Finite Group Databasebeta

← C2×D4×C9⋊A4 #13164824

Subgroups — C2×D4×C9⋊A4 #13164824

Order 1728.

1059 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1059 conjugacy classes of subgroups — too many to draw. Only its 190 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×C9⋊A4C212yesmaximal, complemented
2D4×C9⋊A4C214yesmaximal, direct product factor
3C2×C4×C9⋊A4C211yesmaximal, characteristic, complemented
4C2×C3×D4×A4C311yesmaximal, characteristic
5C23×D4×C9C311yesmaximal, characteristic, complemented
6C2×D4×C22⋊C9C312yesmaximal, complemented
7C22×C9⋊A4—24no—
8C22×C9⋊A4C2211yescharacteristic
9C22×C9⋊A4C2214yescomplemented
10C2×D4×3−1+2—41nomaximal
11C4×C9⋊A4C2212yescomplemented
12C23×C3×A4C2×C312yes—
13C3×D4×A4C2×C314yes—
14C2×C3×C4×A4C2×C311yescharacteristic
15C25×C9C2×C312yescomplemented
16C23×C22⋊C9C2×C314yescomplemented
17C22×D4×C9—31no—
18C22×D4×C9—31no—
19C22×D4×C9—32no—
20C22×D4×C9—34no—
21C22×D4×C9C2×C314yescomplemented
22C23×C4×C9C2×C311yescharacteristic, complemented
23D4×C22⋊C9C2×C318yescomplemented
24C2×C4×C22⋊C9C2×C312yescomplemented
25C23×3−1+2—42no—
26C2×C9⋊A4—24no—
27C2×C9⋊A4D412yesdirect product factor
28C2×C9⋊A4C2311yescharacteristic
29D4×3−1+2—44no—
30C2×C4×3−1+2—41no—
31C23×C3×D4C3211yescharacteristic
32C2×D4×A4—31no—
33C22×C3×A4—24no—
34C22×C3×A4C22×C314yes—
35C22×C3×A4C22×C311yescharacteristic
36C2×C32×D4—41no—
37C3×C4×A4C22×C312yes—
38C24×C9—24no—
39C24×C9—32no—
40C24×C9—34no—
41C24×C9—32no—
42C24×C9—64no—
43C24×C9C22×C311yescharacteristic
44C24×C9C22×C314yescomplemented
45C22×C22⋊C9—28no—
46C22×C22⋊C9C22×C312yes—
47C22×C22⋊C9C22×C318yescomplemented
48C2×D4×C9—38no—
49C2×D4×C9—32no—
50C2×D4×C9—34no—
51C2×D4×C9—32no—
52C2×D4×C9—34no—
53C2×D4×C9—34no—
54C2×D4×C9—34no—
55C2×D4×C9—38no—
56C2×D4×C9—31no—
57C2×D4×C9—42no—
58C2×D4×C9A411yescharacteristic, complemented
59C22×C4×C9—31no—
60C22×C4×C9—31no—
61C22×C4×C9—32no—
62C22×C4×C9C22×C312yescomplemented
63C4×C22⋊C9C22×C314yescomplemented
64C22×3−1+2—41no—
65C22×3−1+2—44no—
66C22×3−1+2—84no—
67C9⋊A4C2×D411yescharacteristic, direct product factor
68C4×3−1+2—42no—
69C25×C3C2×C3212yes—
70C23×A4—32no—
71C22×C3×D4—31no—
72C22×C3×D4—31no—
73C22×C3×D4—32no—
74C22×C3×D4—34no—
75C22×C3×D4C2×C3214yes—
76C23×C3×C4C2×C3211yescharacteristic
77D4×A4—34no—
78C2×C4×A4—31no—
79C23×C32—42no—
80C2×C3×A4—24no—
81C2×C3×A4C3×D412yes—
82C2×C3×A4C23×C311yescharacteristic
83C32×D4—44no—
84C2×C32×C4—41no—
85C23×C9—24no—
86C23×C9—31no—
87C23×C9—32no—
88C23×C9—31no—
89C23×C9—34no—
90C23×C9—34no—
91C23×C9—32no—
92C23×C9—34no—
93C23×C9—34no—
94C23×C9—44no—
95C23×C9—64no—
96C23×C9—64no—
97C23×C9—64no—
98C23×C9—68no—
