Finite Group Databasebeta

← #13146973

Subgroups — #13146973

Order 1728.

311 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 55 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
1Q8×C9×A4C211yesmaximal, characteristic, complemented
2C4⋊(C9⋊S4)C211yesmaximal, characteristic
3C9⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
4#10593651—31nomaximal
5#10589519—32nomaximal
6#10588908—31nomaximal
7C2×C9⋊S4—21no—
8C4×C9×A4—21no—
9C4×C9×A4C2211yescharacteristic
10#88548—41nomaximal
11C3×Q8×A4D311yescharacteristic
12C4⋊(C3⋊S4)—31no—
13C3⋊(A4⋊C8)—31no—
14C22×Q8×C9D311yescharacteristic, complemented
15Q8×C22⋊C9D312yescomplemented
16C4⋊((C22⋊C9)⋊C2)—32no—
17C2×#3675—31no—
18C2×#3675—31no—
19#63492—31no—
20(C22⋊C9)⋊C8—32no—
21#63395—31no—
22#63395—31no—
23#63389—31no—
24C2×C9×A4D411yescharacteristic
25C9⋊S4—41no—
26C3×Q8×C9—41no—
27C4⋊(C3⋊D9)—41no—
28C3⋊(C9⋊C8)—41no—
29#5417—93no—
30#5227—91no—
31C2×C3⋊S4—61no—
32C3×C4×A4—21no—
33C3×C4×A4C2×D311yescharacteristic
34C2×(C22⋊C9)⋊C2—62no—
35#3755—121no—
36C2×Q8×C9—31no—
37C2×Q8×C9—31no—
38C2×Q8×C9—61no—
39C22×C4×C9—21no—
40C22×C4×C9C2×D311yescharacteristic
41C2×#611—61no—
42C2×D36—31no—
43C4×C22⋊C9—22no—
44C4×C22⋊C9C2×D312yes—
45#3675—62no—
46#3675—61no—
47#3675—61no—
48#3675—122no—
49(C2×D9)⋊C4—61no—
50C4⋊Dic9—31no—
51C2×C9⋊C8—31no—
52C2×C9⋊C8—31no—
53C2×C3⋊D9—81no—
54C9×A4SD1611yescharacteristic, complemented
55C3×C4×C9—41no—
56C3×C4×C9—81no—
57C22×C3×Q8C3⋊D311yescharacteristic
58Q8×A4D913yescomplemented
59C4⋊S4—93no—
60C2×#215—91no—
61C2×#215—91no—
62#930—91no—
63A4⋊C8—93no—
64#835—91no—
65#835—91no—
66#830—91no—
67C2×C3×A4#6711yescharacteristic
68C3⋊S4—121no—
69C32×Q8—41no—
70C4⋊(C3⋊D3)—121no—
71C23×C9#6711yescharacteristic
72C22×D9—61no—
73C2×C22⋊C9#6712yes—
74(C22⋊C9)⋊C2—122no—
75C3⋊(C3⋊C8)—121no—
76Q8×C9—32no—
77Q8×C9—31no—
78Q8×C9—42no—
79Q8×C9—61no—
80Q8×C9S411yescharacteristic, complemented
81C2×C4×C9—31no—
82C2×C4×C9—31no—
83C2×C4×C9—61no—
84C2×C4×C9—61no—
85#611—121no—
86#611—121no—
87C2×Dic9—61no—
88D36—61no—
89D36—61no—
90D36—122no—
91C9⋊C8—61no—
92C9⋊C8—61no—
93C9⋊C8—122no—
94#448—271noSylow
95C2×C3×C9—41no—
96C3⋊D9—161no—
97C2×S4—183no—
98C2×C3×Q8—31no—
99C2×C3×Q8—31no—
100C2×C3×Q8—61no—
101C22×C3×C4—21no—
102C22×C3×C4C2×C3⋊D311yescharacteristic
103C2×#67—181no—
104C2×D12—91no—
105C4×A4—23no—
106C4×A4C2×D913yes—
107#215—182no—
108#215—181no—
109#215—181no—
110#215—363no—
111(C2×D3)⋊C4—181no—
112C4⋊Dic3—91no—
113C2×C3⋊C8—91no—
114C2×C3⋊C8—91no—
115C2×C3⋊D3—241no—
116C3×A4#21511yescharacteristic
117C32×C4—41no—
118C32×C4—81no—
119C22×C9—31no—
120C22×C9—31no—
121C22×C9#21511yescharacteristic, complemented
122C2×D9—121no—
123C2×D9—121no—
124C2×D9—121no—
125C2×D9—242no—
126C22⋊C9#21512yescomplemented
127C4×C9—21no—
128C4×C9—31no—
129C4×C9—42no—
130C4×C9—61no—
131C4×C9—82no—
132C4×C9C2×S411yescharacteristic
