Finite Group Databasebeta

← #13147181

Subgroups — #13147181

Order 1728.

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

normal subgroups only

This group has 274 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×#88614C211yesmaximal, characteristic, complemented
3Dic9⋊SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#5834⋊C3—31nomaximal
5#10589936C311yesmaximal, characteristic, complemented
6#88683—21no—
7C2×C9⋊SL(2,𝔽3)C2211yescharacteristic
8#88614—21no—
9(3−1+2⋊C4)⋊C4—41nomaximal
10C2×C3×#41⋊C3D311yescharacteristic
11C2×C3⋊(#41⋊C3)—31no—
12Dic3×SL(2,𝔽3)—31no—
13C2×C9×#41C2×C311yescharacteristic
14C2×#3701C2×C311yescharacteristic
15C2×#41⋊C9—21no—
16Q8×Dic9C2×C311yescharacteristic
17#63502—31no—
18#63500—31no—
19#63492—31no—
20C4×#611—31no—
21C2×C4×3−1+2—41no—
22C2×3−1+2⋊C4—41no—
23C2×3−1+2⋊C4—41no—
24C9⋊SL(2,𝔽3)D411yescharacteristic, complemented
25#5834—31no—
26C4⋊(#41⋊C3)—91no—
27C3×#41⋊C3—21no—
28C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
29C3⋊(#41⋊C3)—61no—
30C3×Dic3⋊C4—121no—
31C9×#41—21no—
32C9×#41—61no—
33C2×Q8×C9C22×C311yescharacteristic
34C2×D4×C9—31no—
35C22×C4×C9—31no—
36C2×#611—32no—
37C2×#611—31no—
38#3701—21no—
39#3701—61no—
40C2×D36—31no—
41C2×C4×D9—31no—
42#41⋊C9—22no—
43C2×Q8⋊C9—21no—
44#3676—31no—
45(C2×D9)⋊C4—31no—
46C4⋊Dic9—31no—
47Dic9⋊C4A411yescharacteristic
48C4×Dic9—31no—
49C22×3−1+2—41no—
50C4×3−1+2—81no—
513−1+2⋊C4—42no—
523−1+2⋊C4—81no—
53C2×C3×#41C3×D311yescharacteristic
54C2×#239—31no—
55C2×#41⋊C3—31no—
56C2×#41⋊C3—91no—
57Q8×Dic3—31no—
58#940—91no—
59#938—91no—
60#930—91no—
61C4×#67—91no—
62C4×SL(2,𝔽3)—91no—
63C2×C32×C4—41no—
64C2×C3×Dic3—121no—
65C2×C3×Dic3—121no—
66C3×SL(2,𝔽3)#6711yescharacteristic
67C23×C9—31no—
68C22×D9—31no—
69Q8×C9—31no—
70Q8×C9C3×D411yescharacteristic
71D4×C9—61no—
72D4×C9—61no—
73C2×C4×C9—31no—
74C2×C4×C9—61no—
75C2×C4×C9—61no—
76C2×C4×C9—81no—
77C2×C4×C9C2×A411yescharacteristic
78#611—62no—
79#611—122no—
80C2×Dic9—31no—
81C2×Dic9C2×A411yescharacteristic
82C2×Dic9C2×A411yescharacteristic
83D36—61no—
84D36—61no—
85C4×D9—61no—
86C4×D9—61no—
87Q8⋊C9—21no—
88#550—91noSylow
89C2×3−1+2—41no—
90C2×3−1+2—41no—
91C2×3−1+2—41no—
92C3×#41—21no—
93C3×#41—61no—
94C2×C3×Q8C2×C3×D311yescharacteristic
95C2×C3×D4—31no—
96C22×C3×C4—31no—
97C2×#67—91no—
98C2×#67—92no—
99#239—61no—
100#239—181no—
101C2×D12—91no—
102C2×C4×D3—91no—
103#41⋊C3—61no—
104#41⋊C3—181no—
105C2×SL(2,𝔽3)—31no—
106C3×C4⋊C4—361no—
107#217—91no—
108(C2×D3)⋊C4—91no—
109C4⋊Dic3—91no—
110Dic3⋊C4—31no—
111C4×Dic3—91no—
112C22×C32—41no—
113C32×C4—81no—
114C3×Dic3—122no—
115C3×Dic3—241no—
116C22×C9—61no—
117C22×C9—61no—
118C22×C9—61no—
119C22×C9—81no—
120C22×C9C22×A411yescharacteristic
121C2×D9—61no—
122C2×D9—61no—
123C2×D9—61no—
