Finite Group Databasebeta

← C2×C4×D9⋊A4 #13164564

Subgroups — C2×C4×D9⋊A4 #13164564

Order 1728.

967 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 967 conjugacy classes of subgroups — too many to draw. Only its 121 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×D9⋊A4C211yesmaximal, characteristic
2C2×C4×C9⋊A4C211yesmaximal, characteristic, complemented
3C2×Dic9⋊A4C211yesmaximal, characteristic, complemented
4C4×D9⋊A4C214yesmaximal, direct product factor
5C2×C4×D3×A4—31nomaximal
6C23×C4×D9C311yesmaximal, characteristic, complemented
7C22×C9⋊A4C2211yescharacteristic
8C2×D9⋊A4C414yesdirect product factor
9C2×D9⋊A4C2212yes—
10C2×C4×AGL(1,ℤ/9)—41nomaximal
11C4×C9⋊A4C2212yescomplemented
12Dic9⋊A4C2212yescomplemented
13C22×D3×A4—31no—
14C2×C3×C4×A4D311yescharacteristic
15C2×Dic3×A4—31no—
16C4×D3×A4—34no—
17C24×D9C2×C311yescharacteristic
18C23×C4×C9C2×C311yescharacteristic, complemented
19C23×Dic9C2×C311yescharacteristic, complemented
20C22×C4×D9—31no—
21C22×C4×D9—31no—
22C22×C4×D9—31no—
23C22×C4×D9—31no—
24C22×C4×D9—34no—
25C22×C4×D9C2×C314yescomplemented
26C2×C4×C22⋊C9—21no—
27C22×AGL(1,ℤ/9)—41no—
28C2×C9⋊A4C2×C412yescomplemented
29C2×C9⋊A4C2311yescharacteristic
30D9⋊A4C2×C414yesdirect product factor
31C2×C4×3−1+2—41no—
32C2×3−1+2⋊C4—41no—
33C4×AGL(1,ℤ/9)—44no—
34C23×C4×D3—31no—
35C22×C4×A4—91no—
36C22×C3×A4C2×D311yescharacteristic
37C2×D3×A4—34no—
38C2×D3×A4—32no—
39C2×C3×C4×D3—121no—
40C3×C4×A4C2×D312yes—
41Dic3×A4—32no—
42C24×C9C22×C311yescharacteristic
43C23×D9—31no—
44C23×D9—34no—
45C23×D9—31no—
46C23×D9—32no—
47C23×D9C3×C414yescomplemented
48C23×D9C22×C312yes—
49C22×C22⋊C9—21no—
50C22×C4×C9—31no—
51C22×C4×C9—31no—
52C22×C4×C9—32no—
53C22×C4×C9C22×C312yescomplemented
54C22×Dic9—31no—
55C22×Dic9—31no—
56C22×Dic9—32no—
57C22×Dic9C22×C312yescomplemented
58C2×C4×D9—32no—
59C2×C4×D9—34no—
60C2×C4×D9—34no—
61C2×C4×D9—34no—
62C2×C4×D9—31no—
63C2×C4×D9—31no—
64C2×C4×D9—31no—
65C2×C4×D9—34no—
66C2×C4×D9—34no—
67C2×C4×D9—34no—
68C2×C4×D9—34no—
69C2×C4×D9—34no—
70C2×C4×D9A411yescharacteristic, complemented
71C4×C22⋊C9—22no—
72C22×3−1+2—41no—
73C2×AGL(1,ℤ/9)—44no—
74C2×AGL(1,ℤ/9)—42no—
75C9⋊A4C22×C411yescharacteristic, complemented
76C4×3−1+2—42no—
773−1+2⋊C4—42no—
78C24×D3—31no—
79C23×A4—91no—
80C23×C3×C4C3×D311yescharacteristic
81C23×Dic3—31no—
82C22×C4×D3—34no—
83C22×C4×D3—91no—
84C22×C4×D3—94no—
85C22×C4×D3—91no—
86C22×C4×D3—91no—
87C22×C4×D3—91no—
88C2×C4×A4—31no—
89C2×C4×A4—94no—
