Finite Group Databasebeta

← C3×#10591069 #13158095

Subgroups — C3×#10591069 #13158095

Order 1728.

1186 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C4×D3C211yesmaximal, characteristic
2C2×C3×#3748C211yesmaximal, characteristic, complemented
3C2×C32×#208C211yesmaximal, characteristic, complemented
4C3×(C2×D3)⋊(C3⋊C8)C214yesmaximal, complemented
5C3×#5207—31nomaximal
6C3×#5126—31nomaximal
7#10591069C311yesmaximal, characteristic, direct product factor
8C23×C32×D3C411yescharacteristic
9C22×C33×C4C2211yescharacteristic
10C22×C32×Dic3C411yescharacteristic
11C2×C32×C4×D3—22no—
12C2×C32×C4×D3—22no—
13C2×C32×C4×D3C412yes—
14C2×C32×C4×D3C2212yes—
15C3×#3748—22no—
16C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
17C32×#208—22no—
18C2×C32×C3⋊C8C2212yescomplemented
19C23×C32×C4—31no—
20C22×C3×C4×D3—21no—
21C22×C3×C4×D3D311yescharacteristic
22C22×C3×C4×D3C2×C311yescharacteristic
23C2×C32×M16—31no—
24C2×#3748C2×C311yescharacteristic, complemented
25C2×C3×#208—21no—
26C2×C3×#208—31no—
27C2×C3×#208—31no—
28C2×C3×#208—61no—
29C2×C3×#208D311yescharacteristic, complemented
30C2×C3×#208C2×C311yescharacteristic, complemented
31C3×#830—34no—
32C3×(C2×D3)⋊C8—34no—
33(C2×D3)⋊(C3⋊C8)C2×C314yescomplemented
34C23×C33C2×C411yescharacteristic
35C22×C32×D3—22no—
36C22×C32×D3—24no—
37C22×C32×D3C2×C412yes—
38C2×C33×C4C2×C411yescharacteristic
39C2×C33×C4D412yes—
40C2×C33×C4D412yes—
41C2×C33×C4C2311yescharacteristic
42C2×C32×Dic3—22no—
43C2×C32×Dic3C2×C412yes—
44C32×C4×D3—24no—
45C32×C4×D3—22no—
46C32×C4×D3—22no—
47C3×C3⋊(C3⋊C8)—22no—
48C32×C3⋊C8—22no—
49C3×#407—91no—
50#5207—31no—
51#5126—31no—
52C24×C32—31no—
53C23×C3×D3—21no—
54C23×C3×D3Dic311yescharacteristic
55C23×C3×D3C3×C411yescharacteristic
56C22×C32×C4—21no—
57C22×C32×C4—21no—
58C22×C32×C4—21no—
59C22×C32×C4—31no—
60C22×C32×C4—32no—
61C22×C32×C4—32no—
62C22×C32×C4—41no—
63C22×C32×C4—62no—
64C22×C32×C4—62no—
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C22×C311yescharacteristic
68C22×C3×Dic3—21no—
69C22×C3×Dic3Dic311yescharacteristic
70C22×C3×Dic3C3×C411yescharacteristic
71C2×C3×C4×D3—24no—
72C2×C3×C4×D3—24no—
73C2×C3×C4×D3—22no—
74C2×C3×C4×D3—22no—
75C2×C3×C4×D3—22no—
76C2×C3×C4×D3—22no—
77C2×C3×C4×D3—22no—
78C2×C3×C4×D3—22no—
79C2×C3×C4×D3Dic312yes—
80C2×C3×C4×D3C3×C412yes—
81C2×C3×C4×D3C2×D312yes—
82C2×C3×C4×D3C22×C312yes—
83C32×M16—62no—
84C2×C32×C8—32no—
85#3748—22no—
86C2×C3⋊(C3⋊C8)C22×C312yescomplemented
87C3×#208—24no—
88C3×#208—22no—
89C3×#208—22no—
90C3×#208—64no—
91C3×#208—62no—
92C3×#208—62no—
93C2×C3×C3⋊C8—22no—
94C2×C3×C3⋊C8—32no—
95C2×C3×C3⋊C8—32no—
96C2×C3×C3⋊C8—62no—
97C2×C3×C3⋊C8C2×D312yescomplemented
