C2semidirect
3. C3×#568C2semidirect
4. #5752C2semidirect
5. #5751C2semidirect
6. #5748C2semidirect
7. C2×#908C2semidirect
10. #1010C22semidirect
12. #1009C22semidirect
13. #1009C22semidirect
19. C2×#235C22semidirect
20. C2×#235C22semidirect
21. C22×D12C22semidirect
22. C3×#131C22semidirect
23. C2×C3×M16C22semidirect
24. #909C22semidirect
25. #909C22semidirect
26. #908C22semidirect
27. #908C22semidirect
28. #908C22semidirect
29. #904C22semidirect
30. C2×D24C22semidirect
31. C2×#204C22semidirect
38. #568D3semidirect
58. #235C23semidirect
63. C2×D12C23semidirect
67. C2×Dic6C23semidirect
69. C3×M16C23semidirect
70. C2×C3×C8C23semidirect
71. Dic12C23semidirect
72. D24C23semidirect
73. #204C23semidirect
74. #204C23semidirect
96. #131C2×D3semidirect
97. C2×M16C2×D3semidirect
161. M16C22×D3semidirect
162. C2×C8C22×D3semidirect
223. C3#3585semidirect

Maximal subgroups

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

subgroupindex
#59642
C2×#10092
C3×#5682
#57522
#57512
#57482
C2×#9082
#35853