Finite Group Databasebeta

← C3×#10590589 #13157586

Subgroups — C3×#10590589 #13157586

Order 1728.

734 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#893C211yesmaximal, characteristic, complemented
2C3×#63894C211yesmaximal, characteristic, complemented
3C2×C3×#3714C211yesmaximal, characteristic
4C3×#63636C211yesmaximal, characteristic, complemented
5C32×#808C211yesmaximal, characteristic, complemented
6C3×#63562C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8C3×#5014—31nomaximal
9C3×#4869—31nomaximal
10#10590589C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3—21no—
14C2×C3×#624—21no—
15C33×C4⋊C4C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×#3722—21no—
22C3×#3714C414yescomplemented
23C32×C4×D4—31no—
24C2×C3×#213—31no—
25C2×C3×D24—31no—
26C3×#893—21no—
27C3×#893—31no—
28C3×#893D311yescharacteristic, complemented
29C3×#893C2×C311yescharacteristic, complemented
30C3×C4×D12—31no—
31#63894C2×C311yescharacteristic, complemented
32C2×#3714C2×C311yescharacteristic
33C32×#106—31no—
34#63636C2×C311yescharacteristic, complemented
35C3×#832—31no—
36C3×#815—31no—
37C3×#809—31no—
38C3×#809—31no—
39C3×#809—31no—
40C3×#809—61no—
41C3×#808—21no—
42C3×#808D311yescharacteristic, complemented
43C3×#808C2×C311yescharacteristic, complemented
44C3×#803—31no—
45#63562C2×C311yescharacteristic, complemented
46#63554C2×C311yescharacteristic, complemented
47C22×C3×C3⋊D3—21no—
48C22×C32×D3—21no—
49C2×C33×C4—21no—
50C2×C33×C4C2311yescharacteristic
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
58C32×C4×D3—42no—
59C3×#624—42no—
60C3×C3⋊(C3⋊C8)—21no—
61C32×C3⋊C8C2×C412yescomplemented
62C3×#442—91no—
63#5014—31no—
64#4869—31no—
65C2×C32×D4—31no—
66C22×C32×C4—61no—
67C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
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—22no—
75C2×C3×C4×D3—21no—
76C2×C3×C4×D3—21no—
77C2×C3×C4×D3—61no—
78C2×C3×C4×D3—61no—
79C2×#624—21no—
80C2×C32×C8—31no—
81C32×C4⋊C4—21no—
82C32×C4⋊C4—21no—
83C32×C4⋊C4—21no—
84C32×C4⋊C4—31no—
85C32×C4⋊C4—41no—
86C32×C4⋊C4C2×D311yescharacteristic
87C32×C4⋊C4C2×D311yescharacteristic
88C32×C4⋊C4C22×C311yescharacteristic
89C32×C22⋊C4—61no—
90C32×C42—31no—
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C3×#213—34no—
93C3×(C2×D3)⋊C4—22no—
94C3×(C2×D3)⋊C4—21no—
95C3×(C2×D3)⋊C4—21no—
96C3×(C2×D3)⋊C4—61no—
97C3×(C2×D3)⋊C4—61no—
98C3×C4⋊Dic3—21no—
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⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—61no—
108C2×C3×C3⋊C8C2×D311yescharacteristic
109C2×C3×C3⋊C8C22×C311yescharacteristic
110C3×D24—34no—
111#3722—21no—
112#3714C3×C414yescomplemented
113C22×C33C2×D411yescharacteristic
114C2×C3×C3⋊D3—22no—
115C2×C3×C3⋊D3—41no—
116C2×C3×C3⋊D3—41no—
117C2×C32×D3—22no—
118C2×C32×D3—41no—
119C2×C32×D3—41no—
120C33×C4—41no—
121C33×C4C22×C411yescharacteristic
122C33×C4#4111yescharacteristic
123C32×Dic3—21no—
124C32×Dic3—21no—
125C32×Dic3—41no—
126C2×C3×D8—91no—
127C3×C4×D4—31no—
128C3×C4×D4—31no—
129C3×C4×D4—61no—
130C3×C4×D4—91no—
131C2×#213—31no—
132C2×D24—31no—
133#893—31no—
134#893C3×D311yescharacteristic, complemented
135C4×D12—31no—
136C3×#106—31no—
137C3×#106—31no—
138C3×#106—61no—
139C3×#102—91no—
140C3×#102—91no—
141C3×#97—91no—
142#832—31no—
143#815—31no—
