Finite Group Databasebeta

← C32×#4989 #13162427

Subgroups — C32×#4989 #13162427

Order 1728.

1772 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C4×D3C211yesmaximal, characteristic
2C2×C33×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C32×(C2×D3)⋊C4C211yesmaximal, characteristic
4C2×C32×(C2×D3)⋊C4C211yesmaximal, characteristic
5C2×C32×Dic3⋊C4C211yesmaximal, characteristic, complemented
6C32×C3⋊#95C211yesmaximal, characteristic, complemented
7C32×C3⋊#95C211yesmaximal, characteristic, complemented
8C32×#385—31nomaximal
9C3×#4989C314yesmaximal, direct product factor
10C23×C32×D3C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C33×C4C2211yescharacteristic
14C22×C32×Dic3C2211yescharacteristic
15C22×C32×Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C2×C32×C4×D3—22no—
18C2×C32×C4×D3—22no—
19C2×C32×C4×D3—22no—
20C33×C4⋊C4—22no—
21C32×(C2×D3)⋊C4—22no—
22C32×(C2×D3)⋊C4C414yescomplemented
23C32×Dic3⋊C4—22no—
24C23×C32×C4—31no—
25C22×C3×C4×D3C2×C314yes—
26C2×C32×C4⋊C4—24no—
27C2×C32×C4⋊C4—31no—
28C2×C32×C4⋊C4D311yescharacteristic, complemented
29C2×C32×C4⋊C4C2×C314yescomplemented
30C2×C32×C22⋊C4—31no—
31C2×C32×C22⋊C4—31no—
32C2×C3×(C2×D3)⋊C4C2×C314yes—
33C2×C3×(C2×D3)⋊C4C2×C314yes—
34C2×C3×Dic3⋊C4C2×C314yescomplemented
35C32×#95—31no—
36C32×#95—31no—
37C3×C3⋊#95C2×C314yescomplemented
38C3×C3⋊#95C2×C314yescomplemented
39C23×C33C2311yescharacteristic
40C22×C32×D3—21no—
41C22×C32×D3—21no—
42C22×C32×D3—21no—
43C22×C32×D3—21no—
44C22×C32×D3C2×C412yes—
45C22×C32×D3D412yes—
46C22×C32×D3Q812yes—
47C2×C33×C4—21no—
48C2×C33×C4—21no—
49C2×C33×C4—21no—
50C2×C33×C4—21no—
51C2×C33×C4—21no—
52C2×C33×C4—21no—
53C2×C33×C4—21no—
54C2×C33×C4C2×C412yescomplemented
55C2×C33×C4D412yes—
56C2×C32×Dic3—21no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3—21no—
61C2×C32×Dic3—21no—
62C2×C32×Dic3—21no—
63C2×C32×Dic3C2×C412yescomplemented
64C2×C32×Dic3D412yes—
65C32×C4×D3—28no—
66C3×#385—34no—
67#4989C3211yescharacteristic, direct product factor
68C24×C32—31no—
69C23×C3×D3C22×C314yes—
70C22×C32×C4—24no—
71C22×C32×C4—24no—
72C22×C32×C4—24no—
73C22×C32×C4—31no—
74C22×C32×C4—31no—
75C22×C32×C4—31no—
76C22×C32×C4—62no—
77C22×C32×C4—62no—
78C22×C32×C4—62no—
79C22×C32×C4C2×D311yescharacteristic
80C22×C32×C4C2×D311yescharacteristic
81C22×C32×C4C2×D311yescharacteristic
82C22×C32×C4C22×C314yes—
83C22×C32×C4C22×C314yes—
84C22×C32×C4C22×C314yes—
85C22×C3×Dic3C22×C314yes—
86C22×C3×Dic3C22×C314yes—
87C22×C3×Dic3C22×C314yes—
88C2×C3×C4×D3—28no—
89C2×C3×C4×D3—28no—
90C2×C3×C4×D3—28no—
91C32×C4⋊C4—216no—
92C32×C4⋊C4—22no—
93C32×C4⋊C4—28no—
94C32×C4⋊C4—62no—
95C32×C22⋊C4—34no—
96C32×C22⋊C4—62no—
97C3×(C2×D3)⋊C4—28no—
98C3×(C2×D3)⋊C4C3×C4116yescomplemented
