Finite Group Databasebeta

← #13151089

Subgroups — #13151089

Order 1728.

864 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721840C211yesmaximal, characteristic, complemented
2C3×#63499C211yesmaximal, characteristic, complemented
3C9×#935C211yesmaximal, characteristic, complemented
4#11721277C212yesmaximal, complemented
5#11721276C212yesmaximal, complemented
6#11721275C212yesmaximal, complemented
7#11721256C212yesmaximal, complemented
8#11721005C212yesmaximal, complemented
9#11720950C212yesmaximal, complemented
10#10595261—31nomaximal
11#10589768—31nomaximal
12C23×C3×D9C2211yescharacteristic
13C2×C3×D4×C9C2211yescharacteristic
14C2×#6233C2212yes—
15C2×#6233C2212yescomplemented
16C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
17C2×C3×#611C2212yes—
18C22×C9×Dic3C2211yescharacteristic, complemented
19C2×#6192C2212yes—
20C2×#6192C2212yes—
21C2×(C3⋊D9)⋊C4C2212yescomplemented
22C4×C3⋊Dic9C2211yescharacteristic, complemented
23C9×#217C2212yes—
24C9×#217C2211yescharacteristic, complemented
25C3×#3676C2211yescharacteristic
26C3×(C2×D9)⋊C4C2212yes—
27C9×Dic3⋊C4C2212yescomplemented
28#88445C2212yescomplemented
29(C2×D9)⋊Dic3C2212yes—
30(C2×D9)⋊Dic3C2212yes—
31Dic3⋊Dic9C2212yescomplemented
32Dic9⋊Dic3C2212yescomplemented
33#64149—31no—
34C3×#937—31no—
35C3×#935D311yescharacteristic
36#63978—32no—
37#63977—32no—
38#63975—32no—
39#63958—32no—
40#63910—32no—
41#63862—32no—
42C9×#123—31no—
43#63502—31no—
44#63502—61no—
45#63499D311yescharacteristic, complemented
46#63492—32no—
47#63460—32no—
48#63457—32no—
49#63446—32no—
50#63445—32no—
51#63444—32no—
52C23×C3×C9C2311yescharacteristic
53C23×C3×C9C2311yescharacteristic
54C22×C3⋊D9C2312yes—
55C22×C3×D9—41no—
56C22×C3×D9—42no—
57C22×C3×D9C2312yes—
58C3×D4×C9—41no—
59C2×C3×C4×C9C2311yescharacteristic
60#6233—42no—
61#6233—42no—
62C2×C3⋊Dic9C2312yes—
63C4⋊(C3⋊D9)—41no—
64C3×#611—42no—
65C2×C9×Dic3—41no—
66C2×C9×Dic3C2312yes—
67C2×C9×Dic3C2312yes—
68C2×C3×Dic9C2312yes—
69#6192—42no—
70#6192—42no—
71(C3⋊D9)⋊C4—42no—
72#5666—91no—
73#5595—91no—
74C23×C3×D3—31no—
75C2×C32×D4C2×D311yescharacteristic
76C2×#638—32no—
77C2×#638—32no—
78C2×C4⋊(C3⋊D3)—31no—
79C2×C3×#67—32no—
80C22×C3×Dic3C2×D311yescharacteristic
81C2×#626—32no—
82C2×#626—32no—
83C2×#624—32no—
84C23×D9C2×D311yescharacteristic
85C4×C3⋊Dic3—31no—
86C3×#217—31no—
87C3×#217C2×D312yes—
88C3×#217C2×D311yescharacteristic
89C3×(C2×D3)⋊C4—32no—
90C3×Dic3⋊C4C2×D312yes—
91Dic3⋊Dic3—32no—
92Dic3⋊Dic3—32no—
93#3722—32no—
94(C2×D3)⋊Dic3—32no—
95(C2×D3)⋊Dic3—32no—
96C2×D4×C9—21no—
97C2×D4×C9C2×D311yescharacteristic
98C22×C4×C9—31no—
99C2×#611—32no—
100C2×#611—32no—
101C2×#611—62no—
102C2×#611—62no—
103C2×#611C2×D312yes—
104C2×D36—32no—
105C2×D36—32no—
106C2×D36—31no—
107C2×D36—61no—
108C2×C4×D9—32no—
