Finite Group Databasebeta

← C32×#4948 #13162388

Subgroups — C32×#4948 #13162388

Order 1728.

1644 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×C22⋊C4C211yesmaximal, characteristic
2C2×C32×#217C211yesmaximal, characteristic
3C2×C32×#217C211yesmaximal, characteristic
4C2×C32×C4⋊Dic3C211yesmaximal, characteristic, complemented
5C2×C32×Dic3⋊C4C211yesmaximal, characteristic, complemented
6C2×C32×Dic3⋊C4C211yesmaximal, characteristic, complemented
7C32×C3⋊#95C211yesmaximal, characteristic, complemented
8C32×#396—31nomaximal
9C3×#4948C314yesmaximal, direct product factor
10C24×C33C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C32×Dic3C2211yescharacteristic
14C22×C32×Dic3C2211yescharacteristic
15C22×C32×Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C33×C22⋊C4—22no—
18C32×#217—22no—
19C32×#217—22no—
20C32×C4⋊Dic3—22no—
21C32×Dic3⋊C4—22no—
22C32×Dic3⋊C4—22no—
23C2×C32×C4⋊C4—31no—
24C2×C32×C4⋊C4—31no—
25C2×C32×C4⋊C4—31no—
26C2×C32×C22⋊C4—24no—
27C2×C32×C22⋊C4—31no—
28C2×C32×C22⋊C4—31no—
29C2×C32×C22⋊C4D311yescharacteristic
30C2×C32×C22⋊C4C2×C314yes—
31C2×C3×#217C2×C314yes—
32C2×C3×#217C2×C314yes—
33C2×C3×C4⋊Dic3C2×C314yescomplemented
34C2×C3×Dic3⋊C4C2×C314yescomplemented
35C2×C3×Dic3⋊C4C2×C314yescomplemented
36C32×#95—31no—
37C3×C3⋊#95C2×C314yescomplemented
38C23×C33—21no—
39C23×C33—21no—
40C23×C33—21no—
41C23×C33—21no—
42C23×C33—21no—
43C23×C33—21no—
44C23×C33Q812yes—
45C23×C33C2311yescharacteristic
46C2×C33×C4—21no—
47C2×C33×C4—21no—
48C2×C33×C4—21no—
49C2×C33×C4—21no—
50C2×C33×C4—21no—
51C2×C33×C4D412yes—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
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×Dic3D412yes—
63C2×C32×Dic3D412yes—
64C3×#396—34no—
65#4948C3211yescharacteristic, direct product factor
66C24×C32—24no—
67C24×C32C2×D311yescharacteristic
68C24×C32C22×C314yes—
69C22×C32×C4—24no—
70C22×C32×C4—24no—
71C22×C32×C4—31no—
72C22×C32×C4—31no—
73C22×C32×C4—31no—
74C22×C32×C4—31no—
75C22×C32×C4C2×D311yescharacteristic
76C22×C32×C4C2×D311yescharacteristic
77C22×C32×C4C22×C314yes—
78C22×C32×C4C22×C314yes—
79C22×C3×Dic3C22×C314yes—
80C22×C3×Dic3C22×C314yes—
81C22×C3×Dic3C22×C314yes—
82C22×C3×Dic3C22×C314yes—
83C32×C4⋊C4—62no—
84C32×C4⋊C4—62no—
85C32×C4⋊C4—62no—
86C32×C22⋊C4—216no—
87C32×C22⋊C4—22no—
88C32×C22⋊C4—28no—
89C32×C22⋊C4—62no—
90C32×C22⋊C4—62no—
91C3×#217—28no—
92C3×#217—28no—
93C3×C4⋊Dic3—28no—
94C3×Dic3⋊C4—28no—
95C3×Dic3⋊C4—28no—
96C22×C33—22no—
97C22×C33—22no—
98C22×C33—22no—
99C22×C33—41no—
100C22×C33—41no—
101C22×C33—41no—
102C22×C33—41no—
103C22×C33C2×D411yescharacteristic
104C22×C33C2×D411yescharacteristic
105C22×C33C2×D411yescharacteristic
106C22×C33C2×Q811yescharacteristic
107C22×C33#4111yescharacteristic
108C22×C33#4111yescharacteristic
109C22×C33#4111yescharacteristic
110C33×C4—22no—
111C33×C4—41no—
112C32×Dic3—22no—
113C32×Dic3—22no—
114C32×Dic3—41no—
115C32×Dic3—41no—
116C2×C3×C4⋊C4—34no—
117C2×C3×C4⋊C4—34no—
118C2×C3×C4⋊C4—34no—
119C2×C3×C22⋊C4—24no—
120C2×C3×C22⋊C4—34no—
121C2×C3×C22⋊C4—34no—
122C2×C3×C22⋊C4C3×D314yes—
123C2×C3×C22⋊C4C2×C3211yescharacteristic
124C2×#217C2×C3211yescharacteristic
