Finite Group Databasebeta

← #13150504

Subgroups — #13150504

Order 1728.

824 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
2C9×#930C211yesmaximal, characteristic, complemented
3C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
4#11721285C212yesmaximal, complemented
5#67×Dic9C212yesmaximal, complemented
6#11721042C212yesmaximal, complemented
7#11721021C212yesmaximal, complemented
8#11721016C212yesmaximal, complemented
9#11720946C212yesmaximal, complemented
10#10594977—31nomaximal
11#10589719—31nomaximal
12C23×C3×D9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C2×#6226C2211yescharacteristic, complemented
15C2×C9×#67C2212yes—
16C22×C3×Dic9C2211yescharacteristic
17C2×C9×D12C2211yescharacteristic, complemented
18C2×#6194C2212yes—
19C2×#6194C2212yes—
20C2×D3×Dic9C2212yescomplemented
21#88551C2212yes—
22C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
23#88534C2212yescomplemented
24C9×(C2×D3)⋊C4C2212yescomplemented
25C3×(C2×D9)⋊C4C2212yes—
26C3×(C2×D9)⋊C4C2212yes—
27C9×C4⋊Dic3C2211yescharacteristic, complemented
28(C2×D3)⋊Dic9C2212yescomplemented
29(C2×D9)⋊Dic3C2212yes—
30(C2×D9)⋊Dic3C2212yes—
31Dic3⋊Dic9C2212yescomplemented
32Dic3×Dic9C2212yescomplemented
33#64133—31no—
34C3×#930D311yescharacteristic
35C2×C3×(C2×D3)⋊C4—31no—
36#63976—32no—
37Dic3×#67—32no—
38#63913—32no—
39#63891—32no—
40#63885—32no—
41#63858—32no—
42C9×#121—31no—
43#63501—32no—
44#63499—32no—
45D4×Dic9—32no—
46C2×(C2×D9)⋊C4D311yescharacteristic, complemented
47#63486—31no—
48#63486—61no—
49#63460—32no—
50#63448—32no—
51#63447—32no—
52C23×C3×C9C2311yescharacteristic
53C22×D3×C9C2312yes—
54C22×C3×D9—22no—
55C22×C3×D9—41no—
56C22×C3×D9—41no—
57C22×C3×D9C2312yes—
58C2×C3×C4×C9—41no—
59C2×C3×C4×C9C2311yescharacteristic
60C2×C3×C4×C9C2311yescharacteristic
61C2×C3⋊Dic9C2312yes—
62C2×C3⋊Dic9C2312yes—
63#6226—41no—
64C9×#67—24no—
65C2×C9×Dic3C2312yes—
66C2×C3×Dic9—41no—
67C2×C3×Dic9C2312yes—
68C9×D12—41no—
69#6194—42no—
70#6194—42no—
71D3×Dic9—42no—
72#5617—91no—
73#5494—91no—
74C23×C3×D3—31no—
75C22×C32×C4C2×D311yescharacteristic
76C2×#634—31no—
77C2×C3×#67C2×D312yes—
78C22×C3×Dic3—31no—
79C2×C3×D12C2×D311yescharacteristic
80C2×#625—32no—
81C2×#625—32no—
82C2×D3×Dic3—32no—
83C23×D9C2×D311yescharacteristic
84#3757—32no—
85C4⋊(C3⋊Dic3)—31no—
86#3750—32no—
87C3×(C2×D3)⋊C4—32no—
88C3×(C2×D3)⋊C4—32no—
89C3×(C2×D3)⋊C4C2×D312yes—
90C3×C4⋊Dic3C2×D311yescharacteristic
91Dic3⋊Dic3—32no—
92(C2×D3)⋊Dic3—32no—
93(C2×D3)⋊Dic3—32no—
94(C2×D3)⋊Dic3—32no—
95Dic32—32no—
96C2×D4×C9—32no—
97C2×D4×C9—31no—
98C22×C4×C9—21no—
99C22×C4×C9C2×D311yescharacteristic
100C2×#611—32no—
101C2×#611—32no—
102C22×Dic9—32no—
103C22×Dic9C2×D311yescharacteristic
104C2×Dic18—31no—
105C2×Dic18—61no—
106C9×C4⋊C4—31no—
107C9×C22⋊C4—32no—
108#3676—32no—
109#3676—32no—
110#3676—62no—
111(C2×D9)⋊C4—32no—
