Finite Group Databasebeta

← C3×#10590745 #13157889

Subgroups — C3×#10590745 #13157889

Order 1728.

702 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63894C211yesmaximal, characteristic, complemented
2C3×Dic3×D12C211yesmaximal, characteristic, complemented
3C2×C3×#3714C211yesmaximal, characteristic
4C3×#63648C211yesmaximal, characteristic, complemented
5C32×#821C211yesmaximal, characteristic, complemented
6C32×C8×Dic3C211yesmaximal, characteristic, complemented
7C3×#63575C211yesmaximal, characteristic, complemented
8C3×#4756—31nomaximal
9C3×C4×D24—31nomaximal
10#10590745C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C3×#624—21no—
14C2×C3×D3×Dic3—21no—
15C2×C33×C8C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3722—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×#3714C414yescomplemented
23C3×D4×Dic3—31no—
24C2×C3×#213—31no—
25C2×C3×D24—31no—
26C3×#893—31no—
27C3×C4×D12—31no—
28C3×C4×D12—31no—
29#63894C2×C311yescharacteristic, complemented
30Dic3×D12C2×C311yescharacteristic, complemented
31C2×#3714C2×C311yescharacteristic
32C32×#102—31no—
33C32×C4×C8—31no—
34#63648C2×C311yescharacteristic, complemented
35C3×#821—21no—
36C3×#821—31no—
37C3×#821—31no—
38C3×#821—61no—
39C3×#821D311yescharacteristic, complemented
40C3×#821C2×C311yescharacteristic, complemented
41C3×#818—31no—
42C3×C8×Dic3—21no—
43C3×C8×Dic3D311yescharacteristic, complemented
44C3×C8×Dic3C2×C311yescharacteristic, complemented
45C3×#807—31no—
46#63575C2×C311yescharacteristic, complemented
47C22×C3×C3⋊D3—21no—
48C22×C32×D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)C2×C412yescomplemented
53C2×C32×Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55C2×C32×Dic3D411yescharacteristic
56C32×D12—21no—
57C32×D12C2×C412yescomplemented
58C3×#624—42no—
59C3×D3×Dic3—42no—
60C33×C8—21no—
61C32×C3⋊C8C2×C412yescomplemented
62C3×C4×D8—91no—
63#4756—31no—
64C4×D24—31no—
65C2×C32×D4—31no—
66C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
67C22×C3×Dic3—61no—
68C2×C3×D12—21no—
69C2×C3×D12—31no—
70C2×C3×D12—31no—
71C2×C3×D12—61no—
72C2×C3×D12C2×D311yescharacteristic
73C2×C3×D12C22×C311yescharacteristic
74C2×C3×C4×D3—61no—
75C2×C3×C4×D3—61no—
76C2×C3×C4×D3—61no—
77C2×#624—21no—
78C2×D3×Dic3—21no—
79C2×C32×C8—21no—
80C2×C32×C8—21no—
81C2×C32×C8—21no—
82C2×C32×C8—31no—
83C2×C32×C8—41no—
84C2×C32×C8C2×D311yescharacteristic
85C2×C32×C8C2×D311yescharacteristic
86C2×C32×C8C22×C311yescharacteristic
87C32×C4⋊C4—31no—
88C32×C42—31no—
89C4⋊(C3⋊Dic3)C22×C311yescharacteristic
90C3×#217—61no—
91C3×#213—34no—
92C3×(C2×D3)⋊C4—61no—
93C3×(C2×D3)⋊C4—61no—
94C3×(C2×D3)⋊C4—61no—
95C3×C4⋊Dic3—21no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3—61no—
99C3×C4⋊Dic3C2×D311yescharacteristic