125C2×#217C2×C3211yescharacteristic
126C2×C4⋊Dic3C2×C3211yescharacteristic, complemented
127C2×Dic3⋊C4C2×C3211yescharacteristic, complemented
128C2×Dic3⋊C4C2×C3211yescharacteristic, complemented
129C3×#95—34no—
130C3⋊#95C2×C3211yescharacteristic, complemented
131C23×C32—28no—
132C23×C32—28no—
133C23×C32—21no—
134C23×C32—21no—
135C23×C32—21no—
136C23×C32—21no—
137C23×C32—21no—
138C23×C32—21no—
139C23×C32—24no—
140C23×C32—24no—
141C23×C32—28no—
142C23×C32—24no—
143C23×C32—24no—
144C23×C32—24no—
145C23×C32—24no—
146C23×C32—24no—
147C23×C32—44no—
148C23×C32—44no—
149C23×C32—44no—
150C23×C32—44no—
151C23×C32Dic612yes—
152C23×C32C3×Q818yes—
153C23×C32C22×D311yescharacteristic
154C23×C32C23×C314yes—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—28no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—24no—
162C2×C32×C4—28no—
163C2×C32×C4—24no—
164C2×C32×C4—24no—
165C2×C32×C4—24no—
166C2×C32×C4—24no—
167C2×C32×C4—32no—
168C2×C32×C4—32no—
169C2×C32×C4—44no—
170C2×C32×C4—44no—
171C2×C32×C4—44no—
172C2×C32×C4—44no—
173C2×C32×C4—61no—
174C2×C32×C4—61no—
175C2×C32×C4—61no—
176C2×C32×C4—61no—
177C2×C32×C4—61no—
178C2×C32×C4—61no—
179C2×C32×C4—61no—
180C2×C32×C4—61no—
181C2×C32×C4—61no—
182C2×C32×C4—61no—
183C2×C32×C4#6712yes—
184C2×C32×C4C3×D418yes—
185C2×C3×Dic3—24no—
186C2×C3×Dic3—24no—
187C2×C3×Dic3—24no—
188C2×C3×Dic3—24no—
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×Dic3C3×D418yes—
196C2×C3×Dic3C3×D418yes—
197#396—31noSylow
198C2×C33—42no—
199C2×C33#12111yescharacteristic
200C2×C33#12111yescharacteristic
201C2×C33#12111yescharacteristic
202C2×C33C22⋊Q811yescharacteristic
203C2×C33C22⋊Q811yescharacteristic
204C2×C33C22⋊Q811yescharacteristic
205C2×C33#12611yescharacteristic
206C24×C3—24no—
207C24×C3C2×C3×D314yes—
208C24×C3C22×C3211yescharacteristic
209C22×C3×C4—24no—
210C22×C3×C4—24no—
211C22×C3×C4—34no—
212C22×C3×C4—34no—
213C22×C3×C4—34no—
214C22×C3×C4—34no—
215C22×C3×C4C2×C3×D314yes—
216C22×C3×C4C2×C3×D314yes—
217C22×C3×C4C22×C3211yescharacteristic
218C22×C3×C4C22×C3211yescharacteristic
219C22×Dic3C22×C3211yescharacteristic
220C22×Dic3C22×C3211yescharacteristic
221C22×Dic3C22×C3211yescharacteristic
222C22×Dic3C22×C3211yescharacteristic
223C3×C4⋊C4—68no—
224C3×C4⋊C4—68no—
225C3×C4⋊C4—68no—
226C3×C22⋊C4—216no—
227C3×C22⋊C4—28no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—68no—
230C3×C22⋊C4—68no—
231#217—22no—
232#217—22no—
233C4⋊Dic3—22no—
234Dic3⋊C4—22no—
235Dic3⋊C4—22no—
236C22×C32—216no—
237C22×C32—22no—
238C22×C32—22no—
239C22×C32—22no—
240C22×C32—28no—
241C22×C32—24no—
242C22×C32—24no—
243C22×C32—24no—
244C22×C32—24no—
245C22×C32—24no—
246C22×C32—24no—
247C22×C32—24no—
248C22×C32—28no—
249C22×C32—28no—
250C22×C32—48no—
251C22×C32—48no—
252C22×C32—48no—
253C22×C32—48no—
254C22×C32—48no—
255C22×C32—48no—
256C22×C32—41no—
257C22×C32—41no—
258C22×C32—41no—
259C22×C32—41no—
260C22×C32—44no—
261C22×C32—44no—
262C22×C32—44no—
263C22×C32—44no—
264C22×C32C2×Dic611yescharacteristic
265C22×C32#23511yescharacteristic
266C22×C32D3×D411yescharacteristic
267C22×C32D3×D411yescharacteristic
268C22×C32#23711yescharacteristic
