Finite Group Databasebeta

← C3×#10590559 #13157546

Subgroups — C3×#10590559 #13157546

Order 1728.

684 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#893C211yesmaximal, characteristic
2C3×#63866C211yesmaximal, characteristic, complemented
3C2×C3×#3716C211yesmaximal, characteristic
4C2×C3×#3715C211yesmaximal, characteristic
5C3×#63637C211yesmaximal, characteristic, complemented
6C32×#810C211yesmaximal, characteristic, complemented
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5019—31nomaximal
9C3×#4818—31nomaximal
10#10590559C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3—21no—
14C2×C32×Dic6C2211yescharacteristic
15C33×C4⋊C4C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×Dic3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22C3×#3716—22no—
23C3×#3715—22no—
24C32×C4×D4—31no—
25C3×#944—31no—
26C2×C3×#215—31no—
27C2×C3×#214—31no—
28C2×C3×#214—31no—
29C2×C3×#204—31no—
30C3×#893—21no—
31C3×#893D311yescharacteristic
32C3×#893C2×C311yescharacteristic
33C3×#869—31no—
34#63866C2×C311yescharacteristic, complemented
35C2×#3716C2×C311yescharacteristic
36C2×#3715C2×C311yescharacteristic
37C32×#103—31no—
38#63637C2×C311yescharacteristic, complemented
39C3×Dic3⋊C8—31no—
40C3×#810—21no—
41C3×#810—31no—
42C3×#810—31no—
43C3×#810—61no—
44C3×#810D311yescharacteristic, complemented
45C3×#810C2×C311yescharacteristic, complemented
46C3×C4⋊(C3⋊C8)—31no—
47Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
48C22×C32×D3—21no—
49C2×C33×C4—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C3×#634—21no—
53C3×#634—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3D411yescharacteristic
56C2×C32×Dic3D411yescharacteristic
57C32×D12—21no—
58C32×D12D412yes—
59C32×C4×D3—42no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64C3×#461—91no—
65#5019—31no—
66#4818—31no—
67C2×C32×Q8—31no—
68C2×C32×D4—31no—
69C22×C32×C4—61no—
70C2×#634C22×C311yescharacteristic
71C2×C3×D12—21no—
72C2×C3×D12C2×D311yescharacteristic
73C2×C3×D12C22×C311yescharacteristic
74C2×C3×C4×D3—22no—
75C2×C3×C4×D3—21no—
76C2×C3×C4×D3—21no—
77C2×C3×Dic6—21no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—61no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×C3×Dic6C22×C311yescharacteristic
83C2×C32×C8—31no—
84C32×C4⋊C4—21no—
85C32×C4⋊C4—21no—
86C32×C4⋊C4—21no—
87C32×C4⋊C4—41no—
88C32×C4⋊C4C2×D311yescharacteristic
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C4⋊C4C22×C311yescharacteristic
91C32×C22⋊C4—61no—
92C32×C42—31no—
93C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
94C3×#215—62no—
95C3×#214—62no—
96C3×#214—62no—
97C3×(C2×D3)⋊C4—22no—
98C3×(C2×D3)⋊C4—21no—
99C3×(C2×D3)⋊C4—21no—
100C3×C4⋊Dic3—61no—
101C3×C4⋊Dic3—61no—
102C3×Dic3⋊C4—61no—
103C3×Dic3⋊C4—61no—
104C3×C4×Dic3—21no—
105C3×C4×Dic3C2×D311yescharacteristic
106C3×C4×Dic3C22×C311yescharacteristic
107C2×C3×C3⋊C8—21no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—31no—
110C2×C3×C3⋊C8—61no—
111C2×C3×C3⋊C8C2×D311yescharacteristic
112C2×C3×C3⋊C8C22×C311yescharacteristic
113C3×#204—62no—
114Dic3⋊Dic3—21no—
115Dic3⋊Dic3—21no—
116#3716—22no—
117#3715—22no—
118C22×C33C2×D411yescharacteristic
119C2×C32×D3—22no—
120C2×C32×D3—41no—
