Finite Group Databasebeta

← #13147697

Subgroups — #13147697

Order 1728.

1098 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×D9C211yesmaximal, characteristic
2C42×C3⋊D9C211yesmaximal, characteristic, complemented
3C9×#872C211yesmaximal, characteristic, complemented
4C3×#63434C211yesmaximal, characteristic, complemented
5#11721048C212yesmaximal
6#11721011C212yesmaximal, complemented
7#11720980C212yesmaximal, complemented
8C4×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
9#11720975C212yesmaximal, complemented
10#11720959C211yesmaximal, characteristic, complemented
11#11720943C211yesmaximal, characteristic, complemented
12#10593992—31nomaximal
13D9×#117—31nomaximal
14C22×D3×D9C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C4×D3×C9C2211yescharacteristic
18C2×C3×C4×D9C2211yescharacteristic
19C2×D3×Dic9C2211yescharacteristic
20C2×(C3⋊D9)⋊C4C2212yes—
21C2×(C3⋊D9)⋊C4C2211yescharacteristic
22C2×Dic3×D9C2211yescharacteristic
23C4×D3×D9C418yescomplemented
24C3×C42×C9C2211yescharacteristic, complemented
25C4×C3⋊Dic9C2212yescomplemented
26C4×C3⋊Dic9C2211yescharacteristic, complemented
27C9×(C2×D3)⋊C4C2212yes—
28C3×(C2×D9)⋊C4C2212yes—
29C9×Dic3⋊C4C2212yescomplemented
30C4×C9×Dic3C2211yescharacteristic, complemented
31C3×Dic9⋊C4C2212yescomplemented
32C3×C4×Dic9C2211yescharacteristic, complemented
33(C2×D3)⋊Dic9C2212yes—
34(C2×D9)⋊Dic3C2212yes—
35Dic3⋊Dic9C2212yescomplemented
36Dic9⋊Dic3C2212yescomplemented
37Dic3×Dic9C2212yescomplemented
38C2×C4×D32—31no—
39C42×C3⋊D3—31no—
40C3×#872—31no—
41C3×#872D311yescharacteristic
42#63921—32no—
43#63883—32no—
44C4×#624—31no—
45#63878—32no—
46#63878—32no—
47#63855—31no—
48#63855—31no—
49C22×C4×D9—31no—
50C9×#117—31no—
51#63488—31no—
52#63454—32no—
53C4⋊C4×D9—32no—
54C22⋊C4×D9—32no—
55#63439—32no—
56#63434—31no—
57#63434D311yescharacteristic, complemented
58C42×D9—31no—
59C42×D9—31no—
60C42×D9—61no—
61C22×C3⋊D9C2311yescharacteristic
62C22×D3×C9C2311yescharacteristic
63C22×C3×D9C2311yescharacteristic
64C2×D3×D9—24no—
65C2×D3×D9C2×C414yes—
66C2×C3×C4×C9C2312yes—
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3⋊Dic9C2312yes—
69C2×C3⋊Dic9C2311yescharacteristic
70C4×C3⋊D9—24no—
71C4×C3⋊D9C2×C414yes—
72C2×C9×Dic3C2312yes—
73C2×C9×Dic3C2311yescharacteristic
74C2×C3×Dic9C2312yes—
75C2×C3×Dic9C2311yescharacteristic
76C4×D3×C9C2×C414yescomplemented
77C3×C4×D9C2×C414yescomplemented
78D3×Dic9C2×C414yescomplemented
79(C3⋊D9)⋊C4—24no—
80(C3⋊D9)⋊C4C2×C414yes—
81Dic3×D9C2×C414yescomplemented
