Finite Group Databasebeta

← C32×#4675 #13162112

Subgroups — C32×#4675 #13162112

Order 1728.

1324 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×(C2×D3)⋊C4C211yesmaximal, characteristic
2C2×C32×(C2×D3)⋊C4C211yesmaximal, characteristic
3C2×C32×(C2×D3)⋊C4C211yesmaximal, characteristic
4C2×C32×Dic3⋊C4C211yesmaximal, characteristic, complemented
5C33×#95C211yesmaximal, characteristic, complemented
6C32×C3⋊#95C211yesmaximal, characteristic, complemented
7C32×C3⋊#95C211yesmaximal, characteristic, complemented
8C32×#397—31nomaximal
9C3×#4675C314yesmaximal, direct product factor
10C23×C32×D3C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C33×C4C2211yescharacteristic
14C22×C32×Dic3C2211yescharacteristic
15C22×C32×Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C32×(C2×D3)⋊C4—22no—
18C32×(C2×D3)⋊C4—22no—
19C32×(C2×D3)⋊C4—22no—
20C32×Dic3⋊C4—22no—
21C2×C32×C4⋊C4—31no—
22C2×C32×C22⋊C4—31no—
23C2×C32×C22⋊C4—31no—
24C2×C32×C22⋊C4—31no—
25C2×C3×(C2×D3)⋊C4C2×C314yes—
26C2×C3×(C2×D3)⋊C4C2×C314yes—
27C2×C3×(C2×D3)⋊C4C2×C314yes—
28C2×C3×Dic3⋊C4C2×C314yescomplemented
29C32×#95—24no—
30C32×#95—31no—
31C32×#95—31no—
32C32×#95D311yescharacteristic, complemented
33C32×#95C2×C314yescomplemented
34C3×C3⋊#95C2×C314yescomplemented
35C3×C3⋊#95C2×C314yescomplemented
36C23×C33C2311yescharacteristic
37C22×C32×D3—21no—
38C22×C32×D3—21no—
39C22×C32×D3—21no—
40C22×C32×D3—21no—
41C22×C32×D3—21no—
42C22×C32×D3—21no—
43C22×C32×D3D412yes—
44C2×C33×C4—21no—
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×C4—21no—
52C2×C33×C4—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×Dic3D412yes—
62C3×#397—34no—
63#4675C3211yescharacteristic, direct product factor
64C24×C32—31no—
65C23×C3×D3C22×C314yes—
66C22×C32×C4—24no—
67C22×C32×C4—24no—
68C22×C32×C4—24no—
69C22×C32×C4—31no—
70C22×C32×C4—31no—
71C22×C32×C4—31no—
72C22×C32×C4C2×D311yescharacteristic
73C22×C32×C4C2×D311yescharacteristic
74C22×C32×C4C2×D311yescharacteristic
75C22×C32×C4C22×C314yes—
76C22×C32×C4C22×C314yes—
77C22×C32×C4C22×C314yes—
78C22×C3×Dic3C22×C314yes—
79C22×C3×Dic3C22×C314yes—
80C22×C3×Dic3C22×C314yes—
81C32×C4⋊C4—62no—
82C32×C22⋊C4—62no—
83C32×C22⋊C4—62no—
84C32×C22⋊C4—62no—
85C3×(C2×D3)⋊C4—28no—
86C3×(C2×D3)⋊C4—28no—
87C3×(C2×D3)⋊C4—28no—
88C3×Dic3⋊C4—28no—
89C22×C33C2×D411yescharacteristic
90C22×C33C2×D411yescharacteristic
91C22×C33#4111yescharacteristic
92C22×C33#4111yescharacteristic
93C22×C33#4111yescharacteristic
94C22×C33#4111yescharacteristic
95C22×C33#4111yescharacteristic
96C2×C32×D3—22no—
97C2×C32×D3—22no—
98C2×C32×D3—22no—
99C2×C32×D3—41no—
100C2×C32×D3—41no—
101C2×C32×D3—41no—
102C2×C32×D3—41no—
103C33×C4—41no—
104C33×C4—41no—
105C33×C4—41no—
106C32×Dic3—22no—
107C32×Dic3—41no—
108C32×Dic3—41no—
109C2×C3×C4⋊C4—34no—
110C2×C3×C22⋊C4—34no—
111C2×C3×C22⋊C4—34no—
112C2×C3×C22⋊C4—34no—
113C2×(C2×D3)⋊C4C2×C3211yescharacteristic
114C2×(C2×D3)⋊C4C2×C3211yescharacteristic