99C23×C9—64no—
100C23×C9—64no—
101C23×C9—68no—
102C23×C9C3×D412yescomplemented
103C23×C9C2×A412yescomplemented
104C23×C9C23×C311yescharacteristic
105C2×C22⋊C9—28no—
106C2×C22⋊C9C3×D414yescomplemented
107C2×C22⋊C9C23×C312yes—
108D4×C9—38no—
109D4×C9—38no—
110D4×C9—34no—
111D4×C9—48no—
112D4×C9C2×A414yescomplemented
113C2×C4×C9—32no—
114C2×C4×C9—31no—
115C2×C4×C9—32no—
116C2×C4×C9—32no—
117C2×C4×C9—32no—
118C2×C4×C9—42no—
119C2×C4×C9C2×A411yescharacteristic, complemented
120C23×D43−1+211yescharacteristic, complemented, Sylow
121C2×3−1+2—42no—
122C2×3−1+2—41no—
123C2×3−1+2—84no—
124C24×C3—24no—
125C24×C3—34no—
126C24×C3—32no—
127C24×C3—32no—
128C24×C3—64no—
129C24×C3C22×C3211yescharacteristic
130C24×C3C22×C3214yes—
131C22×A4—31no—
132C22×A4—34no—
133C22×A4—64no—
134C2×C3×D4—38no—
135C2×C3×D4—32no—
136C2×C3×D4—34no—
137C2×C3×D4—32no—
138C2×C3×D4—34no—
139C2×C3×D4—34no—
140C2×C3×D4—34no—
141C2×C3×D4—38no—
142C2×C3×D4—31no—
143C2×C3×D4—121no—
144C2×C3×D4C3×A411yescharacteristic
145C22×C3×C4—31no—
146C22×C3×C4—31no—
147C22×C3×C4—32no—
148C22×C3×C4C22×C3212yes—
149C4×A4—32no—
150C22×C32—44no—
151C22×C32—41no—
152C22×C32—84no—
153C3×A4C2×C3×D411yescharacteristic
154C32×C4—42no—
155C22×C9—24no—
156C22×C9—32no—
157C22×C9—31no—
158C22×C9—32no—
159C22×C9—32no—
160C22×C9—32no—
161C22×C9—31no—
162C22×C9—31no—
163C22×C9—34no—
164C22×C9—48no—
165C22×C9—42no—
166C22×C9—64no—
167C22×C9—68no—
168C22×C9—64no—
169C22×C9—64no—
170C22×C9—64no—
171C22×C9—68no—
172C22×C9—64no—
173C22×C9—88no—
174C22×C9C2×C3×D411yescharacteristic, complemented
175C22×C9C22×A411yescharacteristic
176C22×C9C22×A414yescomplemented
177C22⋊C9C2×C3×D412yescomplemented
178C4×C9—32no—
179C4×C9—44no—
180C4×C9C22×A412yescomplemented
181C25C2×3−1+212yescomplemented
182C22×D4—31no—
183C22×D4—31no—
184C22×D4—32no—
185C22×D4—34no—
186C22×D4C2×3−1+214yescomplemented
187C23×C4C2×3−1+211yescharacteristic, complemented
1883−1+2—41noSylow
189C23×C3—24no—
190C23×C3—31no—
191C23×C3—32no—
192C23×C3—31no—
193C23×C3—34no—
194C23×C3—34no—
195C23×C3—34no—
196C23×C3—34no—
197C23×C3—32no—
198C23×C3—64no—
199C23×C3—64no—
200C23×C3—64no—
201C23×C3—68no—
202C23×C3—64no—
203C23×C3—64no—
204C23×C3—68no—
205C23×C3—122no—
206C23×C3C32×D412yes—
207C23×C3C2×C3×A412yes—
208C23×C3C23×C3211yescharacteristic
209C2×A4—32no—
210C2×A4—31no—
211C2×A4—64no—
212C3×D4—38no—
213C3×D4—38no—
214C3×D4—34no—
215C3×D4—124no—
216C3×D4C2×C3×A414yes—
217C2×C3×C4—32no—
218C2×C3×C4—31no—
219C2×C3×C4—32no—
220C2×C3×C4—32no—
221C2×C3×C4—32no—
222C2×C3×C4—121no—
223C2×C3×C4C2×C3×A411yescharacteristic
224C2×C32—42no—
225C2×C32—41no—
226C2×C32—84no—
227C2×C9—24no—
228C2×C9—32no—