133Dic9—121no—
134C22×Q8C3⋊D911yescharacteristic, complemented
135C2×SD16—271no—
136C2×SD16—271no—
137#121—271no—
138#103—271no—
139#103—271no—
140C22⋊C8—271no—
141C3×C9—41noSylow
142C23×C3#63811yescharacteristic
143C22×D3—181no—
144C2×A4#61113yes—
145S4—363no—
146C3×Q8—32no—
147C3×Q8—31no—
148C3×Q8—43no—
149C3×Q8—61no—
150C3×Q8C3⋊S411yescharacteristic
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—61no—
154C2×C3×C4—61no—
155#67—361no—
156#67—361no—
157C2×Dic3—181no—
158D12—181no—
159D12—181no—
160D12—363no—
161C3⋊C8—181no—
162C3⋊C8—181no—
163C3⋊C8—363no—
164C2×C32—41no—
165C3⋊D3—481no—
166C2×C9—31no—
167C2×C9—31no—
168C2×C9—42no—
169C2×C9GL(2,ℤ/4)11yescharacteristic
170D9—241no—
171D9—482no—
172C2×Q8—31no—
173C2×Q8—31no—
174C2×Q8—61no—
175C2×D4—271no—
176C2×D4—541no—
177C22×C4—21no—
178C22×C4C2×C3⋊D911yescharacteristic
179SD16—542no—
180SD16—541no—
181SD16—541no—
182C2×C8—271no—
183C2×C8—271no—
184C4⋊C4—271no—
185C22⋊C4—541no—
186C22×C3—31no—
187C22×C3—31no—
188C22×C3#375511yescharacteristic
189C2×D3—361no—
190C2×D3—361no—
191C2×D3—361no—
192C2×D3—723no—
193A4#367513yescomplemented
194C3×C4—21no—
195C3×C4—31no—
196C3×C4—43no—
197C3×C4—61no—
198C3×C4—83no—
199C3×C4C2×C3⋊S411yescharacteristic
200Dic3—361no—
201C32—41no—
202C9—42no—
203C9#541711yescharacteristic, complemented
204C23—541no—
205C23#623311yescharacteristic
206Q8—32no—
207Q8—31no—
208Q8—61no—
209Q8C9⋊S411yescharacteristic, complemented
210D4—541no—
211D4—541no—
212D4—1081no—
213D4—1081no—
214C2×C4—31no—
215C2×C4—31no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—541no—
219C8—541no—
220C8—541no—
221C2×C3—31no—
222C2×C3—31no—
223C2×C3—43no—
224C2×C3#6426911yescharacteristic
225D3—721no—
226D3—1443no—
227C22—31no—
228C22—31no—
229C22—1081no—
230C22—1081no—
231C22—1081no—
232C22#8854811yescharacteristic, complemented
233C4—21no—
234C4—31no—
235C4—61no—
236C4—1081no—
237C4C2×C9⋊S411yescharacteristic
238C3—43no—
239C3#1059365111yescharacteristic
240C2—31no—
241C2—31no—
242C2—2161no—
243C2#1172237611yescharacteristic
subgroupG/H
centerC2#11722376
commutator, cosocleC4×C9×A4C22
FrattiniC3×C4C2×C3⋊S4
FittingC22×Q8×C9D3
socleC23×C3#638
prime radicalC4⋊(C9⋊S4)C2
hypercenterQ8C9⋊S4
hypocenterC9×A4SD16
LCSG ▷ C4×C9×A4 ▷ C2×C9×A4 ▷ C9×A4
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C9×A4C2semidirect
3. C9⋊(A4⋊C8)C2semidirect
14. C22×Q8×C9D3semidirect
15. Q8×C22⋊C9D3semidirect
54. C9×A4SD16semidirect
58. Q8×A4D9semidirect
80. Q8×C9S4semidirect
121. C22×C9#215semidirect
126. C22⋊C9#215semidirect
134. C22×Q8C3⋊D9semidirect
193. A4#3675semidirect
203. C9#5417semidirect
209. Q8C9⋊S4semidirect
232. C22#88548semidirect

Maximal subgroups

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

subgroupindex
Q8×C9×A42
C4⋊(C9⋊S4)2
C9⋊(A4⋊C8)2
#105936513
#105895193
#105889083
#885484