Finite Group Databasebeta

← C3×#10591401 #13158446

Subgroups — C3×#10591401 #13158446

Order 1728.

754 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3754C211yesmaximal, characteristic
2C32×D4×Dic3C211yesmaximal, characteristic
3C3×#63844C211yesmaximal, characteristic, complemented
4C32×#832C211yesmaximal, characteristic
5C3×#63574C211yesmaximal, characteristic, complemented
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8C3×C4×#214—31nomaximal
9C3×#4758—31nomaximal
10#10591401C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C22×C32×Dic3—21no—
14C3×#3754C414yescomplemented
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×#217—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×#3724—21no—
22C3×Dic32—21no—
23C32×C4×D4—31no—
24C2×#3754C2×C311yescharacteristic
25C3×D4×Dic3—21no—
26C3×D4×Dic3D311yescharacteristic
27C3×D4×Dic3C2×C311yescharacteristic
28C2×C3×#214—31no—
29C2×C3×#214—31no—
30C2×C3×#214—61no—
31C3×#887—31no—
32C3×C4×Dic6—31no—
33#63844C2×C311yescharacteristic, complemented
34C32×#102—31no—
35C3×#832—21no—
36C3×#832D311yescharacteristic
37C3×#832C2×C311yescharacteristic
38C3×#816—31no—
39C3×C8×Dic3—31no—
40C3×#810—31no—
41C3×#808—31no—
42C3×#808—31no—
43C3×C4×C3⋊C8—31no—
44#63574C2×C311yescharacteristic, complemented
45#63567C2×C311yescharacteristic, complemented
46Dic3×C3⋊C8C2×C311yescharacteristic, complemented
47C23×C33—21no—
48C33×D4—21no—
49C33×D4C2×C412yes—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C3×#634—21no—
53C3×#634C2×C412yescomplemented
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—42no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C3×C3⋊(C3⋊C8)C2×C412yescomplemented
60C32×C3⋊C8—21no—
61C3×C4×SD16—91no—
62C4×#214—31no—
63#4758—31no—
64C2×C32×D4—21no—
65C2×C32×D4—21no—
66C2×C32×D4—21no—
67C2×C32×D4—41no—
68C2×C32×D4C2×D311yescharacteristic
69C2×C32×D4C2×D311yescharacteristic
70C2×C32×D4C22×C311yescharacteristic
71C22×C32×C4—61no—
72C2×#634C22×C311yescharacteristic
73C22×C3×Dic3—22no—
74C22×C3×Dic3—21no—
75C22×C3×Dic3—21no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—61no—
79C2×C32×C8—31no—
80C32×C4⋊C4—31no—
81C32×C4⋊C4—31no—
82C32×C22⋊C4—61no—
83C32×C42—31no—
84#3754C3×C414yescomplemented
85C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
86C3×#217—22no—
87C3×#217—21no—
88C3×#217—21no—
89C3×#214—34no—
90C3×#214—34no—
91C3×#214—64no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4⋊Dic3C22×C311yescharacteristic
97C3×C4⋊Dic3C22×C311yescharacteristic
98C3×Dic3⋊C4—61no—
99C3×Dic3⋊C4—61no—
100C3×C4×Dic3—21no—
101C3×C4×Dic3—61no—
102C3×C4×Dic3—61no—
103C3×C4×Dic3C2×D311yescharacteristic
104C3×C4×Dic3C22×C311yescharacteristic
105C2×C3×C3⋊C8—21no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—61no—
109C2×C3×C3⋊C8C2×D311yescharacteristic
110C2×C3×C3⋊C8C22×C311yescharacteristic
111#3724—21no—
112Dic32—21no—
113C22×C33—22no—
114C22×C33—41no—
115C22×C33—41no—
116C22×C33C2×D411yescharacteristic
117C33×C4C22×C411yescharacteristic
118C33×C4#4111yescharacteristic
119C3×C3⋊Dic3—22no—
120C32×Dic3—21no—
121C32×Dic3—41no—
122C32×Dic3—41no—
123C32×Dic3SD1612yescomplemented
124C2×C3×SD16—91no—
125C3×C4×Q8—91no—
