Finite Group Databasebeta

← C3×#10588624 #13142452

Subgroups — C3×#10588624 #13142452

Order 1728.

702 conjugacy classes of subgroups.

The subgroup lattice has 538 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×Dic9⋊C4C211yesmaximal, characteristic, complemented
5C2×C3×C4×Dic9C211yesmaximal, characteristic, complemented
6C3×C9⋊#95C211yesmaximal, characteristic, complemented
7C3×C9⋊#95C211yesmaximal, characteristic, complemented
8C3×#4944—31nomaximal
9#10588624C311yesmaximal, 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×#3676C414yescomplemented
20C3×Dic9⋊C4—22no—
21C3×C4×Dic9—22no—
22C2×C32×C22⋊C4D311yescharacteristic
23C2×C3×#217—31no—
24C2×C3×#217—31no—
25C2×C3×Dic3⋊C4—31no—
26C2×C3×C4×Dic3—31no—
27C3×C3⋊#95—31no—
28C3×C3⋊#95—31no—
29C2×C9×C22⋊C4—21no—
30C2×C9×C22⋊C4C2×C311yescharacteristic
31C2×#3676C2×C311yescharacteristic
32C2×#3676C2×C311yescharacteristic
33C2×Dic9⋊C4C2×C311yescharacteristic, complemented
34C2×C4×Dic9C2×C311yescharacteristic, complemented
35C9⋊#95C2×C311yescharacteristic, complemented
36C9⋊#95C2×C311yescharacteristic, complemented
37C23×C3×C9—21no—
38C23×C3×C9—21no—
39C23×C3×C9—21no—
40C23×C3×C9—21no—
41C23×C3×C9—21no—
42C23×C3×C9—21no—
43C23×C3×C9C2×C412yes—
44C23×C3×C9C2311yescharacteristic
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9—21no—
50C2×C3×C4×C9D412yes—
51C2×C3×Dic9—21no—
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×Dic9C2×C412yes—
61C2×C3×Dic9C2×C412yes—
62C2×C3×Dic9D412yes—
63C3×#388—91no—
64#4944—31no—
65C24×C32C2×D311yescharacteristic
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C2×D311yescharacteristic
68C22×C3×Dic3—31no—
69C22×C3×Dic3—31no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C24×C9—21no—
73C24×C9C22×C311yescharacteristic
74C32×C22⋊C4—22no—
75C3×#217—34no—
76C3×#217—62no—
77C3×Dic3⋊C4—62no—
78C3×C4×Dic3—62no—
79C22×C4×C9—21no—
80C22×C4×C9—21no—
81C22×C4×C9C22×C311yescharacteristic
82C22×C4×C9C22×C311yescharacteristic
83C22×Dic9C22×C311yescharacteristic
84C22×Dic9C22×C311yescharacteristic
85C22×Dic9C22×C311yescharacteristic
86C22×Dic9C22×C311yescharacteristic
87C9×C22⋊C4—24no—
88C9×C22⋊C4—22no—
89#3676—22no—
90#3676C3×C414yescomplemented
91Dic9⋊C4—22no—
92C4×Dic9—22no—
93C22×C3×C9—22no—
94C22×C3×C9—22no—
95C22×C3×C9—22no—
96C22×C3×C9—41no—
97C22×C3×C9—41no—
98C22×C3×C9—41no—
99C22×C3×C9—41no—
100C22×C3×C9C22×C411yescharacteristic
101C22×C3×C9C2×D411yescharacteristic
102C22×C3×C9C2×D411yescharacteristic
103C22×C3×C9#4111yescharacteristic
104C22×C3×C9#4111yescharacteristic
105C22×C3×C9#4111yescharacteristic
106C22×C3×C9#4111yescharacteristic
107C3×C4×C9—22no—
108C3×C4×C9—41no—
109C3×Dic9—22no—
110C3×Dic9—22no—
111C3×Dic9—22no—
112C3×Dic9—41no—
113C2×C3×C4⋊C4—91no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4—91no—
116C2×C3×C22⋊C4—91no—
117C2×C3×C22⋊C4D911yescharacteristic
