Finite Group Databasebeta

← #13148803

Subgroups — #13148803

Order 1728.

1000 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
1C2×#88641C211yesmaximal, characteristic
2#11721743C211yesmaximal, characteristic, complemented
3C9×#892C211yesmaximal, characteristic, complemented
4C3×#63455C211yesmaximal, characteristic, complemented
5#11721028C212yesmaximal, complemented
6#11721012C212yesmaximal, complemented
7C4×D3×Dic9C211yesmaximal, characteristic, complemented
8#11720998C211yesmaximal, characteristic, complemented
9#11720974C212yesmaximal, complemented
10C4×Dic3×D9C212yesmaximal, complemented
11#11720913C211yesmaximal, characteristic, complemented
12#10594358—31nomaximal
13#10589624—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C4×D3×C9C2211yescharacteristic
16C2×C3×D36C2211yescharacteristic
17C2×C3×C4×D9C2212yes—
18C2×#6194C2212yes—
19C2×D3×Dic9C2211yescharacteristic
20C2×D3×Dic9C2211yescharacteristic
21C2×Dic3×D9C2212yes—
22C2×Dic3×D9C2212yes—
23#88641C418yescomplemented
24C3×C9×C4⋊C4C2211yescharacteristic
25#88534C2212yes—
26C4×C3⋊Dic9C2212yes—
27C4×C3⋊Dic9C2211yescharacteristic
28C9×(C2×D3)⋊C4C2212yes—
29C3×(C2×D9)⋊C4C2212yes—
30C9×C4⋊Dic3C2211yescharacteristic
31C4×C9×Dic3C2212yes—
32C3×C4×Dic9C2211yescharacteristic
33(C2×D9)⋊Dic3C2212yes—
34#88442C2212yes—
35Dic3×Dic9C2211yescharacteristic
36Dic3×Dic9C2212yes—
37Dic3×Dic9C2211yescharacteristic
38C2×#3795—31no—
39#64096—31no—
40C3×#893—31no—
41C3×#892D311yescharacteristic
42#63902—32no—
43#63898—31no—
44#63879—32no—
45#63877—32no—
46C4×D3×Dic3—31no—
47C4×D3×Dic3—32no—
48#63844—31no—
49C2×#3697—31no—
50C9×#117—31no—
51C2×C4×Dic9—31no—
52#63455—31no—
53#63455D311yescharacteristic, complemented
54#63449—31no—
55#63449—31no—
56#63449—61no—
57#611⋊C4—32no—
58#611⋊C4—32no—
59#63434—32no—
60C42×D9—32no—
61C22×D3×C9C2311yescharacteristic
62C22×C3×D9C2312yes—
63C2×C3×C4×C9C2312yes—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67C2×C3⋊Dic9C2311yescharacteristic
68#6226C2×C414yescomplemented
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C4×D3×C9C2×C414yescomplemented
74C3×D36C2×C414yescomplemented
75C3×C4×D9—24no—
76#6194C2×C418yescomplemented
77D3×Dic9—22no—
78D3×Dic9—22no—
79Dic3×D9—24no—
80Dic3×D9C2×C418yescomplemented
81C4×#237—91no—
82#5522—91no—
83C2×#634—31no—
84C2×C3×D12—31no—
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3C2×D311yescharacteristic
87C2×#625—32no—
88C2×D3×Dic3—32no—
89C2×D3×Dic3—31no—
90C2×D3×Dic3—31no—
91C2×D3×Dic3—32no—
92#3795—38no—
93C32×C4⋊C4C2×D311yescharacteristic
94#3750—32no—
95C4×C3⋊Dic3—31no—
96C4×C3⋊Dic3—32no—
97C3×(C2×D3)⋊C4—32no—
98C3×(C2×D3)⋊C4C2×D312yes—
99C3×C4⋊Dic3C2×D311yescharacteristic
100C3×C4×Dic3—31no—
101C3×C4×Dic3C2×D312yes—
102#3724—32no—
103(C2×D3)⋊Dic3—32no—
104Dic32—32no—
105Dic32—31no—
106Dic32—31no—
107C22×C4×C9—31no—
108C2×#611—32no—
109C22×Dic9—31no—
110C22×Dic9—31no—
111#3697—38no—
112C2×D36C2×D311yescharacteristic
