Finite Group Databasebeta

← C3×#10596001 #13179284

Subgroups — C3×#10596001 #13179284

Order 1728.

1732 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×#117C211yesmaximal, characteristic, complemented
2C3×#64139C211yesmaximal, characteristic, complemented
3C3×C4×#638C211yesmaximal, characteristic, complemented
4C3×#64104C211yesmaximal, characteristic, complemented
5C3×#64102C211yesmaximal, characteristic, complemented
6C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
7C3×#64099C211yesmaximal, characteristic, complemented
8C3×#64093C212yesmaximal, complemented
9C3×#64092C212yesmaximal, complemented
10C3×#64085C212yesmaximal, complemented
11C3×C4×C4⋊(C3⋊D3)C212yesmaximal, complemented
12#10596001C311yesmaximal, characteristic, direct product factor
13C3×#5530—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C2×C3×#638C2211yescharacteristic
16C2×C3×#638C2212yes—
17C2×C3×C4⋊(C3⋊D3)C2212yes—
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
19C2×C3×C4×C3⋊D3C2211yescharacteristic
20C2×C3×C4×C3⋊D3C2212yes—
21C2×C3×C4×C3⋊D3C2212yes—
22C33×C4⋊C4C2211yescharacteristic
23C33×C4⋊C4C2211yescharacteristic
24C33×C22⋊C4C2212yes—
25C33×C42C2212yes—
26C3×#3757C2211yescharacteristic
27C3×#3752C2211yescharacteristic
28C3×#3752C2212yes—
29C3×#3752C2212yes—
30C3×#3752C2212yes—
31C3×C4⋊(C3⋊Dic3)C2212yes—
32C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
33C3×#3750C2211yescharacteristic
34C3×#3750C2212yes—
35C3×#3750C2212yes—
36C3×C4×C3⋊Dic3C2211yescharacteristic
37C32×#117—24no—
38C32×#117D314yescomplemented
39C32×#117C2×C311yescharacteristic, complemented
40#64139C2×C311yescharacteristic, complemented
41C4×#638C2×C311yescharacteristic, complemented
42#64104C2×C311yescharacteristic, complemented
43#64102C2×C311yescharacteristic, complemented
44C4⋊C4×C3⋊D3C2×C311yescharacteristic, complemented
45#64099C2×C311yescharacteristic, complemented
46#64093C2×C312yescomplemented
47#64092C2×C312yescomplemented
48#64085C2×C312yescomplemented
49C4×C4⋊(C3⋊D3)C2×C312yescomplemented
50C3×#930—34no—
51C3×C4×#67—34no—
52C3×#895—34no—
53C3×#893—34no—
54C3×D3×C4⋊C4—34no—
55C3×#890—34no—
56C3×#884—38no—
57C3×#883—38no—
58C3×#876—38no—
59C3×C4×D12—38no—
60C23×C33C2311yescharacteristic
61C22×C3×C3⋊D3C2312yes—
62C22×C3×C3⋊D3C2311yescharacteristic
63C2×C33×C4—22no—
64C2×C33×C4C2312yes—
65C2×C33×C4C2311yescharacteristic
66C2×C33×C4C2312yes—
67C2×C33×C4C2311yescharacteristic
68C3×#638—22no—
69C3×#638—42no—
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2312yes—
74C3×C4⋊(C3⋊D3)—24no—
75C3×C4⋊(C3⋊D3)—22no—
76C3×C4×C3⋊D3—22no—
77C3×C4×C3⋊D3—42no—
78C3×C4×C3⋊D3—42no—
79C3×#553—91no—
80#5530—34no—
81C22×C32×C4—24no—
82C22×C32×C4C2×D314yes—
83C22×C32×C4C22×C311yescharacteristic
84C2×#638C22×C311yescharacteristic
85C2×#638C22×C312yes—
86C2×C4⋊(C3⋊D3)C22×C312yes—
87C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
88C2×C4×C3⋊D3C22×C311yescharacteristic
89C2×C4×C3⋊D3C22×C312yes—
90C2×C4×C3⋊D3C22×C312yes—
91C2×C3×#67—34no—
