Finite Group Databasebeta

← C3×D3×#807 #13157562

Subgroups — C3×D3×#807 #13157562

Order 1728.

710 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×D3×C4⋊C4C211yesmaximal, characteristic
2C3×D3×C4⋊Dic3C211yesmaximal, characteristic
3C2×C3×D3×C3⋊C8C211yesmaximal, characteristic
4C3×#63634C211yesmaximal, characteristic, complemented
5C32×#807C211yesmaximal, characteristic, complemented
6C3×#63576C211yesmaximal, characteristic, complemented
7C3×#63575C211yesmaximal, characteristic, complemented
8C2×C3×#807—31nomaximal
9C3×D3×#107—31nomaximal
10D3×#807C311yesmaximal, characteristic, direct product factor
11C2×C32×C4×D3—21no—
12C2×C32×C4×D3C2211yescharacteristic
13C2×C3×D3×Dic3—21no—
14C33×C4⋊C4C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C3×D3×C3⋊C8C414yescomplemented
23C2×C32×C4⋊C4—31no—
24C2×C3×C4⋊Dic3—31no—
25C22×C3×C3⋊C8—31no—
26C2×C3×D3×C8—31no—
27C3×D3×C4⋊C4—21no—
28C3×D3×C4⋊C4—31no—
29C3×D3×C4⋊C4D311yescharacteristic
30C3×D3×C4⋊C4C2×C311yescharacteristic
31D3×C4⋊Dic3C2×C311yescharacteristic
32C2×D3×C3⋊C8C2×C311yescharacteristic
33C32×#107—31no—
34#63634C2×C311yescharacteristic, complemented
35C3×#818—31no—
36C3×#807—21no—
37C3×#807—31no—
38C3×#807—31no—
39C3×#807—31no—
40C3×#807—31no—
41C3×#807—31no—
42C3×#807—61no—
43C3×#807D311yescharacteristic, direct product factor
44C3×#807C2×C311yescharacteristic, complemented
45#63576C2×C311yescharacteristic, complemented
46#63575C2×C311yescharacteristic, complemented
47C22×C32×D3D411yescharacteristic
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C32×C4×D3—42no—
55C32×C4×D3C2×C412yes—
56C32×C4×D3Q812yes—
57C3×D3×Dic3—42no—
58C3×C3⋊(C3⋊C8)C2×C412yescomplemented
59C32×C3⋊C8C2×C412yescomplemented
60C2×C3×#107—91no—
61C2×#807—31no—
62D3×#107—31no—
63C22×C32×C4—31no—
64C22×C32×C4—61no—
65C22×C3×Dic3—61no—
66C2×C3×C4×D3—21no—
67C2×C3×C4×D3—22no—
68C2×C3×C4×D3—21no—
69C2×C3×C4×D3—21no—
70C2×C3×C4×D3—61no—
71C2×C3×C4×D3C2×D311yescharacteristic
72C2×C3×C4×D3C22×C311yescharacteristic
73C2×D3×Dic3—21no—
74C2×C32×C8—31no—
75C32×C4⋊C4—21no—
76C32×C4⋊C4—21no—
77C32×C4⋊C4—21no—
78C32×C4⋊C4—31no—
79C32×C4⋊C4—31no—
80C32×C4⋊C4—41no—
81C32×C4⋊C4—61no—
82C32×C4⋊C4C2×D311yescharacteristic
83C32×C4⋊C4C2×D311yescharacteristic
84C32×C4⋊C4C22×C311yescharacteristic
85C4⋊(C3⋊Dic3)C22×C311yescharacteristic
86C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4⋊Dic3C22×C311yescharacteristic
96C3×C4⋊Dic3C22×C311yescharacteristic
97C3×Dic3⋊C4—22no—
98C3×Dic3⋊C4—21no—
99C3×Dic3⋊C4—21no—
100C3×Dic3⋊C4—61no—
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—34no—
105C2×C3×C3⋊C8—61no—
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C22×C311yescharacteristic
108C3×D3×C8—34no—
109Dic3⋊Dic3—21no—
110D3×C3⋊C8C3×C414yescomplemented
111C22×C33C2×D411yescharacteristic
112C2×C32×D3C4⋊C412yes—
113C2×C32×D3D812yes—
114C2×C32×D3Q1612yes—
115C33×C4—41no—
116C33×C4C22×C411yescharacteristic
117C33×C4C2×Q811yescharacteristic
118C3×C3⋊Dic3—41no—
119C32×Dic3—41no—
120C32×Dic3—41no—
121C32×Dic3C4⋊C412yes—
122C22×C3×C8—91no—
123C2×C3×C4⋊C4—31no—
124C2×C3×C4⋊C4—31no—
125C2×C3×C4⋊C4—61no—
126C2×C3×C4⋊C4—91no—
