Finite Group Databasebeta

← #13164434

Subgroups — #13164434

Order 1728.

547 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 547 conjugacy classes of subgroups — too many to draw. Only its 65 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#11722576C211yesmaximal, characteristic, complemented
2#63692⋊C3C213yesmaximal, complemented
3#63691⋊C3C213yesmaximal, complemented
4C3×#5835—31nomaximal
5#10593100C311yesmaximal, characteristic, complemented
6#88724—41nomaximal
7#88691C2213yescomplemented
8#3689⋊C3C2213yescomplemented
9#3688⋊C3C2211yescharacteristic, complemented
10C32×H(5,𝔽2)D311yescharacteristic
11C3×#1009—31no—
12C3×#950—33no—
13C3×#949—33no—
14C3×#948—31no—
15C3×#932—33no—
16C2×C3×#213—33no—
17#63700—21no—
18#63700C2×C311yescharacteristic, complemented
19#63692C2×C313yescomplemented
20#63691C2×C313yescomplemented
21Q8×3−1+2—41no—
22C4⋊AGL(1,ℤ/9)—43no—
23C4×AGL(1,ℤ/9)—43no—
24#6204C2311yescharacteristic
25C3×#576—91no—
26#5835—31no—
27C32×#41—31no—
28C32×#41C2×D313yes—
29C2×C32×D4—33no—
30C3×#239—61no—
31C3×D3×D4—63no—
32C3×#235—33no—
33C2×C3×D12—33no—
34C3×#216—31no—
35C3×#215—33no—
36C3×#214—33no—
37C3×#213—33no—
38C3×#213—36no—
39C3×#208—33no—
40C2×C3×C3⋊C8—33no—
41#3701—41no—
42#41⋊C9—23no—
43#41⋊C9C22×C313yescomplemented
44#3689C22×C313yescomplemented
45#3688C22×C311yescharacteristic, complemented
46C2×AGL(1,ℤ/9)—43no—
47C4×3−1+2—43no—
483−1+2⋊C4—41no—
49C3×H(5,𝔽2)—31no—
50C3×H(5,𝔽2)—91no—
51C3×H(5,𝔽2)C3×D311yescharacteristic
52#1009—31no—
53C3×AGL(1,ℤ/8)—93no—
54C3×AGL(1,ℤ/8)—93no—
55C3×#135—93no—
56C2×C3×D8—93no—
57C3×#131—91no—
58#950—33no—
59#949—33no—
60#948—31no—
61#932—33no—
62C2×#213—33no—
63C23×C32—61no—
64C22×C3×D3—63no—
65C32×Q8S411yescharacteristic
66C32×Q8C22×D311yescharacteristic
67C32×D4—33no—
68C32×D4—33no—
69C2×C32×C4—33no—
70C3×#67—63no—
71C3×D12—33no—
72C3×D12—63no—
73C3×C4×D3—63no—
74C3×Dic6—31no—
75C3×C3⋊C8—31no—
76C3×C3⋊C8—33no—
77Q8×C9—41no—
78Q8×C9—81no—
79D36—43no—
80C4×D9—43no—
81Q8⋊C9—21no—
82Q8⋊C9C23×C311yescharacteristic
83#576—91noSylow
84C2×3−1+2—41no—
85AGL(1,ℤ/9)—83no—
86C3×#41—31no—
87C3×#41—31no—
88C3×#41—33no—
89C3×#41—61no—
90C3×#41—93no—
91C3×#41—181no—
92C3×#41C2×C3×D313yes—
93C2×C3×D4—33no—
94C2×C3×D4—33no—
95C2×C3×D4—63no—
96C2×C3×D4—93no—
97C2×C3×D4—183no—
98#239—61no—
99D3×D4—63no—
100#235—33no—
101C2×D12—33no—
102C3×Q16—91no—
103C3×SD16—93no—
104C3×SD16—93no—
105C3×D8—93no—
106C3×D8—96no—
107C3×M16—93no—
108C2×C3×C8—93no—
109#216—31no—
110#215—33no—
111#214—33no—
112#213—33no—
113#213—36no—
114#208—33no—
115C2×C3⋊C8—33no—
116C22×C32—33no—
117C22×C32—241no—
118C2×C3×D3—63no—
119C2×C3×D3—243no—
120C32×C4—31no—
121C32×C4C2×S413yes—
122C3×Dic3—61no—
