Finite Group Databasebeta

← #13148804

Subgroups — #13148804

Order 1728.

952 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88641C211yesmaximal, characteristic
2#11721743C211yesmaximal, characteristic, complemented
3D3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×#63455C211yesmaximal, characteristic, complemented
5C4×#6194C212yesmaximal, complemented
6#11720998C211yesmaximal, characteristic, complemented
7#11720976C212yesmaximal, complemented
8#11720975C212yesmaximal, complemented
9D9×Dic3⋊C4C212yesmaximal, complemented
10#11720959C211yesmaximal, characteristic, complemented
11#11720913C211yesmaximal, characteristic, complemented
12#10594359—31nomaximal
13#10589606—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C4×D3×C9C2212yes—
16C2×C4×D3×C9C2211yescharacteristic
17C2×C3×D36C2211yescharacteristic
18C2×C3×C4×D9C2212yes—
19C2×#6194C2212yes—
20C2×Dic3×D9C2212yes—
21C2×Dic3×D9C2212yes—
22#88641C418yescomplemented
23C3×C9×C4⋊C4C2211yescharacteristic
24#88534C2212yes—
25C4×C3⋊Dic9C2212yes—
26C4×C3⋊Dic9C2211yescharacteristic
27C3×(C2×D9)⋊C4C2212yes—
28C9×C4⋊Dic3C2211yescharacteristic
29C9×Dic3⋊C4C2212yes—
30C3×C4×Dic9C2211yescharacteristic
31(C2×D3)⋊Dic9C2211yescharacteristic
32(C2×D3)⋊Dic9C2211yescharacteristic
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2211yescharacteristic
35Dic3⋊Dic9C2211yescharacteristic
36#88442C2212yes—
37Dic3×Dic9C2212yes—
38C2×#3795—31no—
39#64096—31no—
40C3×#893—31no—
41C3×D3×C4⋊C4D311yescharacteristic
42C4×#625—32no—
43#63898—31no—
44#63879—32no—
45#63878—32no—
46D3×Dic3⋊C4—32no—
47#63855—31no—
48#63844—31no—
49C2×#3697—31no—
50C2×C9×C4⋊C4—31no—
51C4×#611—32no—
52C4×#611—32no—
53#63488—31no—
54#63455—31no—
55#63455D311yescharacteristic, complemented
56#63454—32no—
57C4⋊C4×D9—32no—
58#63449—31no—
59#63449—31no—
60#63449—61no—
61C22×D3×C9C2311yescharacteristic
62C22×C3×D9C2312yes—
63C2×C3×C4×C9C2312yes—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2312yes—
68#6226C2×C414yescomplemented
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C4×D3×C9—42no—
74C4×D3×C9C2×C414yescomplemented
75C3×D36C2×C414yescomplemented
76C3×C4×D9—42no—
77#6194C2×C418yescomplemented
78Dic3×D9—42no—
79Dic3×D9C2×C418yescomplemented
80#5546—91no—
81#5504—91no—
82C2×#634—31no—
83C2×C3×D12—31no—
84C2×C3×C4×D3—32no—
85C2×C3×C4×D3C2×D312yes—
86C2×C3×C4×D3C2×D311yescharacteristic
87C2×#625—32no—
88C2×D3×Dic3—32no—
89C2×D3×Dic3—32no—
90#3795—38no—
91C32×C4⋊C4C2×D311yescharacteristic
92#3750—32no—
93C4×C3⋊Dic3—31no—
94C4×C3⋊Dic3—32no—
95C3×(C2×D3)⋊C4—32no—
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×Dic3⋊C4C2×D312yes—
98C3×C4×Dic3—31no—
99#3724—32no—
100Dic3⋊Dic3—31no—
101Dic3⋊Dic3—31no—
102(C2×D3)⋊Dic3—31no—
103(C2×D3)⋊Dic3—31no—
104(C2×D3)⋊Dic3—32no—
105Dic32—32no—
106C22×C4×C9—31no—
107C22×C4×C9—32no—
108C2×#611—32no—
109#3697—38no—