118C2×C3×C22⋊C4C3×D311yescharacteristic
119C2×C3×C42—91no—
120C2×#217—31no—
121C2×#217—31no—
122C2×Dic3⋊C4—31no—
123C2×C4×Dic3—31no—
124C3×#95—91no—
125C3×#95—91no—
126C3⋊#95—31no—
127C3⋊#95—31no—
128C23×C32—21no—
129C23×C32—21no—
130C23×C32—21no—
131C23×C32—21no—
132C23×C32—21no—
133C23×C32—21no—
134C23×C32C4×D312yes—
135C23×C32C22×D311yescharacteristic
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4#6712yes—
142C2×C3×Dic3—32no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
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—
154C23×C9—22no—
155C23×C9—22no—
156C23×C9—21no—
157C23×C9—21no—
158C23×C9—21no—
159C23×C9—21no—
160C23×C9—21no—
161C23×C9—21no—
162C23×C9—22no—
163C23×C9—21no—
164C23×C9—41no—
165C23×C9—41no—
166C23×C9—41no—
167C23×C9—41no—
168C23×C9C2×C3×C412yes—
169C23×C9C23×C311yescharacteristic
170C2×C4×C9—21no—
171C2×C4×C9—21no—
172C2×C4×C9—22no—
173C2×C4×C9—21no—
174C2×C4×C9—21no—
175C2×C4×C9—21no—
176C2×C4×C9—22no—
177C2×C4×C9—41no—
178C2×C4×C9—41no—
179C2×C4×C9—41no—
180C2×C4×C9—41no—
181C2×C4×C9C3×D412yes—
182C2×Dic9—21no—
183C2×Dic9—21no—
184C2×Dic9—21no—
185C2×Dic9—21no—
186C2×Dic9—21no—
187C2×Dic9—21no—
188C2×Dic9—21no—
189C2×Dic9—21no—
190C2×Dic9—21no—
191C2×Dic9C2×C3×C412yes—
192C2×Dic9C2×C3×C412yes—
193C2×Dic9C3×D412yes—
194#388—91noSylow
195C2×C3×C9—42no—
196C2×C3×C9#11711yescharacteristic
197C2×C3×C9C4×D411yescharacteristic
198C2×C3×C9C4×D411yescharacteristic
199C2×C3×C9#12111yescharacteristic
200C2×C3×C9#12311yescharacteristic
201C2×C3×C9#12411yescharacteristic
202C2×C3×C9#12611yescharacteristic
203C24×C3—21no—
204C24×C3C2×D911yescharacteristic
205C24×C3C2×C3×D311yescharacteristic
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—91no—
209C22×C3×C4—91no—
210C22×C3×C4—91no—
211C22×C3×C4—91no—
212C22×C3×C4C2×D911yescharacteristic
213C22×C3×C4C2×D911yescharacteristic
214C22×C3×C4C2×C3×D311yescharacteristic
215C22×C3×C4C2×C3×D311yescharacteristic
216C22×Dic3—31no—
217C22×Dic3—31no—
218C22×Dic3—31no—
219C22×Dic3—31no—
220C3×C4⋊C4—182no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—94no—
225C3×C22⋊C4—182no—
226C3×C42—182no—
227#217—34no—
228#217—62no—
229Dic3⋊C4—62no—
230C4×Dic3—62no—
231C22×C32—22no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—41no—
235C22×C32—41no—
236C22×C32—41no—
237C22×C32—41no—
238C22×C32C2×C4×D311yescharacteristic
239C22×C32#23511yescharacteristic
240C22×C32D3×D411yescharacteristic
241C22×C32#23711yescharacteristic
242C22×C32#23711yescharacteristic
243C22×C32#23711yescharacteristic
244C22×C32C2×#6711yescharacteristic
245C32×C4—22no—
246C32×C4—41no—
247C3×Dic3—62no—
248C3×Dic3—62no—
249C3×Dic3—62no—
250C3×Dic3—121no—
251C22×C9—24no—
252C22×C9—22no—
253C22×C9—22no—
254C22×C9—22no—
255C22×C9—21no—
256C22×C9—21no—
257C22×C9—21no—
258C22×C9—21no—
259C22×C9—21no—
260C22×C9—21no—
261C22×C9—21no—
262C22×C9—42no—
263C22×C9—42no—
264C22×C9—42no—
