Finite Group Databasebeta

← C3×#10589698 #13152787

Subgroups — C3×#10589698 #13152787

Order 1728.

1024 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3699C212yesmaximal, complemented
2C3×C9×#121C211yesmaximal, characteristic, complemented
3C3×#63502C211yesmaximal, characteristic, complemented
4C3×#63501C212yesmaximal, complemented
5C3×D4×Dic9C211yesmaximal, characteristic, complemented
6C3×#63492C211yesmaximal, characteristic, complemented
7C2×C3×C4×Dic9C211yesmaximal, characteristic, complemented
8C3×#63455C211yesmaximal, characteristic, complemented
9C3×#63450C211yesmaximal, characteristic, complemented
10C3×#63447C212yesmaximal, complemented
11C3×#611⋊C4C212yesmaximal, complemented
12C3×#5596—31nomaximal
13#10589698C311yesmaximal, characteristic, direct product factor
14C2×C3×D4×C9C2212yes—
15C2×C3×D4×C9C2211yescharacteristic
16C22×C3×C4×C9C2211yescharacteristic
17C2×C3×#611C2212yes—
18C2×C3×#611C2212yes—
19C2×C3×#611C2212yes—
20C22×C3×Dic9C2211yescharacteristic
21C22×C3×Dic9C2212yes—
22C22×C3×Dic9C2211yescharacteristic
23C3×#3699—28no—
24C2×C3×D36C2211yescharacteristic
25C2×C3×C4×D9C2212yes—
26C2×C3×Dic18C2212yes—
27C3×C9×C4⋊C4C2211yescharacteristic
28C3×C9×C22⋊C4C2212yes—
29C3×#3676C2212yes—
30C3×(C2×D9)⋊C4C2212yes—
31C3×(C2×D9)⋊C4C2212yes—
32C3×C4⋊Dic9C2211yescharacteristic
33C3×Dic9⋊C4C2212yes—
34C3×C4×Dic9C2211yescharacteristic
35C3×C4×Dic9C2211yescharacteristic
36C3×C4×Dic9C2211yescharacteristic
37C3×C4×Dic9C2211yescharacteristic
38C2×C3×#237—32no—
39C32×#121D311yescharacteristic
40C3×#940—31no—
41C3×#939—32no—
42C3×D4×Dic3—31no—
43C3×#930—31no—
44C2×C3×C4×Dic3—31no—
45C3×#893—31no—
46C3×#888—31no—
47C3×#885—32no—
48C3×#67⋊C4—32no—
49C2×#3699C2×C312yescomplemented
50C9×#121—21no—
51C9×#121C2×C311yescharacteristic, complemented
52#63502C2×C311yescharacteristic, complemented
53#63501C2×C312yescomplemented
54D4×Dic9C2×C311yescharacteristic, complemented
55#63492C2×C311yescharacteristic, complemented
56C2×C4×Dic9C2×C311yescharacteristic, complemented
57#63455C2×C311yescharacteristic, complemented
58#63450C2×C311yescharacteristic, complemented
59#63447C2×C312yescomplemented
60#611⋊C4C2×C312yescomplemented
61C23×C3×C9C2312yes—
62C23×C3×C9C2311yescharacteristic
63C22×C3×D9C2312yes—
64C3×D4×C9—24no—
65C3×D4×C9—22no—
66C2×C3×C4×C9—22no—
67C2×C3×C4×C9C2312yes—
68C2×C3×C4×C9C2311yescharacteristic
69C2×C3×C4×C9C2311yescharacteristic
70C3×#611—24no—
71C3×#611—24no—
72C3×#611—24no—
73C2×C3×Dic9—24no—
74C2×C3×Dic9—22no—
75C2×C3×Dic9D414yescomplemented
76C2×C3×Dic9C2312yes—
77C2×C3×Dic9C2311yescharacteristic
78C2×C3×Dic9C2311yescharacteristic
79C2×C3×Dic9C2311yescharacteristic
80C2×C3×Dic9C2311yescharacteristic
81C3×D36—22no—
82C3×C4×D9—24no—