127C2×C4⋊Dic3—31no—
128C22×C3⋊C8—31no—
129C2×D3×C8—31no—
130D3×C4⋊C4—31no—
131D3×C4⋊C4C3×D311yescharacteristic
132C3×#107—31no—
133C3×#107—31no—
134C3×#107—61no—
135C3×#107—91no—
136C3×#107—91no—
137C3×#107—91no—
138#818—31no—
139#807—31no—
140#807—31no—
141#807—31no—
142#807—31no—
143#807—31no—
144#807—61no—
145#807C3×D311yescharacteristic, direct product factor
146C23×C32—31no—
147C22×C3×D3—21no—
148C22×C3×D3#6711yescharacteristic
149C22×C3×D3C3×D411yescharacteristic
150C2×C32×C4—21no—
151C2×C32×C4—22no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—31no—
158C2×C32×C4—32no—
159C2×C32×C4—32no—
160C2×C32×C4—41no—
161C2×C32×C4—41no—
162C2×C32×C4—41no—
163C2×C32×C4—42no—
164C2×C32×C4—61no—
165C2×C32×C4—61no—
166C2×C32×C4—122no—
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C23×C311yescharacteristic
170C2×C3⋊Dic3—21no—
171C2×C3×Dic3—22no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—41no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—121no—
181C2×C3×Dic3—122no—
182C2×C3×Dic3#6711yescharacteristic
183C2×C3×Dic3C3×D411yescharacteristic
184C3×C4×D3—22no—
185C3×C4×D3—22no—
186C3×C4×D3—44no—
187C3×C4×D3—42no—
188C3×C4×D3—42no—
189C3×C4×D3—122no—
190C3×C4×D3Dic612yes—
191C3×C4×D3C4×D312yes—
192C3×C4×D3C2×C3×C412yes—
193C3×C4×D3C3×Q812yes—
194D3×Dic3—42no—
195C32×C8—32no—
196C3⋊(C3⋊C8)C2×C3×C412yescomplemented
197C3×C3⋊C8—22no—
198C3×C3⋊C8—32no—
199C3×C3⋊C8—32no—
200C3×C3⋊C8—62no—
201C3×C3⋊C8C4×D312yescomplemented
202C3×C3⋊C8C2×C3×C412yescomplemented
203C2×#107—91noSylow
204C2×C33C2×C4⋊C411yescharacteristic
205C2×C33C2×D811yescharacteristic
206C2×C33C2×Q1611yescharacteristic
207C32×D3#10714yescomplemented
208C22×C3×C4—31no—
209C22×C3×C4—31no—
210C22×C3×C4—61no—
211C22×C3×C4—61no—
212C22×C3×C4—61no—
213C22×C3×C4—62no—
214C22×C3×C4—181no—
215C22×Dic3—61no—
216C2×C4×D3—21no—
217C2×C4×D3—61no—
218C2×C4×D3C2×C3×D311yescharacteristic
219C2×C3×C8—31no—
220C2×C3×C8—31no—
221C2×C3×C8—61no—
222C2×C3×C8—94no—
223C2×C3×C8—91no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—31no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—41no—
232C3×C4⋊C4—61no—
233C3×C4⋊C4—61no—
234C3×C4⋊C4—61no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—62no—
237C3×C4⋊C4—91no—
238C3×C4⋊C4—181no—
239C3×C4⋊C4D3211yescharacteristic
240C3×C4⋊C4C2×C3×D311yescharacteristic
241C3×C4⋊C4C2×C3×D311yescharacteristic
242C4⋊Dic3—31no—
243C4⋊Dic3—31no—
244C4⋊Dic3—61no—
245C4⋊Dic3—61no—
246C4⋊Dic3C2×C3×D311yescharacteristic
247C4⋊Dic3C2×C3×D311yescharacteristic
248Dic3⋊C4—21no—
249Dic3⋊C4—61no—
250C2×C3⋊C8—31no—
251C2×C3⋊C8—31no—
252C2×C3⋊C8—34no—
253C2×C3⋊C8—61no—
254C2×C3⋊C8C2×C3×D311yescharacteristic
255D3×C8—34no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—32no—
260C22×C32—32no—
261C22×C32—32no—
262C22×C32—41no—
263C22×C32D3×D411yescharacteristic
264C22×C32C2×#6711yescharacteristic
265C22×C32C2×C3×D411yescharacteristic
266C2×C3×D3—22no—
267C2×C3×D3—22no—
268C2×C3×D3—22no—
269C2×C3×D3Dic3⋊C412yes—
270C2×C3×D3#21312yes—
271C2×C3×D3#21612yes—
272C2×C3×D3C3×C4⋊C412yes—
273C2×C3×D3C3×D812yes—
274C2×C3×D3C3×Q1612yes—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—32no—