110C2×D36C2×D311yescharacteristic
111C2×C4×D9—32no—
112C2×C4×D9—32no—
113C2×C4×D9C2×D312yes—
114C2×Dic18—31no—
115C2×Dic18—61no—
116C9×C4⋊C4—21no—
117C9×C4⋊C4—32no—
118C9×C4⋊C4—31no—
119C9×C4⋊C4C2×D311yescharacteristic
120#3676—31no—
121#3676—31no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4C2×D312yes—
124C4⋊Dic9—31no—
125C4⋊Dic9—31no—
126Dic9⋊C4—32no—
127Dic9⋊C4—32no—
128Dic9⋊C4—62no—
129C4×Dic9—31no—
130C4×Dic9—32no—
131C4×Dic9—32no—
132C4×Dic9—62no—
133C4×Dic9—61no—
134C4×Dic9C2×D311yescharacteristic
135C22×C3×C9C2411yescharacteristic
136C2×D3×C9—21no—
137C2×D3×C9—21no—
138C2×D3×C9C22×C412yes—
139C2×C3×D9—22no—
140C2×C3×D9—22no—
141C2×C3×D9C22×C414yes—
142C3×C4×C9—22no—
143C3×C4×C9C22×C412yes—
144C3⋊Dic9—21no—
145C3⋊Dic9—21no—
146C3⋊Dic9C22×C414yes—
147C9×Dic3—22no—
148C9×Dic3C22×C412yes—
149C3×Dic9—21no—
150C3×Dic9—21no—
151C2×#237—91no—
152C2×#235—91no—
153C3×C4×D4—91no—
154C2×C3×C4⋊C4—31no—
155D4×Dic3—91no—
156C4×#67—92no—
157C4×#67—92no—
158C4×#67—92no—
159#926—91no—
160C2×Dic3⋊C4—92no—
161#893—31no—
162#893—91no—
163#892—92no—
164D3×C4⋊C4—92no—
165D3×C4⋊C4D911yescharacteristic, complemented
166#887—91no—
167#887—91no—
168#887—91no—
169#887—181no—
170#67⋊C4—92no—
171#877—92no—
172#872—91no—
173C4×Dic6—91no—
174C22×C3×D3—32no—
175C22×C3×D3C22×D311yescharacteristic
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C3⋊Dic3—32no—
179C2×C3⋊Dic3—31no—
180C2×C3⋊Dic3—31no—
181#634—34no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3C22×D312yes—
185C2×C3×Dic3C22×D311yescharacteristic
186C3×D12—34no—
187C3×C4×D3—42no—
188C3×C4×D3—122no—
189C3×C4×D3C4×D314yes—
190#625—38no—
191D3×Dic3—38no—
192D3×Dic3—122no—
193C23×C9—31no—
194C22×D9C22×D312yes—
195C2×C4×C9—21no—
196C2×C4×C9—22no—
197C2×C4×C9—31no—
198C2×C4×C9—34no—
199C2×C4×C9—32no—
200C2×C4×C9—122no—
201C2×C4×C9C22×D312yes—
202C2×C4×C9C22×D311yescharacteristic
203#611—38no—
204C2×Dic9—32no—
205C2×Dic9—31no—
206C2×Dic9—31no—
207C2×Dic9—61no—
208C2×Dic9—62no—
209C2×Dic9—61no—
210C2×Dic9C22×D311yescharacteristic
211C2×Dic9C22×D311yescharacteristic
212D36C4×D314yescomplemented
213C4×D9—38no—
214C4×D9—42no—
215C4×D9—122no—
216Dic18—34no—
217Dic18—64no—
218#520—271noSylow
219C2×C3×C9C23×C411yescharacteristic
220C2×C3×C9H(5,𝔽2)11yescharacteristic
221C2×C3×C92−1+411yescharacteristic
222D3×C9—22no—
223C3×D9—24no—
224C2×C3×D4—91no—
225C22×C3×C4—31no—
226C22×C3×C4—32no—
227C22×C3×C4—92no—
228C2×#67—92no—
229C2×#67—92no—
230C22×Dic3—92no—
231C22×Dic3—92no—
232#237—98no—
233#235—98no—
234C2×D12—31no—
235C2×C4×D3—32no—
236C2×C4×D3—92no—
237C2×C4×D3—92no—
238C2×C4×D3C2×D912yes—
239C2×C4×D3C2×D911yescharacteristic
240C2×Dic6—91no—
241C2×Dic6—91no—
242C2×Dic6—181no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—32no—
246C3×C4⋊C4C2×D911yescharacteristic
247C3×C4⋊C4D3211yescharacteristic
248C3×C22⋊C4—92no—
249C3×C42—91no—
250#217—91no—
251#217—91no—
