Finite Group Databasebeta

← C2×C3×(C22⋊C9)⋊C8 #13146866

Subgroups — C2×C3×(C22⋊C9)⋊C8 #13146866

Order 1728.

378 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 86 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×C3×C4×C22⋊C9C211yesmaximal, characteristic
2C3×(C22⋊C9)⋊C8C212yesmaximal, direct product factor
3C2×C3×#830—31nomaximal
4C2×(C22⋊C9)⋊C8C311yesmaximal, characteristic, direct product factor
5C22×C3×C22⋊C9C411yescharacteristic
6C3×C4×C22⋊C9C411yescharacteristic
7C3×C4×C22⋊C9C2211yescharacteristic
8C2×C3×C9⋊C8—41nomaximal
9C23×C32×C4D311yescharacteristic
10C22×C3×C3⋊C8—31no—
11C22×C3×C3⋊C8—31no—
12C2×C4×C22⋊C9—21no—
13C2×C4×C22⋊C9C2×C311yescharacteristic
14C3×#830—32no—
15C3×#830—32no—
16(C22⋊C9)⋊C8C2×C312yesdirect product factor
17C2×C3×C22⋊C9C812yescomplemented
18C2×C3×C22⋊C9C2×C411yescharacteristic
19C2×C3×C4×C9—41no—
20C3×C9⋊C8—42no—
21C2×C3×C22⋊C8—91no—
22C2×#830—31no—
23C24×C32Dic311yescharacteristic
24C22×C32×C4—31no—
25C22×C32×C4—31no—
26C22×C32×C4—31no—
27C22×C32×C4—31no—
28C22×C32×C4Dic311yescharacteristic
29C22×C32×C4C2×D311yescharacteristic
30C22×C22⋊C9—21no—
31C22×C22⋊C9C3×C411yescharacteristic
32C2×C3×C3⋊C8—32no—
33C2×C3×C3⋊C8—32no—
34C2×C3×C3⋊C8—61no—
35C2×C3×C3⋊C8—61no—
36C2×C3×C3⋊C8—61no—
37C2×C3×C3⋊C8—61no—
38C4×C22⋊C9—21no—
39C4×C22⋊C9—21no—
40C4×C22⋊C9C3×C411yescharacteristic
41C4×C22⋊C9C22×C311yescharacteristic
42C2×C9⋊C8—41no—
43C22×C3×C9—41no—
44C3×C22⋊C9C2×C811yescharacteristic, complemented
45C3×C4×C9—41no—
46C3×C4×C9—41no—
47C23×C3×C4—21no—
48C23×C3×C4D911yescharacteristic
49C23×C3×C4C3×D311yescharacteristic
50C22×C3×C8—91no—
51C22×C3×C8—91no—
52C22×C3⋊C8—31no—
53C22×C3⋊C8—31no—
54C3×C22⋊C8—92no—
55C3×C22⋊C8—92no—
56#830—32no—
57#830—32no—
58C23×C32—31no—
59C23×C32—32no—
60C23×C32—31no—
61C23×C32C3⋊C812yes—
62C23×C32C2×Dic311yescharacteristic
63C2×C32×C4—31no—
64C2×C32×C4—31no—
65C2×C32×C4—31no—
66C2×C32×C4—31no—
67C2×C32×C4—31no—
68C2×C32×C4—31no—
69C2×C32×C4—31no—
70C2×C32×C4—61no—
71C2×C32×C4S411yescharacteristic
72C2×C22⋊C9—22no—
73C2×C22⋊C9—21no—
74C2×C22⋊C9C3×C812yescomplemented
75C2×C22⋊C9C2×C3×C411yescharacteristic
76C3×C3⋊C8—62no—
77C3×C3⋊C8—62no—
78C2×C4×C9—41no—
79C2×C4×C9—81no—
80C9⋊C8—42no—
81C2×C22⋊C8—91noSylow
82C2×C3×C9—42no—
83C2×C3×C9—41no—
84C24×C3—21no—
85C24×C3Dic911yescharacteristic
86C24×C3C3×Dic311yescharacteristic
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—31no—
90C22×C3×C4—31no—
91C22×C3×C4—31no—
92C22×C3×C4—31no—
93C22×C3×C4—31no—
94C22×C3×C4—31no—
95C22×C3×C4—31no—