99C3×Dic3⋊C4—28no—
100C22×C33C22×C411yescharacteristic
101C22×C33C2×D411yescharacteristic
102C22×C33C2×D411yescharacteristic
103C22×C33C2×D411yescharacteristic
104C22×C33C2×Q811yescharacteristic
105C22×C33#4111yescharacteristic
106C22×C33#4111yescharacteristic
107C2×C32×D3—22no—
108C2×C32×D3—22no—
109C2×C32×D3—22no—
110C2×C32×D3—22no—
111C2×C32×D3—22no—
112C2×C32×D3—22no—
113C2×C32×D3C4⋊C414yescomplemented
114C33×C4—22no—
115C33×C4—22no—
116C33×C4—41no—
117C32×Dic3—22no—
118C32×Dic3—22no—
119C32×Dic3—41no—
120C23×C3×C4—34no—
121C22×C4×D3C2×C3211yescharacteristic
122C2×C3×C4⋊C4—24no—
123C2×C3×C4⋊C4—34no—
124C2×C3×C4⋊C4C3×D314yescomplemented
125C2×C3×C4⋊C4C2×C3211yescharacteristic, complemented
126C2×C3×C22⋊C4—34no—
127C2×C3×C22⋊C4—34no—
128C2×(C2×D3)⋊C4C2×C3211yescharacteristic
129C2×(C2×D3)⋊C4C2×C3211yescharacteristic
130C2×Dic3⋊C4C2×C3211yescharacteristic, complemented
131C3×#95—34no—
132C3×#95—34no—
133C3⋊#95C2×C3211yescharacteristic, complemented
134C3⋊#95C2×C3211yescharacteristic, complemented
135C23×C32—24no—
136C23×C32—32no—
137C23×C32—32no—
138C23×C32—32no—
139C23×C32—61no—
140C23×C32—61no—
141C23×C32—61no—
142C23×C32—61no—
143C23×C32C22×D311yescharacteristic
144C23×C32C23×C314yes—
145C22×C3×D3—24no—
146C22×C3×D3—24no—
147C22×C3×D3—24no—
148C22×C3×D3—24no—
149C22×C3×D3C2×C3×C418yes—
150C22×C3×D3C3×D418yes—
151C22×C3×D3C3×Q818yes—
152C2×C32×C4—28no—
153C2×C32×C4—28no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—24no—
159C2×C32×C4—24no—
160C2×C32×C4—28no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—24no—
165C2×C32×C4—28no—
166C2×C32×C4—28no—
167C2×C32×C4—24no—
168C2×C32×C4—24no—
169C2×C32×C4—24no—
170C2×C32×C4—24no—
171C2×C32×C4—32no—
172C2×C32×C4—32no—
173C2×C32×C4—44no—
174C2×C32×C4—44no—
175C2×C32×C4—44no—
176C2×C32×C4—44no—
177C2×C32×C4—68no—
178C2×C32×C4—61no—
179C2×C32×C4—61no—
180C2×C32×C4—61no—
181C2×C32×C4—61no—
182C2×C32×C4—61no—
183C2×C32×C4—61no—
184C2×C32×C4—61no—
185C2×C32×C4C4×D312yescomplemented
186C2×C32×C4#6712yes—
187C2×C32×C4C2×C3×C418yescomplemented
188C2×C32×C4C3×D418yes—
189C2×C3×Dic3—24no—
190C2×C3×Dic3—24no—
191C2×C3×Dic3—24no—
192C2×C3×Dic3—24no—
193C2×C3×Dic3—24no—
194C2×C3×Dic3—24no—
195C2×C3×Dic3—24no—
196C2×C3×Dic3C2×C3×C418yescomplemented
197C2×C3×Dic3C3×D418yes—
198C3×C4×D3—232no—
199#385—31noSylow
200C2×C33C2×C4⋊C411yescharacteristic
201C2×C33C4×D411yescharacteristic
202C2×C33C4×D411yescharacteristic
203C2×C33#12011yescharacteristic
204C2×C33C22⋊Q811yescharacteristic
205C2×C33C22⋊Q811yescharacteristic
206C2×C33#12311yescharacteristic
207C32×D3—24no—
208C24×C3—34no—
209C23×D3C22×C3211yescharacteristic
210C22×C3×C4—24no—
211C22×C3×C4—24no—
212C22×C3×C4—24no—
213C22×C3×C4—34no—
214C22×C3×C4—34no—
