Finite Group Databasebeta

← C32×#4994 #13162416

Subgroups — C32×#4994 #13162416

Order 1728.

1484 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C32×(C2×D3)⋊C4C211yesmaximal, characteristic
3C2×C32×(C2×D3)⋊C4C211yesmaximal, characteristic
4C2×C32×(C2×D3)⋊C4C211yesmaximal, characteristic
5C2×C32×C4⋊Dic3C211yesmaximal, characteristic, complemented
6C2×C32×Dic3⋊C4C211yesmaximal, characteristic, complemented
7C32×C3⋊#95C211yesmaximal, characteristic, complemented
8C32×#396—31nomaximal
9C3×#4994C314yesmaximal, 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×C4⋊C4—22no—
18C32×(C2×D3)⋊C4—22no—
19C32×(C2×D3)⋊C4—22no—
20C32×(C2×D3)⋊C4—22no—
21C32×C4⋊Dic3—22no—
22C32×Dic3⋊C4—22no—
23C2×C32×C4⋊C4—24no—
24C2×C32×C4⋊C4—31no—
25C2×C32×C4⋊C4—31no—
26C2×C32×C4⋊C4D311yescharacteristic, complemented
27C2×C32×C4⋊C4C2×C314yescomplemented
28C2×C32×C22⋊C4—31no—
29C2×C32×C22⋊C4—31no—
30C2×C32×C22⋊C4—31no—
31C2×C3×(C2×D3)⋊C4C2×C314yes—
32C2×C3×(C2×D3)⋊C4C2×C314yes—
33C2×C3×(C2×D3)⋊C4C2×C314yes—
34C2×C3×C4⋊Dic3C2×C314yescomplemented
35C2×C3×Dic3⋊C4C2×C314yescomplemented
36C32×#95—31no—
37C3×C3⋊#95C2×C314yescomplemented
38C23×C33C2311yescharacteristic
39C22×C32×D3—21no—
40C22×C32×D3—21no—
41C22×C32×D3—21no—
42C22×C32×D3—21no—
43C22×C32×D3—21no—
44C22×C32×D3—21no—
45C22×C32×D3Q812yes—
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×C33×C4D412yes—
54C2×C33×C4D412yes—
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×Dic3—21no—
63C2×C32×Dic3D412yes—
64C3×#396—34no—
65#4994C3211yescharacteristic, direct product factor
66C24×C32—31no—
67C23×C3×D3C22×C314yes—
68C22×C32×C4—24no—
69C22×C32×C4—24no—
70C22×C32×C4—24no—
71C22×C32×C4—31no—
72C22×C32×C4—31no—
73C22×C32×C4—31no—
74C22×C32×C4C2×D311yescharacteristic
75C22×C32×C4C2×D311yescharacteristic
76C22×C32×C4C2×D311yescharacteristic
77C22×C32×C4C22×C314yes—
78C22×C32×C4C22×C314yes—
79C22×C32×C4C22×C314yes—
80C22×C3×Dic3C22×C314yes—
81C22×C3×Dic3C22×C314yes—
82C22×C3×Dic3C22×C314yes—
83C32×C4⋊C4—216no—
84C32×C4⋊C4—22no—
85C32×C4⋊C4—28no—
86C32×C4⋊C4—62no—
87C32×C4⋊C4—62no—
88C32×C22⋊C4—62no—
89C32×C22⋊C4—62no—
90C32×C22⋊C4—62no—
91C3×(C2×D3)⋊C4—28no—
92C3×(C2×D3)⋊C4—28no—
93C3×(C2×D3)⋊C4—28no—
94C3×C4⋊Dic3—28no—
95C3×Dic3⋊C4—28no—
96C22×C33C2×D411yescharacteristic
97C22×C33C2×D411yescharacteristic
98C22×C33C2×D411yescharacteristic
99C22×C33C2×Q811yescharacteristic
100C22×C33#4111yescharacteristic
101C22×C33#4111yescharacteristic
102C22×C33#4111yescharacteristic
103C2×C32×D3—22no—
104C2×C32×D3—22no—
105C2×C32×D3—22no—
106C2×C32×D3—41no—
107C2×C32×D3—41no—
108C2×C32×D3—41no—
109C2×C32×D3—41no—
110C33×C4—22no—
111C33×C4—22no—
112C33×C4—41no—
113C32×Dic3—22no—
114C32×Dic3—41no—
115C32×Dic3—41no—
116C2×C3×C4⋊C4—24no—
117C2×C3×C4⋊C4—34no—
118C2×C3×C4⋊C4—34no—
119C2×C3×C4⋊C4C3×D314yescomplemented
120C2×C3×C4⋊C4C2×C3211yescharacteristic, complemented
121C2×C3×C22⋊C4—34no—
122C2×C3×C22⋊C4—34no—
123C2×C3×C22⋊C4—34no—
124C2×(C2×D3)⋊C4C2×C3211yescharacteristic
125C2×(C2×D3)⋊C4C2×C3211yescharacteristic