100C3×C4⋊Dic3C22×C311yescharacteristic
101C3×C4×Dic3—21no—
102C3×C4×Dic3C2×D311yescharacteristic
103C3×C4×Dic3C22×C311yescharacteristic
104C2×C3×C3⋊C8—21no—
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C22×C311yescharacteristic
107C3×D24—34no—
108#3722—21no—
109(C2×D3)⋊Dic3—21no—
110#3714C3×C414yescomplemented
111C22×C33C2×D411yescharacteristic
112C2×C3×C3⋊D3—22no—
113C2×C3×C3⋊D3—41no—
114C2×C3×C3⋊D3—41no—
115C2×C32×D3—22no—
116C2×C32×D3—41no—
117C2×C32×D3—41no—
118C33×C4C22×C411yescharacteristic
119C33×C4#4111yescharacteristic
120C3×C3⋊Dic3—41no—
121C32×Dic3—21no—
122C32×Dic3—41no—
123C32×Dic3D812yescomplemented
124C2×C3×D8—91no—
125C3×C4×D4—91no—
126C3×C4×D4—91no—
127D4×Dic3—31no—
128C2×#213—31no—
129C2×D24—31no—
130#893—31no—
131C4×D12—31no—
132C4×D12—31no—
133C3×#107—91no—
134C3×#102—31no—
135C3×#102—31no—
136C3×#102—61no—
137C3×#102—91no—
138C3×C4×C8—31no—
139C3×C4×C8—31no—
140C3×C4×C8—61no—
141#821—31no—
142#821—31no—
143#821—61no—
144#821C3×D311yescharacteristic, complemented
145#818—31no—
146C8×Dic3C3×D311yescharacteristic, complemented
147#807—31no—
148C23×C32—61no—
149C22×C3⋊D3—21no—
150C22×C3×D3—21no—
151C22×C3×D3—21no—
152C22×C3×D3—41no—
153C22×C3×D3—61no—
154C22×C3×D3—61no—
155C22×C3×D3—121no—
156C32×D4—32no—
157C32×D4—61no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—31no—
162C2×C32×C4—31no—
163C2×C32×C4—41no—
164C2×C32×C4—61no—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C23×C311yescharacteristic
168C2×C3⋊Dic3—21no—
169C4⋊(C3⋊D3)—21no—
170C4⋊(C3⋊D3)C2×C3×C412yescomplemented
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—41no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—121no—
179C2×C3×Dic3—122no—
180C2×C3×Dic3D1211yescharacteristic
181C2×C3×Dic3D1211yescharacteristic
182C2×C3×Dic3C3×D411yescharacteristic
183C2×C3×Dic3C3×D411yescharacteristic
184C3×D12—21no—
185C3×D12—21no—
186C3×D12—22no—
187C3×D12—32no—
188C3×D12—32no—
189C3×D12—41no—
190C3×D12—62no—
191C3×D12—61no—
192C3×D12—61no—
193C3×D12—121no—
194C3×D12C4×D312yescomplemented
195C3×D12C2×C3×C412yescomplemented
196C3×C4×D3—122no—
197C3×C4×D3—122no—
198C3×C4×D3—122no—
199#624—42no—
200D3×Dic3—42no—
201C32×C8—21no—
202C32×C8—22no—
203C32×C8—21no—
204C32×C8—21no—
205C32×C8—32no—
206C32×C8—41no—
207C32×C8—41no—
208C32×C8—42no—
209C3×C3⋊C8—22no—
210C3×C3⋊C8C4×D312yescomplemented
211C3×C3⋊C8C2×C3×C412yescomplemented
212C4×D8—91noSylow
213C2×C33C4×D411yescharacteristic
214C2×C33C2×D811yescharacteristic
215C2×C33#13511yescharacteristic
216C3×C3⋊D3—42no—
217C32×D3—42no—
218C2×C3×D4—31no—
219C2×C3×D4—31no—
220C2×C3×D4—61no—
221C2×C3×D4—91no—
222C22×C3×C4—181no—
223C22×C3×C4—181no—
224C22×Dic3—61no—
225C2×D12—31no—
226C2×D12—31no—
227C2×D12—61no—
