Finite Group Databasebeta

← #13150248

Subgroups — #13150248

Order 1728.

776 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721813C211yesmaximal, characteristic, complemented
2C3×#63492C211yesmaximal, characteristic
3C2×C4×C9×Dic3C211yesmaximal, characteristic, complemented
4Dic3×#611C212yesmaximal, complemented
5#11721259C212yesmaximal, complemented
6Dic3×D36C211yesmaximal, characteristic, complemented
7#11720974C212yesmaximal, complemented
8#11720948C211yesmaximal, characteristic, complemented
9#11720947C212yesmaximal, complemented
10#11720927C211yesmaximal, characteristic, complemented
11#11720912C211yesmaximal, characteristic, complemented
12#10594949—31nomaximal
13#10589581—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C3×#611C2212yes—
17C22×C9×Dic3C2211yescharacteristic
18C22×C9×Dic3C2211yescharacteristic
19C2×C3×D36C2211yescharacteristic
20C2×Dic3×D9C2212yes—
21#88551C2212yes—
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23#88534C2212yes—
24C3×(C2×D9)⋊C4C2212yes—
25C4×C9×Dic3C2211yescharacteristic
26C4×C9×Dic3C2211yescharacteristic
27C4×C9×Dic3C2211yescharacteristic
28C4×C9×Dic3C2211yescharacteristic
29C3×C4⋊Dic9C2211yescharacteristic
30(C2×D9)⋊Dic3C2212yes—
31(C2×D9)⋊Dic3C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33#88442C2212yes—
34#88442C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64133—31no—
38C3×#930—31no—
39C2×C3×C4×Dic3D311yescharacteristic
40Dic3×#67—32no—
41#63961—32no—
42Dic3×D12—31no—
43#63877—32no—
44#63869—31no—
45#63860—31no—
46#63859—32no—
47#63844—31no—
48C2×C42×C9—31no—
49#63492D311yescharacteristic
50#63491—32no—
51C4×#611—32no—
52#63486—31no—
53#63486—61no—
54#63438—32no—
55#63437—31no—
56C4×D36—31no—
57#63434—32no—
58#63432—31no—
59C4×Dic18—31no—
60C23×C3×C9C2311yescharacteristic
61C22×C3×D9C2312yes—
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2312yes—
67#6226—22no—
68C3×#611—24no—
69C2×C9×Dic3—22no—
70C2×C9×Dic3—22no—
71C2×C9×Dic3C2311yescharacteristic
72C2×C9×Dic3C2311yescharacteristic
73C2×C9×Dic3C2311yescharacteristic
74C2×C9×Dic3C2311yescharacteristic
75C2×C3×Dic9C2312yes—
76C3×D36—22no—
77Dic3×D9—24no—
78#5600—91no—
79#5479—91no—
80C22×C32×C4C2×D311yescharacteristic
81C2×#634—31no—
82C2×C3×#67—32no—
83C22×C3×Dic3C2×D311yescharacteristic
84C22×C3×Dic3C2×D311yescharacteristic
85C2×C3×D12—31no—
86C2×D3×Dic3—32no—
87#3757—32no—
88C4⋊(C3⋊Dic3)—31no—
89#3750—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×C4⋊Dic3—31no—
92C3×C4×Dic3C2×D311yescharacteristic
93C3×C4×Dic3C2×D311yescharacteristic
94C3×C4×Dic3C2×D311yescharacteristic
95C3×C4×Dic3C2×D311yescharacteristic
96#3724—32no—
97#3724—32no—
98Dic3⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102Dic32—32no—
103C22×C4×C9—21no—
104C22×C4×C9—31no—
105C22×C4×C9—31no—
