Finite Group Databasebeta

← C2×#11724243 #13180947

Subgroups — C2×#11724243 #13180947

Order 1728.

570 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C23×C3×#157C211yesmaximal, characteristic
2C22×C3⋊AGL(1,𝔽9)C212yesmaximal, complemented
3#11724243C214yesmaximal, direct product factor
4C2×C22⋊AGL(1,𝔽9)—31nomaximal
5C23×C3×C3⋊D3C411yescharacteristic
6C22×C3×#157—24no—
7C22×C3×#157C411yescharacteristic
8C22×C3×#157C412yes—
9C22×C3×#157C2212yes—
10C22×C3×#157C2211yescharacteristic
11C2×C3⋊AGL(1,𝔽9)—24no—
12C2×C3⋊AGL(1,𝔽9)C2214yescomplemented
13C23×#157D311yescharacteristic
14C22×AGL(1,𝔽9)—32no—
15C22⋊AGL(1,𝔽9)—34no—
16C23×C33C811yescharacteristic, complemented
17C22×C3×C3⋊D3—22no—
18C22×C3×C3⋊D3—22no—
19C22×C3×C3⋊D3C812yescomplemented
20C22×C3×C3⋊D3C811yescharacteristic, complemented
21C22×C3×C3⋊D3C2×C412yes—
22C22×C3×C3⋊D3C2×C411yescharacteristic
23C2×C3×#157—22no—
24C2×C3×#157—24no—
25C2×C3×#157—24no—
26C2×C3×#157C2×C412yes—
27C2×C3×#157C2×C411yescharacteristic
28C2×C3×#157D414yes—
29C2×C3×#157C2311yescharacteristic
30C3⋊AGL(1,𝔽9)—24no—
31C2×#830—91nomaximal
32C23×C3⋊D3Dic311yescharacteristic
33C23×C3×D3—121no—
34C22×#157—24no—
35C22×#157Dic311yescharacteristic
36C22×#157Dic312yes—
37C22×#157C2×D312yes—
38C22×#157C2×D311yescharacteristic
39C2×AGL(1,𝔽9)—34no—
40C2×AGL(1,𝔽9)—64no—
41C22×C33—22no—
42C22×C33C2×C811yescharacteristic
43C22×C33C2×C812yescomplemented
44C2×C3×C3⋊D3—22no—
45C2×C3×C3⋊D3—24no—
46C2×C3×C3⋊D3—22no—
47C2×C3×C3⋊D3—22no—
48C2×C3×C3⋊D3C22⋊C412yes—
49C2×C3×C3⋊D3C2×C811yescharacteristic
50C2×C3×C3⋊D3C2×C812yescomplemented
51C2×C3×C3⋊D3M1612yes—
52C2×C3×C3⋊D3C22×C411yescharacteristic
53C3×#157—22no—
54C3×#157C22⋊C412yes—
55C3×#157C2×D412yes—
56C23×C3×C4—91no—
57C22×C3⋊C8—92no—
58#830—94no—
59C23×C32—41no—
60C23×C32—81no—
61C23×C32C3⋊C811yescharacteristic, complemented
62C22×C3⋊D3—22no—
63C22×C3⋊D3—22no—
64C22×C3⋊D3C3⋊C812yescomplemented
65C22×C3⋊D3C3⋊C811yescharacteristic, complemented
66C22×C3⋊D3C2×Dic312yes—
67C22×C3⋊D3C2×Dic311yescharacteristic
68C22×C3×D3—121no—
69C22×C3×D3—122no—
70C22×C3×D3—121no—
71C22×C3×D3—122no—
72C22×C3×D3—124no—
73C22×C3×D3—124no—
74C2×#157—22no—
75C2×#157—24no—
76C2×#157—24no—
77C2×#157C2×Dic312yes—
78C2×#157C2×Dic311yescharacteristic
79C2×#157#6714yes—
80C2×#157C22×D311yescharacteristic
81AGL(1,𝔽9)—64no—
82C2×C22⋊C8—271noSylow
83C2×C33—22no—
84C2×C33C22⋊C812yescomplemented