228C2×D12C2×C3×D311yescharacteristic
229C2×C4×D3—61no—
230C2×C4×D3—61no—
231C2×C4×D3—61no—
232C3×D8—94no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—31no—
237C2×C3×C8—31no—
238C2×C3×C8—41no—
239C2×C3×C8—61no—
240C2×C3×C8D3211yescharacteristic
241C2×C3×C8C2×C3×D311yescharacteristic
242C2×C3×C8C2×C3×D311yescharacteristic
243C3×C4⋊C4—31no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—61no—
246C3×C4⋊C4—91no—
247C3×C22⋊C4—181no—
248C3×C22⋊C4—181no—
249C3×C42—31no—
250C3×C42—31no—
251C3×C42—61no—
252#217—61no—
253#213—34no—
254(C2×D3)⋊C4—61no—
255(C2×D3)⋊C4—61no—
256(C2×D3)⋊C4—61no—
257C4⋊Dic3—31no—
258C4⋊Dic3—31no—
259C4⋊Dic3—61no—
260C4⋊Dic3C2×C3×D311yescharacteristic
261C4×Dic3C2×C3×D311yescharacteristic
262C2×C3⋊C8C2×C3×D311yescharacteristic
263D24—34no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32—21no—
267C22×C32—41no—
268C22×C32—62no—
269C22×C32—121no—
270C22×C32—121no—
271C22×C32C2×D1211yescharacteristic
272C22×C32D3×D411yescharacteristic
273C22×C32C2×C3×D411yescharacteristic
274C2×C3⋊D3—22no—
275C2×C3⋊D3—41no—
276C2×C3⋊D3—41no—
277C2×C3×D3—22no—
278C2×C3×D3—22no—
279C2×C3×D3—42no—
280C2×C3×D3—41no—
281C2×C3×D3—41no—
282C2×C3×D3—41no—
283C2×C3×D3—41no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3—81no—
287C2×C3×D3—81no—
288C2×C3×D3—122no—
289C2×C3×D3—121no—
290C2×C3×D3—121no—
291C2×C3×D3—121no—
292C2×C3×D3—121no—
293C2×C3×D3—241no—
294C2×C3×D3—241no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—32no—
302C32×C4—41no—
303C32×C4—41no—
304C32×C4—61no—
305C32×C4—121no—
306C32×C4C2×C4×D311yescharacteristic
307C32×C4C2×C4×D311yescharacteristic
308C32×C4#23511yescharacteristic
309C32×C4#23711yescharacteristic
310C32×C4C22×C3×C411yescharacteristic
311C32×C4C3×#4111yescharacteristic
312C3⋊Dic3—41no—
313C3×Dic3—22no—
314C3×Dic3—21no—
315C3×Dic3—21no—
316C3×Dic3—22no—
317C3×Dic3—41no—
318C3×Dic3—41no—
319C3×Dic3—81no—
320C3×Dic3—121no—
321C3×Dic3—121no—
322C3×Dic3—241no—
323C3×Dic3D2412yescomplemented
324C3×Dic3C3×D812yescomplemented
325C2×D8—91no—
326C4×D4—91no—
327C4×D4—91no—
328#107—91no—
329#102—31no—
330#102—91no—
331C4×C8—31no—
332C33C4×D811yescharacteristic, complemented, Sylow
333C23×C3—61no—
334C23×C3—61no—
335C23×C3—121no—
336C23×C3—181no—
337C22×D3—21no—
338C22×D3—61no—
339C22×D3—61no—
340C22×D3—121no—
341C3×D4—32no—
342C3×D4—32no—
343C3×D4—62no—
344C3×D4—61no—
345C3×D4—61no—
346C3×D4—92no—
347C3×D4—121no—
348C3×D4—181no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—31no—
353C2×C3×C4—31no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—41no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—121no—
362C2×C3×C4—181no—
363C2×C3×C4—362no—
364C2×C3×C4—362no—
365C2×C3×C4C2×D3211yescharacteristic