90C2×C4×A4—91no—
91C22×C3×D3—121no—
92C2×C3×A4C4×D312yes—
93C2×C3×A4C22×D311yescharacteristic
94D3×A4—34no—
95C2×C32×C4—41no—
96C2×C3×Dic3—121no—
97C3×C4×D3—124no—
98C23×C9—31no—
99C23×C9—32no—
100C23×C9—31no—
101C23×C9C2×C3×C412yescomplemented
102C23×C9C23×C311yescharacteristic
103C22×D9—32no—
104C22×D9—34no—
105C22×D9—38no—
106C22×D9—31no—
107C22×D9—34no—
108C22×D9—34no—
109C22×D9—32no—
110C22×D9—32no—
111C22×D9—32no—
112C22×D9—34no—
113C22×D9—38no—
114C22×D9—34no—
115C22×D9C2×C3×C414yescomplemented
116C22×D9C2×A411yescharacteristic
117C2×C22⋊C9—22no—
118C2×C22⋊C9—21no—
119C2×C4×C9—32no—
120C2×C4×C9—31no—
121C2×C4×C9—32no—
122C2×C4×C9—32no—
123C2×C4×C9—32no—
124C2×C4×C9—81no—
125C2×C4×C9C2×A411yescharacteristic, complemented
126C2×Dic9—32no—
127C2×Dic9—31no—
128C2×Dic9—32no—
129C2×Dic9—32no—
130C2×Dic9—32no—
131C2×Dic9C2×A411yescharacteristic, complemented
132C4×D9—38no—
133C4×D9—34no—
134C4×D9—34no—
135C4×D9—34no—
136C4×D9C2×A414yescomplemented
137C24×C4—91noSylow
138C2×3−1+2—42no—
139C2×3−1+2—41no—
140AGL(1,ℤ/9)—44no—
141C24×C3C2×C3×D311yescharacteristic
142C23×D3—34no—
143C23×D3—32no—
144C23×D3—91no—
145C23×D3—92no—
146C23×D3—91no—
147C23×D3—94no—
148C22×A4—31no—
149C22×A4—92no—
150C22×A4—94no—
151C22×C3×C4—31no—
152C22×C3×C4—31no—
153C22×C3×C4—32no—
154C22×C3×C4—361no—
155C22×C3×C4C2×C3×D312yes—
156C22×Dic3—32no—
157C22×Dic3—91no—
158C22×Dic3—92no—
159C22×Dic3—91no—
160C2×C4×D3—31no—
161C2×C4×D3—94no—
162C2×C4×D3—94no—
163C2×C4×D3—91no—
164C2×C4×D3—91no—
165C2×C4×D3—94no—
166C2×C4×D3—94no—
167C2×C4×D3—92no—
168C2×C4×D3—94no—
169C2×C4×D3—91no—
170C2×C4×D3—94no—
171C2×C4×D3—94no—
172C2×C4×D3—94no—
173C4×A4—32no—
174C4×A4—92no—
175C22×C32—41no—
176C2×C3×D3—122no—
177C2×C3×D3—124no—
178C3×A4C2×C4×D311yescharacteristic
179C32×C4—42no—
180C3×Dic3—122no—
181C22×C9—32no—
182C22×C9—31no—
183C22×C9—32no—
184C22×C9—32no—
185C22×C9—32no—
186C22×C9—31no—
187C22×C9—31no—
188C22×C9—81no—
189C22×C9C22×C3×C411yescharacteristic, complemented
190C22×C9C22×A411yescharacteristic
191C2×D9—38no—
192C2×D9—34no—
193C2×D9—34no—
194C2×D9—32no—
195C2×D9—34no—
196C2×D9—38no—
197C2×D9—34no—
198C2×D9—34no—
199C2×D9C4×A414yescomplemented
200C2×D9C22×A412yes—
201C22⋊C9—21no—
202C4×C9—32no—
203C4×C9—82no—
204C4×C9C22×A412yescomplemented
205Dic9—32no—
206Dic9C22×A412yescomplemented
207C25—91no—
208C23×C4—91no—
209C23×C4—94no—
210C23×C4—271no—
211C23×C4—271no—
212C23×C4—274no—
213C23×C4—271no—
214C23×C4—271no—
