Finite Group Databasebeta

← C3×#10590537 #13157513

Subgroups — C3×#10590537 #13157513

Order 1728.

558 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#890C211yesmaximal, characteristic, complemented
2C3×#63864C211yesmaximal, characteristic
3C3×#63636C211yesmaximal, characteristic
4C32×#810C211yesmaximal, characteristic, complemented
5C3×#63565C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8C3×#5068—31nomaximal
9C3×#4800—31nomaximal
10#10590537C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3722—21no—
21C3×#3722—21no—
22C32×#125—31no—
23C3×#947—31no—
24C3×#890—21no—
25C3×#890D311yescharacteristic, complemented
26C3×#890C2×C311yescharacteristic, complemented
27C3×#875—31no—
28#63864C2×C311yescharacteristic
29C32×#103—31no—
30#63636C2×C311yescharacteristic
31C3×#835—31no—
32C3×#821—31no—
33C3×C8×Dic3—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×C4×C3⋊C8—31no—
43#63565C2×C311yescharacteristic, complemented
44#63564C2×C311yescharacteristic
45Dic3×C3⋊C8C2×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—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59C3×#488—91no—
60#5068—31no—
61#4800—31no—
62C2×C32×Q8—31no—
63C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
64C2×C3×D12—31no—
65C2×C3×D12—31no—
66C2×C3×D12—61no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6C2×D311yescharacteristic
69C2×C3×Dic6C22×C311yescharacteristic
70C2×C32×C8—31no—
71C32×C4⋊C4—21no—
72C32×C4⋊C4—21no—
73C32×C4⋊C4—21no—
74C32×C4⋊C4—31no—
75C32×C4⋊C4—41no—
76C32×C4⋊C4—61no—
77C32×C4⋊C4—61no—
78C32×C4⋊C4C2×D311yescharacteristic
79C32×C4⋊C4C2×D311yescharacteristic
80C32×C4⋊C4C22×C311yescharacteristic
81C32×C42—31no—
82C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
83C3×(C2×D3)⋊C4—61no—
84C3×(C2×D3)⋊C4—61no—
85C3×(C2×D3)⋊C4—61no—
86C3×(C2×D3)⋊C4—61no—
87C3×C4⋊Dic3—22no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C22×C311yescharacteristic
93C3×Dic3⋊C4—22no—
94C3×Dic3⋊C4—21no—
95C3×Dic3⋊C4—21no—
96C3×C4×Dic3—21no—
97C3×C4×Dic3C2×D311yescharacteristic
98C3×C4×Dic3C22×C311yescharacteristic
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C22×C311yescharacteristic
105#3722—21no—
106#3722—21no—
107C22×C33C2×D411yescharacteristic
108C2×C3×C3⋊D3—41no—
109C2×C3×C3⋊D3—41no—
110C2×C3×C3⋊D3—41no—
111C33×C4—41no—
112C33×C4#4111yescharacteristic
113C33×C4#4111yescharacteristic
114C32×Dic3—21no—
115C32×Dic3—21no—
116C32×Dic3—41no—
117C32×Dic3—41no—
118C3×#125—31no—
119C3×#125—31no—
120C3×#125—61no—
121C3×#124—91no—
122#947—31no—
123#890C3×D311yescharacteristic, complemented
124#875—31no—
125C3×#103—31no—
126C3×#103—31no—
127C3×#103—61no—
128C3×#103—91no—
129C3×#102—91no—
130C3×#102—91no—
131C3×C4×C8—91no—
132#835—31no—
133#821—31no—
134C8×Dic3—31no—
135#810—31no—
136#810C3×D311yescharacteristic, complemented
137#809—31no—
138#809—31no—
139#809—31no—
140#809—61no—
141C4×C3⋊C8—31no—
142C22×C3⋊D3—21no—
143C22×C3×D3—61no—
144C22×C3×D3—61no—
145C22×C3×D3—121no—
146C32×Q8—61no—
147C32×Q8—61no—
148C2×C32×C4—21no—
149C2×C32×C4—22no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—31no—
156C2×C32×C4—31no—
157C2×C32×C4—41no—
158C2×C32×C4—41no—
159C2×C32×C4—41no—
160C2×C32×C4—42no—
161C2×C32×C4—61no—
162C2×C32×C4—61no—
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C23×C311yescharacteristic
166C4⋊(C3⋊D3)—21no—
167C4⋊(C3⋊D3)—21no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—41no—
176C2×C3×Dic3#6711yescharacteristic
177C2×C3×Dic3#6711yescharacteristic
178C2×C3×Dic3C3×D411yescharacteristic
179C2×C3×Dic3C3×D411yescharacteristic
180C3×D12—62no—
181C3×D12—61no—
182C3×D12—61no—
183C3×D12—61no—
184C3×D12—61no—
185C3×D12—121no—
