Finite Group Databasebeta

← C3×#10596033 #13179313

Subgroups — C3×#10596033 #13179313

Order 1728.

1816 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×D4C211yesmaximal, characteristic, complemented
2C3×#64149C211yesmaximal, characteristic, complemented
3C3×#64147C211yesmaximal, characteristic, complemented
4C3×#64139C212yesmaximal, complemented
5C3×#64138C212yesmaximal, complemented
6C3×#64104C211yesmaximal, characteristic, complemented
7C3×#64099C211yesmaximal, characteristic, complemented
8C3×#64093C212yesmaximal, complemented
9C3×#64092C212yesmaximal, complemented
10C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
11C3×#64081C211yesmaximal, characteristic, complemented
12#10596033C311yesmaximal, characteristic, direct product factor
13C3×#5562—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C33×C4C2212yes—
16C2×C3×#638C2212yes—
17C2×C3×#638C2212yes—
18C2×C3×#638C2212yes—
19C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
20C2×C3×C4⋊(C3⋊D3)C2212yes—
21C2×C3×C4×C3⋊D3C2211yescharacteristic
22C2×C3×C4×C3⋊D3C2212yes—
23C33×C4⋊C4C2211yescharacteristic
24C33×C22⋊C4C2212yes—
25C33×C42C2211yescharacteristic
26C3×#3757C2212yes—
27C3×#3752C2211yescharacteristic
28C3×#3752C2211yescharacteristic
29C3×#3752C2212yes—
30C3×#3752C2212yes—
31C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
32C3×C4⋊(C3⋊Dic3)C2212yes—
33C3×#3750C2211yescharacteristic
34C3×#3750C2211yescharacteristic
35C3×#3750C2212yes—
36C3×C4×C3⋊Dic3C2211yescharacteristic
37C32×C4×D4—24no—
38C32×C4×D4D314yescomplemented
39C32×C4×D4C2×C311yescharacteristic, complemented
40#64149C2×C311yescharacteristic, complemented
41#64147C2×C311yescharacteristic, complemented
42#64139C2×C312yescomplemented
43#64138C2×C312yescomplemented
44#64104C2×C311yescharacteristic, complemented
45#64099C2×C311yescharacteristic, complemented
46#64093C2×C312yescomplemented
47#64092C2×C312yescomplemented
48C4×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
49#64081C2×C311yescharacteristic, complemented
50C3×#940—34no—
51C3×#938—34no—
52C3×#930—38no—
53C3×#929—38no—
54C3×#895—34no—
55C3×#890—34no—
56C3×#884—38no—
57C3×#883—38no—
58C3×C4×D12—34no—
59C3×#872—34no—
60C23×C33C2312yes—
61C22×C3×C3⋊D3C2311yescharacteristic
62C22×C3×C3⋊D3C2312yes—
63C33×D4—22no—
64C2×C33×C4—42no—
65C2×C33×C4C2312yes—
66C2×C33×C4C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68C2×C33×C4C2311yescharacteristic
69C3×#638—42no—
70C3×#638—42no—
71C3×#638—42no—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C3×C3⋊Dic3C2312yes—
76C3×C4⋊(C3⋊D3)—22no—
77C3×C4⋊(C3⋊D3)—42no—
78C3×C4×C3⋊D3—22no—
