Finite Group Databasebeta

← #13151094

Subgroups — #13151094

Order 1728.

886 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721837C211yesmaximal, characteristic, complemented
2C3×#63502C211yesmaximal, characteristic, complemented
3C9×#935C211yesmaximal, characteristic, complemented
4#11721278C212yesmaximal, complemented
5#11721277C212yesmaximal, complemented
6#11721271C212yesmaximal, complemented
7#11721269C212yesmaximal, complemented
8#11720983C212yesmaximal, complemented
9#11720934C212yesmaximal, complemented
10#10595266—31nomaximal
11#10589768—31nomaximal
12C23×C3⋊D9C2211yescharacteristic
13C2×C3×D4×C9C2211yescharacteristic
14C2×#6233C2212yes—
15C2×C3×#611C2212yes—
16C2×C3×#611C2212yescomplemented
17C22×C9×Dic3C2211yescharacteristic, complemented
18C2×C3×D36C2211yescharacteristic, complemented
19C2×#6192C2212yes—
20C2×#6192C2212yes—
21C2×Dic3×D9C2212yescomplemented
22#88551C2211yescharacteristic
23#88536C2212yes—
24C9×#217C2212yes—
25C9×#217C2211yescharacteristic, complemented
26C9×Dic3⋊C4C2212yescomplemented
27C3×C4×Dic9C2211yescharacteristic, complemented
28#88445C2212yes—
29#88445C2212yes—
30(C2×D9)⋊Dic3C2212yescomplemented
31Dic3⋊Dic9C2212yescomplemented
32#88442C2212yescomplemented
33#64146—31no—
34C3×#940—31no—
35C3×#935D311yescharacteristic
36#63981—32no—
37#63978—32no—
38#63971—32no—
39#63969—32no—
40#63886—32no—
41#63868—32no—
42C9×#123—31no—
43#63502D311yescharacteristic, complemented
44#63499—31no—
45#63499—61no—
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⋊D9—42no—
55C22×C3⋊D9—41no—
56C22×C3⋊D9C2312yes—
57C22×C3×D9C2312yes—
58C3×D4×C9—41no—
59C2×C3×C4×C9C2311yescharacteristic
60#6233—42no—
61C2×C3⋊Dic9C2312yes—
62C3×#611—42no—
63C3×#611—42no—
64C2×C9×Dic3—41no—
65C2×C9×Dic3C2312yes—
66C2×C9×Dic3C2312yes—
67C2×C3×Dic9C2312yes—
68C3×D36—41no—
69#6192—42no—
70#6192—42no—
71Dic3×D9—42no—
72#5720—91no—
73#5666—91no—
74C23×C3⋊D3—31no—
75C2×C32×D4C2×D311yescharacteristic
76C2×#638—32no—
77C2×C3×#67—32no—
78C2×C3×#67—32no—
79C22×C3×Dic3C2×D311yescharacteristic
80C2×C3×D12—31no—
81C2×#626—32no—
82C2×#626—32no—
83C2×D3×Dic3—32no—
84C23×D9—31no—
85C23×D9—61no—
86#3757—31no—
87#3752—32no—
88C3×#217C2×D312yes—
89C3×#217C2×D311yescharacteristic
90C3×Dic3⋊C4C2×D312yes—
91C3×C4×Dic3—31no—
92#3724—32no—
93Dic3⋊Dic3—32no—
94#3722—32no—
95#3722—32no—
96(C2×D3)⋊Dic3—32no—
97C2×D4×C9—21no—
98C2×D4×C9C2×D311yescharacteristic
99C22×C4×C9—31no—
100C2×#611—32no—
101C2×#611—62no—
102C2×#611C2×D312yes—
103C2×#611C2×D312yescomplemented
104C2×D36—32no—
105C2×D36—32no—
106C2×D36C2×D311yescharacteristic, complemented
107C2×C4×D9—32no—
108C9×C4⋊C4—32no—
109C9×C22⋊C4—32no—
