Finite Group Databasebeta

← C3×#10589926 #13153699

Subgroups — C3×#10589926 #13153699

Order 1728.

1040 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×C9×#41C211yesmaximal, characteristic, complemented
2C2×C3×#3699C211yesmaximal, characteristic, complemented
3C3×#63508C211yesmaximal, characteristic, complemented
4C3×#63506C211yesmaximal, characteristic, complemented
5C3×#63501C212yesmaximal, complemented
6C3×#63497C212yesmaximal, complemented
7C3×D4×Dic9C211yesmaximal, characteristic, complemented
8C3×#63491C212yesmaximal, complemented
9C3×C4×#611C211yesmaximal, characteristic, complemented
10C3×#63486C211yesmaximal, characteristic, complemented
11C2×C3×Dic9⋊C4C212yesmaximal, complemented
12C3×#5824—31nomaximal
13#10589926C311yesmaximal, characteristic, direct product factor
14C3×C9×#41—24no—
15C2×C3×Q8×C9C2211yescharacteristic
16C2×C3×D4×C9C2212yes—
17C2×C3×D4×C9C2211yescharacteristic
18C22×C3×C4×C9C2212yes—
19C22×C3×C4×C9C2211yescharacteristic
20C2×C3×#611C2211yescharacteristic
21C2×C3×#611C2212yes—
22C22×C3×Dic9C2212yes—
23C22×C3×Dic9C2212yes—
24C3×#3699—24no—
25C2×C3×C4×D9C2211yescharacteristic
26C2×C3×Dic18C2211yescharacteristic
27C3×#3676C2212yes—
28C3×#3676C2211yescharacteristic
29C3×#3676C2212yes—
30C3×(C2×D9)⋊C4C2211yescharacteristic
31C3×(C2×D9)⋊C4C2212yes—
32C3×C4⋊Dic9C2211yescharacteristic
33C3×Dic9⋊C4C2211yescharacteristic
34C3×Dic9⋊C4C2212yes—
35C3×Dic9⋊C4C2212yes—
36C3×Dic9⋊C4C2212yes—
37C3×Dic9⋊C4C2212yes—
38C3×C4×Dic9C2211yescharacteristic
39C2×C32×#41D311yescharacteristic
40C2×C3×#237—31no—
41C3×#946—31no—
42C3×#944—31no—
43C3×#939—32no—
44C3×#935—32no—
45C3×D4×Dic3—31no—
46C3×#929—32no—
47C3×C4×#67—31no—
48C3×#924—31no—
49C2×C3×Dic3⋊C4—32no—
50C2×C9×#41—21no—
51C2×C9×#41C2×C311yescharacteristic, complemented
52C2×#3699C2×C311yescharacteristic, complemented
53#63508C2×C311yescharacteristic, complemented
54#63506C2×C311yescharacteristic, complemented
55#63501C2×C312yescomplemented
56#63497C2×C312yescomplemented
57D4×Dic9C2×C311yescharacteristic, complemented
58#63491C2×C312yescomplemented
59C4×#611C2×C311yescharacteristic, complemented
60#63486C2×C311yescharacteristic, complemented
61C2×Dic9⋊C4C2×C312yescomplemented
62C23×C3×C9C2311yescharacteristic
63C23×C3×C9C2312yes—
64C22×C3×D9C2311yescharacteristic
65C3×Q8×C9—22no—
66C3×D4×C9—24no—
67C3×D4×C9D414yescomplemented
68C2×C3×C4×C9—24no—
69C2×C3×C4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2312yes—
72C2×C3×C4×C9C2311yescharacteristic
73C3×#611—24no—
74C3×#611—22no—
75C2×C3×Dic9—24no—
76C2×C3×Dic9—24no—
77C2×C3×Dic9C2312yes—
78C2×C3×Dic9C2311yescharacteristic
79C2×C3×Dic9C2312yes—
80C2×C3×Dic9C2311yescharacteristic
81C2×C3×Dic9C2311yescharacteristic
82C3×C4×D9—22no—
83C3×Dic18—22no—
84C3×#547—91no—
