Finite Group Databasebeta

← C3×#10590458 #13157597

Subgroups — C3×#10590458 #13157597

Order 1728.

670 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#887C211yesmaximal, characteristic, complemented
2C3×#63894C211yesmaximal, characteristic
3C2×C3×#3717C211yesmaximal, characteristic
4C3×#63636C211yesmaximal, characteristic
5C32×#808C211yesmaximal, characteristic, complemented
6C3×#63565C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8C3×C4×#215—31nomaximal
9C3×#4852—31nomaximal
10#10590458C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C3×#624—21no—
14C33×C4⋊C4C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3—21no—
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×#3722—21no—
22C3×#3717C414yescomplemented
23C32×C4×Q8—31no—
24C2×C3×#215—31no—
25C2×C3×#204—31no—
26C3×#893—31no—
27C3×#887—21no—
28C3×#887D311yescharacteristic, complemented
29C3×#887C2×C311yescharacteristic, complemented
30C3×C4×D12—31no—
31#63894C2×C311yescharacteristic
32C2×#3717C2×C311yescharacteristic
33C32×#106—31no—
34#63636C2×C311yescharacteristic
35C3×#835—31no—
36C3×C8×Dic3—31no—
37C3×#810—31no—
38C3×#809—31no—
39C3×#809—31no—
40C3×#809—61no—
41C3×#808—21no—
42C3×#808D311yescharacteristic, complemented
43C3×#808C2×C311yescharacteristic, complemented
44C3×C4×C3⋊C8—31no—
45#63565C2×C311yescharacteristic, complemented
46Dic3×C3⋊C8C2×C311yescharacteristic, complemented
47C22×C3×C3⋊D3—21no—
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)C2×C412yes—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55C2×C32×Dic3D411yescharacteristic
56C32×Dic6—21no—
57C32×Dic6C2×C412yescomplemented
58C3×#624—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8C2×C412yescomplemented
61C3×C4×SD16—91no—
62C4×#215—31no—
63#4852—31no—
64C2×C32×Q8—31no—
65C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×D12—61no—
69C2×C3×C4×D3—61no—
70C2×C3×C4×D3—61no—
71C2×C3×Dic6—21no—
72C2×C3×Dic6C2×D311yescharacteristic
73C2×C3×Dic6C22×C311yescharacteristic
74C2×#624—21no—
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—
86C32×C42—61no—
87C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
88C3×#215—34no—
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—22no—
98C3×C4×Dic3—21no—
99C3×C4×Dic3—21no—
100C3×C4×Dic3—21no—
101C3×C4×Dic3C2×D311yescharacteristic
102C3×C4×Dic3C22×C311yescharacteristic
103C2×C3×C3⋊C8—21no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—61no—
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C2×C3×C3⋊C8C22×C311yescharacteristic
109C3×#204—34no—
110#3722—21no—
111#3717C3×C414yescomplemented
112C22×C33C2×D411yescharacteristic
113C2×C3×C3⋊D3—22no—
114C2×C3×C3⋊D3—41no—
115C2×C3×C3⋊D3—41no—
116C33×C4—41no—
117C33×C4C22×C411yescharacteristic
118C33×C4#4111yescharacteristic
119C32×Dic3—22no—
120C32×Dic3—21no—
121C32×Dic3—41no—
122C32×Dic3SD1612yescomplemented
123C2×C3×SD16—91no—
124C3×C4×Q8—31no—
125C3×C4×Q8—31no—
126C3×C4×Q8—61no—
127C3×C4×D4—91no—
128C2×#215—31no—
129C2×#204—31no—
130#893—31no—
131#887C3×D311yescharacteristic, complemented
132C4×D12—31no—
133C3×#106—31no—
