Finite Group Databasebeta

← C32×#4940 #13162376

Subgroups — C32×#4940 #13162376

Order 1728.

1572 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×C42C211yesmaximal, characteristic, complemented
2C2×C32×(C2×D3)⋊C4C211yesmaximal, characteristic
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×#388—31nomaximal
9C3×#4940C314yesmaximal, direct product factor
10C23×C32×D3C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C33×C4C2211yescharacteristic
14C22×C32×Dic3C2211yescharacteristic
15C22×C32×Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C33×C42—22no—
18C32×(C2×D3)⋊C4—22no—
19C32×(C2×D3)⋊C4—22no—
20C32×(C2×D3)⋊C4C414yescomplemented
21C32×Dic3⋊C4—22no—
22C2×C32×C4⋊C4—31no—
23C2×C32×C22⋊C4—31no—
24C2×C32×C22⋊C4—31no—
25C2×C32×C22⋊C4—31no—
26C2×C32×C42—24no—
27C2×C32×C42D311yescharacteristic, complemented
28C2×C32×C42C2×C314yescomplemented
29C2×C3×(C2×D3)⋊C4C2×C314yes—
30C2×C3×(C2×D3)⋊C4C2×C314yes—
31C2×C3×(C2×D3)⋊C4C2×C314yes—
32C2×C3×Dic3⋊C4C2×C314yescomplemented
33C32×#95—31no—
34C32×#95—31no—
35C3×C3⋊#95C2×C314yescomplemented
36C3×C3⋊#95C2×C314yescomplemented
37C23×C33C2311yescharacteristic
38C22×C32×D3—21no—
39C22×C32×D3—21no—
40C22×C32×D3—21no—
41C22×C32×D3—21no—
42C22×C32×D3—21no—
43C22×C32×D3—21no—
44C22×C32×D3C2×C412yes—
45C2×C33×C4—21no—
46C2×C33×C4—21no—
47C2×C33×C4—21no—
48C2×C33×C4—21no—
49C2×C33×C4—21no—
50C2×C33×C4—21no—
51C2×C33×C4C2×C412yes—
52C2×C33×C4D412yes—
53C2×C33×C4D412yes—
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×Dic3C2×C412yes—
63C3×#388—34no—
64#4940C3211yescharacteristic, direct product factor
65C24×C32—31no—
66C23×C3×D3C22×C314yes—
67C22×C32×C4—24no—
68C22×C32×C4—24no—
69C22×C32×C4—24no—
70C22×C32×C4—31no—
71C22×C32×C4—31no—
72C22×C32×C4—31no—
73C22×C32×C4C2×D311yescharacteristic
74C22×C32×C4C2×D311yescharacteristic
75C22×C32×C4C2×D311yescharacteristic
76C22×C32×C4C22×C314yes—
77C22×C32×C4C22×C314yes—
78C22×C32×C4C22×C314yes—
79C22×C3×Dic3C22×C314yes—
80C22×C3×Dic3C22×C314yes—
81C22×C3×Dic3C22×C314yes—
82C32×C4⋊C4—62no—
83C32×C22⋊C4—34no—
84C32×C22⋊C4—62no—
85C32×C22⋊C4—62no—
86C32×C42—216no—
87C32×C42—22no—
88C32×C42—28no—
89C3×(C2×D3)⋊C4—28no—
90C3×(C2×D3)⋊C4—28no—
91C3×(C2×D3)⋊C4C3×C4116yescomplemented
92C3×Dic3⋊C4—28no—
93C22×C33C22×C411yescharacteristic
94C22×C33C2×D411yescharacteristic
95C22×C33C2×D411yescharacteristic
96C22×C33#4111yescharacteristic
97C22×C33#4111yescharacteristic
98C22×C33#4111yescharacteristic
99C22×C33#4111yescharacteristic
100C2×C32×D3—22no—
101C2×C32×D3—22no—
102C2×C32×D3—22no—
103C2×C32×D3—41no—
104C2×C32×D3—41no—
105C2×C32×D3—41no—
106C2×C32×D3—41no—
107C33×C4—22no—
108C33×C4—22no—
109C33×C4—22no—
110C32×Dic3—22no—
111C32×Dic3—41no—
112C32×Dic3—41no—
113C2×C3×C4⋊C4—34no—
114C2×C3×C22⋊C4—34no—
115C2×C3×C22⋊C4—34no—
116C2×C3×C22⋊C4—34no—
117C2×C3×C42—24no—
118C2×C3×C42C3×D314yescomplemented
119C2×C3×C42C2×C3211yescharacteristic, complemented
120C2×(C2×D3)⋊C4C2×C3211yescharacteristic
121C2×(C2×D3)⋊C4C2×C3211yescharacteristic
122C2×(C2×D3)⋊C4C2×C3211yescharacteristic
123C2×Dic3⋊C4C2×C3211yescharacteristic, complemented
124C3×#95—34no—
125C3×#95—34no—
126C3⋊#95C2×C3211yescharacteristic, complemented
127C3⋊#95C2×C3211yescharacteristic, complemented
128C23×C32—24no—
129C23×C32—32no—
130C23×C32—61no—
131C23×C32—61no—
132C23×C32—61no—
133C23×C32—61no—
134C23×C32—61no—
135C23×C32—61no—
136C23×C32C22×D311yescharacteristic
137C23×C32C23×C314yes—
138C22×C3×D3—24no—
139C22×C3×D3—24no—
140C22×C3×D3—24no—
141C22×C3×D3—24no—
142C22×C3×D3—24no—
143C22×C3×D3—24no—
144C22×C3×D3C2×C3×C418yes—
145C2×C32×C4—28no—
146C2×C32×C4—28no—
147C2×C32×C4—28no—
148C2×C32×C4—28no—
149C2×C32×C4—28no—
150C2×C32×C4—28no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—24no—
158C2×C32×C4—24no—
159C2×C32×C4—24no—
