Finite Group Databasebeta

← C3×#10591105 #13158136

Subgroups — C3×#10591105 #13158136

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×C3×#3828C211yesmaximal, characteristic
2C32×#926C211yesmaximal, characteristic, complemented
3C22×C32×C3⋊C8C211yesmaximal, characteristic, complemented
4C3×#63567C212yesmaximal, complemented
5C3×#63564C212yesmaximal, complemented
6C3×#5113—31nomaximal
7C3×#5002—31nomaximal
8#10591105C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C2×C3×#638—21no—
11C3×#3828—22no—
12C3×#3828C414yescomplemented
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4×C3⋊D3—21no—
15C2×C3×#634C2211yescharacteristic
16C32×#217—21no—
17C32×C4⋊Dic3C2212yes—
18C32×C4×Dic3—21no—
19C2×C32×C3⋊C8—22no—
20C2×C32×C3⋊C8C2212yes—
21C2×#3828C2×C311yescharacteristic
22C2×C3×#235—31no—
23C2×C3×#235—31no—
24C2×C3×#235—61no—
25C22×C32×C8—31no—
26C32×#117—31no—
27C3×#926—21no—
28C3×#926D311yescharacteristic, complemented
29C3×#926C2×C311yescharacteristic, complemented
30C22×C3×C3⋊C8—21no—
31C22×C3×C3⋊C8D311yescharacteristic, complemented
32C22×C3×C3⋊C8C2×C311yescharacteristic, complemented
33C3×#821—32no—
34C3×#816—32no—
35C3×#810—32no—
36C3×#809—32no—
37#63567C2×C312yescomplemented
38#63564C2×C312yescomplemented
39C23×C33D411yescharacteristic
40C22×C3×C3⋊D3—21no—
41C2×C33×C4C2×C412yes—
42C2×C33×C4D411yescharacteristic
43C2×C33×C4D412yes—
44C2×C33×C4C2311yescharacteristic
45C3×#638—24no—
46C2×C3×C3⋊Dic3—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)C2×C412yes—
49C3×C4×C3⋊D3—24no—
50C3×#634—21no—
51C3×#634C2×C412yes—
52C2×C32×Dic3—22no—
53C32×C3⋊C8—22no—
54C3×#416—91no—
55#5113—31no—
56#5002—31no—
57C22×C32×C4—21no—
58C22×C32×C4—21no—
59C22×C32×C4—21no—
60C22×C32×C4—41no—
61C22×C32×C4C2×D311yescharacteristic
62C22×C32×C4C2×D311yescharacteristic
63C22×C32×C4C22×C311yescharacteristic
64C2×#638—21no—
65#3828—22no—
66#3828C3×C414yescomplemented
67C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
68C2×C4×C3⋊D3—21no—
69C2×#634C22×C311yescharacteristic
70C2×C3×#67—62no—
71C2×C3×#67—61no—
72C2×C3×#67—61no—
73C3×#235—34no—
74C3×#235—34no—
75C3×#235—64no—
76C3×#235—64no—
77C3×#235—62no—
78C3×#235—62no—
79C2×C3×D12—31no—
80C2×C3×D12—31no—
81C2×C3×D12—61no—
82C2×C3×C4×D3—62no—
83C2×C3×C4×D3—61no—
84C2×C3×C4×D3—61no—
85C2×C3×Dic6—31no—
86C2×C3×Dic6—31no—
87C2×C3×Dic6—61no—
88C2×C32×C8—32no—
89C2×C32×C8—62no—
90C32×C4⋊C4—32no—
91C32×C22⋊C4—61no—
92C32×C42—61no—
93C3×#217—22no—
94C3×#217—21no—
95C3×#217—21no—
96C3×C4⋊Dic3—22no—
97C3×C4⋊Dic3C2×D312yes—
98C3×C4⋊Dic3C22×C312yes—
99C3×C4×Dic3—22no—
100C3×C4×Dic3—21no—
101C3×C4×Dic3—21no—
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×C3×C3⋊D3—22no—
112C2×C3×C3⋊D3—42no—
113C33×C4C22⋊C412yes—
114C33×C4C22×C411yescharacteristic
115C33×C4C2×D411yescharacteristic