186C3×Dic6—22no—
187C3×Dic6—21no—
188C3×Dic6—21no—
189C3×Dic6—21no—
190C3×Dic6—21no—
191C3×Dic6—41no—
192C32×C8—61no—
193C3⋊(C3⋊C8)—21no—
194C3×C3⋊C8—21no—
195C3×C3⋊C8—21no—
196C3×C3⋊C8—41no—
197C3×C3⋊C8—61no—
198C3×C3⋊C8—61no—
199C3×C3⋊C8—121no—
200#488—91noSylow
201C2×C33#12411yescharacteristic
202C2×C33#13511yescharacteristic
203C2×C33#13511yescharacteristic
204C3×C3⋊D3—81no—
205C2×C3×Q8—31no—
206C2×C3×Q8—31no—
207C2×C3×Q8—61no—
208C2×C3×D4—91no—
209C2×D12—31no—
210C2×D12—31no—
211C2×D12—61no—
212C2×Dic6C2×C3×D311yescharacteristic
213C2×C3×C8—31no—
214C2×C3×C8—31no—
215C2×C3×C8—61no—
216C2×C3×C8—91no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—31no—
222C3×C4⋊C4—41no—
223C3×C4⋊C4—61no—
224C3×C4⋊C4—61no—
225C3×C4⋊C4—61no—
226C3×C4⋊C4—61no—
227C3×C4⋊C4—61no—
228C3×C4⋊C4—62no—
229C3×C4⋊C4—62no—
230C3×C4⋊C4D3211yescharacteristic
231C3×C4⋊C4C2×C3×D311yescharacteristic
232C3×C4⋊C4C2×C3×D311yescharacteristic
233C3×C22⋊C4—181no—
234C3×C22⋊C4—181no—
235C3×C42—31no—
236C3×C42—31no—
237C3×C42—61no—
238(C2×D3)⋊C4—61no—
239(C2×D3)⋊C4—61no—
240(C2×D3)⋊C4—61no—
241(C2×D3)⋊C4—61no—
242C4⋊Dic3—21no—
243C4⋊Dic3C2×C3×D311yescharacteristic
244Dic3⋊C4—21no—
245C4×Dic3C2×C3×D311yescharacteristic
246C2×C3⋊C8—31no—
247C2×C3⋊C8—31no—
248C2×C3⋊C8—61no—
249C2×C3⋊C8C2×C3×D311yescharacteristic
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—41no—
254C22×C32D3×D411yescharacteristic
255C22×C32C2×#6711yescharacteristic
256C22×C32C2×C3×D411yescharacteristic
257C2×C3⋊D3—41no—
258C2×C3⋊D3—41no—
259C2×C3⋊D3—41no—
260C2×C3×D3—122no—
261C2×C3×D3—121no—
262C2×C3×D3—121no—
263C2×C3×D3—121no—
264C2×C3×D3—121no—
265C2×C3×D3—121no—
266C2×C3×D3—121no—
267C2×C3×D3—241no—
268C2×C3×D3—241no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—41no—
276C32×C4—41no—
277C32×C4—41no—
278C32×C4—42no—
279C32×C4—41no—
280C32×C4—41no—
281C32×C4—61no—
282C32×C4—61no—
283C32×C4—81no—
284C32×C4—81no—
285C32×C4—82no—
286C32×C4—121no—
287C32×C4—121no—
288C32×C4#23511yescharacteristic
289C32×C4#23711yescharacteristic
290C32×C4#23911yescharacteristic
291C32×C4#23911yescharacteristic
292C32×C4C3×#4111yescharacteristic
293C32×C4C3×#4111yescharacteristic
294C3×Dic3—21no—
295C3×Dic3—21no—
296C3×Dic3—21no—
297C3×Dic3—21no—
298C3×Dic3—41no—
299C3×Dic3—41no—
300C3×Dic3—42no—
301C3×Dic3—42no—
302C3×Dic3—41no—
303C3×Dic3—41no—
304C3×Dic3—41no—
305C3×Dic3—41no—
306#125—31no—
307#124—91no—
308#103—31no—
309#103—91no—
310#102—91no—
311#102—91no—
312C4×C8—91no—
313C33#48811yescharacteristic, complemented, Sylow
314C23×C3—181no—
315C22×D3—61no—
316C22×D3—61no—
317C22×D3—121no—
318C3×Q8—62no—
319C3×Q8—61no—
320C3×Q8—61no—
321C3×Q8—61no—
322C3×Q8—61no—
323C3×Q8—121no—
324C3×D4—181no—
325C3×D4—181no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—31no—
336C2×C3×C4—31no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—62no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—121no—
349C2×C3×C4C2×D3211yescharacteristic
350C2×C3×C4C22×C3×D311yescharacteristic
351C2×C3×C4C22×C3×D311yescharacteristic
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3C3×#6711yescharacteristic
355C2×Dic3C3×#6711yescharacteristic
356D12—62no—
357D12—61no—
358D12—61no—
359D12—61no—
360D12—61no—
361D12—121no—
362Dic6—21no—
363Dic6—21no—
364C3×C8—61no—
365C3×C8—61no—
366C3×C8—121no—
367C3×C8—181no—
368C3⋊C8—21no—
369C3⋊C8—61no—
370C3⋊C8—61no—
371C3⋊C8—121no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—41no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32#88511yescharacteristic
385C2×C32#91611yescharacteristic
386C2×C32#91911yescharacteristic
387C2×C32#94711yescharacteristic
388C2×C32#95011yescharacteristic
