Finite Group Databasebeta

← #13147185

Subgroups — #13147185

Order 1728.

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

normal subgroups only

This group has 283 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×#88683C211yesmaximal, characteristic, complemented
2C2×D9⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3Dic9⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#5831⋊C3—31nomaximal
5#10589933C311yesmaximal, characteristic, complemented
6#88683—21no—
7C2×C9⋊SL(2,𝔽3)C2211yescharacteristic
8D9⋊SL(2,𝔽3)—21no—
9(C2×AGL(1,ℤ/9))⋊C4—41nomaximal
10C2×C3×#41⋊C3D311yescharacteristic
11C2×D3×SL(2,𝔽3)—31no—
12Dic3×SL(2,𝔽3)—31no—
13C2×C9×#41C2×C311yescharacteristic, complemented
14C2×Q8×D9C2×C311yescharacteristic
15C2×#41⋊C9—21no—
16Q8×Dic9C2×C311yescharacteristic, complemented
17#63500—31no—
18#63498—31no—
19C4×#611—31no—
20#63486—31no—
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#5831—31no—
26#548⋊C3—91no—
27C3×#41⋊C3—21no—
28C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
29D3×SL(2,𝔽3)—61no—
30C3×(C2×D3)⋊C4—121no—
31C9×#41—21no—
32C9×#41—61no—
33C2×Q8×C9C22×C311yescharacteristic
34C2×D4×C9—31no—
35C22×C4×C9—31no—
36C2×#611—31no—
37Q8×D9—21no—
38Q8×D9—61no—
39C2×C4×D9—31no—
40C2×Dic18—31no—
41#41⋊C9—22no—
42C2×Q8⋊C9—21no—
43#3676—32no—
44#3676—31no—
45(C2×D9)⋊C4A411yescharacteristic
46C4⋊Dic9—31no—
47Dic9⋊C4—31no—
48C4×Dic9—31no—
49C22×3−1+2—41no—
50C2×AGL(1,ℤ/9)—42no—
51C2×AGL(1,ℤ/9)—81no—
52C2×AGL(1,ℤ/9)—81no—
53C4×3−1+2—81no—
543−1+2⋊C4—81no—
55C2×C3×#41C3×D311yescharacteristic
56C2×D3×Q8—31no—
57C2×#41⋊C3—31no—
58C22×SL(2,𝔽3)—91no—
59Q8×Dic3—31no—
60#938—91no—
61#936—91no—
62C4×#67—91no—
63#924—91no—
64C4×SL(2,𝔽3)—91no—
65C22×C3×D3—121no—
66C2×C32×C4—41no—
67C2×C3×Dic3—121no—
68C3×SL(2,𝔽3)#6711yescharacteristic
69C23×C9—31no—
70C22×D9C2×A411yescharacteristic
71Q8×C9—31no—
72Q8×C9C3×D411yescharacteristic
73D4×C9—61no—
74D4×C9—61no—
75C2×C4×C9—31no—
76C2×C4×C9—61no—
77C2×C4×C9—61no—
78C2×C4×C9—81no—
79C2×C4×C9C2×A411yescharacteristic
80#611—62no—
81C2×Dic9—31no—
82C2×Dic9—31no—
83C2×Dic9C2×A411yescharacteristic
84C4×D9—61no—
85C4×D9—61no—
86Dic18—61no—
87Dic18—61no—
88Q8⋊C9—21no—
89#548—91noSylow
90C2×3−1+2—41no—
91C2×3−1+2—41no—
92C2×3−1+2—41no—
93AGL(1,ℤ/9)—82no—
94C3×#41—21no—
95C3×#41—61no—
96C2×C3×Q8C2×C3×D311yescharacteristic
97C2×C3×D4—31no—
98C22×C3×C4—31no—
99C2×#67—91no—
100D3×Q8—61no—
101D3×Q8—181no—
102C2×C4×D3—91no—
103C2×Dic6—91no—
104#41⋊C3—61no—
105C2×SL(2,𝔽3)—31no—
106C2×SL(2,𝔽3)—181no—
107C3×C22⋊C4—361no—
108#217—91no—
109#217—92no—
110(C2×D3)⋊C4—31no—
111C4⋊Dic3—91no—
112Dic3⋊C4—91no—
113C4×Dic3—91no—
114C22×C32—41no—
115C2×C3×D3—122no—
116C2×C3×D3—241no—
117C2×C3×D3—241no—
118C32×C4—81no—
119C3×Dic3—241no—
120C22×C9—61no—
121C22×C9—61no—
122C22×C9—61no—
123C22×C9—81no—
124C22×C9C22×A411yescharacteristic
125C2×D9—21no—
126C2×D9—21no—
127C2×D9#41⋊C312yescomplemented
128C4×C9—21no—
129C4×C9—31no—
130C4×C9—31no—
131C4×C9—82no—