110C9×C22⋊C4—31no—
111#3676—31no—
112#3676—61no—
113(C2×D9)⋊C4—32no—
114(C2×D9)⋊C4—32no—
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4—62no—
118C4⋊Dic9—32no—
119Dic9⋊C4—32no—
120C4×Dic9C2×D311yescharacteristic, complemented
121C22×C3×C9—21no—
122C22×C3×C9—21no—
123C22×C3×C9—22no—
124C22×C3×C9—22no—
125C22×C3×C9C2411yescharacteristic
126C2×C3⋊D9—22no—
127C2×C3⋊D9—22no—
128C2×C3⋊D9—22no—
129C2×C3⋊D9—44no—
130C2×C3×D9—22no—
131C2×C3×D9—22no—
132C2×C3×D9—22no—
133C3×C4×C9—21no—
134C3⋊Dic9—22no—
135C9×Dic3—22no—
136C9×Dic3—22no—
137C3×Dic9—22no—
138C3×#127—91no—
139C3×#123—31no—
140#940—31no—
141#939—92no—
142#939—92no—
143#939—92no—
144#937—91no—
145#937—91no—
146#937—92no—
147#937—181no—
148#935—92no—
149#935D911yescharacteristic, complemented
150#930—92no—
151#898—92no—
152#895—92no—
153#884—92no—
154#883—92no—
155#882—92no—
156#876—92no—
157C23×C32C22×D311yescharacteristic
158C23×C32C22×D311yescharacteristic
159C22×C3⋊D3—32no—
160C22×C3⋊D3—122no—
161C22×C3⋊D3—121no—
162C22×C3×D3—32no—
163C32×D4—41no—
164C2×C32×C4C22×D311yescharacteristic
165#638—122no—
166C2×C3⋊Dic3—32no—
167C3×#67—122no—
168C3×#67—122no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—41no—
171C2×C3×Dic3C22×D312yes—
172C2×C3×Dic3C22×D312yes—
173C3×D12—121no—
174#626—122no—
175#626—122no—
176D3×Dic3—122no—
177C23×C9—21no—
178C23×C9—21no—
179C23×C9C22×D311yescharacteristic
180C23×C9C22×D311yescharacteristic
181C22×D9—32no—
182C22×D9—62no—
183C22×D9—121no—
184C22×D9—122no—
185C22×D9—124no—
186C22×D9—122no—
187C22×D9C22×D312yes—
188D4×C9—41no—
189D4×C9—42no—
190C2×C4×C9—21no—
191C2×C4×C9—32no—
192C2×C4×C9—32no—
193C2×C4×C9—121no—
194C2×C4×C9C22×D311yescharacteristic
195#611—42no—
196#611—42no—
197#611—122no—
198#611—124no—
199C2×Dic9—32no—
200C2×Dic9—62no—
201C2×Dic9C22×D312yes—
202D36—41no—
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—
213C23×D3—91no—
214C23×D3—91no—
215C23×D3—181no—
216C2×C3×D4—21no—
217C2×C3×D4—92no—
218C2×C3×D4—92no—
219C2×C3×D4—91no—
220C2×C3×D4C2×D911yescharacteristic
221C2×C3×D4D3211yescharacteristic
222C22×C3×C4—31no—
223C2×#67—32no—
224C2×#67—32no—
225C2×#67—92no—
226C2×#67—92no—
227C2×#67—92no—
228C2×#67—92no—
229C2×#67—182no—
230C22×Dic3—92no—
231C22×Dic3C2×D911yescharacteristic, complemented
232C2×D12—31no—
233C2×D12—92no—
234C2×D12—92no—
235C2×C4×D3—92no—
236C3×C4⋊C4—32no—
237C3×C22⋊C4—32no—
238C3×C22⋊C4—31no—
239C3×C42—91no—
240#217—91no—
241#217—91no—
242#217—92no—
243#217—181no—
244#217C2×D912yes—
245#217C2×D911yescharacteristic, complemented
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—182no—
254C4⋊Dic3—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4C2×D912yescomplemented
