Finite Group Databasebeta

← C9×#4989 #13142594

Subgroups — C9×#4989 #13142594

Order 1728.

794 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D3×C9C211yesmaximal, characteristic
2C2×C3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C9×(C2×D3)⋊C4C211yesmaximal, characteristic
4C2×C9×(C2×D3)⋊C4C211yesmaximal, characteristic
5C2×C9×Dic3⋊C4C211yesmaximal, characteristic, complemented
6C9×C3⋊#95C211yesmaximal, characteristic, complemented
7C9×C3⋊#95C211yesmaximal, characteristic, complemented
8C3×#4989C311yesmaximal, characteristic
9C9×#385—31nomaximal
10C23×D3×C9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C9×Dic3C2211yescharacteristic
17C2×C4×D3×C9—22no—
18C2×C4×D3×C9—22no—
19C2×C4×D3×C9—22no—
20C3×C9×C4⋊C4—22no—
21C9×(C2×D3)⋊C4—22no—
22C9×(C2×D3)⋊C4C414yescomplemented
23C9×Dic3⋊C4—22no—
24C22×C3×C4×D3C2×C311yescharacteristic
25C2×C32×C4⋊C4C2×C311yescharacteristic
26C2×C3×(C2×D3)⋊C4C2×C311yescharacteristic
27C2×C3×(C2×D3)⋊C4C2×C311yescharacteristic
28C2×C3×Dic3⋊C4C2×C311yescharacteristic
29C23×C4×C9—31no—
30C3×C3⋊#95C2×C311yescharacteristic
31C3×C3⋊#95C2×C311yescharacteristic
32C2×C9×C4⋊C4—21no—
33C2×C9×C4⋊C4—31no—
34C2×C9×C4⋊C4D311yescharacteristic, complemented
35C2×C9×C22⋊C4—31no—
36C2×C9×C22⋊C4—31no—
37C9×#95—31no—
38C9×#95—31no—
39C23×C3×C9C2311yescharacteristic
40C22×D3×C9—21no—
41C22×D3×C9—21no—
42C22×D3×C9—21no—
43C22×D3×C9—21no—
44C22×D3×C9C2×C412yes—
45C22×D3×C9D412yes—
46C22×D3×C9Q812yes—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9—21no—
50C2×C3×C4×C9—21no—
51C2×C3×C4×C9—21no—
52C2×C3×C4×C9—21no—
53C2×C3×C4×C9—21no—
54C2×C3×C4×C9C2×C412yescomplemented
55C2×C3×C4×C9D412yes—
56C2×C9×Dic3—21no—
57C2×C9×Dic3—21no—
58C2×C9×Dic3—21no—
59C2×C9×Dic3—21no—
60C2×C9×Dic3—21no—
61C2×C9×Dic3—21no—
62C2×C9×Dic3—21no—
63C2×C9×Dic3C2×C412yescomplemented
64C2×C9×Dic3D412yes—
65C4×D3×C9—28no—
66C3×#385—31no—
67#4989C911yescharacteristic, direct product factor
68C23×C3×D3C22×C311yescharacteristic
69C22×C32×C4C22×C311yescharacteristic
70C22×C32×C4C22×C311yescharacteristic
71C22×C32×C4C22×C311yescharacteristic
72C22×C3×Dic3C22×C311yescharacteristic
73C22×C3×Dic3C22×C311yescharacteristic
74C22×C3×Dic3C22×C311yescharacteristic
75C2×C3×C4×D3—22no—
76C2×C3×C4×D3—22no—
77C2×C3×C4×D3—22no—
78C24×C9—31no—
79C32×C4⋊C4—22no—
80C3×(C2×D3)⋊C4—22no—
81C3×(C2×D3)⋊C4C3×C414yes—
82C3×Dic3⋊C4—22no—
83C22×C4×C9—21no—
84C22×C4×C9—21no—
85C22×C4×C9—21no—
86C22×C4×C9—31no—
87C22×C4×C9—31no—
88C22×C4×C9—31no—
89C22×C4×C9—62no—
90C22×C4×C9—62no—
91C22×C4×C9—62no—
92C22×C4×C9C2×D311yescharacteristic
93C22×C4×C9C2×D311yescharacteristic
