Finite Group Databasebeta

← C3×#68896 #11787430

Subgroups — C3×#68896 #11787430

Order 1152.

861 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#204C211yesmaximal, characteristic
2C2×C3×#893C211yesmaximal, characteristic
3C3×#4973C211yesmaximal, characteristic, complemented
4C2×C3×#810C211yesmaximal, characteristic, complemented
5C2×C3×#809C211yesmaximal, characteristic
6C32×#340C211yesmaximal, characteristic, complemented
7C3×#4550C211yesmaximal, characteristic, complemented
8C3×#1877—31nomaximal
9#68896C311yesmaximal, characteristic, direct product factor
10C22×C3×D12C2211yescharacteristic
11C22×C3×C4×D3—21no—
12C22×C3×Dic6C2211yescharacteristic
13C22×C32×C8C2211yescharacteristic
14C2×C32×C4⋊C4C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×C3×(C2×D3)⋊C4—21no—
17C2×C3×C4×Dic3C2211yescharacteristic
18C22×C3×C3⋊C8C2211yescharacteristic
19C2×C3×#204—22no—
20C2×C3×#204—22no—
21C2×C3×#204C414yescomplemented
22C3×#893—22no—
23C3×#893—22no—
24C3×C3⋊#95—21no—
25C3×C3⋊#95—21no—
26C3×#810—22no—
27C3×#809—22no—
28C22×C3×SD16—31no—
29C2×C3×C4×D4—31no—
30C22×#204C2×C311yescharacteristic
31C2×#893C2×C311yescharacteristic
32C2×C3×#103—31no—
33C2×C3×#102—31no—
34C3×#391—31no—
35#4973C2×C311yescharacteristic, complemented
36C2×#810C2×C311yescharacteristic, complemented
37C2×#809C2×C311yescharacteristic
38C3×#340—21no—
39C3×#340D311yescharacteristic, complemented
40C3×#340C2×C311yescharacteristic, complemented
41C3×#336—31no—
42#4550C2×C311yescharacteristic, complemented
43C23×C3×D3—21no—
44C22×C32×C4—21no—
45C22×C32×C4—21no—
46C22×C32×C4C2311yescharacteristic
47C22×C3×Dic3—21no—
48C22×C3×Dic3D411yescharacteristic
49C22×C3×Dic3D411yescharacteristic
50C2×C3×D12—21no—
51C2×C3×D12—21no—
52C2×C3×D12—21no—
53C2×C3×D12C2×C412yes—
54C2×C3×D12D412yes—
55C2×C3×D12D412yes—
56C2×C3×C4×D3—24no—
57C2×C3×C4×D3—42no—
58C2×C3×C4×D3—42no—
59C2×C3×Dic6—21no—
60C2×C3×Dic6—21no—
61C2×C3×Dic6—21no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6C2×C412yescomplemented
65C2×C32×C8—21no—
66C2×C32×C8—21no—
67C2×C32×C8C2×C412yescomplemented
68C32×C4⋊C4—21no—
69C32×C4⋊C4—21no—
70C32×C4⋊C4—21no—
71C32×C4⋊C4D412yes—
72C3×(C2×D3)⋊C4—24no—
73C3×C4×Dic3—21no—
74C3×C4×Dic3—21no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8—21no—
78C3×#204—28no—
79#1877—31noSylow
80C22×C3×Q8—31no—
81C22×C3×D4—31no—
82C23×C3×C4—61no—
83C22×D12C22×C311yescharacteristic
84C22×C4×D3—21no—
85C22×Dic6C22×C311yescharacteristic
86C2×C3×SD16—34no—
87C2×C3×SD16—62no—
88C2×C3×SD16—62no—
89C22×C3×C8—21no—
90C22×C3×C8—31no—
91C22×C3×C8C2×D311yescharacteristic
92C22×C3×C8C22×C311yescharacteristic
93C3×C4×D4—62no—
94C3×C4×D4—62no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4C2×D311yescharacteristic
98C2×C3×C4⋊C4C2×D311yescharacteristic
99C2×C3×C4⋊C4C22×C311yescharacteristic
100C2×C3×C4⋊C4C22×C311yescharacteristic
101C2×C3×C22⋊C4—61no—
102C2×C3×C42—31no—
103C2×(C2×D3)⋊C4—21no—
104C2×C4×Dic3C22×C311yescharacteristic
105C22×C3⋊C8C22×C311yescharacteristic
106C2×#204—22no—
107C2×#204—22no—
108C2×#204C3×C414yescomplemented
109#893—22no—
110#893—22no—
111C3×#103—62no—
112C3×#102—62no—
113C3×#95—61no—
114C3×#95—61no—
115C3⋊#95—21no—
116C3⋊#95—21no—
117#810—22no—
118#809—22no—
119C23×C32C2×D411yescharacteristic
120C22×C3×D3—22no—
121C22×C3×D3—22no—
122C22×C3×D3—22no—
123C22×C3×D3—41no—
124C22×C3×D3—41no—
125C22×C3×D3—41no—