215C22×C3×C4—34no—
216C22×C3×C4—68no—
217C22×C3×C4—68no—
218C22×C3×C4—68no—
219C22×C3×C4C2×C3×D314yes—
220C22×C3×C4C2×C3×D314yes—
221C22×C3×C4C2×C3×D314yes—
222C22×C3×C4C22×C3211yescharacteristic
223C22×C3×C4C22×C3211yescharacteristic
224C22×C3×C4C22×C3211yescharacteristic
225C22×Dic3C22×C3211yescharacteristic
226C22×Dic3C22×C3211yescharacteristic
227C22×Dic3C22×C3211yescharacteristic
228C2×C4×D3—22no—
229C2×C4×D3—22no—
230C2×C4×D3—22no—
231C3×C4⋊C4—216no—
232C3×C4⋊C4—28no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—68no—
235C3×C22⋊C4—316no—
236C3×C22⋊C4—68no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4C32×C414yescomplemented
239Dic3⋊C4—22no—
240C22×C32—24no—
241C22×C32—24no—
242C22×C32—24no—
243C22×C32—24no—
244C22×C32—24no—
245C22×C32—24no—
246C22×C32—24no—
247C22×C32—34no—
248C22×C32—62no—
249C22×C32—62no—
250C22×C32—62no—
251C22×C32—62no—
252C22×C32—62no—
253C22×C32—62no—
254C22×C32C2×C4×D311yescharacteristic
255C22×C32#23511yescharacteristic
256C22×C32D3×D411yescharacteristic
257C22×C32D3×D411yescharacteristic
258C22×C32D3×Q811yescharacteristic
259C22×C32#23911yescharacteristic
260C22×C32C2×#6711yescharacteristic
261C22×C32C22×C3×C414yes—
262C22×C32C2×C3×D414yes—
263C22×C32C2×C3×D414yes—
264C22×C32C2×C3×D414yes—
265C22×C32C2×C3×Q814yes—
266C22×C32C3×#4114yes—
267C22×C32C3×#4114yes—
268C2×C3×D3—28no—
269C2×C3×D3—28no—
270C2×C3×D3—28no—
271C2×C3×D3—28no—
272C2×C3×D3—28no—
273C2×C3×D3—28no—
274C2×C3×D3C3×C4⋊C4116yescomplemented
275C32×C4—216no—
276C32×C4—22no—
277C32×C4—22no—
278C32×C4—28no—
279C32×C4—28no—
280C32×C4—48no—
281C32×C4—48no—
282C32×C4—41no—
283C32×C4—44no—
284C32×C4—62no—
285C32×C4—62no—
286C32×C4—121no—
287C3×Dic3—28no—
288C3×Dic3—28no—
289C3×Dic3—44no—
290C23×C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4C32×D311yescharacteristic, complemented
293C2×C22⋊C4—31no—
294C2×C22⋊C4—31no—
295#95—31no—
296#95—31no—
297C33#38511yescharacteristic, complemented, Sylow
298C23×C3—24no—
299C23×C3—38no—
300C23×C3—38no—
301C23×C3—38no—
302C23×C3—64no—
303C23×C3—64no—
304C23×C3—64no—
305C23×C3—64no—
306C23×C3C22×C3×D314yes—
307C23×C3C23×C3211yescharacteristic
308C22×D3—21no—
309C22×D3—21no—
310C22×D3—21no—
311C22×D3—21no—
312C22×D3C2×C32×C412yes—
313C22×D3C32×D412yes—
314C22×D3C32×Q812yes—
315C2×C3×C4—28no—
316C2×C3×C4—28no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—28no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—28no—
333C2×C3×C4—28no—
334C2×C3×C4—38no—
335C2×C3×C4—38no—
336C2×C3×C4—44no—
337C2×C3×C4—44no—
338C2×C3×C4—44no—
339C2×C3×C4—44no—
340C2×C3×C4—632no—
341C2×C3×C4—64no—
342C2×C3×C4—64no—
343C2×C3×C4—64no—
344C2×C3×C4—64no—
345C2×C3×C4—64no—
346C2×C3×C4—64no—
347C2×C3×C4—64no—
