Finite Group Databasebeta

← C3×#10592639 #13162978

Subgroups — C3×#10592639 #13162978

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×C3×#3752C211yesmaximal, characteristic
3C2×C3×#3752C211yesmaximal, characteristic
4C2×C3×#3752C211yesmaximal, characteristic
5C2×C3×#3750C211yesmaximal, characteristic, complemented
6C3×#63658C211yesmaximal, characteristic, complemented
7C3×#63658C211yesmaximal, characteristic, complemented
8#10592639C311yesmaximal, characteristic, direct product factor
9C3×#4940—34nomaximal
10C23×C3×C3⋊D3C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C33×C4C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C33×C42—22no—
18C3×#3752—22no—
19C3×#3752—22no—
20C3×#3752C414yescomplemented
21C3×#3750—22no—
22C2×C32×C42—24no—
23C2×C32×C42D314yescomplemented
24C2×C32×C42C2×C311yescharacteristic, complemented
25C2×#3752C2×C311yescharacteristic
26C2×#3752C2×C311yescharacteristic
27C2×#3752C2×C311yescharacteristic
28C2×#3750C2×C311yescharacteristic, complemented
29C2×C3×(C2×D3)⋊C4—34no—
30C2×C3×(C2×D3)⋊C4—34no—
31C2×C3×(C2×D3)⋊C4—34no—
32C2×C3×Dic3⋊C4—34no—
33#63658C2×C311yescharacteristic, complemented
34#63658C2×C311yescharacteristic, complemented
35C3×C3⋊#95—34no—
36C3×C3⋊#95—34no—
37C23×C33C2311yescharacteristic
38C22×C3×C3⋊D3—21no—
39C22×C3×C3⋊D3—21no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3—21no—
43C22×C3×C3⋊D3—21no—
44C22×C3×C3⋊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×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⋊Dic3C2×C412yes—
63C3×#388—91no—
64#4940—34no—
65C23×C3⋊D3C22×C311yescharacteristic
66C23×C3×D3—34no—
67C22×C32×C4—24no—
68C22×C32×C4—24no—
69C22×C32×C4—24no—
70C22×C32×C4C2×D314yes—
71C22×C32×C4C2×D314yes—
72C22×C32×C4C2×D314yes—
73C22×C32×C4C22×C311yescharacteristic
74C22×C32×C4C22×C311yescharacteristic
75C22×C32×C4C22×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—
82C32×C42—216no—
83C32×C42—28no—
84C32×C42—22no—
85#3752—22no—
86#3752—22no—
87#3752C3×C414yescomplemented
88#3750—22no—
89C3×(C2×D3)⋊C4—316no—
90C3×(C2×D3)⋊C4—68no—
91C3×(C2×D3)⋊C4—68no—
92C3×Dic3⋊C4—68no—
93C22×C33C22×C411yescharacteristic
94C22×C33C2×D411yescharacteristic
95C22×C33C2×D411yescharacteristic
96C22×C33#4111yescharacteristic
97C22×C33#4111yescharacteristic
98C22×C33#4111yescharacteristic
99C22×C33#4111yescharacteristic
100C2×C3×C3⋊D3—22no—
101C2×C3×C3⋊D3—22no—
102C2×C3×C3⋊D3—22no—
103C2×C3×C3⋊D3—41no—
104C2×C3×C3⋊D3—41no—
105C2×C3×C3⋊D3—41no—
106C2×C3×C3⋊D3—41no—
107C33×C4—22no—
108C33×C4—22no—
109C33×C4—22no—
110C3×C3⋊Dic3—22no—
111C3×C3⋊Dic3—41no—
112C3×C3⋊Dic3—41no—
113C2×C3×C4⋊C4—91no—
114C2×C3×C22⋊C4—91no—
115C2×C3×C22⋊C4—91no—
116C2×C3×C22⋊C4—91no—
117C2×C3×C42—24no—
118C2×C3×C42C3×D314yescomplemented
119C2×C3×C42C3⋊D311yescharacteristic, complemented
120C2×(C2×D3)⋊C4—34no—
121C2×(C2×D3)⋊C4—34no—