83C3×Dic18—24no—
84C3×#532—91no—
85#5596—31no—
86C2×C32×D4C2×D311yescharacteristic
87C2×C32×D4C2×D312yes—
88C22×C32×C4C2×D311yescharacteristic
89C2×C3×#67—32no—
90C2×C3×#67—32no—
91C2×C3×#67—32no—
92C22×C3×Dic3—31no—
93C22×C3×Dic3—32no—
94C22×C3×Dic3—31no—
95C3×#237—68no—
96C2×C3×D12—31no—
97C2×C3×C4×D3—32no—
98C2×C3×Dic6—32no—
99C32×C4⋊C4C2×D311yescharacteristic
100C32×C22⋊C4C2×D312yes—
101C3×#217—32no—
102C3×(C2×D3)⋊C4—32no—
103C3×(C2×D3)⋊C4—32no—
104C3×C4⋊Dic3—31no—
105C3×Dic3⋊C4—32no—
106C3×C4×Dic3—31no—
107C3×C4×Dic3—31no—
108C3×C4×Dic3—31no—
109C3×C4×Dic3—31no—
110C2×D4×C9—21no—
111C2×D4×C9—22no—
112C2×D4×C9C22×C311yescharacteristic
113C2×D4×C9C22×C312yes—
114C22×C4×C9—21no—
115C22×C4×C9C22×C311yescharacteristic
116C2×#611C22×C312yes—
117C2×#611C22×C312yes—
118C2×#611C22×C312yes—
119C22×Dic9C22×C312yes—
120C22×Dic9C22×C311yescharacteristic
121C22×Dic9C22×C311yescharacteristic
122#3699—28no—
123C2×D36C22×C311yescharacteristic
124C2×C4×D9C22×C312yes—
125C2×Dic18C22×C312yes—
126C9×C4⋊C4—21no—
127C9×C4⋊C4C22×C311yescharacteristic
128C9×C22⋊C4—22no—
129C9×C22⋊C4C22×C312yes—
130#3676C22×C312yes—
131(C2×D9)⋊C4C22×C312yes—
132(C2×D9)⋊C4C22×C312yes—
133C4⋊Dic9C22×C311yescharacteristic
134Dic9⋊C4C22×C312yes—
135C4×Dic9C22×C311yescharacteristic
136C4×Dic9C22×C311yescharacteristic
137C4×Dic9C22×C311yescharacteristic
138C4×Dic9C22×C311yescharacteristic
139C22×C3×C9—21no—
140C22×C3×C9—21no—
141C22×C3×C9—22no—
142C22×C3×C9—22no—
143C22×C3×C9—22no—
144C22×C3×C9C2×D412yes—
145C22×C3×C9C2411yescharacteristic
146C2×C3×D9—22no—
147C2×C3×D9—22no—
148C2×C3×D9—22no—
149C3×C4×C9—22no—
150C3×C4×C9—21no—
151C3×C4×C9#4112yescomplemented
152C3×Dic9—22no—
153C3×Dic9—21no—
154C3×Dic9C2×D412yes—
155C3×Dic9C2×D412yes—
156C3×Dic9#4112yescomplemented
157C2×C3×#41—92no—
158C2×#237—32no—
159C3×C4⋊Q8—91no—
160C3×#127—91no—
161C3×#124—92no—
162C3×#121—21no—
163C3×#121—92no—
164C3×#121—91no—
165C3×#121D911yescharacteristic, complemented
166C3×#121C3×D311yescharacteristic
167C3×C4×D4—91no—
168C3×C4×D4—91no—
169C3×C4×D4—92no—
170C2×C3×C42—91no—
171#940—31no—
172#939—32no—
173D4×Dic3—31no—
174#930—31no—
175C2×C4×Dic3—31no—
176#893—31no—
177#888—31no—
178#885—32no—
179#67⋊C4—32no—
180C23×C32C22×D312yes—
181C23×C32C22×D311yescharacteristic
182C22×C3×D3—32no—
183C32×D4—24no—
184C32×D4—22no—
185C2×C32×C4—22no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D312yes—
188C2×C32×C4C22×D311yescharacteristic
189C3×#67—64no—
190C3×#67—64no—
191C3×#67—64no—
192C2×C3×Dic3—34no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—64no—
199C2×C3×Dic3—62no—
200C3×D12—62no—
201C3×C4×D3—64no—
202C3×Dic6—64no—
203C23×C9—21no—
204C23×C9—22no—
205C23×C9C23×C312yes—
206C23×C9C23×C311yescharacteristic
