Finite Group Databasebeta

← #11721061

Subgroups — #11721061

Order 864.

258 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 50 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#3828⋊C3C211yesmaximal, characteristic, complemented
2C2×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3#88538C211yesmaximal, characteristic, complemented
4#88432C212yesmaximal, complemented
5#88431C212yesmaximal, complemented
6#63830—31nomaximal
7#63825—31nomaximal
8C22×(C3⋊D3)⋊C3—21no—
9C2×C4×H(3,𝔽3)C2211yescharacteristic
10#638⋊C3—21no—
11C4⋊((C3⋊D3)⋊C3)—21no—
12C4⋊((C3⋊D3)⋊C3)C2212yes—
13C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
14C4×(C3⋊D3)⋊C3—21no—
15#634⋊C3C2211yescharacteristic, complemented
16#6180C2212yescomplemented
17#3828D311yescharacteristic, complemented
18C2×C4⋊(C3⋊D3)D311yescharacteristic, complemented
19C3×#235—31no—
20C2×C3×D12—31no—
21C3×#208—31no—
22C3×#208—31no—
23C3×#208—61no—
24#3717—32no—
25#3716—32no—
26#3714—32no—
27#3714—32no—
28C22×H(3,𝔽3)D411yescharacteristic
29C2×(C3⋊D3)⋊C3—21no—
30C2×(C3⋊D3)⋊C3—22no—
31C2×(C3⋊D3)⋊C3—42no—
32C4×H(3,𝔽3)D411yescharacteristic
33C4×H(3,𝔽3)C2311yescharacteristic
34(C3⋊Dic3)⋊C3—21no—
35#949—91no—
36#932—91no—
37#908—91no—
38C22×C3⋊D3—21no—
39C22×C3×D3—61no—
40C2×C32×C4—21no—
41C2×C32×C4C2×D311yescharacteristic
42C2×C32×C4C2×D311yescharacteristic
43#638—21no—
44C4⋊(C3⋊D3)—21no—
45C4⋊(C3⋊D3)C2×D312yes—
46C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
47C4×C3⋊D3—21no—
48#634C2×D311yescharacteristic, complemented
49C3×#67—61no—
50C3×D12—32no—
51C3×D12—31no—
52C3×D12—61no—
53C3×C4×D3—61no—
54C3×Dic6—31no—
55C3×C3⋊C8—32no—
56C3×C3⋊C8—32no—
57C3×C3⋊C8—62no—
58C2×H(3,𝔽3)—21no—
59C2×H(3,𝔽3)C2×D411yescharacteristic
60(C3⋊D3)⋊C3—41no—
61(C3⋊D3)⋊C3—42no—
62C3×#41—91no—
63C2×C3×D4—91no—
64#235—31no—
65#235—181no—
66C2×D12—31no—
67C2×D12—91no—
68C3×M16—91no—
69#215—92no—
70#214—92no—
71#213—92no—
72#213—92no—
73#208—31no—
74#208—31no—
75#208—61no—
76D24—92no—
77#204—92no—
78C22×C32—21no—
79C22×C32#6711yescharacteristic
80C22×C32#6711yescharacteristic
81C2×C3⋊D3—21no—
82C2×C3⋊D3—22no—
83C2×C3⋊D3—42no—
84C2×C3×D3—61no—
85C2×C3×D3—62no—
86C2×C3×D3—122no—
87C32×C4—21no—
88C32×C4—21no—
89C32×C4#6711yescharacteristic
90C32×C4#6711yescharacteristic
91C32×C4C22×D311yescharacteristic
92C32×C4C22×D311yescharacteristic
93C3⋊Dic3—21no—
94C3×Dic3—61no—
95AGL(1,ℤ/8)—271noSylow
96H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
97C23×C3—181no—
98C22×D3—61no—
99C22×D3—181no—
100C3×Q8—91no—
101C3×D4—91no—
102C3×D4—92no—
103C3×D4—181no—
104C3×D4—181no—
105C2×C3×C4—31no—
106C2×C3×C4—61no—
107C2×C3×C4—61no—
