Finite Group Databasebeta

← C3×#10588628 #13142459

Subgroups — C3×#10588628 #13142459

Order 1728.

690 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×C22⋊C4C211yesmaximal, characteristic
2C2×C3×#3676C211yesmaximal, characteristic
3C2×C3×#3676C211yesmaximal, characteristic
4C2×C3×C4⋊Dic9C211yesmaximal, characteristic, complemented
5C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
6C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
7C3×C9⋊#95C211yesmaximal, characteristic, complemented
8C3×#4948—31nomaximal
9#10588628C311yesmaximal, characteristic, direct product factor
10C24×C3×C9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3×Dic9C2211yescharacteristic
14C22×C3×Dic9C2211yescharacteristic
15C22×C3×Dic9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C3×C9×C22⋊C4—22no—
18C3×#3676—22no—
19C3×#3676—22no—
20C3×C4⋊Dic9—22no—
21C3×Dic9⋊C4—22no—
22C3×Dic9⋊C4—22no—
23C2×C32×C22⋊C4D311yescharacteristic
24C2×C3×#217—31no—
25C2×C3×#217—31no—
26C2×C3×C4⋊Dic3—31no—
27C2×C3×Dic3⋊C4—31no—
28C2×C3×Dic3⋊C4—31no—
29C3×C3⋊#95—31no—
30C2×C9×C22⋊C4—21no—
31C2×C9×C22⋊C4C2×C311yescharacteristic
32C2×#3676C2×C311yescharacteristic
33C2×#3676C2×C311yescharacteristic
34C2×C4⋊Dic9C2×C311yescharacteristic, complemented
35C2×Dic9⋊C4C2×C311yescharacteristic, complemented
36C2×Dic9⋊C4C2×C311yescharacteristic, complemented
37C9⋊#95C2×C311yescharacteristic, complemented
38C23×C3×C9—21no—
39C23×C3×C9—21no—
40C23×C3×C9—21no—
41C23×C3×C9—21no—
42C23×C3×C9—21no—
43C23×C3×C9—21no—
44C23×C3×C9Q812yes—
45C23×C3×C9C2311yescharacteristic
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9—21no—
50C2×C3×C4×C9—21no—
51C2×C3×C4×C9D412yes—
52C2×C3×Dic9—21no—
53C2×C3×Dic9—21no—
54C2×C3×Dic9—21no—
55C2×C3×Dic9—21no—
56C2×C3×Dic9—21no—
57C2×C3×Dic9—21no—
58C2×C3×Dic9—21no—
59C2×C3×Dic9—21no—
60C2×C3×Dic9—21no—
61C2×C3×Dic9—21no—
62C2×C3×Dic9D412yes—
63C2×C3×Dic9D412yes—
64C3×#396—91no—
65#4948—31no—
66C24×C32C2×D311yescharacteristic
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C3×Dic3—31no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3—31no—
73C24×C9—21no—
74C24×C9C22×C311yescharacteristic
75C32×C22⋊C4—22no—
76C3×#217—62no—
77C3×#217—62no—
78C3×C4⋊Dic3—62no—
79C3×Dic3⋊C4—62no—
80C3×Dic3⋊C4—62no—
81C22×C4×C9—21no—
82C22×C4×C9—21no—
83C22×C4×C9C22×C311yescharacteristic
84C22×C4×C9C22×C311yescharacteristic
85C22×Dic9C22×C311yescharacteristic
86C22×Dic9C22×C311yescharacteristic
87C22×Dic9C22×C311yescharacteristic
88C22×Dic9C22×C311yescharacteristic
89C9×C22⋊C4—24no—
90C9×C22⋊C4—22no—
91#3676—22no—
92#3676—22no—
93C4⋊Dic9—22no—
94Dic9⋊C4—22no—
95Dic9⋊C4—22no—
96C22×C3×C9—22no—
97C22×C3×C9—22no—
98C22×C3×C9—22no—
99C22×C3×C9—41no—
100C22×C3×C9—41no—
101C22×C3×C9—41no—
102C22×C3×C9—41no—
103C22×C3×C9C2×D411yescharacteristic
104C22×C3×C9C2×D411yescharacteristic
105C22×C3×C9C2×D411yescharacteristic
106C22×C3×C9C2×Q811yescharacteristic
107C22×C3×C9#4111yescharacteristic