85C2×C33C22×C811yescharacteristic
86C3×C3⋊D3—22no—
87C3×C3⋊D3C22⋊C812yescomplemented
88C3×C3⋊D3C2×C22⋊C411yescharacteristic
89C3×C3⋊D3C2×M1611yescharacteristic
90C24×C3—91no—
91C23×D3—121no—
92C22×C3×C4—91no—
93C22×C3×C4—92no—
94C22×C3×C4—91no—
95C22×C3×C4—92no—
96C22×C3×C4—184no—
97C2×C3⋊C8—94no—
98C2×C3⋊C8—184no—
99C22×C32—22no—
100C22×C32—44no—
101C22×C32—41no—
102C22×C32—42no—
103C22×C32—84no—
104C22×C32—81no—
105C22×C32—82no—
106C22×C32C2×C3⋊C811yescharacteristic
107C22×C32C2×C3⋊C812yescomplemented
108C2×C3⋊D3—22no—
109C2×C3⋊D3—24no—
110C2×C3⋊D3—22no—
111C2×C3⋊D3—22no—
112C2×C3⋊D3C2×C3⋊C811yescharacteristic
113C2×C3⋊D3C2×C3⋊C812yescomplemented
114C2×C3⋊D3#20812yes—
115C2×C3⋊D3#21712yes—
116C2×C3⋊D3C22×Dic311yescharacteristic
117C2×C3×D3—122no—
118C2×C3×D3—121no—
119C2×C3×D3—122no—
120C2×C3×D3—122no—
121C2×C3×D3—121no—
122C2×C3×D3—124no—
123C2×C3×D3—128no—
124C2×C3×D3—124no—
125C2×C3×D3—124no—
126#157—22no—
127#157#21712yes—
128#157C2×#6712yes—
129C23×C4—91no—
130C22×C8—272no—
131C22⋊C8—274no—
132C33C2×C22⋊C811yescharacteristic, complemented, Sylow
133C23×C3—41no—
134C23×C3—81no—
135C23×C3—91no—
136C23×C3—91no—
137C23×C3—92no—
138C23×C3—92no—
139C23×C3—182no—
140C23×C3—182no—
141C23×C3AGL(1,𝔽9)11yescharacteristic, complemented
142C22×D3—121no—
143C22×D3—122no—
144C22×D3—121no—
145C22×D3—122no—
146C22×D3—124no—
147C22×D3—124no—
148C2×C3×C4—91no—
149C2×C3×C4—94no—
150C2×C3×C4—91no—
151C2×C3×C4—92no—
152C2×C3×C4—184no—
153C2×C3×C4—184no—
154C2×C3×C4—182no—
155C3⋊C8—184no—
156C2×C32—22no—
157C2×C32—44no—
158C2×C32—41no—
159C2×C32—42no—
160C2×C32—84no—
161C2×C32—82no—
162C2×C32—81no—
163C2×C32#83012yescomplemented
164C2×C32C22×C3⋊C811yescharacteristic
165C3⋊D3—22no—
166C3⋊D3#83012yescomplemented
167C3⋊D3C2×#20811yescharacteristic
168C3⋊D3C2×#21711yescharacteristic
169C3×D3—122no—
170C3×D3—121no—
171C3×D3—121no—
172C3×D3—124no—
173C24—91no—
174C22×C4—91no—
175C22×C4—92no—
176C22×C4—91no—
177C22×C4—92no—
178C22×C4—184no—
179C2×C8—274no—
180C2×C8—544no—
181C22×C3—22no—
182C22×C3—44no—
183C22×C3—41no—
184C22×C3—42no—
185C22×C3—84no—
186C22×C3—81no—
187C22×C3—82no—
188C22×C3—91no—
189C22×C3—91no—
190C22×C3—92no—
191C22×C3—92no—
192C22×C3—92no—
193C22×C3—182no—
194C22×C3—182no—
195C22×C3—184no—
196C22×C3—182no—
197C22×C3C2×AGL(1,𝔽9)11yescharacteristic
198C22×C3C2×AGL(1,𝔽9)12yescomplemented
199C2×D3—121no—
200C2×D3—122no—
201C2×D3—122no—
