Finite Group Databasebeta

← C3×#10591139 #13158167

Subgroups — C3×#10591139 #13158167

Order 1728.

762 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3748C211yesmaximal, characteristic
2C32×#926C211yesmaximal, characteristic
3C2×C32×C4⋊Dic3C211yesmaximal, characteristic
4C3×#63576C212yesmaximal, complemented
5C3×#63574C212yesmaximal, complemented
6C3×#4995—31nomaximal
7C3×#4964—31nomaximal
8#10591139C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C32×Dic3—21no—
11C3×#3748C414yescomplemented
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×#217—21no—
14C32×C4⋊Dic3—21no—
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2212yes—
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3—21no—
19C32×#117—31no—
20C2×C32×C4⋊C4—31no—
21C2×#3748C2×C311yescharacteristic
22C3×#926—21no—
23C3×#926D311yescharacteristic
24C3×#926C2×C311yescharacteristic
25C2×C3×C4⋊Dic3—21no—
26C2×C3×C4⋊Dic3D311yescharacteristic
27C2×C3×C4⋊Dic3C2×C311yescharacteristic
28C2×C3×#208—31no—
29C2×C3×#208—31no—
30C2×C3×#208—61no—
31C3×#808—32no—
32C3×#808—32no—
33C3×#807—32no—
34C3×#807—32no—
35#63576C2×C312yescomplemented
36#63574C2×C312yescomplemented
37C23×C33D411yescharacteristic
38C2×C33×C4C2×C412yes—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4Q812yes—
41C2×C33×C4C2311yescharacteristic
42C2×C32×Dic3—21no—
43C2×C32×Dic3—21no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3—42no—
46C3×C3⋊(C3⋊C8)C2×C414yescomplemented
47C3×#428—91no—
48#4995—31no—
49#4964—31no—
50C22×C32×C4—21no—
51C22×C32×C4—21no—
52C22×C32×C4—21no—
53C22×C32×C4—41no—
54C22×C32×C4—61no—
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C2×D311yescharacteristic
57C22×C32×C4C22×C311yescharacteristic
58C22×C3×Dic3—22no—
59C22×C3×Dic3—21no—
60C22×C3×Dic3—21no—
61C32×C4⋊C4—31no—
62C32×C4⋊C4—32no—
63C32×C4⋊C4—31no—
64C32×C4⋊C4—61no—
65C32×C22⋊C4—61no—
66C32×C42—61no—
67#3748C3×C414yescomplemented
68C2×C3⋊(C3⋊C8)C22×C312yes—
69C3×#217—22no—
70C3×#217—21no—
71C3×#217—21no—
72C3×C4⋊Dic3—22no—
73C3×C4⋊Dic3—22no—
74C3×C4⋊Dic3—21no—
75C3×C4⋊Dic3—21no—
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3—21no—
78C3×C4⋊Dic3C2×D312yes—
79C3×C4⋊Dic3C2×D311yescharacteristic
80C3×C4⋊Dic3C2×D311yescharacteristic
81C3×C4⋊Dic3C22×C312yes—
82C3×C4⋊Dic3C22×C311yescharacteristic
83C3×C4⋊Dic3C22×C311yescharacteristic
84C3×C4×Dic3—22no—
85C3×C4×Dic3—21no—
86C3×C4×Dic3—21no—
87C3×#208—34no—
88C3×#208—34no—
89C3×#208—64no—
90C2×C3×C3⋊C8—32no—
91C2×C3×C3⋊C8—32no—
92C2×C3×C3⋊C8—62no—
93C22×C33—21no—
94C22×C33—21no—
95C22×C33C4⋊C412yes—
96C22×C33C2×D411yescharacteristic
97C33×C4C4⋊C412yes—
98C33×C4C22×C411yescharacteristic
99C33×C4C2×Q811yescharacteristic
100C32×Dic3—41no—
101C32×Dic3—42no—
102C32×Dic3—41no—
103C2×C3×M16—91no—
104C3×#117—31no—
105C3×#117—31no—
106C3×#117—61no—
107C2×C3×C4⋊C4—31no—
108C2×C3×C4⋊C4—31no—
109C2×C3×C4⋊C4—61no—
110#926C3×D311yescharacteristic
111C2×C4⋊Dic3C3×D311yescharacteristic
112C2×#208—31no—
113C2×#208—31no—
114C2×#208—61no—
115C3×#107—92no—
116C3×#106—92no—
117#808—32no—
118#808—32no—
119#807—32no—
120#807—32no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—41no—
125C23×C32#6711yescharacteristic
126C23×C32#6711yescharacteristic
127C23×C32C3×D411yescharacteristic
128C2×C32×C4—22no—
129C2×C32×C4—21no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—42no—
141C2×C32×C4—41no—
142C2×C32×C4—42no—
143C2×C32×C4—41no—
144C2×C32×C4—61no—
145C2×C32×C4—61no—
146C2×C32×C4—62no—
147C2×C32×C4—122no—
148C2×C32×C4Dic612yes—
149C2×C32×C4Dic612yes—
150C2×C32×C4C4×D312yes—
151C2×C32×C4C4×D312yes—
152C2×C32×C4#6711yescharacteristic
153C2×C32×C4#6711yescharacteristic
154C2×C32×C4C2×C3×C412yes—
