Finite Group Databasebeta

← C3×#10590688 #13157689

Subgroups — C3×#10590688 #13157689

Order 1728.

634 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
1C2×C3×#3743C211yesmaximal, characteristic
2C3×#63900C211yesmaximal, characteristic
3C3×#63849C211yesmaximal, characteristic
4C32×#818C211yesmaximal, characteristic, complemented
5C32×#817C211yesmaximal, characteristic, complemented
6C3×#63576C211yesmaximal, characteristic, complemented
7C3×#63574C211yesmaximal, characteristic, complemented
8C3×#4881—31nomaximal
9C3×#4861—31nomaximal
10#10590688C311yesmaximal, characteristic, direct product factor
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#624—21no—
13C2×C33×C8C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×#3743C414yescomplemented
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C3×#3724—21no—
21C3×#3722—21no—
22C3×Dic32—21no—
23C2×#3743C2×C311yescharacteristic
24C2×C3×ΓL(1,𝔽25)—31no—
25C2×C3×ΓL(1,𝔽25)—31no—
26C2×C3×ΓL(1,𝔽25)—61no—
27C3×#892—31no—
28C3×#892—31no—
29C3×D3×C4⋊C4—31no—
30C3×D3×C4⋊C4—31no—
31#63900C2×C311yescharacteristic
32#63849C2×C311yescharacteristic
33C32×#107—31no—
34C32×#106—31no—
35C3×#818—21no—
36C3×#818D311yescharacteristic, complemented
37C3×#818C2×C311yescharacteristic, complemented
38C3×#817—21no—
39C3×#817D311yescharacteristic, complemented
40C3×#817C2×C311yescharacteristic, complemented
41C3×#808—31no—
42C3×#808—31no—
43C3×#807—31no—
44C3×#807—31no—
45#63576C2×C311yescharacteristic, complemented
46#63574C2×C311yescharacteristic, complemented
47C22×C3×C3⋊D3D411yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3D411yescharacteristic
50C3×C4×C3⋊D3C2×C412yes—
51C3×C4×C3⋊D3Q812yes—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C3×#624—42no—
57C33×C8C2×C412yescomplemented
58C3×C3⋊(C3⋊C8)C2×C412yescomplemented
59C3×#428—91no—
60#4881—31no—
61#4861—31no—
62C2×C4×C3⋊D3C22×C311yescharacteristic
63C2×C3×C4×D3—31no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—61no—
66C2×C3×C4×D3—61no—
67C2×C3×C4×D3—61no—
68C2×#624—21no—
69C2×C32×C8—21no—
70C2×C32×C8—21no—
71C2×C32×C8—21no—
72C2×C32×C8—41no—
73C2×C32×C8C2×D311yescharacteristic
74C2×C32×C8C2×D311yescharacteristic
75C2×C32×C8C22×C311yescharacteristic
76C32×C4⋊C4—31no—
77C32×C4⋊C4—31no—
78C32×C4⋊C4—31no—
79C32×C4⋊C4—31no—
80C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
81#3743C3×C414yescomplemented
82C3×(C2×D3)⋊C4—61no—
83C3×(C2×D3)⋊C4—61no—
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C22×C311yescharacteristic
93C3×C4⋊Dic3C22×C311yescharacteristic
94C3×C4⋊Dic3C22×C311yescharacteristic
95C3×C4⋊Dic3C22×C311yescharacteristic
96C3×Dic3⋊C4—61no—
97C3×Dic3⋊C4—61no—
98C3×C4×Dic3—61no—
99C3×C4×Dic3—61no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103C3×ΓL(1,𝔽25)—34no—
104C3×ΓL(1,𝔽25)—34no—
105C3×ΓL(1,𝔽25)—64no—
106#3724—21no—
107#3722—21no—
108Dic32—21no—
109C22×C33C2×D411yescharacteristic
110C2×C3×C3⋊D3—21no—
111C2×C3×C3⋊D3—21no—
112C2×C3×C3⋊D3C4⋊C412yes—
113C33×C4C22×C411yescharacteristic
114C33×C4C2×Q811yescharacteristic