92C2×C3×#67—38no—
93C2×C3×D12—38no—
94C2×C3×D12—34no—
95C2×C3×C4×D3—34no—
96C2×C3×C4×D3—38no—
97C2×C3×C4×D3—38no—
98C32×C4⋊C4—24no—
99C32×C4⋊C4—24no—
100C32×C4⋊C4C2×D314yes—
101C32×C4⋊C4C2×D314yes—
102C32×C4⋊C4C22×C311yescharacteristic
103C32×C4⋊C4C22×C311yescharacteristic
104C32×C22⋊C4—28no—
105C32×C22⋊C4C2×D318yes—
106C32×C22⋊C4C22×C312yes—
107C32×C42—28no—
108C32×C42C2×D318yes—
109C32×C42C22×C312yes—
110#3757C22×C311yescharacteristic
111#3752C22×C311yescharacteristic
112#3752C22×C312yes—
113#3752C22×C312yes—
114#3752C22×C312yes—
115C4⋊(C3⋊Dic3)C22×C312yes—
116C4⋊(C3⋊Dic3)C22×C311yescharacteristic
117#3750C22×C311yescharacteristic
118#3750C22×C312yes—
119#3750C22×C312yes—
120C4×C3⋊Dic3C22×C311yescharacteristic
121C3×#217—34no—
122C3×(C2×D3)⋊C4—38no—
123C3×(C2×D3)⋊C4—38no—
124C3×(C2×D3)⋊C4—34no—
125C3×(C2×D3)⋊C4—38no—
126C3×C4⋊Dic3—38no—
127C3×C4⋊Dic3—34no—
128C3×Dic3⋊C4—38no—
129C3×Dic3⋊C4—38no—
130C3×Dic3⋊C4—34no—
131C3×C4×Dic3—34no—
132C22×C33—21no—
133C22×C33—21no—
134C22×C33—21no—
135C22×C33C2411yescharacteristic
136C2×C3×C3⋊D3—21no—
137C2×C3×C3⋊D3—21no—
138C2×C3×C3⋊D3—22no—
139C2×C3×C3⋊D3—22no—
140C2×C3×C3⋊D3—22no—
141C2×C3×C3⋊D3#4112yes—
142C33×C4—21no—
143C33×C4—22no—
144C33×C4—22no—
145C33×C4#4112yes—
146C3×C3⋊Dic3—21no—
147C3×C3⋊Dic3—21no—
148C3×C3⋊Dic3—21no—
149C3×C3⋊Dic3—22no—
150C3×#126—92no—
151C3×#125—91no—
152C3×#123—92no—
153C3×#121—91no—
154C3×#121—91no—
155C3×#121—92no—
156C3×C4×D4—91no—
157C3×C4×D4—91no—
158C3×C4×D4—92no—
159C3×#117—24no—
160C3×#117C3×D314yescomplemented
161C3×#117C3⋊D311yescharacteristic, complemented
162C2×C3×C4⋊C4—91no—
163#930—34no—
164C4×#67—34no—
165#895—34no—
166#893—34no—
167D3×C4⋊C4—34no—
168#890—34no—
169#884—38no—
170#883—38no—
171#876—38no—
172C4×D12—38no—
173C23×C32—24no—
174C23×C32C22×D314yes—
175C23×C32C23×C311yescharacteristic
176C22×C3⋊D3C23×C312yes—
177C22×C3⋊D3C23×C311yescharacteristic
178C22×C3×D3—34no—
179C22×C3×D3—38no—
180C2×C32×C4—216no—
181C2×C32×C4—28no—
182C2×C32×C4—22no—
183C2×C32×C4—24no—
184C2×C32×C4—24no—
185C2×C32×C4—28no—
186C2×C32×C4—28no—
187C2×C32×C4C22×D318yes—
188C2×C32×C4C22×D314yes—
189C2×C32×C4C22×D318yes—
190C2×C32×C4C22×D314yes—
191C2×C32×C4C23×C312yes—
192C2×C32×C4C23×C311yescharacteristic
193C2×C32×C4C23×C312yes—
194C2×C32×C4C23×C311yescharacteristic
195#638—22no—
196#638—42no—
197C2×C3⋊Dic3C23×C311yescharacteristic
198C2×C3⋊Dic3C23×C311yescharacteristic
199C2×C3⋊Dic3C23×C311yescharacteristic
200C2×C3⋊Dic3C23×C312yes—
201C4⋊(C3⋊D3)—24no—
202C4⋊(C3⋊D3)—22no—
203C4×C3⋊D3—22no—
204C4×C3⋊D3—42no—
205C4×C3⋊D3—42no—
206C3×#67—68no—
207C3×#67—128no—
208C2×C3×Dic3—34no—
209C2×C3×Dic3—34no—
210C2×C3×Dic3—34no—
211C2×C3×Dic3—38no—
212C3×D12—616no—
213C3×D12—68no—
214C3×C4×D3—68no—
215C3×C4×D3—128no—
216C3×C4×D3—128no—
217#553—91noSylow
218C2×C33—41no—
219C2×C33C2×#4111yescharacteristic
220C2×C33C2×#4111yescharacteristic
221C2×C33H(5,𝔽2)11yescharacteristic
222C3×C3⋊D3—22no—
223C3×C3⋊D3—42no—
224C2×C3×D4—91no—
225C2×C3×D4—92no—
226C2×C3×D4—92no—
