Finite Group Databasebeta

← C3×#10596275 #13179555

Subgroups — C3×#10596275 #13179555

Order 1728.

3020 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C33×D4C211yesmaximal, characteristic, complemented
2C22×C3×#638C211yesmaximal, characteristic, complemented
3C3×#64162C212yesmaximal, complemented
4C2×C3×#3757C211yesmaximal, characteristic, complemented
5C3×#64148C214yesmaximal, complemented
6C3×#64146C212yesmaximal, complemented
7C3×#64144C212yesmaximal, complemented
8C3×#64138C212yesmaximal, complemented
9#10596275C311yesmaximal, characteristic, direct product factor
10C3×#5804—34nomaximal
11C24×C33C2211yescharacteristic
12C24×C33C2211yescharacteristic, complemented
13C23×C3×C3⋊D3C2211yescharacteristic, complemented
14C2×C33×D4—24no—
15C2×C33×D4C2212yes—
16C2×C33×D4C2212yescomplemented
17C22×C33×C4C2211yescharacteristic, complemented
18C2×C3×#638—24no—
19C2×C3×#638C2212yes—
20C2×C3×#638C2212yescomplemented
21C2×C3×#638C2214yescomplemented
22C22×C3×C3⋊Dic3C2211yescharacteristic, complemented
23C22×C3×C3⋊Dic3C2212yescomplemented
24C3×#3757C2212yescomplemented
25C3×#3757C2212yescomplemented
26C3×#3757C2214yescomplemented
27C3×#3752C2214yescomplemented
28C3×#3750C2214yescomplemented
29C22×C32×D4—24no—
30C22×C32×D4D314yescomplemented
31C22×C32×D4C2×C311yescharacteristic, complemented
32C22×#638C2×C311yescharacteristic, complemented
33C22×C3×#67—34no—
34#64162C2×C312yescomplemented
35C2×#3757C2×C311yescharacteristic, complemented
36#64148C2×C314yescomplemented
37#64146C2×C312yescomplemented
38#64144C2×C312yescomplemented
39#64138C2×C312yescomplemented
40C3×#953—38no—
41C2×C3×#217—34no—
42C3×#939—316no—
43C3×#937—38no—
44C3×#935—38no—
45C3×#929—38no—
46C23×C33—22no—
47C23×C33—42no—
48C23×C33—42no—
49C23×C33D414yescomplemented
50C23×C33C2312yes—
51C23×C33C2311yescharacteristic
52C23×C33C2311yescharacteristic
53C23×C33C2311yescharacteristic
54C22×C3×C3⋊D3—22no—
55C22×C3×C3⋊D3—42no—
56C22×C3×C3⋊D3C2312yes—
57C33×D4—24no—
58C33×D4—24no—
59C2×C33×C4—22no—
60C2×C33×C4C2312yes—
61C3×#638—24no—
62C3×#638—24no—
63C3×#638—44no—
64C2×C3×C3⋊Dic3—22no—
65C2×C3×C3⋊Dic3—42no—
66C2×C3×C3⋊Dic3C2312yes—
67C2×C3×C3⋊Dic3C2314yes—
68C3×#534—91no—
69#5804—34no—
70C24×C32—24no—
71C24×C32—24no—
72C24×C32C2×D314yes—
73C24×C32C2×D314yescomplemented
74C24×C32C22×C311yescharacteristic
75C24×C32C22×C311yescharacteristic, complemented
76C23×C3⋊D3C22×C311yescharacteristic, complemented
77C23×C3×D3—34no—
78C2×C32×D4—232no—
79C2×C32×D4—28no—
80C2×C32×D4—28no—
81C2×C32×D4—216no—
82C2×C32×D4—24no—
83C2×C32×D4C2×D318yes—
84C2×C32×D4C2×D318yescomplemented
85C2×C32×D4C22×C312yes—
86C2×C32×D4C22×C312yescomplemented
87C22×C32×C4—24no—
88C22×C32×C4C2×D314yescomplemented
89C22×C32×C4C22×C311yescharacteristic, complemented