109C9×C4⋊C4—32no—
110C9×C22⋊C4—32no—
111C9×C22⋊C4—31no—
112#3676C2×D311yescharacteristic
113(C2×D9)⋊C4—32no—
114(C2×D9)⋊C4—32no—
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4C2×D312yes—
117C4⋊Dic9—32no—
118Dic9⋊C4—32no—
119C4×Dic9—31no—
120C4×Dic9—61no—
121C22×C3×C9—21no—
122C22×C3×C9—22no—
123C22×C3×C9—22no—
124C22×C3×C9—21no—
125C22×C3×C9C2411yescharacteristic
126C2×C3⋊D9—22no—
127C2×C3⋊D9—22no—
128C2×C3⋊D9—22no—
129C2×C3×D9—22no—
130C2×C3×D9—22no—
131C2×C3×D9—22no—
132C2×C3×D9—44no—
133C3×C4×C9—21no—
134C3⋊Dic9—22no—
135C9×Dic3—22no—
136C9×Dic3—22no—
137C3×Dic9—22no—
138C3×#123—31no—
139C3×#120—91no—
140#953—92no—
141#940—91no—
142#940—91no—
143#940—181no—
144#939—92no—
145#938—92no—
146#937—31no—
147#935D911yescharacteristic, complemented
148#930—92no—
149#898—92no—
150#895—92no—
151#884—92no—
152#884—92no—
153#883—92no—
154#883—92no—
155#882—92no—
156#879—92no—
157C23×C32C22×D311yescharacteristic
158C23×C32C22×D311yescharacteristic
159C22×C3⋊D3—32no—
160C22×C3×D3—32no—
161C22×C3×D3—121no—
162C22×C3×D3—122no—
163C32×D4—41no—
164C2×C32×C4C22×D311yescharacteristic
165#638—122no—
166#638—122no—
167C2×C3⋊Dic3—32no—
168C4⋊(C3⋊D3)—121no—
169C3×#67—122no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—41no—
172C2×C3×Dic3C22×D312yes—
173C2×C3×Dic3C22×D312yes—
174#626—122no—
175#626—122no—
176#624—122no—
177C23×C9—21no—
178C23×C9—21no—
179C23×C9C22×D311yescharacteristic
180C23×C9C22×D311yescharacteristic
181C22×D9—32no—
182C22×D9—41no—
183C22×D9—42no—
184C22×D9—62no—
185C22×D9C22×D312yes—
186D4×C9—41no—
187D4×C9—42no—
188C2×C4×C9—21no—
189C2×C4×C9—32no—
190C2×C4×C9—32no—
191C2×C4×C9—121no—
192C2×C4×C9C22×D311yescharacteristic
193#611—42no—
194#611—122no—
195#611—122no—
196#611—124no—
197#611—124no—
198C2×Dic9—32no—
199C2×Dic9—62no—
200C2×Dic9C22×D312yes—
201D36—121no—
202D36—122no—
203D36—122no—
204D36—122no—
205C4×D9—122no—
206#560—271noSylow
207C2×C3×C9—41no—
208C2×C3×C9—41no—
209C2×C3×C9H(5,𝔽2)11yescharacteristic
210C2×C3×C9H(5,𝔽2)12yes—
211C3⋊D9—42no—
212C3×D9—42no—
213C24×C3—91no—
214C23×D3—31no—
215C2×C3×D4—21no—
216C2×C3×D4—92no—
217C2×C3×D4C2×D911yescharacteristic
218C2×C3×D4D3211yescharacteristic
219C22×C3×C4—31no—
220C2×#67—32no—
221C2×#67—92no—
222C2×#67—92no—
223C2×#67—92no—
224C2×#67—92no—
225C2×#67—92no—
226C2×#67—92no—
227C2×#67—182no—
228C2×#67—182no—
229C22×Dic3C2×D911yescharacteristic, complemented
230C2×D12—92no—
231C2×D12—92no—
232C2×D12—91no—
233C2×D12—91no—
234C2×D12—181no—
235C2×C4×D3—92no—
236C2×C4×D3—92no—
237C3×C4⋊C4—32no—
238C3×C22⋊C4—32no—
239C3×C22⋊C4—31no—
240C3×C22⋊C4—91no—
241C3×C22⋊C4—92no—
242#217—31no—
243#217—92no—
244#217—92no—
245#217C2×D912yes—
246#217C2×D911yescharacteristic, complemented
247(C2×D3)⋊C4—32no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252C4⋊Dic3—92no—