229C2×C9—31no—
230C2×C9—31no—
231C2×C9—44no—
232C2×C9—42no—
233C2×C9—64no—
234C2×C9—88no—
235C2×C9D4×A412yescomplemented
236C2×C9C23×A411yescharacteristic
237C24—24no—
238C24—34no—
239C24—32no—
240C24—32no—
241C24—64no—
242C24C22×3−1+211yescharacteristic
243C24C22×3−1+214yescomplemented
244C2×D4—38no—
245C2×D4—32no—
246C2×D4—34no—
247C2×D4—32no—
248C2×D4—34no—
249C2×D4—34no—
250C2×D4—34no—
251C2×D4—38no—
252C2×D4—31no—
253C2×D4C9⋊A411yescharacteristic, direct product factor
254C22×C4—31no—
255C22×C4—31no—
256C22×C4—32no—
257C22×C4C22×3−1+212yescomplemented
258C22×C3—24no—
259C22×C3—32no—
260C22×C3—31no—
261C22×C3—32no—
262C22×C3—32no—
263C22×C3—32no—
264C22×C3—31no—
265C22×C3—31no—
266C22×C3—34no—
267C22×C3—64no—
268C22×C3—68no—
269C22×C3—64no—
270C22×C3—64no—
271C22×C3—64no—
272C22×C3—68no—
273C22×C3—64no—
274C22×C3—121no—
275C22×C3—124no—
276C22×C3—244no—
277C22×C3C2×C32×D411yescharacteristic
278C22×C3C22×C3×A414yes—
279C22×C3C22×C3×A411yescharacteristic
280A4—31no—
281C3×C4—32no—
282C3×C4—122no—
283C3×C4C22×C3×A412yes—
284C32—41no—
285C9—42no—
286C9C2×D4×A411yescharacteristic, complemented
287C23—24no—
288C23—31no—
289C23—32no—
290C23—31no—
291C23—34no—
292C23—34no—
293C23—32no—
294C23—34no—
295C23—34no—
296C23—64no—
297C23—64no—
298C23—64no—
299C23—68no—
300C23—64no—
301C23—64no—
302C23—68no—
303C23D4×3−1+212yescomplemented
304C23C2×C9⋊A412yescomplemented
305C23C23×3−1+211yescharacteristic
306D4—38no—
307D4—38no—
308D4—34no—
309D4C2×C9⋊A414yesdirect product factor
310C2×C4—32no—
311C2×C4—31no—
312C2×C4—32no—
313C2×C4—32no—
314C2×C4—32no—
315C2×C4C2×C9⋊A411yescharacteristic, complemented
316C2×C3—24no—
317C2×C3—31no—
318C2×C3—32no—
319C2×C3—31no—
320C2×C3—64no—
321C2×C3—121no—
322C2×C3—122no—
323C2×C3—244no—
324C2×C3C3×D4×A412yes—
325C2×C3C23×C3×A411yescharacteristic
326C22—24no—
327C22—32no—
328C22—31no—
329C22—32no—
330C22—31no—
331C22—31no—
332C22—32no—
333C22—32no—
334C22—34no—
335C22—64no—
336C22—64no—
337C22—68no—
338C22—64no—
339C22—64no—
340C22—68no—
341C22—64no—
342C22C2×D4×3−1+211yescharacteristic, complemented
343C22C22×C9⋊A411yescharacteristic
344C22C22×C9⋊A414yescomplemented
345C4—32no—
346C4C22×C9⋊A412yescomplemented
347C3—121no—
348C3C2×C3×D4×A411yescharacteristic
349C2—24no—
350C2—31no—
351C2—32no—
352C2—31no—
353C2—64no—
354C2D4×C9⋊A412yesdirect product factor
355C2C23×C9⋊A411yescharacteristic
subgroupG/H
centerC22×C3C22×C3×A4
commutator, cosocleC23×C3C23×C32
FrattiniC2×C3C23×C3×A4
FittingC23×D4×C9C3
socleC24×C3C22×C32
prime radicalC2×C3×D4×A4C3
hypercenterC2×D4×C9A4
hypocenterC22C2×D4×3−1+2
LCSG ▷ C23×C3 ▷ C22
UCSC1 ◁ C22×C3 ◁ C2×D4×C9