116C3×C3⋊Dic3—22no—
117C32×Dic3—42no—
118C2×C3×#41—91no—
119C2×#235—31no—
120C2×#235—31no—
121C2×#235—61no—
122C22×C3×C8—31no—
123C22×C3×C8—31no—
124C22×C3×C8—61no—
125C3×#117—31no—
126C3×#117—31no—
127C3×#117—61no—
128#926C3×D311yescharacteristic, complemented
129C22×C3⋊C8C3×D311yescharacteristic, complemented
130C3×#103—92no—
131C3×#102—92no—
132#821—32no—
133#816—32no—
134#810—32no—
135#809—32no—
136C23×C32—21no—
137C23×C32—21no—
138C23×C32—21no—
139C23×C32—41no—
140C23×C32D1211yescharacteristic
141C23×C32#6711yescharacteristic
142C23×C32C3×D411yescharacteristic
143C22×C3⋊D3—21no—
144C22×C3×D3—62no—
145C22×C3×D3—61no—
146C22×C3×D3—61no—
147C2×C32×C4—22no—
148C2×C32×C4—22no—
149C2×C32×C4—21no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4—21no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—42no—
160C2×C32×C4—41no—
161C2×C32×C4—42no—
162C2×C32×C4—41no—
163C2×C32×C4—62no—
164C2×C32×C4C4×D312yes—
165C2×C32×C4C4×D312yes—
166C2×C32×C4D1212yes—
167C2×C32×C4D1211yescharacteristic
168C2×C32×C4#6712yes—
169C2×C32×C4#6711yescharacteristic
170C2×C32×C4C2×C3×C412yes—
171C2×C32×C4C3×D411yescharacteristic
172C2×C32×C4C3×D412yes—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C23×C311yescharacteristic
176#638—24no—
177C2×C3⋊Dic3—21no—
178C4⋊(C3⋊D3)—21no—
179C4⋊(C3⋊D3)C2×C3×C412yes—
180C4×C3⋊D3—24no—
181#634—21no—
182#634C2×C3×C412yes—
183C3×#67—68no—
184C3×#67—64no—
185C3×#67—64no—
186C2×C3×Dic3—24no—
187C2×C3×Dic3—22no—
188C2×C3×Dic3—22no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C3×D12—32no—
193C3×D12—32no—
194C3×D12—62no—
195C3×D12—62no—
196C3×D12—61no—
197C3×D12—61no—
198C3×C4×D3—68no—
199C3×C4×D3—64no—
200C3×C4×D3—64no—
201C3×Dic6—32no—
202C3×Dic6—32no—
203C3×Dic6—62no—
204C3×Dic6—62no—
205C3×Dic6—61no—
206C3×Dic6—61no—
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—
215C3×C3⋊D3—42no—
216C3×#41—94no—
217C3×#41—182no—
218C2×C3×Q8—91no—
219C2×C3×D4—91no—
220C2×C3×D4—181no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—41no—
225C22×C3×C4—181no—
226C22×C3×C4D3211yescharacteristic
227C22×C3×C4C2×C3×D311yescharacteristic
228C22×C3×C4C2×C3×D311yescharacteristic
229C2×#67—62no—
230C2×#67—61no—
231C2×#67—61no—
232#235—34no—
233#235—34no—
234#235—64no—
235#235—64no—
236#235—62no—
237#235—62no—
238C2×D12—31no—
239C2×D12—31no—
240C2×D12—61no—
241C2×C4×D3—62no—
242C2×C4×D3—61no—
243C2×C4×D3—61no—
244C2×Dic6—31no—
245C2×Dic6—31no—
246C2×Dic6—61no—
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—32no—
254C3×C4⋊C4—32no—
255C3×C4⋊C4—62no—
256C3×C22⋊C4—62no—
257C3×C22⋊C4—61no—
258C3×C22⋊C4—61no—
259C3×C42—62no—
260C3×C42—61no—
261C3×C42—61no—
262#217—21no—
263C4⋊Dic3C2×C3×D312yes—
264C4×Dic3—21no—
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(C2×D3)⋊C412yes—
283C22×C32(C2×D3)⋊C412yes—