94C22×C4×C9C2×D311yescharacteristic
95C9×C4⋊C4—24no—
96C9×C4⋊C4—22no—
97C9×C4⋊C4—62no—
98C9×C22⋊C4—34no—
99C9×C22⋊C4—62no—
100C22×C3×C9C22×C411yescharacteristic
101C22×C3×C9C2×D411yescharacteristic
102C22×C3×C9C2×D411yescharacteristic
103C22×C3×C9C2×D411yescharacteristic
104C22×C3×C9C2×Q811yescharacteristic
105C22×C3×C9#4111yescharacteristic
106C22×C3×C9#4111yescharacteristic
107C2×D3×C9—22no—
108C2×D3×C9—22no—
109C2×D3×C9—22no—
110C2×D3×C9—22no—
111C2×D3×C9—22no—
112C2×D3×C9—22no—
113C2×D3×C9C4⋊C414yescomplemented
114C3×C4×C9—22no—
115C3×C4×C9—22no—
116C3×C4×C9—41no—
117C9×Dic3—22no—
118C9×Dic3—22no—
119C9×Dic3—41no—
120C23×C3×C4—31no—
121C22×C4×D3C2×C911yescharacteristic
122C2×C3×C4⋊C4—21no—
123C2×C3×C4⋊C4—31no—
124C2×C3×C4⋊C4C2×C911yescharacteristic, complemented
125C2×C3×C4⋊C4C3×D311yescharacteristic
126C2×C3×C22⋊C4—31no—
127C2×C3×C22⋊C4—31no—
128C2×(C2×D3)⋊C4C2×C911yescharacteristic
129C2×(C2×D3)⋊C4C2×C911yescharacteristic
130C2×Dic3⋊C4C2×C911yescharacteristic, complemented
131C3×#95—31no—
132C3×#95—31no—
133C3⋊#95C2×C911yescharacteristic, complemented
134C3⋊#95C2×C911yescharacteristic, complemented
135C23×C32C23×C311yescharacteristic
136C22×C3×D3—21no—
137C22×C3×D3—21no—
138C22×C3×D3—21no—
139C22×C3×D3—21no—
140C22×C3×D3C2×C3×C412yes—
141C22×C3×D3C3×D412yes—
142C22×C3×D3C3×Q812yes—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4C2×C3×C412yes—
151C2×C32×C4C3×D412yes—
152C2×C3×Dic3—21no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3—21no—
155C2×C3×Dic3—21no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3C2×C3×C412yes—
160C2×C3×Dic3C3×D412yes—
161C3×C4×D3—28no—
162C23×C9—21no—
163C23×C9—32no—
164C23×C9—32no—
165C23×C9—32no—
166C23×C9—61no—
167C23×C9—61no—
168C23×C9—61no—
169C23×C9—61no—
170C23×C9C22×D311yescharacteristic
171C2×C4×C9—22no—
172C2×C4×C9—22no—
173C2×C4×C9—21no—
174C2×C4×C9—21no—
175C2×C4×C9—21no—
176C2×C4×C9—21no—
177C2×C4×C9—22no—
178C2×C4×C9—21no—
179C2×C4×C9—21no—
180C2×C4×C9—21no—
181C2×C4×C9—22no—
182C2×C4×C9—22no—
183C2×C4×C9—32no—
184C2×C4×C9—32no—
185C2×C4×C9—41no—
186C2×C4×C9—41no—
187C2×C4×C9—41no—
188C2×C4×C9—41no—
189C2×C4×C9—68no—
190C2×C4×C9—61no—
191C2×C4×C9—61no—
192C2×C4×C9—61no—
193C2×C4×C9—61no—
194C2×C4×C9—61no—
195C2×C4×C9—61no—
196C2×C4×C9—61no—
197C2×C4×C9C4×D312yescomplemented
198C2×C4×C9#6712yes—
199#385—31noSylow
200C2×C3×C9C2×C4⋊C411yescharacteristic
201C2×C3×C9C4×D411yescharacteristic
202C2×C3×C9C4×D411yescharacteristic
203C2×C3×C9#12011yescharacteristic
204C2×C3×C9C22⋊Q811yescharacteristic
205C2×C3×C9C22⋊Q811yescharacteristic
206C2×C3×C9#12311yescharacteristic
