Finite Group Databasebeta

← C3×#10588674 #13142560

Subgroups — C3×#10588674 #13142560

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic
3C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic
4C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic
5C2×C3×C4⋊Dic9C211yesmaximal, characteristic, complemented
6C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
7C3×C9⋊#95C211yesmaximal, characteristic, complemented
8C3×#4994—31nomaximal
9#10588674C311yesmaximal, characteristic, direct product factor
10C23×C3×D9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3×Dic9C2211yescharacteristic
15C22×C3×Dic9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C3×C9×C4⋊C4—22no—
18C3×(C2×D9)⋊C4—22no—
19C3×(C2×D9)⋊C4—22no—
20C3×(C2×D9)⋊C4—22no—
21C3×C4⋊Dic9—22no—
22C3×Dic9⋊C4—22no—
23C2×C32×C4⋊C4D311yescharacteristic
24C2×C3×(C2×D3)⋊C4—31no—
25C2×C3×(C2×D3)⋊C4—31no—
26C2×C3×(C2×D3)⋊C4—31no—
27C2×C3×C4⋊Dic3—31no—
28C2×C3×Dic3⋊C4—31no—
29C3×C3⋊#95—31no—
30C2×C9×C4⋊C4—21no—
31C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
32C2×(C2×D9)⋊C4C2×C311yescharacteristic
33C2×(C2×D9)⋊C4C2×C311yescharacteristic
34C2×(C2×D9)⋊C4C2×C311yescharacteristic
35C2×C4⋊Dic9C2×C311yescharacteristic, complemented
36C2×Dic9⋊C4C2×C311yescharacteristic, complemented
37C9⋊#95C2×C311yescharacteristic, complemented
38C23×C3×C9C2311yescharacteristic
39C22×C3×D9—21no—
40C22×C3×D9—21no—
41C22×C3×D9—21no—
42C22×C3×D9—21no—
43C22×C3×D9—21no—
44C22×C3×D9—21no—
45C22×C3×D9Q812yes—
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×C9—21no—
52C2×C3×C4×C9—21no—
53C2×C3×C4×C9D412yes—
54C2×C3×C4×C9D412yes—
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×Dic9—21no—
63C2×C3×Dic9D412yes—
64C3×#396—91no—
65#4994—31no—
66C23×C3×D3—31no—
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C32×C4C2×D311yescharacteristic
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3—31no—
73C23×D9C22×C311yescharacteristic
74C32×C4⋊C4—22no—
75C3×(C2×D3)⋊C4—62no—
76C3×(C2×D3)⋊C4—62no—
77C3×(C2×D3)⋊C4—62no—
78C3×C4⋊Dic3—62no—
79C3×Dic3⋊C4—62no—
80C22×C4×C9—21no—
81C22×C4×C9—21no—
82C22×C4×C9—21no—
83C22×C4×C9C22×C311yescharacteristic
84C22×C4×C9C22×C311yescharacteristic
85C22×C4×C9C22×C311yescharacteristic
86C22×Dic9C22×C311yescharacteristic
87C22×Dic9C22×C311yescharacteristic
88C22×Dic9C22×C311yescharacteristic
89C9×C4⋊C4—24no—
90C9×C4⋊C4—22no—
91(C2×D9)⋊C4—22no—
92(C2×D9)⋊C4—22no—
93(C2×D9)⋊C4—22no—
94C4⋊Dic9—22no—
95Dic9⋊C4—22no—
96C22×C3×C9C2×D411yescharacteristic
97C22×C3×C9C2×D411yescharacteristic
98C22×C3×C9C2×D411yescharacteristic
99C22×C3×C9C2×Q811yescharacteristic
100C22×C3×C9#4111yescharacteristic
101C22×C3×C9#4111yescharacteristic
102C22×C3×C9#4111yescharacteristic
103C2×C3×D9—22no—
104C2×C3×D9—22no—
105C2×C3×D9—22no—
106C2×C3×D9—41no—
107C2×C3×D9—41no—
108C2×C3×D9—41no—
109C2×C3×D9—41no—
110C3×C4×C9—22no—
111C3×C4×C9—22no—
112C3×C4×C9—41no—
113C3×Dic9—22no—
114C3×Dic9—41no—
115C3×Dic9—41no—