113C2×C4×D9—32no—
114C2×C4×D9—32no—
115C2×C4×D9C2×D312yes—
116C2×Dic18—31no—
117C2×Dic18—61no—
118C9×C4⋊C4—21no—
119C9×C4⋊C4—31no—
120C9×C4⋊C4C2×D311yescharacteristic
121C9×C22⋊C4—32no—
122C42×C9—32no—
123(C2×D9)⋊C4—32no—
124(C2×D9)⋊C4C2×D312yes—
125Dic9⋊C4—32no—
126Dic9⋊C4—32no—
127Dic9⋊C4—62no—
128C4×Dic9—32no—
129C4×Dic9—31no—
130C4×Dic9—31no—
131C4×Dic9—32no—
132C4×Dic9—31no—
133C4×Dic9—62no—
134C4×Dic9—61no—
135C4×Dic9C2×D311yescharacteristic
136C22×C3×C9C2411yescharacteristic
137C2×D3×C9—21no—
138C2×D3×C9—21no—
139C2×D3×C9C22×C412yes—
140C2×C3×D9—22no—
141C2×C3×D9—22no—
142C2×C3×D9C22×C414yes—
143C3×C4×C9—22no—
144C3×C4×C9C22×C412yes—
145C3⋊Dic9—21no—
146C3⋊Dic9C22×C414yes—
147C3⋊Dic9#4112yescomplemented
148C9×Dic3—22no—
149C9×Dic3C22×C412yes—
150C3×Dic9—21no—
151C3×Dic9#4112yescomplemented
152C2×#237—91no—
153C2×#235—91no—
154C3×C4×D4—91no—
155C3×#117—31no—
156D4×Dic3—91no—
157C4×#67—92no—
158C2×C4×Dic3—91no—
159C2×C4×Dic3—92no—
160#893—31no—
161#893—91no—
162#892D911yescharacteristic, complemented
163#887—91no—
164#887—91no—
165#887—91no—
166#887—181no—
167#67⋊C4—92no—
168#67⋊C4—92no—
169#67⋊C4—92no—
170#877—92no—
171#872—92no—
172C42×D3—92no—
173C42×D3—91no—
174C4×Dic6—91no—
175C22×C3×D3—32no—
176C22×C3×D3C22×D311yescharacteristic
177C2×C32×C4C22×D312yes—
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—32no—
180C2×C3⋊Dic3—31no—
181C2×C3⋊Dic3—31no—
182#634—34no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3C22×D312yes—
186C2×C3×Dic3C22×D311yescharacteristic
187C3×D12—34no—
188C3×C4×D3—64no—
189C3×C4×D3C4×D314yes—
190#625—38no—
191D3×Dic3—38no—
192D3×Dic3—64no—
193D3×Dic3—62no—
194D3×Dic3—62no—
195C23×C9—31no—
196C22×D9C22×D312yes—
197C2×C4×C9—21no—
198C2×C4×C9—22no—
199C2×C4×C9—31no—
200C2×C4×C9—34no—
201C2×C4×C9—32no—
202C2×C4×C9C22×D312yes—
203C2×C4×C9C22×D311yescharacteristic
204#611—38no—
205C2×Dic9—31no—
206C2×Dic9—32no—
207C2×Dic9—31no—
208C2×Dic9—61no—
209C2×Dic9—62no—
210C2×Dic9—61no—
211C2×Dic9—62no—
212C2×Dic9—62no—
213C2×Dic9C22×D311yescharacteristic
214C2×Dic9C22×D311yescharacteristic
215D36C4×D314yescomplemented
216C4×D9—24no—
217C4×D9—38no—
218C4×D9—64no—
219Dic18—34no—
220Dic18—64no—
221C4×#41—271noSylow
222C2×C3×C9C23×C411yescharacteristic
223C2×C3×C9C2×#4111yescharacteristic
224C2×C3×C9C2×#4111yescharacteristic
225D3×C9—22no—
226C3×D9—24no—
227C2×C3×D4—91no—
228C22×C3×C4—31no—
229C22×C3×C4—92no—
230C2×#67—92no—
231C2×#67—92no—
232C22×Dic3—91no—
233C22×Dic3—91no—
234C22×Dic3—92no—
235C22×Dic3—92no—
236#237—98no—
237#235—98no—
238C2×D12—31no—
239C2×C4×D3—32no—
240C2×C4×D3—92no—
241C2×C4×D3—92no—
242C2×C4×D3—91no—
243C2×C4×D3—91no—
244C2×C4×D3C2×D911yescharacteristic
245C2×Dic6—91no—
246C2×Dic6—91no—
247C2×Dic6—181no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—31no—
250C3×C4⋊C4C2×D911yescharacteristic
251C3×C4⋊C4D3211yescharacteristic
252C3×C22⋊C4—32no—
253C3×C22⋊C4—92no—
254C3×C42—32no—
255C3×C42—91no—
256#217—92no—