124C4×C9—21no—
125C4×C9—31no—
126C4×C9—31no—
127C4×C9—82no—
128Dic9—21no—
129Dic9—61no—
130Dic9#41⋊C312yescomplemented
131C2×#41—91no—
132C2×#41AGL(1,ℤ/9)11yescharacteristic
133#127—271no—
134#121—271no—
135#121—271no—
136C4×Q8—91no—
137C4×D4—271no—
1383−1+2—41noSylow
139C23×C3—31no—
140C22×D3—91no—
141C3×Q8—31no—
142C3×Q8C3×#6711yescharacteristic
143C3×D4—61no—
144C3×D4—61no—
145C2×C3×C4—31no—
146C2×C3×C4—61no—
147C2×C3×C4—61no—
148C2×C3×C4—121no—
149C2×C3×C4—361no—
150C2×C3×C4—361no—
151C2×C3×C4D3×A411yescharacteristic
152#67—182no—
153#67—362no—
154C2×Dic3—31no—
155C2×Dic3—31no—
156C2×Dic3—91no—
157D12—181no—
158D12—181no—
159C4×D3—181no—
160C4×D3—181no—
161SL(2,𝔽3)—31no—
162C2×C32—41no—
163C2×C32—41no—
164C2×C32—41no—
165C2×C9—81no—
166C2×C9—81no—
167C2×C9—81no—
168C2×C9—121no—
169C2×C9D4×A411yescharacteristic
170C2×C9C2×#41⋊C311yescharacteristic
171C2×C9#99411yescharacteristic
172D9—121no—
173#41—21no—
174#41—61no—
175#41—181no—
176#41—541no—
177C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
178C2×D4—31no—
179C2×D4—271no—
180C2×D4—271no—
181C2×D4—272no—
182C22×C4—31no—
183C22×C4—271no—
184C4⋊C4—91no—
185C4⋊C4—271no—
186C22⋊C4—271no—
187C22⋊C4—271no—
188C42—271no—
189C22×C3—61no—
190C22×C3—61no—
191C22×C3—61no—
192C22×C3—121no—
193C22×C3C2×D3×A411yescharacteristic
194C2×D3—181no—
195C2×D3—181no—
196C2×D3—181no—
197C3×C4—21no—
198C3×C4—31no—
199C3×C4—31no—
200C3×C4—241no—
201C3×C4—362no—
202C3×C4—721no—
203Dic3—32no—
204Dic3—61no—
205Dic3—181no—
206C32—41no—
207C9—81no—
208C9C4⋊(#41⋊C3)11yescharacteristic, complemented
209C23—31no—
210C23—271no—
211Q8—31no—
212Q8#611⋊C311yescharacteristic
213D4—61no—
214D4—61no—
215D4—541no—
216D4—541no—
217D4—542no—
218D4—1082no—
219C2×C4—31no—
220C2×C4—61no—
221C2×C4—61no—
222C2×C4—91no—
223C2×C4—91no—
224C2×C4—271no—
225C2×C4—541no—
226C2×C4—541no—
227C2×C4D9⋊A411yescharacteristic
228C2×C3—121no—
229C2×C3—121no—
230C2×C3—121no—
231C2×C3—121no—
232C2×C3C3⋊#99411yescharacteristic
233C2×C3D3×#41⋊C311yescharacteristic
234C2×C3A4×#6711yescharacteristic
235D3—361no—
236C22—61no—
237C22—61no—
238C22—61no—
239C22—541no—
240C22—541no—
241C22—541no—
242C22C2×D9⋊A411yescharacteristic
243C4—21no—
244C4—31no—
245C4—31no—
246C4—92no—
247C4—181no—
248C4—541no—
249C3—121no—
250C3#5834⋊C311yescharacteristic
251C2—121no—
252C2—1081no—
253C2#1172241111yescharacteristic
254C2D9⋊(#41⋊C3)11yescharacteristic
255C2#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×#88614C2semidirect
3. Dic9⋊SL(2,𝔽3)C2semidirect
5. #10589936C3semidirect
24. C9⋊SL(2,𝔽3)D4semidirect
130. Dic9#41⋊C3semidirect
208. C9C4⋊(#41⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C2×#886832
C2×#886142
Dic9⋊SL(2,𝔽3)2
#5834⋊C33
#105899363
(3−1+2⋊C4)⋊C44