144#809—31no—
145#809—31no—
146#809—31no—
147#809—61no—
148#808C3×D311yescharacteristic, complemented
149#803—31no—
150C23×C32—61no—
151C22×C3⋊D3—21no—
152C22×C3×D3—22no—
153C22×C3×D3—21no—
154C22×C3×D3—21no—
155C22×C3×D3—61no—
156C22×C3×D3—61no—
157C22×C3×D3—121no—
158C32×D4—32no—
159C32×D4—61no—
160C2×C32×C4—21no—
161C2×C32×C4—22no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—31no—
168C2×C32×C4—31no—
169C2×C32×C4—41no—
170C2×C32×C4—41no—
171C2×C32×C4—41no—
172C2×C32×C4—42no—
173C2×C32×C4—61no—
174C2×C32×C4—122no—
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C23×C311yescharacteristic
178C4⋊(C3⋊D3)—21no—
179C4⋊(C3⋊D3)C2×C3×C412yescomplemented
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—41no—
185C2×C3×Dic3#6711yescharacteristic
186C2×C3×Dic3#6711yescharacteristic
187C2×C3×Dic3C3×D411yescharacteristic
188C2×C3×Dic3C3×D411yescharacteristic
189C3×D12—22no—
190C3×D12—21no—
191C3×D12—21no—
192C3×D12—22no—
193C3×D12—32no—
194C3×D12—32no—
195C3×D12—62no—
196C3×D12—61no—
197C3×D12—61no—
198C3×D12—121no—
199C3×D12C4×D312yescomplemented
200C3×D12C2×C3×C412yescomplemented
201C3×C4×D3—44no—
202C3×C4×D3—42no—
203C3×C4×D3—42no—
204C3×C4×D3—122no—
205C3×C4×D3—122no—
206#624—42no—
207C32×C8—32no—
208C3⋊(C3⋊C8)—21no—
209C3×C3⋊C8—22no—
210C3×C3⋊C8—61no—
211C3×C3⋊C8—61no—
212C3×C3⋊C8—121no—
213C3×C3⋊C8C4×D312yescomplemented
214C3×C3⋊C8C2×C3×C412yescomplemented
215#442—91noSylow
216C2×C33C4×D411yescharacteristic
217C2×C33AGL(1,ℤ/8)11yescharacteristic
218C2×C33AGL(1,ℤ/8)11yescharacteristic
219C3×C3⋊D3—42no—
220C32×D3—42no—
221C2×C3×D4—31no—
222C2×C3×D4—31no—
223C2×C3×D4—61no—
224C2×C3×D4—91no—
225C22×C3×C4—61no—
226C22×C3×C4—61no—
227C22×C3×C4—62no—
228C22×C3×C4—181no—
229C2×D12—31no—
230C2×D12—31no—
231C2×D12—61no—
232C2×D12C2×C3×D311yescharacteristic
233C2×C4×D3—21no—
234C2×C4×D3—61no—
235C2×C4×D3—61no—
236C3×D8—94no—
237C2×C3×C8—31no—
238C2×C3×C8—31no—
239C2×C3×C8—61no—
240C2×C3×C8—91no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—31no—
246C3×C4⋊C4—41no—
247C3×C4⋊C4—61no—
248C3×C4⋊C4D3211yescharacteristic
249C3×C4⋊C4C2×C3×D311yescharacteristic
250C3×C4⋊C4C2×C3×D311yescharacteristic
251C3×C22⋊C4—61no—
252C3×C22⋊C4—61no—
253C3×C22⋊C4—62no—
254C3×C22⋊C4—181no—
255C3×C42—31no—
256C3×C42—31no—
257C3×C42—61no—
258#213—34no—
259(C2×D3)⋊C4—21no—
260(C2×D3)⋊C4—61no—
261(C2×D3)⋊C4—61no—
262C4⋊Dic3C2×C3×D311yescharacteristic
263C4×Dic3C2×C3×D311yescharacteristic
264C2×C3⋊C8—31no—
265C2×C3⋊C8—31no—
266C2×C3⋊C8—61no—
267C2×C3⋊C8C2×C3×D311yescharacteristic
268D24—34no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32—62no—
274C22×C32—121no—
275C22×C32—121no—
276C22×C32D3×D411yescharacteristic
277C22×C32C2×#6711yescharacteristic
278C22×C32C2×C3×D411yescharacteristic
279C2×C3⋊D3—22no—
280C2×C3⋊D3—41no—
281C2×C3⋊D3—41no—
282C2×C3×D3—24no—
283C2×C3×D3—22no—
284C2×C3×D3—22no—
285C2×C3×D3—42no—
286C2×C3×D3—42no—
287C2×C3×D3—41no—
288C2×C3×D3—41no—
289C2×C3×D3—41no—
290C2×C3×D3—41no—
291C2×C3×D3—62no—
292C2×C3×D3—62no—
293C2×C3×D3—122no—
294C2×C3×D3—121no—
295C2×C3×D3—121no—
296C2×C3×D3—121no—
297C2×C3×D3—121no—
298C2×C3×D3—241no—
299C2×C3×D3—241no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—21no—
306C32×C4—41no—
307C32×C4—41no—
308C32×C4—41no—
309C32×C4—42no—
310C32×C4—41no—