126C3×C4×D4—31no—
127C3×C4×D4—31no—
128C3×C4×D4—61no—
129D4×Dic3C3×D311yescharacteristic
130C2×#214—31no—
131C2×#214—31no—
132C2×#214—61no—
133#887—31no—
134C4×Dic6—31no—
135C3×#106—91no—
136C3×#103—91no—
137C3×#102—31no—
138C3×#102—31no—
139C3×#102—61no—
140C3×C4×C8—91no—
141#832C3×D311yescharacteristic
142#816—31no—
143C8×Dic3—31no—
144#810—31no—
145#808—31no—
146#808—31no—
147C4×C3⋊C8—31no—
148C23×C32—21no—
149C23×C32—22no—
150C23×C32—21no—
151C23×C32—21no—
152C23×C32—41no—
153C23×C32—41no—
154C23×C32—42no—
155C32×D4—21no—
156C32×D4—22no—
157C32×D4—21no—
158C32×D4—21no—
159C32×D4—22no—
160C32×D4—22no—
161C32×D4—22no—
162C32×D4—42no—
163C32×D4—41no—
164C32×D4—41no—
165C32×D4—42no—
166C32×D4C4×D312yes—
167C32×D4C4×D312yes—
168C32×D4C2×C3×C412yes—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—31no—
173C2×C32×C4—31no—
174C2×C32×C4—41no—
175C2×C32×C4—61no—
176C2×C32×C4—61no—
177C2×C32×C4—122no—
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C23×C311yescharacteristic
181C2×C3⋊Dic3—21no—
182#634—21no—
183#634C2×C3×C412yescomplemented
184C2×C3×Dic3—22no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—41no—
192C2×C3×Dic3—44no—
193C2×C3×Dic3—42no—
194C2×C3×Dic3—42no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—121no—
198C2×C3×Dic3#6711yescharacteristic
199C2×C3×Dic3#6711yescharacteristic
200C2×C3×Dic3C3×D411yescharacteristic
201C2×C3×Dic3C3×D411yescharacteristic
202C3×Dic6—32no—
203C3×Dic6—32no—
204C3×Dic6—62no—
205C3×Dic6—61no—
206C3×Dic6—61no—
207C3×Dic6—121no—
208C32×C8—61no—
209C3⋊(C3⋊C8)C2×C3×C412yescomplemented
210C3×C3⋊C8—21no—
211C3×C3⋊C8—21no—
212C3×C3⋊C8—32no—
213C3×C3⋊C8—32no—
214C3×C3⋊C8—41no—
215C3×C3⋊C8—62no—
216C4×SD16—91noSylow
217C2×C33—42no—
218C2×C33C4×D411yescharacteristic
219C2×C33C2×SD1611yescharacteristic
220C2×C33#13511yescharacteristic
221C2×C3×Q8—91no—
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4—21no—
225C2×C3×D4—41no—
226C2×C3×D4D3211yescharacteristic
227C2×C3×D4C2×C3×D311yescharacteristic
228C2×C3×D4C2×C3×D311yescharacteristic
229C22×C3×C4—61no—
230C22×C3×C4—61no—
231C22×C3×C4—62no—
232C22×Dic3—21no—
233C2×Dic6—31no—
234C2×Dic6—31no—
235C2×Dic6—61no—
236C3×SD16—94no—
237C2×C3×C8—31no—
238C2×C3×C8—31no—
239C2×C3×C8—61no—
240C2×C3×C8—91no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—61no—
246C3×C4⋊C4—61no—
247C3×C4⋊C4—181no—
248C3×C22⋊C4—61no—
249C3×C22⋊C4—61no—
250C3×C22⋊C4—62no—
251C3×C42—31no—
252C3×C42—31no—
253C3×C42—61no—
254C3×C42—181no—
255#217—21no—
256#214—34no—
257#214—34no—
258#214—64no—
259C4⋊Dic3C2×C3×D311yescharacteristic
260C4⋊Dic3C2×C3×D311yescharacteristic
261Dic3⋊C4—61no—
262Dic3⋊C4—61no—
263C4×Dic3—61no—
264C4×Dic3—61no—
265C4×Dic3C2×C3×D311yescharacteristic
266C2×C3⋊C8—31no—
267C2×C3⋊C8—31no—
268C2×C3⋊C8—61no—
269C2×C3⋊C8C2×C3×D311yescharacteristic
270C22×C32—22no—
271C22×C32—24no—
272C22×C32—22no—
273C22×C32—22no—
274C22×C32—21no—
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—41no—
278C22×C32—42no—
279C22×C32—42no—
280C22×C32—44no—
281C22×C32—41no—
282C22×C32—41no—
283C22×C32—42no—
284C22×C32—42no—
285C22×C32—41no—
286C22×C32—41no—
287C22×C32—41no—
288C22×C32—41no—
289C22×C32—81no—
290C22×C32—81no—