284C22×C32C3×C22⋊C412yes—
285C22×C32C2×D1211yescharacteristic
286C22×C32C2×#6711yescharacteristic
287C22×C32C2×C3×D411yescharacteristic
288C2×C3⋊D3—22no—
289C2×C3⋊D3—42no—
290C2×C3×D3—64no—
291C2×C3×D3—62no—
292C2×C3×D3—62no—
293C2×C3×D3—124no—
294C2×C3×D3—122no—
295C2×C3×D3—122no—
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—122no—
309C32×C4(C2×D3)⋊C412yes—
310C32×C4(C2×D3)⋊C412yes—
311C32×C4C3×C22⋊C412yes—
312C32×C4C2×C4×D311yescharacteristic
313C32×C4C2×C4×D311yescharacteristic
314C32×C4C2×D1211yescharacteristic
315C32×C4C2×#6711yescharacteristic
316C32×C4C22×C3×C411yescharacteristic
317C32×C4C2×C3×D411yescharacteristic
318C3⋊Dic3—22no—
319C3×Dic3—44no—
320C3×Dic3—42no—
321C3×Dic3—42no—
322C3×Dic3—64no—
323C3×Dic3—62no—
324C3×Dic3—62no—
325C2×#41—91no—
326C22×C8—31no—
327#117—31no—
328#103—92no—
329#102—92no—
330C33#41611yescharacteristic, complemented, Sylow
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—41no—
335C23×C3—181no—
336C23×C3#62611yescharacteristic
337C23×C3C3×D1211yescharacteristic
338C23×C3C3×#6711yescharacteristic
339C22×D3—62no—
340C22×D3—61no—
341C22×D3—61no—
342C3×Q8—92no—
343C3×Q8—181no—
344C3×D4—92no—
345C3×D4—184no—
346C3×D4—181no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—64no—
364C2×C3×C4—62no—
365C2×C3×C4—62no—
366C2×C3×C4—181no—
367C2×C3×C4—184no—
368C2×C3×C4#62412yes—
369C2×C3×C4#62612yes—
370C2×C3×C4#62611yescharacteristic
371C2×C3×C4C3×C4×D312yes—
372C2×C3×C4C3×C4×D312yes—
373C2×C3×C4C3×D1211yescharacteristic
374C2×C3×C4C3×D1212yes—
375C2×C3×C4C3×#6711yescharacteristic
376C2×C3×C4C3×#6712yes—
377C2×C3×C4C2×D3211yescharacteristic
378C2×C3×C4C22×C3×D311yescharacteristic
379C2×C3×C4C22×C3×D311yescharacteristic
380#67—68no—
381#67—64no—
382#67—64no—
383C2×Dic3—22no—
384C2×Dic3—62no—
385C2×Dic3—61no—
386C2×Dic3—61no—
387D12—32no—
388D12—32no—
389D12—62no—
390D12—62no—
391D12—61no—
392D12—61no—
393C4×D3—68no—
394C4×D3—64no—
395C4×D3—64no—
396Dic6—32no—
397Dic6—32no—
398Dic6—62no—
399Dic6—62no—
400Dic6—61no—
401Dic6—61no—
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—21no—
413C2×C32—22no—
414C2×C32—22no—
415C2×C32—21no—
416C2×C32—41no—
417C2×C32—42no—
418C2×C32—42no—
419C2×C32—42no—
420C2×C32—44no—
421C2×C32#90412yes—
422C2×C32C2×(C2×D3)⋊C411yescharacteristic
423C2×C32C2×(C2×D3)⋊C411yescharacteristic
424C2×C32#95012yes—
425C2×C32C2×C3×C22⋊C411yescharacteristic
426C2×C32C3×#13512yes—
427C3⋊D3—42no—
428C3×D3—124no—
429C3×D3—122no—
430C3×D3—122no—
431#41—94no—
432#41—182no—
433C2×Q8—91no—
434C2×D4—91no—
435C2×D4—181no—
436C22×C4—181no—
437C22×C4C3×D3211yescharacteristic
438C2×C8—32no—
439C2×C8—62no—
440C4⋊C4—32no—
441C22⋊C4—61no—
442C42—61no—
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—182no—
459C22×C3—362no—
460C22×C3#372212yes—
