Finite Group Databasebeta

← C3×#10595985 #13179265

Subgroups — C3×#10595985 #13179265

Order 1728.

1796 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×C4⋊C4C211yesmaximal, characteristic, complemented
2C3×#64139C211yesmaximal, characteristic, complemented
3C3×#64139C212yesmaximal, complemented
4C3×C4×#638C211yesmaximal, characteristic, complemented
5C3×C4×#638C212yesmaximal, complemented
6C3×#64107C212yesmaximal, complemented
7C3×#64104C211yesmaximal, characteristic, complemented
8C3×#64103C212yesmaximal, complemented
9C3×#64102C211yesmaximal, characteristic, complemented
10C3×#64101C211yesmaximal, characteristic, complemented
11C3×#64099C211yesmaximal, characteristic, complemented
12#10595985C311yesmaximal, characteristic, direct product factor
13C3×#5514—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C22×C33×C4C2212yes—
16C2×C3×#638C2211yescharacteristic
17C2×C3×#638C2212yes—
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
19C2×C3×C4⋊(C3⋊D3)C2212yes—
20C2×C3×C4×C3⋊D3C2211yescharacteristic
21C2×C3×C4×C3⋊D3C2212yes—
22C33×C4⋊C4C2211yescharacteristic
23C33×C4⋊C4C2211yescharacteristic
24C33×C4⋊C4C2212yes—
25C3×#3757C2211yescharacteristic
26C3×#3752C2212yes—
27C3×#3752C2212yes—
28C3×#3752C2211yescharacteristic
29C3×#3752C2212yes—
30C3×#3752C2212yes—
31C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
32C3×C4⋊(C3⋊Dic3)C2212yes—
33C3×#3750C2212yes—
34C3×#3750C2211yescharacteristic
35C3×C4×C3⋊Dic3C2211yescharacteristic
36C3×C4×C3⋊Dic3C2212yes—
37C2×C32×C4⋊C4—24no—
38C2×C32×C4⋊C4D314yescomplemented
39C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
40#64139C2×C311yescharacteristic, complemented
41#64139C2×C312yescomplemented
42C4×#638C2×C311yescharacteristic, complemented
43C4×#638C2×C312yescomplemented
44#64107C2×C312yescomplemented
45#64104C2×C311yescharacteristic, complemented
46#64103C2×C312yescomplemented
47#64102C2×C311yescharacteristic, complemented
48#64101C2×C311yescharacteristic, complemented
49#64099C2×C311yescharacteristic, complemented
50C3×#930—34no—
51C3×#930—38no—
52C3×C4×#67—34no—
53C3×C4×#67—38no—
54C3×#898—38no—
55C3×#895—34no—
56C3×#894—38no—
57C3×#893—34no—
58C3×#892—34no—
59C3×#890—34no—
60C23×C33C2311yescharacteristic
61C22×C3×C3⋊D3C2311yescharacteristic
62C22×C3×C3⋊D3C2312yes—
63C2×C33×C4—22no—
64C2×C33×C4—42no—
65C2×C33×C4C2312yes—
66C2×C33×C4C2312yes—
67C2×C33×C4C2311yescharacteristic
68C2×C33×C4C2311yescharacteristic
69C3×#638—22no—
70C3×#638—24no—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C2×C3×C3⋊Dic3C2312yes—
75C3×C4⋊(C3⋊D3)—22no—
76C3×C4⋊(C3⋊D3)—42no—
