Finite Group Databasebeta

← #13150282

Subgroups — #13150282

Order 1728.

1226 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C3⋊D9C211yesmaximal, characteristic
2C9×#926C211yesmaximal, characteristic, complemented
3C3×#63488C211yesmaximal, characteristic, complemented
4#11721274C212yesmaximal
5#11721248C212yesmaximal, complemented
6#11721000C212yesmaximal, complemented
7C4×(C3⋊D9)⋊C4C212yesmaximal, complemented
8#11720931C212yesmaximal, complemented
9#11720922C212yesmaximal, complemented
10#10594969—31nomaximal
11D9×#117—31nomaximal
12C23×C3⋊D9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C2×C4×C3⋊D9C418yescomplemented
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C4×C3⋊D9C2211yescharacteristic
18C2×C4×C3⋊D9C2211yescharacteristic
19C2×C4×C3⋊D9C2211yescharacteristic
20C2×(C3⋊D9)⋊C4C2214yes—
21C9×#217C2212yes—
22C3×#3676C2212yes—
23C9×C4⋊Dic3C2212yescomplemented
24C4×C9×Dic3C2212yescomplemented
25C3×C4⋊Dic9C2212yescomplemented
26C3×C4×Dic9C2212yescomplemented
27#88445C2214yes—
28#88442C2214yescomplemented
29Dic3×Dic9C2214yescomplemented
30C22×C4×C3⋊D3—31no—
31C3×#926—31no—
32C3×#926D311yescharacteristic
33#63974—32no—
34#63957—32no—
35#63900—32no—
36C4×#624—32no—
37#63874—32no—
38#63849—32no—
39C22×C4×D9—31no—
40C22×C4×D9—61no—
41C9×#117—31no—
42#63488D311yescharacteristic, complemented
43#63454—32no—
44C4⋊C4×D9—32no—
45C22⋊C4×D9—32no—
46#63439—32no—
47#63434—32no—
48C42×D9—32no—
49C23×C3×C9C2311yescharacteristic
50C22×C3⋊D9—24no—
51C22×C3⋊D9C2×C414yes—
52C22×C3⋊D9C2311yescharacteristic
53C22×C3⋊D9C2311yescharacteristic
54C2×C3×C4×C9C2×C414yescomplemented
55C2×C3×C4×C9C2311yescharacteristic
56C2×C3×C4×C9C2311yescharacteristic
57C2×C3⋊Dic9C2×C414yescomplemented
58C2×C3⋊Dic9C2311yescharacteristic
59C2×C3⋊Dic9C2311yescharacteristic
60C4×C3⋊D9C2×C414yes—
61C4×C3⋊D9C2×C414yes—
62C4×C3⋊D9C2×C414yescomplemented
63C4×C3⋊D9C2×C414yescomplemented
64C2×C9×Dic3C2314yes—
65C2×C3×Dic9C2314yes—
66(C3⋊D9)⋊C4—28no—
67D3×#117—91no—
68D3×#117—91no—
69C23×C3⋊D3—31no—
70C22×C32×C4C2×D311yescharacteristic
71C22×C3⋊Dic3—31no—
72C2×C4×C3⋊D3—38no—
73C2×C4×C3⋊D3—31no—
74C2×C4×C3⋊D3—31no—
75C2×C4×C3⋊D3—31no—
76C2×C4×C3⋊D3—31no—
77C2×#624—34no—
78C23×D9—31no—
79C23×D9—61no—
80C3×#217—32no—
81C3×#217C2×D312yes—
82C3×C4⋊Dic3—32no—
83C3×C4⋊Dic3C2×D312yes—
84C3×C4×Dic3—32no—
85C3×C4×Dic3C2×D312yes—
86#3724—34no—
87#3722—34no—
88Dic32—34no—
89C22×C4×C9—21no—
90C22×C4×C9C2×D311yescharacteristic
91C22×Dic9—31no—
92C22×Dic9—61no—
93C2×C4×D9—38no—
94C2×C4×D9—31no—
95C2×C4×D9—31no—
96C2×C4×D9—31no—
97C2×C4×D9—31no—
98C2×C4×D9—34no—
99C2×C4×D9—68no—
100C2×C4×D9—61no—
101C2×C4×D9—61no—
102C2×C4×D9—61no—
103C2×C4×D9—61no—
104C9×C4⋊C4—32no—
105C9×C22⋊C4—32no—
106C42×C9—32no—
107#3676C2×D312yes—
108(C2×D9)⋊C4—34no—
