Finite Group Databasebeta

← #13173079

Subgroups — #13173079

Order 1728.

756 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 111 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×#994C211yesmaximal, characteristic, complemented
2#11724471C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊GU(2,𝔽3))C211yesmaximal, characteristic, complemented
4#10593663—312nomaximal
5#10592940—31nomaximal
6C32×#41⋊C3—21no—
7C32×#41⋊C3C2211yescharacteristic
8C3⋊(C3⋊GL(2,𝔽3))—21no—
9#88862—41nomaximal
10C32×H(5,𝔽2)D311yescharacteristic, complemented
11C3×#994D3112yescomplemented
12#64267—312no—
13#64158—31no—
14#64158—31no—
15#64149—31no—
16C3⋊GU(2,𝔽3)—312no—
17#63664—31no—
18#63664—31no—
19#63660—31no—
20C33×Q8—41no—
21C4⋊(C3⋊(C3⋊D3))—41no—
22C32×SL(2,𝔽3)D411yescharacteristic, complemented
23C3⋊(C3⋊(C3⋊C8))—41no—
24#5429—99no—
25#5241—94no—
26C32×#41—21no—
27C32×#41—31no—
28C32×#41C2×D311yescharacteristic
29C2×C32×D4—31no—
30C2×C32×D4—61no—
31C2×#638—62no—
32C2×C4⋊(C3⋊D3)—31no—
33C3×#41⋊C3—212no—
34C3×#41⋊C3C2×D3112yes—
35C3⋊GL(2,𝔽3)—612no—
36#3755—61no—
37#3755—61no—
38#3755—1212no—
39#3753—61no—
40#3753—61no—
41C4×C3⋊Dic3—31no—
42#3748—31no—
43#3748—31no—
44C2×C3⋊(C3⋊D3)—81no—
45C33×C4—41no—
46C33×C4—81no—
47C3×H(5,𝔽2)C3⋊D314yescomplemented
48#994C3⋊D319yescomplemented
49#986—99no—
50#949—94no—
51#949—94no—
52#940—94no—
53GU(2,𝔽3)—99no—
54#837—94no—
55#837—94no—
56#833—94no—
57C23×C32—32no—
58C22×C3⋊D3—61no—
59C32×Q8—412no—
60C32×Q8#6711yescharacteristic
61C32×Q8S411yescharacteristic
62C32×D4—31no—
63C32×D4—31no—
64C32×D4—61no—
65C32×D4—61no—
66C2×C32×C4—31no—
67C2×C32×C4—61no—
68#638—124no—
69C2×C3⋊Dic3—61no—
70C4⋊(C3⋊D3)—61no—
71C4⋊(C3⋊D3)—61no—
72C4⋊(C3⋊D3)—1212no—
73C3×SL(2,𝔽3)#67112yescomplemented
74C3⋊(C3⋊C8)—61no—
75C3⋊(C3⋊C8)—61no—
76C3⋊(C3⋊C8)—1212no—
77#453—271noSylow
78C2×C33—41no—
79C3⋊(C3⋊D3)—161no—
80C3×#41—24no—
81C3×#41—34no—
82C3×#41C2×C3⋊D314yes—
83C2×C3×D4—34no—
84C2×C3×D4—64no—
85C2×#67—188no—
86C2×D12—94no—
87#41⋊C3—29no—
88#41⋊C3C2×C3⋊D319yes—
89GL(2,𝔽3)—189no—
90#215—184no—
91#215—184no—
92#215—369no—
93#213—184no—
94#213—184no—
95C4×Dic3—94no—
96#208—94no—
97#208—94no—
98C22×C32—31no—
99C22×C32—61no—
100C22×C32—122no—
101C2×C3⋊D3—122no—
102C2×C3⋊D3—121no—
103C2×C3⋊D3—2412no—
104C32×C4—21no—
105C32×C4—31no—
106C32×C4—412no—
107C32×C4—812no—
108C32×C4C2×S411yescharacteristic
109C3⋊Dic3—62no—
110H(5,𝔽2)C3⋊(C3⋊D3)11yescharacteristic, complemented
111AGL(1,ℤ/8)—271no—
