Finite Group Databasebeta

← C22×C3×C4×D32 #13181537

Subgroups — C22×C3×C4×D32 #13181537

Order 1728.

6210 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×D32C211yesmaximal, characteristic
2C22×C3×C4×C3⋊D3C211yesmaximal, characteristic, complemented
3C22×C32×C4×D3C212yesmaximal, complemented
4C22×C3×#624C211yesmaximal, characteristic, complemented
5C22×C3×D3×Dic3C212yesmaximal, complemented
6C2×C3×C4×D32C2124yesmaximal, direct product factor
7C23×C3×C4×D3—32nomaximal
8C22×C4×D32C311yesmaximal, characteristic, direct product factor
9C23×C3×C3⋊D3C2211yescharacteristic
10C23×C32×D3C2212yes—
11C22×C3×D32C4116yesdirect product factor
12C22×C3×D32C22112yes—
13C22×C33×C4C2211yescharacteristic, complemented
14C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
15C2×C3×C4×C3⋊D3C22112yescomplemented
16C22×C32×Dic3C2212yescomplemented
17C2×C32×C4×D3C22124yescomplemented
18C2×C3×#624C22112yescomplemented
19C2×C3×D3×Dic3C22124yescomplemented
20C3×C4×D32C22164yesdirect product factor
21C24×C3×D3—32no—
22C23×D32C2×C311yescharacteristic
23C23×C32×C4—32no—
24C22×C4×C3⋊D3C2×C311yescharacteristic, complemented
25C23×C3×Dic3—32no—
26C22×C3×C4×D3—22no—
27C22×C3×C4×D3—32no—
28C22×C3×C4×D3—32no—
29C22×C3×C4×D3—348no—
30C22×C3×C4×D3—32no—
31C22×C3×C4×D3—61no—
32C22×C3×C4×D3D312yesdirect product factor
33C22×C3×C4×D3C2×C312yescomplemented
34C22×#624C2×C311yescharacteristic, complemented
35C22×D3×Dic3C2×C312yescomplemented
36C2×C4×D32C2×C3124yesdirect product factor
37C23×C33C2311yescharacteristic
38C22×C3×C3⋊D3C2×C418yescomplemented
39C22×C3×C3⋊D3C2316yes—
40C22×C32×D3C2×C4116yescomplemented
41C22×C32×D3C23112yes—
42C2×C3×D32C2×C4196yesdirect product factor
43C2×C3×D32C23116yes—
44C2×C33×C4C2316yescomplemented
45C2×C3×C3⋊Dic3C2316yescomplemented
46C3×C4×C3⋊D3C23116yescomplemented
47C2×C32×Dic3C23112yescomplemented
48C32×C4×D3C23132yescomplemented
49C3×#624C23116yescomplemented
50C3×D3×Dic3C23132yescomplemented
51C24×C3×C4—91no—
52C23×C4×D3—32no—
53C24×C32—32no—
54C23×C3⋊D3C22×C311yescharacteristic
55C23×C3×D3—22no—
56C23×C3×D3—32no—
57C23×C3×D3—332no—
58C23×C3×D3—324no—
59C23×C3×D3—61no—
60C23×C3×D3C2×D312yes—
61C23×C3×D3C22×C312yes—
62C22×D32C3×C4116yesdirect product factor
63C22×D32C22×C3112yes—
64C22×C32×C4—21no—
65C22×C32×C4—22no—
66C22×C32×C4—32no—
67C22×C32×C4—324no—
68C22×C32×C4—41no—
69C22×C32×C4C2×D312yescomplemented
70C22×C32×C4C22×C311yescharacteristic, complemented
71C22×C3⋊Dic3C22×C311yescharacteristic, complemented
72C2×C4×C3⋊D3C22×C3112yescomplemented
73C22×C3×Dic3—22no—
74C22×C3×Dic3—32no—
75C22×C3×Dic3—324no—
76C22×C3×Dic3—61no—
77C22×C3×Dic3C2×D312yescomplemented
78C22×C3×Dic3C22×C312yescomplemented
79C2×C3×C4×D3—224no—
80C2×C3×C4×D3—324no—
81C2×C3×C4×D3—324no—
82C2×C3×C4×D3—324no—
83C2×C3×C4×D3—3128no—
84C2×C3×C4×D3—612no—
85C2×C3×C4×D3C2×D3124yesdirect product factor
86C2×C3×C4×D3C22×C3124yescomplemented
