Finite Group Databasebeta

← #11723560

Subgroups — #11723560

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×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2#89014C211yesmaximal, characteristic, complemented
3#88861C211yesmaximal, characteristic, complemented
4C32×#213C211yesmaximal, characteristic, complemented
5#88796C211yesmaximal, characteristic, complemented
6#88789C211yesmaximal, characteristic, complemented
7#88787C211yesmaximal, characteristic, complemented
8#64120—31nomaximal
9#63932—34nomaximal
10C22×C3×C3⋊D3—21no—
11C33×D4C2211yescharacteristic
12#6308C2211yescharacteristic, complemented
13C3×#638—21no—
14C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C3×C4×C3⋊D3C2211yescharacteristic
16C32×D12C2211yescharacteristic, complemented
17#6290—21no—
18D3×C3⋊Dic3—21no—
19C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
20C32×C3⋊C8C2211yescharacteristic, complemented
21#3830—31no—
22D4×C3⋊D3D311yescharacteristic, complemented
23C3×D3×D4—34no—
24#3795—34no—
25C32×D8—31no—
26#3754—31no—
27#3754—34no—
28#3754—64no—
29#3753—31no—
30#3744—31no—
31#3743—31no—
32C3×#213D314yescomplemented
33#3716—34no—
34#3713—34no—
35#3711—34no—
36C22×C33—21no—
37C2×C3×C3⋊D3—21no—
38C2×C3×C3⋊D3—42no—
39C2×C3×C3⋊D3D411yescharacteristic
40C2×C32×D3—21no—
41C33×C4C2311yescharacteristic
42C3⋊(C3⋊Dic3)—21no—
43C3×C3⋊Dic3D411yescharacteristic
44#932—91no—
45#911—94no—
46C22×C3⋊D3—21no—
47C22×C3×D3—64no—
48C32×D4—24no—
49C32×D4—31no—
50C32×D4C2×D314yes—
51C32×D4C2×D311yescharacteristic
52#638—21no—
53#638—61no—
54C2×C3⋊Dic3—61no—
55C4⋊(C3⋊D3)C2×D311yescharacteristic
56C4×C3⋊D3C2×D311yescharacteristic
57#634—31no—
58#634—34no—
59#634—64no—
60C3×#67—64no—
61C3×D12—34no—
62C3×D12C2×D314yescomplemented
63C3×C4×D3—34no—
64#625—64no—
65D3×Dic3—64no—
66C32×C8—31no—
67C3⋊(C3⋊C8)—31no—
68C3⋊(C3⋊C8)—34no—
69C3⋊(C3⋊C8)—64no—
70C3×C3⋊C8C2×D314yescomplemented
71C2×C33—41no—
72C2×C33C2×D411yescharacteristic
73C3×C3⋊D3—21no—
74C3×C3⋊D3—41no—
75C32×D3—41no—
76C2×C3×D4—91no—
77#237—94no—
78D3×D4—34no—
79#235—91no—
80C3×D8—34no—
81#214—94no—
82#214—91no—
83#214—91no—
84#214—184no—
85#213—94no—
86#213—91no—
87#213C3⋊D311yescharacteristic, complemented
88#208—91no—
89#204—94no—
90ΓL(1,𝔽25)—94no—
91C22×C32—21no—
92C22×C32—24no—
93C22×C32—44no—
94C22×C32—61no—
95C2×C3⋊D3—21no—
96C2×C3⋊D3—42no—
97C2×C3⋊D3#6711yescharacteristic
98C2×C3×D3—24no—
99C2×C3×D3—34no—
100C2×C3×D3—64no—
101C2×C3×D3—128no—
102C32×C4—24no—
103C32×C4C22×D314yes—
104C32×C4C22×D311yescharacteristic
105C3⋊Dic3—61no—
106C3⋊Dic3—64no—
107C3⋊Dic3—124no—
108C3⋊Dic3#6711yescharacteristic
109C3×Dic3—34no—
110AGL(1,ℤ/8)—271noSylow
111C33AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
112C23×C3—181no—
113C22×D3—64no—
114C3×D4—24no—
115C3×D4—34no—
