Finite Group Databasebeta

← #13150343

Subgroups — #13150343

Order 1728.

844 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721819C211yesmaximal, characteristic, complemented
2C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
3C9×#924C211yesmaximal, characteristic, complemented
4#11721276C212yesmaximal, complemented
5#11721241C212yesmaximal, complemented
6#11721005C212yesmaximal, complemented
7#11721003C212yesmaximal, complemented
8#11720949C212yesmaximal, complemented
9#11720942C212yesmaximal, complemented
10#10595038—31nomaximal
11#10589746—31nomaximal
12C23×C3×D9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C2×#6233C2212yes—
15C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
16C22×C3×Dic9C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic, complemented
18C2×#6192C2212yes—
19C2×#6192C2212yes—
20C2×(C3⋊D9)⋊C4C2212yescomplemented
21#88536C2212yescomplemented
22C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
23C9×#217C2212yes—
24C3×(C2×D9)⋊C4C2212yes—
25C3×(C2×D9)⋊C4C2212yes—
26C9×C4⋊Dic3C2211yescharacteristic, complemented
27C9×Dic3⋊C4C2212yescomplemented
28#88445C2212yescomplemented
29(C2×D9)⋊Dic3C2212yes—
30(C2×D9)⋊Dic3C2212yes—
31#88442C2212yescomplemented
32Dic3×Dic9C2212yescomplemented
33#64139—31no—
34C2×C3×(C2×D3)⋊C4—31no—
35C3×#924D311yescharacteristic
36#63977—32no—
37#63952—32no—
38#63910—32no—
39#63907—32no—
40#63861—32no—
41#63854—32no—
42C9×C22⋊Q8—31no—
43#63509—32no—
44#63492—31no—
45#63492—61no—
46C2×(C2×D9)⋊C4D311yescharacteristic, complemented
47#63460—32no—
48#63457—32no—
49#63456—32no—
50#63444—32no—
51#611⋊C4—32no—
52C23×C3×C9C2311yescharacteristic
53C22×C3⋊D9C2312yes—
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
61#6233—24no—
62C2×C3⋊Dic9C2312yes—
63C4⋊(C3⋊D9)—41no—
64C2×C9×Dic3C2312yes—
65C2×C9×Dic3C2312yes—
66C2×C3×Dic9—41no—
67C2×C3×Dic9C2312yes—
68C9×Dic6—41no—
69#6192—42no—
70#6192—42no—
71(C3⋊D9)⋊C4—42no—
72#5644—91no—
73#5494—91no—
74C23×C3×D3—31no—
75C22×C32×C4C2×D311yescharacteristic
76C2×#638—32no—
77C2×C4⋊(C3⋊D3)—31no—
78C22×C3×Dic3—31no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×#626—32no—
81C2×#626—32no—
82C2×#624—32no—
83C23×D9C2×D311yescharacteristic
84#3752—32no—
85C4⋊(C3⋊Dic3)—31no—
86C3×#217C2×D312yes—
87C3×(C2×D3)⋊C4—32no—
88C3×(C2×D3)⋊C4—32no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×Dic3⋊C4C2×D312yes—
91#3724—32no—
92#3722—32no—
93(C2×D3)⋊Dic3—32no—
94(C2×D3)⋊Dic3—32no—
95Dic32—32no—
96C2×Q8×C9—31no—
97C22×C4×C9—21no—
98C22×C4×C9C2×D311yescharacteristic
99C2×#611—32no—
100C2×#611—62no—
101C22×Dic9C2×D311yescharacteristic
102C2×D36—32no—
103C2×D36—32no—
104C2×D36—31no—
105C2×D36—61no—
106C2×C4×D9—32no—
107C9×C4⋊C4—32no—
108C9×C4⋊C4—31no—
109C9×C22⋊C4—32no—
110(C2×D9)⋊C4—32no—
