Finite Group Databasebeta

← C3×#10590545 #13157524

Subgroups — C3×#10590545 #13157524

Order 1728.

600 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#893C211yesmaximal, characteristic
2C3×#63869C211yesmaximal, characteristic, complemented
3C3×#63634C211yesmaximal, characteristic, complemented
4C32×#808C211yesmaximal, characteristic, complemented
5C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic
7C3×#63562C211yesmaximal, characteristic
8C3×#5009—31nomaximal
9C3×#4871—31nomaximal
10#10590545C311yesmaximal, characteristic, direct product factor
11C2×C32×D12C2211yescharacteristic
12C2×C32×C4×D3—21no—
13C33×C4⋊C4C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×(C2×D3)⋊C4—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3724—21no—
21C3×Dic3⋊Dic3—21no—
22C32×C4×D4—31no—
23C3×#893—21no—
24C3×#893D311yescharacteristic
25C3×#893C2×C311yescharacteristic
26C3×#890—31no—
27C3×#870—31no—
28#63869C2×C311yescharacteristic, complemented
29C32×#106—31no—
30#63634C2×C311yescharacteristic, complemented
31C3×#832—31no—
32C3×#832—31no—
33C3×#821—31no—
34C3×Dic3⋊C8—31no—
35C3×#809—31no—
36C3×#808—21no—
37C3×#808D311yescharacteristic, complemented
38C3×#808C2×C311yescharacteristic, complemented
39C3×#807—31no—
40C3×#807—31no—
41C3×#807—61no—
42C3×C4⋊(C3⋊C8)—31no—
43Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
44#63563C2×C311yescharacteristic
45#63562C2×C311yescharacteristic
46C22×C32×D3—21no—
47C2×C33×C4—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3D411yescharacteristic
52C2×C32×Dic3D411yescharacteristic
53C32×D12—21no—
54C32×D12Q812yes—
55C32×C4×D3—42no—
56C3×C3⋊(C3⋊C8)—21no—
57C32×C3⋊C8—21no—
58C3×#478—91no—
59#5009—31no—
60#4871—31no—
61C2×C32×D4—31no—
62C22×C32×C4—61no—
63C2×C3×D12—21no—
64C2×C3×D12C2×D311yescharacteristic
65C2×C3×D12C22×C311yescharacteristic
66C2×C3×C4×D3—22no—
67C2×C3×C4×D3—21no—
68C2×C3×C4×D3—21no—
69C2×C32×C8—31no—
70C32×C4⋊C4—21no—
71C32×C4⋊C4—21no—
72C32×C4⋊C4—21no—
73C32×C4⋊C4—31no—
74C32×C4⋊C4—41no—
75C32×C4⋊C4C2×D311yescharacteristic
76C32×C4⋊C4C2×D311yescharacteristic
77C32×C4⋊C4C22×C311yescharacteristic
78C32×C22⋊C4—61no—
79C32×C42—31no—
80C4⋊(C3⋊Dic3)C22×C311yescharacteristic
81C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
82C3×(C2×D3)⋊C4—22no—
83C3×(C2×D3)⋊C4—21no—
84C3×(C2×D3)⋊C4—21no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C22×C311yescharacteristic
92C3×Dic3⋊C4—61no—
93C3×Dic3⋊C4—61no—
94C3×Dic3⋊C4—61no—
95C3×C4×Dic3—21no—
96C3×C4×Dic3C2×D311yescharacteristic
97C3×C4×Dic3C22×C311yescharacteristic
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—61no—
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C22×C311yescharacteristic
104#3724—21no—
105Dic3⋊Dic3—21no—
106C22×C33C2×D411yescharacteristic
107C2×C32×D3—22no—
108C2×C32×D3—41no—
109C2×C32×D3—41no—
110C33×C4—41no—