160C2×C32×C4—24no—
161C2×C32×C4—24no—
162C2×C32×C4—24no—
163C2×C32×C4—28no—
164C2×C32×C4—28no—
165C2×C32×C4—28no—
166C2×C32×C4—32no—
167C2×C32×C4—61no—
168C2×C32×C4—61no—
169C2×C32×C4—61no—
170C2×C32×C4—61no—
171C2×C32×C4—61no—
172C2×C32×C4—61no—
173C2×C32×C4—61no—
174C2×C32×C4—61no—
175C2×C32×C4C4×D312yes—
176C2×C32×C4D1212yes—
177C2×C32×C4#6712yes—
178C2×C32×C4C2×C3×C418yes—
179C2×C32×C4C3×D418yes—
180C2×C32×C4C3×D418yes—
181C2×C3×Dic3—24no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—24no—
185C2×C3×Dic3—24no—
186C2×C3×Dic3—24no—
187C2×C3×Dic3—24no—
188C2×C3×Dic3—24no—
189C2×C3×Dic3C2×C3×C418yes—
190#388—31noSylow
191C2×C33#11711yescharacteristic
192C2×C33C4×D411yescharacteristic
193C2×C33C4×D411yescharacteristic
194C2×C33#12111yescharacteristic
195C2×C33#12311yescharacteristic
196C2×C33#12411yescharacteristic
197C2×C33#12611yescharacteristic
198C32×D3—42no—
199C24×C3—34no—
200C23×D3C22×C3211yescharacteristic
201C22×C3×C4—24no—
202C22×C3×C4—24no—
203C22×C3×C4—24no—
204C22×C3×C4—34no—
205C22×C3×C4—34no—
206C22×C3×C4—34no—
207C22×C3×C4C2×C3×D314yes—
208C22×C3×C4C2×C3×D314yes—
209C22×C3×C4C2×C3×D314yes—
210C22×C3×C4C22×C3211yescharacteristic
211C22×C3×C4C22×C3211yescharacteristic
212C22×C3×C4C22×C3211yescharacteristic
213C22×Dic3C22×C3211yescharacteristic
214C22×Dic3C22×C3211yescharacteristic
215C22×Dic3C22×C3211yescharacteristic
216C3×C4⋊C4—68no—
217C3×C22⋊C4—316no—
218C3×C22⋊C4—68no—
219C3×C22⋊C4—68no—
220C3×C42—216no—
221C3×C42—28no—
222C3×C42—22no—
223(C2×D3)⋊C4—22no—
224(C2×D3)⋊C4—22no—
225(C2×D3)⋊C4C32×C414yescomplemented
226Dic3⋊C4—22no—
227C22×C32—24no—
228C22×C32—24no—
229C22×C32—24no—
230C22×C32—24no—
231C22×C32—24no—
232C22×C32—24no—
233C22×C32—24no—
234C22×C32—62no—
235C22×C32—62no—
236C22×C32—62no—
237C22×C32—121no—
238C22×C32—121no—
239C22×C32—121no—
240C22×C32—121no—
241C22×C32C2×C4×D311yescharacteristic
242C22×C32C2×D1211yescharacteristic
243C22×C32#23511yescharacteristic
244C22×C32#23511yescharacteristic
245C22×C32#23511yescharacteristic
246C22×C32#23511yescharacteristic
247C22×C32C2×#6711yescharacteristic
248C22×C32C22×C3×C414yes—
249C22×C32C2×C3×D414yes—
250C22×C32C2×C3×D414yes—
251C22×C32C3×#4114yes—
252C22×C32C3×#4114yes—
253C22×C32C3×#4114yes—
254C22×C32C3×#4114yes—
255C2×C3×D3—28no—
256C2×C3×D3—28no—
257C2×C3×D3—28no—
258C2×C3×D3—44no—
259C2×C3×D3—44no—
260C2×C3×D3—44no—
261C2×C3×D3—44no—
262C32×C4—216no—
263C32×C4—216no—
264C32×C4—216no—
265C32×C4—22no—
266C32×C4—22no—
267C32×C4—22no—
268C32×C4—28no—
269C32×C4—28no—
270C32×C4—28no—
271C32×C4—62no—
272C32×C4—121no—
273C32×C4—121no—
274C3×Dic3—28no—
275C3×Dic3—44no—
276C3×Dic3—44no—
277C2×C4⋊C4—31no—
278C2×C22⋊C4—31no—
279C2×C22⋊C4—31no—
280C2×C22⋊C4—31no—
281C2×C42C32×D311yescharacteristic, complemented
282#95—31no—
283#95—31no—
284C33#38811yescharacteristic, complemented, Sylow
285C23×C3—24no—
286C23×C3—38no—
287C23×C3—64no—
288C23×C3—64no—
289C23×C3—64no—
290C23×C3—64no—
291C23×C3—64no—
292C23×C3—64no—
293C23×C3C22×C3×D314yes—
294C23×C3C23×C3211yescharacteristic
295C22×D3—21no—
296C22×D3—21no—
297C22×D3—21no—
298C22×D3—21no—
299C22×D3—21no—
300C22×D3—21no—
301C22×D3C2×C32×C412yes—
302C2×C3×C4—28no—
303C2×C3×C4—28no—
304C2×C3×C4—28no—
305C2×C3×C4—28no—
306C2×C3×C4—28no—
307C2×C3×C4—28no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—28no—
321C2×C3×C4—28no—
322C2×C3×C4—28no—
323C2×C3×C4—38no—
324C2×C3×C4—64no—
325C2×C3×C4—64no—
326C2×C3×C4—64no—
327C2×C3×C4—64no—
328C2×C3×C4—64no—
329C2×C3×C4—64no—
330C2×C3×C4—64no—
331C2×C3×C4—64no—
332C2×C3×C4C3×C4×D318yes—
333C2×C3×C4C3×D1218yes—
334C2×C3×C4C3×#6718yes—
335C2×C3×C4C2×C32×C412yes—
336C2×C3×C4C32×D412yes—
337C2×C3×C4C32×D412yes—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3C2×C32×C412yes—
347C2×C32—24no—
348C2×C32—24no—