87C2×#624C22×C3112yescomplemented
88C2×D3×Dic3C22×C3124yescomplemented
89C4×D32C22×C3164yesdirect product factor
90C22×C33C22×C414yescomplemented
91C22×C33C2413yes—
92C2×C3×C3⋊D3C22×C4124yescomplemented
93C2×C3×C3⋊D3C2414yes—
94C2×C32×D3C22×C4148yescomplemented
95C2×C32×D3C2418yes—
96C3×D32C22×C4164yesdirect product factor
97C33×C4C2414yescomplemented
98C3×C3⋊Dic3C2414yescomplemented
99C32×Dic3C2418yescomplemented
100C25×C3—91no—
101C24×D3—32no—
102C23×C3×C4—32no—
103C23×C3×C4—32no—
104C23×C3×C4—62no—
105C23×C3×C4—91no—
106C23×C3×C4—924no—
107C23×C3×C4—92no—
108C23×C3×C4—91no—
109C23×Dic3—32no—
110C22×C4×D3—32no—
111C22×C4×D3—32no—
112C22×C4×D3—32no—
113C22×C4×D3—348no—
114C22×C4×D3—61no—
115C22×C4×D3C3×D312yesdirect product factor
116C23×C32—21no—
117C23×C32—22no—
118C23×C32—316no—
119C23×C32—312no—
120C23×C32—41no—
121C23×C32C22×D312yes—
122C23×C32C23×C311yescharacteristic
123C22×C3⋊D3C2×C3×C418yescomplemented
124C22×C3⋊D3C23×C316yes—
125C22×C3×D3—216no—
126C22×C3×D3—212no—
127C22×C3×D3—316no—
128C22×C3×D3—312no—
129C22×C3×D3—3192no—
130C22×C3×D3—332no—
131C22×C3×D3—68no—
132C22×C3×D3—66no—
133C22×C3×D3C4×D3116yesdirect product factor
134C22×C3×D3C2×C3×C4116yescomplemented
135C22×C3×D3C22×D3112yes—
136C22×C3×D3C23×C3112yes—
137C2×D32C2×C3×C4196yesdirect product factor
138C2×D32C23×C3116yes—
139C2×C32×C4—26no—
140C2×C32×C4—212no—
141C2×C32×C4—312no—
142C2×C32×C4—332no—
143C2×C32×C4—46no—
144C2×C32×C4C22×D3112yescomplemented
145C2×C32×C4C23×C316yescomplemented
146C2×C3⋊Dic3C23×C316yescomplemented
147C4×C3⋊D3C23×C3116yescomplemented
148C2×C3×Dic3—212no—
149C2×C3×Dic3—312no—
150C2×C3×Dic3—332no—
151C2×C3×Dic3—66no—
152C2×C3×Dic3C22×D3112yescomplemented
153C2×C3×Dic3C23×C3112yescomplemented
154C3×C4×D3—232no—
155C3×C4×D3—332no—
156C3×C4×D3—332no—
157C3×C4×D3—332no—
158C3×C4×D3—616no—
159C3×C4×D3C22×D3132yesdirect product factor
160C3×C4×D3C23×C3132yescomplemented
161#624C23×C3116yescomplemented
162D3×Dic3C23×C3132yescomplemented
163C24×C4—91noSylow
164C2×C33C23×C416yescomplemented
165C2×C33C2511yescharacteristic
166C3×C3⋊D3C23×C418yescomplemented
167C32×D3C23×C4116yescomplemented
168C24×C3—32no—
169C24×C3—32no—
170C24×C3—62no—
171C24×C3—916no—
172C24×C3—912no—
173C24×C3—91no—
174C23×D3—32no—
175C23×D3—332no—
176C23×D3—324no—
177C23×D3—61no—
178C23×D3C2×C3×D312yes—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—32no—
182C22×C3×C4—32no—
183C22×C3×C4—324no—
184C22×C3×C4—324no—
185C22×C3×C4—41no—
186C22×C3×C4—624no—
187C22×C3×C4—62no—
188C22×C3×C4—912no—
189C22×C3×C4—964no—
190C22×C3×C4—924no—
191C22×C3×C4—91no—
192C22×C3×C4—912no—
193C22×C3×C4D3211yescharacteristic, direct product factor
194C22×C3×C4C2×C3×D312yescomplemented
195C22×Dic3—32no—
196C22×Dic3—324no—
197C22×Dic3—61no—
198C22×Dic3C2×C3×D312yescomplemented
199C2×C4×D3—324no—
200C2×C4×D3—324no—
201C2×C4×D3—324no—