109C4⋊Dic9C2×D312yescomplemented
110Dic9⋊C4—34no—
111C4×Dic9—34no—
112C4×Dic9C2×D312yescomplemented
113C22×C3×C9—22no—
114C22×C3×C9C22×C412yes—
115C22×C3×C9C2411yescharacteristic
116C2×C3⋊D9—22no—
117C2×C3⋊D9—28no—
118C2×C3⋊D9C22×C412yes—
119C2×C3⋊D9C22×C412yes—
120C2×C3⋊D9#4114yes—
121C3×C4×C9C22×C412yes—
122C3×C4×C9C22×C412yes—
123C3⋊Dic9C22×C412yes—
124C3⋊Dic9C22×C412yes—
125C9×Dic3—24no—
126C3×Dic9—24no—
127C22×C4×D3—91no—
128C22×C4×D3—91no—
129C22×C4×D3—181no—
130C3×#117—31no—
131C3×#117—91no—
132#926—31no—
133#926D911yescharacteristic, complemented
134#892—92no—
135#892—92no—
136D3×C4⋊C4—92no—
137D3×C4⋊C4—92no—
138D3×C22⋊C4—92no—
139D3×C22⋊C4—92no—
140#877—92no—
141#877—92no—
142#872—92no—
143#872—92no—
144C42×D3—92no—
145C42×D3—92no—
146C23×C32C22×D311yescharacteristic
147C22×C3⋊D3—31no—
148C22×C3⋊D3—34no—
149C22×C3⋊D3—31no—
150C22×C3⋊D3—64no—
151C2×C32×C4C4×D314yes—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C3⋊Dic3—34no—
155C2×C3⋊Dic3—31no—
156C2×C3⋊Dic3—31no—
157C4×C3⋊D3—34no—
158C4×C3⋊D3—34no—
159C4×C3⋊D3—34no—
160C4×C3⋊D3—34no—
161C2×C3×Dic3—34no—
162C2×C3×Dic3C22×D314yes—
163#624—68no—
164C23×C9—21no—
165C23×C9C22×D311yescharacteristic
166C22×D9—31no—
167C22×D9—34no—
168C22×D9—31no—
169C22×D9—64no—
170C22×D9—61no—
171C22×D9—61no—
172C22×D9—64no—
173C22×D9—68no—
174C2×C4×C9—24no—
175C2×C4×C9—21no—
176C2×C4×C9—21no—
177C2×C4×C9—34no—
178C2×C4×C9C4×D314yescomplemented
179C2×C4×C9C22×D311yescharacteristic
180C2×C4×C9C22×D311yescharacteristic
181C2×Dic9—34no—
182C2×Dic9—31no—
183C2×Dic9—31no—
184C2×Dic9—64no—
185C2×Dic9—61no—
186C2×Dic9—61no—
187C2×Dic9C22×D314yes—
188C4×D9—34no—
189C4×D9—34no—
190C4×D9—34no—
191C4×D9—34no—
192C4×D9—64no—
193C4×D9—64no—
194C4×D9—64no—
195C4×D9—64no—
196C4×D9—68no—
197C2×#117—271noSylow
198C2×C3×C9—22no—
199C2×C3×C9C23×C411yescharacteristic
200C2×C3×C9C2×#4112yes—
201C3⋊D9—22no—
202C3⋊D9#11714yescomplemented
203C23×D3—91no—
204C23×D3—91no—
205C23×D3—181no—
206C22×C3×C4—21no—
207C22×C3×C4C2×D911yescharacteristic
208C22×C3×C4D3211yescharacteristic
209C22×Dic3—91no—
210C22×Dic3—91no—
211C22×Dic3—181no—
212C2×C4×D3—98no—
213C2×C4×D3—91no—
214C2×C4×D3—91no—
215C2×C4×D3—91no—
216C2×C4×D3—91no—
217C2×C4×D3—98no—
218C2×C4×D3—91no—
219C2×C4×D3—91no—
220C2×C4×D3—91no—
221C2×C4×D3—91no—
222C2×C4×D3—94no—
223C2×C4×D3—94no—
224C2×C4×D3—188no—
225C2×C4×D3—181no—
226C2×C4×D3—181no—
227C2×C4×D3—181no—
228C2×C4×D3—181no—
229C3×C4⋊C4—32no—
230C3×C4⋊C4—92no—
231C3×C22⋊C4—32no—
232C3×C22⋊C4—92no—
233C3×C42—32no—
234C3×C42—92no—
235#217—32no—
236#217C2×D912yes—
237(C2×D3)⋊C4—94no—
238(C2×D3)⋊C4—94no—
239C4⋊Dic3—32no—
240C4⋊Dic3C2×D912yescomplemented
241Dic3⋊C4—94no—
242Dic3⋊C4—94no—
243C4×Dic3—32no—
244C4×Dic3—94no—
245C4×Dic3—94no—
246C4×Dic3C2×D912yescomplemented
247C22×C32—22no—