85#5824—31no—
86C32×#41—24no—
87C2×C32×Q8C2×D311yescharacteristic
88C2×C32×D4C2×D312yes—
89C2×C32×D4C2×D311yescharacteristic
90C22×C32×C4C2×D312yes—
91C22×C32×C4C2×D311yescharacteristic
92C2×C3×#67—31no—
93C2×C3×#67—32no—
94C22×C3×Dic3—32no—
95C22×C3×Dic3—32no—
96C3×#237—64no—
97C2×C3×C4×D3—31no—
98C2×C3×Dic6—31no—
99C3×#217—32no—
100C3×#217—31no—
101C3×#217—32no—
102C3×(C2×D3)⋊C4—31no—
103C3×(C2×D3)⋊C4—32no—
104C3×C4⋊Dic3—31no—
105C3×Dic3⋊C4—31no—
106C3×Dic3⋊C4—32no—
107C3×Dic3⋊C4—32no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×C4×Dic3—31no—
111C9×#41—28no—
112C9×#41—24no—
113C2×Q8×C9—21no—
114C2×Q8×C9C22×C311yescharacteristic
115C2×D4×C9—22no—
116C2×D4×C9—21no—
117C2×D4×C9C22×C312yes—
118C2×D4×C9C22×C311yescharacteristic
119C22×C4×C9—22no—
120C22×C4×C9—21no—
121C22×C4×C9C22×C311yescharacteristic
122C22×C4×C9C22×C312yes—
123C2×#611C22×C311yescharacteristic
124C2×#611C22×C312yes—
125C22×Dic9C22×C312yes—
126C22×Dic9C22×C312yes—
127#3699—24no—
128C2×C4×D9C22×C311yescharacteristic
129C2×Dic18C22×C311yescharacteristic
130#3676C22×C311yescharacteristic
131#3676C22×C312yes—
132#3676C22×C312yes—
133(C2×D9)⋊C4C22×C311yescharacteristic
134(C2×D9)⋊C4C22×C312yes—
135C4⋊Dic9C22×C311yescharacteristic
136Dic9⋊C4C22×C311yescharacteristic
137Dic9⋊C4C22×C312yes—
138Dic9⋊C4C22×C312yes—
139Dic9⋊C4C22×C312yes—
140Dic9⋊C4C22×C312yes—
141C4×Dic9C22×C311yescharacteristic
142C22×C3×C9—22no—
143C22×C3×C9—22no—
144C22×C3×C9—21no—
145C22×C3×C9—21no—
146C22×C3×C9—21no—
147C22×C3×C9C2×D414yes—
148C22×C3×C9C2411yescharacteristic
149C2×C3×D9—21no—
150C2×C3×D9—21no—
151C2×C3×D9—21no—
152C3×C4×C9—21no—
153C3×C4×C9—22no—
154C3×C4×C9C2×D412yes—
155C3×Dic9—22no—
156C3×Dic9—21no—
157C3×Dic9—22no—
158C3×Dic9—21no—
159C3×Dic9#4112yescomplemented
160C2×C3×#41—21no—
161C2×C3×#41—91no—
162C2×C3×#41D911yescharacteristic, complemented
163C2×C3×#41C3×D311yescharacteristic
164C2×#237—31no—
165C3×C4⋊Q8—91no—
166C3×#123—92no—
167C3×#123—92no—
168C3×C22⋊Q8—91no—
169C3×C22⋊Q8—91no—
170C3×#121—92no—
171C3×C4×D4—91no—
172C3×C4×D4—91no—
173C2×C3×C4⋊C4—92no—
174#946—31no—
175#944—31no—
176#939—32no—
177#935—32no—
178D4×Dic3—31no—
179#929—32no—
180C4×#67—31no—
181#924—31no—
182C2×Dic3⋊C4—32no—
183C23×C32C22×D311yescharacteristic
184C23×C32C22×D312yes—
185C22×C3×D3—31no—
186C32×Q8—22no—
187C32×D4—24no—
188C32×D4#6714yes—
189C2×C32×C4—24no—
190C2×C32×C4—22no—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D312yes—
193C2×C32×C4C22×D311yescharacteristic
194C3×#67—62no—
195C3×#67—64no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—64no—
202C2×C3×Dic3—64no—
203C3×C4×D3—62no—
204C3×Dic6—62no—
205C23×C9—21no—
206C23×C9—22no—
207C23×C9C23×C311yescharacteristic
208C23×C9C23×C312yes—
209C22×D9C23×C311yescharacteristic