106C22×C4×C9C2×D311yescharacteristic
107C2×#611C2×D312yes—
108C2×D36C2×D311yescharacteristic
109C2×C4×D9—32no—
110C2×Dic18—31no—
111C2×Dic18—61no—
112C42×C9—31no—
113C42×C9—31no—
114C42×C9—31no—
115C42×C9—31no—
116#3676—32no—
117#3676—62no—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4C2×D312yes—
122C4⋊Dic9—31no—
123C4⋊Dic9—61no—
124C4⋊Dic9C2×D311yescharacteristic
125Dic9⋊C4—32no—
126Dic9⋊C4—32no—
127Dic9⋊C4—32no—
128Dic9⋊C4—32no—
129Dic9⋊C4—62no—
130C4×Dic9—32no—
131C22×C3×C9—21no—
132C22×C3×C9—21no—
133C22×C3×C9#4112yes—
134C22×C3×C9C2411yescharacteristic
135C2×C3×D9—22no—
136C2×C3×D9—22no—
137C2×C3×D9—22no—
138C3×C4×C9—21no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—22no—
141C3⋊Dic9—22no—
142C9×Dic3—21no—
143C9×Dic3—21no—
144C9×Dic3—21no—
145C9×Dic3#4112yes—
146C3×Dic9—22no—
147C3×#121—91no—
148C2×C3×C42—31no—
149#936—91no—
150#935—92no—
151D4×Dic3—92no—
152D4×Dic3—91no—
153#930—31no—
154#929—92no—
155C4×#67—92no—
156#924—91no—
157#924—91no—
158#924—181no—
159C2×C4×Dic3D911yescharacteristic, complemented
160#890—91no—
161#887—91no—
162#879—92no—
163#877—92no—
164#876—92no—
165#875—91no—
166C4×D12—91no—
167#872—92no—
168#870—91no—
169C4×Dic6—91no—
170C23×C32C22×D311yescharacteristic
171C22×C3×D3—32no—
172C2×C32×C4—22no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—32no—
176C2×C3⋊Dic3—32no—
177#634—62no—
178C3×#67—64no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3C22×D311yescharacteristic
183C2×C3×Dic3C22×D311yescharacteristic
184C2×C3×Dic3C22×D311yescharacteristic
185C2×C3×Dic3C22×D311yescharacteristic
186C3×D12—62no—
187D3×Dic3—64no—
188C23×C9—21no—
189C23×C9C22×D311yescharacteristic
190C22×D9C22×D312yes—
191C2×C4×C9—22no—
192C2×C4×C9—21no—
193C2×C4×C9—21no—
194C2×C4×C9—31no—
195C2×C4×C9—31no—
196C2×C4×C9—31no—
197C2×C4×C9—31no—
198C2×C4×C9—42no—
199C2×C4×C9—62no—
200C2×C4×C9—62no—
201C2×C4×C9C22×D311yescharacteristic
202C2×C4×C9C22×D311yescharacteristic
203#611—24no—
204C2×Dic9—32no—
205C2×Dic9—32no—
206C2×Dic9—62no—
207C2×Dic9—62no—
208C2×Dic9C22×D312yes—
209D36—22no—
210C4×D9—64no—
211Dic18—62no—
212Dic18—122no—
213#529—271noSylow
214C2×C3×C9—22no—
215C2×C3×C9C2×#4111yescharacteristic
216C2×C3×C9C2×#4111yescharacteristic
217C2×C3×C9C2×#4111yescharacteristic
218C3×D9—42no—
219C2×C3×D4—92no—
220C2×C3×D4—91no—
221C22×C3×C4—21no—
222C22×C3×C4—31no—
223C22×C3×C4—31no—
224C22×C3×C4C2×D911yescharacteristic
225C22×C3×C4D3211yescharacteristic
226C2×#67—32no—
227C22×Dic3—92no—
228C22×Dic3C2×D911yescharacteristic
229C22×Dic3C2×D911yescharacteristic
230C2×D12—31no—
231C2×C4×D3—92no—
232C2×Dic6—91no—
233C2×Dic6—91no—
234C2×Dic6—181no—
235C3×C4⋊C4—91no—
236C3×C22⋊C4—92no—
237C3×C42—31no—
238C3×C42—31no—
239C3×C42—31no—
240C3×C42—31no—
241#217—92no—
242#217—92no—
243#217—92no—
244#217—92no—