108C22×C3×C9#4111yescharacteristic
109C22×C3×C9#4111yescharacteristic
110C3×C4×C9—22no—
111C3×C4×C9—41no—
112C3×Dic9—22no—
113C3×Dic9—22no—
114C3×Dic9—41no—
115C3×Dic9—41no—
116C2×C3×C4⋊C4—91no—
117C2×C3×C4⋊C4—91no—
118C2×C3×C4⋊C4—91no—
119C2×C3×C22⋊C4—21no—
120C2×C3×C22⋊C4—91no—
121C2×C3×C22⋊C4—91no—
122C2×C3×C22⋊C4D911yescharacteristic
123C2×C3×C22⋊C4C3×D311yescharacteristic
124C2×#217—31no—
125C2×#217—31no—
126C2×C4⋊Dic3—31no—
127C2×Dic3⋊C4—31no—
128C2×Dic3⋊C4—31no—
129C3×#95—91no—
130C3⋊#95—31no—
131C23×C32—21no—
132C23×C32—21no—
133C23×C32—21no—
134C23×C32—21no—
135C23×C32—21no—
136C23×C32—21no—
137C23×C32Dic612yes—
138C23×C32C22×D311yescharacteristic
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4#6712yes—
145C2×C3×Dic3—32no—
146C2×C3×Dic3—32no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—61no—
157C23×C9—22no—
158C23×C9—22no—
159C23×C9—21no—
160C23×C9—21no—
161C23×C9—21no—
162C23×C9—21no—
163C23×C9—21no—
164C23×C9—21no—
165C23×C9—22no—
166C23×C9—21no—
167C23×C9—41no—
168C23×C9—41no—
169C23×C9—41no—
170C23×C9—41no—
171C23×C9C3×Q812yes—
172C23×C9C23×C311yescharacteristic
173C2×C4×C9—21no—
174C2×C4×C9—21no—
175C2×C4×C9—22no—
176C2×C4×C9—21no—
177C2×C4×C9—21no—
178C2×C4×C9—21no—
179C2×C4×C9—22no—
180C2×C4×C9—41no—
181C2×C4×C9—41no—
182C2×C4×C9—41no—
183C2×C4×C9—41no—
184C2×C4×C9C3×D412yes—
185C2×Dic9—21no—
186C2×Dic9—21no—
187C2×Dic9—21no—
188C2×Dic9—21no—
189C2×Dic9—21no—
190C2×Dic9—21no—
191C2×Dic9—21no—
192C2×Dic9—21no—
193C2×Dic9—21no—
194C2×Dic9—21no—
195C2×Dic9C3×D412yes—
196C2×Dic9C3×D412yes—
197#396—91noSylow
198C2×C3×C9—42no—
199C2×C3×C9#12111yescharacteristic
200C2×C3×C9#12111yescharacteristic
201C2×C3×C9#12111yescharacteristic
202C2×C3×C9C22⋊Q811yescharacteristic
203C2×C3×C9C22⋊Q811yescharacteristic
204C2×C3×C9C22⋊Q811yescharacteristic
205C2×C3×C9#12611yescharacteristic
206C24×C3—21no—
207C24×C3C2×D911yescharacteristic
208C24×C3C2×C3×D311yescharacteristic
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—91no—
212C22×C3×C4—91no—
213C22×C3×C4—91no—
214C22×C3×C4—91no—
215C22×C3×C4C2×D911yescharacteristic
216C22×C3×C4C2×D911yescharacteristic
217C22×C3×C4C2×C3×D311yescharacteristic
218C22×C3×C4C2×C3×D311yescharacteristic
219C22×Dic3—31no—
220C22×Dic3—31no—
221C22×Dic3—31no—
222C22×Dic3—31no—
223C3×C4⋊C4—182no—
224C3×C4⋊C4—182no—
225C3×C4⋊C4—182no—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—182no—
230C3×C22⋊C4—182no—
231#217—62no—
232#217—62no—
233C4⋊Dic3—62no—
234Dic3⋊C4—62no—
235Dic3⋊C4—62no—
236C22×C32—22no—
237C22×C32—22no—
238C22×C32—22no—
239C22×C32—41no—
240C22×C32—41no—
241C22×C32—41no—
242C22×C32—41no—
243C22×C32C2×Dic611yescharacteristic
244C22×C32#23511yescharacteristic
245C22×C32D3×D411yescharacteristic
246C22×C32D3×D411yescharacteristic
247C22×C32#23711yescharacteristic
248C22×C32#23711yescharacteristic
249C22×C32C2×#6711yescharacteristic
250C32×C4—22no—
251C32×C4—41no—
252C3×Dic3—62no—
253C3×Dic3—62no—
254C3×Dic3—121no—