348C2×C3×C4C3×C4×D318yescomplemented
349C2×C3×C4C3×#6718yes—
350C2×C3×C4C2×C32×C412yescomplemented
351C2×C3×C4C32×D412yes—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3C2×C32×C412yescomplemented
360C2×Dic3C32×D412yes—
361C4×D3—28no—
362C2×C32—24no—
363C2×C32—24no—
364C2×C32—24no—
365C2×C32—24no—
366C2×C32—24no—
367C2×C32—24no—
368C2×C32—24no—
369C2×C32—64no—
370C2×C32D3×C4⋊C411yescharacteristic
371C2×C32#89311yescharacteristic
372C2×C32#89411yescharacteristic
373C2×C32#89611yescharacteristic
374C2×C32C4×#6711yescharacteristic
375C2×C32#93711yescharacteristic
376C2×C32#94611yescharacteristic
377C2×C32C2×C3×C4⋊C414yes—
378C2×C32C3×C4×D414yes—
379C2×C32C3×C4×D414yes—
380C2×C32C3×#12014yes—
381C2×C32C3×C22⋊Q814yes—
382C2×C32C3×C22⋊Q814yes—
383C2×C32C3×#12314yes—
384C3×D3—216no—
385C24—31no—
386C22×C4—31no—
387C22×C4—31no—
388C22×C4—31no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4—62no—
392C22×C4C2×C32×D311yescharacteristic
393C22×C4C2×C32×D311yescharacteristic
394C22×C4C2×C32×D311yescharacteristic
395C4⋊C4—22no—
396C4⋊C4—62no—
397C22⋊C4—34no—
398C22⋊C4—62no—
399C22×C3—24no—
400C22×C3—24no—
401C22×C3—24no—
402C22×C3—24no—
403C22×C3—24no—
404C22×C3—24no—
405C22×C3—24no—
406C22×C3—316no—
407C22×C3—68no—
408C22×C3—68no—
409C22×C3—68no—
410C22×C3—68no—
411C22×C3—68no—
412C22×C3—68no—
413C22×C3C2×C3×C4×D314yes—
414C22×C3C3×#23514yes—
415C22×C3C3×D3×D414yes—
416C22×C3C3×D3×D414yes—
417C22×C3C3×D3×Q814yes—
418C22×C3C3×#23914yes—
419C22×C3C2×C3×#6714yes—
420C22×C3C22×C32×C411yescharacteristic
421C22×C3C2×C32×D411yescharacteristic
422C22×C3C2×C32×D411yescharacteristic
423C22×C3C2×C32×D411yescharacteristic
424C22×C3C2×C32×Q811yescharacteristic
425C22×C3C32×#4111yescharacteristic
426C22×C3C32×#4111yescharacteristic
427C2×D3—22no—
428C2×D3—22no—
429C2×D3—22no—
430C2×D3—22no—
431C2×D3—22no—
432C2×D3—22no—
433C2×D3C32×C4⋊C414yescomplemented
434C3×C4—216no—
435C3×C4—28no—
436C3×C4—28no—
437C3×C4—22no—
438C3×C4—22no—
439C3×C4—48no—
440C3×C4—48no—
441C3×C4—44no—
442C3×C4—41no—
443C3×C4—68no—
444C3×C4—68no—
445C3×C4—124no—
446Dic3—22no—
447Dic3—22no—
448Dic3—41no—
449C32—24no—
450C32#498911yescharacteristic, direct product factor
451C32C3×#38514yescomplemented
452C23—32no—
453C23—32no—
454C23—32no—
455C23—61no—
456C23—61no—
457C23—61no—
458C23—61no—
459C23C22×C32×D311yescharacteristic
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—32no—
468C2×C4—32no—
469C2×C4—68no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4C32×C4×D312yescomplemented
478C2×C4C32×#6712yes—
479C2×C3—24no—
480C2×C3—24no—
481C2×C3—24no—
482C2×C3—24no—
483C2×C3—24no—
484C2×C3—24no—
485C2×C3—24no—
486C2×C3—616no—
487C2×C3C3×D3×C4⋊C414yes—
488C2×C3C3×#89314yes—
489C2×C3C3×#89414yes—