115C3×C3⋊Dic3C4⋊C412yes—
116C32×Dic3—41no—
117C32×Dic3—41no—
118C32×Dic3—41no—
119C32×Dic3—41no—
120C2×C3×M16—91no—
121C3×#117—91no—
122C2×C3×C4⋊C4—91no—
123C2×ΓL(1,𝔽25)—31no—
124C2×ΓL(1,𝔽25)—31no—
125C2×ΓL(1,𝔽25)—61no—
126#892—31no—
127#892—31no—
128D3×C4⋊C4—31no—
129D3×C4⋊C4—31no—
130C3×#107—31no—
131C3×#107—31no—
132C3×#107—61no—
133C3×#107—91no—
134C3×#106—31no—
135C3×#106—31no—
136C3×#106—61no—
137C3×#106—91no—
138#818C3×D311yescharacteristic, complemented
139#817C3×D311yescharacteristic, complemented
140#808—31no—
141#808—31no—
142#807—31no—
143#807—31no—
144C22×C3⋊D3C3×D411yescharacteristic
145C22×C3×D3—31no—
146C22×C3×D3—31no—
147C22×C3×D3—61no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—41no—
152C2×C32×C4—61no—
153C2×C32×C4—61no—
154C2×C32×C4—61no—
155C2×C32×C4—61no—
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C23×C311yescharacteristic
159C2×C3⋊Dic3C3×D411yescharacteristic
160C4×C3⋊D3C2×C3×C412yes—
161C4×C3⋊D3C3×Q812yes—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—31no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—41no—
173C2×C3×Dic3—41no—
174C2×C3×Dic3—41no—
175C2×C3×Dic3—41no—
176C2×C3×Dic3—61no—
177C3×C4×D3—32no—
178C3×C4×D3—32no—
179C3×C4×D3—32no—
180C3×C4×D3—32no—
181C3×C4×D3—62no—
182C3×C4×D3—62no—
183C3×C4×D3—122no—
184C3×C4×D3—122no—
185#624—42no—
186C32×C8—22no—
187C32×C8—22no—
188C32×C8—22no—
189C32×C8—42no—
190C32×C8C4×D312yescomplemented
191C32×C8C4×D312yescomplemented
192C32×C8C2×C3×C412yescomplemented
193C3⋊(C3⋊C8)C2×C3×C412yescomplemented
194C3×C3⋊C8—32no—
195C3×C3⋊C8—32no—
196C3×C3⋊C8—62no—
197#428—91noSylow
198C2×C33C2×C4⋊C411yescharacteristic
199C2×C33AGL(1,ℤ/8)11yescharacteristic
200C2×C33#13711yescharacteristic
201C3×C3⋊D3—22no—
202C22×C3×C4—91no—
203C22×C3×C4—181no—
204C2×C4×D3—31no—
205C2×C4×D3—31no—
206C2×C4×D3—61no—
207C2×C4×D3—61no—
208C2×C4×D3—61no—
209C3×M16—94no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—41no—
214C2×C3×C8—91no—
215C2×C3×C8D3211yescharacteristic
216C2×C3×C8C2×C3×D311yescharacteristic
217C2×C3×C8C2×C3×D311yescharacteristic
218C3×C4⋊C4—31no—
219C3×C4⋊C4—31no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—31no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—31no—
224C3×C4⋊C4—31no—
225C3×C4⋊C4—31no—
226C3×C4⋊C4—61no—
227C3×C4⋊C4—61no—
228C3×C4⋊C4—61no—
229C3×C4⋊C4—61no—
230C3×C4⋊C4—181no—
231C3×C22⋊C4—181no—
232C3×C42—181no—
233(C2×D3)⋊C4—61no—
234(C2×D3)⋊C4—61no—
235C4⋊Dic3C2×C3×D311yescharacteristic
236C4⋊Dic3C2×C3×D311yescharacteristic
237C4⋊Dic3C2×C3×D311yescharacteristic
238C4⋊Dic3C2×C3×D311yescharacteristic
239Dic3⋊C4—61no—
240Dic3⋊C4—61no—
241C4×Dic3—61no—
242C4×Dic3—61no—
243C2×C3⋊C8—31no—
244C2×C3⋊C8—31no—
245C2×C3⋊C8—61no—
246ΓL(1,𝔽25)—34no—
247ΓL(1,𝔽25)—34no—
248ΓL(1,𝔽25)—64no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—41no—
253C22×C32D3×D411yescharacteristic
254C22×C32D3×D411yescharacteristic
255C22×C32C2×C3×D411yescharacteristic
256C2×C3⋊D3—21no—
257C2×C3⋊D3—21no—
258C2×C3⋊D3C3×C4⋊C412yes—