116C3×D4—91no—
117C3×D4—181no—
118C3×D4D3214yes—
119C3×D4C2×C3⋊D311yescharacteristic
120C2×C3×C4—91no—
121#67—64no—
122#67—181no—
123#67—184no—
124C2×Dic3—184no—
125D12—34no—
126D12C2×C3⋊D311yescharacteristic, complemented
127C4×D3—34no—
128C4×D3—181no—
129Dic6—94no—
130Dic6—91no—
131Dic6—184no—
132C3×C8—34no—
133C3⋊C8—94no—
134C3⋊C8—91no—
135C3⋊C8—184no—
136C3⋊C8C2×C3⋊D311yescharacteristic, complemented
137C2×C32—24no—
138C2×C32—41no—
139C2×C32—44no—
140C2×C32—84no—
141C2×C32—121no—
142C2×C32D3×D414yes—
143C2×C32C2×#6711yescharacteristic
144C3⋊D3—21no—
145C3⋊D3—41no—
146C3×D3—44no—
147C3×D3—64no—
148C3×D3—124no—
149#41—271no—
150C2×D4—91no—
151SD16—271no—
152SD16—271no—
153D8—31no—
154D8—271no—
155M16—271no—
156C22×C3—24no—
157C22×C3—21no—
158C22×C3—44no—
159C22×C3—64no—
160C22×C3—91no—
161C22×C3—181no—
162C22×C3—362no—
163C2×D3—21no—
164C2×D3—34no—
165C2×D3—64no—
166C2×D3—128no—
167C3×C4—24no—
168C3×C4—91no—
169C3×C4C2×D3214yes—
170C3×C4C22×C3⋊D311yescharacteristic
171Dic3—34no—
172Dic3—184no—
173Dic3—181no—
174Dic3—364no—
175C32—24no—
176C32#91114yescomplemented
177C32#93211yescharacteristic, complemented
178C23—181no—
179Q8—271no—
180D4—31no—
181D4—91no—
182D4—181no—
183D4—541no—
184D4D3×C3⋊D311yescharacteristic
185C2×C4—91no—
186C2×C4—541no—
187C8—31no—
188C8—271no—
189C2×C3—24no—
190C2×C3—41no—
191C2×C3—44no—
192C2×C3—84no—
193C2×C3—124no—
194C2×C3—181no—
195C2×C3—361no—
196C2×C3D3×#6714yes—
197C2×C3D4×C3⋊D311yescharacteristic
198D3—41no—
199D3—64no—
200D3—124no—
201C22—21no—
202C22—61no—
203C22—91no—
204C22—181no—
205C22—362no—
206C4—91no—
207C4—541no—
208C4C2×D3×C3⋊D311yescharacteristic
209C3—24no—
210C3#6393214yescomplemented
211C3#6412011yescharacteristic, complemented
212C2—41no—
213C2—121no—
214C2—181no—
215C2—361no—
216C2C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC2C3⋊D3×#67
commutator, cosocleC33×C4C23
FrattiniC4C2×D3×C3⋊D3
FittingC33×D4C22
socleC2×C33C2×D4
hypercenterD4D3×C3⋊D3
hypocenterC33AGL(1,ℤ/8)
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C3×D4×C3⋊D3C2semidirect
2. #89014C2semidirect
3. #88861C2semidirect
4. C32×#213C2semidirect
5. #88796C2semidirect
6. #88789C2semidirect
7. #88787C2semidirect
12. #6308C22semidirect
16. C32×D12C22semidirect
19. C3⋊(C3⋊(C3⋊C8))C22semidirect
20. C32×C3⋊C8C22semidirect
22. D4×C3⋊D3D3semidirect
32. C3×#213D3semidirect
62. C3×D12C2×D3semidirect
70. C3×C3⋊C8C2×D3semidirect
87. #213C3⋊D3semidirect
111. C33AGL(1,ℤ/8)semidirect
126. D12C2×C3⋊D3semidirect
136. C3⋊C8C2×C3⋊D3semidirect
176. C32#911semidirect
177. C32#932semidirect
210. C3#63932semidirect
211. C3#64120semidirect

Maximal subgroups

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

subgroupindex
C3×D4×C3⋊D32
#890142
#888612
C32×#2132
#887962
#887892
#887872
#641203
#639323