282C32×C4—41no—
283C32×C4—41no—
284C32×C4—41no—
285C32×C4—42no—
286C32×C4—41no—
287C32×C4—41no—
288C32×C4—81no—
289C32×C4—81no—
290C32×C4—82no—
291C32×C4—121no—
292C32×C4—121no—
293C32×C4C2×Dic611yescharacteristic
294C32×C4C2×C4×D311yescharacteristic
295C32×C4C2×C4×D311yescharacteristic
296C32×C4D3×Q811yescharacteristic
297C32×C4C22×C3×C411yescharacteristic
298C32×C4C2×C3×Q811yescharacteristic
299C3⋊Dic3—41no—
300C3×Dic3—22no—
301C3×Dic3—42no—
302C3×Dic3—41no—
303C3×Dic3—41no—
304C3×Dic3—41no—
305C3×Dic3—41no—
306C3×Dic3—81no—
307C3×Dic3—121no—
308C3×Dic3—121no—
309C3×Dic3—241no—
310C3×Dic3Dic3⋊C412yes—
311C3×Dic3C3×C4⋊C412yes—
312C22×C8—91no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—91no—
315#107—31no—
316#107—91no—
317#107—91no—
318#107—91no—
319C33C2×#10711yescharacteristic, complemented, Sylow
320C23×C3—31no—
321C23×C3—31no—
322C23×C3—61no—
323C22×D3C3×#6711yescharacteristic
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—32no—
334C2×C3×C4—32no—
335C2×C3×C4—32no—
336C2×C3×C4—32no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—62no—
342C2×C3×C4—62no—
343C2×C3×C4—61no—
344C2×C3×C4—62no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—121no—
350C2×C3×C4—122no—
351C2×C3×C4—122no—
352C2×C3×C4—124no—
353C2×C3×C4—181no—
354C2×C3×C4—362no—
355C2×C3×C4C2×D3211yescharacteristic
356C2×C3×C4C22×C3×D311yescharacteristic
357C2×C3×C4C22×C3×D311yescharacteristic
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—61no—
361C2×Dic3—61no—
362C2×Dic3—121no—
363C2×Dic3—122no—
364C2×Dic3C3×#6711yescharacteristic
365C4×D3—42no—
366C4×D3—122no—
367C4×D3C3×Dic612yes—
368C4×D3C3×C4×D312yes—
369C3×C8—32no—
370C3×C8—32no—
371C3×C8—62no—
372C3×C8—92no—
373C3⋊C8—32no—
374C3⋊C8—32no—
375C3⋊C8—62no—
376C3⋊C8C3×C4×D312yescomplemented
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—34no—
387C2×C32—41no—
388C2×C32—41no—
389C2×C32—41no—
390C2×C32D3×C4⋊C411yescharacteristic
391C2×C32D3×D811yescharacteristic
392C2×C32D3×Q1611yescharacteristic
393C2×C32C2×Dic3⋊C411yescharacteristic
394C2×C32C2×#21311yescharacteristic
395C2×C32C2×#21611yescharacteristic
396C2×C32C2×C3×C4⋊C411yescharacteristic
397C2×C32C2×C3×D811yescharacteristic
398C2×C32C2×C3×Q1611yescharacteristic
399C3×D3—24no—
400C3×D3#80714yesdirect product factor
401C3×D3C3×#10714yescomplemented
402C22×C4—31no—
403C22×C4—61no—
404C22×C4—181no—
405C2×C8—31no—
406C2×C8—94no—
407C2×C8—91no—
408C4⋊C4—31no—
409C4⋊C4—31no—
410C4⋊C4—61no—
411C4⋊C4—91no—
412C4⋊C4—181no—
413C4⋊C4C3×D3211yescharacteristic
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—32no—
418C22×C3—32no—
419C22×C3—32no—
420C22×C3—32no—
421C22×C3—32no—
422C22×C3—32no—
423C22×C3—41no—
424C22×C3—62no—
425C22×C3—62no—
426C22×C3—62no—
427C22×C3D3×#6711yescharacteristic
428C22×C3C3×D3×D411yescharacteristic
429C22×C3C2×C3×#6711yescharacteristic
430C2×D3C3×Dic3⋊C412yes—
431C2×D3C3×#21312yes—
432C2×D3C3×#21612yes—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—32no—
440C3×C4—32no—
441C3×C4—42no—
442C3×C4—41no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—62no—
448C3×C4—81no—
449C3×C4—81no—
450C3×C4—82no—
