Finite Group Databasebeta

← #13144222

Subgroups — #13144222

Order 1728.

451 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721808C211yesmaximal, characteristic
2Q16×#272C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×Q32C211yesmaximal, characteristic, complemented
4#11720570C211yesmaximal, characteristic, complemented
5#11720569C211yesmaximal, characteristic, complemented
6#11720513C211yesmaximal, characteristic, complemented
7#11720511C211yesmaximal, characteristic, complemented
8C3×#4918—34nomaximal
9#88748—21no—
10Q8×#272—21no—
11Q16×H(3,𝔽3)C2211yescharacteristic
12Q16×H(3,𝔽3)C2211yescharacteristic
13#88550—21no—
14#88549—21no—
15#88530C2211yescharacteristic
16#88529C2211yescharacteristic
17C8×#272C2211yescharacteristic
18C16×H(3,𝔽3)C2211yescharacteristic
19#88386C2211yescharacteristic
20C3×#919—34no—
21C3×D3×Q16—34no—
22C32×Q32D314yescomplemented
23C3×#829—34no—
24C3×#828—34no—
25C3×#800—34no—
26C3×#798—34no—
27Q8×H(3,𝔽3)—21no—
28Q8×H(3,𝔽3)—21no—
29C4⋊#272—21no—
30C4⋊#272—41no—
31C4×#272—41no—
32C4×#272—41no—
33C4×#272D411yescharacteristic
34#6229—21no—
35#6229—41no—
36C8×H(3,𝔽3)C2311yescharacteristic
37#6180D411yescharacteristic
38C3×#510—91no—
39#4918—34no—
40C3×#239—64no—
41C3×D3×Q8—64no—
42C32×Q16C2×D314yes—
43C32×Q16C2×D314yes—
44C3×#216—64no—
45C3×#215—64no—
46C3×Dic12—34no—
47C3×D24—34no—
48C3×D3×C8—34no—
49C32×C16C2×D314yes—
50C3×C3⋊C16—34no—
51C2×#272—41no—
52C2×#272D811yescharacteristic
53C4×H(3,𝔽3)—41no—
54C4×H(3,𝔽3)—41no—
55C4×H(3,𝔽3)C2×D411yescharacteristic
56#1084—41no—
57#1084D811yescharacteristic
58C3×#135—91no—
59C2×C3×Q16—91no—
60#919—34no—
61D3×Q16—34no—
62C3×Q32—34no—
63C3×Q32—91no—
64C3×Q32C3⋊D311yescharacteristic
65C3×SD32—91no—
66C3×SD32—91no—
67C3×M32—91no—
68#829—34no—
69#828—34no—
70#800—34no—
71#798—34no—
72C32×Q8—24no—
73C32×Q8—24no—
74C3×D12—64no—
75C3×D12—124no—
76C3×C4×D3—34no—
77C3×C4×D3—124no—
78C3×C4×D3—124no—
79C3×Dic6—64no—
80C3×Dic6—124no—
81C32×C8C22×D314yes—
82C3×C3⋊C8—34no—
83#510—91noSylow
84C2×H(3,𝔽3)C2×D811yescharacteristic
85#272—21no—
86#272—81no—
87C3×#41—181no—
88C2×C3×Q8—181no—
89#239—64no—
90D3×Q8—64no—
91C3×Q16—34no—
92C3×Q16—34no—
93C3×Q16—91no—
94C3×Q16—181no—
95C3×Q16C2×C3⋊D311yescharacteristic
96C3×Q16C2×C3⋊D311yescharacteristic
97C3×SD16—181no—
98C3×D8—91no—
99C2×C3×C8—91no—
100#216—64no—
101#215—64no—
102Dic12—34no—
103D24—34no—
104D3×C8—34no—
105C3×C16—34no—
106C3×C16—91no—
107C3×C16C2×C3⋊D311yescharacteristic
108C3⋊C16—34no—
109C2×C3×D3—34no—
110C2×C3×D3—124no—
111C32×C4—44no—
112C32×C4—44no—
113C32×C4D3×D414yes—
114C3×Dic3—34no—
115C3×Dic3—124no—
116#135—91no—
117C2×Q16—91no—
118Q32—91no—
119Q32#27211yescharacteristic, complemented
120SD32—91no—
121SD32—91no—
122M32—91no—
123H(3,𝔽3)#51011yescharacteristic, complemented, Sylow
124C3×Q8—21no—
125C3×Q8—21no—
126C3×Q8—64no—
127C3×Q8—64no—
128C3×Q8—181no—
129C3×Q8—361no—
130C3×D4—181no—
131C3×D4—361no—
132C2×C3×C4—91no—
133C2×C3×C4—361no—
134C2×C3×C4—361no—
135D12—64no—
136D12—124no—
137C4×D3—34no—
138C4×D3—124no—
139C4×D3—124no—
140Dic6—64no—
141Dic6—124no—
142C3×C8—34no—
143C3×C8—91no—
144C3×C8C22×C3⋊D311yescharacteristic
145C3⋊C8—34no—
146C2×C32D3×D814yes—
147C3×D3—64no—
148C3×D3—244no—
149#41—181no—
150C2×Q8—181no—
151Q16—91no—
152Q16—181no—
153Q16C2×#27211yescharacteristic
154Q16C2×#27211yescharacteristic
155SD16—181no—
156D8—91no—
157C2×C8—91no—
158C16—91no—
159C16C2×#27211yescharacteristic
160C22×C3—91no—
161C22×C3—361no—
162C2×D3—34no—
163C2×D3—124no—
164C3×C4—34no—
165C3×C4—41no—
166C3×C4—41no—
167C3×C4—91no—
168C3×C4—124no—
169C3×C4—124no—
170C3×C4—361no—
171C3×C4D4×C3⋊D311yescharacteristic
172Dic3—34no—
173Dic3—124no—
174C32#491814yescomplemented
175Q8—21no—
176Q8—21no—
177Q8—181no—
178Q8—361no—
179D4—181no—
180D4—361no—
181C2×C4—91no—
182C2×C4—361no—
183C2×C4—361no—
184C8—91no—
185C8C22×#27211yescharacteristic
186C2×C3—34no—
187C2×C3—181no—
188C2×C3—721no—
189C2×C3D8×C3⋊D311yescharacteristic
190D3—64no—
191D3—244no—
192C22—91no—
193C22—361no—
194C4—41no—
195C4—41no—
196C4—91no—
197C4—361no—
198C4D4×#27211yescharacteristic
199C3—34no—
200C3#1059261711yescharacteristic
201C2—181no—
202C2—721no—
203C2D8×#27211yescharacteristic
subgroupG/H
center, socleC2×C3D8×C3⋊D3
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C22×C3⋊D3
FittingH(3,𝔽3)×Q32C2
prime radical#11721808C2
hypercenterC3×Q32C3⋊D3
hypocenterH(3,𝔽3)#510
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 ◁ C3×Q32

Complements

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

NH ≅ G/Ntype
2. Q16×#272C2semidirect
3. H(3,𝔽3)×Q32C2semidirect
4. #11720570C2semidirect
5. #11720569C2semidirect
6. #11720513C2semidirect
7. #11720511C2semidirect
22. C32×Q32D3semidirect
119. Q32#272semidirect
123. H(3,𝔽3)#510semidirect
174. C32#4918semidirect

Maximal subgroups

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

subgroupindex
#117218082
Q16×#2722
H(3,𝔽3)×Q322
#117205702
#117205692
#117205132
#117205112
C3×#49183