Finite Group Databasebeta

← C2×C3×D4×C22⋊C9 #13164842

Subgroups — C2×C3×D4×C22⋊C9 #13164842

Order 1728.

1794 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C22⋊C9C212yesmaximal, complemented
2C3×D4×C22⋊C9C214yesmaximal, direct product factor
3C2×C3×C4×C22⋊C9C211yesmaximal, characteristic, complemented
4C23×C32×D4C311yesmaximal, characteristic
5C2×D4×C22⋊C9C313yesmaximal, direct product factor
6C22×C3×C22⋊C9—24no—
7C22×C3×C22⋊C9C2211yescharacteristic
8C22×C3×C22⋊C9C2214yescomplemented
9C2×C3×D4×C9—41nomaximal
10C3×C4×C22⋊C9C2212yescomplemented
11C25×C32C2×C312yes—
12C22×C32×D4—32no—
13C22×C32×D4—31no—
14C22×C32×D4—34no—
15C22×C32×D4—31no—
16C22×C32×D4C2×C314yes—
17C23×C32×C4C2×C311yescharacteristic
18C23×C22⋊C9C2×C316yescomplemented
19D4×C22⋊C9C2×C3112yesdirect product factor
20C2×C4×C22⋊C9C2×C313yescomplemented
21C23×C3×C9—42no—
22C2×C3×C22⋊C9—24no—
23C2×C3×C22⋊C9D412yesdirect product factor
24C2×C3×C22⋊C9C2311yescharacteristic
25C3×D4×C9—44no—
26C2×C3×C4×C9—41no—
27C23×C3×D4C913yescomplemented
28C23×C3×D4C3211yescharacteristic
29C24×C32—24no—
30C24×C32—32no—
31C24×C32—34no—
32C24×C32—32no—
33C24×C32—64no—
34C24×C32C22×C314yes—
35C24×C32C22×C311yescharacteristic
36C2×C32×D4—38no—
37C2×C32×D4—32no—
38C2×C32×D4—34no—
39C2×C32×D4—32no—
40C2×C32×D4—34no—
41C2×C32×D4—34no—
42C2×C32×D4—34no—
43C2×C32×D4—38no—
44C2×C32×D4—31no—
45C2×C32×D4A411yescharacteristic
46C22×C32×C4—31no—
47C22×C32×C4—31no—
48C22×C32×C4—32no—
49C22×C32×C4C22×C312yes—
50C22×C22⋊C9—212no—
51C22×C22⋊C9C22×C313yes—
52C22×C22⋊C9C22×C3112yescomplemented
53C2×D4×C9—43no—
54C4×C22⋊C9C22×C316yescomplemented
55C22×C3×C9—41no—
56C22×C3×C9—44no—
57C22×C3×C9—84no—
58C3×C22⋊C9C2×D411yescharacteristic, direct product factor
59C3×C4×C9—42no—
60C25×C3C2×C916yescomplemented
61C25×C3C2×C3212yes—
62C22×C3×D4—33no—
63C22×C3×D4—31no—
64C22×C3×D4—33no—
65C22×C3×D4—36no—
66C22×C3×D4—312no—
67C22×C3×D4—31no—
68C22×C3×D4—32no—
69C22×C3×D4—34no—
70C22×C3×D4C2×C9112yescomplemented
71C22×C3×D4C2×C3214yes—
72C23×C3×C4C2×C913yescomplemented
73C23×C3×C4C2×C3211yescharacteristic
74C23×C32—24no—
75C23×C32—31no—
76C23×C32—32no—
77C23×C32—31no—
78C23×C32—34no—
79C23×C32—34no—
80C23×C32—34no—
81C23×C32—34no—
82C23×C32—32no—
83C23×C32—64no—
84C23×C32—64no—
85C23×C32—64no—
86C23×C32—68no—
87C23×C32—64no—
88C23×C32—64no—
89C23×C32—68no—
90C23×C32C3×D412yes—
91C23×C32C2×A412yes—
92C23×C32C23×C311yescharacteristic
93C32×D4—38no—
94C32×D4—38no—
95C32×D4—34no—
96C32×D4C2×A414yes—
97C2×C32×C4—32no—
98C2×C32×C4—31no—
99C2×C32×C4—32no—
100C2×C32×C4—32no—
101C2×C32×C4—32no—
102C2×C32×C4C2×A411yescharacteristic
103C23×C9—46no—
104C2×C22⋊C9—212no—
105C2×C22⋊C9C3×D416yesdirect product factor
106C2×C22⋊C9C23×C313yes—
107D4×C9—412no—
108C2×C4×C9—43no—
109C23×D4C3×C911yescharacteristic, complemented, Sylow
110C2×C3×C9—42no—
111C2×C3×C9—41no—
112C2×C3×C9—84no—
113C24×C3—212no—
114C24×C3—24no—
115C24×C3—312no—
