Finite Group Databasebeta

← C3×#10591084 #13158112

Subgroups — C3×#10591084 #13158112

Order 1728.

1002 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×C32×#235C211yesmaximal, characteristic
2C3×#64135C211yesmaximal, characteristic, complemented
3C22×C32×C3⋊C8C211yesmaximal, characteristic, complemented
4C3×#63566C212yesmaximal, complemented
5C3×#63563C212yesmaximal, complemented
6C3×#5233—31nomaximal
7C3×#5113—31nomaximal
8#10591084C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C2×C32×#67—21no—
11C32×#235—22no—
12C32×#235C414yescomplemented
13C2×C32×D12C2211yescharacteristic
14C2×C32×C4×D3—21no—
15C2×C32×Dic6C2211yescharacteristic
16C3×#3757—21no—
17C3×C4⋊(C3⋊Dic3)C2212yes—
18C3×C4×C3⋊Dic3—21no—
19C2×C32×C3⋊C8—22no—
20C2×C32×C3⋊C8C2212yes—
21C2×C32×#41—31no—
22C2×C3×#235—21no—
23C2×C3×#235D311yescharacteristic
24C2×C3×#235C2×C311yescharacteristic
25C22×C32×C8—31no—
26#64135C2×C311yescharacteristic, complemented
27C3×#926—31no—
28C3×#926—31no—
29C3×#926—61no—
30C22×C3×C3⋊C8—21no—
31C22×C3×C3⋊C8D311yescharacteristic, complemented
32C22×C3×C3⋊C8C2×C311yescharacteristic, complemented
33C3×#835—32no—
34C3×#832—32no—
35C3×#821—32no—
36C3×#816—32no—
37#63566C2×C312yescomplemented
38#63563C2×C312yescomplemented
39C23×C33D411yescharacteristic
40C22×C32×D3—21no—
41C2×C33×C4C2×C412yes—
42C2×C33×C4D411yescharacteristic
43C2×C33×C4D412yes—
44C2×C33×C4C2311yescharacteristic
45C2×C3×C3⋊Dic3—22no—
46C32×#67—24no—
47C2×C32×Dic3—21no—
48C32×D12—21no—
49C32×D12C2×C412yes—
50C32×C4×D3—24no—
51C32×Dic6—21no—
52C32×Dic6C2×C412yes—
53C32×C3⋊C8—22no—
54C3×#416—91no—
55#5233—31no—
56#5113—31no—
57C32×#41—34no—
58C32×#41—62no—
59C2×C32×Q8—31no—
60C2×C32×D4—31no—
61C2×C32×D4—61no—
62C22×C32×C4—21no—
63C22×C32×C4—21no—
64C22×C32×C4—21no—
65C22×C32×C4—41no—
66C22×C32×C4—61no—
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C32×C4C22×C311yescharacteristic
70C2×C3×#67—22no—
71C2×C3×#67—21no—
72C2×C3×#67—21no—
73C3×#235—24no—
74C3×#235—24no—
75C3×#235—22no—
76C3×#235—22no—
77C3×#235Dic314yescomplemented
78C3×#235C3×C414yescomplemented
79C2×C3×D12—21no—
80C2×C3×D12C2×D311yescharacteristic
81C2×C3×D12C22×C311yescharacteristic
82C2×C3×C4×D3—22no—
83C2×C3×C4×D3—21no—
84C2×C3×C4×D3—21no—
85C2×C3×Dic6—21no—
86C2×C3×Dic6C2×D311yescharacteristic
87C2×C3×Dic6C22×C311yescharacteristic
88C2×C32×C8—32no—
89C2×C32×C8—62no—
90#3757—21no—
91C4⋊(C3⋊Dic3)C22×C312yes—
92C4×C3⋊Dic3—21no—
93C3×#217—62no—
94C3×#217—61no—
95C3×#217—61no—
96C3×C4⋊Dic3—32no—
97C3×C4⋊Dic3—32no—
98C3×C4⋊Dic3—62no—
99C3×C4×Dic3—62no—
100C3×C4×Dic3—61no—
101C3×C4×Dic3—61no—
102C2×C3×C3⋊C8—24no—
103C2×C3×C3⋊C8—22no—
104C2×C3×C3⋊C8—22no—
105C2×C3×C3⋊C8—22no—
106C2×C3×C3⋊C8C2×D312yes—
107C2×C3×C3⋊C8C22×C312yes—
108C22×C33—22no—
109C22×C33C22⋊C412yes—
110C22×C33C2×D411yescharacteristic
111C2×C32×D3—22no—
112C2×C32×D3—42no—
113C33×C4C22⋊C412yes—
114C33×C4C22×C411yescharacteristic
115C33×C4C2×D411yescharacteristic
116C3×C3⋊Dic3—42no—
117C32×Dic3—22no—
118C2×C3×#41—31no—
119C2×C3×#41—31no—
120C2×C3×#41—61no—
121C2×#235C3×D311yescharacteristic
122C22×C3×C8—31no—
123C22×C3×C8—31no—
124C22×C3×C8—61no—
125C3×#117—91no—
