Finite Group Databasebeta

← C3×#10592647 #13162990

Subgroups — C3×#10592647 #13162990

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×C3×#3757C211yesmaximal, characteristic
3C2×C3×#3757C211yesmaximal, characteristic
4C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
5C2×C3×#3750C211yesmaximal, characteristic, complemented
6C2×C3×#3750C211yesmaximal, characteristic, complemented
7C3×#63658C211yesmaximal, characteristic, complemented
8#10592647C311yesmaximal, characteristic, direct product factor
9C3×#4948—34nomaximal
10C24×C33C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C33×C22⋊C4—22no—
18C3×#3757—22no—
19C3×#3757—22no—
20C3×C4⋊(C3⋊Dic3)—22no—
21C3×#3750—22no—
22C3×#3750—22no—
23C2×C32×C22⋊C4—24no—
24C2×C32×C22⋊C4D314yes—
25C2×C32×C22⋊C4C2×C311yescharacteristic
26C2×#3757C2×C311yescharacteristic
27C2×#3757C2×C311yescharacteristic
28C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
29C2×#3750C2×C311yescharacteristic, complemented
30C2×#3750C2×C311yescharacteristic, complemented
31C2×C3×#217—34no—
32C2×C3×#217—34no—
33C2×C3×C4⋊Dic3—34no—
34C2×C3×Dic3⋊C4—34no—
35C2×C3×Dic3⋊C4—34no—
36#63658C2×C311yescharacteristic, complemented
37C3×C3⋊#95—34no—
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×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3—21no—
57C2×C3×C3⋊Dic3—21no—
58C2×C3×C3⋊Dic3—21no—
59C2×C3×C3⋊Dic3—21no—
60C2×C3×C3⋊Dic3—21no—
61C2×C3×C3⋊Dic3—21no—
62C2×C3×C3⋊Dic3D412yes—
63C2×C3×C3⋊Dic3D412yes—
64C3×#396—91no—
65#4948—34no—
66C24×C32—24no—
67C24×C32C2×D314yes—
68C24×C32C22×C311yescharacteristic
69C22×C32×C4—24no—
70C22×C32×C4—24no—
71C22×C32×C4C2×D314yes—
72C22×C32×C4C2×D314yes—
73C22×C32×C4C22×C311yescharacteristic
74C22×C32×C4C22×C311yescharacteristic
75C22×C3⋊Dic3C22×C311yescharacteristic
76C22×C3⋊Dic3C22×C311yescharacteristic
77C22×C3⋊Dic3C22×C311yescharacteristic
78C22×C3⋊Dic3C22×C311yescharacteristic
79C22×C3×Dic3—34no—
80C22×C3×Dic3—34no—
81C22×C3×Dic3—34no—
82C22×C3×Dic3—34no—
83C32×C22⋊C4—216no—
84C32×C22⋊C4—28no—
85C32×C22⋊C4—22no—
86#3757—22no—
87#3757—22no—
88C4⋊(C3⋊Dic3)—22no—
89#3750—22no—
90#3750—22no—
91C3×#217—68no—
92C3×#217—68no—
93C3×C4⋊Dic3—68no—
94C3×Dic3⋊C4—68no—
95C3×Dic3⋊C4—68no—
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—
112C3×C3⋊Dic3—22no—
113C3×C3⋊Dic3—22no—
114C3×C3⋊Dic3—41no—
115C3×C3⋊Dic3—41no—
116C2×C3×C4⋊C4—91no—
117C2×C3×C4⋊C4—91no—
118C2×C3×C4⋊C4—91no—
119C2×C3×C22⋊C4—24no—
120C2×C3×C22⋊C4—91no—
121C2×C3×C22⋊C4—91no—
122C2×C3×C22⋊C4C3×D314yes—
123C2×C3×C22⋊C4C3⋊D311yescharacteristic
124C2×#217—34no—
125C2×#217—34no—
126C2×C4⋊Dic3—34no—
127C2×Dic3⋊C4—34no—
128C2×Dic3⋊C4—34no—
129C3×#95—91no—
130C3⋊#95—34no—
131C23×C32—28no—
132C23×C32—28no—
133C23×C32—24no—
134C23×C32—24no—
135C23×C32—21no—
136C23×C32—21no—
137C23×C32—21no—
138C23×C32—21no—
139C23×C32—21no—
140C23×C32—21no—
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×C32Dic618yes—
152C23×C32C3×Q812yes—
153C23×C32C22×D314yes—
154C23×C32C23×C311yescharacteristic
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—28no—
158C2×C32×C4—24no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—28no—
163C2×C32×C4—24no—
164C2×C32×C4—24no—
165C2×C32×C4—24no—
166C2×C32×C4—24no—
167C2×C32×C4—44no—
168C2×C32×C4—44no—
169C2×C32×C4—44no—
170C2×C32×C4—44no—