490C2×C3C3×#89614yes—
491C2×C3C3×C4×#6714yes—
492C2×C3C3×#93714yes—
493C2×C3C3×#94614yes—
494C2×C3C2×C32×C4⋊C411yescharacteristic
495C2×C3C32×C4×D411yescharacteristic
496C2×C3C32×C4×D411yescharacteristic
497C2×C3C32×#12011yescharacteristic
498C2×C3C32×C22⋊Q811yescharacteristic
499C2×C3C32×C22⋊Q811yescharacteristic
500C2×C3C32×#12311yescharacteristic
501D3—24no—
502C22—34no—
503C22—62no—
504C22—62no—
505C22—62no—
506C22—62no—
507C22—62no—
508C22—62no—
509C22C2×C32×C4×D311yescharacteristic
510C22C32×#23511yescharacteristic
511C22C32×D3×D411yescharacteristic
512C22C32×D3×D411yescharacteristic
513C22C32×D3×Q811yescharacteristic
514C22C32×#23911yescharacteristic
515C22C2×C32×#6711yescharacteristic
516C4—22no—
517C4—22no—
518C4—41no—
519C4—62no—
520C4—62no—
521C4—121no—
522C3—24no—
523C3C3×#498914yesdirect product factor
524C3C32×#38511yescharacteristic, complemented
525C2—64no—
526C2C32×D3×C4⋊C411yescharacteristic
527C2C32×#89311yescharacteristic
528C2C32×#89411yescharacteristic
529C2C32×#89611yescharacteristic
530C2C32×C4×#6711yescharacteristic
531C2C32×#93711yescharacteristic
532C2C32×#94611yescharacteristic
subgroupG/H
centerC23×C32C22×D3
commutatorC22×C3C22×C32×C4
FrattiniC23C22×C32×D3
FittingC2×C33×C4⋊C4C2
socleC23×C33C23
prime radicalC23×C32×D3C22
cosocleC23×C3C23×C32
hypercenterC2×C32×C4⋊C4D3
hypocenterC3C32×#385
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C32 ◁ C2×C32×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C33×C4⋊C4C2semidirect
5. C2×C32×Dic3⋊C4C2semidirect
6. C32×C3⋊#95C2semidirect
7. C32×C3⋊#95C2semidirect
9. C3×#4989C3direct (N × H)
22. C32×(C2×D3)⋊C4C4semidirect
28. C2×C32×C4⋊C4D3semidirect
29. C2×C32×C4⋊C4C2×C3semidirect
34. C2×C3×Dic3⋊C4C2×C3semidirect
37. C3×C3⋊#95C2×C3semidirect
38. C3×C3⋊#95C2×C3semidirect
54. C2×C33×C4C2×C4semidirect
63. C2×C32×Dic3C2×C4semidirect
67. #4989C32direct (N × H)
98. C3×(C2×D3)⋊C4C3×C4semidirect
113. C2×C32×D3C4⋊C4semidirect
124. C2×C3×C4⋊C4C3×D3semidirect
125. C2×C3×C4⋊C4C2×C32semidirect
130. C2×Dic3⋊C4C2×C32semidirect
133. C3⋊#95C2×C32semidirect
134. C3⋊#95C2×C32semidirect
185. C2×C32×C4C4×D3semidirect
187. C2×C32×C4C2×C3×C4semidirect
196. C2×C3×Dic3C2×C3×C4semidirect
238. (C2×D3)⋊C4C32×C4semidirect
274. C2×C3×D3C3×C4⋊C4semidirect
292. C2×C4⋊C4C32×D3semidirect
297. C33#385semidirect
348. C2×C3×C4C3×C4×D3semidirect
350. C2×C3×C4C2×C32×C4semidirect
359. C2×Dic3C2×C32×C4semidirect
433. C2×D3C32×C4⋊C4semidirect
450. C32#4989direct (N × H)
451. C32C3×#385semidirect
477. C2×C4C32×C4×D3semidirect
523. C3C3×#4989direct (N × H)
524. C3C32×#385semidirect

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×C33×C4⋊C42
C2×C32×(C2×D3)⋊C42
C2×C32×Dic3⋊C42
C32×C3⋊#952
C32×#3853
C3×#49893