121C2×C32×D3—41no—
122C33×C4—41no—
123C33×C4C2×D411yescharacteristic
124C33×C4#4111yescharacteristic
125C3×C3⋊Dic3—41no—
126C32×Dic3—21no—
127C32×Dic3—41no—
128C32×Dic3SD1612yescomplemented
129C2×C3×SD16—91no—
130C2×C3×SD16—91no—
131C3×C4⋊Q8—91no—
132C3×C4×D4—31no—
133C3×C4×D4—31no—
134C3×C4×D4—61no—
135#944—31no—
136C2×#215—31no—
137C2×#214—31no—
138C2×#214—31no—
139C2×#204—31no—
140#893C3×D311yescharacteristic
141#869—31no—
142C3×C4⋊C8—91no—
143C3×#103—31no—
144C3×#103—31no—
145C3×#103—61no—
146C3×#103—91no—
147Dic3⋊C8—31no—
148#810—31no—
149#810—31no—
150#810—61no—
151#810C3×D311yescharacteristic, complemented
152C4⋊(C3⋊C8)—31no—
153C23×C32—61no—
154C22×C3×D3—22no—
155C22×C3×D3—21no—
156C22×C3×D3—21no—
157C32×Q8—61no—
158C32×Q8—61no—
159C32×D4—32no—
160C32×D4—61no—
161C2×C32×C4—21no—
162C2×C32×C4—22no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—31no—
169C2×C32×C4—31no—
170C2×C32×C4—41no—
171C2×C32×C4—41no—
172C2×C32×C4—41no—
173C2×C32×C4—42no—
174C2×C32×C4—61no—
175C2×C32×C4—122no—
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C23×C311yescharacteristic
179C2×C3⋊Dic3—21no—
180#634—21no—
181#634—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—41no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—121no—
190C2×C3×Dic3#6711yescharacteristic
191C2×C3×Dic3#6711yescharacteristic
192C2×C3×Dic3C3×D411yescharacteristic
193C2×C3×Dic3C3×D411yescharacteristic
194C3×D12—22no—
195C3×D12—21no—
196C3×D12—21no—
197C3×D12—22no—
198C3×D12D1212yes—
199C3×D12C3×D412yes—
200C3×C4×D3—44no—
201C3×C4×D3—42no—
202C3×C4×D3—42no—
203C3×Dic6—22no—
204C3×Dic6—21no—
205C3×Dic6—21no—
206C3×Dic6—21no—
207C3×Dic6—21no—
208C3×Dic6—41no—
209C3×Dic6—62no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—61no—
214C3×Dic6—121no—
215C32×C8—61no—
216C3⋊(C3⋊C8)—21no—
217C3×C3⋊C8—21no—
218C3×C3⋊C8—21no—
219C3×C3⋊C8—41no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—121no—
223#461—91noSylow
224C2×C33#12111yescharacteristic
225C2×C33C2×SD1611yescharacteristic
226C2×C33#13711yescharacteristic
227C32×D3—42no—
228C2×C3×Q8—31no—
229C2×C3×Q8—31no—
230C2×C3×Q8—61no—
231C2×C3×Q8—91no—
232C2×C3×D4—31no—
233C2×C3×D4—31no—
234C2×C3×D4—61no—
235C22×C3×C4—61no—
236C22×C3×C4—61no—
237C22×C3×C4—62no—
238C2×D12C2×C3×D311yescharacteristic
239C2×C4×D3—21no—
240C2×Dic6—31no—
241C2×Dic6—31no—
242C2×Dic6—61no—
243C2×Dic6C2×C3×D311yescharacteristic
244C3×SD16—182no—
245C3×SD16—182no—
246C2×C3×C8—31no—
247C2×C3×C8—31no—
248C2×C3×C8—61no—
249C2×C3×C8—91no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—41no—
254C3×C4⋊C4—181no—
255C3×C4⋊C4—181no—
256C3×C4⋊C4D3211yescharacteristic
257C3×C4⋊C4C2×C3×D311yescharacteristic
258C3×C4⋊C4C2×C3×D311yescharacteristic
259C3×C22⋊C4—61no—
260C3×C22⋊C4—61no—
261C3×C22⋊C4—62no—
262C3×C42—31no—
263C3×C42—31no—
264C3×C42—61no—
265#215—62no—
266#214—62no—
267#214—62no—
268(C2×D3)⋊C4—21no—
269C4⋊Dic3—61no—
270C4⋊Dic3—61no—
271Dic3⋊C4—61no—
272Dic3⋊C4—61no—
273C4×Dic3C2×C3×D311yescharacteristic
274C2×C3⋊C8—31no—
275C2×C3⋊C8—31no—
276C2×C3⋊C8—61no—
277C2×C3⋊C8C2×C3×D311yescharacteristic
278#204—62no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—41no—
283C22×C32—62no—
284C22×C32—121no—