227C2×C3×D4—91no—
228C22×C3×C4—24no—
229C22×C3×C4—91no—
230C22×C3×C4—92no—
231C22×C3×C4—92no—
232C22×C3×C4C2×C3×D314yes—
233C22×C3×C4C2×C3⋊D311yescharacteristic
234C2×#67—34no—
235C2×#67—38no—
236C2×D12—38no—
237C2×D12—34no—
238C2×C4×D3—34no—
239C2×C4×D3—38no—
240C2×C4×D3—38no—
241C3×C4⋊C4—24no—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—92no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—92no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—92no—
248C3×C4⋊C4C2×C3×D314yes—
249C3×C4⋊C4C2×C3×D314yes—
250C3×C4⋊C4C2×C3⋊D311yescharacteristic
251C3×C4⋊C4C2×C3⋊D311yescharacteristic
252C3×C22⋊C4—28no—
253C3×C22⋊C4—91no—
254C3×C22⋊C4—91no—
255C3×C22⋊C4—92no—
256C3×C22⋊C4—92no—
257C3×C22⋊C4—92no—
258C3×C22⋊C4C2×C3×D318yes—
259C3×C22⋊C4C2×C3⋊D312yes—
260C3×C42—28no—
261C3×C42—91no—
262C3×C42C2×C3×D318yes—
263C3×C42C2×C3⋊D312yes—
264#217—34no—
265(C2×D3)⋊C4—34no—
266(C2×D3)⋊C4—38no—
267(C2×D3)⋊C4—38no—
268(C2×D3)⋊C4—38no—
269C4⋊Dic3—38no—
270C4⋊Dic3—34no—
271Dic3⋊C4—34no—
272Dic3⋊C4—38no—
273Dic3⋊C4—38no—
274C4×Dic3—34no—
275C22×C32—28no—
276C22×C32—24no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—24no—
281C22×C32—24no—
282C22×C32—24no—
283C22×C32—44no—
284C22×C32—44no—
285C22×C32C23×D314yes—
286C22×C32C24×C311yescharacteristic
287C2×C3⋊D3—21no—
288C2×C3⋊D3—21no—
289C2×C3⋊D3—22no—
290C2×C3⋊D3—22no—
291C2×C3⋊D3—22no—
292C2×C3⋊D3C3×#4112yes—
293C2×C3×D3—38no—
294C2×C3×D3—64no—
295C2×C3×D3—64no—
296C2×C3×D3—68no—
297C2×C3×D3—68no—
298C2×C3×D3—68no—
299C32×C4—28no—
300C32×C4—24no—
301C32×C4—22no—
302C32×C4—21no—
303C32×C4—22no—
304C32×C4—28no—
305C32×C4—28no—
306C32×C4—28no—
307C32×C4—48no—
308C32×C4—48no—
309C32×C4#23518yes—
310C32×C4C3×#4112yes—
311C3⋊Dic3—21no—
312C3⋊Dic3—21no—
313C3⋊Dic3—21no—
314C3⋊Dic3—22no—
315C3×Dic3—64no—
316C3×Dic3—64no—
317C3×Dic3—64no—
318C3×Dic3—68no—
319#126—92no—
320#125—91no—
321#123—92no—
322#121—91no—
323#121—91no—
324#121—92no—
325C4×D4—91no—
326C4×D4—91no—
327C4×D4—92no—
328#117C3×C3⋊D311yescharacteristic, complemented
329C2×C4⋊C4—91no—
330C33#55311yescharacteristic, complemented, Sylow
331C23×C3—24no—
332C23×C3—91no—
333C23×C3—92no—
334C23×C3C22×C3×D314yes—
335C23×C3C22×C3⋊D311yescharacteristic
336C22×D3—34no—
337C22×D3—38no—
338C3×D4—184no—
339C3×D4—182no—
340C3×D4—182no—
341C3×D4—362no—
342C2×C3×C4—216no—
343C2×C3×C4—28no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—28no—
348C2×C3×C4—28no—
349C2×C3×C4—91no—
350C2×C3×C4—91no—
351C2×C3×C4—91no—
352C2×C3×C4—92no—
353C2×C3×C4—182no—
354C2×C3×C4—362no—
355C2×C3×C4—362no—
356C2×C3×C4C22×C3×D318yes—
357C2×C3×C4C22×C3×D314yes—
358C2×C3×C4C22×C3×D318yes—
359C2×C3×C4C22×C3×D314yes—
360C2×C3×C4C22×C3⋊D311yescharacteristic
361C2×C3×C4C22×C3⋊D312yes—
362C2×C3×C4C22×C3⋊D312yes—
363C2×C3×C4C22×C3⋊D311yescharacteristic
364#67—68no—
365#67—128no—
366C2×Dic3—34no—
367C2×Dic3—34no—
368C2×Dic3—34no—
369C2×Dic3—38no—
370D12—616no—
371D12—68no—
372C4×D3—68no—
373C4×D3—128no—
374C4×D3—128no—
375C2×C32—24no—