111C33×C4C2×Q811yescharacteristic
112C33×C4#4111yescharacteristic
113C3×C3⋊Dic3—41no—
114C32×Dic3—21no—
115C32×Dic3—21no—
116C32×Dic3—41no—
117C3×#125—91no—
118C3×C4×D4—31no—
119C3×C4×D4—31no—
120C3×C4×D4—61no—
121#893C3×D311yescharacteristic
122#890—31no—
123#870—31no—
124C3×#107—91no—
125C3×#106—31no—
126C3×#106—31no—
127C3×#106—61no—
128C3×C4⋊C8—91no—
129C3×#102—91no—
130C3×#102—91no—
131#832—31no—
132#832—31no—
133#821—31no—
134Dic3⋊C8—31no—
135#809—31no—
136#808C3×D311yescharacteristic, complemented
137#807—31no—
138#807—31no—
139#807—61no—
140C4⋊(C3⋊C8)—31no—
141C23×C32—61no—
142C22×C3×D3—22no—
143C22×C3×D3—21no—
144C22×C3×D3—21no—
145C32×D4—32no—
146C32×D4—61no—
147C2×C32×C4—21no—
148C2×C32×C4—22no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—31no—
155C2×C32×C4—31no—
156C2×C32×C4—41no—
157C2×C32×C4—41no—
158C2×C32×C4—41no—
159C2×C32×C4—42no—
160C2×C32×C4—61no—
161C2×C32×C4—122no—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C23×C311yescharacteristic
165C2×C3⋊Dic3—21no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—41no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—121no—
174C2×C3×Dic3#6711yescharacteristic
175C2×C3×Dic3#6711yescharacteristic
176C2×C3×Dic3C3×D411yescharacteristic
177C2×C3×Dic3C3×D411yescharacteristic
178C3×D12—22no—
179C3×D12—21no—
180C3×D12—21no—
181C3×D12—22no—
182C3×D12Dic612yes—
183C3×D12C3×Q812yes—
184C3×C4×D3—44no—
185C3×C4×D3—42no—
186C3×C4×D3—42no—
187C32×C8—61no—
188C3⋊(C3⋊C8)—21no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8—21no—
191C3×C3⋊C8—21no—
192C3×C3⋊C8—62no—
193C3×C3⋊C8—61no—
194C3×C3⋊C8—61no—
195#478—91noSylow
196C2×C33C22⋊Q811yescharacteristic
197C2×C33#13511yescharacteristic
198C2×C33AGL(1,ℤ/8)11yescharacteristic
199C32×D3—42no—
200C2×C3×D4—31no—
201C2×C3×D4—31no—
202C2×C3×D4—61no—
203C22×C3×C4—61no—
204C22×C3×C4—61no—
205C22×C3×C4—62no—
206C2×D12C2×C3×D311yescharacteristic
207C2×C4×D3—21no—
208C2×C3×C8—31no—
209C2×C3×C8—31no—
210C2×C3×C8—61no—
211C2×C3×C8—91no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—31no—
216C3×C4⋊C4—31no—
217C3×C4⋊C4—41no—
218C3×C4⋊C4—61no—
219C3×C4⋊C4—91no—
220C3×C4⋊C4—181no—
221C3×C4⋊C4—181no—
222C3×C4⋊C4D3211yescharacteristic
223C3×C4⋊C4C2×C3×D311yescharacteristic
224C3×C4⋊C4C2×C3×D311yescharacteristic
225C3×C22⋊C4—61no—
226C3×C22⋊C4—61no—
227C3×C22⋊C4—62no—
228C3×C42—31no—
229C3×C42—31no—
230C3×C42—61no—
231(C2×D3)⋊C4—21no—
232C4⋊Dic3—31no—
233C4⋊Dic3—31no—
234C4⋊Dic3—61no—
235C4⋊Dic3—61no—
236C4⋊Dic3C2×C3×D311yescharacteristic
237Dic3⋊C4—61no—
238Dic3⋊C4—61no—
239Dic3⋊C4—61no—
240C4×Dic3C2×C3×D311yescharacteristic
241C2×C3⋊C8—31no—
242C2×C3⋊C8—31no—
243C2×C3⋊C8—61no—
244C2×C3⋊C8C2×C3×D311yescharacteristic
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—41no—
249C22×C32—62no—
250C22×C32—121no—
251C22×C32—121no—
252C22×C32D3×D411yescharacteristic