171C2×C32×C4#6718yes—
172C2×C32×C4C3×D412yes—
173C2×C3⋊Dic3—21no—
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3—21no—
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3—21no—
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3—21no—
183C2×C3⋊Dic3C3×D412yes—
184C2×C3⋊Dic3C3×D412yes—
185C2×C3×Dic3—38no—
186C2×C3×Dic3—38no—
187C2×C3×Dic3—64no—
188C2×C3×Dic3—64no—
189C2×C3×Dic3—64no—
190C2×C3×Dic3—64no—
191C2×C3×Dic3—64no—
192C2×C3×Dic3—64no—
193C2×C3×Dic3—64no—
194C2×C3×Dic3—64no—
195C2×C3×Dic3—64no—
196C2×C3×Dic3—64no—
197#396—91noSylow
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×C3C2×C3⋊D311yescharacteristic
209C22×C3×C4—24no—
210C22×C3×C4—24no—
211C22×C3×C4—91no—
212C22×C3×C4—91no—
213C22×C3×C4—91no—
214C22×C3×C4—91no—
215C22×C3×C4C2×C3×D314yes—
216C22×C3×C4C2×C3×D314yes—
217C22×C3×C4C2×C3⋊D311yescharacteristic
218C22×C3×C4C2×C3⋊D311yescharacteristic
219C22×Dic3—34no—
220C22×Dic3—34no—
221C22×Dic3—34no—
222C22×Dic3—34no—
223C3×C4⋊C4—182no—
224C3×C4⋊C4—182no—
225C3×C4⋊C4—182no—
226C3×C22⋊C4—216no—
227C3×C22⋊C4—28no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—182no—
230C3×C22⋊C4—182no—
231#217—68no—
232#217—68no—
233C4⋊Dic3—68no—
234Dic3⋊C4—68no—
235Dic3⋊C4—68no—
236C22×C32—216no—
237C22×C32—28no—
238C22×C32—22no—
239C22×C32—22no—
240C22×C32—22no—
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—44no—
257C22×C32—44no—
258C22×C32—44no—
259C22×C32—44no—
260C22×C32—41no—
261C22×C32—41no—
262C22×C32—41no—
263C22×C32—41no—
264C22×C32C2×Dic614yes—
265C22×C32#23514yes—
266C22×C32D3×D414yes—
267C22×C32D3×D414yes—
268C22×C32#23714yes—
269C22×C32#23714yes—
270C22×C32C2×#6714yes—
271C22×C32C2×C3×D411yescharacteristic
272C22×C32C2×C3×D411yescharacteristic
273C22×C32C2×C3×D411yescharacteristic
274C22×C32C2×C3×Q811yescharacteristic
275C22×C32C3×#4111yescharacteristic
276C22×C32C3×#4111yescharacteristic
277C22×C32C3×#4111yescharacteristic
278C32×C4—22no—
279C32×C4—28no—
280C32×C4—48no—
281C32×C4—48no—
282C32×C4—44no—
283C32×C4—41no—
284C3⋊Dic3—22no—
285C3⋊Dic3—22no—
286C3⋊Dic3—41no—
287C3⋊Dic3—41no—
288C3×Dic3—68no—
289C3×Dic3—68no—
290C3×Dic3—124no—
291C3×Dic3—124no—
292C2×C4⋊C4—91no—
293C2×C4⋊C4—91no—
294C2×C4⋊C4—91no—
295C2×C22⋊C4—91no—
296C2×C22⋊C4—91no—
297C2×C22⋊C4C3×C3⋊D311yescharacteristic
298#95—91no—
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×C3#63412yes—
322C23×C3C22×C3×D314yes—
323C23×C3C22×C3⋊D311yescharacteristic
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—44no—
337C2×C3×C4—44no—
338C2×C3×C4—44no—
339C2×C3×C4—44no—
340C2×C3×C4—92no—
341C2×C3×C4—92no—
342C2×C3×C4—181no—
343C2×C3×C4—181no—
344C2×C3×C4—181no—
345C2×C3×C4—181no—
346C2×C3×C4—181no—
347C2×C3×C4—181no—
348C2×C3×C4—181no—
349C2×C3×C4—181no—
350C2×C3×C4—181no—
351C2×C3×C4—181no—
352C2×C3×C4C3×#6718yes—
353C2×C3×C4#63812yes—
354C2×Dic3—38no—
355C2×Dic3—38no—
356C2×Dic3—64no—
357C2×Dic3—64no—
358C2×Dic3—64no—
359C2×Dic3—64no—
360C2×Dic3—64no—
361C2×Dic3—64no—
362C2×Dic3—64no—
363C2×Dic3—64no—
364C2×Dic3—64no—
365C2×Dic3—64no—
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—48no—
375C2×C32—42no—
376C2×C32#87814yes—
377C2×C32#87814yes—
378C2×C32#87914yes—
379C2×C32#88414yes—
380C2×C32#92414yes—
381C2×C32#93814yes—
382C2×C32#93914yes—
383C2×C32C3×#12111yescharacteristic
384C2×C32C3×#12111yescharacteristic