245#217—92no—
246#217—182no—
247(C2×D3)⋊C4—32no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251C4⋊Dic3—31no—
252C4⋊Dic3—91no—
253C4⋊Dic3—91no—
254C4⋊Dic3—92no—
255C4⋊Dic3—181no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263Dic3⋊C4—182no—
264C4×Dic3—92no—
265C4×Dic3—92no—
266C4×Dic3C2×D911yescharacteristic
267C4×Dic3C2×D911yescharacteristic
268C4×Dic3C2×D911yescharacteristic
269C4×Dic3C2×D911yescharacteristic
270C22×C32—21no—
271C22×C32—21no—
272C22×C32#23512yes—
273C22×C32C23×D311yescharacteristic
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C2×C3×D3—62no—
277C32×C4—21no—
278C32×C4#23512yes—
279C3⋊Dic3—62no—
280C3⋊Dic3—62no—
281C3×Dic3—21no—
282C3×Dic3—21no—
283C3×Dic3—21no—
284C3×Dic3—62no—
285C3×Dic3#23512yes—
286C22×C9—21no—
287C22×C9—21no—
288C22×C9—21no—
289C22×C9—22no—
290C22×C9—41no—
291C22×C9—41no—
292C22×C9#23712yes—
293C22×C9C23×D311yescharacteristic
294C2×D9—22no—
295C2×D9—22no—
296C2×D9—22no—
297C4×C9—21no—
298C4×C9—22no—
299C4×C9—32no—
300C4×C9—41no—
301C4×C9—61no—
302C4×C9—61no—
303C4×C9—61no—
304C4×C9#23712yes—
305Dic9—22no—
306Dic9—62no—
307Dic9—62no—
308Dic9—122no—
309Dic9—122no—
310#126—272no—
311#125—271no—
312#124—271no—
313#123—272no—
314C22⋊Q8—271no—
315#121—91no—
316C4×Q8—271no—
317C4×D4—271no—
318C4×D4—272no—
319#117—272no—
320C2×C42—31no—
321C3×C9#52911yescharacteristic, complemented, Sylow
322C23×C3—21no—
323C23×C3—92no—
324C23×C3C22×D911yescharacteristic
325C23×C3C2×D3211yescharacteristic
326C22×D3—32no—
327C3×D4—184no—
328C3×D4—182no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—31no—
336C2×C3×C4—31no—
337C2×C3×C4—42no—
338C2×C3×C4—62no—
339C2×C3×C4—62no—
340C2×C3×C4—92no—
341C2×C3×C4C22×D911yescharacteristic
342C2×C3×C4C22×D911yescharacteristic
343C2×C3×C4C2×D3211yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345#67—64no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—32no—
349C2×Dic3—92no—
350C2×Dic3—92no—
351C2×Dic3—92no—
352C2×Dic3—92no—
353C2×Dic3—182no—
354C2×Dic3—182no—
355C2×Dic3—184no—
356C2×Dic3C22×D911yescharacteristic
357C2×Dic3C22×D911yescharacteristic
358C2×Dic3C22×D911yescharacteristic
359C2×Dic3C22×D911yescharacteristic
360D12—62no—
361C4×D3—184no—
362Dic6—182no—
363Dic6—182no—
364Dic6—362no—
365C2×C32—22no—
366C2×C32C2×#23511yescharacteristic
367C2×C32C2×#23511yescharacteristic
368C2×C32C2×#23511yescharacteristic
369C3×D3—122no—
370C2×C9—22no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9—21no—
374C2×C9—42no—
375C2×C9C2×#23711yescharacteristic
376C2×C9C2×#23711yescharacteristic
377C2×C9D3×#4111yescharacteristic
378D9—42no—
379C2×Q8—271no—
380C2×D4—92no—
381C2×D4—91no—
382C22×C4—31no—
383C22×C4—31no—
384C22×C4—272no—
385C22×C4D3×D911yescharacteristic
386C4⋊C4—91no—
387C4⋊C4—272no—
388C4⋊C4—272no—
389C4⋊C4—272no—