98C2×C3×C3⋊C8C22×C312yescomplemented
99C22×C33—22no—
100C22×C33C22⋊C412yes—
101C22×C33C22×C411yescharacteristic
102C2×C32×D3—28no—
103C2×C32×D3—22no—
104C2×C32×D3—22no—
105C2×C32×D3M1614yescomplemented
106C33×C4C22⋊C412yes—
107C33×C4C2×D411yescharacteristic
108C33×C4C2×D411yescharacteristic
109C32×Dic3—22no—
110C23×C3×C4—31no—
111C23×C3×C4—31no—
112C23×C3×C4—61no—
113C22×C4×D3C3×D311yescharacteristic
114C2×C3×M16—31no—
115C2×C3×M16—31no—
116C2×C3×M16—61no—
117C2×C3×M16—91no—
118C2×#208—31no—
119C2×#208—31no—
120C2×#208—61no—
121C2×#208C3×D311yescharacteristic, complemented
122C3×C22⋊C8—94no—
123#830—34no—
124(C2×D3)⋊C8—34no—
125C23×C32—21no—
126C23×C32—21no—
127C23×C32—21no—
128C23×C32—32no—
129C23×C32—41no—
130C23×C32—64no—
131C23×C32—62no—
132C23×C32C4×D311yescharacteristic
133C23×C32C2×Dic311yescharacteristic
134C23×C32C2×C3×C411yescharacteristic
135C22×C3×D3—28no—
136C22×C3×D3—24no—
137C22×C3×D3—22no—
138C22×C3×D3—22no—
139C22×C3×D3—24no—
140C22×C3×D3—24no—
141C22×C3×D3—22no—
142C22×C3×D3C2×Dic312yes—
143C22×C3×D3C2×C3×C412yes—
144C2×C32×C4—21no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4—21no—
148C2×C32×C4—22no—
149C2×C32×C4—21no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4—22no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—32no—
157C2×C32×C4—41no—
158C2×C32×C4—42no—
159C2×C32×C4—42no—
160C2×C32×C4—41no—
161C2×C32×C4—62no—
162C2×C32×C4—64no—
163C2×C32×C4—62no—
164C2×C32×C4—62no—
165C2×C32×C4C4×D311yescharacteristic
166C2×C32×C4D1212yes—
167C2×C32×C4C2×Dic311yescharacteristic
168C2×C32×C4#6712yes—
169C2×C32×C4#6712yes—
170C2×C32×C4#6712yes—
171C2×C32×C4C2×C3×C411yescharacteristic
172C2×C32×C4C3×D412yes—
173C2×C32×C4C3×D412yes—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C23×C311yescharacteristic
177C2×C3×Dic3—24no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3C2×Dic312yes—
182C2×C3×Dic3C2×C3×C412yes—
183C3×C4×D3—28no—
184C3×C4×D3—24no—
185C3×C4×D3—24no—
186C3×C4×D3—24no—
187C3×C4×D3—24no—
188C3×C4×D3—22no—
189C3×C4×D3—22no—
190C3×C4×D3—22no—
191C3×C4×D3—22no—
192C32×C8—62no—
193C3⋊(C3⋊C8)—22no—
194C3×C3⋊C8—24no—
195C3×C3⋊C8—22no—
196C3×C3⋊C8—22no—
197C3×C3⋊C8—64no—
198C3×C3⋊C8—62no—
199C3×C3⋊C8—62no—
200#407—91noSylow
201C2×C33—22no—
202C2×C33C2×C22⋊C411yescharacteristic
203C2×C33C2×M1612yes—
204C32×D3—24no—
205C24×C3—31no—
206C24×C3—31no—
207C24×C3—61no—
208C23×D3C3×Dic311yescharacteristic
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—31no—
213C22×C3×C4—31no—
214C22×C3×C4—32no—
215C22×C3×C4—32no—
216C22×C3×C4—32no—
217C22×C3×C4—32no—
218C22×C3×C4—41no—
219C22×C3×C4—62no—
220C22×C3×C4—62no—
