Finite Group Databasebeta

← C3×#10590681 #13157683

Subgroups — C3×#10590681 #13157683

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×C32×ΓL(1,𝔽25)C211yesmaximal, characteristic
2C3×D3×C4⋊Dic3C211yesmaximal, characteristic
3C3×#63841C211yesmaximal, characteristic
4C3×#63645C211yesmaximal, characteristic, complemented
5C32×#817C211yesmaximal, characteristic, complemented
6C3×#63575C211yesmaximal, characteristic, complemented
7C3×#63573C211yesmaximal, characteristic, complemented
8C3×#5121—31nomaximal
9C3×#4861—31nomaximal
10#10590681C311yesmaximal, characteristic, direct product factor
11C2×C32×C4×D3C2211yescharacteristic
12C2×C3×D3×Dic3—21no—
13C2×C33×C8C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C32×ΓL(1,𝔽25)C414yescomplemented
20C3×Dic3⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×Dic32—21no—
23C2×C32×M16—31no—
24C3×#926—31no—
25C2×C3×C4⋊Dic3—31no—
26C2×C3×ΓL(1,𝔽25)—21no—
27C2×C3×ΓL(1,𝔽25)D311yescharacteristic
28C2×C3×ΓL(1,𝔽25)C2×C311yescharacteristic
29C3×#892—31no—
30C3×D3×C4⋊C4—31no—
31D3×C4⋊Dic3C2×C311yescharacteristic
32#63841C2×C311yescharacteristic
33C32×#106—31no—
34#63645C2×C311yescharacteristic, complemented
35C3×#818—31no—
36C3×#818—31no—
37C3×#818—31no—
38C3×#818—61no—
39C3×#817—21no—
40C3×#817—31no—
41C3×#817D311yescharacteristic, complemented
42C3×#817C2×C311yescharacteristic, complemented
43C3×#808—31no—
44C3×#807—31no—
45#63575C2×C311yescharacteristic, complemented
46#63573C2×C311yescharacteristic, complemented
47C22×C32×D3D411yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C32×C4×D3C2×C412yes—
55C32×C4×D3Q812yes—
56C3×D3×Dic3—42no—
57C33×C8C2×C412yescomplemented
58C32×C3⋊C8C2×C412yescomplemented
59C3×#428—91no—
60#5121—31no—
61#4861—31no—
62C22×C32×C4—31no—
63C22×C3×Dic3—61no—
64C2×C3×C4×D3—21no—
65C2×C3×C4×D3—61no—
66C2×C3×C4×D3C2×D311yescharacteristic
67C2×C3×C4×D3C22×C311yescharacteristic
68C2×D3×Dic3—21no—
69C32×M16—34no—
70C2×C32×C8—21no—
71C2×C32×C8—21no—
72C2×C32×C8—21no—
73C2×C32×C8—31no—
74C2×C32×C8—41no—
75C2×C32×C8C2×D311yescharacteristic
76C2×C32×C8C2×D311yescharacteristic
77C2×C32×C8C22×C311yescharacteristic
78C32×C4⋊C4—31no—
79C32×C4⋊C4—31no—
80C4⋊(C3⋊Dic3)C22×C311yescharacteristic
81C4⋊(C3⋊Dic3)C22×C311yescharacteristic
82C3×#217—61no—
83C3×(C2×D3)⋊C4—61no—
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4⋊Dic3C22×C311yescharacteristic
96C3×C4⋊Dic3C22×C311yescharacteristic
97C3×Dic3⋊C4—61no—
98C3×C4×Dic3—61no—
99C3×C4×Dic3—61no—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C22×C311yescharacteristic
103C3×ΓL(1,𝔽25)—24no—
104C3×ΓL(1,𝔽25)Dic314yescomplemented
105C3×ΓL(1,𝔽25)C3×C414yescomplemented
106Dic3⋊Dic3—21no—
107(C2×D3)⋊Dic3—21no—
108Dic32—21no—
109C22×C33C2×D411yescharacteristic
110C2×C32×D3—21no—
111C2×C32×D3—21no—
112C2×C32×D3C4⋊C412yes—
113C33×C4C22×C411yescharacteristic
114C33×C4C2×Q811yescharacteristic
115C3×C3⋊Dic3—41no—
116C3×C3⋊Dic3—41no—
117C32×Dic3—41no—
118C32×Dic3—41no—
119C32×Dic3C4⋊C412yes—
120C2×C3×M16—31no—
121C2×C3×M16—31no—
122C2×C3×M16—61no—