Complements

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

NH ≅ G/Ntype
1. C23×C9⋊A4C2semidirect
2. D4×C9⋊A4C2direct (N × H)
3. C2×C4×C9⋊A4C2semidirect
5. C23×D4×C9C3semidirect
6. C2×D4×C22⋊C9C3semidirect
9. C22×C9⋊A4C22semidirect
11. C4×C9⋊A4C22semidirect
15. C25×C9C2×C3semidirect
16. C23×C22⋊C9C2×C3semidirect
21. C22×D4×C9C2×C3semidirect
22. C23×C4×C9C2×C3semidirect
23. D4×C22⋊C9C2×C3semidirect
24. C2×C4×C22⋊C9C2×C3semidirect
27. C2×C9⋊A4D4direct (N × H)
44. C24×C9C22×C3semidirect
47. C22×C22⋊C9C22×C3semidirect
58. C2×D4×C9A4semidirect
62. C22×C4×C9C22×C3semidirect
63. C4×C22⋊C9C22×C3semidirect
67. C9⋊A4C2×D4direct (N × H)
102. C23×C9C3×D4semidirect
103. C23×C9C2×A4semidirect
106. C2×C22⋊C9C3×D4semidirect
112. D4×C9C2×A4semidirect
119. C2×C4×C9C2×A4semidirect
120. C23×D43−1+2semidirect
174. C22×C9C2×C3×D4semidirect
176. C22×C9C22×A4semidirect
177. C22⋊C9C2×C3×D4semidirect
180. C4×C9C22×A4semidirect
181. C25C2×3−1+2semidirect
186. C22×D4C2×3−1+2semidirect
187. C23×C4C2×3−1+2semidirect
235. C2×C9D4×A4semidirect
243. C24C22×3−1+2semidirect
253. C2×D4C9⋊A4direct (N × H)
257. C22×C4C22×3−1+2semidirect
286. C9C2×D4×A4semidirect
303. C23D4×3−1+2semidirect
304. C23C2×C9⋊A4semidirect
309. D4C2×C9⋊A4direct (N × H)
315. C2×C4C2×C9⋊A4semidirect
342. C22C2×D4×3−1+2semidirect
344. C22C22×C9⋊A4semidirect
346. C4C22×C9⋊A4semidirect
354. C2D4×C9⋊A4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C9⋊A42
D4×C9⋊A42
C2×C4×C9⋊A42
C2×C3×D4×A43
C23×D4×C93
C2×D4×C22⋊C93
C2×D4×3−1+24