116C2×C3×C4⋊C4—21no—
117C2×C3×C4⋊C4—91no—
118C2×C3×C4⋊C4—91no—
119C2×C3×C4⋊C4D911yescharacteristic, complemented
120C2×C3×C4⋊C4C3×D311yescharacteristic
121C2×C3×C22⋊C4—91no—
122C2×C3×C22⋊C4—91no—
123C2×C3×C22⋊C4—91no—
124C2×(C2×D3)⋊C4—31no—
125C2×(C2×D3)⋊C4—31no—
126C2×(C2×D3)⋊C4—31no—
127C2×C4⋊Dic3—31no—
128C2×Dic3⋊C4—31no—
129C3×#95—91no—
130C3⋊#95—31no—
131C23×C32C22×D311yescharacteristic
132C22×C3×D3—32no—
133C22×C3×D3—61no—
134C22×C3×D3—61no—
135C22×C3×D3—61no—
136C22×C3×D3—61no—
137C22×C3×D3—61no—
138C22×C3×D3—61no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4D1212yes—
147C2×C32×C4#6712yes—
148C2×C3×Dic3—32no—
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—21no—
158C23×C9C23×C311yescharacteristic
159C22×D9—21no—
160C22×D9—21no—
161C22×D9—21no—
162C22×D9—21no—
163C22×D9—21no—
164C22×D9—21no—
165C22×D9C3×Q812yes—
166C2×C4×C9—22no—
167C2×C4×C9—22no—
168C2×C4×C9—21no—
169C2×C4×C9—21no—
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—22no—
178C2×C4×C9—41no—
179C2×C4×C9—41no—
180C2×C4×C9—41no—
181C2×C4×C9—41no—
182C2×C4×C9C3×D412yes—
183C2×C4×C9C3×D412yes—
184C2×Dic9—21no—
185C2×Dic9—21no—
186C2×Dic9—21no—
187C2×Dic9—21no—
188C2×Dic9—21no—
189C2×Dic9—21no—
190C2×Dic9—21no—
191C2×Dic9—21no—
192C2×Dic9C3×D412yes—
193#396—91noSylow
194C2×C3×C9#12111yescharacteristic
195C2×C3×C9#12111yescharacteristic
196C2×C3×C9#12111yescharacteristic
197C2×C3×C9C22⋊Q811yescharacteristic
198C2×C3×C9C22⋊Q811yescharacteristic
199C2×C3×C9C22⋊Q811yescharacteristic
200C2×C3×C9#12611yescharacteristic
201C3×D9—42no—
202C24×C3—91no—
203C23×D3—31no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—91no—
208C22×C3×C4—91no—
209C22×C3×C4—91no—
210C22×C3×C4C2×D911yescharacteristic
211C22×C3×C4C2×D911yescharacteristic
212C22×C3×C4C2×D911yescharacteristic
213C22×C3×C4C2×C3×D311yescharacteristic
214C22×C3×C4C2×C3×D311yescharacteristic
215C22×C3×C4C2×C3×D311yescharacteristic
216C22×Dic3—31no—
217C22×Dic3—31no—
218C22×Dic3—31no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—182no—
223C3×C4⋊C4—182no—
224C3×C22⋊C4—182no—
225C3×C22⋊C4—182no—
226C3×C22⋊C4—182no—
227(C2×D3)⋊C4—62no—
228(C2×D3)⋊C4—62no—
229(C2×D3)⋊C4—62no—
230C4⋊Dic3—62no—
231Dic3⋊C4—62no—
232C22×C32C2×D1211yescharacteristic
233C22×C32#23511yescharacteristic
234C22×C32D3×D411yescharacteristic
235C22×C32#23711yescharacteristic
236C22×C32D3×Q811yescharacteristic
237C22×C32#23911yescharacteristic
238C22×C32C2×#6711yescharacteristic
239C2×C3×D3—62no—
240C2×C3×D3—62no—
241C2×C3×D3—62no—
242C2×C3×D3—121no—
243C2×C3×D3—121no—
244C2×C3×D3—121no—
245C2×C3×D3—121no—
246C32×C4—22no—
247C32×C4—22no—
248C32×C4—41no—
249C3×Dic3—62no—
250C3×Dic3—121no—
251C3×Dic3—121no—
252C22×C9—21no—
253C22×C9—21no—
254C22×C9—21no—
255C22×C9—21no—
256C22×C9—21no—
257C22×C9—21no—
258C22×C9—21no—
259C22×C9C2×C3×D411yescharacteristic
260C22×C9C2×C3×D411yescharacteristic
261C22×C9C2×C3×D411yescharacteristic
262C22×C9C2×C3×Q811yescharacteristic