253C4⋊Dic3—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4C2×D912yescomplemented
257C4×Dic3—91no—
258C4×Dic3—91no—
259C4×Dic3—181no—
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32C23×D311yescharacteristic
265C2×C3⋊D3—62no—
266C2×C3⋊D3—62no—
267C2×C3⋊D3—62no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—62no—
271C2×C3×D3—124no—
272C32×C4—21no—
273C3⋊Dic3—62no—
274C3×Dic3—22no—
275C3×Dic3—22no—
276C3×Dic3—62no—
277C22×C9—22no—
278C22×C9—22no—
279C22×C9—21no—
280C22×C9—21no—
281C22×C9—21no—
282C22×C9—42no—
283C22×C9—42no—
284C22×C9—41no—
285C22×C9—41no—
286C22×C9C23×D311yescharacteristic
287C2×D9—22no—
288C2×D9—22no—
289C2×D9—22no—
290C2×D9—44no—
291C2×D9—62no—
292C2×D9—62no—
293C2×D9—62no—
294C2×D9—122no—
295C2×D9—122no—
296C2×D9—122no—
297C4×C9—22no—
298C4×C9—21no—
299C4×C9—62no—
300C4×C9—62no—
301Dic9—22no—
302Dic9—64no—
303Dic9—62no—
304#127—271no—
305#126—272no—
306#123—31no—
307#123—272no—
308#121—272no—
309#121—272no—
310#121—272no—
311#120—91no—
312#120—272no—
313C3×C9#56011yescharacteristic, complemented, Sylow
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—92no—
317C23×C3—361no—
318C23×C3—362no—
319C23×C3C22×D911yescharacteristic
320C23×C3C22×D911yescharacteristic
321C23×C3C2×D3211yescharacteristic
322C23×C3C2×D3211yescharacteristic
323C22×D3—32no—
324C22×D3—92no—
325C22×D3—92no—
326C22×D3—122no—
327C22×D3—121no—
328C22×D3—182no—
329C3×D4—41no—
330C3×D4—41no—
331C3×D4—42no—
332C3×D4—362no—
333C2×C3×C4—21no—
334C2×C3×C4—32no—
335C2×C3×C4—32no—
336C2×C3×C4—92no—
337C2×C3×C4—121no—
338C2×C3×C4C22×D911yescharacteristic
339C2×C3×C4C2×D3211yescharacteristic
340#67—122no—
341#67—362no—
342#67—362no—
343#67—362no—
344#67—362no—
345#67—362no—
346#67—362no—
347#67—364no—
348#67—364no—
349C2×Dic3—32no—
350C2×Dic3—41no—
351C2×Dic3—92no—
352C2×Dic3—92no—
353C2×Dic3—182no—
354C2×Dic3C22×D912yes—
355C2×Dic3C22×D912yes—
356D12—361no—
357D12—361no—
358D12—362no—
359D12—362no—
360D12—362no—
361C4×D3—362no—
362C4×D3—362no—
363C2×C32—41no—
364C2×C32—41no—
365C2×C32#100411yescharacteristic
366C2×C32#100912yes—
367C3⋊D3—122no—
368C3×D3—122no—
369C2×C9—22no—
370C2×C9—21no—
371C2×C9—41no—
372C2×C9—41no—
373C2×C9—81no—
374C2×C9—81no—
375C2×C9#100412yes—
376C2×C9#100411yescharacteristic
377D9—42no—
378D9—122no—
379D9—242no—
380C24—91no—
381C2×D4—92no—
382C2×D4—272no—
383C2×D4—272no—
384C2×D4—272no—
385C2×D4—272no—
386C2×D4—271no—
387C2×D4D3×D911yescharacteristic
388C22×C4—31no—
389C22×C4—272no—
390C4⋊C4—32no—
391C4⋊C4—272no—
392C4⋊C4—272no—
393C22⋊C4—32no—
394C22⋊C4—31no—
395C22⋊C4—91no—
396C22⋊C4—92no—
397C22⋊C4—272no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C42—271no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—21no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3—182no—
