Finite Group Databasebeta

← #13147988

Subgroups — #13147988

Order 1728.

1122 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic
2C22⋊C4×C3⋊D9C211yesmaximal, characteristic
3C9×#877C211yesmaximal, characteristic, complemented
4C3×#63439C211yesmaximal, characteristic, complemented
5#11721274C211yesmaximal, characteristic
6#11721245C211yesmaximal, characteristic, complemented
7#11721244C211yesmaximal, characteristic, complemented
8#11720980C212yesmaximal, complemented
9C4×(C3⋊D9)⋊C4C212yesmaximal, complemented
10#11720899C212yesmaximal, complemented
11#11720898C212yesmaximal, complemented
12#10594052—31nomaximal
13D9×#117—31nomaximal
14C23×C3⋊D9C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C22×C9×Dic3C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C2×(C3⋊D9)⋊C4C418yescomplemented
19C2×(C3⋊D9)⋊C4C2212yes—
20C2×(C3⋊D9)⋊C4C2211yescharacteristic
21C2×(C3⋊D9)⋊C4C2211yescharacteristic
22C2×(C3⋊D9)⋊C4C2211yescharacteristic
23C2×(C3⋊D9)⋊C4C2211yescharacteristic
24C3×C9×C22⋊C4C2211yescharacteristic
25#88551C2211yescharacteristic
26#88536C2212yes—
27C9×#217C2211yescharacteristic
28C3×#3676C2211yescharacteristic
29C9×Dic3⋊C4C2212yescomplemented
30C4×C9×Dic3C2212yescomplemented
31C3×Dic9⋊C4C2212yescomplemented
32C3×C4×Dic9C2212yescomplemented
33#88445C2212yes—
34Dic3⋊Dic9C2212yescomplemented
35Dic9⋊Dic3C2212yescomplemented
36Dic3×Dic9C2212yescomplemented
37Dic3×Dic9C2212yescomplemented
38C22×#624—31no—
39C22⋊C4×C3⋊D3—31no—
40C3×#877—31no—
41C3×#877D311yescharacteristic
42#63974—31no—
43#63955—31no—
44#63955—31no—
45#63883—32no—
46C4×#624—32no—
47#63836—32no—
48#63836—32no—
49C22×C4×D9—31no—
50C9×#117—31no—
51#63488—31no—
52#63454—32no—
53C4⋊C4×D9—32no—
54C22⋊C4×D9—31no—
55C22⋊C4×D9—31no—
56C22⋊C4×D9—61no—
57#63439—31no—
58#63439D311yescharacteristic, complemented
59#63434—32no—
60C42×D9—32no—
61C23×C3×C9C2311yescharacteristic
62C22×C3⋊D9—24no—
63C22×C3⋊D9C2×C414yes—
64C22×C3⋊D9C2311yescharacteristic
65C22×C3⋊D9C2311yescharacteristic
66C2×C3×C4×C9C2312yes—
67C2×C3⋊Dic9C2312yes—
68C4×C3⋊D9—24no—
69C2×C9×Dic3C2×C414yescomplemented
70C2×C9×Dic3C2312yes—
71C2×C9×Dic3C2311yescharacteristic
72C2×C9×Dic3C2311yescharacteristic
73C2×C3×Dic9C2×C414yescomplemented
74C2×C3×Dic9C2312yes—
75C2×C3×Dic9C2311yescharacteristic
76C2×C3×Dic9C2311yescharacteristic
77(C3⋊D9)⋊C4—24no—
78(C3⋊D9)⋊C4C2×C414yes—
79(C3⋊D9)⋊C4C2×C414yes—
80(C3⋊D9)⋊C4C2×C414yescomplemented
81(C3⋊D9)⋊C4C2×C414yescomplemented