112(C2×D9)⋊C4—32no—
113(C2×D9)⋊C4C2×D312yes—
114(C2×D9)⋊C4C2×D312yes—
115C4⋊Dic9—31no—
116C4⋊Dic9—32no—
117C4⋊Dic9—61no—
118Dic9⋊C4—32no—
119Dic9⋊C4—62no—
120C4×Dic9—32no—
121C22×C3×C9—21no—
122C22×C3×C9—21no—
123C22×C3×C9#4112yes—
124C22×C3×C9C2411yescharacteristic
125C2×D3×C9—22no—
126C2×D3×C9—22no—
127C2×D3×C9—22no—
128C2×C3×D9—22no—
129C2×C3×D9—22no—
130C2×C3×D9—22no—
131C2×C3×D9—44no—
132C3×C4×C9—21no—
133C3×C4×C9—21no—
134C3⋊Dic9—22no—
135C3⋊Dic9—22no—
136C9×Dic3—22no—
137C3×Dic9—22no—
138C3×#121—31no—
139C2×C3×C22⋊C4—91no—
140#953—92no—
141#939—92no—
142#937—92no—
143D4×Dic3—92no—
144#930D911yescharacteristic, complemented
145C4×#67—92no—
146C2×(C2×D3)⋊C4—31no—
147#924—91no—
148#924—91no—
149#924—181no—
150#898—92no—
151#886—92no—
152#885—92no—
153#885—92no—
154#884—92no—
155#883—92no—
156#879—92no—
157C23×C32C22×D311yescharacteristic
158C22×C3×D3—32no—
159C22×C3×D3—62no—
160C22×C3×D3—121no—
161C22×C3×D3—121no—
162C22×C3×D3C22×D312yes—
163C2×C32×C4—41no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C3⋊Dic3—32no—
167C2×C3⋊Dic3—32no—
168#634—121no—
169C3×#67—24no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—121no—
172C2×C3×Dic3C22×D312yes—
173C3×D12—41no—
174#625—122no—
175#625—122no—
176D3×Dic3—122no—
177C23×C9—21no—
178C23×C9—32no—
179C23×C9C22×D311yescharacteristic
180C22×D9—22no—
181C22×D9—41no—
182C22×D9—41no—
183C22×D9C22×D312yes—
184D4×C9—64no—
185D4×C9—121no—
186C2×C4×C9—21no—
187C2×C4×C9—21no—
188C2×C4×C9—32no—
189C2×C4×C9—41no—
190C2×C4×C9—42no—
191C2×C4×C9C22×D311yescharacteristic
192C2×C4×C9C22×D311yescharacteristic
193#611—122no—
194#611—122no—
195C2×Dic9—32no—
196C2×Dic9—32no—
197C2×Dic9—41no—
198C2×Dic9—62no—
199C2×Dic9—62no—
200C2×Dic9—122no—
201C2×Dic9C22×D312yes—
202Dic18—121no—
203Dic18—122no—
204#538—271noSylow
205C2×C3×C9—22no—
206C2×C3×C9C2×#4111yescharacteristic
207C2×C3×C9H(5,𝔽2)11yescharacteristic
208C2×C3×C9H(5,𝔽2)11yescharacteristic
209D3×C9—42no—
210C3×D9—42no—
211C24×C3—91no—
212C23×D3—31no—
213C2×C3×D4—32no—
214C2×C3×D4—31no—
215C22×C3×C4—21no—
216C22×C3×C4—91no—
217C22×C3×C4C2×D911yescharacteristic
218C22×C3×C4D3211yescharacteristic
219C2×#67—92no—
220C2×#67—92no—
221C2×#67—92no—
222C2×#67—92no—
223C2×#67C2×D912yes—
224C22×Dic3—31no—
225C22×Dic3—92no—
226C2×D12C2×D911yescharacteristic, complemented
227C2×C4×D3—92no—
228C2×Dic6—91no—
229C2×Dic6—91no—
230C2×Dic6—181no—
231C3×C4⋊C4—31no—
232C3×C22⋊C4—32no—
233C3×C22⋊C4—92no—
234C3×C22⋊C4—92no—
235#217—92no—
236#217—92no—
237#217—92no—
238#217—92no—
239#217—92no—
240#217—182no—
241(C2×D3)⋊C4—32no—
242(C2×D3)⋊C4—32no—
243(C2×D3)⋊C4—92no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4C2×D912yescomplemented
247C4⋊Dic3—92no—
248C4⋊Dic3—91no—
249C4⋊Dic3—91no—
250C4⋊Dic3—181no—
251C4⋊Dic3C2×D911yescharacteristic, complemented