389C2×C32#95011yescharacteristic
390C2×C32C3×#12411yescharacteristic
391C2×C32C3×#13511yescharacteristic
392C2×C32C3×#13511yescharacteristic
393C3⋊D3—81no—
394C3×D3—242no—
395C3×D3—241no—
396C3×D3—241no—
397C2×Q8—31no—
398C2×D4—91no—
399C2×C8—31no—
400C2×C8—91no—
401C4⋊C4—31no—
402C4⋊C4—61no—
403C4⋊C4—61no—
404C4⋊C4C3×D3211yescharacteristic
405C22⋊C4—181no—
406C22⋊C4—181no—
407C42—31no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—41no—
412C22×C3—361no—
413C22×C3—361no—
414C22×C3—361no—
415C22×C3D3×#6711yescharacteristic
416C22×C3C3×D3×D411yescharacteristic
417C22×C3C2×C3×#6711yescharacteristic
418C2×D3—122no—
419C2×D3—121no—
420C2×D3—121no—
421C2×D3—121no—
422C2×D3—121no—
423C2×D3—121no—
424C2×D3—121no—
425C2×D3—241no—
426C2×D3—241no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—42no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—41no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—61no—
440C3×C4—61no—
441C3×C4—61no—
442C3×C4—61no—
443C3×C4—81no—
444C3×C4—81no—
445C3×C4—82no—
446C3×C4—121no—
447C3×C4—121no—
448C3×C4—122no—
449C3×C4—122no—
450C3×C4—121no—
451C3×C4—121no—
452C3×C4—121no—
453C3×C4—121no—
454C3×C4#379611yescharacteristic
455C3×C4#379911yescharacteristic
456C3×C4C3×#23511yescharacteristic
457C3×C4C3×#23711yescharacteristic
458C3×C4C3×#23911yescharacteristic
459C3×C4C3×#23911yescharacteristic
460Dic3—21no—
461Dic3—21no—
462Dic3—41no—
463Dic3—41no—
464C32—21no—
465C32—21no—
466C32—21no—
467C32—41no—
468C32#480011yescharacteristic, complemented
469C32#506811yescharacteristic, complemented
470C32C3×#48811yescharacteristic, complemented
471C23—181no—
472Q8—61no—
473Q8—61no—
474D4—181no—
475D4—181no—
476C2×C4—21no—
477C2×C4—31no—
478C2×C4—31no—
479C2×C4—61no—
480C2×C4—61no—
481C2×C4C2×C3×D3211yescharacteristic
482C8—61no—
483C8—181no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—41no—
494C2×C3—41no—
495C2×C3—41no—
496C2×C3—721no—
497C2×C3#6390711yescharacteristic
498C2×C3#6393511yescharacteristic
499C2×C3#6394911yescharacteristic
500C2×C3C3×#88511yescharacteristic
501C2×C3C3×#91611yescharacteristic
502C2×C3C3×#91911yescharacteristic
503C2×C3C3×#94711yescharacteristic
504C2×C3C3×#95011yescharacteristic
505C2×C3C3×#95011yescharacteristic
506D3—242no—
507D3—241no—
508D3—241no—
509C22—361no—
510C22—361no—
511C22—361no—
512C22C3×D3×#6711yescharacteristic
513C4—41no—
514C4—61no—
515C4—61no—
516C4—121no—
517C4—121no—
518C4C3×#379611yescharacteristic
519C4C3×#379911yescharacteristic
520C3—21no—
521C3—21no—
522C3—21no—
523C3—41no—
524C3#1059053711yescharacteristic, direct product factor
525C3C3×#480011yescharacteristic, complemented
526C3C3×#506811yescharacteristic, complemented
527C2—721no—
528C2C3×#6390711yescharacteristic
529C2C3×#6393511yescharacteristic
530C2C3×#6394911yescharacteristic
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×#488
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×#890C2semidirect
4. C32×#810C2semidirect
5. C3×#63565C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
10. #10590537C3direct (N × H)
25. C3×#890D3semidirect
26. C3×#890C2×C3semidirect
36. C3×#810D3semidirect
37. C3×#810C2×C3semidirect
43. #63565C2×C3semidirect
45. Dic3×C3⋊C8C2×C3semidirect
123. #890C3×D3semidirect
136. #810C3×D3semidirect
313. C33#488semidirect
468. C32#4800semidirect
469. C32#5068semidirect
470. C32C3×#488semidirect
524. C3#10590537direct (N × H)
525. C3C3×#4800semidirect
526. C3C3×#5068semidirect

Maximal subgroups

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

subgroupindex
C32×#8902
C3×#638642
C3×#636362
C32×#8102
C3×#635652
C3×#635642
C3×Dic3×C3⋊C82
C3×#50683
C3×#48003
#105905373