82D3×#117—91no—
83D3×#117—91no—
84C23×C3⋊D3—31no—
85C2×C4×C3⋊D3—32no—
86C22×C3×Dic3—31no—
87C22×C3×Dic3C2×D311yescharacteristic
88C2×#624—38no—
89C2×#624—31no—
90C2×#624—31no—
91C2×#624—31no—
92C2×#624—31no—
93C2×#624—32no—
94C23×D9—31no—
95C23×D9—61no—
96C32×C22⋊C4C2×D311yescharacteristic
97#3757—31no—
98#3752—32no—
99C3×#217—31no—
100C3×#217C2×D311yescharacteristic
101C3×Dic3⋊C4—32no—
102C3×Dic3⋊C4C2×D312yes—
103C3×C4×Dic3—32no—
104C3×C4×Dic3C2×D312yes—
105Dic3⋊Dic3—32no—
106Dic3⋊Dic3—32no—
107#3722—32no—
108Dic32—32no—
109Dic32—32no—
110C22×C4×C9—31no—
111C22×Dic9C2×D311yescharacteristic
112C2×C4×D9—38no—
113C2×C4×D9—31no—
114C2×C4×D9—31no—
115C2×C4×D9—31no—
116C2×C4×D9—31no—
117C2×C4×D9—32no—
118C2×C4×D9—32no—
119C2×C4×D9—62no—
120C9×C4⋊C4—32no—
121C9×C22⋊C4—21no—
122C9×C22⋊C4—31no—
123C9×C22⋊C4C2×D311yescharacteristic
124C42×C9—32no—
125#3676—31no—
126#3676—61no—
127#3676C2×D311yescharacteristic
128(C2×D9)⋊C4—32no—
129(C2×D9)⋊C4—32no—
130(C2×D9)⋊C4—62no—
131C4⋊Dic9—32no—
132Dic9⋊C4—32no—
133Dic9⋊C4C2×D312yescomplemented
134C4×Dic9—32no—
135C4×Dic9—32no—
136C4×Dic9C2×D312yescomplemented
137C22×C3×C9—22no—
138C22×C3×C9C22×C412yes—
139C22×C3×C9C2411yescharacteristic
140C2×C3⋊D9—22no—
141C2×C3⋊D9—28no—
142C2×C3⋊D9C22×C412yes—
143C2×C3⋊D9C22×C412yes—
144C2×C3⋊D9#4114yes—
145C3×C4×C9—22no—
146C3⋊Dic9—22no—
147C9×Dic3—22no—
148C9×Dic3C22×C412yes—
149C9×Dic3C22×C412yes—
150C3×Dic9—22no—
151C3×Dic9C22×C412yes—
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—91no—
164D3×C22⋊C4—91no—
165D3×C22⋊C4—91no—
166D3×C22⋊C4—91no—
167D3×C22⋊C4—181no—
168#877—31no—
169#877—91no—
170#877—91no—
171#877D911yescharacteristic, complemented
172#872—92no—
173#872—92no—
174C42×D3—92no—
175C42×D3—92no—
176C23×C32C22×D311yescharacteristic
177C22×C3⋊D3—31no—
178C22×C3⋊D3—34no—
179C22×C3⋊D3—31no—
180C22×C3⋊D3—64no—
181C2×C32×C4C22×D312yes—
182C2×C3⋊Dic3—32no—
183C4×C3⋊D3—64no—
184C2×C3×Dic3—34no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—31no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3C4×D314yes—
189C2×C3×Dic3C22×D312yes—
190C2×C3×Dic3C22×D311yescharacteristic
191C2×C3×Dic3C22×D311yescharacteristic
192#624—34no—
193#624—34no—
194#624—34no—
195#624—34no—
196#624—64no—
197C23×C9—21no—
198C23×C9C22×D311yescharacteristic
199C22×D9—31no—
200C22×D9—34no—
201C22×D9—31no—
202C22×D9—61no—
203C22×D9—61no—
204C22×D9—64no—
205C22×D9—64no—
206C22×D9—124no—
207C2×C4×C9—22no—
208C2×C4×C9—34no—
209C2×C4×C9—31no—