77C3×C4×C3⋊D3—22no—
78C3×C4×C3⋊D3—42no—
79C3×#553—91no—
80#5514—34no—
81C22×C32×C4—24no—
82C22×C32×C4—28no—
83C22×C32×C4C2×D314yes—
84C22×C32×C4C2×D318yes—
85C22×C32×C4C22×C311yescharacteristic
86C22×C32×C4C22×C312yes—
87C2×#638C22×C311yescharacteristic
88C2×#638C22×C312yes—
89C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
90C2×C4⋊(C3⋊D3)C22×C312yes—
91C2×C4×C3⋊D3C22×C311yescharacteristic
92C2×C4×C3⋊D3C22×C312yes—
93C2×C3×#67—34no—
94C2×C3×#67—38no—
95C2×C3×D12—34no—
96C2×C3×D12—38no—
97C2×C3×C4×D3—34no—
98C2×C3×C4×D3—38no—
99C32×C4⋊C4—24no—
100C32×C4⋊C4—24no—
101C32×C4⋊C4—28no—
102C32×C4⋊C4C2×D314yes—
103C32×C4⋊C4C2×D314yes—
104C32×C4⋊C4C2×D318yes—
105C32×C4⋊C4C22×C311yescharacteristic
106C32×C4⋊C4C22×C311yescharacteristic
107C32×C4⋊C4C22×C312yes—
108#3757C22×C311yescharacteristic
109#3752C22×C312yes—
110#3752C22×C312yes—
111#3752C22×C311yescharacteristic
112#3752C22×C312yes—
113#3752C22×C312yes—
114C4⋊(C3⋊Dic3)C22×C311yescharacteristic
115C4⋊(C3⋊Dic3)C22×C312yes—
116#3750C22×C312yes—
117#3750C22×C311yescharacteristic
118C4×C3⋊Dic3C22×C311yescharacteristic
119C4×C3⋊Dic3C22×C312yes—
120C3×#217—34no—
121C3×(C2×D3)⋊C4—38no—
122C3×(C2×D3)⋊C4—38no—
123C3×(C2×D3)⋊C4—34no—
124C3×(C2×D3)⋊C4—38no—
125C3×(C2×D3)⋊C4—38no—
126C3×C4⋊Dic3—34no—
127C3×C4⋊Dic3—38no—
128C3×Dic3⋊C4—38no—
129C3×Dic3⋊C4—34no—
130C3×C4×Dic3—34no—
131C3×C4×Dic3—38no—
132C22×C33—21no—
133C22×C33—21no—
134C22×C33#4112yes—
135C22×C33C2411yescharacteristic
136C2×C3×C3⋊D3—21no—
137C2×C3×C3⋊D3—22no—
138C2×C3×C3⋊D3—21no—
139C2×C3×C3⋊D3—22no—
140C2×C3×C3⋊D3—22no—
141C2×C3×C3⋊D3—21no—
142C33×C4—21no—
143C33×C4—22no—
144C33×C4—22no—
145C33×C4#4112yes—
146C3×C3⋊Dic3—21no—
147C3×C3⋊Dic3—21no—
148C3×C3⋊Dic3—22no—
149C3×C3⋊Dic3—21no—
150C3×#126—92no—
151C3×#125—91no—
152C3×#123—92no—
153C3×#121—92no—
154C3×#121—91no—
155C3×#121—91no—
156C3×C4×D4—92no—
157C3×C4×D4—91no—
158C3×C4×D4—91no—
159C3×#117—91no—
160C2×C3×C4⋊C4—24no—
161C2×C3×C4⋊C4C3×D314yescomplemented
162C2×C3×C4⋊C4C3⋊D311yescharacteristic, complemented
163#930—34no—
164#930—38no—
165C4×#67—34no—
166C4×#67—38no—
167#898—38no—
168#895—34no—
169#894—38no—
170#893—34no—
171#892—34no—
172#890—34no—
173C23×C32—24no—
174C23×C32C22×D314yes—
175C23×C32C23×C311yescharacteristic
176C22×C3⋊D3C23×C311yescharacteristic
177C22×C3⋊D3C23×C312yes—
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×C4—48no—
188C2×C32×C4—42no—
189C2×C32×C4—416no—
190C2×C32×C4C22×D318yes—
191C2×C32×C4C22×D318yes—
192C2×C32×C4C22×D314yes—
193C2×C32×C4C22×D314yes—
194C2×C32×C4C23×C312yes—
195C2×C32×C4C23×C312yes—
