Finite Group Databasebeta

← #13146987

Subgroups — #13146987

Order 1728.

292 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C9×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2#11722374C211yesmaximal, characteristic, complemented
3C9⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593665—31nomaximal
5#10589533—32nomaximal
6#10588926—31nomaximal
7C9×#41⋊C3C2211yescharacteristic
8C2×C9×SL(2,𝔽3)—21no—
9C9⋊GL(2,𝔽3)—21no—
10#88544—41nomaximal
11C3×2−1+4⋊C3D311yescharacteristic
12#64267—31no—
13C3⋊GU(2,𝔽3)—31no—
14C9×2−1+4D311yescharacteristic, complemented
152−1+4⋊C9D312yescomplemented
16#63692—32no—
17#63512—31no—
18#63512—31no—
19#63509—31no—
20#63422—32no—
21#63396—31no—
22#63396—31no—
23#63394—31no—
24C3×D4×C9—41no—
25C4⋊(C3⋊D9)—41no—
26C9×SL(2,𝔽3)D411yescharacteristic
27C3⋊(C9⋊C8)—41no—
28#5431—93no—
29#5245—91no—
30C3×#41⋊C3C2×D311yescharacteristic
31C2×C3×SL(2,𝔽3)—21no—
32C3⋊GL(2,𝔽3)—61no—
33#3753—121no—
34C9×#41—31no—
35C9×#41—61no—
36C9×#41C2×D311yescharacteristic
37C2×Q8×C9—21no—
38C2×Q8×C9—31no—
39C2×D36—31no—
40#41⋊C9C2×D312yes—
41C2×Q8⋊C9—22no—
42#3689—62no—
43#3675—61no—
44#3675—61no—
45#3673—61no—
46#3673—61no—
47#3673—122no—
48(C2×D9)⋊C4—62no—
49C4×Dic9—31no—
50#3667—31no—
51#3667—31no—
52C22×C3×C9—81no—
53C2×C3⋊D9—81no—
54C3×C4×C9—41no—
55C3×2−1+4C3⋊D311yescharacteristic
562−1+4⋊C3D913yescomplemented
57#986—93no—
58#949—91no—
59#949—91no—
60#947—91no—
61GU(2,𝔽3)—93no—
62#837—91no—
63#837—91no—
64#836—91no—
65C32×D4—41no—
66C4⋊(C3⋊D3)—121no—
67C3×SL(2,𝔽3)#6711yescharacteristic
68C22×D9—61no—
69C3⋊(C3⋊C8)—121no—
70Q8×C9—31no—
71Q8×C9—61no—
72Q8×C9#6711yescharacteristic
73D4×C9—31no—
74D4×C9—42no—
75D4×C9—61no—
76D4×C9S411yescharacteristic
77C2×C4×C9—31no—
78C2×C4×C9—32no—
79C2×C4×C9—61no—
80C2×Dic9—61no—
81D36—61no—
82D36—61no—
83D36—122no—
84Q8⋊C9#6712yes—
85C9⋊C8—61no—
86C9⋊C8—61no—
87C9⋊C8—122no—
88#454—271noSylow
89C2×C3×C9—41no—
90C2×C3×C9—161no—
91C3⋊D9—161no—
92C3×#41—31no—
93C3×#41—61no—
94C3×#41C2×C3⋊D311yescharacteristic
95C2×C3×Q8—21no—
96C2×C3×Q8—31no—
97C2×D12—91no—
98#41⋊C3C2×D913yes—
99C2×SL(2,𝔽3)—23no—
100GL(2,𝔽3)—183no—
101#215—181no—
102#215—181no—
103#213—181no—
104#213—181no—
105#213—363no—
106(C2×D3)⋊C4—182no—
107C4×Dic3—91no—
108#208—91no—
109#208—91no—
110C22×C32—81no—
111C2×C3⋊D3—241no—
112C32×C4—41no—
113C22×C9—21no—
114C22×C9—31no—
115C22×C9—82no—
116C2×D9—122no—
117C2×D9—121no—
118C2×D9—242no—
119C4×C9—31no—
120C4×C9—42no—