210C2×C4×C9—31no—
211C2×C4×C9—32no—
212C2×C4×C9C22×D312yes—
213C2×Dic9—32no—
214C2×Dic9—62no—
215C2×Dic9C4×D314yescomplemented
216C2×Dic9C22×D312yes—
217C2×Dic9C22×D311yescharacteristic
218C2×Dic9C22×D311yescharacteristic
219C4×D9—34no—
220C4×D9—34no—
221C4×D9—34no—
222C4×D9—34no—
223C4×D9—64no—
224C4×D9—64no—
225C4×D9—124no—
226C2×#117—271noSylow
227C2×C3×C9—22no—
228C2×C3×C9C23×C411yescharacteristic
229C2×C3×C9C2×#4112yes—
230C3⋊D9—22no—
231C3⋊D9#11714yescomplemented
232C23×D3—91no—
233C23×D3—91no—
234C23×D3—181no—
235C22×C3×C4—31no—
236C22×C3×C4—91no—
237C22×Dic3—31no—
238C22×Dic3C2×D911yescharacteristic
239C2×C4×D3—91no—
240C2×C4×D3—91no—
241C2×C4×D3—91no—
242C2×C4×D3—91no—
243C2×C4×D3—98no—
244C2×C4×D3—98no—
245C2×C4×D3—91no—
246C2×C4×D3—91no—
247C2×C4×D3—91no—
248C2×C4×D3—91no—
249C2×C4×D3—92no—
250C2×C4×D3—92no—
251C2×C4×D3—92no—
252C2×C4×D3—92no—
253C2×C4×D3—182no—
254C3×C4⋊C4—32no—
255C3×C4⋊C4—92no—
256C3×C22⋊C4—21no—
257C3×C22⋊C4—31no—
258C3×C22⋊C4—91no—
259C3×C22⋊C4C2×D911yescharacteristic
260C3×C22⋊C4D3211yescharacteristic
261C3×C42—32no—
262C3×C42—92no—
263#217—31no—
264#217—91no—
265#217—91no—
266#217—181no—
267#217C2×D911yescharacteristic
268(C2×D3)⋊C4—92no—
269(C2×D3)⋊C4—92no—
270(C2×D3)⋊C4—92no—
271(C2×D3)⋊C4—92no—
272(C2×D3)⋊C4—182no—
273C4⋊Dic3—92no—
274C4⋊Dic3—92no—
275Dic3⋊C4—32no—
276Dic3⋊C4—92no—
277Dic3⋊C4—92no—
278Dic3⋊C4C2×D912yescomplemented
279C4×Dic3—32no—
280C4×Dic3—92no—
281C4×Dic3—92no—
282C4×Dic3—92no—
283C4×Dic3—92no—
284C4×Dic3C2×D912yescomplemented
285C22×C32—22no—
286C22×C32C2×C4×D312yes—
287C22×C32C23×D311yescharacteristic
288C2×C3⋊D3—32no—
289C2×C3⋊D3—34no—
290C2×C3⋊D3—32no—
291C2×C3⋊D3—68no—
292C2×C3⋊D3—62no—
293C32×C4—22no—
294C3⋊Dic3—62no—
295C3×Dic3—22no—
296C3×Dic3—32no—
297C3×Dic3—32no—
298C3×Dic3—62no—
299C3×Dic3C2×C4×D312yes—
300C3×Dic3C2×C4×D312yes—
301C22×C9—22no—
302C22×C9—21no—
303C22×C9—22no—
304C22×C9—42no—
305C22×C9C2×C4×D312yes—
306C22×C9C23×D311yescharacteristic
307C2×D9—32no—
308C2×D9—34no—
309C2×D9—32no—
310C2×D9—62no—
311C2×D9—64no—
312C2×D9—62no—
313C2×D9—68no—
314C2×D9—62no—
315C2×D9—128no—
316C2×D9—122no—
317C4×C9—22no—
318C4×C9—32no—
319C4×C9—32no—
320C4×C9—42no—
321C4×C9—62no—
322Dic9—22no—
323Dic9—62no—
324Dic9—122no—
325Dic9C2×C4×D312yes—
326Dic9C2×C4×D312yes—
327C23×C4—271no—
328#117—31no—
329#117—91no—
330#117—272no—
331#117—272no—
332#117—271no—
333#117—271no—
334C2×C4⋊C4—272no—
335C2×C22⋊C4—271no—
336C2×C22⋊C4—271no—
