Finite Group Databasebeta

← #13144217

Subgroups — #13144217

Order 1728.

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

normal subgroups only

This group has 462 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, complemented
2#11721794C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×SD32C211yesmaximal, characteristic, complemented
4#11720570C211yesmaximal, characteristic, complemented
5#11720565C211yesmaximal, characteristic, complemented
6#11720513C211yesmaximal, characteristic, complemented
7C16×#272C211yesmaximal, characteristic, complemented
8C3×#4916—34nomaximal
9#88748—21no—
10#88744—21no—
11Q16×H(3,𝔽3)C2211yescharacteristic
12D8×H(3,𝔽3)C2211yescharacteristic
13#88549—21no—
14#88546—21no—
15#88530C2211yescharacteristic
16#88529C2211yescharacteristic
17C8×#272C2211yescharacteristic
18C16×H(3,𝔽3)C2211yescharacteristic
19#88386C2211yescharacteristic
20C3×#919—34no—
21C3×#912—34no—
22C32×SD32D314yescomplemented
23C3×#829—34no—
24C3×#826—34no—
25C3×#800—34no—
26C3×D3×C16—34no—
27Q8×H(3,𝔽3)—21no—
28D4×H(3,𝔽3)—21no—
29#6235—41no—
30C2×#1084—41no—
31C4⋊#272—21no—
32C4⋊#272—41no—
33C4×#272—41no—
34C4×#272D411yescharacteristic
35#6229—21no—
36C8×H(3,𝔽3)C2311yescharacteristic
37#6180D411yescharacteristic
38C3×#508—91no—
39#4916—34no—
40C3×#239—64no—
41C3×#237—64no—
42C32×Q16C2×D314yes—
43C32×D8C2×D314yes—
44C3×#215—64no—
45C3×#214—64no—
46C3×Dic12—34no—
47C3×D24—34no—
48C3×D3×C8—34no—
49C32×C16C2×D314yes—
50C3×C3⋊C16—34no—
51C22×H(3,𝔽3)—41no—
52C2×#272—41no—
53C2×#272D811yescharacteristic
54C4×H(3,𝔽3)—41no—
55C4×H(3,𝔽3)C2×D411yescharacteristic
56#1084—41no—
57#1084D811yescharacteristic
58C3×#135—91no—
59C3×#135—91no—
60#919—34no—
61#912—34no—
62C3×Q32—91no—
63C3×SD32—34no—
64C3×SD32—91no—
65C3×SD32C3⋊D311yescharacteristic
66C3×D16—91no—
67C2×C3×C16—91no—
68#829—34no—
69#826—34no—
70#800—34no—
71D3×C16—34no—
72C32×Q8—24no—
73C32×D4—24no—
74C3×#67—124no—
75C2×C3×Dic3—124no—
76C3×D12—64no—
77C3×D12—124no—
78C3×C4×D3—34no—
79C3×C4×D3—124no—
80C3×Dic6—64no—
81C32×C8C22×D314yes—
82C3×C3⋊C8—34no—
83#508—91noSylow
84C2×H(3,𝔽3)—81no—
85C2×H(3,𝔽3)C2×D811yescharacteristic
86#272—21no—
87#272—81no—
88C3×#41—181no—
89C3×#41—181no—
90#239—64no—
91#237—64no—
92C3×Q16—34no—
93C3×Q16—91no—
94C3×Q16C2×C3⋊D311yescharacteristic
95C3×SD16—181no—
96C3×SD16—181no—
97C3×D8—34no—
98C3×D8—91no—
99C3×D8C2×C3⋊D311yescharacteristic
100C2×C3×C8—91no—
101#215—64no—
102#214—64no—
103Dic12—34no—
104D24—34no—
105D3×C8—34no—
106C3×C16—34no—
107C3×C16—91no—
108C3×C16C2×C3⋊D311yescharacteristic
109C3⋊C16—34no—
110C22×C32—44no—
111C2×C3×D3—34no—
112C2×C3×D3—124no—
113C32×C4—44no—
114C32×C4D3×D414yes—
115C3×Dic3—34no—
116C3×Dic3—124no—
117#135—91no—
118#135—91no—
119Q32—91no—
120SD32—91no—
121SD32#27211yescharacteristic, complemented
122D16—91no—
123C2×C16—91no—
124H(3,𝔽3)#50811yescharacteristic, complemented, Sylow
125C3×Q8—21no—
126C3×Q8—64no—
127C3×Q8—181no—
128C3×D4—21no—
129C3×D4—64no—
130C3×D4—181no—
131C3×D4—361no—
132C3×D4—361no—
133C2×C3×C4—91no—
134C2×C3×C4—361no—
135C2×C3×C4—361no—
136#67—124no—
137C2×Dic3—124no—
138D12—64no—
139D12—124no—
140C4×D3—34no—
141C4×D3—124no—
142Dic6—64no—
143C3×C8—34no—
144C3×C8—91no—
145C3×C8C22×C3⋊D311yescharacteristic
146C3⋊C8—34no—
147C2×C32—84no—
148C2×C32D3×D814yes—
149C3×D3—64no—
150C3×D3—244no—
151#41—181no—
152#41—181no—
153Q16—91no—
154Q16C2×#27211yescharacteristic
155SD16—181no—
156SD16—181no—
157D8—91no—
158D8C2×#27211yescharacteristic
159C2×C8—91no—
160C16—91no—
161C16C2×#27211yescharacteristic
162C22×C3—41no—
163C22×C3—91no—
164C22×C3—124no—
165C22×C3—361no—
166C2×D3—34no—
167C2×D3—124no—
168C3×C4—34no—
169C3×C4—41no—
170C3×C4—91no—
171C3×C4—124no—
172C3×C4—361no—
173C3×C4D4×C3⋊D311yescharacteristic
174Dic3—34no—
175Dic3—124no—
176C32#491614yescomplemented
177Q8—21no—
178Q8—181no—
179D4—21no—
180D4—181no—
181D4—361no—
182D4—361no—
183C2×C4—91no—
184C2×C4—361no—
185C2×C4—361no—
186C8—91no—
187C8C22×#27211yescharacteristic
188C2×C3—34no—
189C2×C3—81no—
190C2×C3—181no—
191C2×C3—244no—
192C2×C3—721no—
193C2×C3D8×C3⋊D311yescharacteristic
194D3—64no—
195D3—244no—
196C22—41no—
197C22—91no—
198C22—361no—
199C4—41no—
200C4—91no—
201C4—361no—
202C4D4×#27211yescharacteristic
203C3—34no—
204C3#1059261511yescharacteristic
205C2—81no—
206C2—181no—
207C2—721no—
208C2D8×#27211yescharacteristic
subgroupG/H
center, socleC2×C3D8×C3⋊D3
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C22×C3⋊D3
FittingH(3,𝔽3)×SD32C2
hypercenterC3×SD32C3⋊D3
hypocenterH(3,𝔽3)#508
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 ◁ C3×SD32

Complements

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

NH ≅ G/Ntype
1. #11721808C2semidirect
2. #11721794C2semidirect
3. H(3,𝔽3)×SD32C2semidirect
4. #11720570C2semidirect
5. #11720565C2semidirect
6. #11720513C2semidirect
7. C16×#272C2semidirect
22. C32×SD32D3semidirect
121. SD32#272semidirect
124. H(3,𝔽3)#508semidirect
176. C32#4916semidirect

Maximal subgroups

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

subgroupindex
#117218082
#117217942
H(3,𝔽3)×SD322
#117205702
#117205652
#117205132
C16×#2722
C3×#49163