255C3×Dic3—121no—
256C22×C9—24no—
257C22×C9—22no—
258C22×C9—22no—
259C22×C9—22no—
260C22×C9—21no—
261C22×C9—21no—
262C22×C9—21no—
263C22×C9—21no—
264C22×C9—21no—
265C22×C9—21no—
266C22×C9—21no—
267C22×C9—42no—
268C22×C9—42no—
269C22×C9—42no—
270C22×C9—42no—
271C22×C9—42no—
272C22×C9—42no—
273C22×C9—41no—
274C22×C9—41no—
275C22×C9—41no—
276C22×C9—41no—
277C22×C9C2×C3×D411yescharacteristic
278C22×C9C2×C3×D411yescharacteristic
279C22×C9C2×C3×D411yescharacteristic
280C22×C9C2×C3×Q811yescharacteristic
281C22×C9C3×#4111yescharacteristic
282C22×C9C3×#4111yescharacteristic
283C22×C9C3×#4111yescharacteristic
284C4×C9—22no—
285C4×C9—42no—
286C4×C9—42no—
287C4×C9—41no—
288Dic9—22no—
289Dic9—22no—
290Dic9—41no—
291Dic9—41no—
292C2×C4⋊C4—91no—
293C2×C4⋊C4—91no—
294C2×C4⋊C4—91no—
295C2×C22⋊C4—91no—
296C2×C22⋊C4—91no—
297C2×C22⋊C4C3×D911yescharacteristic
298#95—91no—
299C3×C9#39611yescharacteristic, complemented, Sylow
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—22no—
315C23×C3—21no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3Dic1812yes—
321C23×C3C22×D911yescharacteristic
322C23×C3C3×Dic612yes—
323C23×C3C22×C3×D311yescharacteristic
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—92no—
341C2×C3×C4—92no—
342C2×C3×C4—181no—
343C2×C3×C4—181no—
344C2×C3×C4—181no—
345C2×C3×C4—181no—
346C2×C3×C4—181no—
347C2×C3×C4—181no—
348C2×C3×C4—181no—
349C2×C3×C4—181no—
350C2×C3×C4—181no—
351C2×C3×C4—181no—
352C2×C3×C4#61112yes—
353C2×C3×C4C3×#6712yes—
354C2×Dic3—32no—
355C2×Dic3—32no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360C2×Dic3—61no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365C2×Dic3—61no—
366C2×C32—42no—
367C2×C32#87811yescharacteristic
368C2×C32#87811yescharacteristic
369C2×C32#87911yescharacteristic
370C2×C32#88411yescharacteristic
371C2×C32#92411yescharacteristic
372C2×C32#93811yescharacteristic
373C2×C32#93911yescharacteristic
374C2×C9—21no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—21no—
378C2×C9—21no—
379C2×C9—21no—
380C2×C9—21no—
381C2×C9—44no—
382C2×C9—42no—
383C2×C9C3×#12111yescharacteristic
384C2×C9C3×#12111yescharacteristic
385C2×C9C3×#12111yescharacteristic
386C2×C9C3×C22⋊Q811yescharacteristic
387C2×C9C3×C22⋊Q811yescharacteristic
388C2×C9C3×C22⋊Q811yescharacteristic
389C2×C9C3×#12611yescharacteristic
390C24C2×C3×D911yescharacteristic
391C22×C4—91no—
392C22×C4—91no—
393C22×C4—91no—
394C22×C4—91no—
395C22×C4C2×C3×D911yescharacteristic
396C22×C4C2×C3×D911yescharacteristic
397C4⋊C4—182no—
398C4⋊C4—182no—
399C4⋊C4—182no—
400C22⋊C4—22no—
401C22⋊C4—182no—
402C22⋊C4—182no—
403C22×C3—24no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3C2×Dic1811yescharacteristic
432C22×C3#369711yescharacteristic
433C22×C3D4×D911yescharacteristic
434C22×C3D4×D911yescharacteristic
435C22×C3#369911yescharacteristic
436C22×C3#369911yescharacteristic
437C22×C3C2×#61111yescharacteristic
438C22×C3C2×C3×Dic611yescharacteristic