253C22×C32C2×#6711yescharacteristic
254C22×C32C2×C3×D411yescharacteristic
255C2×C3×D3—24no—
256C2×C3×D3—22no—
257C2×C3×D3—22no—
258C2×C3×D3—42no—
259C2×C3×D3—42no—
260C2×C3×D3—41no—
261C2×C3×D3—41no—
262C2×C3×D3—41no—
263C2×C3×D3—41no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—41no—
271C32×C4—41no—
272C32×C4—41no—
273C32×C4—42no—
274C32×C4—41no—
275C32×C4—41no—
276C32×C4—61no—
277C32×C4—61no—
278C32×C4—81no—
279C32×C4—81no—
280C32×C4—82no—
281C32×C4—121no—
282C32×C4C2×Dic611yescharacteristic
283C32×C4#23511yescharacteristic
284C32×C4#23511yescharacteristic
285C32×C4D3×Q811yescharacteristic
286C32×C4C2×C3×Q811yescharacteristic
287C32×C4C3×#4111yescharacteristic
288C3⋊Dic3—41no—
289C3×Dic3—22no—
290C3×Dic3—21no—
291C3×Dic3—21no—
292C3×Dic3—21no—
293C3×Dic3—21no—
294C3×Dic3—41no—
295C3×Dic3—41no—
296C3×Dic3—41no—
297C3×Dic3—81no—
298C3×Dic3—121no—
299C3×Dic3—121no—
300C3×Dic3—241no—
301#125—91no—
302C4×D4—31no—
303#107—91no—
304#106—31no—
305C4⋊C8—91no—
306#102—91no—
307#102—91no—
308C33#47811yescharacteristic, complemented, Sylow
309C23×C3—62no—
310C23×C3—61no—
311C23×C3—61no—
312C22×D3—21no—
313C3×D4—32no—
314C3×D4—32no—
315C3×D4—62no—
316C3×D4—62no—
317C3×D4—61no—
318C3×D4—61no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—31no—
327C2×C3×C4—31no—
328C2×C3×C4—31no—
329C2×C3×C4—31no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—41no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—121no—
339C2×C3×C4—124no—
340C2×C3×C4—122no—
341C2×C3×C4—122no—
342C2×C3×C4—181no—
343C2×C3×C4C2×D3211yescharacteristic
344C2×C3×C4C22×C3×D311yescharacteristic
345C2×C3×C4C22×C3×D311yescharacteristic
346C2×Dic3—21no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—121no—
350C2×Dic3C3×#6711yescharacteristic
351C2×Dic3C3×#6711yescharacteristic
352D12—21no—
353D12C3×Dic612yes—
354C4×D3—42no—
355C3×C8—62no—
356C3×C8—61no—
357C3×C8—61no—
358C3×C8—181no—
359C3⋊C8—21no—
360C3⋊C8—62no—
361C3⋊C8—61no—
362C3⋊C8—61no—
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—122no—
376C2×C32#89611yescharacteristic
377C2×C32#91411yescharacteristic
378C2×C32#91611yescharacteristic
379C2×C32#92411yescharacteristic
380C2×C32#93211yescharacteristic
381C2×C32#95011yescharacteristic
382C2×C32C3×C22⋊Q811yescharacteristic
383C2×C32C3×#13511yescharacteristic
384C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
385C3×D3—44no—
386C3×D3—42no—
387C3×D3—42no—
388C2×D4—31no—
389C22×C4—61no—
390C2×C8—31no—
391C2×C8—91no—
392C4⋊C4—31no—
393C4⋊C4—91no—
394C4⋊C4—181no—
395C4⋊C4—181no—
396C4⋊C4C3×D3211yescharacteristic
397C22⋊C4—61no—
398C42—31no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—41no—
403C22×C3—64no—
404C22×C3—62no—
405C22×C3—62no—
406C22×C3—122no—
407C22×C3—122no—
408C22×C3—121no—
409C22×C3—121no—
410C22×C3—121no—
411C22×C3—121no—
412C22×C3D3×#6711yescharacteristic