79C3×C4×C3⋊D3—42no—
80C3×#540—91no—
81#5562—34no—
82C2×C32×D4—24no—
83C2×C32×D4C2×D314yes—
84C2×C32×D4C22×C311yescharacteristic
85C22×C32×C4—28no—
86C22×C32×C4C2×D318yes—
87C22×C32×C4C22×C312yes—
88C2×#638C22×C312yes—
89C2×#638C22×C312yes—
90C2×#638C22×C312yes—
91C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
92C2×C4⋊(C3⋊D3)C22×C312yes—
93C2×C4×C3⋊D3C22×C311yescharacteristic
94C2×C4×C3⋊D3C22×C312yes—
95C2×C3×#67—38no—
96C2×C3×#67—38no—
97C2×C3×#67—38no—
98C2×C3×D12—34no—
99C2×C3×D12—38no—
100C2×C3×C4×D3—34no—
101C2×C3×C4×D3—38no—
102C32×C4⋊C4—24no—
103C32×C4⋊C4C2×D314yes—
104C32×C4⋊C4C22×C311yescharacteristic
105C32×C22⋊C4—28no—
106C32×C22⋊C4C2×D318yes—
107C32×C22⋊C4C22×C312yes—
108C32×C42—24no—
109C32×C42C2×D314yes—
110C32×C42C22×C311yescharacteristic
111#3757C22×C312yes—
112#3752C22×C311yescharacteristic
113#3752C22×C311yescharacteristic
114#3752C22×C312yes—
115#3752C22×C312yes—
116C4⋊(C3⋊Dic3)C22×C311yescharacteristic
117C4⋊(C3⋊Dic3)C22×C312yes—
118#3750C22×C311yescharacteristic
119#3750C22×C311yescharacteristic
120#3750C22×C312yes—
121C4×C3⋊Dic3C22×C311yescharacteristic
122C3×#217—38no—
123C3×(C2×D3)⋊C4—38no—
124C3×(C2×D3)⋊C4—34no—
125C3×(C2×D3)⋊C4—38no—
126C3×(C2×D3)⋊C4—34no—
127C3×C4⋊Dic3—38no—
128C3×C4⋊Dic3—34no—
129C3×Dic3⋊C4—34no—
130C3×Dic3⋊C4—38no—
131C3×Dic3⋊C4—34no—
132C3×C4×Dic3—34no—
133C22×C33—22no—
134C22×C33—22no—
135C22×C33—22no—
136C22×C33C2411yescharacteristic
137C2×C3×C3⋊D3—21no—
138C2×C3×C3⋊D3—22no—
139C2×C3×C3⋊D3—22no—
140C2×C3×C3⋊D3—21no—
141C2×C3×C3⋊D3—22no—
142C2×C3×C3⋊D3—21no—
143C33×C4—21no—
144C33×C4—21no—
145C33×C4—22no—
146C33×C4#4112yes—
147C3×C3⋊Dic3—21no—
148C3×C3⋊Dic3—22no—
149C3×C3⋊Dic3—21no—
150C3×C3⋊Dic3—21no—
151C3×#127—91no—
152C3×#125—91no—
153C3×#123—92no—
154C3×#123—92no—
155C3×#121—91no—
156C3×#121—92no—
157C3×#121—91no—
158C3×#121—92no—
159C3×C4×D4—24no—
160C3×C4×D4—91no—
161C3×C4×D4C3×D314yescomplemented
162C3×C4×D4C3⋊D311yescharacteristic, complemented
163C3×#117—91no—
164#940—34no—
165#938—34no—
166#930—38no—
167#929—38no—
168#895—34no—
169#890—34no—
170#884—38no—
171#883—38no—
172C4×D12—34no—
173#872—34no—
174C23×C32—28no—
175C23×C32C22×D318yes—
176C23×C32C23×C312yes—
177C22×C3⋊D3C23×C311yescharacteristic
178C22×C3⋊D3C23×C312yes—
179C22×C3×D3—34no—
180C22×C3×D3—38no—
181C32×D4—216no—
182C32×D4—28no—
183C32×D4—22no—
184C2×C32×C4—24no—
185C2×C32×C4—24no—
186C2×C32×C4—24no—
187C2×C32×C4—28no—
188C2×C32×C4—48no—
189C2×C32×C4—42no—
190C2×C32×C4—416no—
191C2×C32×C4C22×D318yes—
192C2×C32×C4C22×D314yes—
193C2×C32×C4C22×D314yes—
194C2×C32×C4C22×D314yes—
195C2×C32×C4C23×C312yes—
196C2×C32×C4C23×C311yescharacteristic
