Finite Group Databasebeta

← C3×#10590535 #13157511

Subgroups — C3×#10590535 #13157511

Order 1728.

586 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#888C211yesmaximal, characteristic, complemented
2C3×#63864C211yesmaximal, characteristic
3C3×#63636C211yesmaximal, characteristic
4C32×#810C211yesmaximal, characteristic, complemented
5C3×#63565C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic
7C3×#63554C211yesmaximal, characteristic, complemented
8C3×#5089—31nomaximal
9C3×#4794—31nomaximal
10#10590535C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6—21no—
13C2×C32×Dic6C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3722—21no—
21C3×#3722—21no—
22C32×C4⋊Q8—31no—
23C3×#947—31no—
24C3×#888—21no—
25C3×#888D311yescharacteristic, complemented
26C3×#888C2×C311yescharacteristic, complemented
27C3×#875—31no—
28#63864C2×C311yescharacteristic
29C32×#103—31no—
30#63636C2×C311yescharacteristic
31C3×#835—31no—
32C3×#821—31no—
33C3×#815—31no—
34C3×#810—21no—
35C3×#810—31no—
36C3×#810D311yescharacteristic, complemented
37C3×#810C2×C311yescharacteristic, complemented
38C3×#809—31no—
39C3×#809—31no—
40C3×#809—31no—
41C3×#809—61no—
42C3×#803—31no—
43#63565C2×C311yescharacteristic, complemented
44#63564C2×C311yescharacteristic
45#63554C2×C311yescharacteristic, complemented
46C22×C3×C3⋊D3—21no—
47C2×C33×C4—21no—
48C2×C33×C4C2311yescharacteristic
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×Dic6—21no—
56C32×Dic6—21no—
57C32×Dic6—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60C3×#489—91no—
61#5089—31no—
62#4794—31no—
63C2×C32×Q8—31no—
64C2×C32×Q8—61no—
65C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12—61no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6—22no—
71C2×C3×Dic6—21no—
72C2×C3×Dic6—21no—
73C2×C3×Dic6C2×D311yescharacteristic
74C2×C3×Dic6C22×C311yescharacteristic
75C2×C32×C8—31no—
76C32×C4⋊C4—21no—
77C32×C4⋊C4—21no—
78C32×C4⋊C4—21no—
79C32×C4⋊C4—31no—
80C32×C4⋊C4—41no—
81C32×C4⋊C4—61no—
82C32×C4⋊C4C2×D311yescharacteristic
83C32×C4⋊C4C2×D311yescharacteristic
84C32×C4⋊C4C22×C311yescharacteristic
85C32×C42—31no—
86C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
87C3×(C2×D3)⋊C4—61no—
88C3×(C2×D3)⋊C4—61no—
89C3×(C2×D3)⋊C4—61no—
90C3×(C2×D3)⋊C4—61no—
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C22×C311yescharacteristic
94C3×Dic3⋊C4—22no—
95C3×Dic3⋊C4—21no—
96C3×Dic3⋊C4—21no—
97C3×C4×Dic3—21no—
98C3×C4×Dic3C2×D311yescharacteristic
99C3×C4×Dic3C22×C311yescharacteristic
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—61no—
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C22×C311yescharacteristic
106#3722—21no—
107#3722—21no—
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3—41no—
110C2×C3×C3⋊D3—41no—
111C2×C3×C3⋊D3—41no—
112C33×C4—41no—
113C33×C4#4111yescharacteristic
114C33×C4#4111yescharacteristic
115C32×Dic3—21no—