126C22×C3×D3—41no—
127C2×C32×C4—22no—
128C2×C32×C4—41no—
129C2×C32×C4—41no—
130C2×C32×C4—41no—
131C2×C32×C4—41no—
132C2×C32×C4—41no—
133C2×C32×C4C22×C411yescharacteristic
134C2×C32×C4C2×D411yescharacteristic
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4#4111yescharacteristic
138C2×C32×C4#4111yescharacteristic
139C2×C3×Dic3—21no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—21no—
145C2×C3×Dic3—41no—
146C2×C3×Dic3—41no—
147C2×C3×Dic3SD1612yescomplemented
148C3×D12—22no—
149C3×D12—22no—
150C3×D12—22no—
151C3×D12C22⋊C414yes—
152C3×C4×D3—48no—
153C3×Dic6—22no—
154C3×Dic6—22no—
155C3×Dic6—41no—
156C3×Dic6—41no—
157C32×C8—22no—
158C3×C3⋊C8—41no—
159C22×SD16—31no—
160C2×C4×D4—31no—
161C2×#103—31no—
162C2×#102—31no—
163#391—31no—
164#340C3×D311yescharacteristic, complemented
165#336—31no—
166C24×C3—61no—
167C23×D3—21no—
168C2×C3×Q8—32no—
169C2×C3×Q8—61no—
170C2×C3×Q8—61no—
171C2×C3×Q8—61no—
172C2×C3×Q8—61no—
173C2×C3×Q8—61no—
174C2×C3×D4—32no—
175C2×C3×D4—32no—
176C2×C3×D4—32no—
177C2×C3×D4—61no—
178C2×C3×D4—61no—
179C2×C3×D4—61no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—31no—
186C22×C3×C4—31no—
187C22×C3×C4—41no—
188C22×C3×C4—41no—
189C22×C3×C4—64no—
190C22×C3×C4—61no—
191C22×C3×C4—122no—
192C22×C3×C4—122no—
193C22×C3×C4C22×D311yescharacteristic
194C22×C3×C4C23×C311yescharacteristic
195C22×Dic3—21no—
196C22×Dic3C3×D411yescharacteristic
197C22×Dic3C3×D411yescharacteristic
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12C2×C3×C412yes—
202C2×D12C3×D412yes—
203C2×D12C3×D412yes—
204C2×C4×D3—24no—
205C2×C4×D3—42no—
206C2×C4×D3—42no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6C2×C3×C412yescomplemented
213C3×SD16—68no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—22no—
219C2×C3×C8—41no—
220C2×C3×C8—41no—
221C2×C3×C8—61no—
222C2×C3×C8—61no—
223C2×C3×C8—61no—
224C2×C3×C8C4×D312yescomplemented
225C2×C3×C8C2×C3×C412yescomplemented
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—41no—
235C3×C4⋊C4—41no—
236C3×C4⋊C4D1212yes—
237C3×C4⋊C4C3×D412yes—
238C3×C22⋊C4—64no—
239C3×C42—61no—
240C3×C42—61no—
241(C2×D3)⋊C4—24no—
242C4×Dic3—21no—
243C4×Dic3—21no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247#204—28no—
248C22×C32C4×D411yescharacteristic
249C22×C32#12011yescharacteristic
250C22×C32#12111yescharacteristic
251C22×C32C2×SD1611yescharacteristic
252C22×C32#13511yescharacteristic
253C22×C32AGL(1,ℤ/8)11yescharacteristic
254C22×C32#13711yescharacteristic
255C2×C3×D3—24no—
256C2×C3×D3—42no—
257C2×C3×D3—42no—
258C2×C3×D3—42no—
259C2×C3×D3—42no—
260C2×C3×D3—42no—
261C2×C3×D3—42no—
262C32×C4—42no—
263C32×C4—81no—
264C32×C4C2×C22⋊C411yescharacteristic
265C32×C4C4×D411yescharacteristic
266C32×C4#12111yescharacteristic
267C32×C4#12311yescharacteristic
268C3×Dic3—22no—
269C3×Dic3—42no—
270C3×Dic3—41no—
271C22×Q8—31no—
272C22×D4—31no—
273C23×C4—61no—
274C2×SD16—34no—
275C2×SD16—62no—
276C2×SD16—62no—
277C22×C8—31no—
278C22×C8C2×C3×D311yescharacteristic
279C4×D4—62no—
280C4×D4—62no—
281C2×C4⋊C4C2×C3×D311yescharacteristic
282C2×C4⋊C4C2×C3×D311yescharacteristic
283C2×C22⋊C4—61no—
284C2×C42—31no—
285#103—62no—
286#102—62no—
287#95—61no—
288#95—61no—
289C23×C3—21no—
290C23×C3—62no—
291C23×C3—62no—
292C23×C3—62no—
293C23×C3—121no—
294C23×C3—121no—
295C23×C3—121no—
296C23×C3—121no—
297C23×C3D3×D411yescharacteristic
298C23×C3C2×C3×D411yescharacteristic