126#926—31no—
127#926—31no—
128#926—61no—
129C22×C3⋊C8C3×D311yescharacteristic, complemented
130C3×#103—92no—
131C3×#102—92no—
132#835—32no—
133#832—32no—
134#821—32no—
135#816—32no—
136C23×C32—21no—
137C23×C32—21no—
138C23×C32—21no—
139C23×C32—41no—
140C23×C32—61no—
141C23×C32D1211yescharacteristic
142C23×C32#6711yescharacteristic
143C23×C32C3×D411yescharacteristic
144C22×C3×D3—22no—
145C22×C3×D3—21no—
146C22×C3×D3—21no—
147C32×Q8—32no—
148C32×Q8—61no—
149C32×D4—32no—
150C32×D4—64no—
151C32×D4—61no—
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4—21no—
155C2×C32×C4—22no—
156C2×C32×C4—21no—
157C2×C32×C4—22no—
158C2×C32×C4—22no—
159C2×C32×C4—21no—
160C2×C32×C4—22no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—42no—
165C2×C32×C4—42no—
166C2×C32×C4—41no—
167C2×C32×C4—41no—
168C2×C32×C4—61no—
169C2×C32×C4—64no—
170C2×C32×C4C4×D312yes—
171C2×C32×C4D1211yescharacteristic
172C2×C32×C4C2×Dic312yes—
173C2×C32×C4#6712yes—
174C2×C32×C4#6712yes—
175C2×C32×C4#6711yescharacteristic
176C2×C32×C4C2×C3×C412yes—
177C2×C32×C4C3×D411yescharacteristic
178C2×C32×C4C3×D412yes—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C23×C311yescharacteristic
182C2×C3⋊Dic3—22no—
183C3×#67—28no—
184C3×#67—24no—
185C3×#67—24no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—64no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—62no—
192C3×D12—22no—
193C3×D12—21no—
194C3×D12—21no—
195C3×D12—22no—
196C3×D12C2×Dic312yes—
197C3×D12C2×C3×C412yes—
198C3×C4×D3—28no—
199C3×C4×D3—24no—
200C3×C4×D3—24no—
201C3×Dic6—22no—
202C3×Dic6—21no—
203C3×Dic6—21no—
204C3×Dic6—22no—
205C3×Dic6C2×Dic312yes—
206C3×Dic6C2×C3×C412yes—
207C32×C8—62no—
208C3×C3⋊C8—24no—
209C3×C3⋊C8—22no—
210C3×C3⋊C8—22no—
211#416—91noSylow
212C2×C33—22no—
213C2×C33C2×C22⋊C411yescharacteristic
214C2×C33#13512yes—
215C32×D3—42no—
216C3×#41—34no—
217C3×#41—34no—
218C3×#41—64no—
219C3×#41—64no—
220C3×#41—62no—
221C3×#41—62no—
222C2×C3×Q8—31no—
223C2×C3×Q8—31no—
224C2×C3×Q8—61no—
225C2×C3×D4—31no—
226C2×C3×D4—31no—
227C2×C3×D4—61no—
228C2×C3×D4—62no—
229C2×C3×D4—61no—
230C2×C3×D4—61no—
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4—21no—
234C22×C3×C4—41no—
235C22×C3×C4—62no—
236C22×C3×C4—61no—
237C22×C3×C4—61no—
238C22×C3×C4D3211yescharacteristic
239C22×C3×C4C2×C3×D311yescharacteristic
240C22×C3×C4C2×C3×D311yescharacteristic
241C2×#67—21no—
242#235—22no—
243#235C3×Dic314yescomplemented
244C2×D12C2×C3×D311yescharacteristic
245C2×C4×D3—21no—
246C2×Dic6C2×C3×D311yescharacteristic
247C2×C3×C8—32no—
248C2×C3×C8—32no—
249C2×C3×C8—62no—
250C2×C3×C8—64no—
251C2×C3×C8—62no—
252C2×C3×C8—62no—
253C3×C4⋊C4—92no—
254C3×C22⋊C4—181no—
255C3×C42—181no—
256#217—62no—
257#217—61no—
258#217—61no—
259C4⋊Dic3—32no—
260C4⋊Dic3—32no—
261C4⋊Dic3—62no—
262C4×Dic3—62no—
263C4×Dic3—61no—
264C4×Dic3—61no—
265C2×C3⋊C8—22no—
266C2×C3⋊C8C2×C3×D312yes—
267C22×C32—22no—
268C22×C32—24no—
269C22×C32—22no—
270C22×C32—22no—
271C22×C32—22no—
272C22×C32—22no—
273C22×C32—22no—
274C22×C32—21no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—42no—
278C22×C32—41no—
279C22×C32—42no—
280C22×C32—42no—
281C22×C32—44no—
282C22×C32—62no—