285C22×C32—121no—
286C22×C32D3×D411yescharacteristic
287C22×C32C2×#6711yescharacteristic
288C22×C32C2×C3×D411yescharacteristic
289C2×C3×D3—24no—
290C2×C3×D3—22no—
291C2×C3×D3—22no—
292C2×C3×D3—42no—
293C2×C3×D3—42no—
294C2×C3×D3—41no—
295C2×C3×D3—41no—
296C2×C3×D3—41no—
297C2×C3×D3—41no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—32no—
305C32×C4—41no—
306C32×C4—41no—
307C32×C4—41no—
308C32×C4—42no—
309C32×C4—41no—
310C32×C4—41no—
311C32×C4—61no—
312C32×C4—81no—
313C32×C4—81no—
314C32×C4—82no—
315C32×C4—121no—
316C32×C4C2×D1211yescharacteristic
317C32×C4#23511yescharacteristic
318C32×C4#23511yescharacteristic
319C32×C4D3×D411yescharacteristic
320C32×C4C2×C3×D411yescharacteristic
321C32×C4C3×#4111yescharacteristic
322C3⋊Dic3—41no—
323C3×Dic3—21no—
324C3×Dic3—21no—
325C3×Dic3—22no—
326C3×Dic3—41no—
327C3×Dic3—42no—
328C3×Dic3—41no—
329C3×Dic3—41no—
330C3×Dic3—122no—
331C3×Dic3—121no—
332C3×Dic3—121no—
333C3×Dic3#21412yescomplemented
334C3×Dic3C3×SD1612yescomplemented
335C2×SD16—91no—
336C2×SD16—91no—
337C4⋊Q8—91no—
338C4×D4—31no—
339C4⋊C8—91no—
340#103—31no—
341#103—91no—
342C33#46111yescharacteristic, complemented, Sylow
343C23×C3—62no—
344C23×C3—61no—
345C23×C3—61no—
346C22×D3—21no—
347C3×Q8—62no—
348C3×Q8—61no—
349C3×Q8—61no—
350C3×Q8—61no—
351C3×Q8—61no—
352C3×Q8—121no—
353C3×Q8—181no—
354C3×Q8—181no—
355C3×D4—32no—
356C3×D4—32no—
357C3×D4—62no—
358C3×D4—62no—
359C3×D4—61no—
360C3×D4—61no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—31no—
369C2×C3×C4—31no—
370C2×C3×C4—31no—
371C2×C3×C4—31no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—42no—
375C2×C3×C4—41no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—121no—
381C2×C3×C4—122no—
382C2×C3×C4—122no—
383C2×C3×C4—124no—
384C2×C3×C4—181no—
385C2×C3×C4C2×D3211yescharacteristic
386C2×C3×C4C22×C3×D311yescharacteristic
387C2×C3×C4C22×C3×D311yescharacteristic
388C2×Dic3—21no—
389C2×Dic3—61no—
390C2×Dic3—61no—
391C2×Dic3—121no—
392C2×Dic3C3×#6711yescharacteristic
393C2×Dic3C3×#6711yescharacteristic
394D12—21no—
395D12C3×D1212yes—
396C4×D3—42no—
397Dic6—21no—
398Dic6—21no—
399Dic6—62no—
400Dic6—61no—
401Dic6—61no—
402Dic6—61no—
403Dic6—61no—
404Dic6—121no—
405C3×C8—61no—
406C3×C8—61no—
407C3×C8—121no—
408C3×C8—181no—
409C3⋊C8—21no—
410C3⋊C8—61no—
411C3⋊C8—61no—
412C3⋊C8—121no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—41no—
423C2×C32—41no—
424C2×C32—41no—
425C2×C32—122no—
426C2×C32#88411yescharacteristic
427C2×C32D3×SD1611yescharacteristic
428C2×C32#91811yescharacteristic
429C2×C32#93011yescharacteristic
430C2×C32C2×#21411yescharacteristic
431C2×C32#95111yescharacteristic
432C2×C32C3×#12111yescharacteristic
433C2×C32C2×C3×SD1611yescharacteristic
434C2×C32C3×#13711yescharacteristic
435C3×D3—44no—
436C3×D3—42no—
437C3×D3—42no—
438C2×Q8—31no—
439C2×Q8—91no—
440C2×D4—31no—
441C22×C4—61no—
442SD16—182no—
443SD16—182no—
444C2×C8—31no—
445C2×C8—91no—
446C4⋊C4—181no—
447C4⋊C4—181no—
448C4⋊C4C3×D3211yescharacteristic
449C22⋊C4—61no—
450C42—31no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—41no—
455C22×C3—64no—
456C22×C3—62no—
457C22×C3—62no—
458C22×C3—122no—
459C22×C3—122no—
460C22×C3—121no—
461C22×C3—121no—