112AGL(1,ℤ/8)—271no—
113#127—271no—
114#104—271no—
115#104—271no—
116#100—271no—
117C33—41noSylow
118C23×C3—38no—
119C22×D3—184no—
120C3×Q8—49no—
121C3×Q8#63814yes—
122C3×Q8C3⋊S414yes—
123C3×D4—34no—
124C3×D4—34no—
125C3×D4—64no—
126C3×D4—64no—
127C2×C3×C4—34no—
128C2×C3×C4—64no—
129#67—3616no—
130C2×Dic3—184no—
131D12—184no—
132D12—184no—
133D12—369no—
134SL(2,𝔽3)#63819yescomplemented
135C3⋊C8—184no—
136C3⋊C8—184no—
137C3⋊C8—369no—
138C2×C32—412no—
139C2×C32—61no—
140C2×C32—121no—
141C2×C32GL(2,ℤ/4)11yescharacteristic
142C3⋊D3—241no—
143C3⋊D3—4812no—
144#41—21no—
145#41—31no—
146#41C2×C3⋊(C3⋊D3)11yescharacteristic
147C2×D4—31no—
148C2×D4—61no—
149C2×D4—271no—
150C2×D4—542no—
151SD16—541no—
152SD16—541no—
153D8—541no—
154D8—541no—
155M16—271no—
156M16—271no—
157C42—271no—
158C22×C3—34no—
159C22×C3—64no—
160C22×C3—128no—
161C2×D3—368no—
162C2×D3—364no—
163C2×D3—729no—
164C3×C4—24no—
165C3×C4—34no—
166C3×C4—49no—
167C3×C4—89no—
168C3×C4C2×C3⋊S414yes—
169Dic3—188no—
170C32—412no—
171C32#542911yescharacteristic, complemented
172C23—32no—
173C23—541no—
174Q8#631211yescharacteristic
175Q8C3⋊(C3⋊S4)11yescharacteristic
176D4—31no—
177D4—31no—
178D4—61no—
179D4—61no—
180D4—541no—
181D4—541no—
182D4—1084no—
183C2×C4—31no—
184C2×C4—61no—
185C2×C4—541no—
186C8—541no—
187C8—541no—
188C2×C3—49no—
189C2×C3—64no—
190C2×C3—124no—
191C2×C3#6426914yes—
192D3—724no—
193D3—1449no—
194C22—31no—
195C22—61no—
196C22—122no—
197C22—1082no—
198C22—1081no—
199C4—21no—
200C4—31no—
201C4—542no—
202C4C2×C3⋊(C3⋊S4)11yescharacteristic
203C3—49no—
204C3#1059366314yescomplemented
205C2—61no—
206C2—121no—
207C2—2161no—
208C2#1172447311yescharacteristic
subgroupG/H
centerC2#11724473
commutator, cosocleC32×#41⋊C3C22
FrattiniC4C2×C3⋊(C3⋊S4)
FittingC32×H(5,𝔽2)D3
socleC2×C32GL(2,ℤ/4)
hypercenterQ8C3⋊(C3⋊S4)
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C32×#41⋊C3 ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C32×#994C2semidirect
2. #11724471C2semidirect
3. C3⋊(C3⋊GU(2,𝔽3))C2semidirect
10. C32×H(5,𝔽2)D3semidirect
11. C3×#994D3semidirect
22. C32×SL(2,𝔽3)D4semidirect
47. C3×H(5,𝔽2)C3⋊D3semidirect
48. #994C3⋊D3semidirect
73. C3×SL(2,𝔽3)#67semidirect
110. H(5,𝔽2)C3⋊(C3⋊D3)semidirect
134. SL(2,𝔽3)#638semidirect
171. C32#5429semidirect
204. C3#10593663semidirect

Maximal subgroups

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

subgroupindex
C32×#9942
#117244712
C3⋊(C3⋊GU(2,𝔽3))2
#105936633
#105929403
#888624