108C2×C3×C4—181no—
109C2×C3×C4D3211yescharacteristic
110#67—61no—
111#67—181no—
112D12—32no—
113D12—31no—
114D12—61no—
115D12—92no—
116D12—181no—
117D12—181no—
118C4×D3—61no—
119C4×D3—181no—
120Dic6—31no—
121Dic6—181no—
122C3×C8—92no—
123C3⋊C8—32no—
124C3⋊C8—32no—
125C3⋊C8—62no—
126C2×C32—21no—
127C2×C32—21no—
128C2×C32—21no—
129C2×C32—41no—
130C2×C32C2×#6711yescharacteristic
131C2×C32C2×#6711yescharacteristic
132C3⋊D3—42no—
133C3⋊D3—41no—
134C3×D3—121no—
135C3×D3—122no—
136#41—91no—
137C2×D4—91no—
138SD16—272no—
139D8—272no—
140M16—91no—
141C22×C3—31no—
142C22×C3—61no—
143C22×C3—61no—
144C22×C3—181no—
145C22×C3—182no—
146C22×C3—362no—
147C22×C3#62511yescharacteristic
148C2×D3—61no—
149C2×D3—62no—
150C2×D3—122no—
151C2×D3—182no—
152C2×D3—181no—
153C2×D3—362no—
154C3×C4—31no—
155C3×C4—31no—
156C3×C4—61no—
157C3×C4—61no—
158C3×C4—61no—
159C3×C4—61no—
160C3×C4—181no—
161C3×C4#62511yescharacteristic
162C3×C4C2×D3211yescharacteristic
163Dic3—61no—
164Dic3—181no—
165C32—21no—
166C32#93211yescharacteristic, complemented
167C32#94911yescharacteristic, complemented
168C23—181no—
169Q8—91no—
170D4—91no—
171D4—92no—
172D4—181no—
173D4—181no—
174C2×C4—181no—
175C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
176C8—92no—
177C2×C3—21no—
178C2×C3—31no—
179C2×C3—61no—
180C2×C3—61no—
181C2×C3—61no—
182C2×C3—121no—
183C2×C3—121no—
184C2×C3—361no—
185C2×C3—362no—
186C2×C3C2×#62511yescharacteristic
187D3—122no—
188D3—121no—
189D3—362no—
190D3—361no—
191C22—181no—
192C22—182no—
193C22—362no—
194C22#619811yescharacteristic
195C4—181no—
196C4#619811yescharacteristic
197C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
198C3—31no—
199C3—61no—
200C3—61no—
201C3#6382311yescharacteristic
202C2—21no—
203C2—362no—
204C2—361no—
205C2C2×#619811yescharacteristic
subgroupG/H
centerC2C2×#6198
commutator, cosocleC4×H(3,𝔽3)C23
FrattiniC3×C4C2×D32
FittingC2×C4×H(3,𝔽3)C22
socleC2×C3C2×#625
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)AGL(1,ℤ/8)
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #3828⋊C3C2semidirect
2. C2×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. #88538C2semidirect
4. #88432C2semidirect
5. #88431C2semidirect
13. C4⋊((C3⋊D3)⋊C3)C22semidirect
15. #634⋊C3C22semidirect
16. #6180C22semidirect
17. #3828D3semidirect
18. C2×C4⋊(C3⋊D3)D3semidirect
46. C4⋊(C3⋊D3)C2×D3semidirect
48. #634C2×D3semidirect
96. H(3,𝔽3)AGL(1,ℤ/8)semidirect
166. C32#932semidirect
167. C32#949semidirect

Maximal subgroups

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

subgroupindex
#3828⋊C32
C2×C4⋊((C3⋊D3)⋊C3)2
#885382
#884322
#884312
#638303
#638253