96C22×C3×C4—31no—
97C22×C3×C4—61no—
98C22×C3×C4—61no—
99C22×C3×C4—61no—
100C22×C3×C4—61no—
101C22×C3×C4Dic911yescharacteristic
102C22×C3×C4C2×D911yescharacteristic
103C22×C3×C4C3×Dic311yescharacteristic
104C22×C3×C4C2×C3×D311yescharacteristic
105C2×C3×C8—92no—
106C2×C3×C8—92no—
107C2×C3×C8—181no—
108C2×C3×C8—181no—
109C2×C3×C8—181no—
110C2×C3×C8—181no—
111C2×C3⋊C8—32no—
112C2×C3⋊C8—32no—
113C2×C3⋊C8—61no—
114C2×C3⋊C8—61no—
115C2×C3⋊C8—61no—
116C2×C3⋊C8—61no—
117C22×C32—32no—
118C22×C32—31no—
119C22×C32—32no—
120C22×C32—32no—
121C22×C32—31no—
122C22×C32—31no—
123C22×C32—61no—
124C22×C32C2×C3⋊C811yescharacteristic
125C22×C32SL(2,ℤ/4)11yescharacteristic
126C32×C4—31no—
127C32×C4—31no—
128C32×C4SL(2,ℤ/4)11yescharacteristic
129C32×C4C2×S411yescharacteristic
130C22×C9—41no—
131C22×C9—81no—
132C22⋊C9—21no—
133C22⋊C9C2×C3×C811yescharacteristic, complemented
134C4×C9—41no—
135C4×C9—41no—
136C4×C9—81no—
137C4×C9—81no—
138C23×C4C3×D911yescharacteristic
139C22×C8—91no—
140C22×C8—91no—
141C22⋊C8—92no—
142C22⋊C8—92no—
143C3×C9—41noSylow
144C23×C3—22no—
145C23×C3—21no—
146C23×C3—31no—
147C23×C3—31no—
148C23×C3—31no—
149C23×C3—31no—
150C23×C3—32no—
151C23×C3—32no—
152C23×C3—61no—
153C23×C3—61no—
154C23×C3—62no—
155C23×C3C9⋊C812yescomplemented
156C23×C3C2×Dic911yescharacteristic
157C23×C3C3×C3⋊C812yes—
158C23×C3C2×C3×Dic311yescharacteristic
159C2×C3×C4—21no—
160C2×C3×C4—31no—
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—61no—
175C2×C3×C4—61no—
176C2×C3×C4—61no—
177C2×C3×C4—61no—
178C2×C3×C4—61no—
179C2×C3×C4—61no—
180C2×C3×C4—61no—
181C2×C3×C4—61no—
182C2×C3×C4—61no—
183C2×C3×C4—62no—
184C2×C3×C4(C22⋊C9)⋊C211yescharacteristic
185C2×C3×C4C3×S411yescharacteristic
186C3×C8—182no—
187C3×C8—182no—
188C3⋊C8—62no—
189C3⋊C8—62no—
190C2×C32—31no—
191C2×C32—32no—
192C2×C32—31no—
193C2×C32A4⋊C812yes—
194C2×C32C2×SL(2,ℤ/4)11yescharacteristic
195C2×C9—41no—
196C2×C9—42no—
197C2×C9—81no—
198C2×C9—82no—
199C24C3×Dic911yescharacteristic
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4C3×Dic911yescharacteristic
205C22×C4C2×C3×D911yescharacteristic
206C2×C8—92no—
207C2×C8—92no—
208C2×C8—181no—
209C2×C8—181no—
210C2×C8—181no—
211C2×C8—181no—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3—32no—
215C22×C3—31no—
216C22×C3—32no—
217C22×C3—31no—
218C22×C3—31no—
219C22×C3—31no—
220C22×C3—32no—
221C22×C3—32no—
222C22×C3—32no—
223C22×C3—32no—
224C22×C3—31no—
225C22×C3—31no—
226C22×C3—61no—
227C22×C3—61no—
228C22×C3—62no—
229C22×C3—62no—
230C22×C3—62no—
231C22×C3—61no—
