Finite Group Databasebeta

← C3×#24171 #10642778

Subgroups — C3×#24171 #10642778

Order 768.

1106 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3511C211yesmaximal, characteristic, complemented
2C3×#3036C211yesmaximal, characteristic, complemented
3C3×C4×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
4C3×#2563C211yesmaximal, characteristic, complemented
5C3×#2040C212yesmaximal, complemented
6C3×#2006C212yesmaximal, complemented
7C3×#1997C211yesmaximal, characteristic, complemented
8C3×#1980C211yesmaximal, characteristic, complemented
9C3×#1883C212yesmaximal, complemented
10C3×#1860C212yesmaximal, complemented
11C3×#1806C211yesmaximal, characteristic, complemented
12#24171C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
14C2×C3×AGL(1,ℤ/8)C2212yes—
15C3×#556—21no—
16C3×#555—21no—
17C3×#554—21no—
18C3×#553—22no—
19C3×#548—22no—
20C3×#547—21no—
21C3×#542—22no—
22C3×#537C2212yes—
23C3×#533—21no—
24C3×#532—21no—
25C3×#532C2211yescharacteristic
26C2×C3×C4⋊Q8C2211yescharacteristic
27C3×#520C2212yes—
28C3×C4×#41C2211yescharacteristic
29C2×C3×C4×D4C2211yescharacteristic
30C3×#463—22no—
31C3×#461—21no—
32C3×#459—21no—
33C3×#442—21no—
34C3×#440—21no—
35C3×C4×SD16—21no—
36C3×C4×D8—21no—
37C3×#428C2211yescharacteristic
38C3×#423C2211yescharacteristic
39C2×C3×#104C2212yes—
40C2×C3×#104C2212yes—
41C2×C3×#104C2212yes—
42C3×#418C2211yescharacteristic
43C3×#417C2211yescharacteristic
44C3×#417C2212yes—
45C3×C4×M16C2211yescharacteristic
46C3×#393—22no—
47C3×#390C2212yes—
48C3×#382C2212yes—
49C3×#380—22no—
50C3×C4×C4⋊C4C2211yescharacteristic
51C3×#343C2212yes—
52C3×#343C2212yes—
53C3×#339C2212yes—
54#3511C2×C311yescharacteristic, complemented
55#3036C2×C311yescharacteristic, complemented
56C4×AGL(1,ℤ/8)C2×C311yescharacteristic, complemented
57#2563C2×C311yescharacteristic, complemented
58#2040C2×C312yescomplemented
59#2006C2×C312yescomplemented
60#1997C2×C311yescharacteristic, complemented
61#1980C2×C311yescharacteristic, complemented
62#1883C2×C312yescomplemented
63#1860C2×C312yescomplemented
64#1806C2×C311yescharacteristic, complemented
65C2×C3×#41—22no—
66C2×C3×#41C2311yescharacteristic
67C2×C3×#41C2312yes—
68C22×C3×Q8D412yes—
69C22×C3×D4C2311yescharacteristic
70C23×C3×C4—21no—
71C3×AGL(1,ℤ/8)—22no—
72C3×AGL(1,ℤ/8)—24no—
73C3×AGL(1,ℤ/8)—22no—
74C3×AGL(1,ℤ/8)—42no—
75C2×C3×SD16—21no—
76C2×C3×SD16—22no—
77C2×C3×D8—21no—
78C2×C3×D8—22no—
79C2×C3×M16C2312yes—
80C2×C3×M16C2311yescharacteristic
81C2×C3×M16C2311yescharacteristic
82C3×C4⋊Q8—21no—
83C3×C4⋊Q8—21no—
84C3×C4⋊Q8—21no—
85C3×C4⋊Q8—21no—
86C3×C4⋊Q8D411yescharacteristic
87C3×#127D411yescharacteristic
88C3×#126—24no—
89C3×#125—22no—
90C3×#124—22no—
91C3×#124—24no—
92C3×#124D412yes—
93C3×#123—24no—
94C3×C22⋊Q8—22no—
95C3×C22⋊Q8—24no—
96C3×C22⋊Q8—22no—
97C3×C22⋊Q8—22no—
98C3×#121—22no—
99C3×#121—22no—
100C3×#121—22no—
101C3×#121—21no—
102C3×#121—21no—
103C3×#121—24no—
104C3×C4×Q8—22no—
105C3×C4×Q8—22no—
106C3×C4×Q8D411yescharacteristic
107C3×C4×Q8D411yescharacteristic
108C3×C4×D4—22no—
109C3×C4×D4—22no—