123C3×#117—91no—
124C2×C3×C4⋊C4—91no—
125#926—31no—
126C2×C4⋊Dic3—31no—
127C2×ΓL(1,𝔽25)C3×D311yescharacteristic
128#892—31no—
129D3×C4⋊C4—31no—
130C3×#107—91no—
131C3×#107—91no—
132C3×#106—31no—
133C3×#106—31no—
134C3×#106—61no—
135C3×#106—91no—
136#818—31no—
137#818—31no—
138#818—31no—
139#818—61no—
140#817—31no—
141#817C3×D311yescharacteristic, complemented
142#808—31no—
143#807—31no—
144C23×C32—31no—
145C22×C3×D3—21no—
146C22×C3×D3D1211yescharacteristic
147C22×C3×D3C3×D411yescharacteristic
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—31no—
152C2×C32×C4—32no—
153C2×C32×C4—32no—
154C2×C32×C4—41no—
155C2×C32×C4—61no—
156C2×C32×C4—61no—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C23×C311yescharacteristic
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—41no—
168C2×C3×Dic3—41no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—121no—
174C2×C3×Dic3—121no—
175C2×C3×Dic3—122no—
176C2×C3×Dic3D1211yescharacteristic
177C2×C3×Dic3C3×D411yescharacteristic
178C3×C4×D3—22no—
179C3×C4×D3—22no—
180C3×C4×D3—122no—
181C3×C4×D3Dic612yes—
182C3×C4×D3C2×Dic312yes—
183C3×C4×D3C2×C3×C412yes—
184C3×C4×D3C3×Q812yes—
185D3×Dic3—42no—
186C32×C8—22no—
187C32×C8—22no—
188C32×C8—22no—
189C32×C8—32no—
190C32×C8—42no—
191C32×C8C4×D312yescomplemented
192C32×C8C2×Dic312yescomplemented
193C32×C8C2×C3×C412yescomplemented
194C3×C3⋊C8—22no—
195C3×C3⋊C8C2×Dic312yescomplemented
196C3×C3⋊C8C2×C3×C412yescomplemented
197#428—91noSylow
198C2×C33C2×C4⋊C411yescharacteristic
199C2×C33AGL(1,ℤ/8)11yescharacteristic
200C2×C33#13711yescharacteristic
201C32×D3—22no—
202C22×C3×C4—31no—
203C22×C3×C4—31no—
204C22×C3×C4—61no—
205C22×C3×C4—181no—
206C22×Dic3—61no—
207C2×C4×D3—61no—
208C2×C4×D3C2×C3×D311yescharacteristic
209C3×M16—34no—
210C3×M16—34no—
211C3×M16—64no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—31no—
216C2×C3×C8—31no—
217C2×C3×C8—41no—
218C2×C3×C8—61no—
219C2×C3×C8D3211yescharacteristic
220C2×C3×C8C2×C3×D311yescharacteristic
221C2×C3×C8C2×C3×D311yescharacteristic
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—91no—
229C3×C4⋊C4—91no—
230C3×C4⋊C4—181no—
231C3×C22⋊C4—181no—
232C3×C42—181no—
233#217—61no—
234(C2×D3)⋊C4—61no—
235C4⋊Dic3—31no—
236C4⋊Dic3—31no—
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239C4⋊Dic3—61no—
240C4⋊Dic3—61no—
241C4⋊Dic3—61no—
242C4⋊Dic3C2×C3×D311yescharacteristic
243C4⋊Dic3C2×C3×D311yescharacteristic
244Dic3⋊C4—61no—
245C4×Dic3—61no—
246C4×Dic3—61no—
247C2×C3⋊C8C2×C3×D311yescharacteristic
248ΓL(1,𝔽25)C3×Dic314yescomplemented
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—32no—
253C22×C32—41no—
254C22×C32—61no—
255C22×C32—61no—
256C22×C32C2×D1211yescharacteristic
257C22×C32D3×D411yescharacteristic
258C22×C32C2×C3×D411yescharacteristic
259C2×C3×D3—21no—
260C2×C3×D3—21no—
261C2×C3×D3—21no—
262C2×C3×D3—21no—
263C2×C3×D3—22no—
264C2×C3×D3—41no—
265C2×C3×D3—41no—
266C2×C3×D3C4⋊Dic312yes—
267C2×C3×D3C3×C4⋊C412yes—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—32no—
275C32×C4—41no—
276C32×C4—41no—
