Finite Group Databasebeta

← #82030

Subgroups — #82030

Order 384.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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
1D3×#135C212yesmaximal, complemented
2C3×#508C211yesmaximal, characteristic, complemented
3#5194C212yesmaximal, complemented
4#4918C212yesmaximal, complemented
5#4915C212yesmaximal, complemented
6#4914C212yesmaximal, complemented
7#4911C212yesmaximal, complemented
8#4904C211yesmaximal, characteristic, complemented
9C2×#798C211yesmaximal, characteristic, complemented
10#3414—31nomaximal, Sylow
11D3×#41—22no—
12C3×#135C2212yescomplemented
13#950—22no—
14#919C2212yes—
15D3×Q16C2212yescomplemented
16#916—22no—
17D3×SD16—22no—
18#912C2212yes—
19D3×D8C2212yescomplemented
20#904C2212yescomplemented
21C2×D3×C8C2211yescharacteristic
22C3×Q32C2211yescharacteristic, complemented
23C3×SD32C2212yescomplemented
24C3×D16C2211yescharacteristic, complemented
25C2×C3×C16C2211yescharacteristic, complemented
26#829C2212yescomplemented
27#828C2212yescomplemented
28#827C2212yescomplemented
29#826C2212yescomplemented
30C2×C3⋊C16C2211yescharacteristic, complemented
31Dic24C2211yescharacteristic, complemented
32#800C2212yescomplemented
33D48C2211yescharacteristic, complemented
34#798C2212yescomplemented
35#798C2212yescomplemented
36C2×#135—32no—
37#510—32no—
38#510—32no—
39#509—32no—
40#509—32no—
41#508—32no—
42#508—31no—
43#508D311yescharacteristic, complemented
44C2×M32—31no—
45C3×#41—22no—
46#239—22no—
47D3×Q8—22no—
48#237—22no—
49#237—42no—
50D3×D4—22no—
51D3×D4—42no—
52#235—22no—
53#235—42no—
54C2×C4×D3—42no—
55C2×C4×D3D411yescharacteristic
56C3×Q16C2312yes—
57C3×SD16—22no—
58C3×D8C2312yes—
59C2×C3×C8C2311yescharacteristic
60#216—22no—
61#215—22no—
62#214—22no—
63#213—22no—
64C2×C3⋊C8D411yescharacteristic
65Dic12C2312yes—
66D24C2312yes—
67#204—22no—
68D3×C8D412yes—
69D3×C8C2312yes—
70C3×C16C2311yescharacteristic
71C3×C16C2311yescharacteristic
72C3⋊C16C2312yes—
73C2×#41—62no—
74#135—32no—
75#135—32no—
76#135—32no—
77#135—62no—
78#135—62no—
79#135C2×D312yescomplemented
80C2×Q16—32no—
81C2×SD16—62no—
82C2×D8—32no—
83C22×C8—31no—
84Q32—32no—
85Q32—31no—
86Q32C2×D311yescharacteristic, complemented
87SD32—32no—
88SD32—32no—
89SD32—32no—
90SD32C2×D312yescomplemented
91D16—32no—
92D16—31no—
93D16C2×D311yescharacteristic, complemented
94M32—32no—
95M32—32no—
96C2×C16—31no—
97C2×C16C2×D311yescharacteristic, complemented
98C22×D3—42no—
99C22×D3D811yescharacteristic
100C3×Q8—22no—
101C3×D4—22no—
102C3×D4—42no—
103C2×C3×C4—42no—
104C2×C3×C4C2×D411yescharacteristic
105#67—42no—
106#67—42no—
107#67—42no—
108C2×Dic3—42no—
109C2×Dic3D811yescharacteristic
110D12—22no—
111D12—42no—
112C4×D3—42no—
113C4×D3—42no—
114C4×D3—42no—
115C4×D3D812yes—
116C4×D3C2×D412yes—