259C2×C3×D3—32no—
260C2×C3×D3—32no—
261C2×C3×D3—62no—
262C2×C3×D3—61no—
263C2×C3×D3—61no—
264C2×C3×D3—61no—
265C2×C3×D3—61no—
266C2×C3×D3—121no—
267C2×C3×D3—121no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—41no—
275C32×C4—41no—
276C32×C4—121no—
277C32×C4—121no—
278C32×C4—121no—
279C32×C4—121no—
280C32×C4C2×C4×D311yescharacteristic
281C32×C4C2×C4×D311yescharacteristic
282C32×C4D3×Q811yescharacteristic
283C32×C4D3×Q811yescharacteristic
284C32×C4C22×C3×C411yescharacteristic
285C32×C4C2×C3×Q811yescharacteristic
286C3⋊Dic3C3×C4⋊C412yes—
287C3×Dic3—32no—
288C3×Dic3—32no—
289C3×Dic3—41no—
290C3×Dic3—41no—
291C3×Dic3—41no—
292C3×Dic3—41no—
293C3×Dic3—41no—
294C3×Dic3—41no—
295C3×Dic3—41no—
296C3×Dic3—41no—
297C3×Dic3—62no—
298C3×Dic3—81no—
299C3×Dic3—81no—
300C3×Dic3—81no—
301C3×Dic3—81no—
302C2×M16—91no—
303#117—91no—
304C2×C4⋊C4—91no—
305#107—31no—
306#107—91no—
307#106—31no—
308#106—91no—
309C33#42811yescharacteristic, complemented, Sylow
310C23×C3—91no—
311C22×D3—31no—
312C22×D3—31no—
313C22×D3—61no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—41no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—91no—
327C2×C3×C4—92no—
328C2×C3×C4—92no—
329C2×C3×C4—121no—
330C2×C3×C4—121no—
331C2×C3×C4—121no—
332C2×C3×C4—121no—
333C2×C3×C4—362no—
334C2×C3×C4C2×D3211yescharacteristic
335C2×C3×C4C22×C3×D311yescharacteristic
336C2×C3×C4C22×C3×D311yescharacteristic
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—31no—
342C2×Dic3—31no—
343C2×Dic3—61no—
344C4×D3—32no—
345C4×D3—32no—
346C4×D3—32no—
347C4×D3—32no—
348C4×D3—62no—
349C4×D3—62no—
350C4×D3—122no—
351C4×D3—122no—
352C3×C8—22no—
353C3×C8—22no—
354C3×C8—22no—
355C3×C8—42no—
356C3×C8—92no—
357C3×C8#62412yescomplemented
358C3×C8C3×C4×D312yescomplemented
359C3×C8C3×C4×D312yescomplemented
360C3⋊C8—32no—
361C3⋊C8—32no—
362C3⋊C8—62no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—41no—
373C2×C32—41no—
374C2×C32—41no—
375C2×C32D3×C4⋊C411yescharacteristic
376C2×C32D3×C4⋊C411yescharacteristic
377C2×C32#91111yescharacteristic
378C2×C32#91411yescharacteristic
379C2×C32#91511yescharacteristic
380C2×C32#91811yescharacteristic
381C2×C32C2×C3×C4⋊C411yescharacteristic
382C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
383C2×C32C3×#13711yescharacteristic
384C3⋊D3—22no—
385C3×D3—62no—
386C3×D3—62no—
387C3×D3—122no—
388C22×C4—91no—
389C22×C4—181no—
390M16—94no—
391C2×C8—91no—
392C2×C8C3×D3211yescharacteristic
393C4⋊C4—31no—
394C4⋊C4—31no—
395C4⋊C4—31no—
396C4⋊C4—31no—
397C4⋊C4—181no—
398C22⋊C4—181no—
399C42—181no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—41no—
404C22×C3—92no—
405C22×C3—181no—
406C22×C3—181no—
407C22×C3#380211yescharacteristic
408C22×C3C3×D3×D411yescharacteristic
409C22×C3C3×D3×D411yescharacteristic
410C2×D3—32no—
411C2×D3—32no—
412C2×D3—62no—
413C2×D3—61no—
414C2×D3—61no—
415C2×D3—61no—
416C2×D3—61no—
417C2×D3—121no—
418C2×D3—121no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—41no—
426C3×C4—41no—
427C3×C4—92no—
428C3×C4—121no—
429C3×C4—121no—
430C3×C4—121no—