111(C2×D9)⋊C4—32no—
112(C2×D9)⋊C4—32no—
113(C2×D9)⋊C4—32no—
114(C2×D9)⋊C4—62no—
115(C2×D9)⋊C4C2×D312yes—
116(C2×D9)⋊C4C2×D312yes—
117C4⋊Dic9—31no—
118C4⋊Dic9—61no—
119Dic9⋊C4—32no—
120C4×Dic9—32no—
121C22×C3×C9—21no—
122C22×C3×C9—21no—
123C22×C3×C9#4112yes—
124C22×C3×C9C2411yescharacteristic
125C2×C3⋊D9—22no—
126C2×C3⋊D9—22no—
127C2×C3⋊D9—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—
135C9×Dic3—22no—
136C9×Dic3—22no—
137C3×Dic9—22no—
138C3×C22⋊Q8—31no—
139C2×C3×C22⋊C4—91no—
140#953—92no—
141#947—92no—
142#930—91no—
143#930—91no—
144#930—181no—
145C4×#67—92no—
146C2×(C2×D3)⋊C4—31no—
147#924D911yescharacteristic, complemented
148#898—92no—
149#895—92no—
150#894—92no—
151#885—92no—
152#884—92no—
153#883—92no—
154#882—92no—
155#67⋊C4—92no—
156#879—92no—
157C23×C32C22×D311yescharacteristic
158C22×C3⋊D3—32no—
159C22×C3×D3—32no—
160C22×C3×D3—62no—
161C22×C3×D3—121no—
162C22×C3×D3—121no—
163C2×C32×C4—41no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166#638—64no—
167C2×C3⋊Dic3—32no—
168C4⋊(C3⋊D3)—121no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—121no—
171C2×C3×Dic3C22×D312yes—
172C2×C3×Dic3C22×D312yes—
173C3×Dic6—41no—
174#626—122no—
175#626—122no—
176#624—122no—
177C23×C9—21no—
178C23×C9C22×D311yescharacteristic
179C22×D9—22no—
180C22×D9—32no—
181C22×D9—41no—
182C22×D9—41no—
183C22×D9—62no—
184C22×D9C22×D312yes—
185Q8×C9—121no—
186C2×C4×C9—21no—
187C2×C4×C9—21no—
188C2×C4×C9—32no—
189C2×C4×C9—32no—
190C2×C4×C9—41no—
191C2×C4×C9—42no—
192C2×C4×C9C22×D311yescharacteristic
193C2×C4×C9C22×D311yescharacteristic
194#611—64no—
195#611—124no—
196C2×Dic9—32no—
197C2×Dic9—41no—
198C2×Dic9—62no—
199C2×Dic9C22×D312yes—
200D36—121no—
201D36—122no—
202D36—122no—
203D36—122no—
204C4×D9—122no—
205#538—271noSylow
206C2×C3×C9—22no—
207C2×C3×C9C2×#4111yescharacteristic
208C2×C3×C9H(5,𝔽2)11yescharacteristic
209C2×C3×C9H(5,𝔽2)11yescharacteristic
210C3⋊D9—42no—
211C3×D9—42no—
212C24×C3—91no—
213C23×D3—31no—
214C2×C3×Q8—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×#67—182no—
224C22×Dic3—31no—
225C2×D12—92no—
226C2×D12—92no—
227C2×D12—91no—
228C2×D12—91no—
229C2×D12—181no—
230C2×C4×D3—92no—
231C2×C4×D3—92no—
232C2×Dic6C2×D911yescharacteristic, complemented
233C3×C4⋊C4—31no—
234C3×C4⋊C4—32no—
235C3×C22⋊C4—32no—
236C3×C22⋊C4—92no—
237C3×C22⋊C4—92no—
238#217—92no—
239#217—92no—
240#217C2×D912yes—
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)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—182no—
250C4⋊Dic3—91no—
251C4⋊Dic3—91no—
252C4⋊Dic3—181no—
253C4⋊Dic3C2×D911yescharacteristic, complemented
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4C2×D912yescomplemented
257C4×Dic3—92no—
258C4×Dic3—92no—
259C22×C32—21no—
260C22×C32—21no—
261C22×C32#23912yes—