82D3×#117—91no—
83D3×#117—91no—
84C22×D32—31no—
85C2×C4×C3⋊D3—31no—
86C2×C4×C3⋊D3—32no—
87C2×C3×C4×D3—31no—
88C2×C3×C4×D3C2×D311yescharacteristic
89C2×#624—31no—
90C2×#624—32no—
91C2×D3×Dic3—31no—
92C2×D3×Dic3—31no—
93C4×D32—38no—
94C23×D9—31no—
95C32×C42C2×D311yescharacteristic
96C4×C3⋊Dic3—31no—
97C4×C3⋊Dic3—32no—
98C3×(C2×D3)⋊C4—32no—
99C3×(C2×D3)⋊C4C2×D312yes—
100C3×Dic3⋊C4—32no—
101C3×Dic3⋊C4C2×D312yes—
102C3×C4×Dic3—31no—
103C3×C4×Dic3C2×D311yescharacteristic
104Dic3⋊Dic3—32no—
105Dic3⋊Dic3—32no—
106(C2×D3)⋊Dic3—32no—
107(C2×D3)⋊Dic3—32no—
108Dic32—32no—
109C22×C4×C9—31no—
110C22×Dic9—31no—
111C2×C4×D9—31no—
112C2×C4×D9—31no—
113C2×C4×D9—31no—
114C2×C4×D9—38no—
115C2×C4×D9—32no—
116C2×C4×D9—32no—
117C2×C4×D9—61no—
118C2×C4×D9—62no—
119C2×C4×D9C2×D311yescharacteristic
120C9×C4⋊C4—32no—
121C9×C22⋊C4—32no—
122C42×C9—21no—
123C42×C9—31no—
124C42×C9C2×D311yescharacteristic, complemented
125#3676—32no—
126(C2×D9)⋊C4—32no—
127(C2×D9)⋊C4C2×D312yes—
128C4⋊Dic9—32no—
129Dic9⋊C4—32no—
130Dic9⋊C4C2×D312yescomplemented
131C4×Dic9—31no—
132C4×Dic9—32no—
133C4×Dic9—32no—
134C4×Dic9—61no—
135C4×Dic9—62no—
136C4×Dic9C2×D311yescharacteristic, complemented
137C22×C3×C9C2411yescharacteristic
138C2×C3⋊D9C22×C412yes—
139C2×C3⋊D9#4114yes—
140C2×D3×C9—22no—
141C2×D3×C9C22×C412yes—
142C2×C3×D9—22no—
143C2×C3×D9C22×C412yes—
144D3×D9—28no—
145C3×C4×C9—22no—
146C3×C4×C9C22×C412yes—
147C3⋊Dic9—22no—
148C3⋊Dic9C22×C412yes—
149C9×Dic3—22no—
150C9×Dic3C22×C412yes—
151C3×Dic9—22no—
152C3×Dic9C22×C412yes—
153C22×C4×D3—91no—
154C22×C4×D3—91no—
155C3×#117—31no—
156C3×#117—91no—
157#926—91no—
158#926—91no—
159#892—92no—
160#892—92no—
161D3×C4⋊C4—92no—
162D3×C4⋊C4—92no—
163D3×C22⋊C4—92no—
164D3×C22⋊C4—92no—
165#877—92no—
166#877—92no—
167#872—31no—
168#872—91no—
169#872—91no—
170#872D911yescharacteristic, complemented
171C42×D3—91no—
172C42×D3—91no—
173C42×D3—91no—
174C42×D3—91no—
175C42×D3—181no—
176C22×C3⋊D3—31no—
177C22×C3×D3—31no—
178C22×C3×D3C22×D311yescharacteristic
179C2×D32—34no—
180C2×D32—64no—
181C2×C32×C4C22×D312yes—
182C2×C32×C4C22×D311yescharacteristic
183C2×C3⋊Dic3—31no—
184C2×C3⋊Dic3—32no—
185C4×C3⋊D3—34no—
186C4×C3⋊D3—64no—
187C2×C3×Dic3—31no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3C22×D312yes—
190C2×C3×Dic3C22×D311yescharacteristic
191C3×C4×D3—34no—
192C3×C4×D3C4×D314yes—
193#624—34no—
194#624—64no—
195D3×Dic3—34no—
196D3×Dic3—34no—
197C23×C9—31no—
198C22×D9—31no—
199C22×D9—34no—
200C22×D9—61no—
201C22×D9—64no—