277C32×C4—121no—
278C32×C4—121no—
279C32×C4C2×Dic611yescharacteristic
280C32×C4C2×C4×D311yescharacteristic
281C32×C4D3×Q811yescharacteristic
282C32×C4C22×Dic311yescharacteristic
283C32×C4C22×C3×C411yescharacteristic
284C32×C4C2×C3×Q811yescharacteristic
285C3⋊Dic3—41no—
286C3⋊Dic3—41no—
287C3×Dic3—22no—
288C3×Dic3—41no—
289C3×Dic3—41no—
290C3×Dic3—41no—
291C3×Dic3—41no—
292C3×Dic3—81no—
293C3×Dic3—81no—
294C3×Dic3—121no—
295C3×Dic3—121no—
296C3×Dic3—121no—
297C3×Dic3—121no—
298C3×Dic3—241no—
299C3×Dic3—241no—
300C3×Dic3C4⋊Dic312yes—
301C3×Dic3C3×C4⋊C412yes—
302C2×M16—31no—
303#117—91no—
304C2×C4⋊C4—91no—
305#107—91no—
306#107—91no—
307#106—31no—
308#106—91no—
309C33#42811yescharacteristic, complemented, Sylow
310C23×C3—31no—
311C23×C3—31no—
312C23×C3—61no—
313C22×D3C3×D1211yescharacteristic
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—31no—
318C2×C3×C4—31no—
319C2×C3×C4—32no—
320C2×C3×C4—32no—
321C2×C3×C4—32no—
322C2×C3×C4—32no—
323C2×C3×C4—41no—
324C2×C3×C4—62no—
325C2×C3×C4—62no—
326C2×C3×C4—61no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—121no—
332C2×C3×C4—121no—
333C2×C3×C4—181no—
334C2×C3×C4—181no—
335C2×C3×C4—362no—
336C2×C3×C4C2×D3211yescharacteristic
337C2×C3×C4C22×C3×D311yescharacteristic
338C2×C3×C4C22×C3×D311yescharacteristic
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—61no—
345C2×Dic3—121no—
346C2×Dic3—121no—
347C2×Dic3—122no—
348C2×Dic3C3×D1211yescharacteristic
349C4×D3—122no—
350C4×D3C3×Dic612yes—
351C4×D3C2×C3×Dic312yes—
352C3×C8—22no—
353C3×C8—22no—
354C3×C8—22no—
355C3×C8—32no—
356C3×C8—32no—
357C3×C8—42no—
358C3×C8—62no—
359C3×C8D3×Dic312yescomplemented
360C3×C8C3×C4×D312yescomplemented
361C3×C8C2×C3×Dic312yescomplemented
362C3⋊C8C2×C3×Dic312yescomplemented
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×C32—62no—
376C2×C32D3×C4⋊C411yescharacteristic
377C2×C32#90811yescharacteristic
378C2×C32#90911yescharacteristic
379C2×C32#91411yescharacteristic
380C2×C32#91511yescharacteristic
381C2×C32C2×C4⋊Dic311yescharacteristic
382C2×C32C2×C3×C4⋊C411yescharacteristic
383C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
384C2×C32C3×#13711yescharacteristic
385C3×D3—22no—
386C3×D3—22no—
387C3×D3—42no—
388C22×C4—31no—
389C22×C4—181no—
390M16—34no—
391C2×C8—31no—
392C2×C8C3×D3211yescharacteristic
393C4⋊C4—31no—
394C4⋊C4—31no—
395C4⋊C4—91no—
396C4⋊C4—91no—
397C4⋊C4—181no—
398C22⋊C4—181no—
399C42—181no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—32no—
404C22×C3—32no—
405C22×C3—41no—
406C22×C3—62no—
407C22×C3—61no—
408C22×C3—61no—
409C22×C3—61no—
410C22×C3—61no—
411C22×C3—121no—
412C22×C3—121no—
413C22×C3D3×D1211yescharacteristic
414C22×C3C2×C3×D1211yescharacteristic
415C22×C3C3×D3×D411yescharacteristic
416C2×D3—21no—
417C2×D3—21no—
418C2×D3C3×C4⋊Dic312yes—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—32no—
426C3×C4—32no—
427C3×C4—41no—
428C3×C4—41no—
429C3×C4—62no—
430C3×C4—121no—
431C3×C4—121no—
432C3×C4—121no—
433C3×C4—121no—
434C3×C4—241no—
435C3×C4—241no—
436C3×C4—361no—