366C2×C3×C4C22×C3×D311yescharacteristic
367C2×C3×C4C22×C3×D311yescharacteristic
368C2×Dic3—21no—
369C2×Dic3—61no—
370C2×Dic3—61no—
371C2×Dic3—121no—
372C2×Dic3—122no—
373C2×Dic3C3×D1211yescharacteristic
374C2×Dic3C3×D1211yescharacteristic
375D12—21no—
376D12—32no—
377D12—32no—
378D12—62no—
379D12—61no—
380D12—61no—
381D12—121no—
382D12C3×C4×D312yescomplemented
383C4×D3—122no—
384C4×D3—122no—
385C4×D3—122no—
386C3×C8—22no—
387C3×C8—21no—
388C3×C8—21no—
389C3×C8—21no—
390C3×C8—32no—
391C3×C8—32no—
392C3×C8—41no—
393C3×C8—41no—
394C3×C8—42no—
395C3×C8—62no—
396C3⋊C8C3×C4×D312yescomplemented
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—41no—
407C2×C32—41no—
408C2×C32—41no—
409C2×C32—122no—
410C2×C32C4×D1211yescharacteristic
411C2×C32#67⋊C411yescharacteristic
412C2×C32C2×D2411yescharacteristic
413C2×C32#90411yescharacteristic
414C2×C32D3×D811yescharacteristic
415C2×C32#91611yescharacteristic
416C2×C32C3×C4×D411yescharacteristic
417C2×C32C2×C3×D811yescharacteristic
418C2×C32C3×#13511yescharacteristic
419C3⋊D3—42no—
420C3×D3—42no—
421C3×D3—42no—
422C3×D3—82no—
423C3×D3—122no—
424C3×D3—122no—
425C3×D3—242no—
426C2×D4—31no—
427C2×D4—91no—
428C22×C4—181no—
429C22×C4—181no—
430D8—94no—
431C2×C8—31no—
432C2×C8C3×D3211yescharacteristic
433C4⋊C4—31no—
434C4⋊C4—91no—
435C22⋊C4—181no—
436C22⋊C4—181no—
437C42—31no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—41no—
442C22×C3—62no—
443C22×C3—62no—
444C22×C3—122no—
445C22×C3—121no—
446C22×C3—121no—
447C22×C3—121no—
448C22×C3—121no—
449C22×C3—182no—
450C22×C3—241no—
451C22×C3—241no—
452C22×C3—361no—
453C22×C3—361no—
454C22×C3D3×D1211yescharacteristic
455C22×C3C2×C3×D1211yescharacteristic
456C22×C3C3×D3×D411yescharacteristic
457C2×D3—22no—
458C2×D3—41no—
459C2×D3—41no—
460C2×D3—62no—
461C2×D3—62no—
462C2×D3—122no—
463C2×D3—121no—
464C2×D3—121no—
465C2×D3—121no—
466C2×D3—121no—
467C2×D3—241no—
468C2×D3—241no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—32no—
476C3×C4—32no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—62no—
480C3×C4—62no—
481C3×C4—61no—
482C3×C4—61no—
483C3×C4—121no—
484C3×C4—121no—
485C3×C4—241no—
486C3×C4—361no—
487C3×C4C4×D3211yescharacteristic
488C3×C4#380411yescharacteristic
489C3×C4C2×C3×C4×D311yescharacteristic
490C3×C4C2×C3×C4×D311yescharacteristic
491C3×C4C3×#23511yescharacteristic
492C3×C4C3×#23711yescharacteristic
493Dic3—21no—
494Dic3—41no—
495Dic3—121no—
496Dic3—121no—
497Dic3—241no—
498Dic3C3×D2412yescomplemented
499C32—21no—
500C32—21no—
501C32—21no—
502C32—41no—
503C32C4×D2411yescharacteristic, complemented
504C32#475611yescharacteristic, complemented
505C32C3×C4×D811yescharacteristic, complemented
506C23—61no—
507C23—181no—
508D4—32no—
509D4—61no—
510D4—92no—
511D4—181no—
512C2×C4—31no—