196C2×C32×C4C23×C311yescharacteristic
197C2×C32×C4C23×C311yescharacteristic
198#638—22no—
199#638—24no—
200C2×C3⋊Dic3C23×C311yescharacteristic
201C2×C3⋊Dic3C23×C311yescharacteristic
202C2×C3⋊Dic3C23×C311yescharacteristic
203C2×C3⋊Dic3C23×C312yes—
204C4⋊(C3⋊D3)—22no—
205C4⋊(C3⋊D3)—42no—
206C4×C3⋊D3—22no—
207C4×C3⋊D3—42no—
208C3×#67—68no—
209C3×#67—616no—
210C2×C3×Dic3—34no—
211C2×C3×Dic3—34no—
212C2×C3×Dic3—34no—
213C2×C3×Dic3—38no—
214C3×D12—68no—
215C3×D12—128no—
216C3×C4×D3—68no—
217C3×C4×D3—128no—
218#553—91noSylow
219C2×C33—22no—
220C2×C33C2×#4111yescharacteristic
221C2×C33C2×#4111yescharacteristic
222C2×C33H(5,𝔽2)11yescharacteristic
223C3×C3⋊D3—41no—
224C3×C3⋊D3—42no—
225C2×C3×D4—92no—
226C2×C3×D4—91no—
227C2×C3×D4—91no—
228C2×C3×D4—92no—
229C22×C3×C4—24no—
230C22×C3×C4—28no—
231C22×C3×C4—91no—
232C22×C3×C4—92no—
233C22×C3×C4C2×C3×D314yes—
234C22×C3×C4C2×C3×D318yes—
235C22×C3×C4C2×C3⋊D311yescharacteristic
236C22×C3×C4C2×C3⋊D312yes—
237C2×#67—34no—
238C2×#67—38no—
239C2×D12—34no—
240C2×D12—38no—
241C2×C4×D3—34no—
242C2×C4×D3—38no—
243C3×C4⋊C4—24no—
244C3×C4⋊C4—24no—
245C3×C4⋊C4—28no—
246C3×C4⋊C4—92no—
247C3×C4⋊C4—91no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4—92no—
250C3×C4⋊C4C2×C3×D314yes—
251C3×C4⋊C4C2×C3×D314yes—
252C3×C4⋊C4C2×C3×D318yes—
253C3×C4⋊C4C2×C3⋊D311yescharacteristic
254C3×C4⋊C4C2×C3⋊D311yescharacteristic
255C3×C4⋊C4C2×C3⋊D312yes—
256C3×C22⋊C4—91no—
257C3×C22⋊C4—92no—
258C3×C22⋊C4—92no—
259C3×C22⋊C4—92no—
260C3×C22⋊C4—91no—
261C3×C22⋊C4—92no—
262C3×C42—92no—
263C3×C42—91no—
264#217—34no—
265(C2×D3)⋊C4—38no—
266(C2×D3)⋊C4—38no—
267(C2×D3)⋊C4—34no—
268(C2×D3)⋊C4—38no—
269(C2×D3)⋊C4—38no—
270C4⋊Dic3—34no—
271C4⋊Dic3—38no—
272Dic3⋊C4—38no—
273Dic3⋊C4—34no—
274C4×Dic3—34no—
275C4×Dic3—38no—
276C22×C32—28no—
277C22×C32—28no—
278C22×C32—24no—
279C22×C32—24no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—24no—
283C22×C32—28no—
284C22×C32#23918yes—
285C22×C32C3×#4112yes—
286C22×C32C23×D314yes—
287C22×C32C24×C311yescharacteristic
288C2×C3⋊D3—21no—
289C2×C3⋊D3—22no—
290C2×C3⋊D3—21no—
291C2×C3⋊D3—22no—
292C2×C3⋊D3—22no—
293C2×C3⋊D3—21no—
294C2×C3×D3—64no—
295C2×C3×D3—68no—
296C2×C3×D3—68no—
297C2×C3×D3—64no—
298C2×C3×D3—64no—
299C2×C3×D3—68no—
300C32×C4—28no—
301C32×C4—24no—
302C32×C4—21no—
303C32×C4—22no—
304C32×C4—22no—
305C32×C4—28no—
306C32×C4—28no—
307C32×C4—28no—
308C32×C4—48no—
309C32×C4—48no—
310C32×C4#23518yes—
311C32×C4C3×#4112yes—
312C3⋊Dic3—21no—
313C3⋊Dic3—21no—
314C3⋊Dic3—22no—
315C3⋊Dic3—21no—
316C3×Dic3—64no—
317C3×Dic3—64no—
318C3×Dic3—64no—
319C3×Dic3—68no—
320#126—92no—
321#125—91no—