311C32×C4—41no—
312C32×C4—61no—
313C32×C4—61no—
314C32×C4—81no—
315C32×C4—81no—
316C32×C4—82no—
317C32×C4—121no—
318C32×C4C2×C4×D311yescharacteristic
319C32×C4C2×C4×D311yescharacteristic
320C32×C4#23511yescharacteristic
321C32×C4#23911yescharacteristic
322C32×C4C22×C3×C411yescharacteristic
323C32×C4C3×#4111yescharacteristic
324C3×Dic3—22no—
325C3×Dic3—21no—
326C3×Dic3—21no—
327C3×Dic3—21no—
328C3×Dic3—21no—
329C3×Dic3—41no—
330C3×Dic3—41no—
331C3×Dic3—41no—
332C3×Dic3—81no—
333C2×D8—91no—
334C4×D4—31no—
335C4×D4—91no—
336#106—31no—
337#102—91no—
338#102—91no—
339#97—91no—
340C33#44211yescharacteristic, complemented, Sylow
341C23×C3—62no—
342C23×C3—61no—
343C23×C3—61no—
344C23×C3—181no—
345C22×D3—21no—
346C22×D3—61no—
347C22×D3—61no—
348C22×D3—121no—
349C3×D4—32no—
350C3×D4—32no—
351C3×D4—62no—
352C3×D4—62no—
353C3×D4—61no—
354C3×D4—61no—
355C3×D4—92no—
356C3×D4—181no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—31no—
365C2×C3×C4—31no—
366C2×C3×C4—31no—
367C2×C3×C4—31no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—42no—
371C2×C3×C4—41no—
372C2×C3×C4—61no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—121no—
377C2×C3×C4—124no—
378C2×C3×C4—122no—
379C2×C3×C4—122no—
380C2×C3×C4—362no—
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C22×C3×D311yescharacteristic
383C2×C3×C4C22×C3×D311yescharacteristic
384C2×Dic3—21no—
385C2×Dic3C3×#6711yescharacteristic
386C2×Dic3C3×#6711yescharacteristic
387D12—21no—
388D12—32no—
389D12—32no—
390D12—62no—
391D12—61no—
392D12—61no—
393D12—121no—
394D12C3×C4×D312yescomplemented
395C4×D3—42no—
396C4×D3—122no—
397C4×D3—122no—
398C3×C8—32no—
399C3×C8—32no—
400C3×C8—62no—
401C3×C8—181no—
402C3⋊C8—61no—
403C3⋊C8—61no—
404C3⋊C8—121no—
405C3⋊C8C3×C4×D312yescomplemented
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—41no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32—122no—
419C2×C32#89311yescharacteristic
420C2×C32#91411yescharacteristic
421C2×C32#91411yescharacteristic
422C2×C32C4×#6711yescharacteristic
423C2×C32#93211yescharacteristic
424C2×C32#94911yescharacteristic
425C2×C32C3×C4×D411yescharacteristic
426C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
427C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
428C3⋊D3—42no—
429C3×D3—44no—
430C3×D3—42no—
431C3×D3—42no—
432C3×D3—122no—
433C3×D3—122no—
434C3×D3—242no—
435C2×D4—31no—
436C2×D4—91no—
437C22×C4—61no—
438C22×C4—181no—
439D8—94no—
440C2×C8—31no—
441C2×C8—91no—
442C4⋊C4—31no—
443C4⋊C4C3×D3211yescharacteristic
444C22⋊C4—61no—
445C22⋊C4—181no—
446C42—31no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—41no—
451C22×C3—64no—
452C22×C3—62no—
453C22×C3—62no—
454C22×C3—122no—
455C22×C3—122no—
456C22×C3—121no—
457C22×C3—121no—
458C22×C3—121no—
459C22×C3—121no—
460C22×C3—182no—
461C22×C3—361no—
462C22×C3—361no—
463C22×C3D3×#6711yescharacteristic
464C22×C3C3×D3×D411yescharacteristic
465C22×C3C2×C3×#6711yescharacteristic
466C2×D3—22no—
467C2×D3—41no—
468C2×D3—41no—
469C2×D3—62no—
470C2×D3—62no—
471C2×D3—122no—
472C2×D3—121no—
473C2×D3—121no—
474C2×D3—121no—
475C2×D3—121no—
476C2×D3—241no—
477C2×D3—241no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—42no—
485C3×C4—41no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—62no—
491C3×C4—61no—
492C3×C4—61no—