90C2×#638—24no—
91C2×#638C22×C312yes—
92C2×#638C22×C312yescomplemented
93C2×#638C22×C314yescomplemented
94C22×C3⋊Dic3C22×C311yescharacteristic, complemented
95C22×C3⋊Dic3C22×C312yescomplemented
96C2×C3×#67—38no—
97C2×C3×#67—38no—
98C2×C3×#67—316no—
99C2×C3×#67—616no—
100C22×C3×Dic3—34no—
101C22×C3×Dic3—38no—
102#3757C22×C312yescomplemented
103#3757C22×C312yescomplemented
104#3757C22×C314yescomplemented
105#3752C22×C314yescomplemented
106#3750C22×C314yescomplemented
107C3×#217—38no—
108C3×#217—38no—
109C3×#217—316no—
110C3×(C2×D3)⋊C4—316no—
111C3×Dic3⋊C4—316no—
112C22×C33—22no—
113C22×C33—22no—
114C22×C33—22no—
115C22×C33—22no—
116C22×C33—24no—
117C22×C33—44no—
118C22×C33—44no—
119C22×C33C2×D412yes—
120C22×C33C2×D412yes—
121C22×C33C2×D412yes—
122C22×C33C2411yescharacteristic
123C2×C3×C3⋊D3—22no—
124C2×C3×C3⋊D3—24no—
125C2×C3×C3⋊D3—44no—
126C33×C4—22no—
127C3×C3⋊Dic3—22no—
128C3×C3⋊Dic3—24no—
129C22×C3×D4—24no—
130C22×C3×D4—91no—
131C22×C3×D4C3×D314yescomplemented
132C22×C3×D4C3⋊D311yescharacteristic, complemented
133C22×#67—34no—
134C3×#123—92no—
135C3×#123—92no—
136C3×#121—94no—
137C3×#120—92no—
138C3×#120—92no—
139C2×C3×C22⋊C4—91no—
140#953—38no—
141C2×#217—34no—
142#939—316no—
143#937—38no—
144#935—38no—
145#929—38no—
146C23×C32—216no—
147C23×C32—28no—
148C23×C32—22no—
149C23×C32—216no—
150C23×C32—24no—
151C23×C32—28no—
152C23×C32—24no—
153C23×C32—24no—
154C23×C32—48no—
155C23×C32—48no—
156C23×C32—42no—
157C23×C32—42no—
158C23×C32—416no—
159C23×C32—416no—
160C23×C32#67116yescomplemented
161C23×C32C3×D414yescomplemented
162C23×C32C22×D318yes—
163C23×C32C22×D314yes—
164C23×C32C22×D314yes—
165C23×C32C22×D314yes—
166C23×C32C23×C312yes—
167C23×C32C23×C311yescharacteristic
168C23×C32C23×C311yescharacteristic
169C23×C32C23×C311yescharacteristic
170C22×C3⋊D3—22no—
171C22×C3⋊D3—42no—
172C22×C3⋊D3C23×C312yes—
173C22×C3×D3—38no—
174C22×C3×D3—68no—
175C22×C3×D3—128no—
176C32×D4—232no—
177C32×D4—232no—
178C32×D4—216no—
179C32×D4—216no—
180C32×D4—24no—
181C32×D4—24no—
182C2×C32×C4—216no—
183C2×C32×C4—28no—
184C2×C32×C4—22no—
185C2×C32×C4—28no—
186C2×C32×C4C22×D318yes—
187C2×C32×C4C23×C312yes—
188#638—24no—
189#638—24no—
190#638—44no—
191C2×C3⋊Dic3—22no—
192C2×C3⋊Dic3—42no—
193C2×C3⋊Dic3C23×C312yes—
194C2×C3⋊Dic3C23×C314yes—
195C3×#67—616no—
196C3×#67—616no—
197C3×#67—1216no—
198C2×C3×Dic3—38no—
199C2×C3×Dic3—316no—
200C2×C3×Dic3—68no—
201C2×C3×Dic3—128no—
202#534—91noSylow
203C2×C33—22no—
204C2×C33—22no—
205C2×C33—22no—
206C2×C33—42no—
207C2×C33C22×D411yescharacteristic
208C2×C33H(5,𝔽2)12yes—
209C3×C3⋊D3—42no—
210C24×C3—24no—
211C24×C3—24no—
212C24×C3—91no—
213C24×C3C2×C3×D314yes—
214C24×C3C2×C3×D314yescomplemented
215C24×C3C2×C3⋊D311yescharacteristic
216C24×C3C2×C3⋊D311yescharacteristic, complemented