337C2×C42—272no—
338C3×C9C2×#11711yescharacteristic, complemented, Sylow
339C23×C3—21no—
340C23×C3C22×D911yescharacteristic
341C23×C3C2×D3211yescharacteristic
342C22×D3—91no—
343C22×D3—91no—
344C22×D3—91no—
345C22×D3—94no—
346C22×D3—91no—
347C22×D3—94no—
348C22×D3—181no—
349C22×D3—181no—
350C22×D3—184no—
351C22×D3—184no—
352C22×D3—184no—
353C22×D3—364no—
354C2×C3×C4—22no—
355C2×C3×C4—31no—
356C2×C3×C4—31no—
357C2×C3×C4—34no—
358C2×C3×C4—32no—
359C2×C3×C4—94no—
360C2×C3×C4—91no—
361C2×C3×C4—91no—
362C2×C3×C4—92no—
363C2×C3×C4C22×D912yes—
364C2×C3×C4C2×D3212yes—
365C2×Dic3—31no—
366C2×Dic3—31no—
367C2×Dic3—34no—
368C2×Dic3—32no—
369C2×Dic3—92no—
370C2×Dic3—92no—
371C2×Dic3—182no—
372C2×Dic3C4×D914yescomplemented
373C2×Dic3C22×D912yes—
374C2×Dic3C22×D911yescharacteristic
375C2×Dic3C22×D911yescharacteristic
376C4×D3—94no—
377C4×D3—94no—
378C4×D3—94no—
379C4×D3—94no—
380C4×D3—94no—
381C4×D3—94no—
382C4×D3—94no—
383C4×D3—94no—
384C4×D3—184no—
385C4×D3—184no—
386C4×D3—184no—
387C4×D3—184no—
388C4×D3—364no—
389C2×C32—22no—
390C2×C32C22×C4×D311yescharacteristic
391C2×C32D3×#4112yes—
392C3⋊D3—34no—
393C3⋊D3—62no—
394C2×C9—22no—
395C2×C9—21no—
396C2×C9—22no—
397C2×C9—42no—
398C2×C9C22×C4×D311yescharacteristic
399C2×C9D3×#4112yes—
400D9—34no—
401D9—64no—
402D9—62no—
403D9—122no—
404C24—271no—
405C22×C4—31no—
406C22×C4—91no—
407C22×C4—271no—
408C22×C4—271no—
409C22×C4—271no—
410C22×C4—271no—
411C22×C4—278no—
412C22×C4—272no—
413C22×C4—272no—
414C4⋊C4—32no—
415C4⋊C4—92no—
416C4⋊C4—272no—
417C4⋊C4—272no—
418C22⋊C4—31no—
419C22⋊C4—91no—
420C22⋊C4—272no—
421C22⋊C4—271no—
422C22⋊C4—272no—
423C22⋊C4D3×D911yescharacteristic
424C42—32no—
425C42—92no—
426C42—272no—
427C42—272no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—21no—
431C22×C3—22no—
432C22×C3—42no—
433C22×C3C2×C4×D912yes—
434C22×C3C23×D911yescharacteristic
435C22×C3C4×D3212yes—
436C22×C3C22×D3211yescharacteristic
437C2×D3—92no—
438C2×D3—94no—
439C2×D3—92no—
440C2×D3—94no—
441C2×D3—92no—
442C2×D3—92no—
443C2×D3—184no—
444C2×D3—182no—
445C2×D3—182no—
446C2×D3—188no—
447C2×D3—188no—
448C2×D3—182no—
449C2×D3—182no—
450C2×D3—368no—
451C2×D3—362no—
452C3×C4—22no—
453C3×C4—22no—
454C3×C4—32no—
455C3×C4—32no—
456C3×C4—42no—
457C3×C4—62no—
458C3×C4—92no—
459C3×C4—92no—
460C3×C4—182no—
461Dic3—22no—
462Dic3—32no—
463Dic3—32no—
464Dic3—62no—
465Dic3—182no—
466Dic3—182no—
467Dic3—362no—
468Dic3C2×C4×D912yes—
469Dic3C2×C4×D912yes—
470C32D3×#11711yescharacteristic