202C22×D9C22×D311yescharacteristic
203C2×C4×C9—22no—
204C2×C4×C9—21no—
205C2×C4×C9—31no—
206C2×C4×C9—34no—
207C2×C4×C9—32no—
208C2×C4×C9C22×D312yes—
209C2×C4×C9C22×D311yescharacteristic
210C2×Dic9—31no—
211C2×Dic9—34no—
212C2×Dic9—32no—
213C2×Dic9—61no—
214C2×Dic9—62no—
215C2×Dic9C22×D312yes—
216C2×Dic9C22×D311yescharacteristic
217C4×D9—34no—
218C4×D9—34no—
219C4×D9—34no—
220C4×D9—64no—
221C4×D9—64no—
222C4×D9—64no—
223C4×D9—68no—
224C4×D9C4×D314yescomplemented
225C2×#117—271noSylow
226C2×C3×C9C23×C411yescharacteristic
227C2×C3×C9C2×#4112yes—
228C3⋊D9#11714yescomplemented
229D3×C9—22no—
230C3×D9—22no—
231C23×D3—91no—
232C23×D3—91no—
233C22×C3×C4—31no—
234C22×C3×C4—91no—
235C22×Dic3—91no—
236C22×Dic3—91no—
237C2×C4×D3—31no—
238C2×C4×D3—98no—
239C2×C4×D3—91no—
240C2×C4×D3—91no—
241C2×C4×D3—91no—
242C2×C4×D3—91no—
243C2×C4×D3—91no—
244C2×C4×D3—91no—
245C2×C4×D3—98no—
246C2×C4×D3—92no—
247C2×C4×D3—92no—
248C2×C4×D3—92no—
249C2×C4×D3—92no—
250C2×C4×D3—181no—
251C2×C4×D3—182no—
252C2×C4×D3C2×D911yescharacteristic
253C3×C4⋊C4—32no—
254C3×C4⋊C4—92no—
255C3×C22⋊C4—32no—
256C3×C22⋊C4—92no—
257C3×C42—21no—
258C3×C42—31no—
259C3×C42—91no—
260C3×C42C2×D911yescharacteristic, complemented
261C3×C42D3211yescharacteristic
262#217—92no—
263#217—92no—
264(C2×D3)⋊C4—32no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—92no—
267(C2×D3)⋊C4C2×D912yes—
268C4⋊Dic3—92no—
269C4⋊Dic3—92no—
270Dic3⋊C4—32no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273Dic3⋊C4C2×D912yescomplemented
274C4×Dic3—31no—
275C4×Dic3—92no—
276C4×Dic3—91no—
277C4×Dic3—92no—
278C4×Dic3—91no—
279C4×Dic3—92no—
280C4×Dic3—92no—
281C4×Dic3—182no—
282C4×Dic3—181no—
283C4×Dic3C2×D911yescharacteristic, complemented
284C22×C32C23×D311yescharacteristic
285C2×C3⋊D3—32no—
286C2×C3⋊D3—34no—
287C2×C3×D3—22no—
288C2×C3×D3—32no—
289C2×C3×D3—62no—
290C2×C3×D3C2×C4×D312yes—
291D32—68no—
292C32×C4—22no—
293C32×C4C2×C4×D312yes—
294C3⋊Dic3—32no—
295C3⋊Dic3—62no—
296C3×Dic3—22no—
297C3×Dic3—32no—
298C3×Dic3—62no—
299C3×Dic3C2×C4×D312yes—
300C22×C9—21no—
301C22×C9—32no—
302C22×C9—62no—
303C22×C9C23×D311yescharacteristic
304C2×D9—22no—
305C2×D9—32no—
306C2×D9—34no—
307C2×D9—62no—
308C2×D9—64no—
309C2×D9—68no—
310C2×D9C2×C4×D312yes—
311C4×C9—24no—
312C4×C9—22no—
313C4×C9—22no—
314C4×C9—32no—
315C4×C9—62no—
316C4×C9C2×C4×D312yes—
317Dic9—22no—
318Dic9—32no—
319Dic9—62no—
320Dic9—64no—
321Dic9—62no—
322Dic9C2×C4×D312yes—
323C23×C4—271no—
324#117—31no—
325#117—91no—
326#117—271no—
327#117—271no—