322#123—92no—
323#121—92no—
324#121—91no—
325#121—91no—
326C4×D4—92no—
327C4×D4—91no—
328C4×D4—91no—
329#117—91no—
330C2×C4⋊C4C3×C3⋊D311yescharacteristic, complemented
331C33#55311yescharacteristic, complemented, Sylow
332C23×C3—24no—
333C23×C3—91no—
334C23×C3—92no—
335C23×C3C22×C3×D314yes—
336C23×C3C22×C3⋊D311yescharacteristic
337C22×D3—34no—
338C22×D3—38no—
339C3×D4—184no—
340C3×D4—182no—
341C3×D4—182no—
342C3×D4—362no—
343C2×C3×C4—216no—
344C2×C3×C4—28no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—28no—
349C2×C3×C4—28no—
350C2×C3×C4—48no—
351C2×C3×C4—42no—
352C2×C3×C4—416no—
353C2×C3×C4—91no—
354C2×C3×C4—92no—
355C2×C3×C4—91no—
356C2×C3×C4—91no—
357C2×C3×C4—182no—
358C2×C3×C4—362no—
359C2×C3×C4C22×C3×D318yes—
360C2×C3×C4C22×C3×D318yes—
361C2×C3×C4C22×C3×D314yes—
362C2×C3×C4C22×C3×D314yes—
363C2×C3×C4C22×C3⋊D312yes—
364C2×C3×C4C22×C3⋊D311yescharacteristic
365C2×C3×C4C22×C3⋊D312yes—
366C2×C3×C4C22×C3⋊D311yescharacteristic
367#67—68no—
368#67—616no—
369C2×Dic3—34no—
370C2×Dic3—34no—
371C2×Dic3—34no—
372C2×Dic3—38no—
373D12—68no—
374D12—128no—
375C4×D3—68no—
376C4×D3—128no—
377C2×C32—216no—
378C2×C32—28no—
379C2×C32—22no—
380C2×C32—24no—
381C2×C32—24no—
382C2×C32—24no—
383C2×C32C2×#23514yes—
384C2×C32#100414yes—
385C2×C32C2×#23914yes—
386C2×C32C2×C3×#4111yescharacteristic
387C2×C32C2×C3×#4111yescharacteristic
388C2×C32C3×H(5,𝔽2)11yescharacteristic
389C3⋊D3—41no—
390C3⋊D3—42no—
391C3×D3—124no—
392C3×D3—128no—
393C2×D4—92no—
394C2×D4—91no—
395C2×D4—91no—
396C2×D4—92no—
397C22×C4—91no—
398C22×C4—92no—
399C22×C4C2×C3×C3⋊D311yescharacteristic
400C22×C4C2×C3×C3⋊D312yes—
401C4⋊C4—91no—
402C4⋊C4—92no—
403C4⋊C4—91no—
404C4⋊C4—92no—
405C4⋊C4C2×C3×C3⋊D311yescharacteristic
406C4⋊C4C2×C3×C3⋊D311yescharacteristic
407C4⋊C4C2×C3×C3⋊D312yes—
408C22⋊C4—91no—
409C22⋊C4—92no—
410C22⋊C4—92no—
411C22⋊C4—92no—
412C22⋊C4—91no—
413C22⋊C4—92no—
414C42—92no—
415C42—91no—
416C22×C3—28no—
417C22×C3—28no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—24no—
423C22×C3—28no—
424C22×C3—181no—
425C22×C3—182no—
426C22×C3—182no—
427C22×C3—181no—
428C22×C3—181no—
429C22×C3—182no—
430C22×C3C3×#23918yes—
431C22×C3#383212yes—
432C22×C3C23×C3×D314yes—
433C22×C3C23×C3⋊D311yescharacteristic
434C2×D3—64no—
435C2×D3—68no—
436C2×D3—68no—
437C2×D3—64no—
438C2×D3—64no—
439C2×D3—68no—
440C3×C4—28no—
441C3×C4—24no—
442C3×C4—28no—
443C3×C4—28no—
444C3×C4—21no—
445C3×C4—22no—
446C3×C4—22no—
447C3×C4—28no—
448C3×C4—48no—
449C3×C4—48no—
450C3×C4—181no—
451C3×C4—181no—
452C3×C4—182no—
453C3×C4—181no—
454C3×C4C3×#23518yes—
455C3×C4#382812yes—
456Dic3—64no—