265C22×C9—42no—
266C22×C9—42no—
267C22×C9—42no—
268C22×C9—41no—
269C22×C9—41no—
270C22×C9—41no—
271C22×C9—41no—
272C22×C9C22×C3×C411yescharacteristic
273C22×C9C2×C3×D411yescharacteristic
274C22×C9C2×C3×D411yescharacteristic
275C22×C9C3×#4111yescharacteristic
276C22×C9C3×#4111yescharacteristic
277C22×C9C3×#4111yescharacteristic
278C22×C9C3×#4111yescharacteristic
279C4×C9—22no—
280C4×C9—42no—
281C4×C9—42no—
282C4×C9—41no—
283Dic9—22no—
284Dic9—22no—
285Dic9—22no—
286Dic9—41no—
287C2×C4⋊C4—91no—
288C2×C22⋊C4—91no—
289C2×C22⋊C4—91no—
290C2×C22⋊C4C3×D911yescharacteristic
291C2×C42—91no—
292#95—91no—
293#95—91no—
294C3×C9#38811yescharacteristic, complemented, Sylow
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—22no—
310C23×C3—21no—
311C23×C3—41no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3C4×D912yes—
316C23×C3C22×D911yescharacteristic
317C23×C3C3×C4×D312yes—
318C23×C3C22×C3×D311yescharacteristic
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—22no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—92no—
336C2×C3×C4—92no—
337C2×C3×C4—92no—
338C2×C3×C4—181no—
339C2×C3×C4—181no—
340C2×C3×C4—181no—
341C2×C3×C4—181no—
342C2×C3×C4—181no—
343C2×C3×C4—181no—
344C2×C3×C4—181no—
345C2×C3×C4—181no—
346C2×C3×C4—181no—
347C2×C3×C4#61112yes—
348C2×C3×C4C3×#6712yes—
349C2×Dic3—32no—
350C2×Dic3—32no—
351C2×Dic3—32no—
352C2×Dic3—61no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360C2×Dic3—61no—
361C2×C32—42no—
362C2×C32#87711yescharacteristic
363C2×C32#87911yescharacteristic
364C2×C32#67⋊C411yescharacteristic
365C2×C32#88511yescharacteristic
366C2×C32C4×#6711yescharacteristic
367C2×C32#93511yescharacteristic
368C2×C32#93811yescharacteristic
369C2×C9—21no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—21no—
373C2×C9—21no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9—44no—
377C2×C9—42no—
378C2×C9C3×#11711yescharacteristic
379C2×C9C3×C4×D411yescharacteristic
380C2×C9C3×C4×D411yescharacteristic
381C2×C9C3×#12111yescharacteristic
382C2×C9C3×#12311yescharacteristic
383C2×C9C3×#12411yescharacteristic
384C2×C9C3×#12611yescharacteristic
385C24C2×C3×D911yescharacteristic
386C22×C4—91no—
387C22×C4—91no—
388C22×C4—91no—
389C22×C4—91no—
390C22×C4C2×C3×D911yescharacteristic
391C22×C4C2×C3×D911yescharacteristic
392C4⋊C4—182no—
393C22⋊C4—22no—
394C22⋊C4—94no—
395C22⋊C4—182no—
396C42—182no—
397C22×C3—24no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—41no—
418C22×C3—41no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3C2×C4×D911yescharacteristic
426C22×C3#369711yescharacteristic
427C22×C3D4×D911yescharacteristic
428C22×C3#369911yescharacteristic
429C22×C3#369911yescharacteristic
430C22×C3#369911yescharacteristic
431C22×C3C2×#61111yescharacteristic
432C22×C3C2×C3×C4×D311yescharacteristic
433C22×C3C3×#23511yescharacteristic
434C22×C3C3×D3×D411yescharacteristic