259C4×Dic3—31no—
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—124no—
269C2×C3×D3—62no—
270C2×C3×D3—62no—
271C2×C3×D3—62no—
272C32×C4—21no—
273C3⋊Dic3—62no—
274C3×Dic3—22no—
275C3×Dic3—22no—
276C3×Dic3—62no—
277C22×C9—22no—
278C22×C9—22no—
279C22×C9—22no—
280C22×C9—21no—
281C22×C9—21no—
282C22×C9—21no—
283C22×C9—42no—
284C22×C9—42no—
285C22×C9—41no—
286C22×C9C23×D311yescharacteristic
287C2×D9—22no—
288C2×D9—22no—
289C2×D9—22no—
290C2×D9—64no—
291C2×D9—62no—
292C2×D9—62no—
293C2×D9—62no—
294C2×D9—122no—
295C2×D9—122no—
296C2×D9—128no—
297C2×D9—124no—
298C4×C9—21no—
299C4×C9—41no—
300C4×C9—62no—
301C4×C9—62no—
302Dic9—22no—
303Dic9—62no—
304Dic9—122no—
305#127—91no—
306#126—272no—
307#123—31no—
308#123—272no—
309#121—272no—
310#121—272no—
311#121—272no—
312#120—272no—
313#120—271no—
314C3×C9#56011yescharacteristic, complemented, Sylow
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—92no—
318C23×C3C22×D911yescharacteristic
319C23×C3C22×D911yescharacteristic
320C23×C3C2×D3211yescharacteristic
321C23×C3C2×D3211yescharacteristic
322C22×D3—32no—
323C22×D3—92no—
324C22×D3—92no—
325C22×D3—182no—
326C22×D3—362no—
327C22×D3—361no—
328C22×D3—361no—
329C22×D3—362no—
330C22×D3—364no—
331C22×D3—362no—
332C3×D4—41no—
333C3×D4—41no—
334C3×D4—42no—
335C3×D4—361no—
336C3×D4—362no—
337C3×D4—362no—
338C2×C3×C4—21no—
339C2×C3×C4—32no—
340C2×C3×C4—32no—
341C2×C3×C4—92no—
342C2×C3×C4—121no—
343C2×C3×C4C22×D911yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345#67—122no—
346#67—122no—
347#67—362no—
348#67—362no—
349#67—362no—
350#67—362no—
351#67—364no—
352C2×Dic3—32no—
353C2×Dic3—41no—
354C2×Dic3—92no—
355C2×Dic3—92no—
356C2×Dic3—182no—
357C2×Dic3—362no—
358C2×Dic3C22×D912yes—
359C2×Dic3C22×D912yes—
360D12—121no—
361D12—362no—
362D12—362no—
363C4×D3—362no—
364C2×C32—41no—
365C2×C32—41no—
366C2×C32#100411yescharacteristic
367C2×C32#100912yes—
368C3⋊D3—122no—
369C3×D3—122no—
370C2×C9—22no—
371C2×C9—21no—
372C2×C9—42no—
373C2×C9—41no—
374C2×C9—41no—
375C2×C9—81no—
376C2×C9#100412yes—
377C2×C9#100411yescharacteristic
378D9—42no—
379D9—124no—
380D9—122no—
381C24—271no—
382C2×D4—92no—
383C2×D4—92no—
384C2×D4—91no—
385C2×D4—272no—
386C2×D4—272no—
387C2×D4—272no—
388C2×D4D3×D911yescharacteristic
389C22×C4—31no—
390C22×C4—272no—
391C4⋊C4—32no—
392C4⋊C4—272no—
393C4⋊C4—272no—
394C22⋊C4—32no—
395C22⋊C4—31no—
396C22⋊C4—272no—
397C22⋊C4—272no—
398C22⋊C4—272no—
399C22⋊C4—271no—
400C22⋊C4—272no—
401C42—91no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—21no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—41no—
415C22×C3—182no—
416C22×C3—182no—
417C22×C3—182no—