121C4×C9—61no—
122C4×C9C2×S411yescharacteristic
123Dic9—121no—
1242−1+4C3⋊D911yescharacteristic, complemented
125AGL(1,ℤ/8)—271no—
126AGL(1,ℤ/8)—271no—
127#124—271no—
128#104—271no—
129#104—271no—
130#101—271no—
131C3×C9—41noSylow
132C22×D3—181no—
133C3×Q8—31no—
134C3×Q8—61no—
135C3×Q8#63811yescharacteristic
136C3×D4—31no—
137C3×D4—43no—
138C3×D4—61no—
139C3×D4C3⋊S411yescharacteristic
140C2×C3×C4—31no—
141C2×C3×C4—32no—
142C2×C3×C4—61no—
143C2×Dic3—181no—
144D12—181no—
145D12—181no—
146D12—363no—
147SL(2,𝔽3)#61113yes—
148C3⋊C8—181no—
149C3⋊C8—181no—
150C3⋊C8—363no—
151C2×C32—41no—
152C2×C32—161no—
153C3⋊D3—481no—
154C2×C9—41no—
155C2×C9—42no—
156C2×C9—61no—
157C2×C9—162no—
158C2×C9GL(2,ℤ/4)11yescharacteristic
159D9—241no—
160D9—482no—
161#41—31no—
162#41—61no—
163#41C2×C3⋊D911yescharacteristic
164C2×Q8—21no—
165C2×Q8—31no—
166C2×D4—271no—
167SD16—541no—
168SD16—541no—
169D8—541no—
170D8—541no—
171M16—271no—
172M16—271no—
173C22⋊C4—542no—
174C42—271no—
175C22×C3—21no—
176C22×C3—31no—
177C22×C3—83no—
178C2×D3—362no—
179C2×D3—361no—
180C2×D3—723no—
181C3×C4—31no—
182C3×C4—43no—
183C3×C4—61no—
184C3×C4C2×C3⋊S411yescharacteristic
185Dic3—361no—
186C32—41no—
187C9—42no—
188C9#543111yescharacteristic, complemented
189C23—541no—
190Q8—31no—
191Q8—61no—
192Q8#623311yescharacteristic
193D4—31no—
194D4—61no—
195D4—541no—
196D4—541no—
197D4C9⋊S411yescharacteristic
198C2×C4—31no—
199C2×C4—32no—
200C2×C4—61no—
201C2×C4—541no—
202C8—541no—
203C8—541no—
204C2×C3—41no—
205C2×C3—43no—
206C2×C3—61no—
207C2×C3—163no—
208C2×C3#6426911yescharacteristic
209D3—721no—
210D3—1443no—
211C22—21no—
212C22—31no—
213C22—1082no—
214C22—1081no—
215C4—31no—
216C4—61no—
217C4—1081no—
218C4C2×C9⋊S411yescharacteristic
219C3—43no—
220C3#1059366511yescharacteristic
221C2—41no—
222C2—61no—
223C2—2161no—
224C2#1172237611yescharacteristic
subgroupG/H
centerC2#11722376
commutator, cosocleC9×#41⋊C3C22
FrattiniC3×C4C2×C3⋊S4
FittingC9×2−1+4D3
socleC2×C3#64269
hypercenterD4C9⋊S4
hypocenterC9×SL(2,𝔽3)D4
LCSG ▷ C9×#41⋊C3 ▷ C9×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C9×2−1+4⋊C3C2semidirect
2. #11722374C2semidirect
3. C9⋊GU(2,𝔽3)C2semidirect
14. C9×2−1+4D3semidirect
15. 2−1+4⋊C9D3semidirect
56. 2−1+4⋊C3D9semidirect
124. 2−1+4C3⋊D9semidirect
188. C9#5431semidirect

Maximal subgroups

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

subgroupindex
C9×2−1+4⋊C32
#117223742
C9⋊GU(2,𝔽3)2
#105936653
#105895333
#105889263
#885444