349C2×C32—24no—
350C2×C32—24no—
351C2×C32—24no—
352C2×C32—24no—
353C2×C32—24no—
354C2×C32—122no—
355C2×C32#87211yescharacteristic
356C2×C32C4×D1211yescharacteristic
357C2×C32#87511yescharacteristic
358C2×C32#87611yescharacteristic
359C2×C32C4×#6711yescharacteristic
360C2×C32#92911yescharacteristic
361C2×C32#93011yescharacteristic
362C2×C32C3×#11714yes—
363C2×C32C3×C4×D414yes—
364C2×C32C3×C4×D414yes—
365C2×C32C3×#12114yes—
366C2×C32C3×#12314yes—
367C2×C32C3×#12414yes—
368C2×C32C3×#12614yes—
369C3×D3—48no—
370C24—31no—
371C22×C4—31no—
372C22×C4—31no—
373C22×C4—31no—
374C22×C4C2×C32×D311yescharacteristic
375C22×C4C2×C32×D311yescharacteristic
376C22×C4C2×C32×D311yescharacteristic
377C4⋊C4—62no—
378C22⋊C4—34no—
379C22⋊C4—62no—
380C22⋊C4—62no—
381C42—22no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—24no—
385C22×C3—24no—
386C22×C3—24no—
387C22×C3—24no—
388C22×C3—24no—
389C22×C3—68no—
390C22×C3—68no—
391C22×C3—68no—
392C22×C3—124no—
393C22×C3—124no—
394C22×C3—124no—
395C22×C3—124no—
396C22×C3C2×C3×C4×D314yes—
397C22×C3C2×C3×D1214yes—
398C22×C3C3×#23514yes—
399C22×C3C3×#23514yes—
400C22×C3C3×#23514yes—
401C22×C3C3×#23514yes—
402C22×C3C2×C3×#6714yes—
403C22×C3C22×C32×C411yescharacteristic
404C22×C3C2×C32×D411yescharacteristic
405C22×C3C2×C32×D411yescharacteristic
406C22×C3C32×#4111yescharacteristic
407C22×C3C32×#4111yescharacteristic
408C22×C3C32×#4111yescharacteristic
409C22×C3C32×#4111yescharacteristic
410C2×D3—22no—
411C2×D3—22no—
412C2×D3—22no—
413C2×D3—41no—
414C2×D3—41no—
415C2×D3—41no—
416C2×D3—41no—
417C3×C4—216no—
418C3×C4—216no—
419C3×C4—216no—
420C3×C4—28no—
421C3×C4—28no—
422C3×C4—28no—
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—68no—
427C3×C4—124no—
428C3×C4—124no—
429Dic3—22no—
430Dic3—41no—
431Dic3—41no—
432C32—24no—
433C32#494011yescharacteristic, direct product factor
434C32C3×#38814yescomplemented
435C23—32no—
436C23—61no—
437C23—61no—
438C23—61no—
439C23—61no—
440C23—61no—
441C23—61no—
442C23C22×C32×D311yescharacteristic
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—32no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4C32×C4×D312yes—
459C2×C4C32×D1212yes—
460C2×C4C32×#6712yes—
461C2×C3—24no—
462C2×C3—24no—
463C2×C3—24no—
464C2×C3—24no—
465C2×C3—24no—
466C2×C3—24no—
467C2×C3—24no—
468C2×C3—128no—
469C2×C3C3×#87214yes—
470C2×C3C3×C4×D1214yes—
471C2×C3C3×#87514yes—
472C2×C3C3×#87614yes—
473C2×C3C3×C4×#6714yes—
474C2×C3C3×#92914yes—
475C2×C3C3×#93014yes—
476C2×C3C32×#11711yescharacteristic
477C2×C3C32×C4×D411yescharacteristic
478C2×C3C32×C4×D411yescharacteristic
479C2×C3C32×#12111yescharacteristic
480C2×C3C32×#12311yescharacteristic
481C2×C3C32×#12411yescharacteristic
482C2×C3C32×#12611yescharacteristic
483D3—42no—
484C22—62no—
485C22—62no—
486C22—62no—
487C22—121no—
488C22—121no—
489C22—121no—
490C22—121no—
491C22C2×C32×C4×D311yescharacteristic
492C22C2×C32×D1211yescharacteristic
493C22C32×#23511yescharacteristic
494C22C32×#23511yescharacteristic
495C22C32×#23511yescharacteristic
496C22C32×#23511yescharacteristic
497C22C2×C32×#6711yescharacteristic
498C4—22no—
499C4—22no—
500C4—22no—
501C4—62no—
502C4—121no—
503C4—121no—
504C3—24no—
505C3C3×#494014yesdirect product factor
506C3C32×#38811yescharacteristic, complemented
507C2—122no—
508C2C32×#87211yescharacteristic
509C2C32×C4×D1211yescharacteristic
510C2C32×#87511yescharacteristic
511C2C32×#87611yescharacteristic
512C2C32×C4×#6711yescharacteristic
513C2C32×#92911yescharacteristic
514C2C32×#93011yescharacteristic
subgroupG/H
centerC23×C32C22×D3
commutatorC22×C3C22×C32×C4
FrattiniC23C22×C32×D3
FittingC2×C33×C42C2
socleC23×C33C23
prime radicalC23×C32×D3C22
cosocleC23×C3C23×C32
hypercenterC2×C32×C42D3
hypocenterC3C32×#388
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C32 ◁ C2×C32×C42