207C22×D9C23×C312yes—
208D4×C9—24no—
209D4×C9—22no—
210D4×C9—42no—
211D4×C9—44no—
212C2×C4×C9—22no—
213C2×C4×C9—21no—
214C2×C4×C9—21no—
215C2×C4×C9—22no—
216C2×C4×C9—42no—
217C2×C4×C9C23×C311yescharacteristic
218C2×C4×C9C23×C312yes—
219C2×C4×C9C23×C311yescharacteristic
220#611—24no—
221#611—24no—
222#611—24no—
223C2×Dic9—24no—
224C2×Dic9—22no—
225C2×Dic9C3×D414yescomplemented
226C2×Dic9C23×C311yescharacteristic
227C2×Dic9C23×C312yes—
228C2×Dic9C23×C311yescharacteristic
229C2×Dic9C23×C311yescharacteristic
230C2×Dic9C23×C311yescharacteristic
231D36—22no—
232C4×D9—24no—
233Dic18—24no—
234#532—91noSylow
235C2×C3×C9—22no—
236C2×C3×C9—42no—
237C2×C3×C9C22×D411yescharacteristic
238C2×C3×C9C2×#4111yescharacteristic
239C2×C3×C9C2×#4111yescharacteristic
240C3×D9—42no—
241C3×#41—188no—
242C2×C3×Q8—92no—
243C2×C3×D4—21no—
244C2×C3×D4—22no—
245C2×C3×D4—92no—
246C2×C3×D4—92no—
247C2×C3×D4—92no—
248C2×C3×D4—91no—
249C2×C3×D4C2×D911yescharacteristic
250C2×C3×D4C2×D912yes—
251C2×C3×D4C2×C3×D311yescharacteristic
252C2×C3×D4C2×C3×D312yes—
253C22×C3×C4—21no—
254C22×C3×C4—91no—
255C22×C3×C4—91no—
256C22×C3×C4—92no—
257C22×C3×C4—92no—
258C22×C3×C4C2×D911yescharacteristic
259C22×C3×C4C2×C3×D311yescharacteristic
260C2×#67—32no—
261C2×#67—32no—
262C2×#67—32no—
263C22×Dic3—31no—
264C22×Dic3—32no—
265C22×Dic3—31no—
266#237—68no—
267C2×D12—31no—
268C2×C4×D3—32no—
269C2×Dic6—32no—
270C3×C4⋊C4—21no—
271C3×C4⋊C4—92no—
272C3×C4⋊C4—91no—
273C3×C4⋊C4C2×D911yescharacteristic
274C3×C4⋊C4C2×C3×D311yescharacteristic
275C3×C22⋊C4—22no—
276C3×C22⋊C4—92no—
277C3×C22⋊C4—92no—
278C3×C22⋊C4—92no—
279C3×C22⋊C4C2×D912yes—
280C3×C22⋊C4C2×C3×D312yes—
281C3×C42—91no—
282C3×C42—91no—
283C3×C42—91no—
284C3×C42—91no—
285#217—32no—
286(C2×D3)⋊C4—32no—
287(C2×D3)⋊C4—32no—
288C4⋊Dic3—31no—
289Dic3⋊C4—32no—
290C4×Dic3—31no—
291C4×Dic3—31no—
292C4×Dic3—31no—
293C4×Dic3—31no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32—22no—
297C22×C32—22no—
298C22×C32—22no—
299C22×C32D3×D412yes—
300C22×C32C23×D311yescharacteristic
301C2×C3×D3—62no—
302C2×C3×D3—62no—
303C2×C3×D3—62no—
304C32×C4—22no—
305C32×C4—21no—
306C32×C4#23712yes—
307C3×Dic3—32no—
308C3×Dic3—32no—
309C3×Dic3—32no—
310C3×Dic3—62no—
311C3×Dic3—61no—
312C22×C9—21no—
313C22×C9—21no—
314C22×C9—22no—
315C22×C9—22no—
316C22×C9—22no—
317C22×C9—21no—
318C22×C9—22no—
319C22×C9—41no—
320C22×C9—41no—
321C22×C9—42no—
322C22×C9—42no—
323C22×C9—42no—
324C22×C9C2×C3×D412yes—
325C22×C9C24×C311yescharacteristic
326C2×D9—22no—
327C2×D9—22no—
328C2×D9—22no—
329C4×C9—22no—
330C4×C9—21no—
331C4×C9—22no—
332C4×C9—41no—
333C4×C9—42no—
334C4×C9C3×#4112yescomplemented
335Dic9—22no—
336Dic9—21no—
337Dic9C2×C3×D412yes—
338Dic9C2×C3×D412yes—
339Dic9C3×#4112yescomplemented
340C2×#41—92no—