493C3×C4—61no—
494C3×C4—61no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4—82no—
498C3×C4—121no—
499C3×C4—121no—
500C3×C4—121no—
501C3×C4—241no—
502C3×C4#379911yescharacteristic
503C3×C4C4×D3211yescharacteristic
504C3×C4C2×C3×C4×D311yescharacteristic
505C3×C4C2×C3×C4×D311yescharacteristic
506C3×C4C3×#23511yescharacteristic
507C3×C4C3×#23911yescharacteristic
508Dic3—21no—
509Dic3—21no—
510Dic3—41no—
511C32—21no—
512C32—21no—
513C32—21no—
514C32—41no—
515C32#486911yescharacteristic, complemented
516C32#501411yescharacteristic, complemented
517C32C3×#44211yescharacteristic, complemented
518C23—61no—
519C23—181no—
520D4—32no—
521D4—61no—
522D4—92no—
523D4—181no—
524C2×C4—21no—
525C2×C4—31no—
526C2×C4—31no—
527C2×C4—61no—
528C2×C4—122no—
529C2×C4—362no—
530C2×C4C2×C3×D3211yescharacteristic
531C8—32no—
532C8—181no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—41no—
543C2×C3—41no—
544C2×C3—41no—
545C2×C3—124no—
546C2×C3—122no—
547C2×C3—122no—
548C2×C3—362no—
549C2×C3#6390411yescharacteristic
550C2×C3#6393411yescharacteristic
551C2×C3#6393911yescharacteristic
552C2×C3C3×#89311yescharacteristic
553C2×C3C3×#91411yescharacteristic
554C2×C3C3×#91411yescharacteristic
555C2×C3C3×C4×#6711yescharacteristic
556C2×C3C3×#93211yescharacteristic
557C2×C3C3×#94911yescharacteristic
558D3—42no—
559D3—122no—
560D3—122no—
561D3—242no—
562C22—62no—
563C22—121no—
564C22—121no—
565C22—182no—
566C22—361no—
567C22—361no—
568C22C3×D3×#6711yescharacteristic
569C4—41no—
570C4—61no—
571C4—61no—
572C4—121no—
573C4C3×#379911yescharacteristic
574C4C3×C4×D3211yescharacteristic
575C3—21no—
576C3—21no—
577C3—21no—
578C3—41no—
579C3#1059058911yescharacteristic, direct product factor
580C3C3×#486911yescharacteristic, complemented
581C3C3×#501411yescharacteristic, complemented
582C2—122no—
583C2—362no—
584C2C3×#6390411yescharacteristic
585C2C3×#6393411yescharacteristic
586C2C3×#6393911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3714C2
cosocleC2×C32×C4C23×C3
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#442
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×#893C2semidirect
2. C3×#63894C2semidirect
4. C3×#63636C2semidirect
5. C32×#808C2semidirect
6. C3×#63562C2semidirect
7. C3×#63554C2semidirect
10. #10590589C3direct (N × H)
22. C3×#3714C4semidirect
28. C3×#893D3semidirect
29. C3×#893C2×C3semidirect
31. #63894C2×C3semidirect
34. #63636C2×C3semidirect
42. C3×#808D3semidirect
43. C3×#808C2×C3semidirect
45. #63562C2×C3semidirect
46. #63554C2×C3semidirect
52. C3×C4⋊(C3⋊D3)C2×C4semidirect
57. C32×D12C2×C4semidirect
61. C32×C3⋊C8C2×C4semidirect
112. #3714C3×C4semidirect
134. #893C3×D3semidirect
148. #808C3×D3semidirect
179. C4⋊(C3⋊D3)C2×C3×C4semidirect
199. C3×D12C4×D3semidirect
200. C3×D12C2×C3×C4semidirect
213. C3×C3⋊C8C4×D3semidirect
214. C3×C3⋊C8C2×C3×C4semidirect
340. C33#442semidirect
394. D12C3×C4×D3semidirect
405. C3⋊C8C3×C4×D3semidirect
515. C32#4869semidirect
516. C32#5014semidirect
517. C32C3×#442semidirect
579. C3#10590589direct (N × H)
580. C3C3×#4869semidirect
581. C3C3×#5014semidirect

Maximal subgroups

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

subgroupindex
C32×#8932
C3×#638942
C2×C3×#37142
C3×#636362
C32×#8082
C3×#635622
C3×#635542
C3×#50143
C3×#48693
#105905893