115C2×(C2×D3)⋊C4C2×C3211yescharacteristic
116C2×Dic3⋊C4C2×C3211yescharacteristic, complemented
117C3×#95—24no—
118C3×#95—34no—
119C3×#95—34no—
120C3×#95C3×D314yescomplemented
121C3×#95C2×C3211yescharacteristic, complemented
122C3⋊#95C2×C3211yescharacteristic, complemented
123C3⋊#95C2×C3211yescharacteristic, complemented
124C23×C32—24no—
125C23×C32—32no—
126C23×C32—61no—
127C23×C32—61no—
128C23×C32—61no—
129C23×C32—61no—
130C23×C32—61no—
131C23×C32—61no—
132C23×C32C22×D311yescharacteristic
133C23×C32C23×C314yes—
134C22×C3×D3—24no—
135C22×C3×D3—24no—
136C22×C3×D3—24no—
137C22×C3×D3—24no—
138C22×C3×D3—24no—
139C22×C3×D3—24no—
140C22×C3×D3C3×D418yes—
141C2×C32×C4—28no—
142C2×C32×C4—28no—
143C2×C32×C4—28no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—24no—
154C2×C32×C4—24no—
155C2×C32×C4—24no—
156C2×C32×C4—24no—
157C2×C32×C4—24no—
158C2×C32×C4—24no—
159C2×C32×C4—24no—
160C2×C32×C4—24no—
161C2×C32×C4—24no—
162C2×C32×C4—32no—
163C2×C32×C4—44no—
164C2×C32×C4—44no—
165C2×C32×C4—44no—
166C2×C32×C4—44no—
167C2×C32×C4—44no—
168C2×C32×C4—44no—
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×C4—61no—
176C2×C32×C4—61no—
177C2×C3×Dic3—24no—
178C2×C3×Dic3—24no—
179C2×C3×Dic3—24no—
180C2×C3×Dic3—24no—
181C2×C3×Dic3—24no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—24no—
185C2×C3×Dic3C3×D418yes—
186#397—31noSylow
187C2×C33#12111yescharacteristic
188C2×C33#12311yescharacteristic
189C2×C33#12311yescharacteristic
190C2×C33#12311yescharacteristic
191C2×C33#12411yescharacteristic
192C2×C33#12611yescharacteristic
193C2×C33#12611yescharacteristic
194C32×D3—42no—
195C24×C3—34no—
196C23×D3C22×C3211yescharacteristic
197C22×C3×C4—24no—
198C22×C3×C4—24no—
199C22×C3×C4—24no—
200C22×C3×C4—34no—
201C22×C3×C4—34no—
202C22×C3×C4—34no—
203C22×C3×C4C2×C3×D314yes—
204C22×C3×C4C2×C3×D314yes—
205C22×C3×C4C2×C3×D314yes—
206C22×C3×C4C22×C3211yescharacteristic
207C22×C3×C4C22×C3211yescharacteristic
208C22×C3×C4C22×C3211yescharacteristic
209C22×Dic3C22×C3211yescharacteristic
210C22×Dic3C22×C3211yescharacteristic
211C22×Dic3C22×C3211yescharacteristic
212C3×C4⋊C4—68no—
213C3×C22⋊C4—68no—
214C3×C22⋊C4—68no—
215C3×C22⋊C4—68no—
216(C2×D3)⋊C4—22no—
217(C2×D3)⋊C4—22no—
218(C2×D3)⋊C4—22no—
219Dic3⋊C4—22no—
220C22×C32—24no—
221C22×C32—24no—
222C22×C32—24no—
223C22×C32—24no—
224C22×C32—24no—
225C22×C32—24no—
226C22×C32—24no—
227C22×C32—62no—
228C22×C32—62no—
229C22×C32—62no—
230C22×C32—121no—
231C22×C32—121no—
232C22×C32—121no—
233C22×C32—121no—
234C22×C32#23511yescharacteristic
235C22×C32#23511yescharacteristic
236C22×C32#23511yescharacteristic
237C22×C32D3×D411yescharacteristic
238C22×C32D3×D411yescharacteristic
239C22×C32#23711yescharacteristic
240C22×C32#23911yescharacteristic
241C22×C32C2×C3×D414yes—
242C22×C32C2×C3×D414yes—
243C22×C32C3×#4114yes—
244C22×C32C3×#4114yes—
245C22×C32C3×#4114yes—
246C22×C32C3×#4114yes—