117Dic6—22no—
118Dic6—42no—
119C3×C8C2×D411yescharacteristic
120C3×C8C2411yescharacteristic
121C3⋊C8C2×D412yes—
122#41—22no—
123#41—62no—
124#41—62no—
125#41—62no—
126#41—122no—
127#41—122no—
128C2×Q8—62no—
129C2×D4—62no—
130C2×D4—122no—
131C22×C4—31no—
132C22×C4—122no—
133Q16—32no—
134Q16—62no—
135Q16C22×D312yes—
136SD16—22no—
137SD16—62no—
138SD16—62no—
139SD16—62no—
140D8—32no—
141D8—62no—
142D8C22×D312yes—
143C2×C8—31no—
144C2×C8—32no—
145C2×C8—32no—
146C2×C8C22×D311yescharacteristic
147C16—32no—
148C16C22×D311yescharacteristic
149C16C22×D311yescharacteristic
150C22×C3—42no—
151C22×C3C2×D811yescharacteristic
152C2×D3—41no—
153C2×D3—42no—
154C2×D3—84no—
155C2×D3C2×D812yes—
156C3×C4—42no—
157C3×C4C2×D811yescharacteristic
158C3×C4C22×D411yescharacteristic
159Dic3—42no—
160Dic3C2×D812yes—
161C23—31no—
162C23—122no—
163Q8—22no—
164Q8—62no—
165Q8—122no—
166D4—22no—
167D4—42no—
168D4—62no—
169D4—122no—
170D4—122no—
171D4—122no—
172D4—122no—
173C2×C4—31no—
174C2×C4—32no—
175C2×C4—32no—
176C2×C4—42no—
177C2×C4—122no—
178C2×C4—122no—
179C2×C4—122no—
180C2×C4—122no—
181C2×C4D3×D411yescharacteristic
182C8—32no—
183C8D3×D411yescharacteristic
184C8C23×D311yescharacteristic
185C2×C3—21no—
186C2×C3—82no—
187C2×C3C22×D811yescharacteristic
188D3—22no—
189D3—82no—
190C22—32no—
191C22—42no—
192C22—121no—
193C22—122no—
194C22—244no—
195C22D3×D811yescharacteristic
196C4—32no—
197C4—42no—
198C4—122no—
199C4D3×D811yescharacteristic
200C4C2×D3×D411yescharacteristic
201C3#341411yescharacteristic, complemented, Sylow
202C2—21no—
203C2—62no—
204C2—82no—
205C2—242no—
206C2C2×D3×D811yescharacteristic
subgroupG/H
centerC4D3×D8
commutator, cosocleC3×C8C24
FrattiniC8C23×D3
FittingC3×#508C2
socleC2×C3C22×D8
hypercenter#508D3
hypocenterC3#3414
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. D3×#135C2semidirect
2. C3×#508C2semidirect
3. #5194C2semidirect
4. #4918C2semidirect
5. #4915C2semidirect
6. #4914C2semidirect
7. #4911C2semidirect
8. #4904C2semidirect
9. C2×#798C2semidirect
12. C3×#135C22semidirect
15. D3×Q16C22semidirect
19. D3×D8C22semidirect
20. #904C22semidirect
22. C3×Q32C22semidirect
23. C3×SD32C22semidirect
24. C3×D16C22semidirect
25. C2×C3×C16C22semidirect
26. #829C22semidirect
27. #828C22semidirect
28. #827C22semidirect
29. #826C22semidirect
30. C2×C3⋊C16C22semidirect
31. Dic24C22semidirect
32. #800C22semidirect
33. D48C22semidirect
34. #798C22semidirect
35. #798C22semidirect
43. #508D3semidirect
79. #135C2×D3semidirect
86. Q32C2×D3semidirect
90. SD32C2×D3semidirect
93. D16C2×D3semidirect
97. C2×C16C2×D3semidirect
201. C3#3414semidirect

Maximal subgroups

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

subgroupindex
D3×#1352
C3×#5082
#51942
#49182
#49152
#49142
#49112
#49042
C2×#7982
#34143