Finite Group Databasebeta

← #87168

Subgroups — #87168

Order 384.

1412 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1412 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×#1010C211yesmaximal, characteristic, complemented
3C3×#568C211yesmaximal, characteristic, complemented
4#5753C218yesmaximal, complemented
5#5751C218yesmaximal, complemented
6#5748C216yesmaximal, complemented
7C2×#909C216yesmaximal, complemented
8#3586—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10#1010—24no—
11#1010C2218yescomplemented
12#1010C2214yescomplemented
13#1009C2214yescomplemented
14D3×#41—24no—
15#1007—21no—
16C2×D3×Q8—21no—
17#1004—23no—
18C2×#237—23no—
19C2×#235C2216yes—
20C2×#235C2213yescomplemented
21C22×Dic6C2213yescomplemented
22C3×#131C2218yescomplemented
23C2×C3×M16C2216yescomplemented
24#909C22112yescomplemented
25#909C22124yescomplemented
26#909C22112yescomplemented
27#908C2214yescomplemented
28#908C22112yescomplemented
29C2×Dic12C22112yescomplemented
30#904C22124yescomplemented
31C2×#204C22112yescomplemented
32#585—31no—
33C2×2−1+4—31no—
34#578—38no—
35#577—38no—
36#575—36no—
37C2×#137—36no—
38#568D311yescharacteristic, complemented
39C3×#41D414yes—
40C3×#41C2314yes—
41C2×C3×Q8D411yescharacteristic
42C2×C3×D4D413yes—
43C22×C3×C4C2313yes—
44C2×#67—26no—
45C2×#67—23no—
46C22×Dic3—23no—
47#239—24no—
48D3×Q8—28no—
49D3×Q8—24no—
50#237—224no—
51#237—212no—
52D3×D4—212no—
53#235—26no—
54#235—26no—
55#235—26no—
56#235C23112yes—
57#235C23112yes—
58#235C2314yes—
59C2×D12C2316yes—
60C2×D12C2311yescharacteristic
61C2×C4×D3—26no—
62C2×C4×D3—23no—
63C2×Dic6—23no—
64C2×Dic6—26no—
65C2×Dic6C23112yes—
66C2×Dic6C2316yes—
67C2×Dic6C2313yes—
68C3×M16C2314yes—
69C3×M16C23112yes—
70C2×C3×C8C23112yes—
71Dic12C23124yes—
72D24C2318yes—
73#204C23124yes—
74#204C2318yes—
752−1+4—38no—
762−1+4—34no—
772−1+4—64no—
782−1+4—61no—
79H(5,𝔽2)—34no—
80H(5,𝔽2)—63no—
81C2×#41—36no—
82C2×#41—33no—
83C2×#41—64no—
84C2×#41—63no—
85C2×#41C2×D311yescharacteristic
86C22×Q8—33no—
87C22×Q8—61no—
88#137—312no—
89#137—324no—
90#137—312no—
91AGL(1,ℤ/8)—34no—
92AGL(1,ℤ/8)—312no—
93#135—324no—
94C2×Q16—312no—
95C2×SD16—312no—
96#131C2×D318yescomplemented
97C2×M16C2×D316yescomplemented
98C23×C3C2×D413yes—
99C22×D3—21no—
100C22×D3—26no—
101C3×Q8C2×D414yes—
102C3×D4C2×D4112yes—
103C2×C3×C4C2×D413yes—
104C2×C3×C4C2×D416yes—
105C2×C3×C4C2416yes—
106C2×C3×C4C2411yescharacteristic
107#67—24no—
108#67—212no—
109#67—212no—
110C2×Dic3—23no—
111C2×Dic3—212no—
112C2×Dic3—26no—
113D12—21no—
114D12—26no—
115D12C2414yes—
116D12C2412yes—
117C4×D3—24no—
118C4×D3—212no—
119C4×D3—212no—
120Dic6—23no—
121Dic6—212no—
122Dic6—26no—
123Dic6C2416yes—
124Dic6C2414yes—