134C3×#106—31no—
135C3×#106—61no—
136C3×#103—91no—
137C3×#102—91no—
138C3×C4×C8—91no—
139#835—31no—
140C8×Dic3—31no—
141#810—31no—
142#809—31no—
143#809—31no—
144#809—61no—
145#808C3×D311yescharacteristic, complemented
146C4×C3⋊C8—31no—
147C22×C3⋊D3—21no—
148C22×C3×D3—61no—
149C22×C3×D3—61no—
150C22×C3×D3—121no—
151C32×Q8—32no—
152C32×Q8—61no—
153C2×C32×C4—21no—
154C2×C32×C4—22no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—31no—
161C2×C32×C4—31no—
162C2×C32×C4—41no—
163C2×C32×C4—41no—
164C2×C32×C4—41no—
165C2×C32×C4—42no—
166C2×C32×C4—61no—
167C2×C32×C4—61no—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C23×C311yescharacteristic
171C4⋊(C3⋊D3)—21no—
172C4⋊(C3⋊D3)C2×C3×C412yes—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—41no—
181C2×C3×Dic3#6711yescharacteristic
182C2×C3×Dic3#6711yescharacteristic
183C2×C3×Dic3C3×D411yescharacteristic
184C2×C3×Dic3C3×D411yescharacteristic
185C3×D12—32no—
186C3×D12—32no—
187C3×D12—62no—
188C3×D12—61no—
189C3×D12—61no—
190C3×D12—121no—
191C3×C4×D3—122no—
192C3×C4×D3—122no—
193C3×Dic6—22no—
194C3×Dic6—21no—
195C3×Dic6—21no—
196C3×Dic6—22no—
197C3×Dic6C4×D312yescomplemented
198C3×Dic6C2×C3×C412yescomplemented
199#624—42no—
200C32×C8—32no—
201C3⋊(C3⋊C8)—21no—
202C3×C3⋊C8—22no—
203C3×C3⋊C8—61no—
204C3×C3⋊C8—61no—
205C3×C3⋊C8—121no—
206C3×C3⋊C8C4×D312yescomplemented
207C3×C3⋊C8C2×C3×C412yescomplemented
208C4×SD16—91noSylow
209C2×C33C4×D411yescharacteristic
210C2×C33C2×SD1611yescharacteristic
211C2×C33#13511yescharacteristic
212C3×C3⋊D3—42no—
213C2×C3×Q8—31no—
214C2×C3×Q8—31no—
215C2×C3×Q8—61no—
216C2×C3×D4—91no—
217C22×C3×C4—181no—
218C2×D12—31no—
219C2×D12—31no—
220C2×D12—61no—
221C2×C4×D3—61no—
222C2×C4×D3—61no—
223C2×Dic6C2×C3×D311yescharacteristic
224C3×SD16—94no—
225C2×C3×C8—31no—
226C2×C3×C8—31no—
227C2×C3×C8—61no—
228C2×C3×C8—91no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—41no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—61no—
237C3×C4⋊C4—61no—
238C3×C4⋊C4—62no—
239C3×C4⋊C4D3211yescharacteristic
240C3×C4⋊C4C2×C3×D311yescharacteristic
241C3×C4⋊C4C2×C3×D311yescharacteristic
242C3×C22⋊C4—181no—
243C3×C42—31no—
244C3×C42—31no—
245C3×C42—61no—
246C3×C42—61no—
247C3×C42—61no—
248C3×C42—62no—
249#215—34no—
250(C2×D3)⋊C4—61no—
251(C2×D3)⋊C4—61no—
252C4⋊Dic3C2×C3×D311yescharacteristic
253Dic3⋊C4—21no—
254C4×Dic3—21no—
255C4×Dic3C2×C3×D311yescharacteristic
256C2×C3⋊C8—31no—
257C2×C3⋊C8—31no—
258C2×C3⋊C8—61no—
259C2×C3⋊C8C2×C3×D311yescharacteristic
260#204—34no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—41no—
265C22×C32D3×D411yescharacteristic
266C22×C32C2×#6711yescharacteristic
267C22×C32C2×C3×D411yescharacteristic
268C2×C3⋊D3—22no—
269C2×C3⋊D3—41no—
270C2×C3⋊D3—41no—
271C2×C3×D3—62no—
272C2×C3×D3—62no—
273C2×C3×D3—122no—
274C2×C3×D3—121no—
275C2×C3×D3—121no—
276C2×C3×D3—121no—
277C2×C3×D3—121no—
278C2×C3×D3—241no—
279C2×C3×D3—241no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—32no—
287C32×C4—41no—
288C32×C4—41no—
289C32×C4—41no—
290C32×C4—42no—
291C32×C4—41no—
292C32×C4—41no—
293C32×C4—62no—
294C32×C4—61no—