217C23×D3—34no—
218C2×C3×D4—232no—
219C2×C3×D4—216no—
220C2×C3×D4—24no—
221C2×C3×D4—28no—
222C2×C3×D4—28no—
223C2×C3×D4—92no—
224C2×C3×D4—92no—
225C2×C3×D4—94no—
226C2×C3×D4—184no—
227C2×C3×D4C2×C3×D318yes—
228C2×C3×D4C2×C3×D318yescomplemented
229C2×C3×D4C2×C3⋊D312yes—
230C2×C3×D4C2×C3⋊D312yescomplemented
231C22×C3×C4—24no—
232C22×C3×C4—91no—
233C22×C3×C4—92no—
234C22×C3×C4C2×C3×D314yescomplemented
235C22×C3×C4C2×C3⋊D311yescharacteristic, complemented
236C2×#67—38no—
237C2×#67—38no—
238C2×#67—316no—
239C2×#67—616no—
240C22×Dic3—34no—
241C22×Dic3—38no—
242C3×C4⋊C4—94no—
243C3×C22⋊C4—92no—
244C3×C22⋊C4—92no—
245C3×C22⋊C4—94no—
246C3×C22⋊C4—94no—
247#217—38no—
248#217—38no—
249#217—316no—
250(C2×D3)⋊C4—316no—
251Dic3⋊C4—316no—
252C22×C32—216no—
253C22×C32—28no—
254C22×C32—22no—
255C22×C32—22no—
256C22×C32—22no—
257C22×C32—216no—
258C22×C32—28no—
259C22×C32—22no—
260C22×C32—24no—
261C22×C32—24no—
262C22×C32—28no—
263C22×C32—28no—
264C22×C32—28no—
265C22×C32—28no—
266C22×C32—28no—
267C22×C32—216no—
268C22×C32—48no—
269C22×C32—48no—
270C22×C32—416no—
271C22×C32—416no—
272C22×C32—44no—
273C22×C32—432no—
274C22×C32—432no—
275C22×C32—416no—
276C22×C32—44no—
277C22×C32C2×#6718yes—
278C22×C32C2×#6718yes—
279C22×C32C2×#6718yes—
280C22×C32C2×C3×D412yes—
281C22×C32C2×C3×D412yes—
282C22×C32C2×C3×D412yes—
283C22×C32C23×D314yes—
284C22×C32C24×C311yescharacteristic
285C2×C3⋊D3—22no—
286C2×C3⋊D3—24no—
287C2×C3⋊D3—44no—
288C2×C3×D3—616no—
289C2×C3×D3—68no—
290C2×C3×D3—1216no—
291C32×C4—216no—
292C32×C4—28no—
293C32×C4—22no—
294C3⋊Dic3—22no—
295C3⋊Dic3—24no—
296C3×Dic3—616no—
297C3×Dic3—68no—
298C22×D4—91no—
299C22×D4C3×C3⋊D311yescharacteristic, complemented
300#123—92no—
301#123—92no—
302#121—94no—
303#120—92no—
304#120—92no—
305C2×C22⋊C4—91no—
306C33#53411yescharacteristic, complemented, Sylow
307C23×C3—216no—
308C23×C3—28no—
309C23×C3—22no—
310C23×C3—216no—
311C23×C3—24no—
312C23×C3—28no—
313C23×C3—24no—
314C23×C3—24no—
315C23×C3—48no—
316C23×C3—48no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—416no—
320C23×C3—416no—
321C23×C3—92no—
322C23×C3—182no—
323C23×C3—362no—
324C23×C3C3×#67116yescomplemented
325C23×C3#63814yescomplemented
326C23×C3C22×C3×D318yes—
327C23×C3C22×C3×D314yes—
328C23×C3C22×C3×D314yes—
329C23×C3C22×C3×D314yes—
330C23×C3C22×C3⋊D312yes—
331C23×C3C22×C3⋊D311yescharacteristic
332C23×C3C22×C3⋊D311yescharacteristic
333C23×C3C22×C3⋊D311yescharacteristic
334C22×D3—38no—
335C22×D3—68no—
336C22×D3—128no—
337C3×D4—232no—
338C3×D4—232no—
339C3×D4—216no—
340C3×D4—216no—
341C3×D4—24no—
342C3×D4—24no—
343C3×D4—184no—
344C3×D4—184no—
345C3×D4—364no—
346C2×C3×C4—216no—
347C2×C3×C4—28no—
348C2×C3×C4—22no—
349C2×C3×C4—28no—
350C2×C3×C4—94no—
351C2×C3×C4—92no—
352C2×C3×C4—182no—
353C2×C3×C4—362no—