197C2×C32×C4C23×C311yescharacteristic
198C2×C32×C4C23×C311yescharacteristic
199#638—42no—
200#638—42no—
201#638—42no—
202C2×C3⋊Dic3C23×C311yescharacteristic
203C2×C3⋊Dic3C23×C311yescharacteristic
204C2×C3⋊Dic3C23×C311yescharacteristic
205C2×C3⋊Dic3C23×C312yes—
206C4⋊(C3⋊D3)—22no—
207C4⋊(C3⋊D3)—42no—
208C4×C3⋊D3—22no—
209C4×C3⋊D3—42no—
210C3×#67—128no—
211C3×#67—128no—
212C3×#67—128no—
213C2×C3×Dic3—34no—
214C2×C3×Dic3—34no—
215C2×C3×Dic3—34no—
216C2×C3×Dic3—38no—
217C3×D12—68no—
218C3×D12—128no—
219C3×C4×D3—68no—
220C3×C4×D3—128no—
221#540—91noSylow
222C2×C33—42no—
223C2×C33C2×#4111yescharacteristic
224C2×C33H(5,𝔽2)11yescharacteristic
225C2×C33H(5,𝔽2)11yescharacteristic
226C3×C3⋊D3—41no—
227C3×C3⋊D3—42no—
228C2×C3×D4—24no—
229C2×C3×D4—92no—
230C2×C3×D4—92no—
231C2×C3×D4—92no—
232C2×C3×D4—91no—
233C2×C3×D4—92no—
234C2×C3×D4C2×C3×D314yes—
235C2×C3×D4C2×C3⋊D311yescharacteristic
236C22×C3×C4—28no—
237C22×C3×C4—91no—
238C22×C3×C4—92no—
239C22×C3×C4C2×C3×D318yes—
240C22×C3×C4C2×C3⋊D312yes—
241C2×#67—38no—
242C2×#67—38no—
243C2×#67—38no—
244C2×D12—34no—
245C2×D12—38no—
246C2×C4×D3—34no—
247C2×C4×D3—38no—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—91no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4—92no—
252C3×C4⋊C4—91no—
253C3×C4⋊C4—92no—
254C3×C4⋊C4C2×C3×D314yes—
255C3×C4⋊C4C2×C3⋊D311yescharacteristic
256C3×C22⋊C4—28no—
257C3×C22⋊C4—92no—
258C3×C22⋊C4—91no—
259C3×C22⋊C4—91no—
260C3×C22⋊C4—92no—
261C3×C22⋊C4—92no—
262C3×C22⋊C4C2×C3×D318yes—
263C3×C22⋊C4C2×C3⋊D312yes—
264C3×C42—24no—
265C3×C42—91no—
266C3×C42C2×C3×D314yes—
267C3×C42C2×C3⋊D311yescharacteristic
268#217—38no—
269(C2×D3)⋊C4—34no—
270(C2×D3)⋊C4—34no—
271(C2×D3)⋊C4—38no—
272(C2×D3)⋊C4—38no—
273C4⋊Dic3—34no—
274C4⋊Dic3—38no—
275Dic3⋊C4—34no—
276Dic3⋊C4—34no—
277Dic3⋊C4—38no—
278C4×Dic3—34no—
279C22×C32—216no—
280C22×C32—28no—
281C22×C32—22no—
282C22×C32—22no—
283C22×C32—22no—
284C22×C32—24no—
285C22×C32—28no—
286C22×C32—28no—
287C22×C32—48no—
288C22×C32—48no—
289C22×C32C23×D314yes—
290C22×C32C24×C311yescharacteristic
291C2×C3⋊D3—21no—
292C2×C3⋊D3—22no—
293C2×C3⋊D3—22no—
294C2×C3⋊D3—21no—
295C2×C3⋊D3—22no—
296C2×C3⋊D3—21no—
297C2×C3×D3—64no—
298C2×C3×D3—68no—
299C2×C3×D3—64no—
300C2×C3×D3—68no—
301C2×C3×D3—68no—
302C2×C3×D3—64no—
303C32×C4—28no—
304C32×C4—24no—
305C32×C4—21no—
306C32×C4—22no—
307C32×C4—21no—
308C32×C4—28no—
309C32×C4—28no—
310C32×C4—24no—
311C32×C4—44no—
312C32×C4—48no—
313C32×C4#23518yes—
314C32×C4C3×#4112yes—
315C3⋊Dic3—21no—
316C3⋊Dic3—22no—
317C3⋊Dic3—21no—
318C3⋊Dic3—21no—
319C3×Dic3—64no—
320C3×Dic3—64no—
321C3×Dic3—68no—