418C22×C3C23×D911yescharacteristic
419C22×C3C22×D3211yescharacteristic
420C2×D3—62no—
421C2×D3—62no—
422C2×D3—62no—
423C2×D3—184no—
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—364no—
433C2×D3—368no—
434C2×D3—364no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—41no—
438C3×C4—62no—
439C3×C4—62no—
440C3×C4—182no—
441Dic3—22no—
442Dic3—22no—
443Dic3—62no—
444Dic3—182no—
445Dic3—182no—
446Dic3—362no—
447C32#566611yescharacteristic
448C9—21no—
449C9#572011yescharacteristic, complemented
450C23—92no—
451C23—272no—
452C23—1082no—
453C23—1081no—
454C23C2×D3×D911yescharacteristic
455C23C2×D3×D911yescharacteristic
456D4—41no—
457D4—361no—
458D4—362no—
459D4—362no—
460D4—1082no—
461D4—1082no—
462D4—1082no—
463C2×C4—32no—
464C2×C4—32no—
465C2×C4—92no—
466C2×C4—121no—
467C2×C4—272no—
468C2×C4—1082no—
469C2×C4C2×D3×D911yescharacteristic
470C2×C3—22no—
471C2×C3—21no—
472C2×C3—42no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—41no—
476C2×C3—41no—
477C2×C3—81no—
478C2×C3—362no—
479C2×C3#6371011yescharacteristic
480C2×C3#6371512yes—
481C2×C3#6431412yes—
482C2×C3#6433011yescharacteristic
483D3—122no—
484D3—364no—
485D3—362no—
486D3—362no—
487C22—22no—
488C22—22no—
489C22—21no—
490C22—21no—
491C22—182no—
492C22—182no—
493C22—182no—
494C22—542no—
495C22—542no—
496C22—542no—
497C22—1084no—
498C22C22×D3×D911yescharacteristic
499C4—21no—
500C4—62no—
501C4—62no—
502C4—182no—
503C4—542no—
504C3—21no—
505C3#1058976811yescharacteristic, complemented
506C3#1059526611yescharacteristic
507C2—41no—
508C2—41no—
509C2—362no—
510C2—1082no—
511C2#1172250112yes—
512C2#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. #11721837C2semidirect
2. C3×#63502C2semidirect
3. C9×#935C2semidirect
4. #11721278C2semidirect
5. #11721277C2semidirect
6. #11721271C2semidirect
7. #11721269C2semidirect
8. #11720983C2semidirect
9. #11720934C2semidirect
16. C2×C3×#611C22semidirect
17. C22×C9×Dic3C22semidirect
18. C2×C3×D36C22semidirect
21. C2×Dic3×D9C22semidirect
25. C9×#217C22semidirect
26. C9×Dic3⋊C4C22semidirect
27. C3×C4×Dic9C22semidirect
30. (C2×D9)⋊Dic3C22semidirect
31. Dic3⋊Dic9C22semidirect
32. #88442C22semidirect
43. #63502D3semidirect
103. C2×#611C2×D3semidirect
106. C2×D36C2×D3semidirect
120. C4×Dic9C2×D3semidirect
149. #935D9semidirect
231. C22×Dic3C2×D9semidirect
245. #217C2×D9semidirect
258. Dic3⋊C4C2×D9semidirect
314. C3×C9#560semidirect
449. C9#5720semidirect
505. C3#10589768semidirect

Maximal subgroups

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

subgroupindex
#117218372
C3×#635022
C9×#9352
#117212782
#117212772
#117212712
#117212692
#117209832
#117209342
#105952663
#105897683