431C3×C4—121no—
432C3×C4—121no—
433C3×C4—121no—
434C3×C4—121no—
435C3×C4—121no—
436C3×C4—241no—
437C3×C4—241no—
438C3×C4—241no—
439C3×C4—241no—
440C3×C4#379711yescharacteristic
441C3×C4C2×#62411yescharacteristic
442C3×C4C2×C3×C4×D311yescharacteristic
443C3×C4C2×C3×C4×D311yescharacteristic
444C3×C4C3×D3×Q811yescharacteristic
445C3×C4C3×D3×Q811yescharacteristic
446Dic3—32no—
447Dic3—32no—
448Dic3—41no—
449Dic3—41no—
450Dic3—41no—
451Dic3—41no—
452Dic3—62no—
453C32—21no—
454C32—21no—
455C32—21no—
456C32—41no—
457C32#486111yescharacteristic, complemented
458C32#488111yescharacteristic, complemented
459C32C3×#42811yescharacteristic, complemented
460C23—91no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—91no—
466C2×C4—92no—
467C2×C4—92no—
468C2×C4—362no—
469C2×C4C2×C3×D3211yescharacteristic
470C8—92no—
471C8C3×#62412yescomplemented
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—41no—
482C2×C3—41no—
483C2×C3—41no—
484C2×C3—182no—
485C2×C3#6379811yescharacteristic
486C2×C3#6381111yescharacteristic
487C2×C3#6390011yescharacteristic
488C2×C3C3×D3×C4⋊C411yescharacteristic
489C2×C3C3×D3×C4⋊C411yescharacteristic
490C2×C3C3×#91111yescharacteristic
491C2×C3C3×#91411yescharacteristic
492C2×C3C3×#91511yescharacteristic
493C2×C3C3×#91811yescharacteristic
494D3—62no—
495D3—62no—
496D3—122no—
497C22—92no—
498C22—181no—
499C22—181no—
500C22C3×#380211yescharacteristic
501C4—92no—
502C4—121no—
503C4—121no—
504C4—121no—
505C4—121no—
506C4C3×#379711yescharacteristic
507C4C2×C3×#62411yescharacteristic
508C3—21no—
509C3—21no—
510C3—21no—
511C3—41no—
512C3#1059068811yescharacteristic, direct product factor
513C3C3×#486111yescharacteristic, complemented
514C3C3×#488111yescharacteristic, complemented
515C2—182no—
516C2C3×#6379811yescharacteristic
517C2C3×#6381111yescharacteristic
518C2C3×#6390011yescharacteristic
subgroupG/H
centerC22×C3#3802
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×C8D32
hypocenterC32C3×#428
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
4. C32×#818C2semidirect
5. C32×#817C2semidirect
6. C3×#63576C2semidirect
7. C3×#63574C2semidirect
10. #10590688C3direct (N × H)
15. C3×#3743C4semidirect
36. C3×#818D3semidirect
37. C3×#818C2×C3semidirect
39. C3×#817D3semidirect
40. C3×#817C2×C3semidirect
45. #63576C2×C3semidirect
46. #63574C2×C3semidirect
57. C33×C8C2×C4semidirect
58. C3×C3⋊(C3⋊C8)C2×C4semidirect
81. #3743C3×C4semidirect
138. #818C3×D3semidirect
139. #817C3×D3semidirect
190. C32×C8C4×D3semidirect
191. C32×C8C4×D3semidirect
192. C32×C8C2×C3×C4semidirect
193. C3⋊(C3⋊C8)C2×C3×C4semidirect
309. C33#428semidirect
357. C3×C8#624semidirect
358. C3×C8C3×C4×D3semidirect
359. C3×C8C3×C4×D3semidirect
457. C32#4861semidirect
458. C32#4881semidirect
459. C32C3×#428semidirect
471. C8C3×#624semidirect
512. C3#10590688direct (N × H)
513. C3C3×#4861semidirect
514. C3C3×#4881semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37432
C3×#639002
C3×#638492
C32×#8182
C32×#8172
C3×#635762
C3×#635742
C3×#48813
C3×#48613
#105906883