122C2×(C2×D3)⋊C4—34no—
123C2×Dic3⋊C4—34no—
124C3×#95—91no—
125C3×#95—91no—
126C3⋊#95—34no—
127C3⋊#95—34no—
128C23×C32—24no—
129C23×C32C22×D314yes—
130C23×C32C23×C311yescharacteristic
131C22×C3⋊D3—21no—
132C22×C3⋊D3—21no—
133C22×C3⋊D3—21no—
134C22×C3⋊D3—21no—
135C22×C3⋊D3—21no—
136C22×C3⋊D3—21no—
137C22×C3⋊D3C2×C3×C412yes—
138C22×C3×D3—38no—
139C22×C3×D3—64no—
140C22×C3×D3—64no—
141C22×C3×D3—64no—
142C22×C3×D3—64no—
143C22×C3×D3—64no—
144C22×C3×D3—64no—
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—24no—
152C2×C32×C4—24no—
153C2×C32×C4—24no—
154C2×C32×C4—24no—
155C2×C32×C4—24no—
156C2×C32×C4—24no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—28no—
164C2×C32×C4—28no—
165C2×C32×C4—28no—
166C2×C32×C4C4×D318yes—
167C2×C32×C4D1218yes—
168C2×C32×C4#6718yes—
169C2×C32×C4C2×C3×C412yes—
170C2×C32×C4C3×D412yes—
171C2×C32×C4C3×D412yes—
172C2×C3⋊Dic3—21no—
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⋊Dic3C2×C3×C412yes—
181C2×C3×Dic3—38no—
182C2×C3×Dic3—64no—
183C2×C3×Dic3—64no—
184C2×C3×Dic3—64no—
185C2×C3×Dic3—64no—
186C2×C3×Dic3—64no—
187C2×C3×Dic3—64no—
188C2×C3×Dic3—64no—
189C2×C3×Dic3—64no—
190#388—91noSylow
191C2×C33#11711yescharacteristic
192C2×C33C4×D411yescharacteristic
193C2×C33C4×D411yescharacteristic
194C2×C33#12111yescharacteristic
195C2×C33#12311yescharacteristic
196C2×C33#12411yescharacteristic
197C2×C33#12611yescharacteristic
198C3×C3⋊D3—42no—
199C24×C3—91no—
200C23×D3—34no—
201C22×C3×C4—24no—
202C22×C3×C4—24no—
203C22×C3×C4—24no—
204C22×C3×C4—91no—
205C22×C3×C4—91no—
206C22×C3×C4—91no—
207C22×C3×C4C2×C3×D314yes—
208C22×C3×C4C2×C3×D314yes—
209C22×C3×C4C2×C3×D314yes—
210C22×C3×C4C2×C3⋊D311yescharacteristic
211C22×C3×C4C2×C3⋊D311yescharacteristic
212C22×C3×C4C2×C3⋊D311yescharacteristic
213C22×Dic3—34no—
214C22×Dic3—34no—
215C22×Dic3—34no—
216C3×C4⋊C4—182no—
217C3×C22⋊C4—94no—
218C3×C22⋊C4—182no—
219C3×C22⋊C4—182no—
220C3×C42—216no—
221C3×C42—28no—
222C3×C42—22no—
223(C2×D3)⋊C4—316no—
224(C2×D3)⋊C4—68no—
225(C2×D3)⋊C4—68no—
226Dic3⋊C4—68no—
227C22×C32—24no—
228C22×C32—24no—
229C22×C32—24no—
230C22×C32—24no—
231C22×C32—24no—
232C22×C32—24no—
233C22×C32—24no—
234C22×C32C2×C4×D314yes—
235C22×C32C2×D1214yes—
236C22×C32#23514yes—
237C22×C32#23514yes—
238C22×C32#23514yes—
239C22×C32#23514yes—
240C22×C32C2×#6714yes—
241C22×C32C22×C3×C411yescharacteristic
242C22×C32C2×C3×D411yescharacteristic
243C22×C32C2×C3×D411yescharacteristic
244C22×C32C3×#4111yescharacteristic
245C22×C32C3×#4111yescharacteristic
246C22×C32C3×#4111yescharacteristic
247C22×C32C3×#4111yescharacteristic
248C2×C3⋊D3—22no—
249C2×C3⋊D3—22no—
250C2×C3⋊D3—22no—
251C2×C3⋊D3—41no—
252C2×C3⋊D3—41no—
253C2×C3⋊D3—41no—
254C2×C3⋊D3—41no—
255C2×C3×D3—68no—
256C2×C3×D3—68no—
257C2×C3×D3—68no—
258C2×C3×D3—124no—
259C2×C3×D3—124no—
260C2×C3×D3—124no—
261C2×C3×D3—124no—
262C32×C4—216no—