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—182no—
256C4×Dic3—92no—
257C4×Dic3—92no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32#23712yes—
261C22×C32C23×D311yescharacteristic
262C2×C3×D3—22no—
263C2×C3×D3—22no—
264C2×C3×D3—22no—
265C2×C3×D3—62no—
266C2×C3×D3—62no—
267C2×C3×D3—62no—
268C2×C3×D3—124no—
269C32×C4—21no—
270C32×C4—21no—
271C3⋊Dic3—62no—
272C3⋊Dic3—62no—
273C3×Dic3—22no—
274C3×Dic3—62no—
275C22×C9—21no—
276C22×C9—21no—
277C22×C9—21no—
278C22×C9—22no—
279C22×C9—41no—
280C22×C9—41no—
281C22×C9—62no—
282C22×C9—62no—
283C22×C9—62no—
284C22×C9#23512yes—
285C22×C9C23×D311yescharacteristic
286C2×D9—22no—
287C2×D9—22no—
288C2×D9—22no—
289C2×D9—44no—
290C4×C9—22no—
291C4×C9—21no—
292C4×C9—21no—
293C4×C9—41no—
294C4×C9—62no—
295Dic9—22no—
296Dic9—62no—
297Dic9—62no—
298Dic9—122no—
299Dic9—122no—
300#126—272no—
301#124—272no—
302#123—272no—
303C22⋊Q8—271no—
304#121—31no—
305#121—272no—
306#120—272no—
307C4×D4—272no—
308C2×C22⋊C4—91no—
309C3×C9#53811yescharacteristic, complemented, Sylow
310C23×C3—21no—
311C23×C3—32no—
312C23×C3—92no—
313C23×C3—182no—
314C23×C3—361no—
315C23×C3—361no—
316C23×C3C22×D911yescharacteristic
317C23×C3C2×D3211yescharacteristic
318C22×D3—32no—
319C22×D3—62no—
320C22×D3—121no—
321C22×D3—121no—
322C22×D3C22×D912yes—
323C3×D4—64no—
324C3×D4—121no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—32no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—92no—
332C2×C3×C4—361no—
333C2×C3×C4C22×D911yescharacteristic
334C2×C3×C4C22×D911yescharacteristic
335C2×C3×C4C2×D3211yescharacteristic
336C2×C3×C4C2×D3211yescharacteristic
337#67—24no—
338#67—362no—
339#67—362no—
340#67—362no—
341#67—362no—
342C2×Dic3—32no—
343C2×Dic3—92no—
344C2×Dic3—92no—
345C2×Dic3—92no—
346C2×Dic3—92no—
347C2×Dic3—121no—
348C2×Dic3—182no—
349C2×Dic3—182no—
350C2×Dic3—362no—
351C2×Dic3C22×D912yes—
352D12—41no—
353C4×D3—362no—
354Dic6—361no—
355Dic6—361no—
356Dic6—362no—
357C2×C32—22no—
358C2×C32C2×#23711yescharacteristic
359C2×C32#100411yescharacteristic
360C2×C32#100911yescharacteristic
361C3×D3—42no—
362C3×D3—122no—
363C2×C9—22no—
364C2×C9—21no—
365C2×C9—21no—
366C2×C9—21no—
367C2×C9—42no—
368C2×C9—122no—
369C2×C9C2×#23511yescharacteristic
370C2×C9#100411yescharacteristic
371C2×C9#100411yescharacteristic
372D9—42no—
373C24—91no—
374C2×Q8—271no—
375C2×D4—32no—
376C2×D4—31no—
377C2×D4—272no—
378C2×D4—272no—
379C22×C4—91no—
380C22×C4—272no—
381C22×C4D3×D911yescharacteristic
382C4⋊C4—31no—
383C4⋊C4—272no—
384C4⋊C4—272no—
385C4⋊C4—271no—
386C22⋊C4—32no—
387C22⋊C4—92no—
388C22⋊C4—92no—
389C22⋊C4—272no—
390C22⋊C4—272no—
391C22⋊C4—272no—
392C22⋊C4—272no—
393C42—272no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—22no—
399C22×C3—21no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—62no—
403C22×C3—62no—
404C22×C3—62no—