247C22×C32C3×#4114yes—
248C2×C3×D3—28no—
249C2×C3×D3—28no—
250C2×C3×D3—28no—
251C2×C3×D3—44no—
252C2×C3×D3—44no—
253C2×C3×D3—44no—
254C2×C3×D3—44no—
255C32×C4—48no—
256C32×C4—48no—
257C32×C4—48no—
258C32×C4—41no—
259C32×C4—41no—
260C32×C4—41no—
261C32×C4—44no—
262C32×C4—44no—
263C32×C4—44no—
264C32×C4—62no—
265C32×C4—121no—
266C32×C4—121no—
267C3×Dic3—28no—
268C3×Dic3—44no—
269C3×Dic3—44no—
270C2×C4⋊C4—31no—
271C2×C22⋊C4—31no—
272C2×C22⋊C4—31no—
273C2×C22⋊C4—31no—
274#95—31no—
275#95—31no—
276#95C32×D311yescharacteristic, complemented
277C33#39711yescharacteristic, complemented, Sylow
278C23×C3—24no—
279C23×C3—38no—
280C23×C3—64no—
281C23×C3—64no—
282C23×C3—64no—
283C23×C3—64no—
284C23×C3—64no—
285C23×C3—64no—
286C23×C3C22×C3×D314yes—
287C23×C3C23×C3211yescharacteristic
288C22×D3—21no—
289C22×D3—21no—
290C22×D3—21no—
291C22×D3—21no—
292C22×D3—21no—
293C22×D3—21no—
294C22×D3C32×D412yes—
295C2×C3×C4—28no—
296C2×C3×C4—28no—
297C2×C3×C4—28no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—38no—
317C2×C3×C4—44no—
318C2×C3×C4—44no—
319C2×C3×C4—44no—
320C2×C3×C4—44no—
321C2×C3×C4—44no—
322C2×C3×C4—44no—
323C2×C3×C4—64no—
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×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3C32×D412yes—
340C2×C32—24no—
341C2×C32—24no—
342C2×C32—24no—
343C2×C32—24no—
344C2×C32—24no—
345C2×C32—24no—
346C2×C32—24no—
347C2×C32—122no—
348C2×C32#87611yescharacteristic
349C2×C32#88311yescharacteristic
350C2×C32#88411yescharacteristic
351C2×C32#88511yescharacteristic
352C2×C32#89411yescharacteristic
353C2×C32#89411yescharacteristic
354C2×C32#89811yescharacteristic
355C2×C32C3×#12114yes—
356C2×C32C3×#12314yes—
357C2×C32C3×#12314yes—
358C2×C32C3×#12314yes—
359C2×C32C3×#12414yes—
360C2×C32C3×#12614yes—
361C2×C32C3×#12614yes—
362C3×D3—48no—
363C24—31no—
364C22×C4—31no—
365C22×C4—31no—
366C22×C4—31no—
367C22×C4C2×C32×D311yescharacteristic
368C22×C4C2×C32×D311yescharacteristic
369C22×C4C2×C32×D311yescharacteristic
370C4⋊C4—62no—
371C22⋊C4—62no—
372C22⋊C4—62no—
373C22⋊C4—62no—
374C22×C3—24no—
375C22×C3—24no—
376C22×C3—24no—
377C22×C3—24no—
378C22×C3—24no—
379C22×C3—24no—
380C22×C3—24no—
381C22×C3—68no—
382C22×C3—68no—
383C22×C3—68no—
384C22×C3—124no—
385C22×C3—124no—
386C22×C3—124no—
387C22×C3—124no—
388C22×C3C3×#23514yes—
389C22×C3C3×#23514yes—
390C22×C3C3×#23514yes—
391C22×C3C3×D3×D414yes—
392C22×C3C3×D3×D414yes—
393C22×C3C3×#23714yes—
394C22×C3C3×#23914yes—
395C22×C3C2×C32×D411yescharacteristic
396C22×C3C2×C32×D411yescharacteristic
397C22×C3C32×#4111yescharacteristic
398C22×C3C32×#4111yescharacteristic
399C22×C3C32×#4111yescharacteristic
400C22×C3C32×#4111yescharacteristic
401C22×C3C32×#4111yescharacteristic
402C2×D3—22no—
403C2×D3—22no—
404C2×D3—22no—
405C2×D3—41no—
406C2×D3—41no—
407C2×D3—41no—
408C2×D3—41no—
409C3×C4—48no—
410C3×C4—48no—
411C3×C4—48no—
412C3×C4—44no—
413C3×C4—44no—
414C3×C4—44no—