132Dic9—21no—
133Dic9—61no—
134Dic9—61no—
135C2×#41AGL(1,ℤ/9)11yescharacteristic, complemented
136C22×Q8—91no—
137#124—271no—
138C22⋊Q8—271no—
139#121—271no—
140C4×Q8—91no—
141C4×D4—271no—
1423−1+2—41noSylow
143C23×C3—31no—
144C23×C3—361no—
145C22×D3—31no—
146C3×Q8—31no—
147C3×Q8C3×#6711yescharacteristic
148C3×D4—61no—
149C3×D4—61no—
150C2×C3×C4—31no—
151C2×C3×C4—61no—
152C2×C3×C4—61no—
153C2×C3×C4—121no—
154C2×C3×C4—361no—
155C2×C3×C4D3×A411yescharacteristic
156#67—182no—
157C2×Dic3—31no—
158C2×Dic3—91no—
159C2×Dic3—91no—
160C4×D3—181no—
161C4×D3—181no—
162Dic6—181no—
163Dic6—181no—
164SL(2,𝔽3)—31no—
165C2×C32—41no—
166C2×C32—41no—
167C2×C32—41no—
168C3×D3—242no—
169C2×C9—81no—
170C2×C9—81no—
171C2×C9—81no—
172C2×C9—121no—
173C2×C9D4×A411yescharacteristic
174C2×C9C2×#41⋊C311yescharacteristic
175C2×C92−1+4⋊C311yescharacteristic
176D9—22no—
177#41—21no—
178#41—61no—
179C2×Q8—181no—
180C2×Q8—271no—
181C2×Q8—541no—
182C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
183C2×D4—31no—
184C2×D4—271no—
185C22×C4—31no—
186C22×C4—271no—
187C4⋊C4—271no—
188C4⋊C4—271no—
189C22⋊C4—91no—
190C22⋊C4—271no—
191C22⋊C4—272no—
192C42—271no—
193C22×C3—61no—
194C22×C3—61no—
195C22×C3—61no—
196C22×C3—121no—
197C22×C3—362no—
198C22×C3—721no—
199C22×C3—721no—
200C22×C3C2×D3×A411yescharacteristic
201C2×D3—32no—
202C2×D3—61no—
203C2×D3—61no—
204C3×C4—21no—
205C3×C4—31no—
206C3×C4—31no—
207C3×C4—241no—
208C3×C4—721no—
209Dic3—61no—
210Dic3—181no—
211Dic3—181no—
212C32—41no—
213C9—81no—
214C9#548⋊C311yescharacteristic, complemented
215C23—31no—
216C23—91no—
217Q8—31no—
218Q8—541no—
219Q8—541no—
220Q8#611⋊C311yescharacteristic
221D4—61no—
222D4—61no—
223D4—542no—
224C2×C4—31no—
225C2×C4—61no—
226C2×C4—61no—
227C2×C4—91no—
228C2×C4—271no—
229C2×C4—271no—
230C2×C4—541no—
231C2×C4—541no—
232C2×C4D9⋊A411yescharacteristic
233C2×C3—121no—
234C2×C3—121no—
235C2×C3—121no—
236C2×C3—121no—
237C2×C3—722no—
238C2×C3D3×#41⋊C311yescharacteristic
239C2×C3#1010⋊C311yescharacteristic
240C2×C3A4×#6711yescharacteristic
241D3—62no—
242C22—61no—
243C22—61no—
244C22—61no—
245C22—92no—
246C22—181no—
247C22—181no—
248C22C2×D9⋊A411yescharacteristic
249C4—21no—
250C4—31no—
251C4—31no—
252C4—181no—
253C4—541no—
254C4—541no—
255C3—121no—
256C3#5831⋊C311yescharacteristic
257C2—121no—
258C2—182no—
259C2D9⋊(#41⋊C3)11yescharacteristic
260C2#1172241411yescharacteristic
261C2#611⋊A411yescharacteristic
subgroupG/H
centerC22C2×D9⋊A4
commutator, cosocleC2×Q8×C9C22×C3
Frattini, socleC22×C3C2×D3×A4
FittingC2×C9×#41C2×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×#88683C2semidirect
2. C2×D9⋊SL(2,𝔽3)C2semidirect
3. Dic9⋊SL(2,𝔽3)C2semidirect
5. #10589933C3semidirect
13. C2×C9×#41C2×C3semidirect
16. Q8×Dic9C2×C3semidirect
24. C9⋊SL(2,𝔽3)D4semidirect
127. C2×D9#41⋊C3semidirect
135. C2×#41AGL(1,ℤ/9)semidirect
214. C9#548⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#886832
C2×D9⋊SL(2,𝔽3)2
Dic9⋊SL(2,𝔽3)2
#5831⋊C33
#105899333
(C2×AGL(1,ℤ/9))⋊C44