110C3×C4×D4—21no—
111C3×C4×D4—21no—
112C3×C4×D4—24no—
113C3×C4×D4—21no—
114C3×C4×D4—21no—
115C3×C4×D4—21no—
116C3×C4×D4—21no—
117C3×C4×D4—21no—
118C3×C4×D4—21no—
119C3×C4×D4D411yescharacteristic
120C3×C4×D4D411yescharacteristic
121C3×#117—22no—
122C3×#117—22no—
123C3×#117—21no—
124C3×#117C2311yescharacteristic
125C2×C3×C4⋊C4—22no—
126C2×C3×C4⋊C4—21no—
127C2×C3×C4⋊C4D412yes—
128C2×C3×C4⋊C4C2311yescharacteristic
129C2×C3×C4⋊C4C2312yes—
130C2×C3×C22⋊C4—21no—
131C2×C3×C22⋊C4—24no—
132C2×C3×C42—21no—
133C2×C3×C42C2312yes—
134C2×C3×C42C2311yescharacteristic
135C3×#107—21no—
136C3×#106—21no—
137C3×C4⋊C8—21no—
138C3×C4⋊C8—21no—
139C3×#104—24no—
140C3×#104—42no—
141C3×#104—42no—
142C3×#103—21no—
143C3×#103—22no—
144C3×#102—21no—
145C3×#102—22no—
146C3×#102—21no—
147C3×#102—21no—
148C3×#97—21no—
149C3×C4×C8—21no—
150C3×#95—21no—
151C3×#95—24no—
152C2×AGL(1,ℤ/8)C22×C311yescharacteristic
153C2×AGL(1,ℤ/8)C22×C312yes—
154#556—21no—
155#555—21no—
156#554—21no—
157#553—22no—
158#548—22no—
159#547—21no—
160#542—22no—
161#537C22×C312yes—
162#533—21no—
163#532—21no—
164#532C22×C311yescharacteristic
165C2×C4⋊Q8C22×C311yescharacteristic
166#520C22×C312yes—
167C4×#41C22×C311yescharacteristic
168C2×C4×D4C22×C311yescharacteristic
169#463—22no—
170#461—21no—
171#459—21no—
172#442—21no—
173#440—21no—
174C4×SD16—21no—
175C4×D8—21no—
176#428C22×C311yescharacteristic
177#423C22×C311yescharacteristic
178C2×#104C22×C312yes—
179C2×#104C22×C312yes—
180C2×#104C22×C312yes—
181#418C22×C311yescharacteristic
182#417C22×C312yes—
183#417C22×C311yescharacteristic
184C4×M16C22×C311yescharacteristic
185#393—22no—
186#390C22×C312yes—
187#382C22×C312yes—
188#380—22no—
189C4×C4⋊C4C22×C311yescharacteristic
190#343C22×C312yes—
191#343C22×C312yes—
192#339C22×C312yes—
193C24×C3—21no—
194C3×#41—22no—
195C3×#41—22no—
196C3×#41—21no—
197C3×#41—21no—
198C3×#41—21no—
199C3×#41—22no—
200C3×#41—22no—
201C3×#41—21no—
202C3×#41—84no—
203C2×C3×Q8—22no—
204C2×C3×Q8—22no—
205C2×C3×Q8—82no—
206C2×C3×Q8C2×D412yes—
207C2×C3×Q8C2×D411yescharacteristic
208C2×C3×D4—21no—
209C2×C3×D4—22no—
210C2×C3×D4—24no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4C2×D412yes—
219C2×C3×D4C2×D411yescharacteristic
220C2×C3×D4#4112yescomplemented
221C22×C3×C4—24no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—22no—
226C22×C3×C4—21no—
227C22×C3×C4—22no—
228C22×C3×C4—44no—
229C22×C3×C4—41no—
230C22×C3×C4—41no—
231C22×C3×C4C2×D411yescharacteristic
232C22×C3×C4C2×D414yes—
233C22×C3×C4C2×D411yescharacteristic
234C22×C3×C4C2411yescharacteristic
235C3×SD16—24no—
236C3×SD16—44no—
237C3×D8—24no—
238C3×D8—44no—
239C3×M16—21no—
240C3×M16—22no—
241C3×M16—21no—
242C3×M16—22no—
243C3×M16—21no—
244C3×M16#4112yescomplemented
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—22no—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—22no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—22no—
257C3×C4⋊C4—22no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—42no—
261C3×C4⋊C4C2×D411yescharacteristic
262C3×C4⋊C4C2×D411yescharacteristic