155C2×C32×C4C3×D411yescharacteristic
156C2×C32×C4C3×Q812yes—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C23×C311yescharacteristic
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—24no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—44no—
170C2×C3×Dic3—42no—
171C2×C3×Dic3—42no—
172C3⋊(C3⋊C8)C2×C3×C414yescomplemented
173C3×C3⋊C8—34no—
174C3×C3⋊C8—34no—
175C3×C3⋊C8—64no—
176#428—91noSylow
177C2×C33—22no—
178C2×C33C2×C4⋊C411yescharacteristic
179C2×C33AGL(1,ℤ/8)11yescharacteristic
180C2×C33#13711yescharacteristic
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—41no—
185C22×C3×C4—62no—
186C22×C3×C4—61no—
187C22×C3×C4—61no—
188C22×C3×C4D3211yescharacteristic
189C22×C3×C4C2×C3×D311yescharacteristic
190C22×C3×C4C2×C3×D311yescharacteristic
191C22×Dic3—21no—
192C3×M16—94no—
193C2×C3×C8—92no—
194C3×C4⋊C4—31no—
195C3×C4⋊C4—32no—
196C3×C4⋊C4—31no—
197C3×C4⋊C4—31no—
198C3×C4⋊C4—32no—
199C3×C4⋊C4—31no—
200C3×C4⋊C4—62no—
201C3×C4⋊C4—61no—
202C3×C4⋊C4—61no—
203C3×C4⋊C4—62no—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—61no—
206C3×C22⋊C4—62no—
207C3×C22⋊C4—61no—
208C3×C22⋊C4—61no—
209C3×C42—62no—
210C3×C42—61no—
211C3×C42—61no—
212#217—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3C2×C3×D311yescharacteristic
215C4⋊Dic3C2×C3×D312yes—
216C4⋊Dic3C2×C3×D311yescharacteristic
217C4×Dic3—21no—
218#208—34no—
219#208—34no—
220#208—64no—
221C2×C3⋊C8—32no—
222C2×C3⋊C8—32no—
223C2×C3⋊C8—62no—
224C22×C32—21no—
225C22×C32—21no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—22no—
235C22×C32—21no—
236C22×C32—21no—
237C22×C32—21no—
238C22×C32—42no—
239C22×C32—41no—
240C22×C32—41no—
241C22×C32—41no—
242C22×C32—41no—
243C22×C32—41no—
244C22×C32—42no—
245C22×C32—42no—
246C22×C32Dic3⋊C412yes—
247C22×C32Dic3⋊C412yes—
248C22×C32C3×C4⋊C412yes—
249C22×C32C2×#6711yescharacteristic
250C22×C32C2×#6711yescharacteristic
251C22×C32C2×C3×D411yescharacteristic
252C32×C4—22no—
253C32×C4—22no—
254C32×C4—22no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—42no—
262C32×C4—41no—
263C32×C4—41no—
264C32×C4—121no—
265C32×C4—121no—
266C32×C4—122no—
267C32×C4Dic3⋊C412yes—
268C32×C4Dic3⋊C412yes—
269C32×C4C3×C4⋊C412yes—
270C32×C4C2×Dic611yescharacteristic
271C32×C4C2×Dic611yescharacteristic
272C32×C4C2×C4×D311yescharacteristic
273C32×C4C2×C4×D311yescharacteristic
274C32×C4C22×C3×C411yescharacteristic
275C32×C4C2×C3×Q811yescharacteristic
276C3×Dic3—42no—
277C3×Dic3—42no—
278C3×Dic3—44no—
279C3×Dic3—42no—
280C3×Dic3—41no—
281C3×Dic3—41no—
282C3×Dic3—41no—
283C3×Dic3—42no—
284C3×Dic3—41no—
285C2×M16—91no—
286#117—31no—
287C2×C4⋊C4—31no—
288#107—92no—
289#106—92no—
290C33#42811yescharacteristic, complemented, Sylow
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3—41no—
295C23×C3#62511yescharacteristic
296C23×C3C3×#6711yescharacteristic
297C23×C3C3×#6711yescharacteristic
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—62no—
315C2×C3×C4—62no—
316C2×C3×C4—64no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—62no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—62no—
323C2×C3×C4—124no—
324C2×C3×C4—122no—
325C2×C3×C4—122no—
326C2×C3×C4#62412yes—
327C2×C3×C4#62511yescharacteristic
328C2×C3×C4#62712yes—
329C2×C3×C4C3×Dic612yes—
330C2×C3×C4C3×Dic612yes—
331C2×C3×C4C3×C4×D312yes—
332C2×C3×C4C3×C4×D312yes—
333C2×C3×C4C3×#6711yescharacteristic
334C2×C3×C4C3×#6711yescharacteristic
335C2×C3×C4C2×D3211yescharacteristic
336C2×C3×C4C22×C3×D311yescharacteristic
337C2×C3×C4C22×C3×D311yescharacteristic
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—22no—
341C2×Dic3—42no—
342C3×C8—94no—
343C3⋊C8—34no—
344C3⋊C8—34no—
345C3⋊C8—64no—
346C2×C32—22no—
347C2×C32—24no—
348C2×C32—22no—