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—41no—
305C22×D3—41no—
306C3×Q8—62no—
307C3×Q8—62no—
308C3×Q8—121no—
309C3×Q8—121no—
310C3×D4—34no—
311C3×D4—62no—
312C3×D4—62no—
313C3×D4—62no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—32no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—42no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—62no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—81no—
342C2×C3×C4—81no—
343C2×C3×C4—81no—
344C2×C3×C4—81no—
345C2×C3×C4—128no—
346C2×C3×C4—121no—
347C2×C3×C4—121no—
348C2×C3×C4C2×C4×D311yescharacteristic
349C2×C3×C4C2×D1211yescharacteristic
350C2×C3×C4#23511yescharacteristic
351C2×C3×C4D3×D411yescharacteristic
352C2×C3×C4D3×D411yescharacteristic
353C2×C3×C4#23911yescharacteristic
354C2×C3×C4C22×C3×C411yescharacteristic
355C2×C3×C4C2×C3×D411yescharacteristic
356C2×C3×C4C2×C3×D411yescharacteristic
357C2×C3×C4C2×C3×D411yescharacteristic
358C2×C3×C4C3×#4111yescharacteristic
359C2×C3×C4C3×#4111yescharacteristic
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—22no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3C3×SD1612yescomplemented
369D12—22no—
370D12—22no—
371D12—22no—
372D12C3×C22⋊C414yes—
373C4×D3—48no—
374Dic6—22no—
375Dic6—22no—
376Dic6—41no—
377Dic6—41no—
378C3×C8—22no—
379C3×C8—22no—
380C3×C8—42no—
381C3×C8—121no—
382C3⋊C8—41no—
383C2×C32#38611yescharacteristic
384C2×C32C4×SD1611yescharacteristic
385C2×C32#44011yescharacteristic
386C2×C32#45011yescharacteristic
387C2×C32#45211yescharacteristic
388C2×C32#46111yescharacteristic
389C2×C32#46311yescharacteristic
390C3×D3—44no—
391C24—61no—
392C2×Q8—32no—
393C2×Q8—61no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×D4—32no—
399C2×D4—32no—
400C2×D4—32no—
401C2×D4—61no—
402C2×D4—61no—
403C2×D4—61no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—31no—
407C22×C4—31no—
408C22×C4—64no—
409C22×C4—61no—
410C22×C4—122no—
411C22×C4—122no—
412C22×C4C22×C3×D311yescharacteristic
413SD16—68no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8C3×C4×D312yescomplemented
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4C3×D1212yes—
424C22⋊C4—64no—
425C42—61no—
426C42—61no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—64no—
435C22×C3—122no—
436C22×C3—122no—
437C22×C3—122no—
438C22×C3—122no—
439C22×C3—122no—
440C22×C3—122no—
441C22×C3#88211yescharacteristic
442C22×C3#88411yescharacteristic
443C22×C3#89311yescharacteristic
444C22×C3#91111yescharacteristic
445C22×C3D3×SD1611yescharacteristic
446C22×C3#91611yescharacteristic
447C22×C3#91811yescharacteristic
448C22×C3C3×C4×D411yescharacteristic
449C22×C3C3×#12011yescharacteristic
450C22×C3C3×#12111yescharacteristic
451C22×C3C2×C3×SD1611yescharacteristic
452C22×C3C3×#13511yescharacteristic
453C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
454C22×C3C3×#13711yescharacteristic
455C2×D3—24no—
456C2×D3—42no—
457C2×D3—42no—
458C2×D3—42no—
459C2×D3—42no—
460C2×D3—42no—
461C2×D3—42no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—42no—
467C3×C4—42no—
468C3×C4—62no—
469C3×C4—82no—
470C3×C4—82no—
471C3×C4—81no—
472C3×C4—81no—
473C3×C4—122no—
474C3×C4—121no—
475C3×C4C4×D1211yescharacteristic
476C3×C4D3×C22⋊C411yescharacteristic
477C3×C4#89411yescharacteristic
478C3×C4#89511yescharacteristic
479C3×C4C2×C3×C22⋊C411yescharacteristic
480C3×C4C3×C4×D411yescharacteristic
481C3×C4C3×#12111yescharacteristic