376C2×C32—24no—
377C2×C32—24no—
378C2×C32—48no—
379C2×C32—44no—
380C2×C32—41no—
381C2×C32C2×#23514yes—
382C2×C32D3×#4114yes—
383C2×C32#100914yes—
384C2×C32C2×C3×#4111yescharacteristic
385C2×C32C2×C3×#4111yescharacteristic
386C2×C32C3×H(5,𝔽2)11yescharacteristic
387C3⋊D3—22no—
388C3⋊D3—42no—
389C3×D3—68no—
390C3×D3—128no—
391C2×D4—91no—
392C2×D4—92no—
393C2×D4—92no—
394C2×D4—91no—
395C22×C4—91no—
396C22×C4—92no—
397C22×C4—92no—
398C22×C4C2×C3×C3⋊D311yescharacteristic
399C4⋊C4—91no—
400C4⋊C4—92no—
401C4⋊C4—92no—
402C4⋊C4—91no—
403C4⋊C4—92no—
404C4⋊C4C2×C3×C3⋊D311yescharacteristic
405C4⋊C4C2×C3×C3⋊D311yescharacteristic
406C22⋊C4—91no—
407C22⋊C4—91no—
408C22⋊C4—92no—
409C22⋊C4—92no—
410C22⋊C4—92no—
411C22⋊C4C2×C3×C3⋊D312yes—
412C42—91no—
413C42C2×C3×C3⋊D312yes—
414C22×C3—28no—
415C22×C3—24no—
416C22×C3—24no—
417C22×C3—24no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—24no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—92no—
425C22×C3—181no—
426C22×C3—181no—
427C22×C3—182no—
428C22×C3—182no—
429C22×C3—182no—
430C22×C3C23×C3×D314yes—
431C22×C3C23×C3⋊D311yescharacteristic
432C2×D3—38no—
433C2×D3—64no—
434C2×D3—64no—
435C2×D3—68no—
436C2×D3—68no—
437C2×D3—68no—
438C3×C4—28no—
439C3×C4—24no—
440C3×C4—28no—
441C3×C4—28no—
442C3×C4—21no—
443C3×C4—22no—
444C3×C4—22no—
445C3×C4—28no—
446C3×C4—48no—
447C3×C4—48no—
448C3×C4—181no—
449C3×C4—181no—
450C3×C4—181no—
451C3×C4—182no—
452C3×C4C3×#23518yes—
453C3×C4#382812yes—
454Dic3—64no—
455Dic3—64no—
456Dic3—64no—
457Dic3—68no—
458C32—24no—
459C32#553014yescomplemented
460C32C3×#55311yescharacteristic, complemented
461C23—91no—
462C23—92no—
463C23C22×C3×C3⋊D311yescharacteristic
464D4—184no—
465D4—182no—
466D4—182no—
467D4—362no—
468C2×C4—22no—
469C2×C4—91no—
470C2×C4—91no—
471C2×C4—91no—
472C2×C4—92no—
473C2×C4—182no—
474C2×C4—362no—
475C2×C4—362no—
476C2×C4C22×C3×C3⋊D311yescharacteristic
477C2×C4C22×C3×C3⋊D312yes—
478C2×C4C22×C3×C3⋊D312yes—
479C2×C4C22×C3×C3⋊D311yescharacteristic
480C2×C3—24no—
481C2×C3—24no—
482C2×C3—24no—
483C2×C3—48no—
484C2×C3—44no—
485C2×C3—41no—
486C2×C3—182no—
487C2×C3—362no—
488C2×C3C2×C3×#23514yes—
489C2×C3C3×D3×#4114yes—
490C2×C3C3×#100914yes—
491C2×C3C2×#382811yescharacteristic
492C2×C3#41×C3⋊D311yescharacteristic
493C2×C3#6436611yescharacteristic
494D3—68no—
495D3—128no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—92no—
500C22—181no—
501C22—181no—
502C22—182no—
503C22—182no—
504C22—182no—
505C22C23×C3×C3⋊D311yescharacteristic
506C4—21no—
507C4—22no—
508C4—22no—
509C4—181no—
510C4—181no—
511C4—181no—
512C4—182no—
513C4C3×#382812yes—
514C3—24no—
515C3C3×#553014yescomplemented
516C3#1059600111yescharacteristic, direct product factor
517C2—41no—
518C2—182no—
519C2—362no—
520C2C2×C3×#382811yescharacteristic
521C2C3×#41×C3⋊D311yescharacteristic
522C2C3×#6436611yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×#117C2
socleC22×C33C24
hypercenterC3×#117C3⋊D3
hypocenterC32C3×#553
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×#117