513C2×C4—31no—
514C2×C4—61no—
515C2×C4—181no—
516C2×C4—362no—
517C2×C4—362no—
518C2×C4C2×C3×D3211yescharacteristic
519C8—21no—
520C8—32no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—41no—
531C2×C3—41no—
532C2×C3—41no—
533C2×C3—122no—
534C2×C3—122no—
535C2×C3—242no—
536C2×C3—362no—
537C2×C3D3×D2411yescharacteristic
538C2×C3#6380611yescharacteristic
539C2×C3#6388011yescharacteristic
540C2×C3C3×C4×D1211yescharacteristic
541C2×C3C3×#67⋊C411yescharacteristic
542C2×C3C2×C3×D2411yescharacteristic
543C2×C3C3×#90411yescharacteristic
544C2×C3C3×D3×D811yescharacteristic
545C2×C3C3×#91611yescharacteristic
546D3—42no—
547D3—122no—
548D3—122no—
549D3—242no—
550C22—62no—
551C22—121no—
552C22—121no—
553C22—182no—
554C22—361no—
555C22—361no—
556C22C3×D3×D1211yescharacteristic
557C4—32no—
558C4—61no—
559C4—121no—
560C4—361no—
561C4C3×C4×D3211yescharacteristic
562C4C3×#380411yescharacteristic
563C3—21no—
564C3—21no—
565C3—21no—
566C3—41no—
567C3#1059074511yescharacteristic, direct product factor
568C3C3×C4×D2411yescharacteristic, complemented
569C3C3×#475611yescharacteristic, complemented
570C2—122no—
571C2—362no—
572C2C3×D3×D2411yescharacteristic
573C2C3×#6380611yescharacteristic
574C2C3×#6388011yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3×#3714C2
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×C8D32
hypocenterC32C3×C4×D8
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
1. C3×#63894C2semidirect
2. C3×Dic3×D12C2semidirect
4. C3×#63648C2semidirect
5. C32×#821C2semidirect
6. C32×C8×Dic3C2semidirect
7. C3×#63575C2semidirect
10. #10590745C3direct (N × H)
22. C3×#3714C4semidirect
29. #63894C2×C3semidirect
30. Dic3×D12C2×C3semidirect
34. #63648C2×C3semidirect
39. C3×#821D3semidirect
40. C3×#821C2×C3semidirect
43. C3×C8×Dic3D3semidirect
44. C3×C8×Dic3C2×C3semidirect
46. #63575C2×C3semidirect
52. C3×C4⋊(C3⋊D3)C2×C4semidirect
57. C32×D12C2×C4semidirect
61. C32×C3⋊C8C2×C4semidirect
110. #3714C3×C4semidirect
123. C32×Dic3D8semidirect
144. #821C3×D3semidirect
146. C8×Dic3C3×D3semidirect
170. C4⋊(C3⋊D3)C2×C3×C4semidirect
194. C3×D12C4×D3semidirect
195. C3×D12C2×C3×C4semidirect
210. C3×C3⋊C8C4×D3semidirect
211. C3×C3⋊C8C2×C3×C4semidirect
323. C3×Dic3D24semidirect
324. C3×Dic3C3×D8semidirect
332. C33C4×D8semidirect
382. D12C3×C4×D3semidirect
396. C3⋊C8C3×C4×D3semidirect
498. Dic3C3×D24semidirect
503. C32C4×D24semidirect
504. C32#4756semidirect
505. C32C3×C4×D8semidirect
567. C3#10590745direct (N × H)
568. C3C3×C4×D24semidirect
569. C3C3×#4756semidirect

Maximal subgroups

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

subgroupindex
C3×#638942
C3×Dic3×D122
C2×C3×#37142
C3×#636482
C32×#8212
C32×C8×Dic32
C3×#635752
C3×#47563
C3×C4×D243
#105907453