328#117—272no—
329#117—272no—
330C2×C4⋊C4—272no—
331C2×C22⋊C4—272no—
332C2×C42—271no—
333C2×C42—271no—
334C3×C9C2×#11711yescharacteristic, complemented, Sylow
335C23×C3—31no—
336C23×C3—91no—
337C22×D3—31no—
338C22×D3—94no—
339C22×D3—91no—
340C22×D3—91no—
341C22×D3—94no—
342C22×D3—181no—
343C22×D3—184no—
344C22×D3—184no—
345C22×D3C22×D911yescharacteristic
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—31no—
349C2×C3×C4—34no—
350C2×C3×C4—32no—
351C2×C3×C4—91no—
352C2×C3×C4—94no—
353C2×C3×C4—92no—
354C2×C3×C4C22×D912yes—
355C2×C3×C4C22×D911yescharacteristic
356C2×C3×C4C2×D3212yes—
357C2×C3×C4C2×D3211yescharacteristic
358C2×Dic3—31no—
359C2×Dic3—32no—
360C2×Dic3—94no—
361C2×Dic3—91no—
362C2×Dic3—91no—
363C2×Dic3—94no—
364C2×Dic3—92no—
365C2×Dic3—92no—
366C2×Dic3—181no—
367C2×Dic3—182no—
368C2×Dic3C22×D912yes—
369C2×Dic3C22×D911yescharacteristic
370C4×D3—34no—
371C4×D3—94no—
372C4×D3—94no—
373C4×D3—94no—
374C4×D3—94no—
375C4×D3—94no—
376C4×D3—94no—
377C4×D3—184no—
378C4×D3—184no—
379C4×D3—184no—
380C4×D3—184no—
381C4×D3—184no—
382C4×D3—188no—
383C4×D3C4×D914yescomplemented
384C2×C32C22×C4×D311yescharacteristic
385C2×C32D3×#4112yes—
386C3⋊D3—34no—
387C3×D3—22no—
388C3×D3—62no—
389C2×C9—21no—
390C2×C9—22no—
391C2×C9—62no—
392C2×C9C22×C4×D311yescharacteristic
393C2×C9D3×#4112yes—
394D9—22no—
395D9—34no—
396D9—64no—
397C24—271no—
398C22×C4—31no—
399C22×C4—91no—
400C22×C4—271no—
401C22×C4—278no—
402C22×C4—272no—
403C22×C4—271no—
404C22×C4—271no—
405C22×C4—271no—
406C22×C4—272no—
407C4⋊C4—32no—
408C4⋊C4—92no—
409C4⋊C4—272no—
410C4⋊C4—272no—
411C22⋊C4—32no—
412C22⋊C4—92no—
413C22⋊C4—272no—
414C22⋊C4—272no—
415C42—31no—
416C42—91no—
417C42—272no—
418C42—272no—
419C42—271no—
420C42D3×D911yescharacteristic, complemented
421C22×C3—21no—
422C22×C3—32no—
423C22×C3—62no—
424C22×C3—92no—
425C22×C3—182no—
426C22×C3C23×D911yescharacteristic
427C22×C3C22×D3211yescharacteristic
428C2×D3—22no—
429C2×D3—32no—
430C2×D3—62no—
431C2×D3—92no—
432C2×D3—94no—
433C2×D3—92no—
434C2×D3—94no—
435C2×D3—182no—
436C2×D3—184no—
437C2×D3—188no—
438C2×D3—188no—
439C2×D3C2×C4×D912yes—
440C3×C4—24no—
441C3×C4—22no—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—32no—
445C3×C4—62no—
446C3×C4—92no—
447C3×C4—182no—
448C3×C4C2×C4×D912yes—
449C3×C4C4×D3212yes—
450Dic3—22no—
451Dic3—32no—
452Dic3—62no—
453Dic3—92no—
454Dic3—92no—
455Dic3—182no—
456Dic3—184no—
457Dic3—182no—
458Dic3—182no—
459Dic3C2×C4×D912yes—
460C32D3×#11711yescharacteristic
461C9—21no—