482C3×C4C3×#12311yescharacteristic
483Dic3—22no—
484Dic3—42no—
485Dic3—41no—
486C32#187711yescharacteristic, complemented, Sylow
487C23—62no—
488C23—62no—
489C23—62no—
490C23—121no—
491C23—121no—
492C23—121no—
493C23—121no—
494C23C3×D3×D411yescharacteristic
495Q8—62no—
496Q8—62no—
497Q8—121no—
498Q8—121no—
499D4—34no—
500D4—62no—
501D4—62no—
502D4—62no—
503C2×C4—22no—
504C2×C4—32no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—62no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—128no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4C2×C3×C4×D311yescharacteristic
520C2×C4C2×C3×D1211yescharacteristic
521C2×C4C3×#23511yescharacteristic
522C2×C4C3×D3×D411yescharacteristic
523C2×C4C3×D3×D411yescharacteristic
524C2×C4C3×#23911yescharacteristic
525C8—22no—
526C8—121no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—124no—
535C2×C3#466511yescharacteristic
536C2×C3#477611yescharacteristic
537C2×C3#478711yescharacteristic
538C2×C3#480811yescharacteristic
539C2×C3#481811yescharacteristic
540C2×C3#485211yescharacteristic
541C2×C3#488911yescharacteristic
542C2×C3C3×#38611yescharacteristic
543C2×C3C3×C4×SD1611yescharacteristic
544C2×C3C3×#44011yescharacteristic
545C2×C3C3×#45011yescharacteristic
546C2×C3C3×#45211yescharacteristic
547C2×C3C3×#46111yescharacteristic
548C2×C3C3×#46311yescharacteristic
549D3—44no—
550C22—64no—
551C22—122no—
552C22—122no—
553C22—122no—
554C22—122no—
555C22—122no—
556C22—122no—
557C22C3×#88211yescharacteristic
558C22C3×#88411yescharacteristic
559C22C3×#89311yescharacteristic
560C22C3×#91111yescharacteristic
561C22C3×D3×SD1611yescharacteristic
562C22C3×#91611yescharacteristic
563C22C3×#91811yescharacteristic
564C4—42no—
565C4—62no—
566C4—81no—
567C4—122no—
568C4—121no—
569C4C3×C4×D1211yescharacteristic
570C4C3×D3×C22⋊C411yescharacteristic
571C4C3×#89411yescharacteristic
572C4C3×#89511yescharacteristic
573C3—21no—
574C3#6889611yescharacteristic, direct product factor
575C3C3×#187711yescharacteristic, complemented
576C2—124no—
577C2C3×#466511yescharacteristic
578C2C3×#477611yescharacteristic
579C2C3×#478711yescharacteristic
580C2C3×#480811yescharacteristic
581C2C3×#481811yescharacteristic
582C2C3×#485211yescharacteristic
583C2C3×#488911yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#340C2
socleC23×C32C2×D4
prime radicalC22×C3×D12C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#340D3
hypocenterC3C3×#1877
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C3×#340

Complements

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

NH ≅ G/Ntype
3. C3×#4973C2semidirect
4. C2×C3×#810C2semidirect
6. C32×#340C2semidirect
7. C3×#4550C2semidirect
9. #68896C3direct (N × H)
21. C2×C3×#204C4semidirect
35. #4973C2×C3semidirect
36. C2×#810C2×C3semidirect
39. C3×#340D3semidirect
40. C3×#340C2×C3semidirect
42. #4550C2×C3semidirect
64. C2×C3×Dic6C2×C4semidirect
67. C2×C32×C8C2×C4semidirect
108. C2×#204C3×C4semidirect
147. C2×C3×Dic3SD16semidirect
164. #340C3×D3semidirect
212. C2×Dic6C2×C3×C4semidirect
224. C2×C3×C8C4×D3semidirect
225. C2×C3×C8C2×C3×C4semidirect
368. C2×Dic3C3×SD16semidirect
419. C2×C8C3×C4×D3semidirect
486. C32#1877semidirect
574. C3#68896direct (N × H)
575. C3C3×#1877semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#2042
C2×C3×#8932
C3×#49732
C2×C3×#8102
C2×C3×#8092
C32×#3402
C3×#45502
C3×#18773
#688963