415C22×C3—182no—
416C22×C3—182no—
417C22×C3—364no—
418C22×C3C23×D911yescharacteristic
419C22×C3C22×D3211yescharacteristic
420C2×D3—62no—
421C2×D3—62no—
422C2×D3—62no—
423C2×D3—124no—
424C2×D3—182no—
425C2×D3—182no—
426C2×D3—182no—
427C2×D3—182no—
428C2×D3—182no—
429C2×D3—182no—
430C2×D3—362no—
431C2×D3—362no—
432C2×D3—362no—
433C3×C4—22no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—62no—
437C3×C4—62no—
438C3×C4—182no—
439Dic3—22no—
440Dic3—22no—
441Dic3—62no—
442Dic3—184no—
443Dic3—182no—
444Dic3—182no—
445C32#566611yescharacteristic
446C9—21no—
447C9#559511yescharacteristic, complemented
448C23—92no—
449C23—272no—
450C23—362no—
451C23—361no—
452C23C2×D3×D911yescharacteristic
453C23C2×D3×D911yescharacteristic
454D4—41no—
455D4—362no—
456D4—1081no—
457D4—1082no—
458D4—1082no—
459D4—1082no—
460D4—1082no—
461C2×C4—32no—
462C2×C4—32no—
463C2×C4—92no—
464C2×C4—121no—
465C2×C4—272no—
466C2×C4—1082no—
467C2×C4C2×D3×D911yescharacteristic
468C2×C3—22no—
469C2×C3—21no—
470C2×C3—41no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3—41no—
474C2×C3—81no—
475C2×C3—81no—
476C2×C3—362no—
477C2×C3#6371011yescharacteristic
478C2×C3#6371512yes—
479C2×C3#6431412yes—
480C2×C3#6433011yescharacteristic
481D3—122no—
482D3—362no—
483D3—362no—
484D3—722no—
485C22—22no—
486C22—22no—
487C22—21no—
488C22—21no—
489C22—182no—
490C22—182no—
491C22—182no—
492C22—364no—
493C22—542no—
494C22—542no—
495C22—542no—
496C22C22×D3×D911yescharacteristic
497C4—21no—
498C4—62no—
499C4—62no—
500C4—182no—
501C4—542no—
502C3—21no—
503C3#1058976811yescharacteristic, complemented
504C3#1059526111yescharacteristic
505C2—41no—
506C2—41no—
507C2—362no—
508C2—1082no—
509C2#1172250112yes—
510C2#1172253511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#560
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11721840C2semidirect
2. C3×#63499C2semidirect
3. C9×#935C2semidirect
4. #11721277C2semidirect
5. #11721276C2semidirect
6. #11721275C2semidirect
7. #11721256C2semidirect
8. #11721005C2semidirect
9. #11720950C2semidirect
15. C2×#6233C22semidirect
16. C2×C4⋊(C3⋊D9)C22semidirect
18. C22×C9×Dic3C22semidirect
21. C2×(C3⋊D9)⋊C4C22semidirect
22. C4×C3⋊Dic9C22semidirect
24. C9×#217C22semidirect
27. C9×Dic3⋊C4C22semidirect
28. #88445C22semidirect
31. Dic3⋊Dic9C22semidirect
32. Dic9⋊Dic3C22semidirect
45. #63499D3semidirect
147. #935D9semidirect
229. C22×Dic3C2×D9semidirect
246. #217C2×D9semidirect
256. Dic3⋊C4C2×D9semidirect
313. C3×C9#560semidirect
447. C9#5595semidirect
503. C3#10589768semidirect

Maximal subgroups

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

subgroupindex
#117218402
C3×#634992
C9×#9352
#117212772
#117212762
#117212752
#117212562
#117210052
#117209502
#105952613
#105897683