451C3×C4—122no—
452C3×C4—121no—
453C3×C4—121no—
454C3×C4—121no—
455C3×C4—121no—
456C3×C4—241no—
457C3×C4—361no—
458C3×C4D3×Dic611yescharacteristic
459C3×C4C4×D3211yescharacteristic
460C3×C4C2×C3×Dic611yescharacteristic
461C3×C4C2×C3×C4×D311yescharacteristic
462C3×C4C2×C3×C4×D311yescharacteristic
463C3×C4C3×D3×Q811yescharacteristic
464Dic3—41no—
465Dic3—41no—
466Dic3—121no—
467Dic3—121no—
468Dic3—241no—
469Dic3C3×Dic3⋊C412yes—
470C32—21no—
471C32—21no—
472C32—21no—
473C32—41no—
474C32D3×#10711yescharacteristic, complemented
475C32C2×#80711yescharacteristic, complemented
476C32C2×C3×#10711yescharacteristic, complemented
477C23—31no—
478C2×C4—21no—
479C2×C4—31no—
480C2×C4—32no—
481C2×C4—32no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—122no—
485C2×C4—181no—
486C2×C4—362no—
487C2×C4C2×C3×D3211yescharacteristic
488C8—32no—
489C8—92no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—34no—
500C2×C3—34no—
501C2×C3—41no—
502C2×C3—41no—
503C2×C3—41no—
504C2×C3—64no—
505C2×C3D3×Dic3⋊C411yescharacteristic
506C2×C3D3×#21311yescharacteristic
507C2×C3D3×#21611yescharacteristic
508C2×C3C3×D3×C4⋊C411yescharacteristic
509C2×C3C3×D3×D811yescharacteristic
510C2×C3C3×D3×Q1611yescharacteristic
511C2×C3C2×C3×Dic3⋊C411yescharacteristic
512C2×C3C2×C3×#21311yescharacteristic
513C2×C3C2×C3×#21611yescharacteristic
514D3C3×#80714yesdirect product factor
515C22—32no—
516C22—32no—
517C22—32no—
518C22C3×D3×#6711yescharacteristic
519C4—32no—
520C4—41no—
521C4—121no—
522C4—121no—
523C4—361no—
524C4C3×D3×Dic611yescharacteristic
525C4C3×C4×D3211yescharacteristic
526C3—21no—
527C3—21no—
528C3—21no—
529C3—41no—
530C3D3×#80711yescharacteristic, direct product factor
531C3C3×D3×#10711yescharacteristic, complemented
532C3C2×C3×#80711yescharacteristic, complemented
533C2—34no—
534C2C3×D3×Dic3⋊C411yescharacteristic
535C2C3×D3×#21311yescharacteristic
536C2C3×D3×#21611yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
cosocleC2×C32×C4C23×C3
hypercenterC3×C4⋊C4D32
hypocenterC32C2×C3×#107
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C3×#63634C2semidirect
5. C32×#807C2semidirect
6. C3×#63576C2semidirect
7. C3×#63575C2semidirect
10. D3×#807C3direct (N × H)
22. C3×D3×C3⋊C8C4semidirect
34. #63634C2×C3semidirect
43. C3×#807D3direct (N × H)
44. C3×#807C2×C3semidirect
45. #63576C2×C3semidirect
46. #63575C2×C3semidirect
58. C3×C3⋊(C3⋊C8)C2×C4semidirect
59. C32×C3⋊C8C2×C4semidirect
110. D3×C3⋊C8C3×C4semidirect
145. #807C3×D3direct (N × H)
196. C3⋊(C3⋊C8)C2×C3×C4semidirect
201. C3×C3⋊C8C4×D3semidirect
202. C3×C3⋊C8C2×C3×C4semidirect
207. C32×D3#107semidirect
319. C33C2×#107semidirect
376. C3⋊C8C3×C4×D3semidirect
400. C3×D3#807direct (N × H)
401. C3×D3C3×#107semidirect
474. C32D3×#107semidirect
475. C32C2×#807semidirect
476. C32C2×C3×#107semidirect
514. D3C3×#807direct (N × H)
530. C3D3×#807direct (N × H)
531. C3C3×D3×#107semidirect
532. C3C2×C3×#807semidirect

Maximal subgroups

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

subgroupindex
C32×D3×C4⋊C42
C3×D3×C4⋊Dic32
C2×C3×D3×C3⋊C82
C3×#636342
C32×#8072
C3×#635762
C3×#635752
C2×C3×#8073
C3×D3×#1073
D3×#8073