248C22×C32C2×C4×D312yes—
249C22×C32C23×D311yescharacteristic
250C2×C3⋊D3—32no—
251C2×C3⋊D3—34no—
252C2×C3⋊D3—32no—
253C2×C3⋊D3—68no—
254C2×C3⋊D3—62no—
255C32×C4C2×C4×D312yes—
256C32×C4C2×C4×D312yes—
257C3⋊Dic3—32no—
258C3⋊Dic3—32no—
259C3×Dic3—24no—
260C3×Dic3—64no—
261C22×C9—24no—
262C22×C9—22no—
263C22×C9—21no—
264C22×C9—22no—
265C22×C9C2×C4×D312yes—
266C22×C9C23×D311yescharacteristic
267C2×D9—32no—
268C2×D9—34no—
269C2×D9—32no—
270C2×D9—62no—
271C2×D9—62no—
272C2×D9—64no—
273C2×D9—68no—
274C2×D9—616no—
275C2×D9—64no—
276C2×D9—62no—
277C4×C9—22no—
278C4×C9—22no—
279C4×C9—64no—
280C4×C9C2×C4×D312yes—
281C4×C9C2×C4×D312yes—
282Dic9—24no—
283Dic9—32no—
284Dic9—32no—
285Dic9—62no—
286Dic9—62no—
287C23×C4—271no—
288#117—31no—
289#117—91no—
290#117—272no—
291#117—272no—
292#117—272no—
293C2×C4⋊C4—272no—
294C2×C22⋊C4—272no—
295C2×C42—272no—
296C3×C9C2×#11711yescharacteristic, complemented, Sylow
297C23×C3—21no—
298C23×C3C22×D911yescharacteristic
299C23×C3C2×D3211yescharacteristic
300C22×D3—91no—
301C22×D3—91no—
302C22×D3—91no—
303C22×D3—94no—
304C22×D3—91no—
305C22×D3—94no—
306C22×D3—181no—
307C22×D3—181no—
308C22×D3—184no—
309C22×D3—184no—
310C22×D3—184no—
311C22×D3—188no—
312C2×C3×C4—24no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—34no—
316C2×C3×C4—94no—
317C2×C3×C4C4×D914yescomplemented
318C2×C3×C4C22×D911yescharacteristic
319C2×C3×C4C22×D911yescharacteristic
320C2×C3×C4#62414yes—
321C2×C3×C4C2×D3211yescharacteristic
322C2×C3×C4C2×D3211yescharacteristic
323C2×Dic3—34no—
324C2×Dic3—91no—
325C2×Dic3—91no—
326C2×Dic3—94no—
327C2×Dic3—91no—
328C2×Dic3—91no—
329C2×Dic3—94no—
330C2×Dic3—181no—
331C2×Dic3—181no—
332C2×Dic3—184no—
333C2×Dic3C22×D914yes—
334C4×D3—94no—
335C4×D3—94no—
336C4×D3—94no—
337C4×D3—94no—
338C4×D3—94no—
339C4×D3—94no—
340C4×D3—94no—
341C4×D3—94no—
342C4×D3—184no—
343C4×D3—184no—
344C4×D3—184no—
345C4×D3—184no—
346C4×D3—188no—
347C4×D3—188no—
348C2×C32—22no—
349C2×C32C22×C4×D311yescharacteristic
350C2×C32D3×#4112yes—
351C3⋊D3—34no—
352C3⋊D3—62no—
353C2×C9—24no—
354C2×C9—22no—
355C2×C9—22no—
356C2×C9—21no—
357C2×C9C22×C4×D311yescharacteristic
358C2×C9D3×#4112yes—
359D9—34no—
360D9—64no—
361D9—64no—
362D9—62no—
363C24—271no—
364C22×C4—274no—
365C22×C4—271no—
366C22×C4—278no—
367C22×C4—271no—
368C22×C4—271no—
369C22×C4—271no—
370C22×C4—271no—
371C22×C4D3×D911yescharacteristic
372C4⋊C4—32no—
373C4⋊C4—92no—
374C4⋊C4—274no—
375C22⋊C4—32no—
376C22⋊C4—92no—
377C22⋊C4—274no—
378C42—32no—
379C42—92no—
380C42—274no—
381C22×C3—24no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—21no—
385C22×C3—22no—
386C22×C3C2×C4×D912yes—
387C22×C3C23×D911yescharacteristic
388C22×C3C2×#62412yes—
389C22×C3C22×D3211yescharacteristic
390C2×D3—94no—
391C2×D3—92no—
392C2×D3—92no—
393C2×D3—94no—
394C2×D3—92no—