126C2×(C2×D3)⋊C4C2×C3211yescharacteristic
127C2×C4⋊Dic3C2×C3211yescharacteristic, complemented
128C2×Dic3⋊C4C2×C3211yescharacteristic, complemented
129C3×#95—34no—
130C3⋊#95C2×C3211yescharacteristic, complemented
131C23×C32—24no—
132C23×C32—32no—
133C23×C32—61no—
134C23×C32—61no—
135C23×C32—61no—
136C23×C32—61no—
137C23×C32—61no—
138C23×C32—61no—
139C23×C32C22×D311yescharacteristic
140C23×C32C23×C314yes—
141C22×C3×D3—24no—
142C22×C3×D3—24no—
143C22×C3×D3—24no—
144C22×C3×D3—24no—
145C22×C3×D3—24no—
146C22×C3×D3—24no—
147C22×C3×D3C3×Q818yes—
148C2×C32×C4—28no—
149C2×C32×C4—28no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—24no—
155C2×C32×C4—24no—
156C2×C32×C4—28no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—24no—
161C2×C32×C4—28no—
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—44no—
169C2×C32×C4—44no—
170C2×C32×C4—44no—
171C2×C32×C4—44no—
172C2×C32×C4—61no—
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×C4D1212yes—
181C2×C32×C4#6712yes—
182C2×C32×C4C3×D418yes—
183C2×C32×C4C3×D418yes—
184C2×C3×Dic3—24no—
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×Dic3C3×D418yes—
193#396—31noSylow
194C2×C33#12111yescharacteristic
195C2×C33#12111yescharacteristic
196C2×C33#12111yescharacteristic
197C2×C33C22⋊Q811yescharacteristic
198C2×C33C22⋊Q811yescharacteristic
199C2×C33C22⋊Q811yescharacteristic
200C2×C33#12611yescharacteristic
201C32×D3—42no—
202C24×C3—34no—
203C23×D3C22×C3211yescharacteristic
204C22×C3×C4—24no—
205C22×C3×C4—24no—
206C22×C3×C4—24no—
207C22×C3×C4—34no—
208C22×C3×C4—34no—
209C22×C3×C4—34no—
210C22×C3×C4C2×C3×D314yes—
211C22×C3×C4C2×C3×D314yes—
212C22×C3×C4C2×C3×D314yes—
213C22×C3×C4C22×C3211yescharacteristic
214C22×C3×C4C22×C3211yescharacteristic
215C22×C3×C4C22×C3211yescharacteristic
216C22×Dic3C22×C3211yescharacteristic
217C22×Dic3C22×C3211yescharacteristic
218C22×Dic3C22×C3211yescharacteristic
219C3×C4⋊C4—216no—
220C3×C4⋊C4—28no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—68no—
223C3×C4⋊C4—68no—
224C3×C22⋊C4—68no—
225C3×C22⋊C4—68no—
226C3×C22⋊C4—68no—
227(C2×D3)⋊C4—22no—
228(C2×D3)⋊C4—22no—
229(C2×D3)⋊C4—22no—
230C4⋊Dic3—22no—
231Dic3⋊C4—22no—
232C22×C32—24no—
233C22×C32—24no—
234C22×C32—24no—
235C22×C32—24no—
236C22×C32—24no—
237C22×C32—24no—
238C22×C32—24no—
239C22×C32—62no—
240C22×C32—62no—
241C22×C32—62no—
242C22×C32—121no—
243C22×C32—121no—
244C22×C32—121no—
245C22×C32—121no—
246C22×C32C2×D1211yescharacteristic
247C22×C32#23511yescharacteristic
248C22×C32D3×D411yescharacteristic
249C22×C32#23711yescharacteristic
250C22×C32D3×Q811yescharacteristic
251C22×C32#23911yescharacteristic
252C22×C32C2×#6711yescharacteristic
253C22×C32C2×C3×D414yes—
254C22×C32C2×C3×D414yes—
255C22×C32C2×C3×D414yes—
256C22×C32C2×C3×Q814yes—
257C22×C32C3×#4114yes—
258C22×C32C3×#4114yes—
259C22×C32C3×#4114yes—
260C2×C3×D3—28no—
261C2×C3×D3—28no—
262C2×C3×D3—28no—
263C2×C3×D3—44no—
264C2×C3×D3—44no—
265C2×C3×D3—44no—
266C2×C3×D3—44no—
267C32×C4—216no—
268C32×C4—22no—