116C24×C3—36no—
117C24×C3—34no—
118C24×C3—32no—
119C24×C3—36no—
120C24×C3—32no—
121C24×C3—612no—
122C24×C3—64no—
123C24×C3C22×C913yes—
124C24×C3C22×C9112yescomplemented
125C24×C3C22×C3211yescharacteristic
126C24×C3C22×C3214yes—
127C2×C3×D4—38no—
128C2×C3×D4—324no—
129C2×C3×D4—32no—
130C2×C3×D4—36no—
131C2×C3×D4—34no—
132C2×C3×D4—312no—
133C2×C3×D4—36no—
134C2×C3×D4—32no—
135C2×C3×D4—312no—
136C2×C3×D4—312no—
137C2×C3×D4—34no—
138C2×C3×D4—34no—
139C2×C3×D4—312no—
140C2×C3×D4—324no—
141C2×C3×D4—34no—
142C2×C3×D4—38no—
143C2×C3×D4—33no—
144C2×C3×D4—31no—
145C2×C3×D4C22⋊C913yesdirect product factor
146C2×C3×D4C3×A411yescharacteristic
147C22×C3×C4—33no—
148C22×C3×C4—31no—
149C22×C3×C4—33no—
150C22×C3×C4—36no—
151C22×C3×C4—31no—
152C22×C3×C4—32no—
153C22×C3×C4C22×C916yescomplemented
154C22×C3×C4C22×C3212yes—
155C22×C32—24no—
156C22×C32—32no—
157C22×C32—31no—
158C22×C32—32no—
159C22×C32—32no—
160C22×C32—32no—
161C22×C32—31no—
162C22×C32—31no—
163C22×C32—34no—
164C22×C32—64no—
165C22×C32—64no—
166C22×C32—68no—
167C22×C32—64no—
168C22×C32—64no—
169C22×C32—68no—
170C22×C32—64no—
171C22×C32C2×C3×D411yescharacteristic
172C22×C32C22×A414yes—
173C22×C32C22×A411yescharacteristic
174C32×C4—32no—
175C32×C4C22×A412yes—
176C22×C9—43no—
177C22×C9—412no—
178C22×C9—812no—
179C22⋊C9C2×C3×D413yesdirect product factor
180C4×C9—46no—
181C25C2×C3×C912yescomplemented
182C22×D4—31no—
183C22×D4—31no—
184C22×D4—32no—
185C22×D4—34no—
186C22×D4C2×C3×C914yescomplemented
187C23×C4C2×C3×C911yescharacteristic, complemented
188C3×C9—41noSylow
189C23×C3—212no—
190C23×C3—24no—
191C23×C3—33no—
192C23×C3—31no—
193C23×C3—36no—
194C23×C3—32no—
195C23×C3—31no—
196C23×C3—33no—
197C23×C3—34no—
198C23×C3—312no—
199C23×C3—312no—
200C23×C3—34no—
201C23×C3—32no—
202C23×C3—34no—
203C23×C3—34no—
204C23×C3—36no—
205C23×C3—312no—
206C23×C3—312no—
207C23×C3—64no—
208C23×C3—612no—
209C23×C3—64no—
210C23×C3—612no—
211C23×C3—612no—
212C23×C3—64no—
213C23×C3—624no—
214C23×C3—68no—
215C23×C3—64no—
216C23×C3—612no—
217C23×C3—64no—
218C23×C3—612no—
219C23×C3—624no—
220C23×C3—68no—
221C23×C3D4×C916yescomplemented
222C23×C3C2×C22⋊C916yescomplemented
223C23×C3C23×C913yes—
224C23×C3C32×D412yes—
225C23×C3C2×C3×A412yes—
226C23×C3C23×C3211yescharacteristic
227C3×D4—38no—
228C3×D4—324no—
229C3×D4—38no—
230C3×D4—324no—
231C3×D4—34no—
232C3×D4—312no—
233C3×D4C2×C22⋊C9112yesdirect product factor
234C3×D4C2×C3×A414yes—
235C2×C3×C4—32no—
236C2×C3×C4—36no—
237C2×C3×C4—31no—
238C2×C3×C4—33no—
239C2×C3×C4—32no—
240C2×C3×C4—32no—
241C2×C3×C4—36no—
242C2×C3×C4—36no—
243C2×C3×C4—36no—
244C2×C3×C4—32no—
245C2×C3×C4C2×C22⋊C913yescomplemented
246C2×C3×C4C2×C3×A411yescharacteristic
247C2×C32—24no—
248C2×C32—32no—
249C2×C32—31no—
250C2×C32—31no—
251C2×C32—64no—
252C2×C32D4×A412yes—
253C2×C32C23×A411yescharacteristic
254C2×C9—46no—
255C2×C9—43no—
256C2×C9—812no—
257C24—24no—
258C24—34no—
259C24—32no—