415C3×C4—41no—
416C3×C4—41no—
417C3×C4—41no—
418C3×C4—68no—
419C3×C4—124no—
420C3×C4—124no—
421Dic3—22no—
422Dic3—41no—
423Dic3—41no—
424C32—24no—
425C32#467511yescharacteristic, direct product factor
426C32C3×#39714yescomplemented
427C23—32no—
428C23—61no—
429C23—61no—
430C23—61no—
431C23—61no—
432C23—61no—
433C23—61no—
434C23C22×C32×D311yescharacteristic
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—32no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C3—24no—
454C2×C3—24no—
455C2×C3—24no—
456C2×C3—24no—
457C2×C3—24no—
458C2×C3—24no—
459C2×C3—24no—
460C2×C3—128no—
461C2×C3C3×#87614yes—
462C2×C3C3×#88314yes—
463C2×C3C3×#88414yes—
464C2×C3C3×#88514yes—
465C2×C3C3×#89414yes—
466C2×C3C3×#89414yes—
467C2×C3C3×#89814yes—
468C2×C3C32×#12111yescharacteristic
469C2×C3C32×#12311yescharacteristic
470C2×C3C32×#12311yescharacteristic
471C2×C3C32×#12311yescharacteristic
472C2×C3C32×#12411yescharacteristic
473C2×C3C32×#12611yescharacteristic
474C2×C3C32×#12611yescharacteristic
475D3—42no—
476C22—62no—
477C22—62no—
478C22—62no—
479C22—121no—
480C22—121no—
481C22—121no—
482C22—121no—
483C22C32×#23511yescharacteristic
484C22C32×#23511yescharacteristic
485C22C32×#23511yescharacteristic
486C22C32×D3×D411yescharacteristic
487C22C32×D3×D411yescharacteristic
488C22C32×#23711yescharacteristic
489C22C32×#23911yescharacteristic
490C4—41no—
491C4—41no—
492C4—41no—
493C4—62no—
494C4—121no—
495C4—121no—
496C3—24no—
497C3C3×#467514yesdirect product factor
498C3C32×#39711yescharacteristic, complemented
499C2—122no—
500C2C32×#87611yescharacteristic
501C2C32×#88311yescharacteristic
502C2C32×#88411yescharacteristic
503C2C32×#88511yescharacteristic
504C2C32×#89411yescharacteristic
505C2C32×#89411yescharacteristic
506C2C32×#89811yescharacteristic
subgroupG/H
centerC23×C32C22×D3
commutator, cosocleC23×C3C23×C32
FrattiniC23C22×C32×D3
FittingC33×#95C2
socleC23×C33C23
prime radicalC23×C32×D3C22
hypercenterC32×#95D3
hypocenterC3C32×#397
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23×C32 ◁ C32×#95

Complements

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

NH ≅ G/Ntype
4. C2×C32×Dic3⋊C4C2semidirect
5. C33×#95C2semidirect
6. C32×C3⋊#95C2semidirect
7. C32×C3⋊#95C2semidirect
9. C3×#4675C3direct (N × H)
28. C2×C3×Dic3⋊C4C2×C3semidirect
32. C32×#95D3semidirect
33. C32×#95C2×C3semidirect
34. C3×C3⋊#95C2×C3semidirect
35. C3×C3⋊#95C2×C3semidirect
63. #4675C32direct (N × H)
116. C2×Dic3⋊C4C2×C32semidirect
120. C3×#95C3×D3semidirect
121. C3×#95C2×C32semidirect
122. C3⋊#95C2×C32semidirect
123. C3⋊#95C2×C32semidirect
276. #95C32×D3semidirect
277. C33#397semidirect
425. C32#4675direct (N × H)
426. C32C3×#397semidirect
497. C3C3×#4675direct (N × H)
498. C3C32×#397semidirect

Maximal subgroups

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

subgroupindex
C2×C32×(C2×D3)⋊C42
C2×C32×Dic3⋊C42
C33×#952
C32×C3⋊#952
C32×#3973
C3×#46753