221C22×C3×C4—61no—
222C22×C3×C4—64no—
223C22×C3×C4—64no—
224C22×C3×C4—62no—
225C22×C3×C4—62no—
226C22×C3×C4—62no—
227C22×C3×C4—62no—
228C22×C3×C4D3211yescharacteristic
229C22×C3×C4C2×C3×D311yescharacteristic
230C22×C3×C4C2×C3×D311yescharacteristic
231C22×Dic3C3×Dic311yescharacteristic
232C2×C4×D3—22no—
233C2×C4×D3—22no—
234C2×C4×D3C3×Dic312yes—
235C2×C4×D3C2×C3×D312yes—
236C3×M16—64no—
237C3×M16—62no—
238C3×M16—62no—
239C3×M16—182no—
240C2×C3×C8—32no—
241C2×C3×C8—32no—
242C2×C3×C8—62no—
243C2×C3×C8—92no—
244#208—22no—
245#208—64no—
246#208—62no—
247#208—62no—
248C2×C3⋊C8—32no—
249C2×C3⋊C8—32no—
250C2×C3⋊C8—62no—
251C2×C3⋊C8C2×C3×D312yescomplemented
252C22×C32—22no—
253C22×C32—24no—
254C22×C32—22no—
255C22×C32—22no—
256C22×C32—22no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—21no—
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—34no—
263C22×C32—42no—
264C22×C32—41no—
265C22×C32—42no—
266C22×C32—42no—
267C22×C32—44no—
268C22×C32—68no—
269C22×C32—62no—
270C22×C32—62no—
271C22×C32(C2×D3)⋊C412yes—
272C22×C32#21712yes—
273C22×C32C3×C22⋊C412yes—
274C22×C32C2×C4×D311yescharacteristic
275C22×C32C22×Dic311yescharacteristic
276C22×C32C22×C3×C411yescharacteristic
277C2×C3×D3—24no—
278C2×C3×D3—216no—
279C2×C3×D3—28no—
280C2×C3×D3—28no—
281C2×C3×D3—24no—
282C2×C3×D3—22no—
283C2×C3×D3—22no—
284C2×C3×D3—22no—
285C2×C3×D3—22no—
286C2×C3×D3—24no—
287C2×C3×D3#20814yescomplemented
288C2×C3×D3C3×M1614yescomplemented
289C32×C4—22no—
290C32×C4—22no—
291C32×C4—22no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—42no—
299C32×C4—41no—
300C32×C4—41no—
301C32×C4—62no—
302C32×C4(C2×D3)⋊C412yes—
303C32×C4#21712yes—
304C32×C4C3×C22⋊C412yes—
305C32×C4C2×D1211yescharacteristic
306C32×C4C2×#6711yescharacteristic
307C32×C4C2×#6711yescharacteristic
308C32×C4C2×#6711yescharacteristic
309C32×C4C2×C3×D411yescharacteristic
310C32×C4C2×C3×D411yescharacteristic
311C3×Dic3—24no—
312C3×Dic3—22no—
313C3×Dic3—22no—
314C23×C4—31no—
315C2×M16—31no—
316C2×M16—91no—
317C22⋊C8—94no—
318C33#40711yescharacteristic, complemented, Sylow
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—21no—
322C23×C3—32no—
323C23×C3—32no—
324C23×C3—41no—
325C23×C3—62no—
326C23×C3—64no—
327C23×C3—64no—
328C23×C3—64no—
329C23×C3—68no—
330C23×C3—62no—
331C23×C3—62no—
332C23×C3D3×Dic311yescharacteristic
333C23×C3C3×C4×D311yescharacteristic
334C23×C3C2×C3×Dic311yescharacteristic
335C22×D3—22no—
336C22×D3—24no—
337C22×D3C2×C3×Dic312yes—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—32no—
351C2×C3×C4—32no—
352C2×C3×C4—41no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—41no—
356C2×C3×C4—62no—
357C2×C3×C4—64no—