269C32×C4—22no—
270C32×C4—28no—
271C32×C4—28no—
272C32×C4—48no—
273C32×C4—48no—
274C32×C4—41no—
275C32×C4—44no—
276C32×C4—62no—
277C32×C4—121no—
278C32×C4—121no—
279C3×Dic3—28no—
280C3×Dic3—44no—
281C3×Dic3—44no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4C32×D311yescharacteristic, complemented
285C2×C22⋊C4—31no—
286C2×C22⋊C4—31no—
287C2×C22⋊C4—31no—
288#95—31no—
289C33#39611yescharacteristic, complemented, Sylow
290C23×C3—24no—
291C23×C3—38no—
292C23×C3—64no—
293C23×C3—64no—
294C23×C3—64no—
295C23×C3—64no—
296C23×C3—64no—
297C23×C3—64no—
298C23×C3C22×C3×D314yes—
299C23×C3C23×C3211yescharacteristic
300C22×D3—21no—
301C22×D3—21no—
302C22×D3—21no—
303C22×D3—21no—
304C22×D3—21no—
305C22×D3—21no—
306C22×D3C32×Q812yes—
307C2×C3×C4—28no—
308C2×C3×C4—28no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—28no—
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—28no—
325C2×C3×C4—28no—
326C2×C3×C4—38no—
327C2×C3×C4—44no—
328C2×C3×C4—44no—
329C2×C3×C4—44no—
330C2×C3×C4—44no—
331C2×C3×C4—64no—
332C2×C3×C4—64no—
333C2×C3×C4—64no—
334C2×C3×C4—64no—
335C2×C3×C4—64no—
336C2×C3×C4—64no—
337C2×C3×C4—64no—
338C2×C3×C4—64no—
339C2×C3×C4C3×D1218yes—
340C2×C3×C4C3×#6718yes—
341C2×C3×C4C32×D412yes—
342C2×C3×C4C32×D412yes—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3C32×D412yes—
352C2×C32—24no—
353C2×C32—24no—
354C2×C32—24no—
355C2×C32—24no—
356C2×C32—24no—
357C2×C32—24no—
358C2×C32—24no—
359C2×C32—122no—
360C2×C32#89511yescharacteristic
361C2×C32#89611yescharacteristic
362C2×C32#89711yescharacteristic
363C2×C32#89811yescharacteristic
364C2×C32#93011yescharacteristic
365C2×C32#93911yescharacteristic
366C2×C32#94611yescharacteristic
367C2×C32C3×#12114yes—
368C2×C32C3×#12114yes—
369C2×C32C3×#12114yes—
370C2×C32C3×C22⋊Q814yes—
371C2×C32C3×C22⋊Q814yes—
372C2×C32C3×C22⋊Q814yes—
373C2×C32C3×#12614yes—
374C3×D3—48no—
375C24—31no—
376C22×C4—31no—
377C22×C4—31no—
378C22×C4—31no—
379C22×C4C2×C32×D311yescharacteristic
380C22×C4C2×C32×D311yescharacteristic
381C22×C4C2×C32×D311yescharacteristic
382C4⋊C4—22no—
383C4⋊C4—62no—
384C4⋊C4—62no—
385C22⋊C4—62no—
386C22⋊C4—62no—
387C22⋊C4—62no—
388C22×C3—24no—
389C22×C3—24no—
390C22×C3—24no—
391C22×C3—24no—
392C22×C3—24no—
393C22×C3—24no—
394C22×C3—24no—
395C22×C3—68no—
396C22×C3—68no—
397C22×C3—68no—
398C22×C3—124no—
399C22×C3—124no—
400C22×C3—124no—
401C22×C3—124no—
402C22×C3C2×C3×D1214yes—
403C22×C3C3×#23514yes—
404C22×C3C3×D3×D414yes—
405C22×C3C3×#23714yes—
406C22×C3C3×D3×Q814yes—
407C22×C3C3×#23914yes—
408C22×C3C2×C3×#6714yes—
409C22×C3C2×C32×D411yescharacteristic
410C22×C3C2×C32×D411yescharacteristic
411C22×C3C2×C32×D411yescharacteristic
412C22×C3C2×C32×Q811yescharacteristic
413C22×C3C32×#4111yescharacteristic
414C22×C3C32×#4111yescharacteristic
415C22×C3C32×#4111yescharacteristic
416C2×D3—22no—
417C2×D3—22no—
418C2×D3—22no—
419C2×D3—41no—
420C2×D3—41no—
421C2×D3—41no—
422C2×D3—41no—
423C3×C4—216no—
424C3×C4—28no—
425C3×C4—28no—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4—48no—
