Finite Group Databasebeta

← #13147111

Subgroups — #13147111

Order 1728.

281 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 44 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
1C2×D9⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#88614C211yesmaximal, characteristic, complemented
3C4×C9⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#5576⋊C3—31nomaximal
5#10589678C311yesmaximal, characteristic, complemented
6C2×C9⋊SL(2,𝔽3)C2211yescharacteristic
7D9⋊SL(2,𝔽3)—21no—
8#88614—21no—
9(C2×AGL(1,ℤ/9))⋊C4—41nomaximal
10C2×D3×SL(2,𝔽3)—31no—
11C2×C3⋊(#41⋊C3)—31no—
12C3×C4×SL(2,𝔽3)D311yescharacteristic
13C2×#3701C2×C311yescharacteristic, complemented
14C2×Q8×D9C2×C311yescharacteristic
15C4×Q8×C9C2×C311yescharacteristic, complemented
16#63459—31no—
17#63457—31no—
18#63437—31no—
19C4×D36—31no—
20C4×Q8⋊C9—21no—
21C22×AGL(1,ℤ/9)—41no—
22C2×C4×3−1+2—41no—
23C2×3−1+2⋊C4—41no—
24C9⋊SL(2,𝔽3)D411yescharacteristic, complemented
25#5576—31no—
26#548⋊C3—91no—
27C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
28D3×SL(2,𝔽3)—61no—
29C3⋊(#41⋊C3)—61no—
30C3×(C2×D3)⋊C4—121no—
31C2×Q8×C9C22×C311yescharacteristic
32#3701—21no—
33#3701—61no—
34Q8×D9—21no—
35Q8×D9—61no—
36C2×D36—31no—
37C2×D36—31no—
38C2×C4×D9—31no—
39C2×C4×D9—31no—
40C2×Dic18—31no—
41C2×Q8⋊C9—21no—
42C9×C4⋊C4—31no—
43C42×C9—31no—
44(C2×D9)⋊C4—32no—
45(C2×D9)⋊C4—31no—
46(C2×D9)⋊C4A411yescharacteristic
47C4⋊Dic9—31no—
48C22×3−1+2—41no—
49C2×AGL(1,ℤ/9)—42no—
50C2×AGL(1,ℤ/9)—81no—
51C2×AGL(1,ℤ/9)—81no—
52C4×3−1+2—81no—
533−1+2⋊C4—81no—
54C2×#239—31no—
55C2×D3×Q8—31no—
56C2×#41⋊C3—91no—
57C22×SL(2,𝔽3)—91no—
58C3×C4×Q8C3×D311yescharacteristic
59#897—91no—
60#895—91no—
61#875—91no—
62C4×D12—91no—
63C4×SL(2,𝔽3)—31no—
64C22×C3×D3—121no—
65C2×C32×C4—41no—
66C2×C3×Dic3—121no—
67C3×SL(2,𝔽3)D1211yescharacteristic
68C22×D9—31no—
69C22×D9C2×A411yescharacteristic
70Q8×C9—31no—
71Q8×C9C3×D411yescharacteristic
72C2×C4×C9—31no—
73C2×C4×C9—31no—
74C2×C4×C9—81no—
75C2×C4×C9C2×A411yescharacteristic
76C2×Dic9—31no—
77C2×Dic9C2×A411yescharacteristic
78D36—61no—
79D36—61no—
80D36—62no—
81C4×D9—61no—
82C4×D9—61no—
83C4×D9—61no—
84C4×D9—61no—
85Dic18—61no—
86Dic18—61no—
87Q8⋊C9—21no—
88#548—91noSylow
89C2×3−1+2—41no—
90C2×3−1+2—41no—
91C2×3−1+2—41no—
92AGL(1,ℤ/9)—82no—
93C2×C3×Q8C2×C3×D311yescharacteristic
94#239—61no—
95#239—181no—
96D3×Q8—61no—
97D3×Q8—181no—
98C2×D12—91no—
99C2×D12—91no—
100C2×C4×D3—91no—
101C2×C4×D3—91no—
102C2×Dic6—91no—
103#41⋊C3—181no—
104C2×SL(2,𝔽3)—31no—
105C2×SL(2,𝔽3)—181no—
106C3×C4⋊C4—31no—
107C3×C22⋊C4—361no—
108C3×C42—31no—
109(C2×D3)⋊C4—31no—
110(C2×D3)⋊C4—92no—
111(C2×D3)⋊C4—91no—
112C4⋊Dic3—91no—
113C22×C32—41no—
114C2×C3×D3—122no—
115C2×C3×D3—241no—
116C2×C3×D3—241no—
117C32×C4—81no—
118C3×Dic3—241no—
119C22×C9—81no—
120C22×C9C22×A411yescharacteristic
121C2×D9—21no—
122C2×D9—21no—
123C2×D9—61no—
124C2×D9—61no—
125C2×D9—61no—
126C2×D9#41⋊C312yescomplemented
127C4×C9—21no—
128C4×C9—31no—
129C4×C9—31no—
130C4×C9—61no—
131C4×C9—82no—
132Dic9—21no—
133Dic9—61no—