263C32×C4—216no—
264C32×C4—216no—
265C32×C4—28no—
266C32×C4—28no—
267C32×C4—28no—
268C32×C4—22no—
269C32×C4—22no—
270C32×C4—22no—
271C3⋊Dic3—22no—
272C3⋊Dic3—41no—
273C3⋊Dic3—41no—
274C3×Dic3—68no—
275C3×Dic3—124no—
276C3×Dic3—124no—
277C2×C4⋊C4—91no—
278C2×C22⋊C4—91no—
279C2×C22⋊C4—91no—
280C2×C22⋊C4—91no—
281C2×C42C3×C3⋊D311yescharacteristic, complemented
282#95—91no—
283#95—91no—
284C33#38811yescharacteristic, complemented, Sylow
285C23×C3—24no—
286C23×C3—92no—
287C23×C3—181no—
288C23×C3—181no—
289C23×C3—181no—
290C23×C3—181no—
291C23×C3—181no—
292C23×C3—181no—
293C23×C3C22×C3×D314yes—
294C23×C3C22×C3⋊D311yescharacteristic
295C22×D3—38no—
296C22×D3—64no—
297C22×D3—64no—
298C22×D3—64no—
299C22×D3—64no—
300C22×D3—64no—
301C22×D3—64no—
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—92no—
324C2×C3×C4—181no—
325C2×C3×C4—181no—
326C2×C3×C4—181no—
327C2×C3×C4—181no—
328C2×C3×C4—181no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×C3×C4—181no—
332C2×C3×C4C3×C4×D318yes—
333C2×C3×C4C3×D1218yes—
334C2×C3×C4C3×#6718yes—
335C2×C3×C4C4×C3⋊D312yes—
336C2×C3×C4C4⋊(C3⋊D3)12yes—
337C2×C3×C4#63812yes—
338C2×Dic3—38no—
339C2×Dic3—64no—
340C2×Dic3—64no—
341C2×Dic3—64no—
342C2×Dic3—64no—
343C2×Dic3—64no—
344C2×Dic3—64no—
345C2×Dic3—64no—
346C2×Dic3—64no—
347C2×C32—24no—
348C2×C32—24no—
349C2×C32—24no—
350C2×C32—24no—
351C2×C32—24no—
352C2×C32—24no—
353C2×C32—24no—
354C2×C32#87214yes—
355C2×C32C4×D1214yes—
356C2×C32#87514yes—
357C2×C32#87614yes—
358C2×C32C4×#6714yes—
359C2×C32#92914yes—
360C2×C32#93014yes—
361C2×C32C3×#11711yescharacteristic
362C2×C32C3×C4×D411yescharacteristic
363C2×C32C3×C4×D411yescharacteristic
364C2×C32C3×#12111yescharacteristic
365C2×C32C3×#12311yescharacteristic
366C2×C32C3×#12411yescharacteristic
367C2×C32C3×#12611yescharacteristic
368C3⋊D3—42no—
369C3×D3—128no—
370C24—91no—
371C22×C4—91no—
372C22×C4—91no—
373C22×C4—91no—
374C22×C4C2×C3×C3⋊D311yescharacteristic
375C22×C4C2×C3×C3⋊D311yescharacteristic
376C22×C4C2×C3×C3⋊D311yescharacteristic
377C4⋊C4—182no—
378C22⋊C4—94no—
379C22⋊C4—182no—
380C22⋊C4—182no—
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—182no—
390C22×C3—182no—
391C22×C3—182no—
392C22×C3—361no—
393C22×C3—361no—
394C22×C3—361no—
395C22×C3—361no—
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×C3C2×C4×C3⋊D311yescharacteristic
404C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
405C22×C3#382811yescharacteristic
406C22×C3#382811yescharacteristic
407C22×C3#382811yescharacteristic
408C22×C3#382811yescharacteristic
409C22×C3C2×#63811yescharacteristic
410C2×D3—68no—
411C2×D3—68no—
412C2×D3—68no—
413C2×D3—124no—
414C2×D3—124no—
415C2×D3—124no—
416C2×D3—124no—
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—182no—
427C3×C4—361no—
428C3×C4—361no—
429Dic3—68no—
430Dic3—124no—
431Dic3—124no—
432C32—24no—
433C32#494014yescomplemented