341C4⋊Q8—91no—
342#127—91no—
343#124—92no—
344#121—92no—
345#121—91no—
346#121C3×D911yescharacteristic, complemented
347C4×D4—91no—
348C4×D4—91no—
349C4×D4—92no—
350C2×C42—91no—
351C3×C9#53211yescharacteristic, complemented, Sylow
352C23×C3—21no—
353C23×C3—22no—
354C23×C3—92no—
355C23×C3C22×D912yes—
356C23×C3C22×D911yescharacteristic
357C23×C3C22×C3×D312yes—
358C23×C3C22×C3×D311yescharacteristic
359C22×D3—32no—
360C3×Q8—184no—
361C3×D4—24no—
362C3×D4—24no—
363C3×D4—22no—
364C3×D4—22no—
365C3×D4—42no—
366C3×D4—44no—
367C3×D4—184no—
368C3×D4—184no—
369C3×D4—184no—
370C3×D4—182no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—42no—
377C2×C3×C4—94no—
378C2×C3×C4—91no—
379C2×C3×C4—91no—
380C2×C3×C4—91no—
381C2×C3×C4—91no—
382C2×C3×C4—92no—
383C2×C3×C4—184no—
384C2×C3×C4—182no—
385C2×C3×C4—184no—
386C2×C3×C4C22×D911yescharacteristic
387C2×C3×C4C22×D912yes—
388C2×C3×C4C22×D911yescharacteristic
389C2×C3×C4C22×C3×D311yescharacteristic
390C2×C3×C4C22×C3×D312yes—
391C2×C3×C4C22×C3×D311yescharacteristic
392#67—64no—
393#67—64no—
394#67—64no—
395C2×Dic3—31no—
396C2×Dic3—31no—
397C2×Dic3—31no—
398C2×Dic3—34no—
399C2×Dic3—31no—
400C2×Dic3—32no—
401C2×Dic3—64no—
402C2×Dic3—62no—
403D12—62no—
404C4×D3—64no—
405Dic6—64no—
406C2×C32—22no—
407C2×C32—42no—
408C2×C32C2×D3×D411yescharacteristic
409C2×C32C2×#23711yescharacteristic
410C2×C32D3×#4111yescharacteristic
411C3×D3—122no—
412C2×C9—22no—
413C2×C9—21no—
414C2×C9—21no—
415C2×C9—21no—
416C2×C9—42no—
417C2×C9—42no—
418C2×C9—82no—
419C2×C9C22×C3×D411yescharacteristic
420C2×C9C2×C3×#4111yescharacteristic
421C2×C9C2×C3×#4111yescharacteristic
422D9—42no—
423#41—188no—
424C2×Q8—92no—
425C2×D4—92no—
426C2×D4—92no—
427C2×D4—92no—
428C2×D4—91no—
429C2×D4C2×C3×D911yescharacteristic
430C2×D4C2×C3×D912yes—
431C22×C4—91no—
432C22×C4—91no—
433C22×C4—92no—
434C22×C4—92no—
435C22×C4C2×C3×D911yescharacteristic
436C4⋊C4—92no—
437C4⋊C4—91no—
438C4⋊C4C2×C3×D911yescharacteristic
439C22⋊C4—92no—
440C22⋊C4—92no—
441C22⋊C4—92no—
442C22⋊C4C2×C3×D912yes—
443C42—91no—
444C42—91no—
445C42—91no—
446C42—91no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—21no—
458C22×C3—22no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—182no—
465C22×C3—182no—
466C22×C3—182no—
467C22×C3D4×D912yes—
468C22×C3C23×D911yescharacteristic
469C22×C3C3×D3×D412yes—
470C22×C3C23×C3×D311yescharacteristic
471C2×D3—62no—
472C2×D3—62no—
473C2×D3—62no—
474C3×C4—22no—
475C3×C4—21no—
476C3×C4—22no—
477C3×C4—21no—
478C3×C4—22no—
479C3×C4—41no—
480C3×C4—42no—
481C3×C4—92no—
482C3×C4—92no—
483C3×C4—92no—
484C3×C4—182no—
485C3×C4—181no—
486C3×C4#369912yescomplemented
487C3×C4C3×#23712yes—
488Dic3—32no—
489Dic3—32no—
490Dic3—32no—
491Dic3—62no—
492Dic3—61no—