202C2×D3—122no—
203C2×D3—121no—
204C2×D3—124no—
205C2×D3—128no—
206C2×D3—124no—
207C2×D3—124no—
208C3×C4—92no—
209C3×C4—92no—
210C3×C4—182no—
211C32—41no—
212C32—81no—
213C32C2×#83011yescharacteristic, complemented
214C23—91no—
215C23—91no—
216C23—92no—
217C23—92no—
218C23—182no—
219C23—182no—
220C23C3⋊AGL(1,𝔽9)11yescharacteristic, complemented
221C2×C4—91no—
222C2×C4—94no—
223C2×C4—91no—
224C2×C4—92no—
225C2×C4—184no—
226C2×C4—184no—
227C2×C4—182no—
228C8—544no—
229C2×C3—22no—
230C2×C3—44no—
231C2×C3—41no—
232C2×C3—42no—
233C2×C3—84no—
234C2×C3—81no—
235C2×C3—82no—
236C2×C3—91no—
237C2×C3—91no—
238C2×C3—92no—
239C2×C3—182no—
240C2×C3C22⋊AGL(1,𝔽9)12yescomplemented
241C2×C3C22×AGL(1,𝔽9)11yescharacteristic
242D3—122no—
243D3—121no—
244D3—121no—
245D3—124no—
246C22—22no—
247C22—91no—
248C22—91no—
249C22—92no—
250C22—92no—
251C22—92no—
252C22—182no—
253C22—182no—
254C22—184no—
255C22—182no—
256C22C2×C3⋊AGL(1,𝔽9)11yescharacteristic
257C22C2×C3⋊AGL(1,𝔽9)12yescomplemented
258C4—92no—
259C4—92no—
260C4—182no—
261C3—41no—
262C3—81no—
263C3C2×C22⋊AGL(1,𝔽9)11yescharacteristic, complemented
264C2—22no—
265C2—91no—
266C2—91no—
267C2—92no—
268C2—182no—
269C2#1172424312yesdirect product factor
270C2C22×C3⋊AGL(1,𝔽9)11yescharacteristic
subgroupG/H
centerC22C2×C3⋊AGL(1,𝔽9)
commutatorC2×C33C22×C8
FrattiniC2C22×C3⋊AGL(1,𝔽9)
FittingC23×C33C8
socleC22×C33C2×C8
prime radicalC23×C3×C3⋊D3C4
cosocleC2×C3×#157C23
hypercenterC23C3⋊AGL(1,𝔽9)
hypocenterC33C2×C22⋊C8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊AGL(1,𝔽9)C2semidirect
3. #11724243C2direct (N × H)
12. C2×C3⋊AGL(1,𝔽9)C22semidirect
16. C23×C33C8semidirect
19. C22×C3×C3⋊D3C8semidirect
20. C22×C3×C3⋊D3C8semidirect
43. C22×C33C2×C8semidirect
50. C2×C3×C3⋊D3C2×C8semidirect
61. C23×C32C3⋊C8semidirect
64. C22×C3⋊D3C3⋊C8semidirect
65. C22×C3⋊D3C3⋊C8semidirect
84. C2×C33C22⋊C8semidirect
87. C3×C3⋊D3C22⋊C8semidirect
107. C22×C32C2×C3⋊C8semidirect
113. C2×C3⋊D3C2×C3⋊C8semidirect
132. C33C2×C22⋊C8semidirect
141. C23×C3AGL(1,𝔽9)semidirect
163. C2×C32#830semidirect
166. C3⋊D3#830semidirect
198. C22×C3C2×AGL(1,𝔽9)semidirect
213. C32C2×#830semidirect
220. C23C3⋊AGL(1,𝔽9)semidirect
240. C2×C3C22⋊AGL(1,𝔽9)semidirect
257. C22C2×C3⋊AGL(1,𝔽9)semidirect
263. C3C2×C22⋊AGL(1,𝔽9)semidirect
269. C2#11724243direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×#1572
C22×C3⋊AGL(1,𝔽9)2
#117242432
C2×C22⋊AGL(1,𝔽9)3
C2×#8309