439C22×C3C3×#23511yescharacteristic
440C22×C3C3×D3×D411yescharacteristic
441C22×C3C3×D3×D411yescharacteristic
442C22×C3C3×#23711yescharacteristic
443C22×C3C3×#23711yescharacteristic
444C22×C3C2×C3×#6711yescharacteristic
445C3×C4—22no—
446C3×C4—22no—
447C3×C4—42no—
448C3×C4—42no—
449C3×C4—41no—
450C3×C4—41no—
451C3×C4—182no—
452C3×C4—182no—
453C3×C4—361no—
454C3×C4—361no—
455Dic3—62no—
456Dic3—62no—
457Dic3—121no—
458Dic3—121no—
459C32#494811yescharacteristic
460C9—21no—
461C9C3×#39611yescharacteristic, complemented
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23C3×Dic1812yes—
469C23C22×C3×D911yescharacteristic
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—92no—
476C2×C4—92no—
477C2×C4—181no—
478C2×C4—181no—
479C2×C4—181no—
480C2×C4—181no—
481C2×C4—181no—
482C2×C4—181no—
483C2×C4—181no—
484C2×C4—181no—
485C2×C4—181no—
486C2×C4—181no—
487C2×C4C3×#61112yes—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—44no—
496C2×C3—42no—
497C2×C3—42no—
498C2×C3#6344011yescharacteristic
499C2×C3#6344011yescharacteristic
500C2×C3#6344111yescharacteristic
501C2×C3#6344611yescharacteristic
502C2×C3#6348611yescharacteristic
503C2×C3#6350011yescharacteristic
504C2×C3#6350111yescharacteristic
505C2×C3C3×#87811yescharacteristic
506C2×C3C3×#87811yescharacteristic
507C2×C3C3×#87911yescharacteristic
508C2×C3C3×#88411yescharacteristic
509C2×C3C3×#92411yescharacteristic
510C2×C3C3×#93811yescharacteristic
511C2×C3C3×#93911yescharacteristic
512C22—22no—
513C22—22no—
514C22—22no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22C2×C3×Dic1811yescharacteristic
520C22C3×#369711yescharacteristic
521C22C3×D4×D911yescharacteristic
522C22C3×D4×D911yescharacteristic
523C22C3×#369911yescharacteristic
524C22C3×#369911yescharacteristic
525C22C2×C3×#61111yescharacteristic
526C4—22no—
527C4—41no—
528C4—182no—
529C4—182no—
530C4—361no—
531C4—361no—
532C3—21no—
533C3#1058862811yescharacteristic, direct product factor
534C3C3×#494811yescharacteristic
535C2—42no—
536C2C3×#6344011yescharacteristic
537C2C3×#6344011yescharacteristic
538C2C3×#6344111yescharacteristic
539C2C3×#6344611yescharacteristic
540C2C3×#6348611yescharacteristic
541C2C3×#6350011yescharacteristic
542C2C3×#6350111yescharacteristic
subgroupG/H
centerC23×C3C22×D9
commutator, cosocleC23×C9C23×C3
FrattiniC23×C3C22×C3×D3
FittingC2×C3×C9×C22⋊C4C2
socleC23×C32C22×D3
prime radicalC24×C32C2×D3
hypercenterC2×C3×C22⋊C4D9
hypocenterC9C3×#396
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×C3×C4⋊Dic9C2semidirect
5. C2×C3×Dic9⋊C4C2semidirect
6. C2×C3×Dic9⋊C4C2semidirect
7. C3×C9⋊#95C2semidirect
9. #10588628C3direct (N × H)
34. C2×C4⋊Dic9C2×C3semidirect
35. C2×Dic9⋊C4C2×C3semidirect
36. C2×Dic9⋊C4C2×C3semidirect
37. C9⋊#95C2×C3semidirect
299. C3×C9#396semidirect
461. C9C3×#396semidirect
533. C3#10588628direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C9×C22⋊C42
C2×C3×#36762
C2×C3×C4⋊Dic92
C2×C3×Dic9⋊C42
C3×C9⋊#952
C3×#49483
#105886283