260C24—32no—
261C24—64no—
262C24C22×C3×C911yescharacteristic
263C24C22×C3×C914yescomplemented
264C2×D4—38no—
265C2×D4—32no—
266C2×D4—34no—
267C2×D4—32no—
268C2×D4—34no—
269C2×D4—34no—
270C2×D4—34no—
271C2×D4—38no—
272C2×D4—31no—
273C2×D4C3×C22⋊C911yescharacteristic, direct product factor
274C22×C4—31no—
275C22×C4—31no—
276C22×C4—32no—
277C22×C4C22×C3×C912yescomplemented
278C22×C3—24no—
279C22×C3—212no—
280C22×C3—36no—
281C22×C3—33no—
282C22×C3—32no—
283C22×C3—31no—
284C22×C3—36no—
285C22×C3—32no—
286C22×C3—32no—
287C22×C3—32no—
288C22×C3—31no—
289C22×C3—31no—
290C22×C3—33no—
291C22×C3—33no—
292C22×C3—36no—
293C22×C3—36no—
294C22×C3—34no—
295C22×C3—312no—
296C22×C3—612no—
297C22×C3—64no—
298C22×C3—624no—
299C22×C3—612no—
300C22×C3—68no—
301C22×C3—64no—
302C22×C3—612no—
303C22×C3—64no—
304C22×C3—612no—
305C22×C3—64no—
306C22×C3—624no—
307C22×C3—68no—
308C22×C3—64no—
309C22×C3—612no—
310C22×C3C2×D4×C913yescomplemented
311C22×C3C22×C22⋊C913yes—
312C22×C3C22×C22⋊C9112yescomplemented
313C22×C3C2×C32×D411yescharacteristic
314C22×C3C22×C3×A411yescharacteristic
315C22×C3C22×C3×A414yes—
316C3×C4—32no—
317C3×C4—36no—
318C3×C4C22×C22⋊C916yescomplemented
319C3×C4C22×C3×A412yes—
320C32C2×D4×A411yescharacteristic
321C9—43no—
322C23—24no—
323C23—31no—
324C23—32no—
325C23—31no—
326C23—34no—
327C23—34no—
328C23—32no—
329C23—34no—
330C23—34no—
331C23—64no—
332C23—64no—
333C23—64no—
334C23—68no—
335C23—64no—
336C23—64no—
337C23—68no—
338C23C3×D4×C912yescomplemented
339C23C2×C3×C22⋊C912yescomplemented
340C23C23×C3×C911yescharacteristic
341D4—38no—
342D4—38no—
343D4—34no—
344D4C2×C3×C22⋊C914yesdirect product factor
345C2×C4—32no—
346C2×C4—31no—
347C2×C4—32no—
348C2×C4—32no—
349C2×C4—32no—
350C2×C4C2×C3×C22⋊C911yescharacteristic, complemented
351C2×C3—212no—
352C2×C3—24no—
353C2×C3—33no—
354C2×C3—36no—
355C2×C3—33no—
356C2×C3—31no—
357C2×C3—32no—
358C2×C3—31no—
359C2×C3—612no—
360C2×C3—64no—
361C2×C3D4×C22⋊C916yesdirect product factor
362C2×C3C23×C22⋊C913yes—
363C2×C3C3×D4×A412yes—
364C2×C3C23×C3×A411yescharacteristic
365C22—24no—
366C22—32no—
367C22—31no—
368C22—32no—
369C22—31no—
370C22—31no—
371C22—32no—
372C22—32no—
373C22—34no—
374C22—64no—
375C22—64no—
376C22—68no—
377C22—64no—
378C22—64no—
379C22—68no—
380C22—64no—
381C22C2×C3×D4×C911yescharacteristic, complemented
382C22C22×C3×C22⋊C911yescharacteristic
383C22C22×C3×C22⋊C914yescomplemented
384C4—32no—
385C4C22×C3×C22⋊C912yescomplemented
386C3C2×D4×C22⋊C913yesdirect product factor
387C3C2×C3×D4×A411yescharacteristic
388C2—24no—
389C2—31no—
390C2—32no—
391C2—31no—
392C2—64no—
393C2C3×D4×C22⋊C912yesdirect product factor
394C2C23×C3×C22⋊C911yescharacteristic
subgroupG/H
centerC22×C32C22×A4
commutatorC23C23×C3×C9
FrattiniC2×C3C23×C3×A4
Fitting, prime radicalC23×C32×D4C3
socleC24×C32C22×C3
cosocleC23×C3C23×C32
hypercenterC2×C32×D4A4
hypocenterC22C2×C3×D4×C9
LCSG ▷ C23 ▷ C22
UCSC1 ◁ C22×C32 ◁ C2×C32×D4