295C32×C4—81no—
296C32×C4—81no—
297C32×C4—82no—
298C32×C4—121no—
299C32×C4C2×C4×D311yescharacteristic
300C32×C4C2×C4×D311yescharacteristic
301C32×C4#23511yescharacteristic
302C32×C4#23911yescharacteristic
303C32×C4C22×C3×C411yescharacteristic
304C32×C4C3×#4111yescharacteristic
305C3×Dic3—24no—
306C3×Dic3—22no—
307C3×Dic3—21no—
308C3×Dic3—22no—
309C3×Dic3—21no—
310C3×Dic3—22no—
311C3×Dic3—41no—
312C3×Dic3—41no—
313C3×Dic3—41no—
314C3×Dic3—81no—
315C3×Dic3#21512yescomplemented
316C3×Dic3C3×SD1612yescomplemented
317C2×SD16—91no—
318C4×Q8—31no—
319C4×D4—91no—
320#106—31no—
321#103—91no—
322#102—91no—
323C4×C8—91no—
324C33C4×SD1611yescharacteristic, complemented, Sylow
325C23×C3—181no—
326C22×D3—61no—
327C22×D3—61no—
328C22×D3—121no—
329C3×Q8—32no—
330C3×Q8—32no—
331C3×Q8—62no—
332C3×Q8—62no—
333C3×Q8—61no—
334C3×Q8—61no—
335C3×D4—92no—
336C3×D4—181no—
337C2×C3×C4—22no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—31no—
345C2×C3×C4—31no—
346C2×C3×C4—31no—
347C2×C3×C4—31no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—41no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—62no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—121no—
360C2×C3×C4—362no—
361C2×C3×C4C2×D3211yescharacteristic
362C2×C3×C4C22×C3×D311yescharacteristic
363C2×C3×C4C22×C3×D311yescharacteristic
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3C3×#6711yescharacteristic
367C2×Dic3C3×#6711yescharacteristic
368D12—32no—
369D12—32no—
370D12—62no—
371D12—61no—
372D12—61no—
373D12—121no—
374C4×D3—122no—
375C4×D3—122no—
376Dic6—21no—
377Dic6C3×C4×D312yescomplemented
378C3×C8—32no—
379C3×C8—32no—
380C3×C8—62no—
381C3×C8—181no—
382C3⋊C8—61no—
383C3⋊C8—61no—
384C3⋊C8—121no—
385C3⋊C8C3×C4×D312yescomplemented
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—41no—
396C2×C32—41no—
397C2×C32—41no—
398C2×C32#89311yescharacteristic
399C2×C32D3×SD1611yescharacteristic
400C2×C32#91611yescharacteristic
401C2×C32C4×#6711yescharacteristic
402C2×C32C2×#21511yescharacteristic
403C2×C32#95011yescharacteristic
404C2×C32C3×C4×D411yescharacteristic
405C2×C32C2×C3×SD1611yescharacteristic
406C2×C32C3×#13511yescharacteristic
407C3⋊D3—42no—
408C3×D3—122no—
409C3×D3—122no—
410C3×D3—242no—
411C2×Q8—31no—
412C2×D4—91no—
413C22×C4—181no—
414SD16—94no—
415C2×C8—31no—
416C2×C8—91no—
417C4⋊C4—31no—
418C4⋊C4—61no—
419C4⋊C4C3×D3211yescharacteristic
420C22⋊C4—181no—
421C42—31no—
422C42—61no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—41no—
427C22×C3—182no—
428C22×C3—361no—
429C22×C3—361no—
430C22×C3D3×#6711yescharacteristic
431C22×C3C3×D3×D411yescharacteristic
432C22×C3C2×C3×#6711yescharacteristic
433C2×D3—62no—
434C2×D3—62no—
435C2×D3—122no—
436C2×D3—121no—
437C2×D3—121no—
438C2×D3—121no—
439C2×D3—121no—
440C2×D3—241no—
441C2×D3—241no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—32no—
449C3×C4—32no—
450C3×C4—42no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—62no—
457C3×C4—64no—
458C3×C4—62no—
459C3×C4—61no—
460C3×C4—62no—
461C3×C4—61no—
462C3×C4—81no—
463C3×C4—81no—
464C3×C4—82no—
465C3×C4—121no—