291C22×C32—81no—
292C22×C32—81no—
293C22×C32—82no—
294C22×C32—82no—
295C22×C32D3×D411yescharacteristic
296C22×C32C2×#6711yescharacteristic
297C22×C32C2×C3×D411yescharacteristic
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—32no—
305C32×C4—41no—
306C32×C4—41no—
307C32×C4—61no—
308C32×C4—121no—
309C32×C4—121no—
310C32×C4C2×C4×D311yescharacteristic
311C32×C4C2×C4×D311yescharacteristic
312C32×C4#23511yescharacteristic
313C32×C4#23711yescharacteristic
314C32×C4C22×C3×C411yescharacteristic
315C32×C4C3×#4111yescharacteristic
316C3⋊Dic3—22no—
317C3×Dic3—21no—
318C3×Dic3—21no—
319C3×Dic3—22no—
320C3×Dic3—41no—
321C3×Dic3—42no—
322C3×Dic3—41no—
323C3×Dic3—41no—
324C3×Dic3—41no—
325C3×Dic3—41no—
326C3×Dic3—62no—
327C3×Dic3—62no—
328C3×Dic3—81no—
329C3×Dic3—122no—
330C3×Dic3#21412yescomplemented
331C3×Dic3C3×SD1612yescomplemented
332C2×SD16—91no—
333C4×Q8—91no—
334C4×D4—31no—
335#106—91no—
336#103—91no—
337#102—31no—
338C4×C8—91no—
339C33C4×SD1611yescharacteristic, complemented, Sylow
340C23×C3—22no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—42no—
347C3×Q8—92no—
348C3×Q8—181no—
349C3×D4—22no—
350C3×D4—21no—
351C3×D4—21no—
352C3×D4—21no—
353C3×D4—22no—
354C3×D4—22no—
355C3×D4—22no—
356C3×D4—41no—
357C3×D4—41no—
358C3×D4—42no—
359C3×D4—42no—
360C3×D4#62412yes—
361C3×D4C3×C4×D312yes—
362C3×D4C3×C4×D312yes—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—31no—
367C2×C3×C4—31no—
368C2×C3×C4—31no—
369C2×C3×C4—31no—
370C2×C3×C4—41no—
371C2×C3×C4—61no—
372C2×C3×C4—61no—
373C2×C3×C4—62no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—121no—
379C2×C3×C4—122no—
380C2×C3×C4—122no—
381C2×C3×C4—124no—
382C2×C3×C4—181no—
383C2×C3×C4C2×D3211yescharacteristic
384C2×C3×C4C22×C3×D311yescharacteristic
385C2×C3×C4C22×C3×D311yescharacteristic
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—42no—
389C2×Dic3—61no—
390C2×Dic3—61no—
391C2×Dic3—121no—
392C2×Dic3C3×#6711yescharacteristic
393C2×Dic3C3×#6711yescharacteristic
394Dic6—32no—
395Dic6—32no—
396Dic6—62no—
397Dic6—61no—
398Dic6—61no—
399Dic6—121no—
400C3×C8—61no—
401C3×C8—61no—
402C3×C8—92no—
403C3×C8—121no—
404C3⋊C8—21no—
405C3⋊C8—32no—
406C3⋊C8—32no—
407C3⋊C8—62no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—41no—
418C2×C32—41no—
419C2×C32—41no—
420C2×C32—42no—
421C2×C32—44no—
422C2×C32—42no—
423C2×C32—42no—
424C2×C32—82no—
425C2×C32—82no—
426C2×C32—84no—
427C2×C32#67⋊C411yescharacteristic
428C2×C32#91211yescharacteristic
429C2×C32D3×SD1611yescharacteristic
430C2×C32C4×#6711yescharacteristic
431C2×C32C2×#21411yescharacteristic
432C2×C32#95011yescharacteristic
433C2×C32C3×C4×D411yescharacteristic
434C2×C32C2×C3×SD1611yescharacteristic
435C2×C32C3×#13511yescharacteristic
436C2×Q8—91no—
437C2×D4C3×D3211yescharacteristic
438C22×C4—61no—
439SD16—94no—
440C2×C8—31no—
441C2×C8—91no—
442C4⋊C4—31no—
443C4⋊C4—31no—
444C4⋊C4—181no—
445C22⋊C4—61no—
446C42—31no—
447C42—181no—
448C22×C3—24no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—44no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3—41no—
466C22×C3—41no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3—82no—
472C22×C3—82no—
473C22×C3D3×#6711yescharacteristic
474C22×C3C3×D3×D411yescharacteristic