462C9D3×#11711yescharacteristic, complemented
463C23—31no—
464C23—91no—
465C23—274no—
466C23—271no—
467C23—544no—
468C2×C4—31no—
469C2×C4—34no—
470C2×C4—32no—
471C2×C4—91no—
472C2×C4—94no—
473C2×C4—92no—
474C2×C4—274no—
475C2×C4—274no—
476C2×C4—274no—
477C2×C4—271no—
478C2×C4—274no—
479C2×C4—272no—
480C2×C4—544no—
481C2×C4—544no—
482C2×C4C2×D3×D912yes—
483C2×C4C2×D3×D911yescharacteristic
484C2×C3—21no—
485C2×C3—22no—
486C2×C3—62no—
487C2×C3—182no—
488C2×C3C22×C4×D911yescharacteristic
489C2×C3#41×D912yes—
490C2×C3C2×C4×D3211yescharacteristic
491C2×C3#6430612yes—
492D3—22no—
493D3—62no—
494D3—94no—
495D3—94no—
496D3—184no—
497C22—32no—
498C22—62no—
499C22—92no—
500C22—182no—
501C22—272no—
502C22—274no—
503C22—548no—
504C22C22×D3×D911yescharacteristic
505C4—22no—
506C4—32no—
507C4—62no—
508C4—92no—
509C4—182no—
510C4—272no—
511C4—542no—
512C4C4×D3×D912yes—
513C3—21no—
514C3D9×#11711yescharacteristic, complemented
515C3#1059399211yescharacteristic
516C2—62no—
517C2—182no—
518C2—274no—
519C2C2×C4×D3×D911yescharacteristic
520C2#1172247512yes—
subgroupG/H
centerC2×C4C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC22×D3×D9C22
cosocleC22×C3×C9C24
hypercenterC42D3×D9
hypocenterC3×C9C2×#117
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C42×C3⋊D9C2semidirect
3. C9×#872C2semidirect
4. C3×#63434C2semidirect
6. #11721011C2semidirect
7. #11720980C2semidirect
8. C4×(C3⋊D9)⋊C4C2semidirect
9. #11720975C2semidirect
10. #11720959C2semidirect
11. #11720943C2semidirect
23. C4×D3×D9C4semidirect
24. C3×C42×C9C22semidirect
25. C4×C3⋊Dic9C22semidirect
26. C4×C3⋊Dic9C22semidirect
29. C9×Dic3⋊C4C22semidirect
30. C4×C9×Dic3C22semidirect
31. C3×Dic9⋊C4C22semidirect
32. C3×C4×Dic9C22semidirect
35. Dic3⋊Dic9C22semidirect
36. Dic9⋊Dic3C22semidirect
37. Dic3×Dic9C22semidirect
57. #63434D3semidirect
76. C4×D3×C9C2×C4semidirect
77. C3×C4×D9C2×C4semidirect
78. D3×Dic9C2×C4semidirect
81. Dic3×D9C2×C4semidirect
124. C42×C9C2×D3semidirect
130. Dic9⋊C4C2×D3semidirect
136. C4×Dic9C2×D3semidirect
170. #872D9semidirect
224. C4×D9C4×D3semidirect
228. C3⋊D9#117semidirect
260. C3×C42C2×D9semidirect
273. Dic3⋊C4C2×D9semidirect
283. C4×Dic3C2×D9semidirect
334. C3×C9C2×#117semidirect
383. C4×D3C4×D9semidirect
420. C42D3×D9semidirect
462. C9D3×#117semidirect
514. C3D9×#117semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×D92
C42×C3⋊D92
C9×#8722
C3×#634342
#117210482
#117210112
#117209802
C4×(C3⋊D9)⋊C42
#117209752
#117209592
#117209432
#105939923
D9×#1173