358C2×C3×C4—64no—
359C2×C3×C4—68no—
360C2×C3×C4—64no—
361C2×C3×C4—62no—
362C2×C3×C4—62no—
363C2×C3×C4—64no—
364C2×C3×C4—64no—
365C2×C3×C4—62no—
366C2×C3×C4—62no—
367C2×C3×C4—62no—
368C2×C3×C4—62no—
369C2×C3×C4D3×Dic311yescharacteristic
370C2×C3×C4#62512yes—
371C2×C3×C4#62612yes—
372C2×C3×C4C3×C4×D311yescharacteristic
373C2×C3×C4C3×D1212yes—
374C2×C3×C4C2×C3×Dic311yescharacteristic
375C2×C3×C4C3×#6712yes—
376C2×C3×C4C3×#6712yes—
377C2×C3×C4C3×#6712yes—
378C2×C3×C4C2×D3211yescharacteristic
379C2×C3×C4C22×C3×D311yescharacteristic
380C2×C3×C4C22×C3×D311yescharacteristic
381C2×Dic3—22no—
382C2×Dic3C2×C3×Dic312yes—
383C4×D3—24no—
384C4×D3—22no—
385C4×D3—22no—
386C3×C8—64no—
387C3×C8—62no—
388C3×C8—62no—
389C3×C8—182no—
390C3⋊C8—22no—
391C3⋊C8—64no—
392C3⋊C8—62no—
393C3⋊C8—62no—
394C2×C32—22no—
395C2×C32—24no—
396C2×C32—22no—
397C2×C32—22no—
398C2×C32—22no—
399C2×C32—22no—
400C2×C32—21no—
401C2×C32—22no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—41no—
405C2×C32—42no—
406C2×C32—42no—
407C2×C32—42no—
408C2×C32—44no—
409C2×C32—64no—
410C2×C32D3×M1612yes—
411C2×C32C2×#20812yes—
412C2×C32C2×(C2×D3)⋊C411yescharacteristic
413C2×C32C2×#21711yescharacteristic
414C2×C32C2×C3×C22⋊C411yescharacteristic
415C2×C32C2×C3×M1612yes—
416C3×D3—28no—
417C3×D3—24no—
418C3×D3—24no—
419C24—31no—
420C22×C4—31no—
421C22×C4—32no—
422C22×C4—32no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4C3×D3211yescharacteristic
426M16—62no—
427M16—182no—
428C2×C8—32no—
429C2×C8—92no—
430C22×C3—24no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—34no—
441C22×C3—34no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—44no—
445C22×C3—41no—
446C22×C3—42no—
447C22×C3—64no—
448C22×C3—64no—
449C22×C3—616no—
450C22×C3—68no—
451C22×C3—68no—
452C22×C3—64no—
453C22×C3—62no—
454C22×C3—62no—
455C22×C3—62no—
456C22×C3—62no—
457C22×C3(C2×D3)⋊Dic312yes—
458C22×C3C3×(C2×D3)⋊C412yes—
459C22×C3C3×#21712yes—
460C22×C3C2×D3×Dic311yescharacteristic
461C22×C3C2×C3×C4×D311yescharacteristic
462C22×C3C22×C3×Dic311yescharacteristic
463C2×D3—28no—
464C2×D3—22no—
465C2×D3—22no—
466C2×D3C3×#20814yescomplemented
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—22no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—42no—
479C3×C4—64no—
480C3×C4—62no—
481C3×C4—62no—
482C3×C4(C2×D3)⋊Dic312yes—
483C3×C4C3×(C2×D3)⋊C412yes—
484C3×C4C3×#21712yes—
485C3×C4C2×#62511yescharacteristic
486C3×C4C2×#62611yescharacteristic
487C3×C4C2×C3×D1211yescharacteristic
488C3×C4C2×C3×#6711yescharacteristic
489C3×C4C2×C3×#6711yescharacteristic
490C3×C4C2×C3×#6711yescharacteristic
491Dic3—22no—
492C32—21no—
493C32—21no—