232C22×C3—61no—
233C22×C3—61no—
234C22×C3—62no—
235C22×C3C2×C9⋊C811yescharacteristic, complemented
236C22×C3(C22⋊C9)⋊C411yescharacteristic
237C22×C3C2×C3×C3⋊C811yescharacteristic
238C22×C3C3×SL(2,ℤ/4)11yescharacteristic
239C3×C4—21no—
240C3×C4—21no—
241C3×C4—31no—
242C3×C4—31no—
243C3×C4—31no—
244C3×C4—31no—
245C3×C4—61no—
246C3×C4—61no—
247C3×C4(C22⋊C9)⋊C411yescharacteristic
248C3×C4C2×(C22⋊C9)⋊C211yescharacteristic
249C3×C4C3×SL(2,ℤ/4)11yescharacteristic
250C3×C4C2×C3×S411yescharacteristic
251C32C2×A4⋊C811yescharacteristic
252C9—41no—
253C9—81no—
254C23—31no—
255C23—31no—
256C23—32no—
257C23C3×C9⋊C812yescomplemented
258C23C2×C3×Dic911yescharacteristic
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—31no—
262C2×C4—31no—
263C2×C4—31no—
264C2×C4—31no—
265C2×C4—31no—
266C2×C4—61no—
267C2×C4C3×(C22⋊C9)⋊C211yescharacteristic
268C8—182no—
269C8—182no—
270C2×C3—22no—
271C2×C3—21no—
272C2×C3—31no—
273C2×C3—32no—
274C2×C3—31no—
275C2×C3—31no—
276C2×C3—32no—
277C2×C3—31no—
278C2×C3—61no—
279C2×C3—61no—
280C2×C3—62no—
281C2×C3(C22⋊C9)⋊C812yesdirect product factor
282C2×C3C2×(C22⋊C9)⋊C411yescharacteristic
283C2×C3C3×A4⋊C812yes—
284C2×C3C2×C3×SL(2,ℤ/4)11yescharacteristic
285C22—32no—
286C22—31no—
287C22—32no—
288C22—32no—
289C22—31no—
290C22—31no—
291C22—61no—
292C22C2×C3×C9⋊C811yescharacteristic, complemented
293C22C3×(C22⋊C9)⋊C411yescharacteristic
294C4—31no—
295C4—31no—
296C4C3×(C22⋊C9)⋊C411yescharacteristic
297C4C2×C3×(C22⋊C9)⋊C211yescharacteristic
298C3—21no—
299C3C2×(C22⋊C9)⋊C811yescharacteristic, direct product factor
300C3C2×C3×A4⋊C811yescharacteristic
301C2—31no—
302C2—32no—
303C2—31no—
304C2C3×(C22⋊C9)⋊C812yesdirect product factor
305C2C2×C3×(C22⋊C9)⋊C411yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4(C22⋊C9)⋊C2
commutator, hypocenterC22⋊C9C2×C3×C8
FrattiniC3×C4C2×C3×S4
FittingC23×C32×C4D3
socle, prime radicalC24×C32Dic3
cosocleC4×C22⋊C9C22×C3
LCSG ▷ C22⋊C9
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×(C22⋊C9)⋊C8C2direct (N × H)
4. C2×(C22⋊C9)⋊C8C3direct (N × H)
16. (C22⋊C9)⋊C8C2×C3direct (N × H)
17. C2×C3×C22⋊C9C8semidirect
44. C3×C22⋊C9C2×C8semidirect
74. C2×C22⋊C9C3×C8semidirect
133. C22⋊C9C2×C3×C8semidirect
155. C23×C3C9⋊C8semidirect
235. C22×C3C2×C9⋊C8semidirect
257. C23C3×C9⋊C8semidirect
281. C2×C3(C22⋊C9)⋊C8direct (N × H)
292. C22C2×C3×C9⋊C8semidirect
299. C3C2×(C22⋊C9)⋊C8direct (N × H)
304. C2C3×(C22⋊C9)⋊C8direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×C22⋊C92
C3×(C22⋊C9)⋊C82
C2×C3×#8303
C2×(C22⋊C9)⋊C83
C2×C3×C9⋊C84