263C3×C22⋊C4—24no—
264C3×C22⋊C4—24no—
265C3×C22⋊C4—24no—
266C3×C22⋊C4—21no—
267C3×C22⋊C4—21no—
268C3×C22⋊C4—21no—
269C3×C22⋊C4—22no—
270C3×C22⋊C4—22no—
271C3×C22⋊C4—41no—
272C3×C22⋊C4—41no—
273C3×C22⋊C4—84no—
274C3×C42—24no—
275C3×C42—21no—
276C3×C42—21no—
277C3×C42—21no—
278C3×C42—22no—
279C3×C42—22no—
280C3×C42C2×D411yescharacteristic
281C3×C42C2×D411yescharacteristic
282C3×C42C2×D411yescharacteristic
283C3×C42C2×D411yescharacteristic
284C2×#41—22no—
285C2×#41C23×C311yescharacteristic
286C2×#41C23×C312yes—
287C22×Q8C3×D412yes—
288C22×D4C23×C311yescharacteristic
289C23×C4—21no—
290AGL(1,ℤ/8)—22no—
291AGL(1,ℤ/8)—24no—
292AGL(1,ℤ/8)—22no—
293AGL(1,ℤ/8)—42no—
294C2×SD16—21no—
295C2×SD16—22no—
296C2×D8—21no—
297C2×D8—22no—
298C2×M16C23×C312yes—
299C2×M16C23×C311yescharacteristic
300C2×M16C23×C311yescharacteristic
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8C3×D411yescharacteristic
306#127C3×D411yescharacteristic
307#126—24no—
308#125—22no—
309#124—22no—
310#124—24no—
311#124C3×D412yes—
312#123—24no—
313C22⋊Q8—22no—
314C22⋊Q8—24no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317#121—22no—
318#121—22no—
319#121—22no—
320#121—21no—
321#121—21no—
322#121—24no—
323C4×Q8—22no—
324C4×Q8—22no—
325C4×Q8C3×D411yescharacteristic
326C4×Q8C3×D411yescharacteristic
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—21no—
330C4×D4—21no—
331C4×D4—24no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4C3×D411yescharacteristic
339C4×D4C3×D411yescharacteristic
340#117—22no—
341#117—22no—
342#117—21no—
343#117C23×C311yescharacteristic
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4C3×D412yes—
347C2×C4⋊C4C23×C311yescharacteristic
348C2×C4⋊C4C23×C312yes—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—24no—
351C2×C42—21no—
352C2×C42C23×C312yes—
353C2×C42C23×C311yescharacteristic
354#107—21no—
355#106—21no—
356C4⋊C8—21no—
357C4⋊C8—21no—
358#104—24no—
359#104—42no—
360#104—42no—
361#103—21no—
362#103—22no—
363#102—21no—
364#102—22no—
365#102—21no—
366#102—21no—
367#97—21no—
368C4×C8—21no—
369#95—21no—
370#95—24no—
371C23×C3—21no—
372C23×C3—22no—
373C23×C3—22no—
374C23×C3—42no—
375C23×C3—42no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3C22×D411yescharacteristic
379C3×Q8—21no—
380C3×Q8—21no—
381C3×Q8—22no—
382C3×Q8—22no—
383C3×Q8—44no—
384C3×Q8—82no—
385C3×D4—21no—
386C3×D4—22no—
387C3×D4—21no—
388C3×D4—22no—
389C3×D4—22no—
390C3×D4—22no—
391C3×D4—42no—
392C3×D4—42no—
393C3×D4—41no—
394C3×D4—42no—
395C3×D4—41no—
396C3×D4—41no—
397C3×D4—41no—
398C3×D4—42no—
399C3×D4—84no—
400C2×C3×C4—22no—
401C2×C3×C4—21no—
402C2×C3×C4—22no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—24no—
406C2×C3×C4—41no—
407C2×C3×C4—42no—
408C2×C3×C4—42no—
409C2×C3×C4—42no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—42no—
413C2×C3×C4—44no—
414C2×C3×C4—42no—
415C2×C3×C4—42no—
416C2×C3×C4—42no—
417C2×C3×C4—41no—
418C2×C3×C4—42no—
419C2×C3×C4—82no—
420C2×C3×C4—84no—
421C2×C3×C4—82no—
422C2×C3×C4#12011yescharacteristic
423C2×C3×C4#12011yescharacteristic