269C22×C32#23711yescharacteristic
270C22×C32C2×#6711yescharacteristic
271C22×C32C2×C3×D414yes—
272C22×C32C2×C3×D414yes—
273C22×C32C2×C3×D414yes—
274C22×C32C2×C3×Q814yes—
275C22×C32C3×#4114yes—
276C22×C32C3×#4114yes—
277C22×C32C3×#4114yes—
278C32×C4—22no—
279C32×C4—28no—
280C32×C4—48no—
281C32×C4—48no—
282C32×C4—41no—
283C32×C4—44no—
284C32×C4—62no—
285C32×C4—62no—
286C32×C4—121no—
287C32×C4—121no—
288C3×Dic3—28no—
289C3×Dic3—28no—
290C3×Dic3—44no—
291C3×Dic3—44no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C22⋊C4—31no—
296C2×C22⋊C4—31no—
297C2×C22⋊C4C32×D311yescharacteristic
298#95—31no—
299C33#39611yescharacteristic, complemented, Sylow
300C23×C3—28no—
301C23×C3—28no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—24no—
306C23×C3—24no—
307C23×C3—24no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—28no—
315C23×C3—24no—
316C23×C3—44no—
317C23×C3—44no—
318C23×C3—44no—
319C23×C3—44no—
320C23×C3C3×Dic618yes—
321C23×C3C32×Q812yes—
322C23×C3C22×C3×D314yes—
323C23×C3C23×C3211yescharacteristic
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—28no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—28no—
336C2×C3×C4—38no—
337C2×C3×C4—38no—
338C2×C3×C4—44no—
339C2×C3×C4—44no—
340C2×C3×C4—44no—
341C2×C3×C4—44no—
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×C4—64no—
349C2×C3×C4—64no—
350C2×C3×C4—64no—
351C2×C3×C4—64no—
352C2×C3×C4C3×#6718yes—
353C2×C3×C4C32×D412yes—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3C32×D412yes—
365C2×Dic3C32×D412yes—
366C2×C32—24no—
367C2×C32—24no—
368C2×C32—24no—
369C2×C32—24no—
370C2×C32—24no—
371C2×C32—24no—
372C2×C32—24no—
373C2×C32—416no—
374C2×C32—42no—
375C2×C32—48no—
376C2×C32#87811yescharacteristic
377C2×C32#87811yescharacteristic
378C2×C32#87911yescharacteristic
379C2×C32#88411yescharacteristic
380C2×C32#92411yescharacteristic
381C2×C32#93811yescharacteristic
382C2×C32#93911yescharacteristic
383C2×C32C3×#12114yes—
384C2×C32C3×#12114yes—
385C2×C32C3×#12114yes—
386C2×C32C3×C22⋊Q814yes—
387C2×C32C3×C22⋊Q814yes—
388C2×C32C3×C22⋊Q814yes—
389C2×C32C3×#12614yes—
390C24C2×C32×D311yescharacteristic
391C22×C4—31no—
392C22×C4—31no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4C2×C32×D311yescharacteristic
396C22×C4C2×C32×D311yescharacteristic
397C4⋊C4—62no—
398C4⋊C4—62no—
399C4⋊C4—62no—
400C22⋊C4—22no—
401C22⋊C4—62no—
402C22⋊C4—62no—
403C22×C3—216no—
404C22×C3—28no—
405C22×C3—28no—
406C22×C3—28no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—24no—
411C22×C3—24no—
412C22×C3—24no—
413C22×C3—24no—
414C22×C3—24no—
415C22×C3—24no—
416C22×C3—24no—
417C22×C3—48no—
418C22×C3—48no—
419C22×C3—48no—
420C22×C3—48no—
421C22×C3—48no—
422C22×C3—48no—
423C22×C3—44no—
424C22×C3—44no—
425C22×C3—44no—
426C22×C3—44no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3C2×C3×Dic614yes—
432C22×C3C3×#23514yes—
433C22×C3C3×D3×D414yes—
434C22×C3C3×D3×D414yes—
435C22×C3C3×#23714yes—
436C22×C3C3×#23714yes—
437C22×C3C2×C3×#6714yes—
438C22×C3C2×C32×D411yescharacteristic
439C22×C3C2×C32×D411yescharacteristic
440C22×C3C2×C32×D411yescharacteristic
441C22×C3C2×C32×Q811yescharacteristic