252#217—92no—
253(C2×D3)⋊C4—32no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—91no—
256(C2×D3)⋊C4—91no—
257C4⋊Dic3—91no—
258C4⋊Dic3—91no—
259C4⋊Dic3C2×D911yescharacteristic
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264Dic3⋊C4—91no—
265Dic3⋊C4—91no—
266Dic3⋊C4—182no—
267Dic3⋊C4C2×D912yes—
268C4×Dic3—31no—
269C4×Dic3—92no—
270C4×Dic3—91no—
271C4×Dic3—92no—
272C4×Dic3—92no—
273C4×Dic3—91no—
274C4×Dic3—92no—
275C4×Dic3—182no—
276C4×Dic3—181no—
277C22×C32C23×D311yescharacteristic
278C2×C3×D3—21no—
279C2×C3×D3—21no—
280C2×C3×D3—34no—
281C2×C3×D3—62no—
282C2×C3×D3—62no—
283C2×C3×D3C2×C4×D312yes—
284C32×C4—22no—
285C32×C4C2×C4×D312yes—
286C3⋊Dic3—34no—
287C3⋊Dic3—61no—
288C3⋊Dic3—61no—
289C3×Dic3—22no—
290C3×Dic3—61no—
291C3×Dic3—61no—
292C3×Dic3C2×C4×D312yes—
293C22×C9—21no—
294C22×C9—32no—
295C22×C9—61no—
296C22×C9—61no—
297C22×C9C23×D311yescharacteristic
298C2×D9—22no—
299C2×D9—22no—
300C2×D9C2×C4×D314yes—
301C4×C9—22no—
302C4×C9—22no—
303C4×C9—32no—
304C4×C9—42no—
305C4×C9—62no—
306C4×C9C2×C4×D312yes—
307Dic9—21no—
308Dic9—21no—
309Dic9—34no—
310Dic9—64no—
311Dic9—62no—
312Dic9—61no—
313Dic9—61no—
314Dic9—121no—
315C2×#41—271no—
316C4×Q8—271no—
317C4×Q8—271no—
318C4×D4—91no—
319C4×D4—272no—
320C4×D4—271no—
321C4×D4—272no—
322#117—271no—
323#117—272no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—272no—
326C3×C9#52011yescharacteristic, complemented, Sylow
327C23×C3—31no—
328C23×C3—92no—
329C22×D3—32no—
330C22×D3C22×D911yescharacteristic
331C3×D4—94no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—31no—
335C2×C3×C4—34no—
336C2×C3×C4—32no—
337C2×C3×C4—91no—
338C2×C3×C4—91no—
339C2×C3×C4—122no—
340C2×C3×C4—362no—
341C2×C3×C4C22×D912yes—
342C2×C3×C4C22×D911yescharacteristic
343C2×C3×C4C2×D3212yes—
344C2×C3×C4C2×D3211yescharacteristic
345#67—98no—
346#67—98no—
347C2×Dic3—31no—
348C2×Dic3—31no—
349C2×Dic3—92no—
350C2×Dic3—92no—
351C2×Dic3—98no—
352C2×Dic3—91no—
353C2×Dic3—91no—
354C2×Dic3—91no—
355C2×Dic3—91no—
356C2×Dic3—181no—
357C2×Dic3—182no—
358C2×Dic3—181no—
359C2×Dic3—362no—
360C2×Dic3C22×D912yes—
361C2×Dic3C22×D911yescharacteristic
362D12—34no—
363C4×D3—42no—
364C4×D3—98no—
365C4×D3—122no—
366C4×D3—362no—
367C4×D3C4×D914yescomplemented
368Dic6—94no—
369Dic6—94no—
370Dic6—184no—
371C2×C32C22×C4×D311yescharacteristic
372C2×C32#100411yescharacteristic
373C2×C32#100711yescharacteristic
374C3×D3—22no—
375C3×D3—64no—
376C2×C9—21no—
377C2×C9—21no—
378C2×C9—21no—
379C2×C9—62no—
380C2×C9C22×C4×D311yescharacteristic
381C2×C9#100411yescharacteristic
382C2×C9#101011yescharacteristic
383D9—24no—
384#41—278no—
385C2×Q8—271no—
386C2×D4—91no—
387C2×D4—272no—
388C22×C4—31no—
389C22×C4—32no—
390C22×C4—92no—
391C22×C4—272no—
392C22×C4—272no—
393C4⋊C4—32no—
394C4⋊C4—31no—
395C4⋊C4—271no—
396C4⋊C4—272no—
397C4⋊C4—271no—
398C4⋊C4—272no—
399C4⋊C4D3×D911yescharacteristic