116C32×Dic3—21no—
117C32×Dic3—41no—
118C32×Dic3—41no—
119C3×C4⋊Q8—31no—
120C3×C4⋊Q8—31no—
121C3×C4⋊Q8—61no—
122C3×#124—91no—
123#947—31no—
124#888C3×D311yescharacteristic, complemented
125#875—31no—
126C3×#103—31no—
127C3×#103—31no—
128C3×#103—61no—
129C3×#103—91no—
130C3×#102—91no—
131C3×#102—91no—
132C3×#97—91no—
133#835—31no—
134#821—31no—
135#815—31no—
136#810—31no—
137#810C3×D311yescharacteristic, complemented
138#809—31no—
139#809—31no—
140#809—31no—
141#809—61no—
142#803—31no—
143C22×C3⋊D3—21no—
144C22×C3×D3—61no—
145C22×C3×D3—61no—
146C22×C3×D3—121no—
147C32×Q8—61no—
148C32×Q8—61no—
149C32×Q8—122no—
150C2×C32×C4—21no—
151C2×C32×C4—22no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—31no—
158C2×C32×C4—31no—
159C2×C32×C4—41no—
160C2×C32×C4—41no—
161C2×C32×C4—41no—
162C2×C32×C4—42no—
163C2×C32×C4—61no—
164C2×C32×C4—61no—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C23×C311yescharacteristic
168C4⋊(C3⋊D3)—21no—
169C4⋊(C3⋊D3)—21no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—41no—
178C2×C3×Dic3#6711yescharacteristic
179C2×C3×Dic3#6711yescharacteristic
180C2×C3×Dic3C3×D411yescharacteristic
181C2×C3×Dic3C3×D411yescharacteristic
182C3×D12—62no—
183C3×D12—61no—
184C3×D12—61no—
185C3×D12—61no—
186C3×D12—61no—
187C3×D12—121no—
188C3×Dic6—22no—
189C3×Dic6—21no—
190C3×Dic6—21no—
191C3×Dic6—21no—
192C3×Dic6—21no—
193C3×Dic6—41no—
194C3×Dic6—44no—
195C3×Dic6—42no—
196C3×Dic6—42no—
197C32×C8—61no—
198C3⋊(C3⋊C8)—21no—
199C3×C3⋊C8—21no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—41no—
202C3×C3⋊C8—61no—
203C3×C3⋊C8—61no—
204C3×C3⋊C8—121no—
205#489—91noSylow
206C2×C33#12411yescharacteristic
207C2×C33AGL(1,ℤ/8)11yescharacteristic
208C2×C33#13711yescharacteristic
209C3×C3⋊D3—81no—
210C2×C3×Q8—31no—
211C2×C3×Q8—31no—
212C2×C3×Q8—61no—
213C2×C3×Q8—61no—
214C2×C3×Q8—61no—
215C2×C3×Q8—62no—
216C2×C3×D4—91no—
217C2×D12—31no—
218C2×D12—31no—
219C2×D12—61no—
220C2×Dic6—21no—
221C2×Dic6C2×C3×D311yescharacteristic
222C2×C3×C8—31no—
223C2×C3×C8—31no—
224C2×C3×C8—61no—
225C2×C3×C8—91no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—41no—
232C3×C4⋊C4—61no—
233C3×C4⋊C4—61no—
234C3×C4⋊C4—61no—
235C3×C4⋊C4—62no—
236C3×C4⋊C4D3211yescharacteristic
237C3×C4⋊C4C2×C3×D311yescharacteristic
238C3×C4⋊C4C2×C3×D311yescharacteristic
239C3×C22⋊C4—181no—
240C3×C22⋊C4—181no—
241C3×C42—31no—
242C3×C42—31no—
243C3×C42—61no—
244(C2×D3)⋊C4—61no—
245(C2×D3)⋊C4—61no—
246(C2×D3)⋊C4—61no—
247(C2×D3)⋊C4—61no—
248C4⋊Dic3C2×C3×D311yescharacteristic
249Dic3⋊C4—21no—
250C4×Dic3C2×C3×D311yescharacteristic
251C2×C3⋊C8—31no—
252C2×C3⋊C8—31no—
253C2×C3⋊C8—61no—
254C2×C3⋊C8C2×C3×D311yescharacteristic
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—41no—
259C22×C32D3×D411yescharacteristic
260C22×C32C2×#6711yescharacteristic
261C22×C32C2×C3×D411yescharacteristic
262C2×C3⋊D3—41no—
263C2×C3⋊D3—41no—
