Finite Group Databasebeta

← #13164432

Subgroups — #13164432

Order 1728.

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

normal subgroups only

This group has 525 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#63691⋊C3C213yesmaximal, complemented
3#63690⋊C3C213yesmaximal, complemented
4C3×#5836—31nomaximal
5#10593099C311yesmaximal, characteristic, complemented
6Q8×AGL(1,ℤ/9)—41nomaximal
7#88691C2213yes—
8#3689⋊C3C2211yescharacteristic, complemented
9#3688⋊C3C2213yescomplemented
10C32×H(5,𝔽2)D311yescharacteristic
11C3×#1010—31no—
12C3×#951—33no—
13C3×#950—33no—
14C3×#948—31no—
15C2×C3×#214—33no—
16C3×#932—33no—
17#63700—21no—
18#63700C2×C311yescharacteristic, complemented
19#63691C2×C313yescomplemented
20#63690C2×C313yescomplemented
21Q8×3−1+2—41no—
22C4×AGL(1,ℤ/9)—43no—
233−1+2⋊Q8—43no—
24#6204C2311yescharacteristic
25C3×#577—91no—
26#5836—31no—
27C32×#41—31no—
28C32×#41C2×D313yes—
29C2×C32×D4—33no—
30C3×D3×Q8—61no—
31C3×#237—63no—
32C3×#235—33no—
33C2×C3×Dic6—33no—
34C3×#216—33no—
35C3×#215—31no—
36C3×#214—36no—
37C3×#214—33no—
38C3×#213—33no—
39C3×#208—33no—
40C2×C3×C3⋊C8—33no—
41Q8×D9—41no—
42#41⋊C9—23no—
43#41⋊C9C22×C313yes—
44#3689C22×C311yescharacteristic, complemented
45#3688C22×C313yescomplemented
46C2×AGL(1,ℤ/9)—41no—
47C4×3−1+2—43no—
483−1+2⋊C4—43no—
49C3×2−1+4—91no—
50C3×H(5,𝔽2)—31no—
51C3×H(5,𝔽2)C3×D311yescharacteristic
52#1010—31no—
53C3×#137—93no—
54C3×AGL(1,ℤ/8)—93no—
55C3×#135—93no—
56C2×C3×SD16—93no—
57C3×#131—91no—
58#951—33no—
59#950—33no—
60#948—31no—
61C2×#214—33no—
62#932—33no—
63C23×C32—61no—
64C32×Q8S411yescharacteristic
65C32×Q8C22×D311yescharacteristic
66C32×D4—33no—
67C32×D4—33no—
68C2×C32×C4—33no—
69C3×#67—63no—
70C2×C3×Dic3—63no—
71C3×D12—31no—
72C3×C4×D3—63no—
73C3×Dic6—33no—
74C3×Dic6—63no—
75C3×C3⋊C8—31no—
76C3×C3⋊C8—33no—
77Q8×C9—41no—
78Q8×C9—81no—
79C4×D9—43no—
80Dic18—43no—
81Q8⋊C9—21no—
82Q8⋊C9C23×C311yescharacteristic
83#577—91noSylow
84C2×3−1+2—41no—
85AGL(1,ℤ/9)—42no—
86C3×#41—31no—
87C3×#41—31no—
88C3×#41—33no—
89C3×#41—61no—
90C3×#41—93no—
91C3×#41—183no—
92C3×#41C2×C3×D313yes—
93C2×C3×Q8—93no—
94C2×C3×Q8—181no—
95C2×C3×D4—33no—
96C2×C3×D4—33no—
97C2×C3×D4—63no—
98D3×Q8—61no—
99#237—63no—
100#235—33no—
101C2×Dic6—33no—
102C3×Q16—93no—
103C3×SD16—93no—
104C3×SD16—96no—
105C3×SD16—91no—
106C3×D8—93no—
107C3×M16—93no—
108C2×C3×C8—93no—
109#216—33no—
110#215—31no—
111#214—36no—
112#214—33no—
113#213—33no—
114#208—33no—
115C2×C3⋊C8—33no—