400C22⋊C4—92no—
401C22⋊C4—272no—
402C22⋊C4—271no—
403C22⋊C4—271no—
404C42—91no—
405C42—272no—
406C42—272no—
407C42—271no—
408C22×C3—21no—
409C22×C3—32no—
410C22×C3—61no—
411C22×C3—61no—
412C22×C3—94no—
413C22×C3—182no—
414C22×C3—182no—
415C22×C3C23×D911yescharacteristic
416C22×C3C22×D3211yescharacteristic
417C2×D3—21no—
418C2×D3—21no—
419C2×D3—34no—
420C2×D3—62no—
421C2×D3—62no—
422C2×D3C2×C4×D912yes—
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—32no—
427C3×C4—42no—
428C3×C4—62no—
429C3×C4—181no—
430C3×C4—181no—
431C3×C4C2×C4×D912yes—
432C3×C4C4×D3212yes—
433Dic3—22no—
434Dic3—61no—
435Dic3—61no—
436Dic3—94no—
437Dic3—94no—
438Dic3—184no—
439Dic3—182no—
440Dic3—181no—
441Dic3—181no—
442Dic3—181no—
443Dic3—181no—
444Dic3—361no—
445Dic3C2×C4×D912yes—
446C32#550411yescharacteristic
447C9—21no—
448C9#554611yescharacteristic, complemented
449C23—31no—
450C23—92no—
451Q8—274no—
452D4—94no—
453D4—278no—
454C2×C4—31no—
455C2×C4—34no—
456C2×C4—32no—
457C2×C4—91no—
458C2×C4—91no—
459C2×C4—122no—
460C2×C4—278no—
461C2×C4—272no—
462C2×C4—271no—
463C2×C4—271no—
464C2×C4—362no—
465C2×C4—1082no—
466C2×C4C2×D3×D912yes—
467C2×C4C2×D3×D911yescharacteristic
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—62no—
472C2×C3—184no—
473C2×C3C22×C4×D911yescharacteristic
474C2×C3#6371011yescharacteristic
475C2×C3#6371311yescharacteristic
476C2×C3C2×C4×D3211yescharacteristic
477C2×C3#6431311yescharacteristic
478C2×C3#6431511yescharacteristic
479D3—22no—
480D3—64no—
481C22—32no—
482C22—61no—
483C22—61no—
484C22—94no—
485C22—182no—
486C22—182no—
487C22C22×D3×D911yescharacteristic
488C4—22no—
489C4—32no—
490C4—62no—
491C4—181no—
492C4—181no—
493C4—274no—
494C4—541no—
495C4—541no—
496C4C4×D3×D912yes—
497C3—21no—
498C3#1058960611yescharacteristic, complemented
499C3#1059435911yescharacteristic
500C2—62no—
501C2—184no—
502C2C2×C4×D3×D911yescharacteristic
503C2#1172249911yescharacteristic
504C2#1172251011yescharacteristic
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×C9#520
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. D3×C9×C4⋊C4C2semidirect
4. C3×#63455C2semidirect
5. C4×#6194C2semidirect
6. #11720998C2semidirect
7. #11720976C2semidirect
8. #11720975C2semidirect
9. D9×Dic3⋊C4C2semidirect
10. #11720959C2semidirect
11. #11720913C2semidirect
22. #88641C4semidirect
55. #63455D3semidirect
68. #6226C2×C4semidirect
74. C4×D3×C9C2×C4semidirect
75. C3×D36C2×C4semidirect
77. #6194C2×C4semidirect
79. Dic3×D9C2×C4semidirect
165. D3×C4⋊C4D9semidirect
212. D36C4×D3semidirect
326. C3×C9#520semidirect
367. C4×D3C4×D9semidirect
448. C9#5546semidirect
498. C3#10589606semidirect

Maximal subgroups

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

subgroupindex
C2×#886412
#117217432
D3×C9×C4⋊C42
C3×#634552
C4×#61942
#117209982
#117209762
#117209752
D9×Dic3⋊C42
#117209592
#117209132
#105943593
#105896063