390C4⋊C4—272no—
391C4⋊C4—271no—
392C22⋊C4—92no—
393C22⋊C4—272no—
394C22⋊C4—272no—
395C22⋊C4—272no—
396C22⋊C4—272no—
397C42—31no—
398C42—31no—
399C42—31no—
400C42—31no—
401C42—272no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—22no—
407C22×C3—21no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—182no—
411C22×C3—182no—
412C22×C3—182no—
413C22×C3#369712yes—
414C22×C3C23×D911yescharacteristic
415C22×C3#379512yes—
416C22×C3C22×D3211yescharacteristic
417C2×D3—62no—
418C2×D3—62no—
419C2×D3—62no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—22no—
423C3×C4—32no—
424C3×C4—41no—
425C3×C4—61no—
426C3×C4—61no—
427C3×C4—61no—
428C3×C4—182no—
429C3×C4#369712yes—
430C3×C4#380412yes—
431Dic3—21no—
432Dic3—21no—
433Dic3—21no—
434Dic3—62no—
435Dic3—182no—
436Dic3—182no—
437Dic3—182no—
438Dic3—182no—
439Dic3—362no—
440Dic3—362no—
441Dic3#369712yes—
442C32#547911yescharacteristic
443C9—21no—
444C9#560011yescharacteristic, complemented
445C23—92no—
446C23C2×D3×D911yescharacteristic
447Q8—542no—
448D4—184no—
449D4—182no—
450C2×C4—22no—
451C2×C4—31no—
452C2×C4—31no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—92no—
458C2×C4—272no—
459C2×C4—272no—
460C2×C4—544no—
461C2×C4C2×D3×D911yescharacteristic
462C2×C4C2×D3×D911yescharacteristic
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—42no—
469C2×C3—362no—
470C2×C3C2×#369711yescharacteristic
471C2×C3C2×#369711yescharacteristic
472C2×C3C2×#369711yescharacteristic
473C2×C3C2×#379511yescharacteristic
474C2×C3D3×#23511yescharacteristic
475C2×C3C2×#380411yescharacteristic
476D3—122no—
477C22—21no—
478C22—21no—
479C22—182no—
480C22—182no—
481C22—182no—
482C22#8864112yes—
483C22C22×D3×D911yescharacteristic
484C4—21no—
485C4—32no—
486C4—61no—
487C4—61no—
488C4—61no—
489C4—182no—
490C4—542no—
491C4—542no—
492C4#8865812yes—
493C3—21no—
494C3#1058958111yescharacteristic, complemented
495C3#1059494911yescharacteristic
496C2—22no—
497C2—362no—
498C2C2×#8864111yescharacteristic
499C2D3×#369711yescharacteristic
500C2C2×#8865811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC3×#63492C2
hypercenterC22×C4D3×D9
hypocenterC3×C9#529
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11721813C2semidirect
3. C2×C4×C9×Dic3C2semidirect
4. Dic3×#611C2semidirect
5. #11721259C2semidirect
6. Dic3×D36C2semidirect
7. #11720974C2semidirect
8. #11720948C2semidirect
9. #11720947C2semidirect
10. #11720927C2semidirect
11. #11720912C2semidirect
159. C2×C4×Dic3D9semidirect
321. C3×C9#529semidirect
444. C9#5600semidirect
494. C3#10589581semidirect

Maximal subgroups

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

subgroupindex
#117218132
C3×#634922
C2×C4×C9×Dic32
Dic3×#6112
#117212592
Dic3×D362
#117209742
#117209482
#117209472
#117209272
#117209122
#105949493
#105895813