322C3×Dic3—64no—
323#127—91no—
324#125—91no—
325#123—92no—
326#123—92no—
327#121—91no—
328#121—92no—
329#121—91no—
330#121—92no—
331C4×D4—91no—
332C4×D4C3×C3⋊D311yescharacteristic, complemented
333#117—91no—
334C33#54011yescharacteristic, complemented, Sylow
335C23×C3—28no—
336C23×C3—91no—
337C23×C3—92no—
338C23×C3C22×C3×D318yes—
339C23×C3C22×C3⋊D312yes—
340C22×D3—34no—
341C22×D3—38no—
342C3×D4—216no—
343C3×D4—28no—
344C3×D4—22no—
345C3×D4—182no—
346C3×D4—362no—
347C3×D4—362no—
348C3×D4—362no—
349C3×D4—362no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—28no—
354C2×C3×C4—48no—
355C2×C3×C4—42no—
356C2×C3×C4—416no—
357C2×C3×C4—91no—
358C2×C3×C4—91no—
359C2×C3×C4—91no—
360C2×C3×C4—92no—
361C2×C3×C4—182no—
362C2×C3×C4—362no—
363C2×C3×C4C22×C3×D318yes—
364C2×C3×C4C22×C3×D314yes—
365C2×C3×C4C22×C3×D314yes—
366C2×C3×C4C22×C3×D314yes—
367C2×C3×C4C22×C3⋊D311yescharacteristic
368C2×C3×C4C22×C3⋊D311yescharacteristic
369C2×C3×C4C22×C3⋊D312yes—
370C2×C3×C4C22×C3⋊D311yescharacteristic
371#67—128no—
372#67—128no—
373#67—128no—
374C2×Dic3—34no—
375C2×Dic3—34no—
376C2×Dic3—34no—
377C2×Dic3—38no—
378D12—68no—
379D12—128no—
380C4×D3—68no—
381C4×D3—128no—
382C2×C32—24no—
383C2×C32—24no—
384C2×C32—24no—
385C2×C32—416no—
386C2×C32—48no—
387C2×C32—42no—
388C2×C32C2×#23514yes—
389C2×C32#100414yes—
390C2×C32#100914yes—
391C2×C32C2×C3×#4111yescharacteristic
392C2×C32C3×H(5,𝔽2)11yescharacteristic
393C2×C32C3×H(5,𝔽2)11yescharacteristic
394C3⋊D3—41no—
395C3⋊D3—42no—
396C3×D3—124no—
397C3×D3—128no—
398C2×D4—92no—
399C2×D4—92no—
400C2×D4—92no—
401C2×D4—91no—
402C2×D4—92no—
403C2×D4C2×C3×C3⋊D311yescharacteristic
404C22×C4—91no—
405C22×C4—92no—
406C22×C4C2×C3×C3⋊D312yes—
407C4⋊C4—91no—
408C4⋊C4—91no—
409C4⋊C4—91no—
410C4⋊C4—92no—
411C4⋊C4—92no—
412C4⋊C4C2×C3×C3⋊D311yescharacteristic
413C22⋊C4—92no—
414C22⋊C4—91no—
415C22⋊C4—91no—
416C22⋊C4—92no—
417C22⋊C4—92no—
418C22⋊C4C2×C3×C3⋊D312yes—
419C42—91no—
420C42C2×C3×C3⋊D311yescharacteristic
421C22×C3—216no—
422C22×C3—28no—
423C22×C3—28no—
424C22×C3—28no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—24no—
429C22×C3—48no—
430C22×C3—48no—
431C22×C3—181no—
432C22×C3—182no—
433C22×C3—181no—
434C22×C3—182no—
435C22×C3—182no—
436C22×C3—181no—
437C22×C3C23×C3×D314yes—
438C22×C3C23×C3⋊D311yescharacteristic
439C2×D3—64no—
440C2×D3—68no—
441C2×D3—64no—
442C2×D3—68no—
443C2×D3—68no—
444C2×D3—64no—
445C3×C4—28no—
446C3×C4—28no—
447C3×C4—24no—
448C3×C4—24no—
449C3×C4—21no—
450C3×C4—22no—
451C3×C4—21no—
452C3×C4—28no—
453C3×C4—48no—
454C3×C4—44no—
455C3×C4—181no—
456C3×C4—181no—
457C3×C4—182no—
458C3×C4—181no—
459C3×C4C3×#23518yes—