262C22×C32C23×D311yescharacteristic
263C2×C3⋊D3—62no—
264C2×C3⋊D3—62no—
265C2×C3⋊D3—62no—
266C2×C3×D3—62no—
267C2×C3×D3—62no—
268C2×C3×D3—62no—
269C2×C3×D3—124no—
270C32×C4—21no—
271C32×C4—21no—
272C3⋊Dic3—62no—
273C3×Dic3—22no—
274C3×Dic3—22no—
275C3×Dic3—62no—
276C22×C9—21no—
277C22×C9—21no—
278C22×C9—21no—
279C22×C9—22no—
280C22×C9—41no—
281C22×C9—41no—
282C22×C9#23512yes—
283C22×C9C23×D311yescharacteristic
284C2×D9—22no—
285C2×D9—22no—
286C2×D9—22no—
287C2×D9—44no—
288C2×D9—62no—
289C2×D9—62no—
290C2×D9—62no—
291C2×D9—122no—
292C2×D9—122no—
293C2×D9—122no—
294C4×C9—22no—
295C4×C9—21no—
296C4×C9—21no—
297C4×C9—41no—
298C4×C9—62no—
299C4×C9—62no—
300Dic9—22no—
301Dic9—62no—
302Dic9—122no—
303#126—272no—
304#124—272no—
305#123—272no—
306C22⋊Q8—31no—
307#121—272no—
308#121—271no—
309#120—272no—
310C4×D4—272no—
311C2×C22⋊C4—91no—
312C3×C9#53811yescharacteristic, complemented, Sylow
313C23×C3—21no—
314C23×C3—92no—
315C23×C3—182no—
316C23×C3—361no—
317C23×C3—361no—
318C23×C3C22×D911yescharacteristic
319C23×C3C2×D3211yescharacteristic
320C22×D3—32no—
321C22×D3—62no—
322C22×D3—92no—
323C22×D3—92no—
324C22×D3—121no—
325C22×D3—121no—
326C22×D3—182no—
327C3×Q8—121no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—32no—
331C2×C3×C4—32no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4—92no—
336C2×C3×C4—361no—
337C2×C3×C4C22×D911yescharacteristic
338C2×C3×C4C22×D911yescharacteristic
339C2×C3×C4C2×D3211yescharacteristic
340C2×C3×C4C2×D3211yescharacteristic
341#67—184no—
342#67—184no—
343#67—364no—
344#67—362no—
345#67—362no—
346C2×Dic3—32no—
347C2×Dic3—92no—
348C2×Dic3—92no—
349C2×Dic3—121no—
350C2×Dic3—182no—
351C2×Dic3C22×D912yes—
352C2×Dic3C22×D912yes—
353D12—361no—
354D12—361no—
355D12—362no—
356D12—362no—
357D12—362no—
358C4×D3—362no—
359C4×D3—362no—
360Dic6—41no—
361C2×C32—22no—
362C2×C32#100411yescharacteristic
363C2×C32C2×#23911yescharacteristic
364C2×C32#100911yescharacteristic
365C3⋊D3—122no—
366C3×D3—122no—
367C2×C9—22no—
368C2×C9—21no—
369C2×C9—21no—
370C2×C9—21no—
371C2×C9—42no—
372C2×C9C2×#23511yescharacteristic
373C2×C9#100411yescharacteristic
374C2×C9#100411yescharacteristic
375D9—42no—
376D9—122no—
377D9—242no—
378C24—91no—
379C2×Q8—31no—
380C2×D4—272no—
381C2×D4—272no—
382C2×D4—272no—
383C2×D4—271no—
384C22×C4—91no—
385C22×C4—272no—
386C22×C4D3×D911yescharacteristic
387C4⋊C4—32no—
388C4⋊C4—31no—
389C4⋊C4—272no—
390C4⋊C4—271no—
391C22⋊C4—32no—
392C22⋊C4—92no—
393C22⋊C4—92no—
394C22⋊C4—272no—
395C22⋊C4—272no—
396C22⋊C4—272no—
397C22⋊C4—272no—
398C42—272no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—22no—
404C22×C3—21no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3—182no—
408C22×C3—182no—
409C22×C3—182no—
410C22×C3—364no—
411C22×C3#370112yes—