263C22×C9C3×#4111yescharacteristic
264C22×C9C3×#4111yescharacteristic
265C22×C9C3×#4111yescharacteristic
266C2×D9—22no—
267C2×D9—22no—
268C2×D9—22no—
269C2×D9—41no—
270C2×D9—41no—
271C2×D9—41no—
272C2×D9—41no—
273C4×C9—24no—
274C4×C9—22no—
275C4×C9—22no—
276C4×C9—42no—
277C4×C9—42no—
278C4×C9—41no—
279Dic9—22no—
280Dic9—41no—
281Dic9—41no—
282C2×C4⋊C4—91no—
283C2×C4⋊C4—91no—
284C2×C4⋊C4C3×D911yescharacteristic, complemented
285C2×C22⋊C4—91no—
286C2×C22⋊C4—91no—
287C2×C22⋊C4—91no—
288#95—91no—
289C3×C9#39611yescharacteristic, complemented, Sylow
290C23×C3—21no—
291C23×C3—92no—
292C23×C3—181no—
293C23×C3—181no—
294C23×C3—181no—
295C23×C3—181no—
296C23×C3—181no—
297C23×C3—181no—
298C23×C3C22×D911yescharacteristic
299C23×C3C22×C3×D311yescharacteristic
300C22×D3—32no—
301C22×D3—61no—
302C22×D3—61no—
303C22×D3—61no—
304C22×D3—61no—
305C22×D3—61no—
306C22×D3—61no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
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—22no—
325C2×C3×C4—22no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—92no—
331C2×C3×C4—181no—
332C2×C3×C4—181no—
333C2×C3×C4—181no—
334C2×C3×C4—181no—
335C2×C3×C4—181no—
336C2×C3×C4—181no—
337C2×C3×C4—181no—
338C2×C3×C4—181no—
339C2×C3×C4D3612yes—
340C2×C3×C4#61112yes—
341C2×C3×C4C3×D1212yes—
342C2×C3×C4C3×#6712yes—
343C2×Dic3—32no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352C2×C32#89511yescharacteristic
353C2×C32#89611yescharacteristic
354C2×C32#89711yescharacteristic
355C2×C32#89811yescharacteristic
356C2×C32#93011yescharacteristic
357C2×C32#93911yescharacteristic
358C2×C32#94611yescharacteristic
359C3×D3—122no—
360C2×C9—21no—
361C2×C9—21no—
362C2×C9—21no—
363C2×C9—21no—
364C2×C9—21no—
365C2×C9—21no—
366C2×C9—21no—
367C2×C9C3×#12111yescharacteristic
368C2×C9C3×#12111yescharacteristic
369C2×C9C3×#12111yescharacteristic
370C2×C9C3×C22⋊Q811yescharacteristic
371C2×C9C3×C22⋊Q811yescharacteristic
372C2×C9C3×C22⋊Q811yescharacteristic
373C2×C9C3×#12611yescharacteristic
374D9—42no—
375C24—91no—
376C22×C4—91no—
377C22×C4—91no—
378C22×C4—91no—
379C22×C4C2×C3×D911yescharacteristic
380C22×C4C2×C3×D911yescharacteristic
381C22×C4C2×C3×D911yescharacteristic
382C4⋊C4—22no—
383C4⋊C4—182no—
384C4⋊C4—182no—
385C22⋊C4—182no—
386C22⋊C4—182no—
387C22⋊C4—182no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—182no—
396C22×C3—182no—
397C22×C3—182no—
398C22×C3—361no—
399C22×C3—361no—
400C22×C3—361no—
401C22×C3—361no—
402C22×C3C2×D3611yescharacteristic
403C22×C3#369711yescharacteristic
404C22×C3D4×D911yescharacteristic
405C22×C3#369911yescharacteristic
406C22×C3Q8×D911yescharacteristic
407C22×C3#370111yescharacteristic
408C22×C3C2×#61111yescharacteristic
409C22×C3C2×C3×D1211yescharacteristic
410C22×C3C3×#23511yescharacteristic
411C22×C3C3×D3×D411yescharacteristic
412C22×C3C3×#23711yescharacteristic
413C22×C3C3×D3×Q811yescharacteristic
414C22×C3C3×#23911yescharacteristic
415C22×C3C2×C3×#6711yescharacteristic