354C2×C3×C4C22×C3×D318yes—
355C2×C3×C4C22×C3⋊D312yes—
356#67—616no—
357#67—616no—
358#67—1216no—
359C2×Dic3—38no—
360C2×Dic3—316no—
361C2×Dic3—68no—
362C2×Dic3—128no—
363C2×C32—216no—
364C2×C32—28no—
365C2×C32—22no—
366C2×C32—22no—
367C2×C32—22no—
368C2×C32—28no—
369C2×C32—24no—
370C2×C32—28no—
371C2×C32—28no—
372C2×C32—48no—
373C2×C32—48no—
374C2×C32—416no—
375C2×C32—48no—
376C2×C32—42no—
377C2×C32#100418yes—
378C2×C32C22×#6714yes—
379C2×C32C22×C3×D411yescharacteristic
380C2×C32C3×H(5,𝔽2)12yes—
381C3⋊D3—42no—
382C3×D3—128no—
383C24—91no—
384C24C2×C3×C3⋊D311yescharacteristic
385C24C2×C3×C3⋊D311yescharacteristic, complemented
386C2×D4—24no—
387C2×D4—92no—
388C2×D4—92no—
389C2×D4—94no—
390C2×D4—184no—
391C2×D4C2×C3×C3⋊D312yes—
392C2×D4C2×C3×C3⋊D312yescomplemented
393C22×C4—91no—
394C22×C4—92no—
395C22×C4C2×C3×C3⋊D311yescharacteristic, complemented
396C4⋊C4—94no—
397C22⋊C4—92no—
398C22⋊C4—92no—
399C22⋊C4—94no—
400C22⋊C4—94no—
401C22×C3—216no—
402C22×C3—28no—
403C22×C3—28no—
404C22×C3—28no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—216no—
409C22×C3—216no—
410C22×C3—28no—
411C22×C3—24no—
412C22×C3—22no—
413C22×C3—24no—
414C22×C3—28no—
415C22×C3—28no—
416C22×C3—28no—
417C22×C3—48no—
418C22×C3—48no—
419C22×C3—416no—
420C22×C3—416no—
421C22×C3—44no—
422C22×C3—432no—
423C22×C3—432no—
424C22×C3—416no—
425C22×C3—44no—
426C22×C3—184no—
427C22×C3—182no—
428C22×C3—364no—
429C22×C3C2×C3×#6718yes—
430C22×C3C2×C3×#6718yes—
431C22×C3C2×C3×#6718yes—
432C22×C3C2×#63812yes—
433C22×C3C2×#63812yes—
434C22×C3C2×#63812yes—
435C22×C3C23×C3×D314yes—
436C22×C3C23×C3⋊D311yescharacteristic
437C2×D3—616no—
438C2×D3—68no—
439C2×D3—1216no—
440C3×C4—216no—
441C3×C4—28no—
442C3×C4—22no—
443C3×C4—184no—
444C3×C4—182no—
445Dic3—616no—
446Dic3—68no—
447C32—24no—
448C32#580414yescomplemented
449C32C3×#53411yescharacteristic, complemented
450C23—22no—
451C23—42no—
452C23—42no—
453C23—92no—
454C23—182no—
455C23—362no—
456C23C3×#63814yescomplemented
457C23C22×C3×C3⋊D312yes—
458C23C22×C3×C3⋊D311yescharacteristic
459C23C22×C3×C3⋊D311yescharacteristic
460C23C22×C3×C3⋊D311yescharacteristic
461D4—24no—
462D4—24no—
463D4—184no—
464D4—184no—
465D4—364no—
466C2×C4—22no—
467C2×C4—94no—
468C2×C4—92no—
469C2×C4—182no—
470C2×C4—362no—
471C2×C4C22×C3×C3⋊D312yes—
472C2×C3—216no—
473C2×C3—28no—
474C2×C3—28no—
475C2×C3—28no—
476C2×C3—22no—
477C2×C3—22no—
478C2×C3—22no—
479C2×C3—28no—
480C2×C3—24no—
481C2×C3—48no—
482C2×C3—48no—
483C2×C3—416no—
484C2×C3—48no—
485C2×C3—42no—
486C2×C3—362no—
487C2×C3C3×#100418yes—
488C2×C3C22×C3×#6714yes—
489C2×C3#6436112yes—
490C2×C3C22×#63811yescharacteristic
491D3—128no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—24no—
496C22—22no—
497C22—44no—
498C22—44no—
499C22—184no—
500C22—182no—