125C3×C8C2418yes—
126#41—312no—
127#41—312no—
128#41—34no—
129#41—64no—
130#41—612no—
131#41—624no—
132#41—66no—
133#41—66no—
134#41—66no—
135#41D1214yes—
136#41C22×D314yes—
137C2×Q8—312no—
138C2×Q8—36no—
139C2×Q8—33no—
140C2×Q8—68no—
141C2×Q8—64no—
142C2×Q8—63no—
143C2×Q8—66no—
144C2×Q8D1211yescharacteristic
145C2×D4—36no—
146C2×D4—31no—
147C2×D4—63no—
148C2×D4—66no—
149C2×D4—612no—
150C2×D4D1213yes—
151C22×C4—63no—
152C22×C4—63no—
153C22×C4—66no—
154C22×C4C22×D313yes—
155Q16—324no—
156SD16—324no—
157SD16—38no—
158D8—38no—
159M16C22×D314yes—
160M16C22×D3112yes—
161C2×C8C22×D3112yes—
162C22×C3—43no—
163C22×C3C22×D411yescharacteristic
164C22×C3C22×D416yes—
165C2×D3—22no—
166C2×D3—24no—
167C2×D3—42no—
168C2×D3—86no—
169C3×C4C22×D411yescharacteristic
170C3×C4C22×D416yes—
171C3×C4C2511yescharacteristic
172Dic3—26no—
173Dic3—24no—
174C23—61no—
175C23—66no—
176C23C2×D1213yes—
177Q8—36no—
178Q8—34no—
179Q8—63no—
180Q8—612no—
181Q8—66no—
182Q8C2×D1214yes—
183D4—32no—
184D4—34no—
185D4—64no—
186D4—612no—
187D4—612no—
188D4—66no—
189D4—61no—
190D4C2×D12112yes—
191C2×C4—63no—
192C2×C4—66no—
193C2×C4—612no—
194C2×C4—64no—
195C2×C4—612no—
196C2×C4—612no—
197C2×C4C2×D1213yes—
198C2×C4C2×D1216yes—
199C2×C4C23×D316yes—
200C2×C4C23×D311yescharacteristic
201C8C23×D318yes—
202C2×C3—21no—
203C2×C3—26no—
204C2×C3C23×D411yescharacteristic
205D3—42no—
206D3—44no—
207C22—43no—
208C22—62no—
209C22—64no—
210C22—122no—
211C22—246no—
212C22C22×D1211yescharacteristic
213C22C22×D1216yes—
214C4—64no—
215C4—66no—
216C4C22×D1211yescharacteristic
217C4C22×D1216yes—
218C4C24×D311yescharacteristic
219C3#358611yescharacteristic, complemented, Sylow
220C2—21no—
221C2—26no—
222C2—122no—
223C2—124no—
224C2C23×D1211yescharacteristic
subgroupG/H
centerC2C23×D12
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC3×#568C2
socleC2×C3C23×D4
hypercenter#568D3
hypocenterC3#3586
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×#1010C2semidirect
3. C3×#568C2semidirect
4. #5753C2semidirect
5. #5751C2semidirect
6. #5748C2semidirect
7. C2×#909C2semidirect
11. #1010C22semidirect
12. #1010C22semidirect
13. #1009C22semidirect
20. C2×#235C22semidirect
21. C22×Dic6C22semidirect
22. C3×#131C22semidirect
23. C2×C3×M16C22semidirect
24. #909C22semidirect
25. #909C22semidirect
26. #909C22semidirect
27. #908C22semidirect
28. #908C22semidirect
29. C2×Dic12C22semidirect
30. #904C22semidirect
31. C2×#204C22semidirect
38. #568D3semidirect
96. #131C2×D3semidirect
97. C2×M16C2×D3semidirect
219. C3#3586semidirect

Maximal subgroups

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

subgroupindex
#59642
C2×#10102
C3×#5682
#57532
#57512
#57482
C2×#9092
#35863