442C22×C3C32×#4111yescharacteristic
443C22×C3C32×#4111yescharacteristic
444C22×C3C32×#4111yescharacteristic
445C3×C4—28no—
446C3×C4—22no—
447C3×C4—48no—
448C3×C4—48no—
449C3×C4—44no—
450C3×C4—41no—
451C3×C4—68no—
452C3×C4—68no—
453C3×C4—124no—
454C3×C4—124no—
455Dic3—22no—
456Dic3—22no—
457Dic3—41no—
458Dic3—41no—
459C32—24no—
460C32#494811yescharacteristic, direct product factor
461C32C3×#39614yescomplemented
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23C32×Dic612yes—
469C23C22×C32×D311yescharacteristic
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—32no—
476C2×C4—32no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—61no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—61no—
487C2×C4C32×#6712yes—
488C2×C3—24no—
489C2×C3—24no—
490C2×C3—24no—
491C2×C3—24no—
492C2×C3—24no—
493C2×C3—24no—
494C2×C3—24no—
495C2×C3—416no—
496C2×C3—48no—
497C2×C3—42no—
498C2×C3C3×#87814yes—
499C2×C3C3×#87814yes—
500C2×C3C3×#87914yes—
501C2×C3C3×#88414yes—
502C2×C3C3×#92414yes—
503C2×C3C3×#93814yes—
504C2×C3C3×#93914yes—
505C2×C3C32×#12111yescharacteristic
506C2×C3C32×#12111yescharacteristic
507C2×C3C32×#12111yescharacteristic
508C2×C3C32×C22⋊Q811yescharacteristic
509C2×C3C32×C22⋊Q811yescharacteristic
510C2×C3C32×C22⋊Q811yescharacteristic
511C2×C3C32×#12611yescharacteristic
512C22—22no—
513C22—22no—
514C22—22no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22C2×C32×Dic611yescharacteristic
520C22C32×#23511yescharacteristic
521C22C32×D3×D411yescharacteristic
522C22C32×D3×D411yescharacteristic
523C22C32×#23711yescharacteristic
524C22C32×#23711yescharacteristic
525C22C2×C32×#6711yescharacteristic
526C4—22no—
527C4—41no—
528C4—62no—
529C4—62no—
530C4—121no—
531C4—121no—
532C3—24no—
533C3C3×#494814yesdirect product factor
534C3C32×#39611yescharacteristic, complemented
535C2—42no—
536C2C32×#87811yescharacteristic
537C2C32×#87811yescharacteristic
538C2C32×#87911yescharacteristic
539C2C32×#88411yescharacteristic
540C2C32×#92411yescharacteristic
541C2C32×#93811yescharacteristic
542C2C32×#93911yescharacteristic
subgroupG/H
centerC23×C32C22×D3
commutator, cosocleC23×C3C23×C32
FrattiniC23C22×C32×D3
FittingC2×C33×C22⋊C4C2
socleC23×C33C23
prime radicalC24×C33C22
hypercenterC2×C32×C22⋊C4D3
hypocenterC3C32×#396
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23×C32 ◁ C2×C32×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×C32×C4⋊Dic3C2semidirect
5. C2×C32×Dic3⋊C4C2semidirect
6. C2×C32×Dic3⋊C4C2semidirect
7. C32×C3⋊#95C2semidirect
9. C3×#4948C3direct (N × H)
33. C2×C3×C4⋊Dic3C2×C3semidirect
34. C2×C3×Dic3⋊C4C2×C3semidirect
35. C2×C3×Dic3⋊C4C2×C3semidirect
37. C3×C3⋊#95C2×C3semidirect
65. #4948C32direct (N × H)
126. C2×C4⋊Dic3C2×C32semidirect
127. C2×Dic3⋊C4C2×C32semidirect
128. C2×Dic3⋊C4C2×C32semidirect
130. C3⋊#95C2×C32semidirect
299. C33#396semidirect
460. C32#4948direct (N × H)
461. C32C3×#396semidirect
533. C3C3×#4948direct (N × H)
534. C3C32×#396semidirect

Maximal subgroups

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

subgroupindex
C2×C33×C22⋊C42
C2×C32×#2172
C2×C32×C4⋊Dic32
C2×C32×Dic3⋊C42
C32×C3⋊#952
C32×#3963
C3×#49483