457Dic3—64no—
458Dic3—64no—
459Dic3—68no—
460C32—24no—
461C32#551414yescomplemented
462C32C3×#55311yescharacteristic, complemented
463C23—91no—
464C23—92no—
465C23C22×C3×C3⋊D311yescharacteristic
466D4—184no—
467D4—182no—
468D4—182no—
469D4—362no—
470C2×C4—22no—
471C2×C4—42no—
472C2×C4—91no—
473C2×C4—92no—
474C2×C4—91no—
475C2×C4—91no—
476C2×C4—182no—
477C2×C4—362no—
478C2×C4C22×C3×C3⋊D312yes—
479C2×C4C22×C3×C3⋊D311yescharacteristic
480C2×C4C22×C3×C3⋊D312yes—
481C2×C4C22×C3×C3⋊D311yescharacteristic
482C2×C3—216no—
483C2×C3—28no—
484C2×C3—22no—
485C2×C3—24no—
486C2×C3—24no—
487C2×C3—24no—
488C2×C3—361no—
489C2×C3—362no—
490C2×C3C2×C3×#23514yes—
491C2×C3C3×#100414yes—
492C2×C3C2×C3×#23914yes—
493C2×C3C2×#382811yescharacteristic
494C2×C3#6436111yescharacteristic
495C2×C3C2×#383211yescharacteristic
496D3—124no—
497D3—128no—
498C22—21no—
499C22—21no—
500C22—181no—
501C22—182no—
502C22—182no—
503C22—181no—
504C22—181no—
505C22—182no—
506C22C3×#383212yes—
507C22C23×C3×C3⋊D311yescharacteristic
508C4—21no—
509C4—22no—
510C4—22no—
511C4—181no—
512C4—181no—
513C4—182no—
514C4—181no—
515C4C3×#382812yes—
516C3—24no—
517C3C3×#551414yescomplemented
518C3#1059598511yescharacteristic, direct product factor
519C2—22no—
520C2—361no—
521C2—362no—
522C2C2×C3×#382811yescharacteristic
523C2C3×#6436111yescharacteristic
524C2C2×C3×#383211yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC2×C33×C4⋊C4C2
socleC22×C33C24
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterC32C3×#553
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C33×C4⋊C4C2semidirect
2. C3×#64139C2semidirect
3. C3×#64139C2semidirect
4. C3×C4×#638C2semidirect
5. C3×C4×#638C2semidirect
6. C3×#64107C2semidirect
7. C3×#64104C2semidirect
8. C3×#64103C2semidirect
9. C3×#64102C2semidirect
10. C3×#64101C2semidirect
11. C3×#64099C2semidirect
12. #10595985C3direct (N × H)
38. C2×C32×C4⋊C4D3semidirect
39. C2×C32×C4⋊C4C2×C3semidirect
40. #64139C2×C3semidirect
41. #64139C2×C3semidirect
42. C4×#638C2×C3semidirect
43. C4×#638C2×C3semidirect
44. #64107C2×C3semidirect
45. #64104C2×C3semidirect
46. #64103C2×C3semidirect
47. #64102C2×C3semidirect
48. #64101C2×C3semidirect
49. #64099C2×C3semidirect
161. C2×C3×C4⋊C4C3×D3semidirect
162. C2×C3×C4⋊C4C3⋊D3semidirect
330. C2×C4⋊C4C3×C3⋊D3semidirect
331. C33#553semidirect
461. C32#5514semidirect
462. C32C3×#553semidirect
517. C3C3×#5514semidirect
518. C3#10595985direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×C4⋊C42
C3×#641392
C3×C4×#6382
C3×#641072
C3×#641042
C3×#641032
C3×#641022
C3×#641012
C3×#640992
#105959853
C3×#55143