437C3×C4—361no—
438C3×C4D3×Dic611yescharacteristic
439C3×C4C2×D3×Dic311yescharacteristic
440C3×C4C2×C3×Dic611yescharacteristic
441C3×C4C2×C3×C4×D311yescharacteristic
442C3×C4C3×D3×Q811yescharacteristic
443C3×C4C22×C3×Dic311yescharacteristic
444Dic3—41no—
445Dic3—41no—
446Dic3—121no—
447Dic3—121no—
448Dic3—121no—
449Dic3—121no—
450Dic3—241no—
451Dic3—241no—
452Dic3C3×C4⋊Dic312yes—
453C32—21no—
454C32—21no—
455C32—21no—
456C32—41no—
457C32#486111yescharacteristic, complemented
458C32#512111yescharacteristic, complemented
459C32C3×#42811yescharacteristic, complemented
460C23—31no—
461C2×C4—31no—
462C2×C4—32no—
463C2×C4—32no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—181no—
467C2×C4—181no—
468C2×C4—362no—
469C2×C4C2×C3×D3211yescharacteristic
470C8—32no—
471C8C3×D3×Dic312yescomplemented
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—62no—
485C2×C3—62no—
486C2×C3—122no—
487C2×C3#6379411yescharacteristic
488C2×C3#6380011yescharacteristic
489C2×C3D3×C4⋊Dic311yescharacteristic
490C2×C3C3×D3×C4⋊C411yescharacteristic
491C2×C3C3×#90811yescharacteristic
492C2×C3C3×#90911yescharacteristic
493C2×C3C3×#91411yescharacteristic
494C2×C3C3×#91511yescharacteristic
495C2×C3C2×C3×C4⋊Dic311yescharacteristic
496D3—22no—
497C22—32no—
498C22—61no—
499C22—61no—
500C22C3×D3×D1211yescharacteristic
501C4—32no—
502C4—121no—
503C4—121no—
504C4—361no—
505C4—361no—
506C4C3×D3×Dic611yescharacteristic
507C4C2×C3×D3×Dic311yescharacteristic
508C3—21no—
509C3—21no—
510C3—21no—
511C3—41no—
512C3#1059068111yescharacteristic, direct product factor
513C3C3×#486111yescharacteristic, complemented
514C3C3×#512111yescharacteristic, complemented
515C2—62no—
516C2C3×#6379411yescharacteristic
517C2C3×#6380011yescharacteristic
518C2C3×D3×C4⋊Dic311yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C32×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. C3×#63645C2semidirect
5. C32×#817C2semidirect
6. C3×#63575C2semidirect
7. C3×#63573C2semidirect
10. #10590681C3direct (N × H)
19. C32×ΓL(1,𝔽25)C4semidirect
34. #63645C2×C3semidirect
41. C3×#817D3semidirect
42. C3×#817C2×C3semidirect
45. #63575C2×C3semidirect
46. #63573C2×C3semidirect
57. C33×C8C2×C4semidirect
58. C32×C3⋊C8C2×C4semidirect
104. C3×ΓL(1,𝔽25)Dic3semidirect
105. C3×ΓL(1,𝔽25)C3×C4semidirect
141. #817C3×D3semidirect
191. C32×C8C4×D3semidirect
192. C32×C8C2×Dic3semidirect
193. C32×C8C2×C3×C4semidirect
195. C3×C3⋊C8C2×Dic3semidirect
196. C3×C3⋊C8C2×C3×C4semidirect
248. ΓL(1,𝔽25)C3×Dic3semidirect
309. C33#428semidirect
359. C3×C8D3×Dic3semidirect
360. C3×C8C3×C4×D3semidirect
361. C3×C8C2×C3×Dic3semidirect
362. C3⋊C8C2×C3×Dic3semidirect
457. C32#4861semidirect
458. C32#5121semidirect
459. C32C3×#428semidirect
471. C8C3×D3×Dic3semidirect
512. C3#10590681direct (N × H)
513. C3C3×#4861semidirect
514. C3C3×#5121semidirect

Maximal subgroups

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

subgroupindex
C2×C32×ΓL(1,𝔽25)2
C3×D3×C4⋊Dic32
C3×#638412
C3×#636452
C32×#8172
C3×#635752
C3×#635732
C3×#51213
C3×#48613
#105906813