Complements

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

NH ≅ G/Ntype
1. C2×C33×C42C2semidirect
5. C2×C32×Dic3⋊C4C2semidirect
6. C32×C3⋊#95C2semidirect
7. C32×C3⋊#95C2semidirect
9. C3×#4940C3direct (N × H)
20. C32×(C2×D3)⋊C4C4semidirect
27. C2×C32×C42D3semidirect
28. C2×C32×C42C2×C3semidirect
32. C2×C3×Dic3⋊C4C2×C3semidirect
35. C3×C3⋊#95C2×C3semidirect
36. C3×C3⋊#95C2×C3semidirect
64. #4940C32direct (N × H)
91. C3×(C2×D3)⋊C4C3×C4semidirect
118. C2×C3×C42C3×D3semidirect
119. C2×C3×C42C2×C32semidirect
123. C2×Dic3⋊C4C2×C32semidirect
126. C3⋊#95C2×C32semidirect
127. C3⋊#95C2×C32semidirect
225. (C2×D3)⋊C4C32×C4semidirect
281. C2×C42C32×D3semidirect
284. C33#388semidirect
433. C32#4940direct (N × H)
434. C32C3×#388semidirect
505. C3C3×#4940direct (N × H)
506. C3C32×#388semidirect

Maximal subgroups

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

subgroupindex
C2×C33×C422
C2×C32×(C2×D3)⋊C42
C2×C32×Dic3⋊C42
C32×C3⋊#952
C32×#3883
C3×#49403