283C22×C32—122no—
284C22×C32(C2×D3)⋊C412yes—
285C22×C32#21712yes—
286C22×C32C3×C22⋊C412yes—
287C22×C32C2×D1211yescharacteristic
288C22×C32C2×#6711yescharacteristic
289C22×C32C2×C3×D411yescharacteristic
290C2×C3×D3—24no—
291C2×C3×D3—22no—
292C2×C3×D3—22no—
293C2×C3×D3—44no—
294C2×C3×D3—42no—
295C2×C3×D3—42no—
296C32×C4—22no—
297C32×C4—22no—
298C32×C4—22no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—42no—
306C32×C4—41no—
307C32×C4—41no—
308C32×C4—62no—
309C32×C4(C2×D3)⋊C412yes—
310C32×C4#21712yes—
311C32×C4C3×C22⋊C412yes—
312C32×C4C2×C4×D311yescharacteristic
313C32×C4C22×Dic311yescharacteristic
314C32×C4C2×#6711yescharacteristic
315C32×C4C2×#6711yescharacteristic
316C32×C4C22×C3×C411yescharacteristic
317C32×C4C2×C3×D411yescharacteristic
318C3⋊Dic3—42no—
319C3×Dic3—24no—
320C3×Dic3—22no—
321C3×Dic3—22no—
322C3×Dic3—124no—
323C3×Dic3—122no—
324C3×Dic3—122no—
325C2×#41—31no—
326C22×C8—31no—
327#117—91no—
328#103—92no—
329#102—92no—
330C33#41611yescharacteristic, complemented, Sylow
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—41no—
335C23×C3—62no—
336C23×C3—61no—
337C23×C3—61no—
338C23×C3#62611yescharacteristic
339C23×C3C3×D1211yescharacteristic
340C23×C3C3×#6711yescharacteristic
341C22×D3—21no—
342C3×Q8—32no—
343C3×Q8—32no—
344C3×Q8—62no—
345C3×Q8—62no—
346C3×Q8—61no—
347C3×Q8—61no—
348C3×D4—32no—
349C3×D4—32no—
350C3×D4—62no—
351C3×D4—62no—
352C3×D4—68no—
353C3×D4—64no—
354C3×D4—64no—
355C3×D4—61no—
356C3×D4—61no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—68no—
374C2×C3×C4—62no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—64no—
378C2×C3×C4—64no—
379C2×C3×C4—182no—
380C2×C3×C4D3×Dic312yes—
381C2×C3×C4#62512yes—
382C2×C3×C4#62611yescharacteristic
383C2×C3×C4C3×C4×D312yes—
384C2×C3×C4C3×D1211yescharacteristic
385C2×C3×C4C2×C3×Dic312yes—
386C2×C3×C4C3×#6712yes—
387C2×C3×C4C3×#6712yes—
388C2×C3×C4C3×#6711yescharacteristic
389C2×C3×C4C2×D3211yescharacteristic
390C2×C3×C4C22×C3×D311yescharacteristic
391C2×C3×C4C22×C3×D311yescharacteristic
392#67—24no—
393C2×Dic3—21no—
394C2×Dic3—64no—
395C2×Dic3—62no—
396C2×Dic3—62no—
397D12—21no—
398D12C2×C3×Dic312yes—
399C4×D3—24no—
400Dic6—21no—
401Dic6C2×C3×Dic312yes—
402C3×C8—64no—
403C3×C8—62no—
404C3×C8—62no—
405C3⋊C8—22no—
406C2×C32—22no—
407C2×C32—24no—
408C2×C32—22no—
409C2×C32—22no—
410C2×C32—22no—
411C2×C32—21no—
412C2×C32—22no—
413C2×C32—21no—
414C2×C32—22no—
415C2×C32—21no—
416C2×C32—41no—
417C2×C32—42no—
418C2×C32—42no—
419C2×C32—42no—
420C2×C32—44no—
421C2×C32—122no—
422C2×C32#90412yes—
423C2×C32C2×(C2×D3)⋊C411yescharacteristic
424C2×C32#95012yes—
425C2×C32C2×#21711yescharacteristic
426C2×C32C2×C3×C22⋊C411yescharacteristic
427C2×C32C3×#13512yes—
428C3×D3—44no—
429C3×D3—42no—
430C3×D3—42no—
431#41—34no—
432#41—62no—
433C2×Q8—31no—
434C2×D4—31no—
435C2×D4—61no—
436C22×C4—61no—
437C22×C4C3×D3211yescharacteristic
438C2×C8—32no—
439C2×C8—62no—
440C4⋊C4—92no—
441C22⋊C4—181no—
442C42—181no—
443C22×C3—24no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—44no—
456C22×C3—41no—
457C22×C3—42no—
458C22×C3—64no—
459C22×C3—62no—
460C22×C3—62no—
461C22×C3—124no—
462C22×C3—122no—
463C22×C3—122no—
