Finite Group Databasebeta

← C32×#5931 #13181415

Subgroups — C32×#5931 #13181415

Order 1728.

510 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 54 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
1C32×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C3×#5931C314yesmaximal, direct product factor
3C32×#448—31nomaximal
4C2×C32×S4—41nomaximal
5C32×GL(2,𝔽3)—42nomaximal
6C22×C32×Q8D311yescharacteristic, complemented
7C3×C22⋊SL(2,𝔽3)—24no—
8C3×C22⋊SL(2,𝔽3)C2×C314yescomplemented
9C2×C32×SD16—32no—
10C32×#121—31no—
11C32×#103—32no—
12C32×C22⋊C8—31no—
13C2×C32×A4—41no—
14C32×S4—42no—
15C32×SL(2,𝔽3)—42no—
16C32×SL(2,𝔽3)—81no—
17#5931C3211yescharacteristic, direct product factor
18C3×#448—34no—
19C2×C3×S4—44no—
20C2×C32×Q8—32no—
21C2×C32×Q8—61no—
22C2×C32×D4—31no—
23C2×C32×D4—61no—
24C22×C32×C4—31no—
25C3×GL(2,𝔽3)—48no—
26C32×SD16—62no—
27C32×SD16—62no—
28C2×C32×C8—32no—
29C32×C4⋊C4—31no—
30C32×C22⋊C4—61no—
31C2×C32×D3—161no—
32C32×A4—41no—
33C22×C3×Q8C3×D314yescomplemented
34C22⋊SL(2,𝔽3)—24no—
35C22⋊SL(2,𝔽3)C2×C3211yescharacteristic, complemented
36C2×C3×SD16—38no—
37C3×#121—34no—
38C3×#103—38no—
39C3×C22⋊C8—34no—
40C23×C32—61no—
41C23×C32S411yescharacteristic
42C2×C3×A4—44no—
43C2×C3×A4—84no—
44C3×S4—48no—
45C32×Q8—21no—
46C32×Q8—32no—
47C32×Q8—61no—
48C32×Q8S412yescomplemented
49C32×D4—62no—
50C32×D4—121no—
51C32×D4—121no—
52C2×C32×C4—31no—
53C2×C32×C4—31no—
54C2×C32×C4—62no—
55C2×C32×C4—61no—
56C3×SL(2,𝔽3)—48no—
57C3×SL(2,𝔽3)—84no—
58C3×SL(2,𝔽3)—88no—
59C3×SL(2,𝔽3)—88no—
60C32×C8—62no—
61#448—31noSylow
62C2×C33—161no—
63C32×D3—162no—
64C2×S4—41no—
65C2×C3×Q8—38no—
66C2×C3×Q8—64no—
67C2×C3×D4—34no—
68C2×C3×D4—64no—
69C22×C3×C4—34no—
70GL(2,𝔽3)—42no—
71C3×SD16—68no—
72C3×SD16—68no—
73C2×C3×C8—38no—
74C3×C4⋊C4—34no—
75C3×C22⋊C4—64no—
76C22×C32—31no—
77C22×C32—31no—
78C22×C32—121no—
79C22×C32—121no—
80C22×C32—121no—
81C22×C32GL(2,𝔽3)11yescharacteristic, complemented
82C2×C3×D3—164no—
83C3×A4—44no—
84C3×A4—84no—
85C32×C4—32no—
86C32×C4—61no—
87C32×C4—121no—
88C22×Q8C32×D311yescharacteristic, complemented
89C2×SD16—32no—
90#121—31no—
91#103—32no—
92C22⋊C8—31no—
93C33—161noSylow
94C23×C3—64no—
95C23×C3C3×S414yes—
96C2×A4—41no—
97C2×A4—84no—
98S4—42no—
99C3×Q8—24no—
100C3×Q8—38no—
101C3×Q8—64no—
102C3×Q8C3×S418yescomplemented
103C3×D4—68no—
104C3×D4—124no—
105C3×D4—124no—
106C2×C3×C4—34no—
107C2×C3×C4—34no—
108C2×C3×C4—68no—
109C2×C3×C4—64no—
110SL(2,𝔽3)—42no—
111SL(2,𝔽3)—81no—
112SL(2,𝔽3)—88no—
113SL(2,𝔽3)—88no—
114C3×C8—68no—
115C2×C32—31no—
116C2×C32—31no—
117C2×C32—164no—
118C2×C32—241no—
119C2×C32—324no—
120C2×C32#102011yescharacteristic
121C3×D3—168no—
122C2×Q8—32no—
123C2×Q8—61no—
124C2×D4—31no—
125C2×D4—61no—
126C22×C4—31no—
127SD16—62no—
128SD16—62no—
129C2×C8—32no—
130C4⋊C4—31no—
131C22⋊C4—61no—
132C22×C3—34no—
133C22×C3—34no—
134C22×C3—124no—
135C22×C3—124no—
136C22×C3—124no—
137C22×C3C3×GL(2,𝔽3)14yescomplemented
138C2×D3—161no—
139A4—41no—
140A4—84no—
141C3×C4—38no—
142C3×C4—64no—
143C3×C4—124no—
144C32—164no—
145C32—324no—
146C32#593111yescharacteristic, direct product factor
147C23—61no—
148C23C32×S411yescharacteristic
149Q8—21no—
150Q8—32no—
151Q8—61no—
152Q8C32×S412yescomplemented
153D4—62no—
154D4—121no—
155D4—121no—
156C2×C4—31no—
157C2×C4—31no—
158C2×C4—62no—
159C2×C4—61no—
160C8—62no—
161C2×C3—34no—
162C2×C3—34no—
163C2×C3—161no—
164C2×C3—244no—
165C2×C3—324no—
166C2×C3C3×#102014yes—
167D3—162no—
168C22—31no—
169C22—31no—
170C22—121no—
171C22—121no—
172C22—121no—
173C22C32×GL(2,𝔽3)11yescharacteristic, complemented
174C4—32no—
175C4—61no—
176C4—121no—
177C3—161no—
178C3—324no—
179C3C3×#593114yesdirect product factor
180C2—31no—
181C2—31no—
182C2—241no—
183C2C32×#102011yescharacteristic
subgroupG/H
center, hypercenterC2×C32#1020
commutator, cosocle, hypocenterC22⋊SL(2,𝔽3)C2×C32
FrattiniC2C32×#1020
FittingC22×C32×Q8D3
socleC23×C32S4
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
1. C32×C22⋊SL(2,𝔽3)C2semidirect
2. C3×#5931C3direct (N × H)
6. C22×C32×Q8D3semidirect
8. C3×C22⋊SL(2,𝔽3)C2×C3semidirect
17. #5931C32direct (N × H)
33. C22×C3×Q8C3×D3semidirect
35. C22⋊SL(2,𝔽3)C2×C32semidirect
48. C32×Q8S4semidirect
81. C22×C32GL(2,𝔽3)semidirect
88. C22×Q8C32×D3semidirect
102. C3×Q8C3×S4semidirect
137. C22×C3C3×GL(2,𝔽3)semidirect
146. C32#5931direct (N × H)
152. Q8C32×S4semidirect
173. C22C32×GL(2,𝔽3)semidirect
179. C3C3×#5931direct (N × H)

Maximal subgroups

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

subgroupindex
C32×C22⋊SL(2,𝔽3)2
C3×#59313
C32×#4483
C2×C32×S44
C32×GL(2,𝔽3)4