413C22×C3C3×D3×D411yescharacteristic
414C22×C3C2×C3×#6711yescharacteristic
415C2×D3—22no—
416C2×D3—41no—
417C2×D3—41no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—42no—
425C3×C4—41no—
426C3×C4—41no—
427C3×C4—41no—
428C3×C4—41no—
429C3×C4—41no—
430C3×C4—62no—
431C3×C4—61no—
432C3×C4—61no—
433C3×C4—61no—
434C3×C4—61no—
435C3×C4—81no—
436C3×C4—81no—
437C3×C4—82no—
438C3×C4—121no—
439C3×C4—121no—
440C3×C4—121no—
441C3×C4—241no—
442C3×C4—361no—
443C3×C4D3×Dic611yescharacteristic
444C3×C4#379811yescharacteristic
445C3×C4C2×C3×Dic611yescharacteristic
446C3×C4C3×#23511yescharacteristic
447C3×C4C3×#23511yescharacteristic
448C3×C4C3×D3×Q811yescharacteristic
449Dic3—21no—
450Dic3—21no—
451Dic3—41no—
452Dic3—121no—
453Dic3—121no—
454Dic3—241no—
455C32—21no—
456C32—21no—
457C32—21no—
458C32—41no—
459C32#487111yescharacteristic, complemented
460C32#500911yescharacteristic, complemented
461C32C3×#47811yescharacteristic, complemented
462C23—61no—
463D4—32no—
464D4—61no—
465C2×C4—21no—
466C2×C4—31no—
467C2×C4—31no—
468C2×C4—61no—
469C2×C4—122no—
470C2×C4—181no—
471C2×C4C2×C3×D3211yescharacteristic
472C8—61no—
473C8—181no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—124no—
487C2×C3—122no—
488C2×C3—122no—
489C2×C3#6385111yescharacteristic
490C2×C3#6393411yescharacteristic
491C2×C3#6394711yescharacteristic
492C2×C3C3×#89611yescharacteristic
493C2×C3C3×#91411yescharacteristic
494C2×C3C3×#91611yescharacteristic
495C2×C3C3×#92411yescharacteristic
496C2×C3C3×#93211yescharacteristic
497C2×C3C3×#95011yescharacteristic
498D3—42no—
499C22—62no—
500C22—121no—
501C22—121no—
502C22C3×D3×#6711yescharacteristic
503C4—41no—
504C4—61no—
505C4—61no—
506C4—121no—
507C4—361no—
508C4C3×D3×Dic611yescharacteristic
509C4C3×#379811yescharacteristic
510C3—21no—
511C3—21no—
512C3—21no—
513C3—41no—
514C3#1059054511yescharacteristic, direct product factor
515C3C3×#487111yescharacteristic, complemented
516C3C3×#500911yescharacteristic, complemented
517C2—122no—
518C2C3×#6385111yescharacteristic
519C2C3×#6393411yescharacteristic
520C2C3×#6394711yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#478
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#63869C2semidirect
3. C3×#63634C2semidirect
4. C32×#808C2semidirect
5. C3×Dic3⋊(C3⋊C8)C2semidirect
10. #10590545C3direct (N × H)
28. #63869C2×C3semidirect
30. #63634C2×C3semidirect
37. C3×#808D3semidirect
38. C3×#808C2×C3semidirect
43. Dic3⋊(C3⋊C8)C2×C3semidirect
136. #808C3×D3semidirect
308. C33#478semidirect
459. C32#4871semidirect
460. C32#5009semidirect
461. C32C3×#478semidirect
514. C3#10590545direct (N × H)
515. C3C3×#4871semidirect
516. C3C3×#5009semidirect

Maximal subgroups

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

subgroupindex
C32×#8932
C3×#638692
C3×#636342
C32×#8082
C3×Dic3⋊(C3⋊C8)2
C3×#635632
C3×#635622
C3×#50093
C3×#48713
#105905453