405C22×C3—182no—
406C22×C3—182no—
407C22×C3—182no—
408C22×C3—364no—
409C22×C3#369912yes—
410C22×C3C23×D911yescharacteristic
411C22×C3#379512yes—
412C22×C3C22×D3211yescharacteristic
413C2×D3—22no—
414C2×D3—22no—
415C2×D3—22no—
416C2×D3—62no—
417C2×D3—62no—
418C2×D3—62no—
419C2×D3—124no—
420C3×C4—22no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—41no—
426C3×C4—62no—
427C3×C4—182no—
428Dic3—22no—
429Dic3—62no—
430Dic3—182no—
431Dic3—182no—
432Dic3—182no—
433Dic3—182no—
434Dic3—362no—
435Dic3—362no—
436C32#561711yescharacteristic
437C9—21no—
438C9#549411yescharacteristic, complemented
439C23—32no—
440C23—92no—
441C23—182no—
442C23—361no—
443C23—361no—
444C23C2×D3×D911yescharacteristic
445Q8—1081no—
446D4—64no—
447D4—121no—
448D4—1082no—
449D4—1082no—
450C2×C4—32no—
451C2×C4—41no—
452C2×C4—92no—
453C2×C4—272no—
454C2×C4—272no—
455C2×C4—361no—
456C2×C4—1082no—
457C2×C4C2×D3×D911yescharacteristic
458C2×C4C2×D3×D911yescharacteristic
459C2×C3—22no—
460C2×C3—22no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—42no—
465C2×C3—122no—
466C2×C3—362no—
467C2×C3C2×#369911yescharacteristic
468C2×C3#6371011yescharacteristic
469C2×C3#6371511yescharacteristic
470C2×C3C2×#379511yescharacteristic
471C2×C3#6430711yescharacteristic
472C2×C3#6433011yescharacteristic
473D3—42no—
474D3—122no—
475C22—21no—
476C22—21no—
477C22—62no—
478C22—62no—
479C22—62no—
480C22—182no—
481C22—182no—
482C22—182no—
483C22—364no—
484C22#8863812yes—
485C22C22×D3×D911yescharacteristic
486C4—21no—
487C4—21no—
488C4—62no—
489C4—182no—
490C4—542no—
491C4—542no—
492C3—21no—
493C3#1058971911yescharacteristic, complemented
494C3#1059497711yescharacteristic
495C2—22no—
496C2—122no—
497C2—362no—
498C2C2×#8863811yescharacteristic
499C2#1172247711yescharacteristic
500C2#1172253511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#538
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
2. C9×#930C2semidirect
3. C2×C3×(C2×D9)⋊C4C2semidirect
4. #11721285C2semidirect
5. #67×Dic9C2semidirect
6. #11721042C2semidirect
7. #11721021C2semidirect
8. #11721016C2semidirect
9. #11720946C2semidirect
14. C2×#6226C22semidirect
17. C2×C9×D12C22semidirect
20. C2×D3×Dic9C22semidirect
22. C4⋊(C3⋊Dic9)C22semidirect
23. #88534C22semidirect
24. C9×(C2×D3)⋊C4C22semidirect
27. C9×C4⋊Dic3C22semidirect
28. (C2×D3)⋊Dic9C22semidirect
31. Dic3⋊Dic9C22semidirect
32. Dic3×Dic9C22semidirect
46. C2×(C2×D9)⋊C4D3semidirect
144. #930D9semidirect
226. C2×D12C2×D9semidirect
246. (C2×D3)⋊C4C2×D9semidirect
251. C4⋊Dic3C2×D9semidirect
309. C3×C9#538semidirect
438. C9#5494semidirect
493. C3#10589719semidirect

Maximal subgroups

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

subgroupindex
#117218132
C9×#9302
C2×C3×(C2×D9)⋊C42
#117212852
#67×Dic92
#117210422
#117210212
#117210162
#117209462
#105949773
#105897193