210Q8×C9—24no—
211Q8×C9—22no—
212D4×C9—28no—
213D4×C9—24no—
214D4×C9—24no—
215D4×C9C3×D414yescomplemented
216C2×C4×C9—28no—
217C2×C4×C9—24no—
218C2×C4×C9—24no—
219C2×C4×C9—22no—
220C2×C4×C9—21no—
221C2×C4×C9—22no—
222C2×C4×C9—21no—
223C2×C4×C9C23×C311yescharacteristic
224C2×C4×C9C23×C312yes—
225C2×C4×C9C23×C311yescharacteristic
226#611—24no—
227#611—22no—
228C2×Dic9—24no—
229C2×Dic9—24no—
230C2×Dic9C23×C311yescharacteristic
231C2×Dic9C23×C312yes—
232C2×Dic9C23×C311yescharacteristic
233C2×Dic9C23×C312yes—
234C2×Dic9C23×C311yescharacteristic
235C4×D9—22no—
236Dic18—22no—
237#547—91noSylow
238C2×C3×C9—24no—
239C2×C3×C9—41no—
240C2×C3×C9C22×D411yescharacteristic
241C2×C3×C9C2×#4111yescharacteristic
242C2×C3×C92−1+411yescharacteristic
243C3×D9—41no—
244C3×#41—28no—
245C3×#41—24no—
246C3×#41—24no—
247C3×#41—184no—
248C2×C3×Q8—21no—
249C2×C3×Q8—91no—
250C2×C3×Q8C2×D911yescharacteristic
251C2×C3×Q8C2×C3×D311yescharacteristic
252C2×C3×D4—22no—
253C2×C3×D4—21no—
254C2×C3×D4—91no—
255C2×C3×D4—92no—
256C2×C3×D4C2×D912yes—
257C2×C3×D4C2×D911yescharacteristic
258C2×C3×D4C2×C3×D312yes—
259C2×C3×D4C2×C3×D311yescharacteristic
260C22×C3×C4—22no—
261C22×C3×C4—21no—
262C22×C3×C4—92no—
263C22×C3×C4—92no—
264C22×C3×C4—91no—
265C22×C3×C4C2×D912yes—
266C22×C3×C4C2×D911yescharacteristic
267C22×C3×C4C2×C3×D312yes—
268C22×C3×C4C2×C3×D311yescharacteristic
269C2×#67—31no—
270C2×#67—32no—
271C22×Dic3—32no—
272C22×Dic3—32no—
273#237—64no—
274C2×C4×D3—31no—
275C2×Dic6—31no—
276C3×C4⋊C4—91no—
277C3×C4⋊C4—92no—
278C3×C4⋊C4—92no—
279C3×C4⋊C4—92no—
280C3×C4⋊C4—92no—
281C3×C4⋊C4—91no—
282C3×C22⋊C4—92no—
283C3×C22⋊C4—91no—
284C3×C22⋊C4—92no—
285C3×C22⋊C4—91no—
286C3×C22⋊C4—92no—
287C3×C42—91no—
288#217—32no—
289#217—31no—
290#217—32no—
291(C2×D3)⋊C4—31no—
292(C2×D3)⋊C4—32no—
293C4⋊Dic3—31no—
294Dic3⋊C4—31no—
295Dic3⋊C4—32no—
296Dic3⋊C4—32no—
297Dic3⋊C4—32no—
298Dic3⋊C4—32no—
299C4×Dic3—31no—
300C22×C32—21no—
301C22×C32—22no—
302C22×C32—22no—
303C22×C32—21no—
304C22×C32—21no—
305C22×C32C2×#6714yes—
306C22×C32C23×D311yescharacteristic
307C2×C3×D3—61no—
308C2×C3×D3—61no—
309C2×C3×D3—61no—
310C32×C4—21no—
311C32×C4—22no—
312C32×C4C2×#6712yes—
313C3×Dic3—32no—
314C3×Dic3—62no—
315C3×Dic3—61no—
316C3×Dic3—62no—
317C3×Dic3—61no—
318C22×C9—22no—
319C22×C9—22no—
320C22×C9—22no—
321C22×C9—21no—
322C22×C9—21no—
323C22×C9—21no—
324C22×C9—21no—
325C22×C9—24no—
326C22×C9—42no—
327C22×C9—42no—
328C22×C9—41no—
329C22×C9—41no—
330C22×C9C2×C3×D414yes—
331C22×C9C24×C311yescharacteristic
332C2×D9—21no—
333C2×D9—21no—
334C2×D9—21no—
335C4×C9—22no—
336C4×C9—24no—
337C4×C9—21no—
338C4×C9—22no—
339C4×C9—22no—
340C4×C9C2×C3×D412yes—
341Dic9—22no—
342Dic9—21no—
343Dic9—22no—
344Dic9—21no—
345Dic9C3×#4112yescomplemented
346C2×#41—91no—
