Finite Group Databasebeta

← C3×#10589707 #13152808

Subgroups — C3×#10589707 #13152808

Order 1728.

1132 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×D9C212yesmaximal, complemented
2C3×C9×#121C211yesmaximal, characteristic, complemented
3C3×#63501C212yesmaximal, complemented
4C3×#63499C212yesmaximal, complemented
5C3×#63492C211yesmaximal, characteristic, complemented
6C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
7C3×#63456C212yesmaximal, complemented
8C3×#63445C212yesmaximal, complemented
9C3×#63444C212yesmaximal, complemented
10C3×#5605—31nomaximal
11#10589707C311yesmaximal, characteristic, direct product factor
12C23×C3×D9C2211yescharacteristic
13C23×C3×D9C2212yescomplemented
14C2×C3×D4×C9C2212yescomplemented
15C2×C3×D4×C9C2211yescharacteristic, complemented
16C22×C3×C4×C9C2211yescharacteristic, complemented
17C2×C3×#611C2212yes—
18C2×C3×#611C2212yescomplemented
19C2×C3×#611C2212yescomplemented
20C22×C3×Dic9C2211yescharacteristic, complemented
21C3×D4×D9—28no—
22C2×C3×D36C2212yes—
23C2×C3×D36C2211yescharacteristic, complemented
24C2×C3×C4×D9C2212yescomplemented
25C3×C9×C4⋊C4C2211yescharacteristic, complemented
26C3×C9×C22⋊C4C2212yescomplemented
27C3×#3676C2212yescomplemented
28C3×(C2×D9)⋊C4C2212yescomplemented
29C3×(C2×D9)⋊C4C2212yescomplemented
30C3×(C2×D9)⋊C4C2212yescomplemented
31C3×(C2×D9)⋊C4C2212yescomplemented
32C3×C4⋊Dic9C2211yescharacteristic, complemented
33C3×Dic9⋊C4C2212yescomplemented
34C2×C3×D3×D4—32no—
35C32×#121D311yescharacteristic
36C3×#939—32no—
37C3×#937—32no—
38C3×#930—31no—
39C2×C3×(C2×D3)⋊C4—31no—
40C3×#894—32no—
41C3×#883—32no—
42C3×#882—32no—
43C2×D4×D9C2×C312yescomplemented
44C9×#121—21no—
45C9×#121C2×C311yescharacteristic, complemented
46#63501C2×C312yescomplemented
47#63499C2×C312yescomplemented
48#63492C2×C311yescharacteristic, complemented
49C2×(C2×D9)⋊C4C2×C311yescharacteristic, complemented
50#63456C2×C312yescomplemented
51#63445C2×C312yescomplemented
52#63444C2×C312yescomplemented
53C23×C3×C9C2312yes—
54C23×C3×C9C2311yescharacteristic
55C22×C3×D9—24no—
56C22×C3×D9—41no—
57C22×C3×D9—41no—
58C22×C3×D9—42no—
59C22×C3×D9—42no—
60C22×C3×D9D414yescomplemented
61C22×C3×D9C2312yes—
62C22×C3×D9C2312yes—
63C3×D4×C9—24no—
64C3×D4×C9—41no—
65C2×C3×C4×C9—41no—
66C2×C3×C4×C9C2312yes—
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3×C4×C9C2311yescharacteristic
69C3×#611—24no—
70C3×#611—24no—
71C3×#611—42no—
72C2×C3×Dic9—41no—
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2312yes—
75C3×D36—24no—
76C3×D36—41no—
77C3×C4×D9—24no—
78C3×#534—91no—
79#5605—31no—
80C23×C3×D3—31no—
81C23×C3×D3—32no—
82C2×C32×D4C2×D312yes—
83C2×C32×D4C2×D311yescharacteristic
84C22×C32×C4C2×D311yescharacteristic
85C2×C3×#67—32no—
86C2×C3×#67—32no—
87C2×C3×#67—32no—
88C22×C3×Dic3—31no—
89C3×D3×D4—68no—
90C2×C3×D12—32no—
91C2×C3×D12—31no—
92C2×C3×C4×D3—32no—
93C23×D9C22×C311yescharacteristic
94C23×D9C22×C312yescomplemented
95C32×C4⋊C4C2×D311yescharacteristic
96C32×C22⋊C4C2×D312yes—
97C3×#217—32no—