123C2×D9—43no—
124C4×C9—43no—
125C4×C9—83no—
126Dic9—41no—
127H(5,𝔽2)—91no—
128H(5,𝔽2)AGL(1,ℤ/9)11yescharacteristic, complemented
129AGL(1,ℤ/8)—93no—
130AGL(1,ℤ/8)—93no—
131#135—93no—
132C2×D8—93no—
133#131—91no—
1343−1+2—41noSylow
135C23×C3—61no—
136C23×C3—63no—
137C23×C3—183no—
138C22×D3—63no—
139C3×Q8—31no—
140C3×Q8—31no—
141C3×Q8—91no—
142C3×Q8C3×S411yescharacteristic
143C3×Q8C22×C3×D311yescharacteristic
144C3×D4—33no—
145C3×D4—33no—
146C3×D4—33no—
147C3×D4—33no—
148C3×D4—63no—
149C3×D4—63no—
150C3×D4—93no—
151C3×D4—183no—
152C3×D4—183no—
153C2×C3×C4—33no—
154C2×C3×C4—33no—
155C2×C3×C4—63no—
156C2×C3×C4—183no—
157#67—63no—
158D12—33no—
159D12—63no—
160C4×D3—63no—
161Dic6—31no—
162C3×C8—91no—
163C3×C8—93no—
164C3⋊C8—31no—
165C3⋊C8—33no—
166C2×C32—63no—
167C2×C32C22×S411yescharacteristic
168C3×D3—123no—
169C2×C9—41no—
170C2×C9—81no—
171D9—83no—
172#41—31no—
173#41—93no—
174#41—181no—
175#41C2×AGL(1,ℤ/9)13yes—
176C2×D4—33no—
177C2×D4—93no—
178C2×D4—183no—
179Q16—91no—
180SD16—93no—
181SD16—93no—
182D8—93no—
183D8—96no—
184M16—93no—
185C2×C8—93no—
186C22×C3—33no—
187C22×C3—33no—
188C22×C3—63no—
189C22×C3—183no—
190C22×C3—241no—
191C22×C3—243no—
192C22×C3—723no—
193C2×D3—63no—
194C2×D3—243no—
195C3×C4—31no—
196C3×C4—31no—
197C3×C4—33no—
198C3×C4—61no—
199C3×C4—181no—
200C3×C4C2×C3×S413yes—
201Dic3—61no—
202C32#592511yescharacteristic
203C9—41no—
204C9—81no—
205C23—61no—
206C23—183no—
207Q8—91no—
208Q8((C22⋊C9)⋊C2)⋊C311yescharacteristic
209Q8C22×AGL(1,ℤ/9)11yescharacteristic
210D4—33no—
211D4—33no—
212D4—93no—
213D4—183no—
214D4—183no—
215C2×C4—33no—
216C2×C4—183no—
217C8—91no—
218C8—93no—
219C2×C3—31no—
220C2×C3—63no—
221C2×C3—63no—
222C2×C3—123no—
223C2×C3—363no—
224C2×C3C22×C3×S411yescharacteristic
225D3—123no—
226C22—33no—
227C22—183no—
228C22—241no—
229C22—723no—
230C4—31no—
231C4—181no—
232C4C2×((C22⋊C9)⋊C2)⋊C313yes—
233C3—31no—
234C3C3×#592511yescharacteristic
235C2—63no—
236C2—363no—
237C2C22×((C22⋊C9)⋊C2)⋊C311yescharacteristic
subgroupG/H
centerC2C22×((C22⋊C9)⋊C2)⋊C3
commutator, cosocle, hypocenterQ8⋊C9C23×C3
Frattini, socleC2×C3C22×C3×S4
FittingC32×H(5,𝔽2)D3
hypercenterQ8((C22⋊C9)⋊C2)⋊C3
LCSG ▷ Q8⋊C9
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. #11722576C2semidirect
2. #63692⋊C3C2semidirect
3. #63691⋊C3C2semidirect
5. #10593100C3semidirect
7. #88691C22semidirect
8. #3689⋊C3C22semidirect
9. #3688⋊C3C22semidirect
18. #63700C2×C3semidirect
19. #63692C2×C3semidirect
20. #63691C2×C3semidirect
43. #41⋊C9C22×C3semidirect
44. #3689C22×C3semidirect
45. #3688C22×C3semidirect
128. H(5,𝔽2)AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
#117225762
#63692⋊C32
#63691⋊C32
C3×#58353
#105931003
#887244