494C32—21no—
495C32—41no—
496C32#512611yescharacteristic, complemented
497C32#520711yescharacteristic, complemented
498C32C3×#40711yescharacteristic, complemented
499C23—32no—
500C23—64no—
501C23—62no—
502C23C3×D3×Dic311yescharacteristic
503C2×C4—32no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—64no—
508C2×C4C3×D3×Dic311yescharacteristic
509C2×C4C3×#62512yes—
510C2×C4C3×#62612yes—
511C2×C4C2×C3×D3211yescharacteristic
512C8—62no—
513C8—182no—
514C2×C3—24no—
515C2×C3—22no—
516C2×C3—22no—
517C2×C3—22no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—22no—
522C2×C3—22no—
523C2×C3—22no—
524C2×C3—42no—
525C2×C3—42no—
526C2×C3—44no—
527C2×C3—41no—
528C2×C3—42no—
529C2×C3—68no—
530C2×C3—64no—
531C2×C3—64no—
532C2×C3D3×#20812yes—
533C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
534C2×C3C3×D3×M1612yes—
535C2×C3C2×C3×#20812yes—
536C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
537C2×C3C2×C3×#21711yescharacteristic
538D3—24no—
539C22—22no—
540C22—34no—
541C22—68no—
542C22—62no—
543C22—62no—
544C22C3×(C2×D3)⋊Dic312yes—
545C22C2×C3×D3×Dic311yescharacteristic
546C4—62no—
547C4C3×(C2×D3)⋊Dic312yes—
548C4C2×C3×#62511yescharacteristic
549C4C2×C3×#62611yescharacteristic
550C3—21no—
551C3—21no—
552C3—21no—
553C3—41no—
554C3#1059106911yescharacteristic, direct product factor
555C3C3×#512611yescharacteristic, complemented
556C3C3×#520711yescharacteristic, complemented
557C2—22no—
558C2—64no—
559C2C3×D3×#20812yes—
560C2C2×C3×(C2×D3)⋊Dic311yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C22×C4
prime radicalC23×C32×D3C4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#407
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3748C2semidirect
3. C2×C32×#208C2semidirect
4. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
7. #10591069C3direct (N × H)
16. C2×C3×C3⋊(C3⋊C8)C22semidirect
18. C2×C32×C3⋊C8C22semidirect
24. C2×#3748C2×C3semidirect
29. C2×C3×#208D3semidirect
30. C2×C3×#208C2×C3semidirect
33. (C2×D3)⋊(C3⋊C8)C2×C3semidirect
86. C2×C3⋊(C3⋊C8)C22×C3semidirect
97. C2×C3×C3⋊C8C2×D3semidirect
98. C2×C3×C3⋊C8C22×C3semidirect
105. C2×C32×D3M16semidirect
121. C2×#208C3×D3semidirect
251. C2×C3⋊C8C2×C3×D3semidirect
287. C2×C3×D3#208semidirect
288. C2×C3×D3C3×M16semidirect
318. C33#407semidirect
466. C2×D3C3×#208semidirect
496. C32#5126semidirect
497. C32#5207semidirect
498. C32C3×#407semidirect
554. C3#10591069direct (N × H)
555. C3C3×#5126semidirect
556. C3C3×#5207semidirect

Maximal subgroups

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

subgroupindex
C22×C32×C4×D32
C2×C3×#37482
C2×C32×#2082
C3×(C2×D3)⋊(C3⋊C8)2
C3×#52073
C3×#51263
#105910693