98C3×(C2×D3)⋊C4—32no—
99C3×(C2×D3)⋊C4—32no—
100C3×(C2×D3)⋊C4—32no—
101C3×(C2×D3)⋊C4—32no—
102C3×C4⋊Dic3—31no—
103C3×Dic3⋊C4—32no—
104C2×D4×C9—21no—
105C2×D4×C9—22no—
106C2×D4×C9C22×C312yescomplemented
107C2×D4×C9C22×C311yescharacteristic, complemented
108C22×C4×C9—21no—
109C22×C4×C9C22×C311yescharacteristic, complemented
110C2×#611C22×C312yes—
111C2×#611C22×C312yescomplemented
112C2×#611C22×C312yescomplemented
113C22×Dic9C22×C311yescharacteristic, complemented
114D4×D9—28no—
115C2×D36C22×C312yes—
116C2×D36C22×C311yescharacteristic, complemented
117C2×C4×D9C22×C312yescomplemented
118C9×C4⋊C4—21no—
119C9×C4⋊C4C22×C311yescharacteristic, complemented
120C9×C22⋊C4—22no—
121C9×C22⋊C4C22×C312yescomplemented
122#3676C22×C312yescomplemented
123(C2×D9)⋊C4C22×C312yescomplemented
124(C2×D9)⋊C4C22×C312yescomplemented
125(C2×D9)⋊C4C22×C312yescomplemented
126(C2×D9)⋊C4C22×C312yescomplemented
127C4⋊Dic9C22×C311yescharacteristic, complemented
128Dic9⋊C4C22×C312yescomplemented
129C22×C3×C9—21no—
130C22×C3×C9—21no—
131C22×C3×C9—22no—
132C22×C3×C9—22no—
133C22×C3×C9—22no—
134C22×C3×C9C2×D412yes—
135C22×C3×C9C2411yescharacteristic
136C2×C3×D9—22no—
137C2×C3×D9—22no—
138C2×C3×D9—22no—
139C2×C3×D9—22no—
140C2×C3×D9—22no—
141C2×C3×D9—44no—
142C2×C3×D9—48no—
143C2×C3×D9C2×D414yes—
144C3×C4×C9—22no—
145C3×C4×C9—21no—
146C3×C4×C9—21no—
147C3×Dic9—22no—
148C3×Dic9—22no—
149C22×C3×D4—92no—
150C2×D3×D4—32no—
151C3×#123—92no—
152C3×#123—92no—
153C3×#121—21no—
154C3×#121—92no—
155C3×#121—91no—
156C3×#121D911yescharacteristic, complemented
157C3×#121C3×D311yescharacteristic
158C3×#120—92no—
159C3×#120—92no—
160C2×C3×C22⋊C4—91no—
161#939—32no—
162#937—32no—
163#930—31no—
164C2×(C2×D3)⋊C4—31no—
165#894—32no—
166#883—32no—
167#882—32no—
168C23×C32C22×D312yes—
169C23×C32C22×D311yescharacteristic
170C22×C3×D3—32no—
171C22×C3×D3—34no—
172C22×C3×D3—32no—
173C22×C3×D3—64no—
174C22×C3×D3—121no—
175C22×C3×D3—122no—
176C22×C3×D3—121no—
177C22×C3×D3—122no—
178C32×D4—24no—
179C32×D4—41no—
180C2×C32×C4—41no—
181C2×C32×C4C22×D312yes—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C3×#67—64no—
185C3×#67—64no—
186C3×#67—122no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—121no—
190C3×D12—64no—
191C3×D12—121no—
192C3×C4×D3—64no—
193C23×C9—21no—
194C23×C9—22no—
195C23×C9C23×C312yes—
196C23×C9C23×C311yescharacteristic
197C22×D9—24no—
198C22×D9—41no—
199C22×D9—41no—
200C22×D9—42no—
201C22×D9—42no—
202C22×D9C3×D414yescomplemented
203C22×D9C23×C312yes—
204C22×D9C23×C312yes—
205D4×C9—24no—
206D4×C9—44no—
207D4×C9—41no—
208D4×C9—42no—
209C2×C4×C9—21no—
210C2×C4×C9—21no—
211C2×C4×C9—22no—
212C2×C4×C9—41no—
213C2×C4×C9—42no—
214C2×C4×C9C23×C312yes—
215C2×C4×C9C23×C311yescharacteristic
216C2×C4×C9C23×C311yescharacteristic
217#611—24no—
218#611—24no—
219#611—42no—
220C2×Dic9—41no—
221C2×Dic9C23×C312yes—
222C2×Dic9C23×C312yes—
223D36—24no—
224D36—41no—
225C4×D9—24no—
226#534—91noSylow
227C2×C3×C9—22no—
228C2×C3×C9—42no—
229C2×C3×C9C22×D411yescharacteristic