416C2×D3—62no—
417C2×D3—62no—
418C2×D3—62no—
419C2×D3—121no—
420C2×D3—121no—
421C2×D3—121no—
422C2×D3—121no—
423C3×C4—24no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4—42no—
429C3×C4—42no—
430C3×C4—41no—
431C3×C4—41no—
432C3×C4—182no—
433C3×C4—361no—
434C3×C4—361no—
435Dic3—62no—
436Dic3—121no—
437Dic3—121no—
438C32#499411yescharacteristic
439C9—21no—
440C9C3×#39611yescharacteristic, complemented
441C23—92no—
442C23—181no—
443C23—181no—
444C23—181no—
445C23—181no—
446C23—181no—
447C23—181no—
448C23C22×C3×D911yescharacteristic
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—92no—
457C2×C4—181no—
458C2×C4—181no—
459C2×C4—181no—
460C2×C4—181no—
461C2×C4—181no—
462C2×C4—181no—
463C2×C4—181no—
464C2×C4—181no—
465C2×C4C3×D3612yes—
466C2×C4C3×#61112yes—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—362no—
475C2×C3#6345711yescharacteristic
476C2×C3#6345811yescharacteristic
477C2×C3#6345911yescharacteristic
478C2×C3#6346011yescharacteristic
479C2×C3#6349211yescharacteristic
480C2×C3#6350111yescharacteristic
481C2×C3#6350811yescharacteristic
482C2×C3C3×#89511yescharacteristic
483C2×C3C3×#89611yescharacteristic
484C2×C3C3×#89711yescharacteristic
485C2×C3C3×#89811yescharacteristic
486C2×C3C3×#93011yescharacteristic
487C2×C3C3×#93911yescharacteristic
488C2×C3C3×#94611yescharacteristic
489D3—122no—
490C22—182no—
491C22—182no—
492C22—182no—
493C22—361no—
494C22—361no—
495C22—361no—
496C22—361no—
497C22C2×C3×D3611yescharacteristic
498C22C3×#369711yescharacteristic
499C22C3×D4×D911yescharacteristic
500C22C3×#369911yescharacteristic
501C22C3×Q8×D911yescharacteristic
502C22C3×#370111yescharacteristic
503C22C2×C3×#61111yescharacteristic
504C4—22no—
505C4—22no—
506C4—41no—
507C4—182no—
508C4—361no—
509C4—361no—
510C3—21no—
511C3#1058867411yescharacteristic, direct product factor
512C3C3×#499411yescharacteristic
513C2—362no—
514C2C3×#6345711yescharacteristic
515C2C3×#6345811yescharacteristic
516C2C3×#6345911yescharacteristic
517C2C3×#6346011yescharacteristic
518C2C3×#6349211yescharacteristic
519C2C3×#6350111yescharacteristic
520C2C3×#6350811yescharacteristic
subgroupG/H
centerC23×C3C22×D9
commutator, cosocleC23×C9C23×C3
FrattiniC23×C3C22×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socleC23×C32C22×D3
prime radicalC23×C3×D9C22
hypercenterC2×C3×C4⋊C4D9
hypocenterC9C3×#396
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×C4⋊C4C2semidirect
5. C2×C3×C4⋊Dic9C2semidirect
6. C2×C3×Dic9⋊C4C2semidirect
7. C3×C9⋊#95C2semidirect
9. #10588674C3direct (N × H)
31. C2×C9×C4⋊C4C2×C3semidirect
35. C2×C4⋊Dic9C2×C3semidirect
36. C2×Dic9⋊C4C2×C3semidirect
37. C9⋊#95C2×C3semidirect
119. C2×C3×C4⋊C4D9semidirect
284. C2×C4⋊C4C3×D9semidirect
289. C3×C9#396semidirect
440. C9C3×#396semidirect
511. C3#10588674direct (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×C4⋊C42
C2×C3×(C2×D9)⋊C42
C2×C3×C4⋊Dic92
C2×C3×Dic9⋊C42
C3×C9⋊#952
C3×#49943
#105886743