207D3×C9—24no—
208C24×C3—31no—
209C23×D3C22×C911yescharacteristic
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—31no—
214C22×C3×C4—31no—
215C22×C3×C4—31no—
216C22×C3×C4—62no—
217C22×C3×C4—62no—
218C22×C3×C4—62no—
219C22×C3×C4C22×C911yescharacteristic
220C22×C3×C4C22×C911yescharacteristic
221C22×C3×C4C22×C911yescharacteristic
222C22×C3×C4C2×C3×D311yescharacteristic
223C22×C3×C4C2×C3×D311yescharacteristic
224C22×C3×C4C2×C3×D311yescharacteristic
225C22×Dic3C22×C911yescharacteristic
226C22×Dic3C22×C911yescharacteristic
227C22×Dic3C22×C911yescharacteristic
228C2×C4×D3—22no—
229C2×C4×D3—22no—
230C2×C4×D3—22no—
231C3×C4⋊C4—24no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—62no—
235C3×C22⋊C4—34no—
236C3×C22⋊C4—62no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4C4×C914yescomplemented
239Dic3⋊C4—22no—
240C22×C32C22×C3×C411yescharacteristic
241C22×C32C2×C3×D411yescharacteristic
242C22×C32C2×C3×D411yescharacteristic
243C22×C32C2×C3×D411yescharacteristic
244C22×C32C2×C3×Q811yescharacteristic
245C22×C32C3×#4111yescharacteristic
246C22×C32C3×#4111yescharacteristic
247C2×C3×D3—22no—
248C2×C3×D3—22no—
249C2×C3×D3—22no—
250C2×C3×D3—22no—
251C2×C3×D3—22no—
252C2×C3×D3—22no—
253C2×C3×D3C3×C4⋊C414yes—
254C32×C4—22no—
255C32×C4—22no—
256C32×C4—41no—
257C3×Dic3—22no—
258C3×Dic3—22no—
259C3×Dic3—41no—
260C22×C9—21no—
261C22×C9—21no—
262C22×C9—21no—
263C22×C9—21no—
264C22×C9—21no—
265C22×C9—21no—
266C22×C9—21no—
267C22×C9—34no—
268C22×C9—62no—
269C22×C9—62no—
270C22×C9—62no—
271C22×C9—62no—
272C22×C9—62no—
273C22×C9—62no—
274C22×C9C2×C4×D311yescharacteristic
275C22×C9#23511yescharacteristic
276C22×C9D3×D411yescharacteristic
277C22×C9D3×D411yescharacteristic
278C22×C9D3×Q811yescharacteristic
279C22×C9#23911yescharacteristic
280C22×C9C2×#6711yescharacteristic
281C4×C9—24no—
282C4×C9—22no—
283C4×C9—22no—
284C4×C9—42no—
285C4×C9—42no—
286C4×C9—41no—
287C4×C9—62no—
288C4×C9—62no—
289C4×C9—121no—
290C23×C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4D3×C911yescharacteristic, complemented
293C2×C22⋊C4—31no—
294C2×C22⋊C4—31no—
295#95—31no—
296#95—31no—
297C3×C9#38511yescharacteristic, complemented, Sylow
298C23×C3—21no—
299C23×C3—32no—
300C23×C3—32no—
301C23×C3—32no—
302C23×C3—61no—
303C23×C3—61no—
304C23×C3—61no—
305C23×C3—61no—
306C23×C3C23×C911yescharacteristic
307C23×C3C22×C3×D311yescharacteristic
308C22×D3—21no—
309C22×D3—21no—
310C22×D3—21no—
311C22×D3—21no—
312C22×D3C2×C4×C912yes—
313C22×D3D4×C912yes—
314C22×D3Q8×C912yes—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—32no—