Complements

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

NH ≅ G/Ntype
1. C23×C3×C22⋊C9C2semidirect
2. C3×D4×C22⋊C9C2direct (N × H)
3. C2×C3×C4×C22⋊C9C2semidirect
5. C2×D4×C22⋊C9C3direct (N × H)
8. C22×C3×C22⋊C9C22semidirect
10. C3×C4×C22⋊C9C22semidirect
18. C23×C22⋊C9C2×C3semidirect
19. D4×C22⋊C9C2×C3direct (N × H)
20. C2×C4×C22⋊C9C2×C3semidirect
23. C2×C3×C22⋊C9D4direct (N × H)
27. C23×C3×D4C9semidirect
52. C22×C22⋊C9C22×C3semidirect
54. C4×C22⋊C9C22×C3semidirect
58. C3×C22⋊C9C2×D4direct (N × H)
60. C25×C3C2×C9semidirect
70. C22×C3×D4C2×C9semidirect
72. C23×C3×C4C2×C9semidirect
105. C2×C22⋊C9C3×D4direct (N × H)
109. C23×D4C3×C9semidirect
124. C24×C3C22×C9semidirect
145. C2×C3×D4C22⋊C9direct (N × H)
153. C22×C3×C4C22×C9semidirect
179. C22⋊C9C2×C3×D4direct (N × H)
181. C25C2×C3×C9semidirect
186. C22×D4C2×C3×C9semidirect
187. C23×C4C2×C3×C9semidirect
221. C23×C3D4×C9semidirect
222. C23×C3C2×C22⋊C9semidirect
233. C3×D4C2×C22⋊C9direct (N × H)
245. C2×C3×C4C2×C22⋊C9semidirect
263. C24C22×C3×C9semidirect
273. C2×D4C3×C22⋊C9direct (N × H)
277. C22×C4C22×C3×C9semidirect
310. C22×C3C2×D4×C9semidirect
312. C22×C3C22×C22⋊C9semidirect
318. C3×C4C22×C22⋊C9semidirect
338. C23C3×D4×C9semidirect
339. C23C2×C3×C22⋊C9semidirect
344. D4C2×C3×C22⋊C9direct (N × H)
350. C2×C4C2×C3×C22⋊C9semidirect
361. C2×C3D4×C22⋊C9direct (N × H)
381. C22C2×C3×D4×C9semidirect
383. C22C22×C3×C22⋊C9semidirect
385. C4C22×C3×C22⋊C9semidirect
386. C3C2×D4×C22⋊C9direct (N × H)
393. C2C3×D4×C22⋊C9direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C22⋊C92
C3×D4×C22⋊C92
C2×C3×C4×C22⋊C92
C23×C32×D43
C2×D4×C22⋊C93
C2×C3×D4×C94