202C2×C4×D3—3128no—
203C2×C4×D3—612no—
204C2×C4×D3C2×C3×D3124yesdirect product factor
205C22×C32—23no—
206C22×C32—26no—
207C22×C32—24no—
208C22×C32—28no—
209C22×C32—348no—
210C22×C32—38no—
211C22×C32—44no—
212C22×C32—43no—
213C22×C32C2×C4×D318yescomplemented
214C22×C32C22×C3×C414yescomplemented
215C22×C32C23×D316yes—
216C22×C32C24×C313yes—
217C2×C3⋊D3C22×C3×C4124yescomplemented
218C2×C3⋊D3C24×C314yes—
219C2×C3×D3—28no—
220C2×C3×D3—248no—
221C2×C3×D3—348no—
222C2×C3×D3—38no—
223C2×C3×D3—3128no—
224C2×C3×D3—624no—
225C2×C3×D3—64no—
226C2×C3×D3C2×C4×D3148yesdirect product factor
227C2×C3×D3C22×C3×C4148yescomplemented
228C2×C3×D3C23×D318yes—
229C2×C3×D3C24×C318yes—
230D32C22×C3×C4164yesdirect product factor
231C32×C4—24no—
232C32×C4—28no—
233C32×C4—38no—
234C32×C4—44no—
235C32×C4C23×D318yescomplemented
236C32×C4C24×C314yescomplemented
237C3⋊Dic3C24×C314yescomplemented
238C3×Dic3—28no—
239C3×Dic3—38no—
240C3×Dic3—64no—
241C3×Dic3C23×D318yescomplemented
242C3×Dic3C24×C318yescomplemented
243C25—91no—
244C23×C4—32no—
245C23×C4—91no—
246C23×C4—924no—
247C23×C4—92no—
248C23×C4—91no—
249C33C24×C411yescharacteristic, complemented, Sylow
250C23×C3—22no—
251C23×C3—21no—
252C23×C3—312no—
253C23×C3—312no—
254C23×C3—316no—
255C23×C3—316no—
256C23×C3—41no—
257C23×C3—612no—
258C23×C3—616no—
259C23×C3—916no—
260C23×C3—996no—
261C23×C3—96no—
262C23×C3—98no—
263C23×C3C2×D3211yescharacteristic
264C23×C3C22×C3×D312yes—
265C22×D3—312no—
266C22×D3—316no—
267C22×D3—332no—
268C22×D3—3192no—
269C22×D3—68no—
270C22×D3—66no—
271C22×D3C3×C4×D3116yesdirect product factor
272C22×D3C22×C3×D3112yes—
273C2×C3×C4—212no—
274C2×C3×C4—26no—
275C2×C3×C4—312no—
276C2×C3×C4—312no—
277C2×C3×C4—332no—
278C2×C3×C4—332no—
279C2×C3×C4—46no—
280C2×C3×C4—632no—
281C2×C3×C4—612no—
282C2×C3×C4—916no—
283C2×C3×C4—932no—
284C2×C3×C4—96no—
285C2×C3×C4—916no—
286C2×C3×C4C2×D3216yesdirect product factor
287C2×C3×C4C22×C3×D3112yescomplemented
288C2×Dic3—312no—
289C2×Dic3—332no—
290C2×Dic3—66no—
291C2×Dic3C22×C3×D3112yescomplemented
292C4×D3—332no—
293C4×D3—332no—
294C4×D3—332no—
295C4×D3—616no—
296C4×D3C22×C3×D3132yesdirect product factor
297C2×C32—26no—
298C2×C32—212no—
299C2×C32—21no—
300C2×C32—22no—
301C2×C32—316no—
302C2×C32—46no—
303C2×C32—41no—
304C2×C32C22×C4×D3112yescomplemented
305C2×C32C23×C3×C416yescomplemented
306C2×C32C24×D312yes—
307C2×C32C25×C311yescharacteristic
308C3⋊D3C23×C3×C418yescomplemented
309C3×D3—216no—
310C3×D3—316no—
311C3×D3—68no—
312C3×D3C22×C4×D3116yesdirect product factor
313C3×D3C23×C3×C4116yescomplemented
314C24—32no—
315C24—912no—
316C24—916no—
317C24—91no—
318C22×C4—32no—
319C22×C4—324no—
320C22×C4—912no—
321C22×C4—964no—
322C22×C4—924no—
323C22×C4—91no—
324C22×C4—912no—
325C22×C4C3×D3211yescharacteristic, direct product factor
326C22×C3—23no—
327C22×C3—26no—
328C22×C3—24no—
329C22×C3—28no—
330C22×C3—348no—
331C22×C3—348no—