493C32#559611yescharacteristic
494C9—21no—
495C9C3×#53211yescharacteristic, complemented
496C23—92no—
497C23C22×C3×D912yes—
498C23C22×C3×D911yescharacteristic
499Q8—184no—
500D4—24no—
501D4—22no—
502D4—184no—
503D4—184no—
504D4—184no—
505D4—182no—
506C2×C4—22no—
507C2×C4—91no—
508C2×C4—91no—
509C2×C4—91no—
510C2×C4—94no—
511C2×C4—91no—
512C2×C4—92no—
513C2×C4—184no—
514C2×C4—184no—
515C2×C4—182no—
516C2×C4C22×C3×D911yescharacteristic
517C2×C4C22×C3×D912yes—
518C2×C4C22×C3×D911yescharacteristic
519C2×C3—22no—
520C2×C3—22no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—42no—
525C2×C3—42no—
526C2×C3—42no—
527C2×C3—82no—
528C2×C3—362no—
529C2×C3C2×D4×D911yescharacteristic
530C2×C3C2×#369911yescharacteristic
531C2×C3#41×D911yescharacteristic
532C2×C3C2×C3×D3×D411yescharacteristic
533C2×C3C2×C3×#23711yescharacteristic
534C2×C3C3×D3×#4111yescharacteristic
535D3—122no—
536C22—21no—
537C22—21no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—182no—
542C22—182no—
543C22—182no—
544C22C3×D4×D912yes—
545C22C23×C3×D911yescharacteristic
546C4—22no—
547C4—21no—
548C4—92no—
549C4—92no—
550C4—92no—
551C4—182no—
552C4—181no—
553C4C3×#369912yescomplemented
554C3—21no—
555C3#1058969811yescharacteristic, direct product factor
556C3C3×#559611yescharacteristic
557C2—22no—
558C2—42no—
559C2—362no—
560C2C2×C3×D4×D911yescharacteristic
561C2C2×C3×#369911yescharacteristic
562C2C3×#41×D911yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC3×C9×#121C2
socleC22×C32C23×D3
hypercenterC3×#121D9
hypocenterC9C3×#532
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×#3699C2semidirect
2. C3×C9×#121C2semidirect
3. C3×#63502C2semidirect
4. C3×#63501C2semidirect
5. C3×D4×Dic9C2semidirect
6. C3×#63492C2semidirect
7. C2×C3×C4×Dic9C2semidirect
8. C3×#63455C2semidirect
9. C3×#63450C2semidirect
10. C3×#63447C2semidirect
11. C3×#611⋊C4C2semidirect
13. #10589698C3direct (N × H)
49. C2×#3699C2×C3semidirect
51. C9×#121C2×C3semidirect
52. #63502C2×C3semidirect
53. #63501C2×C3semidirect
54. D4×Dic9C2×C3semidirect
55. #63492C2×C3semidirect
56. C2×C4×Dic9C2×C3semidirect
57. #63455C2×C3semidirect
58. #63450C2×C3semidirect
59. #63447C2×C3semidirect
60. #611⋊C4C2×C3semidirect
75. C2×C3×Dic9D4semidirect
151. C3×C4×C9#41semidirect
156. C3×Dic9#41semidirect
165. C3×#121D9semidirect
225. C2×Dic9C3×D4semidirect
334. C4×C9C3×#41semidirect
339. Dic9C3×#41semidirect
346. #121C3×D9semidirect
351. C3×C9#532semidirect
486. C3×C4#3699semidirect
495. C9C3×#532semidirect
553. C4C3×#3699semidirect
555. C3#10589698direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#36992
C3×C9×#1212
C3×#635022
C3×#635012
C3×D4×Dic92
C3×#634922
C2×C3×C4×Dic92
C3×#634552
C3×#634502
C3×#634472
C3×#611⋊C42
C3×#55963
#105896983