134C2×#41—91no—
135C22×Q8—91no—
136#124—271no—
137C22⋊Q8—271no—
138#121—271no—
139C4×Q8AGL(1,ℤ/9)11yescharacteristic, complemented
140C4×D4—271no—
1413−1+2—41noSylow
142C23×C3—361no—
143C22×D3—31no—
144C22×D3—91no—
145C3×Q8—31no—
146C3×Q8C3×D1211yescharacteristic
147C2×C3×C4—31no—
148C2×C3×C4—31no—
149C2×C3×C4—121no—
150C2×C3×C4—361no—
151C2×C3×C4D3×A411yescharacteristic
152C2×Dic3—31no—
153C2×Dic3—91no—
154D12—181no—
155D12—181no—
156D12—182no—
157C4×D3—181no—
158C4×D3—181no—
159C4×D3—181no—
160C4×D3—181no—
161Dic6—181no—
162Dic6—181no—
163SL(2,𝔽3)—31no—
164C2×C32—41no—
165C2×C32—41no—
166C2×C32—41no—
167C3×D3—242no—
168C2×C9—81no—
169C2×C9—81no—
170C2×C9—81no—
171C2×C9D4×A411yescharacteristic
172C2×C9C2×#41⋊C311yescharacteristic
173C2×C92−1+4⋊C311yescharacteristic
174D9—22no—
175D9—121no—
176#41—181no—
177#41—541no—
178C2×Q8—181no—
179C2×Q8—271no—
180C2×Q8—541no—
181C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
182C2×D4—271no—
183C2×D4—271no—
184C22×C4—271no—
185C22×C4—271no—
186C4⋊C4—31no—
187C4⋊C4—271no—
188C22⋊C4—91no—
189C22⋊C4—272no—
190C22⋊C4—271no—
191C42—31no—
192C22×C3—121no—
193C22×C3—362no—
194C22×C3—721no—
195C22×C3—721no—
196C22×C3C2×D3×A411yescharacteristic
197C2×D3—32no—
198C2×D3—61no—
199C2×D3—61no—
200C2×D3—181no—
201C2×D3—181no—
202C2×D3—181no—
203C3×C4—21no—
204C3×C4—31no—
205C3×C4—31no—
206C3×C4—61no—
207C3×C4—241no—
208C3×C4—721no—
209Dic3—61no—
210Dic3—181no—
211C32—41no—
212C9—81no—
213C9#548⋊C311yescharacteristic, complemented
214C23—91no—
215C23—271no—
216Q8—31no—
217Q8—541no—
218Q8—541no—
219Q8C4⋊AGL(1,ℤ/9)11yescharacteristic
220D4—541no—
221D4—541no—
222D4—542no—
223C2×C4—31no—
224C2×C4—31no—
225C2×C4—91no—
226C2×C4—271no—
227C2×C4—541no—
228C2×C4—541no—
229C2×C4—541no—
230C2×C4—541no—
231C2×C4D9⋊A411yescharacteristic
232C2×C3—121no—
233C2×C3—121no—
234C2×C3—121no—
235C2×C3—722no—
236C2×C3A4×D1211yescharacteristic
237C2×C3#1007⋊C311yescharacteristic
238C2×C3D3×#41⋊C311yescharacteristic
239D3—62no—
240D3—361no—
241C22—92no—
242C22—181no—
243C22—181no—
244C22—541no—
245C22—541no—
246C22—541no—
247C22C2×D9⋊A411yescharacteristic
248C4—21no—
249C4—31no—
250C4—31no—
251C4—61no—
252C4—181no—
253C4—541no—
254C3—121no—
255C3#5576⋊C311yescharacteristic
256C2—182no—
257C2—1081no—
258C2C4⋊(D9⋊A4)11yescharacteristic
259C2#1172240811yescharacteristic
260C2D9⋊(#41⋊C3)11yescharacteristic
subgroupG/H
centerC22C2×D9⋊A4
commutator, cosocleC2×Q8×C9C22×C3
Frattini, socleC22×C3C2×D3×A4
FittingC4×Q8×C9C2×C3
hypercenterC2×C4D9⋊A4
hypocenterQ8×C9C3×D4
LCSG ▷ C2×Q8×C9 ▷ Q8×C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×D9⋊SL(2,𝔽3)C2semidirect
2. C2×#88614C2semidirect
3. C4×C9⋊SL(2,𝔽3)C2semidirect
5. #10589678C3semidirect
13. C2×#3701C2×C3semidirect
15. C4×Q8×C9C2×C3semidirect
24. C9⋊SL(2,𝔽3)D4semidirect
126. C2×D9#41⋊C3semidirect
139. C4×Q8AGL(1,ℤ/9)semidirect
213. C9#548⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D9⋊SL(2,𝔽3)2
C2×#886142
C4×C9⋊SL(2,𝔽3)2
#5576⋊C33
#105896783
(C2×AGL(1,ℤ/9))⋊C44