230C2×C3×C9H(5,𝔽2)11yescharacteristic
231C2×C3×C9H(5,𝔽2)11yescharacteristic
232C3×D9—24no—
233C3×D9—42no—
234C24×C3—91no—
235C24×C3—92no—
236C23×D3—31no—
237C23×D3—32no—
238C2×C3×D4—22no—
239C2×C3×D4—21no—
240C2×C3×D4—92no—
241C2×C3×D4—92no—
242C2×C3×D4—92no—
243C2×C3×D4—92no—
244C2×C3×D4—91no—
245C2×C3×D4—188no—
246C2×C3×D4C2×D912yescomplemented
247C2×C3×D4C2×D911yescharacteristic, complemented
248C2×C3×D4C2×C3×D312yes—
249C2×C3×D4C2×C3×D311yescharacteristic
250C22×C3×C4—21no—
251C22×C3×C4—91no—
252C22×C3×C4—92no—
253C22×C3×C4C2×D911yescharacteristic, complemented
254C22×C3×C4C2×C3×D311yescharacteristic
255C2×#67—32no—
256C2×#67—32no—
257C2×#67—32no—
258C22×Dic3—31no—
259D3×D4—68no—
260C2×D12—32no—
261C2×D12—31no—
262C2×C4×D3—32no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4—92no—
265C3×C4⋊C4—91no—
266C3×C4⋊C4C2×D911yescharacteristic, complemented
267C3×C4⋊C4C2×C3×D311yescharacteristic
268C3×C22⋊C4—22no—
269C3×C22⋊C4—92no—
270C3×C22⋊C4—92no—
271C3×C22⋊C4—92no—
272C3×C22⋊C4—92no—
273C3×C22⋊C4—92no—
274C3×C22⋊C4C2×D912yescomplemented
275C3×C22⋊C4C2×C3×D312yes—
276#217—32no—
277(C2×D3)⋊C4—32no—
278(C2×D3)⋊C4—32no—
279(C2×D3)⋊C4—32no—
280(C2×D3)⋊C4—32no—
281C4⋊Dic3—31no—
282Dic3⋊C4—32no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—22no—
286C22×C32—22no—
287C22×C32—22no—
288C22×C32D3×D412yes—
289C22×C32C23×D311yescharacteristic
290C2×C3×D3—34no—
291C2×C3×D3—62no—
292C2×C3×D3—62no—
293C2×C3×D3—62no—
294C2×C3×D3—62no—
295C2×C3×D3—62no—
296C2×C3×D3—124no—
297C2×C3×D3—128no—
298C32×C4—21no—
299C32×C4—22no—
300C32×C4—21no—
301C3×Dic3—62no—
302C3×Dic3—62no—
303C22×C9—21no—
304C22×C9—21no—
305C22×C9—22no—
306C22×C9—22no—
307C22×C9—22no—
308C22×C9—21no—
309C22×C9—22no—
310C22×C9—41no—
311C22×C9—41no—
312C22×C9—42no—
313C22×C9—42no—
314C22×C9—42no—
315C22×C9C2×C3×D412yes—
316C22×C9C24×C311yescharacteristic
317C2×D9—22no—
318C2×D9—22no—
319C2×D9—22no—
320C2×D9—22no—
321C2×D9—22no—
322C2×D9—44no—
323C2×D9—48no—
324C2×D9C2×C3×D414yes—
325C4×C9—22no—
326C4×C9—22no—
327C4×C9—21no—
328C4×C9—21no—
329C4×C9—41no—
330C4×C9—42no—
331Dic9—22no—
332Dic9—22no—
333C22×D4—92no—
334#123—92no—
335#123—92no—
336#121—92no—
337#121—91no—
338#121C3×D911yescharacteristic, complemented
339#120—92no—
340#120—92no—
341C2×C22⋊C4—91no—
342C3×C9#53411yescharacteristic, complemented, Sylow
343C23×C3—21no—
344C23×C3—22no—
345C23×C3—92no—
346C23×C3—94no—
347C23×C3—92no—
348C23×C3—184no—
349C23×C3—361no—
350C23×C3—361no—
351C23×C3—362no—
352C23×C3—362no—
353C23×C3C22×D912yes—
354C23×C3C22×D911yescharacteristic
355C23×C3C22×C3×D312yes—
356C23×C3C22×C3×D311yescharacteristic
357C22×D3—32no—
358C22×D3—34no—
359C22×D3—32no—
360C22×D3—64no—
361C22×D3—121no—
362C22×D3—122no—
363C22×D3—121no—
364C22×D3—122no—
365C3×D4—24no—
366C3×D4—24no—
367C3×D4—44no—
368C3×D4—41no—
369C3×D4—41no—
370C3×D4—42no—
371C3×D4—184no—
372C3×D4—184no—
373C3×D4—184no—
374C3×D4—361no—
375C3×D4—362no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—22no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—42no—