385C2×C32C3×#12111yescharacteristic
386C2×C32C3×C22⋊Q811yescharacteristic
387C2×C32C3×C22⋊Q811yescharacteristic
388C2×C32C3×C22⋊Q811yescharacteristic
389C2×C32C3×#12611yescharacteristic
390C24C2×C3×C3⋊D311yescharacteristic
391C22×C4—91no—
392C22×C4—91no—
393C22×C4—91no—
394C22×C4—91no—
395C22×C4C2×C3×C3⋊D311yescharacteristic
396C22×C4C2×C3×C3⋊D311yescharacteristic
397C4⋊C4—182no—
398C4⋊C4—182no—
399C4⋊C4—182no—
400C22⋊C4—22no—
401C22⋊C4—182no—
402C22⋊C4—182no—
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×#63411yescharacteristic
439C22×C3#382811yescharacteristic
440C22×C3D4×C3⋊D311yescharacteristic
441C22×C3D4×C3⋊D311yescharacteristic
442C22×C3#383011yescharacteristic
443C22×C3#383011yescharacteristic
444C22×C3C2×#63811yescharacteristic
445C3×C4—28no—
446C3×C4—22no—
447C3×C4—48no—
448C3×C4—48no—
449C3×C4—44no—
450C3×C4—41no—
451C3×C4—182no—
452C3×C4—182no—
453C3×C4—361no—
454C3×C4—361no—
455Dic3—68no—
456Dic3—68no—
457Dic3—124no—
458Dic3—124no—
459C32—24no—
460C32#494814yescomplemented
461C32C3×#39611yescharacteristic, complemented
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23C3×#63412yes—
469C23C22×C3×C3⋊D311yescharacteristic
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—92no—
476C2×C4—92no—
477C2×C4—181no—
478C2×C4—181no—
479C2×C4—181no—
480C2×C4—181no—
481C2×C4—181no—
482C2×C4—181no—
483C2×C4—181no—
484C2×C4—181no—
485C2×C4—181no—
486C2×C4—181no—
487C2×C4C3×#63812yes—
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×C3#6408711yescharacteristic
506C2×C3#6408711yescharacteristic
507C2×C3#6408811yescharacteristic
508C2×C3#6409311yescharacteristic
509C2×C3#6413311yescharacteristic
510C2×C3#6414711yescharacteristic
511C2×C3#6414811yescharacteristic
512C22—22no—
513C22—22no—
514C22—22no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22C2×C3×#63411yescharacteristic
520C22C3×#382811yescharacteristic
521C22C3×D4×C3⋊D311yescharacteristic
522C22C3×D4×C3⋊D311yescharacteristic
523C22C3×#383011yescharacteristic
524C22C3×#383011yescharacteristic
525C22C2×C3×#63811yescharacteristic
526C4—22no—
527C4—41no—
528C4—182no—
529C4—182no—
530C4—361no—
531C4—361no—
532C3—24no—
533C3C3×#494814yescomplemented
534C3#1059264711yescharacteristic, direct product factor
535C2—42no—
536C2C3×#6408711yescharacteristic
537C2C3×#6408711yescharacteristic
538C2C3×#6408811yescharacteristic
539C2C3×#6409311yescharacteristic
540C2C3×#6413311yescharacteristic
541C2C3×#6414711yescharacteristic
542C2C3×#6414811yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutator, cosocleC23×C32C23×C3
FrattiniC23C22×C3×C3⋊D3
FittingC2×C33×C22⋊C4C2
socleC23×C33C23
prime radicalC24×C33C22
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterC32C3×#396
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
5. C2×C3×#3750C2semidirect
6. C2×C3×#3750C2semidirect
7. C3×#63658C2semidirect
8. #10592647C3direct (N × H)
28. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
29. C2×#3750C2×C3semidirect
30. C2×#3750C2×C3semidirect
36. #63658C2×C3semidirect
299. C33#396semidirect
460. C32#4948semidirect
461. C32C3×#396semidirect
533. C3C3×#4948semidirect
534. C3#10592647direct (N × H)

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×C3×#37572
C2×C3×C4⋊(C3⋊Dic3)2
C2×C3×#37502
C3×#636582
#105926473
C3×#49483