Finite Group Databasebeta

← #13180427

Subgroups — #13180427

Order 1728.

386 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C22×C3×PSU(3,𝔽2)C211yesmaximal, characteristic, complemented
2C2×#89092C212yesmaximal, complemented
3#11724355C211yesmaximal, characteristic
4#11724243C211yesmaximal, characteristic, complemented
5#11724102C212yesmaximal, complemented
6#10596418—31nomaximal
7C22×C3×#157—21no—
8C22×C3×#157C2211yescharacteristic
9C2×C3×PSU(3,𝔽2)—21no—
10C2×C3×PSU(3,𝔽2)—24no—
11C2×C3×PSU(3,𝔽2)C2212yes—
12C2×#157⋊D3C2211yescharacteristic
13C2×C3⋊AGL(1,𝔽9)C2212yes—
14#89092—24no—
15#89049—21no—
16#88940C2211yescharacteristic
17#88939—21no—
18C22×PSU(3,𝔽2)D311yescharacteristic, complemented
19C2×AΓL(1,𝔽9)—32no—
20#64242—31no—
21C22⋊AGL(1,𝔽9)—31no—
22PSU(3,𝔽2)⋊C4—32no—
23C22×C3×C3⋊D3D411yescharacteristic
24C2×#1113—21no—
25C2×C3×#157—24no—
26C2×C3×#157—22no—
27C2×C3×#157—22no—
28C2×C3×#157D411yescharacteristic
29C2×C3×#157C2311yescharacteristic
30C3×PSU(3,𝔽2)—24no—
31C3×PSU(3,𝔽2)—22no—
32C3×PSU(3,𝔽2)D414yescomplemented
33#157⋊D3—22no—
34C3⋊AGL(1,𝔽9)—22no—
35C3×#638—41no—
36#6295—41no—
37#6294—21no—
38#5227—91nomaximal
39C22×#157—21no—
40C22×#157C2×D311yescharacteristic
41C2×PSU(3,𝔽2)—21no—
42C2×PSU(3,𝔽2)—24no—
43C2×PSU(3,𝔽2)C2×D312yes—
44C2×GO+(4,𝔽2)—31no—
45C2×AGL(1,𝔽9)—32no—
46AΓL(1,𝔽9)—64no—
47#3811—61no—
48D3×#67—121no—
49#3773—31no—
50D32⋊C4—61no—
51C22×C33SD1611yescharacteristic, complemented
52C2×C3×C3⋊D3—21no—
53C2×C3×C3⋊D3—21no—
54C2×C3×C3⋊D3—41no—
55C2×C3×C3⋊D3SD1611yescharacteristic, complemented
56C2×C3×C3⋊D3C2×D411yescharacteristic
57#1113—41no—
58#1113—41no—
59C3×#157—21no—
60C3×#157—24no—
61C3×#157C2×D412yes—
62C3×C3⋊Dic3—41no—
63C22×C3×Q8—91no—
64C2×#215—92no—
65#930—91no—
66#835—92no—
67#830—91no—
68C22×C3⋊D3#6711yescharacteristic
69C22×C3×D3—121no—
70C2×D32—61no—
71C2×D32—121no—
72C2×#157—24no—
73C2×#157—22no—
74C2×#157—22no—
75C2×#157#6711yescharacteristic
76C2×#157C22×D311yescharacteristic
77PSU(3,𝔽2)—24no—
78PSU(3,𝔽2)—22no—
79PSU(3,𝔽2)#6714yescomplemented
80GO+(4,𝔽2)—62no—
81AGL(1,𝔽9)—62no—
82#638—41no—
83C3×#67—121no—
84C3×#67—121no—
85C3×#67—241no—
86#626—121no—
87#626—121no—
88#625—121no—
89#624—61no—
90D3×Dic3—121no—
91#448—271noSylow
92C2×C33—21no—
93C2×C33C2×SD1611yescharacteristic
94C3×C3⋊D3—21no—
95C3×C3⋊D3—81no—
96C3×C3⋊D3#12011yescharacteristic
97C3×C3⋊D3#13711yescharacteristic
98C2×C3×Q8—92no—
99C2×C3×Q8—181no—
100C2×C3×Q8—184no—
101C22×C3×C4—91no—
102C22×C3×C4—181no—
103C2×#67—181no—
104D3×D4—361no—
105C2×D12—91no—
106#215—184no—
107(C2×D3)⋊C4—181no—
108C4⋊Dic3—91no—
109C2×C3⋊C8—92no—
110C22×C32—41no—
111C22×C32—81no—
112C22×C32#21511yescharacteristic, complemented
113C2×C3⋊D3—21no—
114C2×C3⋊D3—21no—
115C2×C3⋊D3—41no—
116C2×C3⋊D3#21511yescharacteristic, complemented
117C2×C3⋊D3C2×#6711yescharacteristic
118C2×C3×D3—121no—
119C2×C3×D3—121no—
120C2×C3×D3—121no—
121C2×C3×D3—121no—
122C2×C3×D3—241no—
123C2×C3×D3—241no—
124C2×C3×D3—241no—
125D32—121no—
126D32—121no—
127D32—241no—
128D32—241no—
129#157—21no—
130#157—24no—
131#157C2×#6712yes—
132C3⋊Dic3—41no—
133C3×Dic3—121no—
134C3×Dic3—121no—
135C3×Dic3—241no—
136C22×Q8—91no—
137C2×SD16—272no—