471C9—21no—
472C9D3×#11711yescharacteristic, complemented
473C23—271no—
474C23—271no—
475C23—274no—
476C23—544no—
477C23C2×D3×D911yescharacteristic
478C2×C4—31no—
479C2×C4—31no—
480C2×C4—34no—
481C2×C4—32no—
482C2×C4—91no—
483C2×C4—91no—
484C2×C4—94no—
485C2×C4—92no—
486C2×C4—274no—
487C2×C4—274no—
488C2×C4—274no—
489C2×C4—274no—
490C2×C4—272no—
491C2×C4—544no—
492C2×C4—544no—
493C2×C4C2×D3×D912yes—
494C2×C3—22no—
495C2×C3—22no—
496C2×C3—22no—
497C2×C3—21no—
498C2×C3—42no—
499C2×C3C22×C4×D911yescharacteristic
500C2×C3#41×D912yes—
501C2×C3C2×C4×D3211yescharacteristic
502C2×C3#6431212yes—
503D3—94no—
504D3—94no—
505D3—184no—
506D3—182no—
507D3—182no—
508D3—362no—
509C22—22no—
510C22—274no—
511C22—272no—
512C22—272no—
513C22—548no—
514C22—542no—
515C22C4×D3×D912yes—
516C22C22×D3×D911yescharacteristic
517C4—22no—
518C4—32no—
519C4—32no—
520C4—62no—
521C4—92no—
522C4—92no—
523C4—182no—
524C4—542no—
525C3—21no—
526C3D9×#11711yescharacteristic, complemented
527C3#1059405211yescharacteristic
528C2—22no—
529C2—274no—
530C2—542no—
531C2C2×C4×D3×D911yescharacteristic
532C2#1172249812yes—
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
prime radicalC23×C3⋊D9C22
cosocleC22×C3×C9C24
hypercenterC22⋊C4D3×D9
hypocenterC3×C9C2×#117
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C9×#877C2semidirect
4. C3×#63439C2semidirect
6. #11721245C2semidirect
7. #11721244C2semidirect
8. #11720980C2semidirect
9. C4×(C3⋊D9)⋊C4C2semidirect
10. #11720899C2semidirect
11. #11720898C2semidirect
18. C2×(C3⋊D9)⋊C4C4semidirect
29. C9×Dic3⋊C4C22semidirect
30. C4×C9×Dic3C22semidirect
31. C3×Dic9⋊C4C22semidirect
32. C3×C4×Dic9C22semidirect
34. Dic3⋊Dic9C22semidirect
35. Dic9⋊Dic3C22semidirect
36. Dic3×Dic9C22semidirect
37. Dic3×Dic9C22semidirect
58. #63439D3semidirect
69. C2×C9×Dic3C2×C4semidirect
73. C2×C3×Dic9C2×C4semidirect
80. (C3⋊D9)⋊C4C2×C4semidirect
81. (C3⋊D9)⋊C4C2×C4semidirect
133. Dic9⋊C4C2×D3semidirect
136. C4×Dic9C2×D3semidirect
171. #877D9semidirect
215. C2×Dic9C4×D3semidirect
231. C3⋊D9#117semidirect
278. Dic3⋊C4C2×D9semidirect
284. C4×Dic3C2×D9semidirect
338. C3×C9C2×#117semidirect
372. C2×Dic3C4×D9semidirect
472. C9D3×#117semidirect
526. C3D9×#117semidirect

Maximal subgroups

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

subgroupindex
C22×(C3⋊D9)⋊C42
C22⋊C4×C3⋊D92
C9×#8772
C3×#634392
#117212742
#117212452
#117212442
#117209802
C4×(C3⋊D9)⋊C42
#117208992
#117208982
#105940523
D9×#1173