257(C2×D3)⋊C4—32no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4C2×D912yes—
260C4⋊Dic3C2×D911yescharacteristic
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264Dic3⋊C4—92no—
265Dic3⋊C4—182no—
266C4×Dic3—31no—
267C4×Dic3—91no—
268C4×Dic3—92no—
269C4×Dic3—91no—
270C4×Dic3—91no—
271C4×Dic3—92no—
272C4×Dic3—92no—
273C4×Dic3—91no—
274C4×Dic3—92no—
275C4×Dic3—91no—
276C4×Dic3—91no—
277C4×Dic3—182no—
278C4×Dic3—181no—
279C4×Dic3C2×D912yes—
280C22×C32C23×D311yescharacteristic
281C2×C3×D3—21no—
282C2×C3×D3—21no—
283C2×C3×D3—34no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3C2×C4×D312yes—
287C32×C4—22no—
288C32×C4C2×C4×D312yes—
289C3⋊Dic3—34no—
290C3⋊Dic3—32no—
291C3⋊Dic3—61no—
292C3×Dic3—22no—
293C3×Dic3—32no—
294C3×Dic3—61no—
295C3×Dic3C2×C4×D312yes—
296C22×C9—21no—
297C22×C9—32no—
298C22×C9—61no—
299C22×C9—61no—
300C22×C9C23×D311yescharacteristic
301C2×D9—22no—
302C2×D9—22no—
303C2×D9C2×C4×D314yes—
304C4×C9—22no—
305C4×C9—22no—
306C4×C9—32no—
307C4×C9—42no—
308C4×C9—62no—
309C4×C9C2×C4×D312yes—
310Dic9—21no—
311Dic9—34no—
312Dic9—32no—
313Dic9—64no—
314Dic9—62no—
315Dic9—61no—
316Dic9—121no—
317Dic9#23712yescomplemented
318C2×#41—271no—
319C4×Q8—271no—
320C4×Q8—271no—
321C4×D4—91no—
322C4×D4—272no—
323C4×D4—272no—
324C4×D4—271no—
325#117—31no—
326#117—272no—
327C2×C42—272no—
328C2×C42—271no—
329C3×C9C4×#4111yescharacteristic, complemented, Sylow
330C23×C3—31no—
331C23×C3—92no—
332C22×D3—32no—
333C22×D3C22×D911yescharacteristic
334C3×D4—94no—
335C2×C3×C4—21no—
336C2×C3×C4—22no—
337C2×C3×C4—31no—
338C2×C3×C4—34no—
339C2×C3×C4—32no—
340C2×C3×C4—91no—
341C2×C3×C4—91no—
342C2×C3×C4—184no—
343C2×C3×C4C22×D912yes—
344C2×C3×C4C22×D911yescharacteristic
345C2×C3×C4C2×D3212yes—
346C2×C3×C4C2×D3211yescharacteristic
347#67—98no—
348#67—98no—
349C2×Dic3—31no—
350C2×Dic3—31no—
351C2×Dic3—91no—
352C2×Dic3—92no—
353C2×Dic3—91no—
354C2×Dic3—92no—
355C2×Dic3—98no—
356C2×Dic3—91no—
357C2×Dic3—91no—
358C2×Dic3—181no—
359C2×Dic3—182no—
360C2×Dic3—181no—
361C2×Dic3—184no—
362C2×Dic3—182no—
363C2×Dic3—182no—
364C2×Dic3C22×D912yes—
365C2×Dic3C22×D911yescharacteristic
366D12—34no—
367C4×D3—64no—
368C4×D3—98no—
369C4×D3—184no—
370C4×D3—182no—
371C4×D3—182no—
372C4×D3C4×D914yescomplemented
373Dic6—94no—
374Dic6—94no—
375Dic6—184no—
376C2×C32C22×C4×D311yescharacteristic
377C2×C32D3×#4111yescharacteristic
378C2×C32D3×#4111yescharacteristic
379C3×D3—22no—
380C3×D3—64no—
381C2×C9—21no—
382C2×C9—21no—
383C2×C9—21no—
384C2×C9—62no—
385C2×C9C22×C4×D311yescharacteristic
386C2×C9C2×#23711yescharacteristic
387C2×C9D3×#4111yescharacteristic
388D9—24no—
389#41—278no—
390C2×Q8—271no—
391C2×D4—91no—
392C2×D4—272no—
393C22×C4—31no—
394C22×C4—92no—
395C22×C4—272no—
396C22×C4—272no—
397C22×C4—271no—
398C22×C4—271no—
399C4⋊C4—31no—
400C4⋊C4—272no—
401C4⋊C4—272no—
402C4⋊C4D3×D911yescharacteristic
403C22⋊C4—32no—
404C22⋊C4—92no—
405C22⋊C4—272no—
406C42—32no—
407C42—91no—
408C42—271no—
409C42—271no—
