Finite Group Databasebeta

← #13144778

Subgroups — #13144778

Order 1728.

462 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 95 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
1H(3,𝔽3)×#135C211yesmaximal, characteristic, complemented
2#11721779C211yesmaximal, characteristic, complemented
3#11720570C211yesmaximal, characteristic, complemented
4#11720569C211yesmaximal, characteristic, complemented
5#11720566C211yesmaximal, characteristic, complemented
6#11720565C211yesmaximal, characteristic, complemented
7C2×#88386C211yesmaximal, characteristic, complemented
8C3×#5194—34nomaximal
9#41×H(3,𝔽3)—21no—
10#88740—21no—
11Q16×H(3,𝔽3)C2211yescharacteristic
12SD16×H(3,𝔽3)—21no—
13D8×H(3,𝔽3)C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15#88530C2211yescharacteristic
16#88529C2211yescharacteristic
17#88528—21no—
18#88386C2211yescharacteristic
19#88386C2211yescharacteristic
20C32×#135D314yescomplemented
21C3×#904—34no—
22C3×#829—34no—
23C3×#828—34no—
24C3×#827—34no—
25C3×#826—34no—
26C2×C3×C3⋊C16—34no—
27Q8×H(3,𝔽3)—21no—
28D4×H(3,𝔽3)—21no—
29D4×H(3,𝔽3)—41no—
30C2×C4×H(3,𝔽3)—41no—
31C2×C4×H(3,𝔽3)D411yescharacteristic
32#6235—41no—
33C4⋊#272—21no—
34C4×#272—41no—
35#6229—21no—
36C8×H(3,𝔽3)D411yescharacteristic
37C8×H(3,𝔽3)C2311yescharacteristic
38C3×#508—91no—
39#5194—34no—
40C32×#41—24no—
41C3×#235—64no—
42C32×Q16C2×D314yes—
43C32×SD16—24no—
44C32×D8C2×D314yes—
45C2×C32×C8C2×D314yes—
46C3×Dic12—34no—
47C3×D24—34no—
48C3×#204—64no—
49C3×C3⋊C16—34no—
50C3×C3⋊C16—34no—
51C22×H(3,𝔽3)—41no—
52C22×H(3,𝔽3)D811yescharacteristic
53C2×#272—41no—
54C4×H(3,𝔽3)—41no—
55C4×H(3,𝔽3)D811yescharacteristic
56C4×H(3,𝔽3)C2×D411yescharacteristic
57#1084—41no—
58C3×#135—34no—
59C3×#135—91no—
60C3×#135C3⋊D311yescharacteristic
61#904—34no—
62C3×Q32—91no—
63C3×SD32—91no—
64C3×SD32—91no—
65C3×D16—91no—
66C2×C3×C16—91no—
67#829—34no—
68#828—34no—
69#827—34no—
70#826—34no—
71C2×C3⋊C16—34no—
72C32×Q8—24no—
73C32×D4—24no—
74C32×D4—44no—
75C2×C32×C4—44no—
76C2×C32×C4#6714yes—
77C3×#67—124no—
78C3×D12—64no—
79C3×C4×D3—124no—
80C3×Dic6—64no—
81C32×C8#6714yes—
82C32×C8C22×D314yes—
83#508—91noSylow
84C2×H(3,𝔽3)—21no—
85C2×H(3,𝔽3)—81no—
86C2×H(3,𝔽3)C2×D811yescharacteristic
87#272—81no—
88C3×#41—21no—
89C3×#41—64no—
90C3×#41—181no—
91#235—64no—
92C3×Q16—34no—
93C3×Q16—91no—
94C3×Q16C2×C3⋊D311yescharacteristic
95C3×SD16—21no—
96C3×SD16—64no—
97C3×SD16—181no—
98C3×D8—34no—
99C3×D8—91no—
100C3×D8C2×C3⋊D311yescharacteristic
101C2×C3×C8—34no—
102C2×C3×C8C2×C3⋊D311yescharacteristic
103Dic12—34no—
104D24—34no—
105#204—64no—
106C3×C16—91no—
107C3×C16—91no—
108C3⋊C16—34no—
109C3⋊C16—34no—
110C22×C32—44no—
111C22×C32#21314yes—
112C2×C3×D3—124no—
113C32×C4—44no—
114C32×C4#21314yes—
115C32×C4C2×#6714yes—
116C3×Dic3—124no—
117#135—91no—
118#135#27211yescharacteristic, complemented
119Q32—91no—
120SD32—91no—
121SD32—91no—
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—41no—
130C3×D4—64no—
131C3×D4—124no—
132C3×D4—181no—
133C3×D4—361no—
134C2×C3×C4—34no—
135C2×C3×C4—41no—
136C2×C3×C4—124no—
137C2×C3×C4—361no—
138C2×C3×C4#63811yescharacteristic
139#67—124no—
140D12—64no—
141C4×D3—124no—
142Dic6—64no—
143C3×C8—34no—
144C3×C8—34no—
145C3×C8#63811yescharacteristic
146C3×C8C22×C3⋊D311yescharacteristic
147C2×C32—24no—
148C2×C32—84no—
149C2×C32C2×#21314yes—
150C3×D3—244no—
151#41—21no—
152#41—181no—
153Q16—91no—
154Q16C2×#27211yescharacteristic
155SD16—21no—
156SD16—181no—
157D8—91no—
158D8C2×#27211yescharacteristic
159C2×C8C2×#27211yescharacteristic
160C16—91no—
161C16—91no—
162C22×C3—34no—
163C22×C3—41no—
164C22×C3—124no—
165C22×C3—361no—
166C22×C3#375311yescharacteristic
167C2×D3—124no—
168C3×C4—34no—
169C3×C4—34no—
170C3×C4—41no—
171C3×C4—124no—
172C3×C4—361no—
173C3×C4#375311yescharacteristic
174C3×C4C2×#63811yescharacteristic
175Dic3—124no—
176C32#519414yescomplemented
177Q8—21no—
178Q8—181no—
179D4—21no—
180D4—41no—
181D4—181no—
182D4—361no—
183C2×C4—41no—
184C2×C4—361no—
185C2×C4#623511yescharacteristic
186C8#623511yescharacteristic
187C8C22×#27211yescharacteristic
188C2×C3—21no—
189C2×C3—34no—
190C2×C3—64no—
191C2×C3—81no—
192C2×C3—244no—
193C2×C3—721no—
194C2×C3C2×#375311yescharacteristic
195D3—244no—
196C22—41no—
197C22—361no—
198C22#8854511yescharacteristic
199C4—41no—
200C4—361no—
201C4#8854511yescharacteristic
202C4C2×#623511yescharacteristic
203C3—34no—
204C3#1059289311yescharacteristic
205C2—21no—
206C2—81no—
207C2—721no—
208C2C2×#8854511yescharacteristic
subgroupG/H
centerC3×C4#3753
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C22×C3⋊D3
FittingH(3,𝔽3)×#135C2
socleC2×C3C2×#3753
hypercenterC3×#135C3⋊D3
hypocenterH(3,𝔽3)#508
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C3×#135

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#135C2semidirect
2. #11721779C2semidirect
3. #11720570C2semidirect
4. #11720569C2semidirect
5. #11720566C2semidirect
6. #11720565C2semidirect
7. C2×#88386C2semidirect
20. C32×#135D3semidirect
118. #135#272semidirect
124. H(3,𝔽3)#508semidirect
176. C32#5194semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1352
#117217792
#117205702
#117205692
#117205662
#117205652
C2×#883862
C3×#51943