382C2×C3×C4—92no—
383C2×C3×C4—92no—
384C2×C3×C4—184no—
385C2×C3×C4—361no—
386C2×C3×C4C22×D911yescharacteristic
387C2×C3×C4C22×D911yescharacteristic
388C2×C3×C4C22×D912yes—
389C2×C3×C4C22×C3×D312yes—
390C2×C3×C4C22×C3×D311yescharacteristic
391C2×C3×C4C22×C3×D311yescharacteristic
392#67—64no—
393#67—64no—
394#67—122no—
395C2×Dic3—32no—
396C2×Dic3—32no—
397C2×Dic3—121no—
398D12—64no—
399D12—121no—
400C4×D3—64no—
401C2×C32—22no—
402C2×C32—42no—
403C2×C32C2×D3×D411yescharacteristic
404C2×C32#100411yescharacteristic
405C2×C32#100911yescharacteristic
406C3×D3—64no—
407C3×D3—122no—
408C2×C9—22no—
409C2×C9—21no—
410C2×C9—21no—
411C2×C9—21no—
412C2×C9—42no—
413C2×C9—42no—
414C2×C9—82no—
415C2×C9C22×C3×D411yescharacteristic
416C2×C9C3×H(5,𝔽2)11yescharacteristic
417C2×C9C3×H(5,𝔽2)11yescharacteristic
418D9—24no—
419D9—42no—
420C24—91no—
421C24—92no—
422C2×D4—92no—
423C2×D4—92no—
424C2×D4—92no—
425C2×D4—92no—
426C2×D4—91no—
427C2×D4—188no—
428C2×D4C2×C3×D912yescomplemented
429C2×D4C2×C3×D911yescharacteristic, complemented
430C22×C4—91no—
431C22×C4—92no—
432C22×C4C2×C3×D911yescharacteristic, complemented
433C4⋊C4—92no—
434C4⋊C4—91no—
435C4⋊C4C2×C3×D911yescharacteristic, complemented
436C22⋊C4—92no—
437C22⋊C4—92no—
438C22⋊C4—92no—
439C22⋊C4—92no—
440C22⋊C4—92no—
441C22⋊C4C2×C3×D912yescomplemented
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—21no—
453C22×C3—22no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—42no—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—94no—
460C22×C3—182no—
461C22×C3—182no—
462C22×C3—182no—
463C22×C3—182no—
464C22×C3—182no—
465C22×C3—364no—
466C22×C3—368no—
467C22×C3D4×D912yes—
468C22×C3C23×D911yescharacteristic
469C22×C3C3×D3×D412yes—
470C22×C3C23×C3×D311yescharacteristic
471C2×D3—34no—
472C2×D3—62no—
473C2×D3—62no—
474C2×D3—62no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—124no—
478C2×D3—128no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—22no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—41no—
487C3×C4—42no—
488C3×C4—182no—
489C3×C4—182no—
490Dic3—62no—
491Dic3—62no—
492C32#560511yescharacteristic
493C9—21no—
494C9C3×#53411yescharacteristic, complemented
495C23—92no—
496C23—94no—
497C23—92no—
498C23—184no—
499C23—361no—
500C23—361no—
501C23—362no—
502C23—362no—
503C23C22×C3×D912yes—
504C23C22×C3×D911yescharacteristic
505D4—24no—
506D4—41no—
507D4—184no—
508D4—184no—
509D4—184no—
510D4—361no—
511D4—362no—
512C2×C4—41no—
513C2×C4—92no—
514C2×C4—92no—
515C2×C4—184no—
516C2×C4—361no—
517C2×C4C22×C3×D911yescharacteristic
518C2×C4C22×C3×D911yescharacteristic
519C2×C4C22×C3×D912yes—
520C2×C3—22no—
521C2×C3—22no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—42no—
526C2×C3—42no—
527C2×C3—42no—
528C2×C3—82no—
529C2×C3—184no—
530C2×C3—362no—
531C2×C3C2×D4×D911yescharacteristic
532C2×C3#6371011yescharacteristic
533C2×C3#6371511yescharacteristic
534C2×C3C2×C3×D3×D411yescharacteristic
535C2×C3C3×#100411yescharacteristic