501C22—364no—
502C22C2×C3×#63812yes—
503C22C2×C3×#63812yes—
504C22C2×C3×#63812yes—
505C22C23×C3×C3⋊D311yescharacteristic
506C4—22no—
507C4—184no—
508C4—182no—
509C3—24no—
510C3C3×#580414yescomplemented
511C3#1059627511yescharacteristic, direct product factor
512C2—22no—
513C2—22no—
514C2—22no—
515C2—42no—
516C2—362no—
517C2C3×#6436112yes—
518C2C22×C3×#63811yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC22×C33×D4C2
socleC22×C33C24
hypercenterC22×C3×D4C3⋊D3
hypocenterC32C3×#534
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×D4

Complements

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

NH ≅ G/Ntype
1. C22×C33×D4C2semidirect
2. C22×C3×#638C2semidirect
3. C3×#64162C2semidirect
4. C2×C3×#3757C2semidirect
5. C3×#64148C2semidirect
6. C3×#64146C2semidirect
7. C3×#64144C2semidirect
8. C3×#64138C2semidirect
9. #10596275C3direct (N × H)
12. C24×C33C22semidirect
13. C23×C3×C3⋊D3C22semidirect
16. C2×C33×D4C22semidirect
17. C22×C33×C4C22semidirect
20. C2×C3×#638C22semidirect
21. C2×C3×#638C22semidirect
22. C22×C3×C3⋊Dic3C22semidirect
23. C22×C3×C3⋊Dic3C22semidirect
24. C3×#3757C22semidirect
25. C3×#3757C22semidirect
26. C3×#3757C22semidirect
27. C3×#3752C22semidirect
28. C3×#3750C22semidirect
30. C22×C32×D4D3semidirect
31. C22×C32×D4C2×C3semidirect
32. C22×#638C2×C3semidirect
34. #64162C2×C3semidirect
35. C2×#3757C2×C3semidirect
36. #64148C2×C3semidirect
37. #64146C2×C3semidirect
38. #64144C2×C3semidirect
39. #64138C2×C3semidirect
49. C23×C33D4semidirect
73. C24×C32C2×D3semidirect
75. C24×C32C22×C3semidirect
76. C23×C3⋊D3C22×C3semidirect
84. C2×C32×D4C2×D3semidirect
86. C2×C32×D4C22×C3semidirect
88. C22×C32×C4C2×D3semidirect
89. C22×C32×C4C22×C3semidirect
92. C2×#638C22×C3semidirect
93. C2×#638C22×C3semidirect
94. C22×C3⋊Dic3C22×C3semidirect
95. C22×C3⋊Dic3C22×C3semidirect
102. #3757C22×C3semidirect
103. #3757C22×C3semidirect
104. #3757C22×C3semidirect
105. #3752C22×C3semidirect
106. #3750C22×C3semidirect
131. C22×C3×D4C3×D3semidirect
132. C22×C3×D4C3⋊D3semidirect
160. C23×C32#67semidirect
161. C23×C32C3×D4semidirect
214. C24×C3C2×C3×D3semidirect
216. C24×C3C2×C3⋊D3semidirect
228. C2×C3×D4C2×C3×D3semidirect
230. C2×C3×D4C2×C3⋊D3semidirect
234. C22×C3×C4C2×C3×D3semidirect
235. C22×C3×C4C2×C3⋊D3semidirect
299. C22×D4C3×C3⋊D3semidirect
306. C33#534semidirect
324. C23×C3C3×#67semidirect
325. C23×C3#638semidirect
385. C24C2×C3×C3⋊D3semidirect
392. C2×D4C2×C3×C3⋊D3semidirect
395. C22×C4C2×C3×C3⋊D3semidirect
448. C32#5804semidirect
449. C32C3×#534semidirect
456. C23C3×#638semidirect
510. C3C3×#5804semidirect
511. C3#10596275direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C33×D42
C22×C3×#6382
C3×#641622
C2×C3×#37572
C3×#641482
C3×#641462
C3×#641442
C3×#641382
#105962753
C3×#58043