215C23×C4AGL(1,ℤ/9)11yescharacteristic, complemented
2163−1+2—41noSylow
217C23×C3—31no—
218C23×C3—32no—
219C23×C3—31no—
220C23×C3—361no—
221C23×C3C3×C4×D312yes—
222C23×C3C22×C3×D311yescharacteristic
223C22×D3—31no—
224C22×D3—34no—
225C22×D3—92no—
226C22×D3—92no—
227C22×D3—94no—
228C22×D3—94no—
229C22×D3—92no—
230C22×D3—98no—
231C22×D3—94no—
232C22×D3—91no—
233C22×D3—92no—
234C22×D3—98no—
235C22×D3—94no—
236C22×D3—94no—
237C2×A4—32no—
238C2×A4—31no—
239C2×A4—94no—
240C2×C3×C4—32no—
241C2×C3×C4—31no—
242C2×C3×C4—32no—
243C2×C3×C4—32no—
244C2×C3×C4—32no—
245C2×C3×C4—121no—
246C2×C3×C4—364no—
247C2×C3×C4—361no—
248C2×C3×C4D3×A411yescharacteristic
249C2×Dic3—31no—
250C2×Dic3—92no—
251C2×Dic3—92no—
252C2×Dic3—92no—
253C2×Dic3—91no—
254C2×Dic3—92no—
255C4×D3—34no—
256C4×D3—94no—
257C4×D3—94no—
258C4×D3—98no—
259C4×D3—94no—
260C2×C32—42no—
261C2×C32—41no—
262C3×D3—124no—
263C2×C9—32no—
264C2×C9—31no—
265C2×C9—31no—
266C2×C9—82no—
267C2×C9—81no—
268C2×C9C2×C4×A412yescomplemented
269C2×C9C23×A411yescharacteristic
270D9—34no—
271D9C2×C4×A414yescomplemented
272C24—94no—
273C24—92no—
274C24—271no—
275C24—272no—
276C24—271no—
277C24—274no—
278C24C2×AGL(1,ℤ/9)11yescharacteristic
279C22×C4—31no—
280C22×C4—31no—
281C22×C4—32no—
282C22×C4—91no—
283C22×C4—92no—
284C22×C4—274no—
285C22×C4—271no—
286C22×C4—274no—
287C22×C4—271no—
288C22×C4—274no—
289C22×C4—274no—
290C22×C4—272no—
291C22×C4—274no—
292C22×C4—274no—
293C22×C4—274no—
294C22×C4—274no—
295C22×C4—271no—
296C22×C4—271no—
297C22×C4—272no—
298C22×C4—271no—
299C22×C4C2×AGL(1,ℤ/9)12yescomplemented
300C22×C3—32no—
301C22×C3—31no—
302C22×C3—32no—
303C22×C3—32no—
304C22×C3—32no—
305C22×C3—31no—
306C22×C3—31no—
307C22×C3—121no—
308C22×C3—362no—
309C22×C3—364no—
310C22×C3C2×C3×C4×D311yescharacteristic
311C22×C3C2×D3×A411yescharacteristic
312C2×D3—34no—
313C2×D3—32no—
314C2×D3—94no—
315C2×D3—94no—
316C2×D3—98no—
317C2×D3—98no—
318C2×D3—94no—
319C2×D3—94no—
320C2×D3—92no—
321C2×D3—94no—
322A4—31no—
323C3×C4—32no—
324C3×C4—122no—
325C3×C4—362no—
326C3×C4C2×D3×A412yes—
327Dic3—32no—
328Dic3—92no—
329C32—41no—
330C9—81no—
331C9C22×C4×A411yescharacteristic, complemented
332C23—31no—
333C23—32no—
334C23—31no—
335C23—91no—
336C23—94no—
337C23—272no—
338C23—272no—
339C23—274no—
340C23—274no—
341C23—272no—
342C23—278no—
343C23—274no—
344C23—274no—
345C23—272no—
346C23—278no—
347C23—274no—
348C23—271no—
349C23C4×AGL(1,ℤ/9)12yescomplemented
350C23C22×AGL(1,ℤ/9)11yescharacteristic
351C2×C4—32no—