412C22×C3C23×D911yescharacteristic
413C22×C3#379912yes—
414C22×C3C22×D3211yescharacteristic
415C2×D3—62no—
416C2×D3—62no—
417C2×D3—62no—
418C2×D3—124no—
419C2×D3—182no—
420C2×D3—182no—
421C2×D3—182no—
422C2×D3—182no—
423C2×D3—182no—
424C2×D3—182no—
425C2×D3—362no—
426C2×D3—362no—
427C2×D3—362no—
428C3×C4—22no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—41no—
434C3×C4—62no—
435C3×C4—62no—
436C3×C4—182no—
437Dic3—22no—
438Dic3—22no—
439Dic3—62no—
440Dic3—182no—
441Dic3—182no—
442Dic3—362no—
443C32#564411yescharacteristic
444C9—21no—
445C9#549411yescharacteristic, complemented
446C23—92no—
447C23—182no—
448C23—272no—
449C23—361no—
450C23—361no—
451C23C2×D3×D911yescharacteristic
452Q8—121no—
453D4—544no—
454D4—1081no—
455D4—1082no—
456D4—1082no—
457C2×C4—32no—
458C2×C4—32no—
459C2×C4—41no—
460C2×C4—92no—
461C2×C4—272no—
462C2×C4—361no—
463C2×C4—1082no—
464C2×C4C2×D3×D911yescharacteristic
465C2×C4C2×D3×D911yescharacteristic
466C2×C3—22no—
467C2×C3—22no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—42no—
472C2×C3—362no—
473C2×C3#6371011yescharacteristic
474C2×C3C2×#370111yescharacteristic
475C2×C3#6371511yescharacteristic
476C2×C3C2×#379911yescharacteristic
477C2×C3#6430711yescharacteristic
478C2×C3#6433011yescharacteristic
479D3—122no—
480D3—362no—
481D3—362no—
482D3—722no—
483C22—21no—
484C22—21no—
485C22—182no—
486C22—182no—
487C22—182no—
488C22—364no—
489C22—542no—
490C22—542no—
491C22—542no—
492C22#8863512yes—
493C22C22×D3×D911yescharacteristic
494C4—21no—
495C4—21no—
496C4—62no—
497C4—62no—
498C4—182no—
499C4—542no—
500C3—21no—
501C3#1058974611yescharacteristic, complemented
502C3#1059503811yescharacteristic
503C2—22no—
504C2—362no—
505C2—1082no—
506C2C2×#8863511yescharacteristic
507C2#1172247711yescharacteristic
508C2#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. #11721819C2semidirect
2. C2×C3×(C2×D9)⋊C4C2semidirect
3. C9×#924C2semidirect
4. #11721276C2semidirect
5. #11721241C2semidirect
6. #11721005C2semidirect
7. #11721003C2semidirect
8. #11720949C2semidirect
9. #11720942C2semidirect
15. C2×C4⋊(C3⋊D9)C22semidirect
17. C2×C9×Dic6C22semidirect
20. C2×(C3⋊D9)⋊C4C22semidirect
21. #88536C22semidirect
22. C4⋊(C3⋊Dic9)C22semidirect
26. C9×C4⋊Dic3C22semidirect
27. C9×Dic3⋊C4C22semidirect
28. #88445C22semidirect
31. #88442C22semidirect
32. Dic3×Dic9C22semidirect
46. C2×(C2×D9)⋊C4D3semidirect
147. #924D9semidirect
232. C2×Dic6C2×D9semidirect
253. C4⋊Dic3C2×D9semidirect
256. Dic3⋊C4C2×D9semidirect
312. C3×C9#538semidirect
445. C9#5494semidirect
501. C3#10589746semidirect

Maximal subgroups

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

subgroupindex
#117218192
C2×C3×(C2×D9)⋊C42
C9×#9242
#117212762
#117212412
#117210052
#117210032
#117209492
#117209422
#105950383
#105897463