395C2×D3—92no—
396C2×D3—182no—
397C2×D3—182no—
398C2×D3—184no—
399C2×D3—188no—
400C2×D3—188no—
401C2×D3—1816no—
402C2×D3—184no—
403C2×D3—182no—
404C2×D3—182no—
405C3×C4—22no—
406C3×C4—22no—
407C3×C4—64no—
408C3×C4—184no—
409C3×C4C2×C4×D912yes—
410C3×C4C2×C4×D912yes—
411C3×C4C2×#62412yes—
412C3×C4C2×#62412yes—
413Dic3—24no—
414Dic3—64no—
415Dic3—92no—
416Dic3—92no—
417Dic3—92no—
418Dic3—92no—
419Dic3—182no—
420Dic3—182no—
421C32D3×#11711yescharacteristic
422C9—21no—
423C9D3×#11711yescharacteristic, complemented
424C23—271no—
425C23—271no—
426C23—274no—
427C23—544no—
428C23C2×D3×D911yescharacteristic
429C2×C4—34no—
430C2×C4—94no—
431C2×C4—271no—
432C2×C4—271no—
433C2×C4—274no—
434C2×C4—274no—
435C2×C4—274no—
436C2×C4—274no—
437C2×C4—274no—
438C2×C4—548no—
439C2×C4(C3⋊D9)⋊C414yescomplemented
440C2×C4C2×D3×D911yescharacteristic
441C2×C4C2×D3×D911yescharacteristic
442C2×C3—24no—
443C2×C3—22no—
444C2×C3—22no—
445C2×C3—21no—
446C2×C3—22no—
447C2×C3C22×C4×D911yescharacteristic
448C2×C3#41×D912yes—
449C2×C3#6430612yes—
450C2×C3C22×#62411yescharacteristic
451D3—94no—
452D3—94no—
453D3—184no—
454D3—184no—
455D3—182no—
456D3—182no—
457C22—22no—
458C22—272no—
459C22—274no—
460C22—272no—
461C22—548no—
462C22—542no—
463C22C2×(C3⋊D9)⋊C412yes—
464C22C22×D3×D911yescharacteristic
465C4—64no—
466C4—184no—
467C4—272no—
468C4—272no—
469C4C2×(C3⋊D9)⋊C412yes—
470C4C2×(C3⋊D9)⋊C412yes—
471C3—21no—
472C3D9×#11711yescharacteristic, complemented
473C3#1059496911yescharacteristic
474C2—22no—
475C2—274no—
476C2—542no—
477C2#1172247512yes—
478C2C22×(C3⋊D9)⋊C411yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC23×C3⋊D9C22
cosocleC22×C3×C9C24
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×#117
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C9×#926C2semidirect
3. C3×#63488C2semidirect
5. #11721248C2semidirect
6. #11721000C2semidirect
7. C4×(C3⋊D9)⋊C4C2semidirect
8. #11720931C2semidirect
9. #11720922C2semidirect
15. C2×C4×C3⋊D9C4semidirect
23. C9×C4⋊Dic3C22semidirect
24. C4×C9×Dic3C22semidirect
25. C3×C4⋊Dic9C22semidirect
26. C3×C4×Dic9C22semidirect
28. #88442C22semidirect
29. Dic3×Dic9C22semidirect
42. #63488D3semidirect
54. C2×C3×C4×C9C2×C4semidirect
57. C2×C3⋊Dic9C2×C4semidirect
62. C4×C3⋊D9C2×C4semidirect
63. C4×C3⋊D9C2×C4semidirect
109. C4⋊Dic9C2×D3semidirect
112. C4×Dic9C2×D3semidirect
133. #926D9semidirect
178. C2×C4×C9C4×D3semidirect
202. C3⋊D9#117semidirect
240. C4⋊Dic3C2×D9semidirect
246. C4×Dic3C2×D9semidirect
296. C3×C9C2×#117semidirect
317. C2×C3×C4C4×D9semidirect
423. C9D3×#117semidirect
439. C2×C4(C3⋊D9)⋊C4semidirect
472. C3D9×#117semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊D92
C9×#9262
C3×#634882
#117212742
#117212482
#117210002
C4×(C3⋊D9)⋊C42
#117209312
#117209222
#105949693
D9×#1173