332C22×C3—38no—
333C22×C3—38no—
334C22×C3—43no—
335C22×C3—44no—
336C22×C3—68no—
337C22×C3—648no—
338C22×C3—964no—
339C22×C3—924no—
340C22×C3—94no—
341C22×C3C4×D3214yesdirect product factor
342C22×C3C2×C3×C4×D318yescomplemented
343C22×C3C22×D3213yes—
344C22×C3C23×C3×D316yes—
345C2×D3—348no—
346C2×D3—38no—
347C2×D3—3128no—
348C2×D3—64no—
349C2×D3—624no—
350C2×D3C2×C3×C4×D3148yesdirect product factor
351C2×D3C23×C3×D318yes—
352C3×C4—24no—
353C3×C4—28no—
354C3×C4—38no—
355C3×C4—38no—
356C3×C4—44no—
357C3×C4—68no—
358C3×C4—94no—
359C3×C4C22×D3214yesdirect product factor
360C3×C4C23×C3×D318yescomplemented
361Dic3—38no—
362Dic3—64no—
363Dic3C23×C3×D318yescomplemented
364C32—21no—
365C32—22no—
366C32—41no—
367C32C23×C4×D312yescomplemented
368C32C24×C3×C411yescharacteristic, complemented
369C23—312no—
370C23—316no—
371C23—916no—
372C23—996no—
373C23—96no—
374C23—98no—
375C23C2×C3×D3211yescharacteristic
376C2×C4—312no—
377C2×C4—332no—
378C2×C4—916no—
379C2×C4—932no—
380C2×C4—96no—
381C2×C4—916no—
382C2×C4C2×C3×D3216yesdirect product factor
383C2×C3—21no—
384C2×C3—22no—
385C2×C3—26no—
386C2×C3—212no—
387C2×C3—316no—
388C2×C3—316no—
389C2×C3—41no—
390C2×C3—46no—
391C2×C3—616no—
392C2×C3—98no—
393C2×C3C2×C4×D3216yesdirect product factor
394C2×C3C22×C3×C4×D3112yescomplemented
395C2×C3C23×D3211yescharacteristic
396C2×C3C24×C3×D312yes—
397D3—316no—
398D3—68no—
399D3C22×C3×C4×D3116yesdirect product factor
400C22—38no—
401C22—348no—
402C22—964no—
403C22—94no—
404C22—924no—
405C22C3×C4×D3214yesdirect product factor
406C22C22×C3×D3213yes—
407C4—38no—
408C4—94no—
409C4C22×C3×D3214yesdirect product factor
410C3—21no—
411C3—22no—
412C3—41no—
413C3C22×C4×D3211yescharacteristic, direct product factor
414C3C23×C3×C4×D312yescomplemented
415C2—316no—
416C2—98no—
417C2C2×C3×C4×D3216yesdirect product factor
418C2C23×C3×D3211yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4D32
commutator, hypocenterC32C24×C3×C4
FrattiniC2C23×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3×D32C2
cosocleC2×C32C25×C3
LCSG ▷ C32
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C22×C3×C4×C3⋊D3C2semidirect
3. C22×C32×C4×D3C2semidirect
4. C22×C3×#624C2semidirect
5. C22×C3×D3×Dic3C2semidirect
6. C2×C3×C4×D32C2direct (N × H)
8. C22×C4×D32C3direct (N × H)
11. C22×C3×D32C4direct (N × H)
13. C22×C33×C4C22semidirect
14. C22×C3×C3⋊Dic3C22semidirect
15. C2×C3×C4×C3⋊D3C22semidirect
16. C22×C32×Dic3C22semidirect
17. C2×C32×C4×D3C22semidirect
18. C2×C3×#624C22semidirect
19. C2×C3×D3×Dic3C22semidirect
20. C3×C4×D32C22direct (N × H)
24. C22×C4×C3⋊D3C2×C3semidirect
32. C22×C3×C4×D3D3direct (N × H)
33. C22×C3×C4×D3C2×C3semidirect
34. C22×#624C2×C3semidirect
35. C22×D3×Dic3C2×C3semidirect
36. C2×C4×D32C2×C3direct (N × H)
38. C22×C3×C3⋊D3C2×C4semidirect
40. C22×C32×D3C2×C4semidirect
42. C2×C3×D32C2×C4direct (N × H)
44. C2×C33×C4C23semidirect