352C2×C4—31no—
353C2×C4—32no—
354C2×C4—32no—
355C2×C4—32no—
356C2×C4—91no—
357C2×C4—94no—
358C2×C4—274no—
359C2×C4—274no—
360C2×C4—278no—
361C2×C4—272no—
362C2×C4—272no—
363C2×C4—272no—
364C2×C4—271no—
365C2×C4—274no—
366C2×C4—272no—
367C2×C4D9⋊A411yescharacteristic, direct product factor
368C2×C3—31no—
369C2×C3—32no—
370C2×C3—31no—
371C2×C3—121no—
372C2×C3—122no—
373C2×C3—364no—
374C2×C3C4×D3×A412yes—
375C2×C3C22×D3×A411yescharacteristic
376D3—34no—
377D3—94no—
378C22—32no—
379C22—31no—
380C22—32no—
381C22—31no—
382C22—31no—
383C22—32no—
384C22—32no—
385C22—92no—
386C22—94no—
387C22—274no—
388C22—274no—
389C22—278no—
390C22—274no—
391C22—274no—
392C22—278no—
393C22—272no—
394C22—274no—
395C22C2×C4×AGL(1,ℤ/9)11yescharacteristic, complemented
396C22C2×D9⋊A411yescharacteristic
397C4—32no—
398C4—92no—
399C4—272no—
400C4C2×D9⋊A412yesdirect product factor
401C3—121no—
402C3C2×C4×D3×A411yescharacteristic
403C2—31no—
404C2—32no—
405C2—31no—
406C2—94no—
407C2—274no—
408C2C4×D9⋊A412yesdirect product factor
409C2C22×D9⋊A411yescharacteristic
subgroupG/H
center, hypercenterC2×C4D9⋊A4
commutator, hypocenterC22×C9C22×C3×C4
FrattiniC2×C3C22×D3×A4
FittingC23×C4×C9C2×C3
socleC24×C3C2×C3×D3
prime radicalC22×D9⋊A4C2
cosocleC23×C9C23×C3
LCSG ▷ C22×C9
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C9⋊A4C2semidirect
3. C2×Dic9⋊A4C2semidirect
4. C4×D9⋊A4C2direct (N × H)
6. C23×C4×D9C3semidirect
8. C2×D9⋊A4C4direct (N × H)
11. C4×C9⋊A4C22semidirect
12. Dic9⋊A4C22semidirect
18. C23×C4×C9C2×C3semidirect
19. C23×Dic9C2×C3semidirect
25. C22×C4×D9C2×C3semidirect
28. C2×C9⋊A4C2×C4semidirect
30. D9⋊A4C2×C4direct (N × H)
47. C23×D9C3×C4semidirect
53. C22×C4×C9C22×C3semidirect
57. C22×Dic9C22×C3semidirect
70. C2×C4×D9A4semidirect
75. C9⋊A4C22×C4semidirect
101. C23×C9C2×C3×C4semidirect
115. C22×D9C2×C3×C4semidirect
125. C2×C4×C9C2×A4semidirect
131. C2×Dic9C2×A4semidirect
136. C4×D9C2×A4semidirect
189. C22×C9C22×C3×C4semidirect
199. C2×D9C4×A4semidirect
204. C4×C9C22×A4semidirect
206. Dic9C22×A4semidirect
215. C23×C4AGL(1,ℤ/9)semidirect
268. C2×C9C2×C4×A4semidirect
271. D9C2×C4×A4semidirect
299. C22×C4C2×AGL(1,ℤ/9)semidirect
331. C9C22×C4×A4semidirect
349. C23C4×AGL(1,ℤ/9)semidirect
367. C2×C4D9⋊A4direct (N × H)
395. C22C2×C4×AGL(1,ℤ/9)semidirect
400. C4C2×D9⋊A4direct (N × H)
408. C2C4×D9⋊A4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D9⋊A42
C2×C4×C9⋊A42
C2×Dic9⋊A42
C4×D9⋊A42
C2×C4×D3×A43
C23×C4×D93
C2×C4×AGL(1,ℤ/9)4