536C2×C3C3×#100911yescharacteristic
537D3—64no—
538D3—122no—
539C22—21no—
540C22—21no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—94no—
545C22—182no—
546C22—182no—
547C22—182no—
548C22—182no—
549C22—182no—
550C22—364no—
551C22—368no—
552C22C3×D4×D912yes—
553C22C23×C3×D911yescharacteristic
554C4—22no—
555C4—21no—
556C4—21no—
557C4—182no—
558C4—182no—
559C3—21no—
560C3#1058970711yescharacteristic, direct product factor
561C3C3×#560511yescharacteristic
562C2—22no—
563C2—42no—
564C2—184no—
565C2—362no—
566C2C2×C3×D4×D911yescharacteristic
567C2C3×#6371011yescharacteristic
568C2C3×#6371511yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#121C2
socleC22×C32C23×D3
hypercenterC3×#121D9
hypocenterC9C3×#534
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×D9C2semidirect
2. C3×C9×#121C2semidirect
3. C3×#63501C2semidirect
4. C3×#63499C2semidirect
5. C3×#63492C2semidirect
6. C2×C3×(C2×D9)⋊C4C2semidirect
7. C3×#63456C2semidirect
8. C3×#63445C2semidirect
9. C3×#63444C2semidirect
11. #10589707C3direct (N × H)
13. C23×C3×D9C22semidirect
14. C2×C3×D4×C9C22semidirect
15. C2×C3×D4×C9C22semidirect
16. C22×C3×C4×C9C22semidirect
18. C2×C3×#611C22semidirect
19. C2×C3×#611C22semidirect
20. C22×C3×Dic9C22semidirect
23. C2×C3×D36C22semidirect
24. C2×C3×C4×D9C22semidirect
25. C3×C9×C4⋊C4C22semidirect
26. C3×C9×C22⋊C4C22semidirect
27. C3×#3676C22semidirect
28. C3×(C2×D9)⋊C4C22semidirect
29. C3×(C2×D9)⋊C4C22semidirect
30. C3×(C2×D9)⋊C4C22semidirect
31. C3×(C2×D9)⋊C4C22semidirect
32. C3×C4⋊Dic9C22semidirect
33. C3×Dic9⋊C4C22semidirect
43. C2×D4×D9C2×C3semidirect
45. C9×#121C2×C3semidirect
46. #63501C2×C3semidirect
47. #63499C2×C3semidirect
48. #63492C2×C3semidirect
49. C2×(C2×D9)⋊C4C2×C3semidirect
50. #63456C2×C3semidirect
51. #63445C2×C3semidirect
52. #63444C2×C3semidirect
60. C22×C3×D9D4semidirect
94. C23×D9C22×C3semidirect
106. C2×D4×C9C22×C3semidirect
107. C2×D4×C9C22×C3semidirect
109. C22×C4×C9C22×C3semidirect
111. C2×#611C22×C3semidirect
112. C2×#611C22×C3semidirect
113. C22×Dic9C22×C3semidirect
116. C2×D36C22×C3semidirect
117. C2×C4×D9C22×C3semidirect
119. C9×C4⋊C4C22×C3semidirect
121. C9×C22⋊C4C22×C3semidirect
122. #3676C22×C3semidirect
123. (C2×D9)⋊C4C22×C3semidirect
124. (C2×D9)⋊C4C22×C3semidirect
125. (C2×D9)⋊C4C22×C3semidirect
126. (C2×D9)⋊C4C22×C3semidirect
127. C4⋊Dic9C22×C3semidirect
128. Dic9⋊C4C22×C3semidirect
156. C3×#121D9semidirect
202. C22×D9C3×D4semidirect
246. C2×C3×D4C2×D9semidirect
247. C2×C3×D4C2×D9semidirect
253. C22×C3×C4C2×D9semidirect
266. C3×C4⋊C4C2×D9semidirect
274. C3×C22⋊C4C2×D9semidirect
338. #121C3×D9semidirect
342. C3×C9#534semidirect
428. C2×D4C2×C3×D9semidirect
429. C2×D4C2×C3×D9semidirect
432. C22×C4C2×C3×D9semidirect
435. C4⋊C4C2×C3×D9semidirect
441. C22⋊C4C2×C3×D9semidirect
494. C9C3×#534semidirect
560. C3#10589707direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×D92
C3×C9×#1212
C3×#635012
C3×#634992
C3×#634922
C2×C3×(C2×D9)⋊C42
C3×#634562
C3×#634452
C3×#634442
C3×#56053
#105897073