45. C2×C3×C3⋊Dic3C23semidirect
46. C3×C4×C3⋊D3C23semidirect
47. C2×C32×Dic3C23semidirect
48. C32×C4×D3C23semidirect
49. C3×#624C23semidirect
50. C3×D3×Dic3C23semidirect
62. C22×D32C3×C4direct (N × H)
69. C22×C32×C4C2×D3semidirect
70. C22×C32×C4C22×C3semidirect
71. C22×C3⋊Dic3C22×C3semidirect
72. C2×C4×C3⋊D3C22×C3semidirect
77. C22×C3×Dic3C2×D3semidirect
78. C22×C3×Dic3C22×C3semidirect
85. C2×C3×C4×D3C2×D3direct (N × H)
86. C2×C3×C4×D3C22×C3semidirect
87. C2×#624C22×C3semidirect
88. C2×D3×Dic3C22×C3semidirect
89. C4×D32C22×C3direct (N × H)
90. C22×C33C22×C4semidirect
92. C2×C3×C3⋊D3C22×C4semidirect
94. C2×C32×D3C22×C4semidirect
96. C3×D32C22×C4direct (N × H)
97. C33×C4C24semidirect
98. C3×C3⋊Dic3C24semidirect
99. C32×Dic3C24semidirect
115. C22×C4×D3C3×D3direct (N × H)
123. C22×C3⋊D3C2×C3×C4semidirect
133. C22×C3×D3C4×D3direct (N × H)
134. C22×C3×D3C2×C3×C4semidirect
137. C2×D32C2×C3×C4direct (N × H)
144. C2×C32×C4C22×D3semidirect
145. C2×C32×C4C23×C3semidirect
146. C2×C3⋊Dic3C23×C3semidirect
147. C4×C3⋊D3C23×C3semidirect
152. C2×C3×Dic3C22×D3semidirect
153. C2×C3×Dic3C23×C3semidirect
159. C3×C4×D3C22×D3direct (N × H)
160. C3×C4×D3C23×C3semidirect
161. #624C23×C3semidirect
162. D3×Dic3C23×C3semidirect
164. C2×C33C23×C4semidirect
166. C3×C3⋊D3C23×C4semidirect
167. C32×D3C23×C4semidirect
193. C22×C3×C4D32direct (N × H)
194. C22×C3×C4C2×C3×D3semidirect
198. C22×Dic3C2×C3×D3semidirect
204. C2×C4×D3C2×C3×D3direct (N × H)
213. C22×C32C2×C4×D3semidirect
214. C22×C32C22×C3×C4semidirect
217. C2×C3⋊D3C22×C3×C4semidirect
226. C2×C3×D3C2×C4×D3direct (N × H)
227. C2×C3×D3C22×C3×C4semidirect
230. D32C22×C3×C4direct (N × H)
235. C32×C4C23×D3semidirect
236. C32×C4C24×C3semidirect
237. C3⋊Dic3C24×C3semidirect
241. C3×Dic3C23×D3semidirect
242. C3×Dic3C24×C3semidirect
249. C33C24×C4semidirect
271. C22×D3C3×C4×D3direct (N × H)
286. C2×C3×C4C2×D32direct (N × H)
287. C2×C3×C4C22×C3×D3semidirect
291. C2×Dic3C22×C3×D3semidirect
296. C4×D3C22×C3×D3direct (N × H)
304. C2×C32C22×C4×D3semidirect
305. C2×C32C23×C3×C4semidirect
308. C3⋊D3C23×C3×C4semidirect
312. C3×D3C22×C4×D3direct (N × H)
313. C3×D3C23×C3×C4semidirect
325. C22×C4C3×D32direct (N × H)
341. C22×C3C4×D32direct (N × H)
342. C22×C3C2×C3×C4×D3semidirect
350. C2×D3C2×C3×C4×D3direct (N × H)
359. C3×C4C22×D32direct (N × H)
360. C3×C4C23×C3×D3semidirect
363. Dic3C23×C3×D3semidirect
367. C32C23×C4×D3semidirect
368. C32C24×C3×C4semidirect
382. C2×C4C2×C3×D32direct (N × H)
393. C2×C3C2×C4×D32direct (N × H)
394. C2×C3C22×C3×C4×D3semidirect
399. D3C22×C3×C4×D3direct (N × H)
405. C22C3×C4×D32direct (N × H)
409. C4C22×C3×D32direct (N × H)
413. C3C22×C4×D32direct (N × H)
414. C3C23×C3×C4×D3semidirect
417. C2C2×C3×C4×D32direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×D322
C22×C3×C4×C3⋊D32
C22×C32×C4×D32
C22×C3×#6242
C22×C3×D3×Dic32
C2×C3×C4×D322
C23×C3×C4×D33
C22×C4×D323