434C32C3×#38811yescharacteristic, complemented
435C23—92no—
436C23—181no—
437C23—181no—
438C23—181no—
439C23—181no—
440C23—181no—
441C23—181no—
442C23C22×C3×C3⋊D311yescharacteristic
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—92no—
450C2×C4—181no—
451C2×C4—181no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4—181no—
455C2×C4—181no—
456C2×C4—181no—
457C2×C4—181no—
458C2×C4C3×C4×C3⋊D312yes—
459C2×C4C3×C4⋊(C3⋊D3)12yes—
460C2×C4C3×#63812yes—
461C2×C3—24no—
462C2×C3—24no—
463C2×C3—24no—
464C2×C3—24no—
465C2×C3—24no—
466C2×C3—24no—
467C2×C3—24no—
468C2×C3—362no—
469C2×C3C3×#87214yes—
470C2×C3C3×C4×D1214yes—
471C2×C3C3×#87514yes—
472C2×C3C3×#87614yes—
473C2×C3C3×C4×#6714yes—
474C2×C3C3×#92914yes—
475C2×C3C3×#93014yes—
476C2×C3#6408111yescharacteristic
477C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
478C2×C3#6408411yescharacteristic
479C2×C3#6408511yescharacteristic
480C2×C3C4×#63811yescharacteristic
481C2×C3#6413811yescharacteristic
482C2×C3#6413911yescharacteristic
483D3—128no—
484C22—182no—
485C22—182no—
486C22—182no—
487C22—361no—
488C22—361no—
489C22—361no—
490C22—361no—
491C22C2×C3×C4×C3⋊D311yescharacteristic
492C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
493C22C3×#382811yescharacteristic
494C22C3×#382811yescharacteristic
495C22C3×#382811yescharacteristic
496C22C3×#382811yescharacteristic
497C22C2×C3×#63811yescharacteristic
498C4—22no—
499C4—22no—
500C4—22no—
501C4—182no—
502C4—361no—
503C4—361no—
504C3—24no—
505C3C3×#494014yescomplemented
506C3#1059263911yescharacteristic, direct product factor
507C2—362no—
508C2C3×#6408111yescharacteristic
509C2C3×C4×C4⋊(C3⋊D3)11yescharacteristic
510C2C3×#6408411yescharacteristic
511C2C3×#6408511yescharacteristic
512C2C3×C4×#63811yescharacteristic
513C2C3×#6413811yescharacteristic
514C2C3×#6413911yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC23C22×C3×C3⋊D3
FittingC2×C33×C42C2
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
cosocleC23×C32C23×C3
hypercenterC2×C3×C42C3⋊D3
hypocenterC32C3×#388
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C2×C3×C42

Complements

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

NH ≅ G/Ntype
1. C2×C33×C42C2semidirect
5. C2×C3×#3750C2semidirect
6. C3×#63658C2semidirect
7. C3×#63658C2semidirect
8. #10592639C3direct (N × H)
20. C3×#3752C4semidirect
23. C2×C32×C42D3semidirect
24. C2×C32×C42C2×C3semidirect
28. C2×#3750C2×C3semidirect
33. #63658C2×C3semidirect
34. #63658C2×C3semidirect
87. #3752C3×C4semidirect
118. C2×C3×C42C3×D3semidirect
119. C2×C3×C42C3⋊D3semidirect
281. C2×C42C3×C3⋊D3semidirect
284. C33#388semidirect
433. C32#4940semidirect
434. C32C3×#388semidirect
505. C3C3×#4940semidirect
506. C3#10592639direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×C422
C2×C3×#37522
C2×C3×#37502
C3×#636582
#105926393
C3×#49403