464C22×C3(C2×D3)⋊Dic312yes—
465C22×C3C3×(C2×D3)⋊C412yes—
466C22×C3C3×#21712yes—
467C22×C3C2×#62611yescharacteristic
468C22×C3C2×C3×D1211yescharacteristic
469C22×C3C2×C3×#6711yescharacteristic
470C2×D3—22no—
471C2×D3—42no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—42no—
484C3×C4—64no—
485C3×C4—62no—
486C3×C4—62no—
487C3×C4—362no—
488C3×C4(C2×D3)⋊Dic312yes—
489C3×C4C3×(C2×D3)⋊C412yes—
490C3×C4C3×#21712yes—
491C3×C4C2×D3×Dic311yescharacteristic
492C3×C4C2×#62511yescharacteristic
493C3×C4C2×C3×C4×D311yescharacteristic
494C3×C4C22×C3×Dic311yescharacteristic
495C3×C4C2×C3×#6711yescharacteristic
496C3×C4C2×C3×#6711yescharacteristic
497Dic3—22no—
498Dic3—124no—
499Dic3—122no—
500Dic3—122no—
501C32—21no—
502C32—21no—
503C32—21no—
504C32—41no—
505C32#511311yescharacteristic, complemented
506C32#523311yescharacteristic, complemented
507C32C3×#41611yescharacteristic, complemented
508C23—61no—
509C23C3×#62611yescharacteristic
510Q8—32no—
511Q8—61no—
512D4—32no—
513D4—64no—
514D4—61no—
515C2×C4—61no—
516C2×C4—64no—
517C2×C4—182no—
518C2×C4C3×D3×Dic312yes—
519C2×C4C3×#62512yes—
520C2×C4C3×#62611yescharacteristic
521C2×C4C2×C3×D3211yescharacteristic
522C8—62no—
523C2×C3—24no—
524C2×C3—22no—
525C2×C3—22no—
526C2×C3—22no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—22no—
531C2×C3—22no—
532C2×C3—22no—
533C2×C3—42no—
534C2×C3—42no—
535C2×C3—44no—
536C2×C3—41no—
537C2×C3—42no—
538C2×C3—124no—
539C2×C3—122no—
540C2×C3—122no—
541C2×C3#6382912yes—
542C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
543C2×C3C3×#90412yes—
544C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
545C2×C3C3×#95012yes—
546C2×C3C2×C3×#21711yescharacteristic
547D3—42no—
548C22—22no—
549C22—62no—
550C22—122no—
551C22C3×(C2×D3)⋊Dic312yes—
552C22C2×C3×#62611yescharacteristic
553C4—62no—
554C4—362no—
555C4C3×(C2×D3)⋊Dic312yes—
556C4C2×C3×D3×Dic311yescharacteristic
557C4C2×C3×#62511yescharacteristic
558C3—21no—
559C3—21no—
560C3—21no—
561C3—41no—
562C3#1059108411yescharacteristic, direct product factor
563C3C3×#511311yescharacteristic, complemented
564C3C3×#523311yescharacteristic, complemented
565C2—22no—
566C2—122no—
567C2C3×#6382912yes—
568C2C2×C3×(C2×D3)⋊Dic311yescharacteristic
subgroupG/H
centerC2×C3×C4#626
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC2×C32×#235C2
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#416
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64135C2semidirect
3. C22×C32×C3⋊C8C2semidirect
4. C3×#63566C2semidirect
5. C3×#63563C2semidirect
8. #10591084C3direct (N × H)
12. C32×#235C4semidirect
26. #64135C2×C3semidirect
31. C22×C3×C3⋊C8D3semidirect
32. C22×C3×C3⋊C8C2×C3semidirect
37. #63566C2×C3semidirect
38. #63563C2×C3semidirect
77. C3×#235Dic3semidirect
78. C3×#235C3×C4semidirect
129. C22×C3⋊C8C3×D3semidirect
243. #235C3×Dic3semidirect
330. C33#416semidirect
505. C32#5113semidirect
506. C32#5233semidirect
507. C32C3×#416semidirect
562. C3#10591084direct (N × H)
563. C3C3×#5113semidirect
564. C3C3×#5233semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2352
C3×#641352
C22×C32×C3⋊C82
C3×#635662
C3×#635632
C3×#52333
C3×#51133
#105910843