Finite Group Databasebeta

← #11723320

Subgroups — #11723320

Order 864.

488 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C3⋊D3×D12C211yesmaximal, characteristic, complemented
2#89014C211yesmaximal, characteristic, complemented
3#88851C211yesmaximal, characteristic, complemented
4C3×#3743C211yesmaximal, characteristic, complemented
5C32×D24C211yesmaximal, characteristic, complemented
6#88794C211yesmaximal, characteristic, complemented
7#88790C211yesmaximal, characteristic, complemented
8#64120—31nomaximal
9#63795—34nomaximal
10C2×D3×C3⋊D3—21no—
11C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12#6308C2211yescharacteristic, complemented
13C3×C4×C3⋊D3C2211yescharacteristic
14C32×D12C2211yescharacteristic
15C32×D12C2211yescharacteristic, complemented
16#6291—21no—
17#6290—21no—
18D3×C3⋊Dic3—21no—
19C33×C8C2211yescharacteristic, complemented
20C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
21#3830—31no—
22D4×C3⋊D3—31no—
23D3×D12—34no—
24#3795—34no—
25C32×D8—31no—
26#3754—31no—
27#3753—31no—
28#3744—31no—
29#3744—34no—
30#3744—64no—
31#3743D311yescharacteristic, complemented
32C3×D24D314yescomplemented
33C3×ΓL(1,𝔽25)—34no—
34#3716—34no—
35#3714—34no—
36C2×C3⋊(C3⋊D3)—21no—
37C2×C3×C3⋊D3D411yescharacteristic
38C2×C32×D3—21no—
39C2×C32×D3—21no—
40D3×C3⋊D3—42no—
41C33×C4C2311yescharacteristic
42C3⋊(C3⋊Dic3)—21no—
43C3×C3⋊Dic3D411yescharacteristic
44#911—94no—
45#908—91no—
46C22×C3⋊D3—61no—
47C2×D32—64no—
48C32×D4—31no—
49C32×D4—31no—
50#638—61no—
51#638—61no—
52C2×C3⋊Dic3—61no—
53C4⋊(C3⋊D3)—31no—
54C4⋊(C3⋊D3)—34no—
55C4⋊(C3⋊D3)—64no—
56C4×C3⋊D3C2×D311yescharacteristic
57#634—31no—
58#634—34no—
59#634—64no—
60C3×D12C2×D314yes—
61C3×D12C2×D314yescomplemented
62C3×C4×D3—34no—
63#626—64no—
64#625—64no—
65D3×Dic3—64no—
66C32×C8—24no—
67C32×C8C2×D314yescomplemented
68C32×C8C2×D311yescharacteristic, complemented
69C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
70C3×C3⋊C8—34no—
71C2×C33C2×D411yescharacteristic
72C3⋊(C3⋊D3)—41no—
73C3×C3⋊D3—21no—
74C32×D3—41no—
75C32×D3—41no—
76#237—94no—
77D3×D4—94no—
78#235—91no—
79C2×D12—91no—
80C3×D8—34no—
81C3×M16—91no—
82#214—94no—
83#213—94no—
84D24—91no—
85D24C3⋊D311yescharacteristic, complemented
86#204—91no—
87#204—94no—
88#204—91no—
89#204—184no—
90ΓL(1,𝔽25)—34no—
91C22×C32—61no—
92C22×C32—61no—
93C2×C3⋊D3—61no—
94C2×C3⋊D3—64no—
95C2×C3⋊D3—124no—
96C2×C3⋊D3—122no—
97C2×C3⋊D3D1211yescharacteristic
98C2×C3×D3—24no—
99C2×C3×D3—24no—
100C2×C3×D3—34no—
101D32—128no—
102C32×C4—24no—
103C32×C4C22×D314yes—
104C32×C4C22×D311yescharacteristic
105C3⋊Dic3—61no—
106C3⋊Dic3—64no—
107C3⋊Dic3—124no—
108C3⋊Dic3D1211yescharacteristic
109C3×Dic3—34no—
110AGL(1,ℤ/8)—271noSylow
111C33AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
112C22×D3—181no—
113C22×D3—184no—
114C3×D4—34no—
115C3×D4—34no—