264C2×C3⋊D3—41no—
265C2×C3×D3—122no—
266C2×C3×D3—121no—
267C2×C3×D3—121no—
268C2×C3×D3—121no—
269C2×C3×D3—121no—
270C2×C3×D3—121no—
271C2×C3×D3—121no—
272C2×C3×D3—241no—
273C2×C3×D3—241no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—41no—
281C32×C4—41no—
282C32×C4—41no—
283C32×C4—42no—
284C32×C4—41no—
285C32×C4—41no—
286C32×C4—61no—
287C32×C4—61no—
288C32×C4—81no—
289C32×C4—81no—
290C32×C4—82no—
291C32×C4—121no—
292C32×C4—121no—
293C32×C4#23511yescharacteristic
294C32×C4#23711yescharacteristic
295C32×C4#23911yescharacteristic
296C32×C4#23911yescharacteristic
297C32×C4C3×#4111yescharacteristic
298C32×C4C3×#4111yescharacteristic
299C3×Dic3—22no—
300C3×Dic3—22no—
301C3×Dic3—21no—
302C3×Dic3—21no—
303C3×Dic3—21no—
304C3×Dic3—21no—
305C3×Dic3—42no—
306C3×Dic3—42no—
307C3×Dic3—41no—
308C3×Dic3—41no—
309C3×Dic3—41no—
310C3×Dic3—41no—
311C4⋊Q8—31no—
312#124—91no—
313#103—31no—
314#103—91no—
315#102—91no—
316#102—91no—
317#97—91no—
318C33#48911yescharacteristic, complemented, Sylow
319C23×C3—181no—
320C22×D3—61no—
321C22×D3—61no—
322C22×D3—121no—
323C3×Q8—62no—
324C3×Q8—61no—
325C3×Q8—61no—
326C3×Q8—61no—
327C3×Q8—61no—
328C3×Q8—121no—
329C3×Q8—124no—
330C3×Q8—122no—
331C3×Q8—122no—
332C3×D4—181no—
333C3×D4—181no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—62no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—121no—
357C2×C3×C4C2×D3211yescharacteristic
358C2×C3×C4C22×C3×D311yescharacteristic
359C2×C3×C4C22×C3×D311yescharacteristic
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3C3×#6711yescharacteristic
363C2×Dic3C3×#6711yescharacteristic
364D12—62no—
365D12—61no—
366D12—61no—
367D12—61no—
368D12—61no—
369D12—121no—
370Dic6—21no—
371Dic6—21no—
372Dic6—42no—
373C3×C8—61no—
374C3×C8—61no—
375C3×C8—121no—
376C3×C8—181no—
377C3⋊C8—21no—
378C3⋊C8—61no—
379C3⋊C8—61no—
380C3⋊C8—121no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—41no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32#88511yescharacteristic
394C2×C32#91411yescharacteristic
395C2×C32#91811yescharacteristic
396C2×C32#93211yescharacteristic
397C2×C32#94211yescharacteristic
398C2×C32#94711yescharacteristic
399C2×C32C3×#12411yescharacteristic
400C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
401C2×C32C3×#13711yescharacteristic
402C3⋊D3—81no—
403C3×D3—242no—
404C3×D3—241no—
405C3×D3—241no—
406C2×Q8—31no—
407C2×Q8—61no—
408C2×D4—91no—
409C2×C8—31no—
410C2×C8—91no—
411C4⋊C4—31no—
412C4⋊C4—61no—
413C4⋊C4C3×D3211yescharacteristic
414C22⋊C4—181no—
415C22⋊C4—181no—
416C42—31no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—41no—
421C22×C3—361no—
422C22×C3—361no—
423C22×C3—361no—
424C22×C3D3×#6711yescharacteristic
425C22×C3C3×D3×D411yescharacteristic
426C22×C3C2×C3×#6711yescharacteristic
427C2×D3—122no—
428C2×D3—121no—
429C2×D3—121no—
430C2×D3—121no—
431C2×D3—121no—
432C2×D3—121no—