335C2×C3×C4—32no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—68no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4C2×C4×C912yescomplemented
349C2×C3×C4D4×C912yes—
350C2×C3×C4C3×C4×D312yes—
351C2×C3×C4C3×#6712yes—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3C2×C4×C912yescomplemented
360C2×Dic3D4×C912yes—
361C4×D3—28no—
362C2×C32C2×C3×C4⋊C411yescharacteristic
363C2×C32C3×C4×D411yescharacteristic
364C2×C32C3×C4×D411yescharacteristic
365C2×C32C3×#12011yescharacteristic
366C2×C32C3×C22⋊Q811yescharacteristic
367C2×C32C3×C22⋊Q811yescharacteristic
368C2×C32C3×#12311yescharacteristic
369C3×D3—24no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9—21no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—64no—
378C2×C9D3×C4⋊C411yescharacteristic
379C2×C9#89311yescharacteristic
380C2×C9#89411yescharacteristic
381C2×C9#89611yescharacteristic
382C2×C9C4×#6711yescharacteristic
383C2×C9#93711yescharacteristic
384C2×C9#94611yescharacteristic
385C24—31no—
386C22×C4—31no—
387C22×C4—31no—
388C22×C4—31no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4C2×D3×C911yescharacteristic
393C22×C4C2×D3×C911yescharacteristic
394C22×C4C2×D3×C911yescharacteristic
395C4⋊C4—22no—
396C4⋊C4—62no—
397C22⋊C4—34no—
398C22⋊C4—62no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—34no—
407C22×C3—62no—
408C22×C3—62no—
409C22×C3—62no—
410C22×C3—62no—
411C22×C3—62no—
412C22×C3—62no—
413C22×C3C22×C4×C911yescharacteristic
414C22×C3C2×D4×C911yescharacteristic
415C22×C3C2×D4×C911yescharacteristic
416C22×C3C2×D4×C911yescharacteristic
417C22×C3C2×Q8×C911yescharacteristic
418C22×C3C9×#4111yescharacteristic
419C22×C3C9×#4111yescharacteristic
420C22×C3C2×C3×C4×D311yescharacteristic
421C22×C3C3×#23511yescharacteristic
422C22×C3C3×D3×D411yescharacteristic
423C22×C3C3×D3×D411yescharacteristic
424C22×C3C3×D3×Q811yescharacteristic
425C22×C3C3×#23911yescharacteristic
426C22×C3C2×C3×#6711yescharacteristic
427C2×D3—22no—
428C2×D3—22no—
429C2×D3—22no—
430C2×D3—22no—
431C2×D3—22no—
432C2×D3—22no—
433C2×D3C9×C4⋊C414yescomplemented
434C3×C4—24no—
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—22no—
438C3×C4—22no—
439C3×C4—42no—
440C3×C4—42no—
441C3×C4—41no—
442C3×C4—41no—
443C3×C4—62no—
444C3×C4—62no—
445C3×C4—121no—
446Dic3—22no—
447Dic3—22no—
448Dic3—41no—
449C32C3×#38511yescharacteristic
450C9—21no—
451C9#498911yescharacteristic, direct product factor
452C23—32no—
453C23—32no—
454C23—32no—
455C23—61no—
456C23—61no—
457C23—61no—
458C23—61no—
459C23C22×D3×C911yescharacteristic
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—32no—
468C2×C4—32no—