347C2×#41C3×D911yescharacteristic, complemented
348C4⋊Q8—91no—
349#123—92no—
350#123—92no—
351C22⋊Q8—91no—
352C22⋊Q8—91no—
353#121—92no—
354C4×D4—91no—
355C4×D4—91no—
356C2×C4⋊C4—92no—
357C3×C9#54711yescharacteristic, complemented, Sylow
358C23×C3—21no—
359C23×C3—22no—
360C23×C3—91no—
361C23×C3C22×D911yescharacteristic
362C23×C3C22×D912yes—
363C23×C3C22×C3×D311yescharacteristic
364C23×C3C22×C3×D312yes—
365C22×D3—31no—
366C3×Q8—24no—
367C3×Q8—22no—
368C3×Q8—22no—
369C3×Q8—182no—
370C3×D4—28no—
371C3×D4—24no—
372C3×D4—24no—
373C3×D4—24no—
374C3×D4—182no—
375C3×D4—184no—
376C3×D4#61114yescomplemented
377C3×D4C3×#6714yes—
378C2×C3×C4—28no—
379C2×C3×C4—24no—
380C2×C3×C4—24no—
381C2×C3×C4—24no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—91no—
388C2×C3×C4—91no—
389C2×C3×C4—92no—
390C2×C3×C4—91no—
391C2×C3×C4—92no—
392C2×C3×C4—184no—
393C2×C3×C4—184no—
394C2×C3×C4—182no—
395C2×C3×C4C22×D911yescharacteristic
396C2×C3×C4C22×D912yes—
397C2×C3×C4C22×D911yescharacteristic
398C2×C3×C4C22×C3×D311yescharacteristic
399C2×C3×C4C22×C3×D312yes—
400C2×C3×C4C22×C3×D311yescharacteristic
401#67—62no—
402#67—64no—
403C2×Dic3—31no—
404C2×Dic3—31no—
405C2×Dic3—32no—
406C2×Dic3—31no—
407C2×Dic3—32no—
408C2×Dic3—64no—
409C2×Dic3—64no—
410C4×D3—62no—
411Dic6—62no—
412C2×C32—24no—
413C2×C32—41no—
414C2×C32D3×#4111yescharacteristic
415C2×C32#101011yescharacteristic
416C2×C32C22×#6711yescharacteristic
417C3×D3—121no—
418C2×C9—24no—
419C2×C9—21no—
420C2×C9—21no—
421C2×C9—21no—
422C2×C9—44no—
423C2×C9—42no—
424C2×C9—41no—
425C2×C9C22×C3×D411yescharacteristic
426C2×C9C2×C3×#4111yescharacteristic
427C2×C9C3×2−1+411yescharacteristic
428D9—41no—
429#41—24no—
430#41—184no—
431C2×Q8—91no—
432C2×Q8C2×C3×D911yescharacteristic
433C2×D4—91no—
434C2×D4—92no—
435C2×D4C2×C3×D912yes—
436C2×D4C2×C3×D911yescharacteristic
437C22×C4—92no—
438C22×C4—92no—
439C22×C4—91no—
440C22×C4C2×C3×D912yes—
441C22×C4C2×C3×D911yescharacteristic
442C4⋊C4—91no—
443C4⋊C4—92no—
444C4⋊C4—92no—
445C4⋊C4—92no—
446C4⋊C4—91no—
447C4⋊C4—92no—
448C22⋊C4—92no—
449C22⋊C4—91no—
450C22⋊C4—92no—
451C22⋊C4—91no—
452C22⋊C4—92no—
453C42—91no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—24no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—41no—
470C22×C3—41no—
471C22×C3—181no—
472C22×C3—181no—
473C22×C3—181no—
474C22×C3C2×#61114yes—
475C22×C3C23×D911yescharacteristic
476C22×C3C2×C3×#6714yes—
477C22×C3C23×C3×D311yescharacteristic
478C2×D3—61no—
479C2×D3—61no—
480C2×D3—61no—
481C3×C4—22no—
482C3×C4—24no—
483C3×C4—21no—
484C3×C4—22no—
485C3×C4—21no—
486C3×C4—22no—
487C3×C4—22no—
488C3×C4—92no—
489C3×C4—182no—
490C3×C4—181no—
491C3×C4—181no—
492C3×C4—182no—
493C3×C4C2×#61112yes—
494C3×C4C2×C3×#6712yes—
495Dic3—32no—
496Dic3—62no—
497Dic3—61no—
498Dic3—62no—
499Dic3—61no—