116C2×C3×C4—91no—
117#67—184no—
118#67—181no—
119#67—184no—
120C2×Dic3—184no—
121D12—94no—
122D12—91no—
123D12—184no—
124D12—181no—
125D12C2×C3⋊D311yescharacteristic
126D12C2×C3⋊D311yescharacteristic, complemented
127C4×D3—34no—
128C4×D3—181no—
129Dic6—94no—
130Dic6—91no—
131Dic6—184no—
132C3×C8—24no—
133C3×C8—91no—
134C3×C8D3214yescomplemented
135C3×C8C2×C3⋊D311yescharacteristic, complemented
136C3⋊C8—34no—
137C2×C32—24no—
138C2×C32—121no—
139C2×C32—121no—
140C2×C32C2×D1211yescharacteristic
141C2×C32D3×D414yes—
142C3⋊D3—21no—
143C3⋊D3—121no—
144C3⋊D3—124no—
145C3⋊D3—244no—
146C3×D3—44no—
147C3×D3—44no—
148C3×D3—64no—
149#41—271no—
150C2×D4—271no—
151SD16—271no—
152SD16—271no—
153D8—31no—
154D8—271no—
155M16—91no—
156C22×C3—64no—
157C22×C3—64no—
158C22×C3—91no—
159C2×D3—21no—
160C2×D3—21no—
161C2×D3—34no—
162C2×D3—184no—
163C2×D3—181no—
164C2×D3—364no—
165C2×D3—362no—
166C2×D3—368no—
167C3×C4—24no—
168C3×C4—91no—
169C3×C4C2×D3214yes—
170C3×C4C22×C3⋊D311yescharacteristic
171Dic3—34no—
172Dic3—184no—
173Dic3—181no—
174Dic3—364no—
175C32—24no—
176C32#90811yescharacteristic, complemented
177C32#91114yescomplemented
178C23—541no—
179Q8—271no—
180D4—31no—
181D4—31no—
182D4—271no—
183D4—541no—
184D4—541no—
185C2×C4—91no—
186C2×C4—541no—
187C8—91no—
188C8D3×C3⋊D311yescharacteristic, complemented
189C2×C3—24no—
190C2×C3—124no—
191C2×C3—124no—
192C2×C3—181no—
193C2×C3D3×D1214yes—
194C2×C3D4×C3⋊D311yescharacteristic
195D3—41no—
196D3—41no—
197D3—64no—
198D3—364no—
199D3—361no—
200D3—724no—
201C22—61no—
202C22—61no—
203C22—91no—
204C22—541no—
205C22—1082no—
206C4—91no—
207C4—541no—
208C4C2×D3×C3⋊D311yescharacteristic
209C3—24no—
210C3#6379514yescomplemented
211C3#6412011yescharacteristic, complemented
212C2—121no—
213C2—121no—
214C2—181no—
215C2—1081no—
216C2C3⋊D3×D1211yescharacteristic
subgroupG/H
centerC2C3⋊D3×D12
commutator, cosocleC33×C4C23
FrattiniC4C2×D3×C3⋊D3
FittingC33×C8C22
socleC2×C33C2×D4
hypercenterC8D3×C3⋊D3
hypocenterC33AGL(1,ℤ/8)
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. C3⋊D3×D12C2semidirect
2. #89014C2semidirect
3. #88851C2semidirect
4. C3×#3743C2semidirect
5. C32×D24C2semidirect
6. #88794C2semidirect
7. #88790C2semidirect
12. #6308C22semidirect
15. C32×D12C22semidirect
19. C33×C8C22semidirect
20. C3×C3⋊(C3⋊C8)C22semidirect
31. #3743D3semidirect
32. C3×D24D3semidirect
61. C3×D12C2×D3semidirect
67. C32×C8C2×D3semidirect
68. C32×C8C2×D3semidirect
69. C3⋊(C3⋊C8)C2×D3semidirect
85. D24C3⋊D3semidirect
111. C33AGL(1,ℤ/8)semidirect
126. D12C2×C3⋊D3semidirect
134. C3×C8D32semidirect
135. C3×C8C2×C3⋊D3semidirect
176. C32#908semidirect
177. C32#911semidirect
188. C8D3×C3⋊D3semidirect
210. C3#63795semidirect
211. C3#64120semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×D122
#890142
#888512
C3×#37432
C32×D242
#887942
#887902
#641203
#637953