475C22×C3C2×C3×#6711yescharacteristic
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—32no—
483C3×C4—32no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—62no—
487C3×C4—61no—
488C3×C4—61no—
489C3×C4—121no—
490C3×C4—122no—
491C3×C4—121no—
492C3×C4—121no—
493C3×C4—121no—
494C3×C4—121no—
495C3×C4—182no—
496C3×C4—241no—
497C3×C4#380411yescharacteristic
498C3×C4C2×#62411yescharacteristic
499C3×C4C2×C3×C4×D311yescharacteristic
500C3×C4C2×C3×C4×D311yescharacteristic
501C3×C4C3×#23511yescharacteristic
502C3×C4C3×#23711yescharacteristic
503Dic3—21no—
504Dic3—41no—
505Dic3—41no—
506Dic3—62no—
507Dic3—62no—
508Dic3—122no—
509Dic3C3×#21412yescomplemented
510C32—21no—
511C32—21no—
512C32—21no—
513C32—41no—
514C32#475811yescharacteristic, complemented
515C32C4×#21411yescharacteristic, complemented
516C32C3×C4×SD1611yescharacteristic, complemented
517C23—21no—
518Q8—92no—
519Q8—181no—
520D4—21no—
521D4C3×#62412yes—
522C2×C4—31no—
523C2×C4—31no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—122no—
527C2×C4—181no—
528C2×C4C2×C3×D3211yescharacteristic
529C8—61no—
530C8—92no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—44no—
541C2×C3—42no—
542C2×C3—42no—
543C2×C3—42no—
544C2×C3—41no—
545C2×C3—41no—
546C2×C3—41no—
547C2×C3—82no—
548C2×C3—82no—
549C2×C3—84no—
550C2×C3D3×#21411yescharacteristic
551C2×C3#6393311yescharacteristic
552C2×C3#6395211yescharacteristic
553C2×C3C3×#67⋊C411yescharacteristic
554C2×C3C3×#91211yescharacteristic
555C2×C3C3×D3×SD1611yescharacteristic
556C2×C3C3×C4×#6711yescharacteristic
557C2×C3C2×C3×#21411yescharacteristic
558C2×C3C3×#95011yescharacteristic
559C22—22no—
560C22—41no—
561C22—41no—
562C22C3×D3×#6711yescharacteristic
563C4—32no—
564C4—61no—
565C4—121no—
566C4—121no—
567C4—182no—
568C4C3×#380411yescharacteristic
569C4C2×C3×#62411yescharacteristic
570C3—21no—
571C3—21no—
572C3—21no—
573C3—41no—
574C3#1059140111yescharacteristic, direct product factor
575C3C3×#475811yescharacteristic, complemented
576C3C3×C4×#21411yescharacteristic, complemented
577C2—42no—
578C2C3×D3×#21411yescharacteristic
579C2C3×#6393311yescharacteristic
580C2C3×#6395211yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×D4D32
hypocenterC32C3×C4×SD16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
3. C3×#63844C2semidirect
5. C3×#63574C2semidirect
6. C3×#63567C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
10. #10591401C3direct (N × H)
14. C3×#3754C4semidirect
33. #63844C2×C3semidirect
44. #63574C2×C3semidirect
45. #63567C2×C3semidirect
46. Dic3×C3⋊C8C2×C3semidirect
53. C3×#634C2×C4semidirect
59. C3×C3⋊(C3⋊C8)C2×C4semidirect
84. #3754C3×C4semidirect
123. C32×Dic3SD16semidirect
183. #634C2×C3×C4semidirect
209. C3⋊(C3⋊C8)C2×C3×C4semidirect
330. C3×Dic3#214semidirect
331. C3×Dic3C3×SD16semidirect
339. C33C4×SD16semidirect
509. Dic3C3×#214semidirect
514. C32#4758semidirect
515. C32C4×#214semidirect
516. C32C3×C4×SD16semidirect
574. C3#10591401direct (N × H)
575. C3C3×#4758semidirect
576. C3C3×C4×#214semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37542
C32×D4×Dic32
C3×#638442
C32×#8322
C3×#635742
C3×#635672
C3×Dic3×C3⋊C82
C3×C4×#2143
C3×#47583
#105914013