433C2×D3—121no—
434C2×D3—241no—
435C2×D3—241no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—42no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—62no—
449C3×C4—62no—
450C3×C4—61no—
451C3×C4—61no—
452C3×C4—61no—
453C3×C4—61no—
454C3×C4—81no—
455C3×C4—81no—
456C3×C4—82no—
457C3×C4—122no—
458C3×C4—122no—
459C3×C4—121no—
460C3×C4—121no—
461C3×C4—121no—
462C3×C4—121no—
463C3×C4#379611yescharacteristic
464C3×C4#379911yescharacteristic
465C3×C4C3×#23511yescharacteristic
466C3×C4C3×#23711yescharacteristic
467C3×C4C3×#23911yescharacteristic
468C3×C4C3×#23911yescharacteristic
469Dic3—21no—
470Dic3—21no—
471Dic3—41no—
472Dic3—41no—
473C32—21no—
474C32—21no—
475C32—21no—
476C32—41no—
477C32#479411yescharacteristic, complemented
478C32#508911yescharacteristic, complemented
479C32C3×#48911yescharacteristic, complemented
480C23—181no—
481Q8—61no—
482Q8—61no—
483Q8—122no—
484D4—181no—
485D4—181no—
486C2×C4—21no—
487C2×C4—31no—
488C2×C4—31no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4C2×C3×D3211yescharacteristic
492C8—61no—
493C8—181no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—41no—
504C2×C3—41no—
505C2×C3—41no—
506C2×C3—721no—
507C2×C3#6390711yescharacteristic
508C2×C3#6393411yescharacteristic
509C2×C3#6394811yescharacteristic
510C2×C3C3×#88511yescharacteristic
511C2×C3C3×#91411yescharacteristic
512C2×C3C3×#91811yescharacteristic
513C2×C3C3×#93211yescharacteristic
514C2×C3C3×#94211yescharacteristic
515C2×C3C3×#94711yescharacteristic
516D3—242no—
517D3—241no—
518D3—241no—
519C22—361no—
520C22—361no—
521C22—361no—
522C22C3×D3×#6711yescharacteristic
523C4—41no—
524C4—61no—
525C4—61no—
526C4—121no—
527C4—121no—
528C4C3×#379611yescharacteristic
529C4C3×#379911yescharacteristic
530C3—21no—
531C3—21no—
532C3—21no—
533C3—41no—
534C3#1059053511yescharacteristic, direct product factor
535C3C3×#479411yescharacteristic, complemented
536C3C3×#508911yescharacteristic, complemented
537C2—721no—
538C2C3×#6390711yescharacteristic
539C2C3×#6393411yescharacteristic
540C2C3×#6394811yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#489
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
1. C32×#888C2semidirect
4. C32×#810C2semidirect
5. C3×#63565C2semidirect
7. C3×#63554C2semidirect
10. #10590535C3direct (N × H)
25. C3×#888D3semidirect
26. C3×#888C2×C3semidirect
36. C3×#810D3semidirect
37. C3×#810C2×C3semidirect
43. #63565C2×C3semidirect
45. #63554C2×C3semidirect
124. #888C3×D3semidirect
137. #810C3×D3semidirect
318. C33#489semidirect
477. C32#4794semidirect
478. C32#5089semidirect
479. C32C3×#489semidirect
534. C3#10590535direct (N × H)
535. C3C3×#4794semidirect
536. C3C3×#5089semidirect

Maximal subgroups

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

subgroupindex
C32×#8882
C3×#638642
C3×#636362
C32×#8102
C3×#635652
C3×#635642
C3×#635542
C3×#50893
C3×#47943
#105905353