500C32#582411yescharacteristic
501C9—21no—
502C9C3×#54711yescharacteristic, complemented
503C23—91no—
504C23C22×C3×D911yescharacteristic
505C23C22×C3×D912yes—
506Q8—22no—
507Q8—182no—
508D4—24no—
509D4—182no—
510D4—184no—
511D4C3×#61114yescomplemented
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—91no—
515C2×C4—91no—
516C2×C4—92no—
517C2×C4—91no—
518C2×C4—92no—
519C2×C4—184no—
520C2×C4—184no—
521C2×C4—182no—
522C2×C4C22×C3×D911yescharacteristic
523C2×C4C22×C3×D912yes—
524C2×C4C22×C3×D911yescharacteristic
525C2×C3—24no—
526C2×C3—24no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—44no—
531C2×C3—42no—
532C2×C3—41no—
533C2×C3—41no—
534C2×C3—361no—
535C2×C3#41×D911yescharacteristic
536C2×C3#6371611yescharacteristic
537C2×C3C22×#61111yescharacteristic
538C2×C3C3×D3×#4111yescharacteristic
539C2×C3C3×#101011yescharacteristic
540C2×C3C22×C3×#6711yescharacteristic
541D3—121no—
542C22—22no—
543C22—22no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—181no—
548C22—181no—
549C22—181no—
550C22C2×C3×#61114yes—
551C22C23×C3×D911yescharacteristic
552C4—21no—
553C4—22no—
554C4—92no—
555C4—182no—
556C4—181no—
557C4—181no—
558C4—182no—
559C4C2×C3×#61112yes—
560C3—21no—
561C3#1058992611yescharacteristic, direct product factor
562C3C3×#582411yescharacteristic
563C2—24no—
564C2—41no—
565C2—361no—
566C2C3×#41×D911yescharacteristic
567C2C3×#6371611yescharacteristic
568C2C22×C3×#61111yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C9×#41C2
socleC22×C32C23×D3
hypercenterC2×C3×#41D9
hypocenterC9C3×#547
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C2×C3×#41

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×#41C2semidirect
2. C2×C3×#3699C2semidirect
3. C3×#63508C2semidirect
4. C3×#63506C2semidirect
5. C3×#63501C2semidirect
6. C3×#63497C2semidirect
7. C3×D4×Dic9C2semidirect
8. C3×#63491C2semidirect
9. C3×C4×#611C2semidirect
10. C3×#63486C2semidirect
11. C2×C3×Dic9⋊C4C2semidirect
13. #10589926C3direct (N × H)
51. C2×C9×#41C2×C3semidirect
52. C2×#3699C2×C3semidirect
53. #63508C2×C3semidirect
54. #63506C2×C3semidirect
55. #63501C2×C3semidirect
56. #63497C2×C3semidirect
57. D4×Dic9C2×C3semidirect
58. #63491C2×C3semidirect
59. C4×#611C2×C3semidirect
60. #63486C2×C3semidirect
61. C2×Dic9⋊C4C2×C3semidirect
67. C3×D4×C9D4semidirect
159. C3×Dic9#41semidirect
162. C2×C3×#41D9semidirect
215. D4×C9C3×D4semidirect
345. Dic9C3×#41semidirect
347. C2×#41C3×D9semidirect
357. C3×C9#547semidirect
376. C3×D4#611semidirect
502. C9C3×#547semidirect
511. D4C3×#611semidirect
561. C3#10589926direct (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×#412
C2×C3×#36992
C3×#635082
C3×#635062
C3×#635012
C3×#634972
C3×D4×Dic92
C3×#634912
C3×C4×#6112
C3×#634862
C2×C3×Dic9⋊C42
C3×#58243
#105899263