Complements

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

NH ≅ G/Ntype
1. C33×#117C2semidirect
2. C3×#64139C2semidirect
3. C3×C4×#638C2semidirect
4. C3×#64104C2semidirect
5. C3×#64102C2semidirect
6. C3×C4⋊C4×C3⋊D3C2semidirect
7. C3×#64099C2semidirect
8. C3×#64093C2semidirect
9. C3×#64092C2semidirect
10. C3×#64085C2semidirect
11. C3×C4×C4⋊(C3⋊D3)C2semidirect
12. #10596001C3direct (N × H)
38. C32×#117D3semidirect
39. C32×#117C2×C3semidirect
40. #64139C2×C3semidirect
41. C4×#638C2×C3semidirect
42. #64104C2×C3semidirect
43. #64102C2×C3semidirect
44. C4⋊C4×C3⋊D3C2×C3semidirect
45. #64099C2×C3semidirect
46. #64093C2×C3semidirect
47. #64092C2×C3semidirect
48. #64085C2×C3semidirect
49. C4×C4⋊(C3⋊D3)C2×C3semidirect
160. C3×#117C3×D3semidirect
161. C3×#117C3⋊D3semidirect
328. #117C3×C3⋊D3semidirect
330. C33#553semidirect
459. C32#5530semidirect
460. C32C3×#553semidirect
515. C3C3×#5530semidirect
516. C3#10596001direct (N × H)

Maximal subgroups

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

subgroupindex
C33×#1172
C3×#641392
C3×C4×#6382
C3×#641042
C3×#641022
C3×C4⋊C4×C3⋊D32
C3×#640992
C3×#640932
C3×#640922
C3×#640852
C3×C4×C4⋊(C3⋊D3)2
#105960013
C3×#55303