435C22×C3C3×#23711yescharacteristic
436C22×C3C3×#23711yescharacteristic
437C22×C3C3×#23711yescharacteristic
438C22×C3C2×C3×#6711yescharacteristic
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—182no—
446C3×C4—182no—
447C3×C4—182no—
448C3×C4—361no—
449Dic3—62no—
450Dic3—62no—
451Dic3—62no—
452Dic3—121no—
453C32#494411yescharacteristic
454C9—21no—
455C9C3×#38811yescharacteristic, complemented
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23C3×C4×D912yes—
463C23C22×C3×D911yescharacteristic
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—92no—
470C2×C4—92no—
471C2×C4—92no—
472C2×C4—181no—
473C2×C4—181no—
474C2×C4—181no—
475C2×C4—181no—
476C2×C4—181no—
477C2×C4—181no—
478C2×C4—181no—
479C2×C4—181no—
480C2×C4—181no—
481C2×C4C3×#61112yes—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—44no—
490C2×C3—42no—
491C2×C3—42no—
492C2×C3#6343911yescharacteristic
493C2×C3#6344111yescharacteristic
494C2×C3#611⋊C411yescharacteristic
495C2×C3#6344711yescharacteristic
496C2×C3C4×#61111yescharacteristic
497C2×C3#6349711yescharacteristic
498C2×C3#6350011yescharacteristic
499C2×C3C3×#87711yescharacteristic
500C2×C3C3×#87911yescharacteristic
501C2×C3C3×#67⋊C411yescharacteristic
502C2×C3C3×#88511yescharacteristic
503C2×C3C3×C4×#6711yescharacteristic
504C2×C3C3×#93511yescharacteristic
505C2×C3C3×#93811yescharacteristic
506C22—22no—
507C22—22no—
508C22—22no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22C2×C3×C4×D911yescharacteristic
514C22C3×#369711yescharacteristic
515C22C3×D4×D911yescharacteristic
516C22C3×#369911yescharacteristic
517C22C3×#369911yescharacteristic
518C22C3×#369911yescharacteristic
519C22C2×C3×#61111yescharacteristic
520C4—22no—
521C4—41no—
522C4—182no—
523C4—182no—
524C4—182no—
525C4—361no—
526C3—21no—
527C3#1058862411yescharacteristic, direct product factor
528C3C3×#494411yescharacteristic
529C2—42no—
530C2C3×#6343911yescharacteristic
531C2C3×#6344111yescharacteristic
532C2C3×#611⋊C411yescharacteristic
533C2C3×#6344711yescharacteristic
534C2C3×C4×#61111yescharacteristic
535C2C3×#6349711yescharacteristic
536C2C3×#6350011yescharacteristic
subgroupG/H
centerC23×C3C22×D9
commutatorC22×C9C22×C3×C4
FrattiniC23×C3C22×C3×D3
FittingC2×C3×C9×C22⋊C4C2
socleC23×C32C22×D3
prime radicalC24×C32C2×D3
cosocleC23×C9C23×C3
hypercenterC2×C3×C22⋊C4D9
hypocenterC9C3×#388
LCSG ▷ C22×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×Dic9⋊C4C2semidirect
5. C2×C3×C4×Dic9C2semidirect
6. C3×C9⋊#95C2semidirect
7. C3×C9⋊#95C2semidirect
9. #10588624C3direct (N × H)
19. C3×#3676C4semidirect
33. C2×Dic9⋊C4C2×C3semidirect
34. C2×C4×Dic9C2×C3semidirect
35. C9⋊#95C2×C3semidirect
36. C9⋊#95C2×C3semidirect
90. #3676C3×C4semidirect
294. C3×C9#388semidirect
455. C9C3×#388semidirect
527. C3#10588624direct (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×Dic9⋊C42
C2×C3×C4×Dic92
C3×C9⋊#952
C3×#49443
#105886243