429C3×C4—48no—
430C3×C4—44no—
431C3×C4—41no—
432C3×C4—68no—
433C3×C4—124no—
434C3×C4—124no—
435Dic3—22no—
436Dic3—41no—
437Dic3—41no—
438C32—24no—
439C32#499411yescharacteristic, direct product factor
440C32C3×#39614yescomplemented
441C23—32no—
442C23—61no—
443C23—61no—
444C23—61no—
445C23—61no—
446C23—61no—
447C23—61no—
448C23C22×C32×D311yescharacteristic
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—32no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4C32×D1212yes—
466C2×C4C32×#6712yes—
467C2×C3—24no—
468C2×C3—24no—
469C2×C3—24no—
470C2×C3—24no—
471C2×C3—24no—
472C2×C3—24no—
473C2×C3—24no—
474C2×C3—128no—
475C2×C3C3×#89514yes—
476C2×C3C3×#89614yes—
477C2×C3C3×#89714yes—
478C2×C3C3×#89814yes—
479C2×C3C3×#93014yes—
480C2×C3C3×#93914yes—
481C2×C3C3×#94614yes—
482C2×C3C32×#12111yescharacteristic
483C2×C3C32×#12111yescharacteristic
484C2×C3C32×#12111yescharacteristic
485C2×C3C32×C22⋊Q811yescharacteristic
486C2×C3C32×C22⋊Q811yescharacteristic
487C2×C3C32×C22⋊Q811yescharacteristic
488C2×C3C32×#12611yescharacteristic
489D3—42no—
490C22—62no—
491C22—62no—
492C22—62no—
493C22—121no—
494C22—121no—
495C22—121no—
496C22—121no—
497C22C2×C32×D1211yescharacteristic
498C22C32×#23511yescharacteristic
499C22C32×D3×D411yescharacteristic
500C22C32×#23711yescharacteristic
501C22C32×D3×Q811yescharacteristic
502C22C32×#23911yescharacteristic
503C22C2×C32×#6711yescharacteristic
504C4—22no—
505C4—22no—
506C4—41no—
507C4—62no—
508C4—121no—
509C4—121no—
510C3—24no—
511C3C3×#499414yesdirect product factor
512C3C32×#39611yescharacteristic, complemented
513C2—122no—
514C2C32×#89511yescharacteristic
515C2C32×#89611yescharacteristic
516C2C32×#89711yescharacteristic
517C2C32×#89811yescharacteristic
518C2C32×#93011yescharacteristic
519C2C32×#93911yescharacteristic
520C2C32×#94611yescharacteristic
subgroupG/H
centerC23×C32C22×D3
commutator, cosocleC23×C3C23×C32
FrattiniC23C22×C32×D3
FittingC2×C33×C4⋊C4C2
socleC23×C33C23
prime radicalC23×C32×D3C22
hypercenterC2×C32×C4⋊C4D3
hypocenterC3C32×#396
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23×C32 ◁ C2×C32×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C33×C4⋊C4C2semidirect
5. C2×C32×C4⋊Dic3C2semidirect
6. C2×C32×Dic3⋊C4C2semidirect
7. C32×C3⋊#95C2semidirect
9. C3×#4994C3direct (N × H)
26. C2×C32×C4⋊C4D3semidirect
27. C2×C32×C4⋊C4C2×C3semidirect
34. C2×C3×C4⋊Dic3C2×C3semidirect
35. C2×C3×Dic3⋊C4C2×C3semidirect
37. C3×C3⋊#95C2×C3semidirect
65. #4994C32direct (N × H)
119. C2×C3×C4⋊C4C3×D3semidirect
120. C2×C3×C4⋊C4C2×C32semidirect
127. C2×C4⋊Dic3C2×C32semidirect
128. C2×Dic3⋊C4C2×C32semidirect
130. C3⋊#95C2×C32semidirect
284. C2×C4⋊C4C32×D3semidirect
289. C33#396semidirect
439. C32#4994direct (N × H)
440. C32C3×#396semidirect
511. C3C3×#4994direct (N × H)
512. C3C32×#396semidirect

Maximal subgroups

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

subgroupindex
C2×C33×C4⋊C42
C2×C32×(C2×D3)⋊C42
C2×C32×C4⋊Dic32
C2×C32×Dic3⋊C42
C32×C3⋊#952
C32×#3963
C3×#49943