116C22×C32—33no—
117C22×C32—241no—
118C2×C3×D3—61no—
119C32×C4—31no—
120C32×C4C2×S413yes—
121C3×Dic3—63no—
122C2×D9—41no—
123C4×C9—43no—
124C4×C9—83no—
125Dic9—43no—
1262−1+4—91no—
127H(5,𝔽2)AGL(1,ℤ/9)11yescharacteristic, complemented
128#137—93no—
129AGL(1,ℤ/8)—93no—
130#135—93no—
131C2×SD16—93no—
132#131—91no—
1333−1+2—41noSylow
134C23×C3—61no—
135C23×C3—63no—
136C3×Q8—31no—
137C3×Q8—31no—
138C3×Q8—93no—
139C3×Q8—183no—
140C3×Q8C3×S411yescharacteristic
141C3×Q8C22×C3×D311yescharacteristic
142C3×D4—33no—
143C3×D4—33no—
144C3×D4—33no—
145C3×D4—33no—
146C3×D4—63no—
147C3×D4—63no—
148C3×D4—91no—
149C3×D4—183no—
150C2×C3×C4—33no—
151C2×C3×C4—33no—
152C2×C3×C4—63no—
153C2×C3×C4—183no—
154C2×C3×C4—183no—
155#67—63no—
156C2×Dic3—63no—
157D12—31no—
158C4×D3—63no—
159Dic6—33no—
160Dic6—63no—
161C3×C8—91no—
162C3×C8—93no—
163C3⋊C8—31no—
164C3⋊C8—33no—
165C2×C32—63no—
166C2×C32C22×S411yescharacteristic
167C3×D3—121no—
168C2×C9—41no—
169C2×C9—81no—
170D9—42no—
171#41—31no—
172#41—93no—
173#41—183no—
174#41C2×AGL(1,ℤ/9)13yes—
175C2×Q8—93no—
176C2×Q8—181no—
177C2×D4—33no—
178Q16—93no—
179SD16—93no—
180SD16—96no—
181SD16—91no—
182D8—93no—
183M16—93no—
184C2×C8—93no—
185C22×C3—33no—
186C22×C3—33no—
187C22×C3—63no—
188C22×C3—181no—
189C22×C3—241no—
190C22×C3—243no—
191C2×D3—61no—
192C3×C4—31no—
193C3×C4—31no—
194C3×C4—33no—
195C3×C4—61no—
196C3×C4—183no—
197C3×C4C2×C3×S413yes—
198Dic3—63no—
199C32#592411yescharacteristic
200C9—41no—
201C9—81no—
202C23—61no—
203Q8—93no—
204Q8—183no—
205Q8((C22⋊C9)⋊C2)⋊C311yescharacteristic
206Q8C22×AGL(1,ℤ/9)11yescharacteristic
207D4—33no—
208D4—33no—
209D4—91no—
210D4—183no—
211C2×C4—33no—
212C2×C4—183no—
213C2×C4—183no—
214C8—91no—
215C8—93no—
216C2×C3—31no—
217C2×C3—63no—
218C2×C3—63no—
219C2×C3—123no—
220C2×C3—361no—
221C2×C3C22×C3×S411yescharacteristic
222D3—121no—
223C22—33no—
224C22—181no—
225C22—241no—
226C4—31no—
227C4—183no—
228C4C2×((C22⋊C9)⋊C2)⋊C313yes—
229C3—31no—
230C3C3×#592411yescharacteristic
231C2—63no—
232C2—361no—
233C2C22×((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. #63691⋊C3C2semidirect
3. #63690⋊C3C2semidirect
5. #10593099C3semidirect
8. #3689⋊C3C22semidirect
9. #3688⋊C3C22semidirect
18. #63700C2×C3semidirect
19. #63691C2×C3semidirect
20. #63690C2×C3semidirect
44. #3689C22×C3semidirect
45. #3688C22×C3semidirect
127. 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
#63691⋊C32
#63690⋊C32
C3×#58363
#105930993
Q8×AGL(1,ℤ/9)4