Finite Group Databasebeta

← #11721055

Subgroups — #11721055

Order 864.

239 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 66 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
2C2×#6180C211yesmaximal, characteristic, complemented
3#3748⋊C3C211yesmaximal, characteristic, complemented
4#88430C212yesmaximal, complemented
5((C3⋊D3)⋊C3)⋊C8C212yesmaximal, complemented
6#63817—31nomaximal
7#63814—31nomaximal
8C2×C4×H(3,𝔽3)C2211yescharacteristic
9#638⋊C3C412yes—
10C4⋊((C3⋊D3)⋊C3)C411yescharacteristic
11C4×(C3⋊D3)⋊C3C2212yes—
12#634⋊C3C411yescharacteristic
13#6180C2211yescharacteristic
14#6180C2211yescharacteristic
15(C3⋊(C3⋊C8))⋊C3C2212yes—
16#3828D311yescharacteristic
17C3×#235—31no—
18#3748D311yescharacteristic, complemented
19C3×#208—31no—
20C2×C3×C3⋊C8—31no—
21C2×C3×C3⋊C8—31no—
22C2×C3×C3⋊C8—61no—
23#3712—32no—
24#3711—32no—
25#3710—32no—
26D3×C3⋊C8—32no—
27C22×H(3,𝔽3)C2×C411yescharacteristic
28C2×(C3⋊D3)⋊C3C2×C412yes—
29C4×H(3,𝔽3)C2×C411yescharacteristic
30C4×H(3,𝔽3)C2311yescharacteristic
31(C3⋊Dic3)⋊C3C2×C412yes—
32#948—91no—
33#907—91no—
34#901—91no—
35C2×C32×C4—21no—
36C2×C32×C4C2×D311yescharacteristic
37C2×C32×C4C2×D311yescharacteristic
38#638Dic312yes—
39C4⋊(C3⋊D3)Dic311yescharacteristic
40C4×C3⋊D3C2×D312yes—
41#634Dic311yescharacteristic
42C3×#67—32no—
43C3×D12—31no—
44C3×C4×D3—32no—
45C3×Dic6—31no—
46C3⋊(C3⋊C8)C2×D312yes—
47C3×C3⋊C8—32no—
48C3×C3⋊C8—31no—
49C3×C3⋊C8—31no—
50C3×C3⋊C8—31no—
51C3×C3⋊C8—31no—
52C3×C3⋊C8—61no—
53C3×C3⋊C8—61no—
54C2×H(3,𝔽3)—21no—
55C2×H(3,𝔽3)C22×C411yescharacteristic
56(C3⋊D3)⋊C3—22no—
57C3×#41—91no—
58#235—31no—
59#235—181no—
60C3×M16—91no—
61C2×C3×C8—91no—
62#208—31no—
63#208—91no—
64#208—92no—
65C2×C3⋊C8—31no—
66C2×C3⋊C8—31no—
67C2×C3⋊C8—61no—
68C2×C3⋊C8—92no—
69ΓL(1,𝔽25)—92no—
70ΓL(1,𝔽25)—92no—
71D3×C8—92no—
72D3×C8—92no—
73C22×C32—21no—
74C22×C32C4×D311yescharacteristic
75C22×C32C2×Dic311yescharacteristic
76C2×C3⋊D3C2×Dic312yes—
77C2×C3×D3—32no—
78C32×C4—21no—
79C32×C4—21no—
80C32×C4C4×D311yescharacteristic
81C32×C4C2×Dic311yescharacteristic
82C32×C4C22×D311yescharacteristic
83C32×C4C22×D311yescharacteristic
84C3⋊Dic3C2×Dic312yes—
85C3×Dic3—32no—
86#131—271noSylow
87H(3,𝔽3)#13111yescharacteristic, complemented, Sylow
88C3×Q8—91no—
89C3×D4—91no—
90C3×D4—92no—
91C2×C3×C4—31no—
92C2×C3×C4—61no—
93C2×C3×C4—61no—
94C2×C3×C4—92no—
95C2×C3×C4D3211yescharacteristic
96#67—32no—
97#67—182no—
98D12—31no—
99D12—181no—
100C4×D3—32no—
101C4×D3—182no—
102Dic6—31no—
103Dic6—181no—
104C3×C8—92no—
105C3×C8—91no—
106C3×C8—91no—
107C3⋊C8—31no—
108C3⋊C8—31no—
109C3⋊C8—31no—
110C3⋊C8—31no—
111C3⋊C8—32no—
112C3⋊C8—61no—
113C3⋊C8—61no—
114C3⋊C8—92no—
115C2×C32—21no—
116C2×C32—21no—
117C2×C32—21no—
118C2×C32—21no—
119C2×C32—21no—
120C2×C32C2×C4×D311yescharacteristic
121C2×C32C22×Dic311yescharacteristic
122C3⋊D3—22no—
123C3×D3—62no—
124#41—91no—
125M16—91no—
126M16—272no—
127C2×C8—91no—
128C2×C8—272no—
129C22×C3—31no—
130C22×C3—61no—
131C22×C3—61no—
132C22×C3—92no—
133C22×C3D3×Dic311yescharacteristic
134C2×D3—32no—
135C2×D3—182no—
136C3×C4—31no—
137C3×C4—31no—
138C3×C4—61no—
139C3×C4—61no—
140C3×C4—61no—
141C3×C4—61no—
142C3×C4—92no—
143C3×C4D3×Dic311yescharacteristic
144C3×C4C2×D3211yescharacteristic
145Dic3—32no—
146Dic3—182no—
147C32—21no—
148C32#90711yescharacteristic, complemented
149C32#94811yescharacteristic, complemented
150Q8—91no—
151D4—92no—
152D4—91no—
153C2×C4—92no—
154C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
155C8—91no—
156C8—91no—
157C8—92no—
158C2×C3—21no—
159C2×C3—31no—
160C2×C3—61no—
161C2×C3—61no—
162C2×C3—61no—
163C2×C3—61no—
164C2×C3—121no—
165C2×C3—121no—
166C2×C3—182no—
167C2×C3C2×D3×Dic311yescharacteristic
168D3—62no—
169D3—362no—
170C22—92no—
171C22((C3⋊D3)⋊C3)⋊C411yescharacteristic
172C4—92no—
173C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
174C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
175C3—31no—
176C3—61no—
177C3—61no—
178C3#6381511yescharacteristic
179C2—21no—
180C2—182no—
181C2C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
subgroupG/H
centerC4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C4
FrattiniC3×C4C2×D32
FittingC2×C4×H(3,𝔽3)C22
socleC2×C3C2×D3×Dic3
prime radical#3828⋊C3C2
cosocleC4×H(3,𝔽3)C23
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#131
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#6180C2semidirect
3. #3748⋊C3C2semidirect
4. #88430C2semidirect
5. ((C3⋊D3)⋊C3)⋊C8C2semidirect
18. #3748D3semidirect
87. H(3,𝔽3)#131semidirect
148. C32#907semidirect
149. C32#948semidirect

Maximal subgroups

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

subgroupindex
#3828⋊C32
C2×#61802
#3748⋊C32
#884302
((C3⋊D3)⋊C3)⋊C82
#638173
#638143