410C42—272no—
411C42—272no—
412C42—271no—
413C22×C3—21no—
414C22×C3—32no—
415C22×C3—61no—
416C22×C3—61no—
417C22×C3—94no—
418C22×C3—182no—
419C22×C3—182no—
420C22×C3C23×D911yescharacteristic
421C22×C3C22×D3211yescharacteristic
422C2×D3—21no—
423C2×D3—21no—
424C2×D3—34no—
425C2×D3—62no—
426C2×D3—62no—
427C2×D3C2×C4×D912yes—
428C3×C4—22no—
429C3×C4—22no—
430C3×C4—22no—
431C3×C4—32no—
432C3×C4—42no—
433C3×C4—62no—
434C3×C4—92no—
435C3×C4—181no—
436C3×C4C2×C4×D912yes—
437C3×C4C4×D3212yes—
438Dic3—22no—
439Dic3—32no—
440Dic3—61no—
441Dic3—94no—
442Dic3—92no—
443Dic3—94no—
444Dic3—92no—
445Dic3—184no—
446Dic3—182no—
447Dic3—181no—
448Dic3—181no—
449Dic3—361no—
450Dic3C2×C4×D912yes—
451C32#552211yescharacteristic
452C9—21no—
453C9C4×#23711yescharacteristic, complemented
454C23—31no—
455C23—92no—
456Q8—274no—
457D4—94no—
458D4—278no—
459C2×C4—31no—
460C2×C4—34no—
461C2×C4—32no—
462C2×C4—91no—
463C2×C4—91no—
464C2×C4—184no—
465C2×C4—278no—
466C2×C4—271no—
467C2×C4—272no—
468C2×C4—271no—
469C2×C4—544no—
470C2×C4—542no—
471C2×C4—542no—
472C2×C4C2×D3×D912yes—
473C2×C4C2×D3×D911yescharacteristic
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—62no—
478C2×C3—184no—
479C2×C3C22×C4×D911yescharacteristic
480C2×C3#41×D911yescharacteristic
481C2×C3#41×D911yescharacteristic
482C2×C3C2×C4×D3211yescharacteristic
483C2×C3D3×#23711yescharacteristic
484C2×C3#6431911yescharacteristic
485D3—22no—
486D3—64no—
487C22—32no—
488C22—61no—
489C22—61no—
490C22—94no—
491C22—182no—
492C22—182no—
493C22C22×D3×D911yescharacteristic
494C4—22no—
495C4—32no—
496C4—62no—
497C4—92no—
498C4—181no—
499C4—272no—
500C4—274no—
501C4—541no—
502C4C4×D3×D912yes—
503C3—21no—
504C3#1058962411yescharacteristic, complemented
505C3#1059435811yescharacteristic
506C2—62no—
507C2—184no—
508C2C2×C4×D3×D911yescharacteristic
509C2D9×#23711yescharacteristic
510C2#1172251411yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#88641C2
cosocleC22×C3×C9C24
hypercenterC4⋊C4D3×D9
hypocenterC3×C9C4×#41
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721743C2semidirect
3. C9×#892C2semidirect
4. C3×#63455C2semidirect
5. #11721028C2semidirect
6. #11721012C2semidirect
7. C4×D3×Dic9C2semidirect
8. #11720998C2semidirect
9. #11720974C2semidirect
10. C4×Dic3×D9C2semidirect
11. #11720913C2semidirect
23. #88641C4semidirect
53. #63455D3semidirect
68. #6226C2×C4semidirect
73. C4×D3×C9C2×C4semidirect
74. C3×D36C2×C4semidirect
76. #6194C2×C4semidirect
80. Dic3×D9C2×C4semidirect
147. C3⋊Dic9#41semidirect
151. C3×Dic9#41semidirect
162. #892D9semidirect
215. D36C4×D3semidirect
317. Dic9#237semidirect
329. C3×C9C4×#41semidirect
372. C4×D3C4×D9semidirect
453. C9C4×#237semidirect
504. C3#10589624semidirect

Maximal subgroups

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

subgroupindex
C2×#886412
#117217432
C9×#8922
C3×#634552
#117210282
#117210122
C4×D3×Dic92
#117209982
#117209742
C4×Dic3×D92
#117209132
#105943583
#105896243