461C22×C3C3×(C2×D3)⋊C412yes—
462C22×C3C3×(C2×D3)⋊C412yes—
463C22×C3C2×#62611yescharacteristic
464C22×C3C2×C3×D1211yescharacteristic
465C22×C3C2×C3×#6711yescharacteristic
466C2×D3—64no—
467C2×D3—62no—
468C2×D3—62no—
469C2×D3—124no—
470C2×D3—122no—
471C2×D3—122no—
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—124no—
485C3×C4—122no—
486C3×C4—122no—
487C3×C4—182no—
488C3×C4#372212yes—
489C3×C4C3×(C2×D3)⋊C412yes—
490C3×C4C3×(C2×D3)⋊C412yes—
491C3×C4C2×#62411yescharacteristic
492C3×C4C2×#62611yescharacteristic
493C3×C4C2×C3×C4×D311yescharacteristic
494C3×C4C2×C3×C4×D311yescharacteristic
495C3×C4C2×C3×D1211yescharacteristic
496C3×C4C2×C3×#6711yescharacteristic
497Dic3—42no—
498Dic3—64no—
499Dic3—62no—
500Dic3—62no—
501C32—21no—
502C32—21no—
503C32—21no—
504C32—41no—
505C32#500211yescharacteristic, complemented
506C32#511311yescharacteristic, complemented
507C32C3×#41611yescharacteristic, complemented
508C23—181no—
509C23C3×#62611yescharacteristic
510Q8—92no—
511Q8—181no—
512D4—92no—
513D4—184no—
514D4—181no—
515C2×C4—62no—
516C2×C4—181no—
517C2×C4—184no—
518C2×C4C3×#62412yes—
519C2×C4C3×#62612yes—
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—362no—
539C2×C3#6382912yes—
540C2×C3C2×#372211yescharacteristic
541C2×C3C3×#90412yes—
542C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
543C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
544C2×C3C3×#95012yes—
545D3—124no—
546D3—122no—
547D3—122no—
548C22—22no—
549C22—182no—
550C22—362no—
551C22C3×#372212yes—
552C22C2×C3×#62611yescharacteristic
553C4—122no—
554C4—182no—
555C4C3×#372212yes—
556C4C2×C3×#62411yescharacteristic
557C4C2×C3×#62611yescharacteristic
558C3—21no—
559C3—21no—
560C3—21no—
561C3—41no—
562C3#1059110511yescharacteristic, direct product factor
563C3C3×#500211yescharacteristic, complemented
564C3C3×#511311yescharacteristic, complemented
565C2—22no—
566C2—362no—
567C2C3×#6382912yes—
568C2C2×C3×#372211yescharacteristic
subgroupG/H
centerC2×C3×C4#626
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3828C2
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. C32×#926C2semidirect
3. C22×C32×C3⋊C8C2semidirect
4. C3×#63567C2semidirect
5. C3×#63564C2semidirect
8. #10591105C3direct (N × H)
12. C3×#3828C4semidirect
28. C3×#926D3semidirect
29. C3×#926C2×C3semidirect
31. C22×C3×C3⋊C8D3semidirect
32. C22×C3×C3⋊C8C2×C3semidirect
37. #63567C2×C3semidirect
38. #63564C2×C3semidirect
66. #3828C3×C4semidirect
128. #926C3×D3semidirect
129. C22×C3⋊C8C3×D3semidirect
330. C33#416semidirect
505. C32#5002semidirect
506. C32#5113semidirect
507. C32C3×#416semidirect
562. C3#10591105direct (N × H)
563. C3C3×#5002semidirect
564. C3C3×#5113semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C32×#9262
C22×C32×C3⋊C82
C3×#635672
C3×#635642
C3×#51133
C3×#50023
#105911053