460C3×C4#382812yes—
461Dic3—64no—
462Dic3—64no—
463Dic3—68no—
464Dic3—64no—
465C32—24no—
466C32#556214yescomplemented
467C32C3×#54011yescharacteristic, complemented
468C23—91no—
469C23—92no—
470C23C22×C3×C3⋊D312yes—
471D4—22no—
472D4—182no—
473D4—362no—
474D4—362no—
475D4—362no—
476D4—362no—
477C2×C4—42no—
478C2×C4—91no—
479C2×C4—91no—
480C2×C4—91no—
481C2×C4—92no—
482C2×C4—182no—
483C2×C4—362no—
484C2×C4C22×C3×C3⋊D311yescharacteristic
485C2×C4C22×C3×C3⋊D311yescharacteristic
486C2×C4C22×C3×C3⋊D312yes—
487C2×C4C22×C3×C3⋊D311yescharacteristic
488C2×C3—24no—
489C2×C3—24no—
490C2×C3—24no—
491C2×C3—416no—
492C2×C3—48no—
493C2×C3—42no—
494C2×C3—361no—
495C2×C3—362no—
496C2×C3C2×C3×#23514yes—
497C2×C3C3×#100414yes—
498C2×C3C3×#100914yes—
499C2×C3C2×#382811yescharacteristic
500C2×C3#6436111yescharacteristic
501C2×C3#6436611yescharacteristic
502D3—124no—
503D3—128no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—181no—
508C22—182no—
509C22—181no—
510C22—182no—
511C22—182no—
512C22—181no—
513C22C23×C3×C3⋊D311yescharacteristic
514C4—21no—
515C4—22no—
516C4—21no—
517C4—181no—
518C4—181no—
519C4—182no—
520C4—181no—
521C4C3×#382812yes—
522C3—24no—
523C3C3×#556214yescomplemented
524C3#1059603311yescharacteristic, direct product factor
525C2—42no—
526C2—361no—
527C2—362no—
528C2C2×C3×#382811yescharacteristic
529C2C3×#6436111yescharacteristic
530C2C3×#6436611yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4×D4C2
socleC22×C33C24
hypercenterC3×C4×D4C3⋊D3
hypocenterC32C3×#540
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
1. C33×C4×D4C2semidirect
2. C3×#64149C2semidirect
3. C3×#64147C2semidirect
4. C3×#64139C2semidirect
5. C3×#64138C2semidirect
6. C3×#64104C2semidirect
7. C3×#64099C2semidirect
8. C3×#64093C2semidirect
9. C3×#64092C2semidirect
10. C3×C4×C4⋊(C3⋊D3)C2semidirect
11. C3×#64081C2semidirect
12. #10596033C3direct (N × H)
38. C32×C4×D4D3semidirect
39. C32×C4×D4C2×C3semidirect
40. #64149C2×C3semidirect
41. #64147C2×C3semidirect
42. #64139C2×C3semidirect
43. #64138C2×C3semidirect
44. #64104C2×C3semidirect
45. #64099C2×C3semidirect
46. #64093C2×C3semidirect
47. #64092C2×C3semidirect
48. C4×C4⋊(C3⋊D3)C2×C3semidirect
49. #64081C2×C3semidirect
161. C3×C4×D4C3×D3semidirect
162. C3×C4×D4C3⋊D3semidirect
332. C4×D4C3×C3⋊D3semidirect
334. C33#540semidirect
466. C32#5562semidirect
467. C32C3×#540semidirect
523. C3C3×#5562semidirect
524. C3#10596033direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4×D42
C3×#641492
C3×#641472
C3×#641392
C3×#641382
C3×#641042
C3×#640992
C3×#640932
C3×#640922
C3×C4×C4⋊(C3⋊D3)2
C3×#640812
#105960333
C3×#55623