466C3×C4—121no—
467C3×C4—121no—
468C3×C4—241no—
469C3×C4#379911yescharacteristic
470C3×C4C4×D3211yescharacteristic
471C3×C4C2×C3×C4×D311yescharacteristic
472C3×C4C2×C3×C4×D311yescharacteristic
473C3×C4C3×#23511yescharacteristic
474C3×C4C3×#23911yescharacteristic
475Dic3—22no—
476Dic3—21no—
477Dic3—41no—
478Dic3C3×#21512yescomplemented
479C32—21no—
480C32—21no—
481C32—21no—
482C32—41no—
483C32#485211yescharacteristic, complemented
484C32C4×#21511yescharacteristic, complemented
485C32C3×C4×SD1611yescharacteristic, complemented
486C23—181no—
487Q8—32no—
488Q8—61no—
489D4—92no—
490D4—181no—
491C2×C4—21no—
492C2×C4—31no—
493C2×C4—31no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—362no—
497C2×C4C2×C3×D3211yescharacteristic
498C8—32no—
499C8—181no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—41no—
510C2×C3—41no—
511C2×C3—41no—
512C2×C3—362no—
513C2×C3#6390411yescharacteristic
514C2×C3#6393511yescharacteristic
515C2×C3D3×#21511yescharacteristic
516C2×C3C3×#89311yescharacteristic
517C2×C3C3×D3×SD1611yescharacteristic
518C2×C3C3×#91611yescharacteristic
519C2×C3C3×C4×#6711yescharacteristic
520C2×C3C2×C3×#21511yescharacteristic
521C2×C3C3×#95011yescharacteristic
522D3—122no—
523D3—122no—
524D3—242no—
525C22—182no—
526C22—361no—
527C22—361no—
528C22C3×D3×#6711yescharacteristic
529C4—32no—
530C4—41no—
531C4—62no—
532C4—61no—
533C4—121no—
534C4C3×#379911yescharacteristic
535C4C3×C4×D3211yescharacteristic
536C3—21no—
537C3—21no—
538C3—21no—
539C3—41no—
540C3#1059045811yescharacteristic, direct product factor
541C3C3×#485211yescharacteristic, complemented
542C3C3×C4×#21511yescharacteristic, complemented
543C2—362no—
544C2C3×#6390411yescharacteristic
545C2C3×#6393511yescharacteristic
546C2C3×D3×#21511yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
cosocleC2×C32×C4C23×C3
hypercenterC3×C4⋊C4D32
hypocenterC32C3×C4×SD16
LCSG ▷ 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×#887C2semidirect
5. C32×#808C2semidirect
6. C3×#63565C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
10. #10590458C3direct (N × H)
22. C3×#3717C4semidirect
28. C3×#887D3semidirect
29. C3×#887C2×C3semidirect
42. C3×#808D3semidirect
43. C3×#808C2×C3semidirect
45. #63565C2×C3semidirect
46. Dic3×C3⋊C8C2×C3semidirect
57. C32×Dic6C2×C4semidirect
60. C32×C3⋊C8C2×C4semidirect
111. #3717C3×C4semidirect
122. C32×Dic3SD16semidirect
131. #887C3×D3semidirect
145. #808C3×D3semidirect
197. C3×Dic6C4×D3semidirect
198. C3×Dic6C2×C3×C4semidirect
206. C3×C3⋊C8C4×D3semidirect
207. C3×C3⋊C8C2×C3×C4semidirect
315. C3×Dic3#215semidirect
316. C3×Dic3C3×SD16semidirect
324. C33C4×SD16semidirect
377. Dic6C3×C4×D3semidirect
385. C3⋊C8C3×C4×D3semidirect
478. Dic3C3×#215semidirect
483. C32#4852semidirect
484. C32C4×#215semidirect
485. C32C3×C4×SD16semidirect
540. C3#10590458direct (N × H)
541. C3C3×#4852semidirect
542. C3C3×C4×#215semidirect

Maximal subgroups

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

subgroupindex
C32×#8872
C3×#638942
C2×C3×#37172
C3×#636362
C32×#8082
C3×#635652
C3×Dic3×C3⋊C82
C3×C4×#2153
C3×#48523
#105904583