349C2×C32—22no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—41no—
360C2×C32—41no—
361C2×C32—41no—
362C2×C32—42no—
363C2×C32—42no—
364C2×C32—44no—
365C2×C32C2×Dic3⋊C411yescharacteristic
366C2×C32C2×Dic3⋊C411yescharacteristic
367C2×C32#93211yescharacteristic
368C2×C32#94211yescharacteristic
369C2×C32#94911yescharacteristic
370C2×C32#95111yescharacteristic
371C2×C32C2×C3×C4⋊C411yescharacteristic
372C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
373C2×C32C3×#13711yescharacteristic
374C22×C4—61no—
375C22×C4C3×D3211yescharacteristic
376M16—94no—
377C2×C8—92no—
378C4⋊C4—31no—
379C4⋊C4—32no—
380C4⋊C4—31no—
381C4⋊C4—61no—
382C22⋊C4—61no—
383C42—61no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—41no—
405C22×C3—42no—
406C22×C3#372412yes—
407C22×C3C3×Dic3⋊C412yes—
408C22×C3C3×Dic3⋊C412yes—
409C22×C3C2×#62511yescharacteristic
410C22×C3C2×C3×#6711yescharacteristic
411C22×C3C2×C3×#6711yescharacteristic
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—22no—
421C3×C4—41no—
422C3×C4—41no—
423C3×C4—42no—
424C3×C4—124no—
425C3×C4—122no—
426C3×C4—122no—
427C3×C4—121no—
428C3×C4—121no—
429C3×C4—122no—
430C3×C4—122no—
431C3×C4—121no—
432C3×C4—121no—
433C3×C4#372412yes—
434C3×C4C3×Dic3⋊C412yes—
435C3×C4C3×Dic3⋊C412yes—
436C3×C4C2×#62411yescharacteristic
437C3×C4C2×#62711yescharacteristic
438C3×C4C2×C3×Dic611yescharacteristic
439C3×C4C2×C3×Dic611yescharacteristic
440C3×C4C2×C3×C4×D311yescharacteristic
441C3×C4C2×C3×C4×D311yescharacteristic
442Dic3—42no—
443Dic3—41no—
444Dic3—41no—
445C32—21no—
446C32—21no—
447C32—21no—
448C32—41no—
449C32#496411yescharacteristic, complemented
450C32#499511yescharacteristic, complemented
451C32C3×#42811yescharacteristic, complemented
452C23C3×#62511yescharacteristic
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—62no—
456C2×C4—122no—
457C2×C4C3×#62412yes—
458C2×C4C3×#62511yescharacteristic
459C2×C4C3×#62712yes—
460C2×C4C2×C3×D3211yescharacteristic
461C8—94no—
462C2×C3—24no—
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—22no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—42no—
476C2×C3—42no—
477C2×C3—44no—
478C2×C3—41no—
479C2×C3—41no—
480C2×C3—41no—
481C2×C3#6382311yescharacteristic
482C2×C3#6382711yescharacteristic
483C2×C3C2×#372411yescharacteristic
484C2×C3C2×C3×Dic3⋊C411yescharacteristic
485C2×C3C2×C3×Dic3⋊C411yescharacteristic
486C2×C3C3×#93211yescharacteristic
487C2×C3C3×#94211yescharacteristic
488C2×C3C3×#94911yescharacteristic
489C2×C3C3×#95111yescharacteristic
490C22—21no—
491C22—21no—
492C22C3×#372412yes—
493C22C2×C3×#62511yescharacteristic
494C4—122no—
495C4—121no—
496C4—121no—
497C4C3×#372412yes—
498C4C2×C3×#62411yescharacteristic
499C4C2×C3×#62711yescharacteristic
500C3—21no—
501C3—21no—
502C3—21no—
503C3—41no—
504C3#1059113911yescharacteristic, direct product factor
505C3C3×#496411yescharacteristic, complemented
506C3C3×#499511yescharacteristic, complemented
507C2—22no—
508C2C3×#6382311yescharacteristic
509C2C3×#6382711yescharacteristic
510C2C2×C3×#372411yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#428
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63576C2semidirect
5. C3×#63574C2semidirect
8. #10591139C3direct (N × H)
11. C3×#3748C4semidirect
35. #63576C2×C3semidirect
36. #63574C2×C3semidirect
46. C3×C3⋊(C3⋊C8)C2×C4semidirect
67. #3748C3×C4semidirect
172. C3⋊(C3⋊C8)C2×C3×C4semidirect
290. C33#428semidirect
449. C32#4964semidirect
450. C32#4995semidirect
451. C32C3×#428semidirect
504. C3#10591139direct (N × H)
505. C3C3×#4964semidirect
506. C3C3×#4995semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37482
C32×#9262
C2×C32×C4⋊Dic32
C3×#635762
C3×#635742
C3×#49953
C3×#49643
#105911393