138#121—271no—
139#103—272no—
140C22⋊C8—271no—
141C33#44811yescharacteristic, complemented, Sylow
142C23×C3—91no—
143C22×D3—121no—
144C22×D3—181no—
145C22×D3—361no—
146C3×Q8—94no—
147C3×Q8—182no—
148C3×Q8—184no—
149C3×D4—361no—
150C2×C3×C4—91no—
151C2×C3×C4—91no—
152C2×C3×C4—182no—
153C2×C3×C4—182no—
154C2×C3×C4—184no—
155#67—121no—
156#67—121no—
157#67—241no—
158#67—361no—
159#67—361no—
160C2×Dic3—181no—
161D12—182no—
162D12—361no—
163C4×D3—361no—
164C3⋊C8—182no—
165C2×C32—21no—
166C2×C32—41no—
167C2×C32—81no—
168C2×C32—81no—
169C2×C32—161no—
170C2×C32C2×#21511yescharacteristic
171C3⋊D3—21no—
172C3⋊D3—81no—
173C3⋊D3#94211yescharacteristic
174C3⋊D3#95311yescharacteristic
175C3×D3—121no—
176C3×D3—121no—
177C3×D3—241no—
178C3×D3—241no—
179C3×D3—241no—
180C3×D3—481no—
181C2×Q8—92no—
182C2×Q8—181no—
183C2×Q8—184no—
184C2×D4—271no—
185C2×D4—541no—
186C22×C4—91no—
187C22×C4—181no—
188SD16—544no—
189C2×C8—272no—
190C4⋊C4—271no—
191C22⋊C4—541no—
192C22×C3—41no—
193C22×C3—81no—
194C22×C3—91no—
195C22×C3—91no—
196C22×C3—181no—
197C22×C3—181no—
198C22×C3—361no—
199C22×C3AΓL(1,𝔽9)11yescharacteristic, complemented
200C2×D3—121no—
201C2×D3—121no—
202C2×D3—121no—
203C2×D3—121no—
204C2×D3—241no—
205C2×D3—241no—
206C2×D3—241no—
207C2×D3—361no—
208C2×D3—361no—
209C2×D3—721no—
210C2×D3—721no—
211C3×C4—92no—
212C3×C4—181no—
213C3×C4—184no—
214C3×C4—361no—
215Dic3—121no—
216Dic3—121no—
217Dic3—241no—
218C32—41no—
219C32—81no—
220C32#522711yescharacteristic, complemented
221C23—91no—
222C23—541no—
223Q8—94no—
224Q8—182no—
225Q8—184no—
226D4—361no—
227D4—542no—
228D4—1081no—
229C2×C4—91no—
230C2×C4—91no—
231C2×C4—182no—
232C2×C4—182no—
233C2×C4—184no—
234C2×C4—541no—
235C8—542no—
236C2×C3—21no—
237C2×C3—41no—
238C2×C3—81no—
239C2×C3—81no—
240C2×C3—91no—
241C2×C3—91no—
242C2×C3—161no—
243C2×C3—181no—
244C2×C3—721no—
245C2×C3C2×AΓL(1,𝔽9)11yescharacteristic
246D3—121no—
247D3—121no—
248D3—241no—
249D3—241no—
250D3—241no—
251D3—481no—
252C22—91no—
253C22—91no—
254C22—181no—
255C22—181no—
256C22—361no—
257C22—1081no—
258C22—1081no—
259C22#8909211yescharacteristic, complemented
260C4—92no—
261C4—181no—
262C4—184no—
263C4—361no—
264C3—41no—
265C3—81no—
266C3#1059641811yescharacteristic, complemented
267C2—21no—
268C2—91no—
269C2—91no—
270C2—181no—
271C2—721no—
272C2C2×#8909211yescharacteristic
subgroupG/H
center, FrattiniC2C2×#89092
commutator, cosocleC2×C3×#157C23
FittingC22×C33SD16
socleC2×C33C2×SD16
prime radical#11724355C2
hypercenterC22#89092
hypocenterC33#448
LCSG ▷ C2×C3×#157 ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×PSU(3,𝔽2)C2semidirect
2. C2×#89092C2semidirect
4. #11724243C2semidirect
5. #11724102C2semidirect
18. C22×PSU(3,𝔽2)D3semidirect
32. C3×PSU(3,𝔽2)D4semidirect
51. C22×C33SD16semidirect
55. C2×C3×C3⋊D3SD16semidirect
79. PSU(3,𝔽2)#67semidirect
112. C22×C32#215semidirect
116. C2×C3⋊D3#215semidirect
141. C33#448semidirect
199. C22×C3AΓL(1,𝔽9)semidirect
220. C32#5227semidirect
259. C22#89092semidirect
266. C3#10596418semidirect

Maximal subgroups

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

subgroupindex
C22×C3×PSU(3,𝔽2)2
C2×#890922
#117243552
#117242432
#117241022
#105964183
#52279