469C2×C4—68no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4C4×D3×C912yescomplemented
478C2×C4C9×#6712yes—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—64no—
487C2×C3C2×C9×C4⋊C411yescharacteristic
488C2×C3C4×D4×C911yescharacteristic
489C2×C3C4×D4×C911yescharacteristic
490C2×C3C9×#12011yescharacteristic
491C2×C3C9×C22⋊Q811yescharacteristic
492C2×C3C9×C22⋊Q811yescharacteristic
493C2×C3C9×#12311yescharacteristic
494C2×C3C3×D3×C4⋊C411yescharacteristic
495C2×C3C3×#89311yescharacteristic
496C2×C3C3×#89411yescharacteristic
497C2×C3C3×#89611yescharacteristic
498C2×C3C3×C4×#6711yescharacteristic
499C2×C3C3×#93711yescharacteristic
500C2×C3C3×#94611yescharacteristic
501D3—24no—
502C22—34no—
503C22—62no—
504C22—62no—
505C22—62no—
506C22—62no—
507C22—62no—
508C22—62no—
509C22C2×C4×D3×C911yescharacteristic
510C22C9×#23511yescharacteristic
511C22D3×D4×C911yescharacteristic
512C22D3×D4×C911yescharacteristic
513C22D3×Q8×C911yescharacteristic
514C22C9×#23911yescharacteristic
515C22C2×C9×#6711yescharacteristic
516C4—22no—
517C4—22no—
518C4—41no—
519C4—62no—
520C4—62no—
521C4—121no—
522C3—21no—
523C3C9×#38511yescharacteristic, complemented
524C3C3×#498911yescharacteristic
525C2—64no—
526C2D3×C9×C4⋊C411yescharacteristic
527C2C9×#89311yescharacteristic
528C2C9×#89411yescharacteristic
529C2C9×#89611yescharacteristic
530C2C4×C9×#6711yescharacteristic
531C2C9×#93711yescharacteristic
532C2C9×#94611yescharacteristic
subgroupG/H
centerC23×C9C22×D3
commutatorC22×C3C22×C4×C9
FrattiniC23×C3C22×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socle, cosocleC23×C32C23×C3
prime radicalC23×C3×D3C22×C3
hypercenterC2×C9×C4⋊C4D3
hypocenterC3C9×#385
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C9 ◁ C2×C9×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C9×C4⋊C4C2semidirect
5. C2×C9×Dic3⋊C4C2semidirect
6. C9×C3⋊#95C2semidirect
7. C9×C3⋊#95C2semidirect
22. C9×(C2×D3)⋊C4C4semidirect
34. C2×C9×C4⋊C4D3semidirect
54. C2×C3×C4×C9C2×C4semidirect
63. C2×C9×Dic3C2×C4semidirect
67. #4989C9direct (N × H)
113. C2×D3×C9C4⋊C4semidirect
124. C2×C3×C4⋊C4C2×C9semidirect
130. C2×Dic3⋊C4C2×C9semidirect
133. C3⋊#95C2×C9semidirect
134. C3⋊#95C2×C9semidirect
197. C2×C4×C9C4×D3semidirect
238. (C2×D3)⋊C4C4×C9semidirect
292. C2×C4⋊C4D3×C9semidirect
297. C3×C9#385semidirect
348. C2×C3×C4C2×C4×C9semidirect
359. C2×Dic3C2×C4×C9semidirect
433. C2×D3C9×C4⋊C4semidirect
451. C9#4989direct (N × H)
477. C2×C4C4×D3×C9semidirect
523. C3C9×#385semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D3×C92
C2×C3×C9×C4⋊C42
C2×C9×(C2×D3)⋊C42
C2×C9×Dic3⋊C42
C9×C3⋊#952
C3×#49893
C9×#3853