462C22×C3—121no—
463C22×C3—121no—
464C22×C3D3×#6711yescharacteristic
465C22×C3C3×D3×D411yescharacteristic
466C22×C3C2×C3×#6711yescharacteristic
467C2×D3—22no—
468C2×D3—41no—
469C2×D3—41no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—32no—
477C3×C4—32no—
478C3×C4—42no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—41no—
484C3×C4—62no—
485C3×C4—61no—
486C3×C4—61no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4—82no—
490C3×C4—121no—
491C3×C4—122no—
492C3×C4—121no—
493C3×C4—121no—
494C3×C4—361no—
495C3×C4#379811yescharacteristic
496C3×C4D3×D1211yescharacteristic
497C3×C4C2×C3×D1211yescharacteristic
498C3×C4C3×#23511yescharacteristic
499C3×C4C3×#23511yescharacteristic
500C3×C4C3×D3×D411yescharacteristic
501Dic3—21no—
502Dic3—41no—
503Dic3—122no—
504Dic3—121no—
505Dic3—121no—
506Dic3C3×#21412yescomplemented
507C32—21no—
508C32—21no—
509C32—21no—
510C32—41no—
511C32#481811yescharacteristic, complemented
512C32#501911yescharacteristic, complemented
513C32C3×#46111yescharacteristic, complemented
514C23—61no—
515Q8—61no—
516Q8—61no—
517Q8—181no—
518Q8—181no—
519D4—32no—
520D4—61no—
521C2×C4—21no—
522C2×C4—31no—
523C2×C4—31no—
524C2×C4—61no—
525C2×C4—122no—
526C2×C4—181no—
527C2×C4C2×C3×D3211yescharacteristic
528C8—61no—
529C8—181no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—41no—
540C2×C3—41no—
541C2×C3—41no—
542C2×C3—124no—
543C2×C3—122no—
544C2×C3—122no—
545C2×C3#6390611yescharacteristic
546C2×C3D3×#21411yescharacteristic
547C2×C3#6394311yescharacteristic
548C2×C3C3×#88411yescharacteristic
549C2×C3C3×D3×SD1611yescharacteristic
550C2×C3C3×#91811yescharacteristic
551C2×C3C3×#93011yescharacteristic
552C2×C3C2×C3×#21411yescharacteristic
553C2×C3C3×#95111yescharacteristic
554D3—42no—
555C22—62no—
556C22—121no—
557C22—121no—
558C22C3×D3×#6711yescharacteristic
559C4—32no—
560C4—41no—
561C4—61no—
562C4—121no—
563C4—361no—
564C4C3×#379811yescharacteristic
565C4C3×D3×D1211yescharacteristic
566C3—21no—
567C3—21no—
568C3—21no—
569C3—41no—
570C3#1059055911yescharacteristic, direct product factor
571C3C3×#481811yescharacteristic, complemented
572C3C3×#501911yescharacteristic, complemented
573C2—122no—
574C2C3×#6390611yescharacteristic
575C2C3×D3×#21411yescharacteristic
576C2C3×#6394311yescharacteristic
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×#461
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×#63866C2semidirect
5. C3×#63637C2semidirect
6. C32×#810C2semidirect
7. C3×Dic3⋊(C3⋊C8)C2semidirect
10. #10590559C3direct (N × H)
34. #63866C2×C3semidirect
38. #63637C2×C3semidirect
44. C3×#810D3semidirect
45. C3×#810C2×C3semidirect
47. Dic3⋊(C3⋊C8)C2×C3semidirect
128. C32×Dic3SD16semidirect
151. #810C3×D3semidirect
333. C3×Dic3#214semidirect
334. C3×Dic3C3×SD16semidirect
342. C33#461semidirect
506. Dic3C3×#214semidirect
511. C32#4818semidirect
512. C32#5019semidirect
513. C32C3×#461semidirect
570. C3#10590559direct (N × H)
571. C3C3×#4818semidirect
572. C3C3×#5019semidirect

Maximal subgroups

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

subgroupindex
C32×#8932
C3×#638662
C2×C3×#37162
C2×C3×#37152
C3×#636372
C32×#8102
C3×Dic3⋊(C3⋊C8)2
C3×#50193
C3×#48183
#105905593