424C2×C3×C4#12011yescharacteristic
425C2×C3×C4#12011yescharacteristic
426C2×C3×C4C22×D411yescharacteristic
427C2×C3×C4C22×D411yescharacteristic
428C2×C3×C4C2×#4111yescharacteristic
429C2×C3×C4C2×#4111yescharacteristic
430C2×C3×C4H(5,𝔽2)11yescharacteristic
431C2×C3×C42−1+411yescharacteristic
432C3×C8—22no—
433C3×C8—41no—
434C3×C8—42no—
435C24—21no—
436#41—22no—
437#41—22no—
438#41—21no—
439#41—21no—
440#41—21no—
441#41—22no—
442#41—22no—
443#41—21no—
444#41—84no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—82no—
448C2×Q8C2×C3×D412yes—
449C2×Q8C2×C3×D411yescharacteristic
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—24no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4C2×C3×D412yes—
461C2×D4C2×C3×D411yescharacteristic
462C2×D4C3×#4112yescomplemented
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4C2×C3×D411yescharacteristic
474C22×C4C2×C3×D414yes—
475C22×C4C2×C3×D411yescharacteristic
476C22×C4C24×C311yescharacteristic
477SD16—24no—
478SD16—44no—
479D8—24no—
480D8—44no—
481M16—21no—
482M16—22no—
483M16—21no—
484M16—22no—
485M16—21no—
486M16C3×#4112yescomplemented
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—22no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—42no—
503C4⋊C4C2×C3×D411yescharacteristic
504C4⋊C4C2×C3×D411yescharacteristic
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—84no—
516C42—24no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—22no—
521C42—22no—
522C42C2×C3×D411yescharacteristic
523C42C2×C3×D411yescharacteristic
524C42C2×C3×D411yescharacteristic
525C42C2×C3×D411yescharacteristic
526C22×C3—21no—
527C22×C3—21no—
528C22×C3—44no—
529C22×C3—42no—
530C22×C3—41no—
531C22×C3—42no—
532C22×C3—41no—
533C22×C3—42no—
534C22×C3—41no—
535C22×C3—42no—
536C22×C3—82no—
537C22×C3—81no—
538C22×C3—81no—
539C22×C3C2×#12011yescharacteristic
540C22×C3#54811yescharacteristic
541C22×C3#55011yescharacteristic
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—21no—
545C3×C4—42no—
546C3×C4—41no—
547C3×C4—41no—
548C3×C4—41no—
549C3×C4—44no—
550C3×C4—82no—
551C3×C4—81no—
552C3×C4#45412yes—
553C3×C4#52511yescharacteristic
554C3×C4#53711yescharacteristic
555C3×C4#54611yescharacteristic
556C3×C4#54711yescharacteristic
557C23—21no—
558C23—22no—
559C23—22no—
560C23—42no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23C22×C3×D411yescharacteristic
565Q8—21no—
566Q8—21no—
567Q8—22no—
568Q8—22no—
569Q8—44no—
570Q8—82no—
571D4—21no—
572D4—22no—
573D4—21no—
574D4—22no—
575D4—22no—
576D4—22no—
577D4—42no—
578D4—42no—
579D4—41no—
580D4—42no—
581D4—41no—
582D4—42no—
583D4—41no—
584D4—41no—
585D4—84no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—82no—
606C2×C4—84no—
607C2×C4—82no—
608C2×C4C3×#12011yescharacteristic
609C2×C4C3×#12011yescharacteristic
610C2×C4C3×#12011yescharacteristic
611C2×C4C3×#12011yescharacteristic
612C2×C4C22×C3×D411yescharacteristic
613C2×C4C22×C3×D411yescharacteristic
614C2×C4C2×C3×#4111yescharacteristic
615C2×C4C2×C3×#4111yescharacteristic
616C2×C4C3×H(5,𝔽2)11yescharacteristic
617C2×C4C3×2−1+411yescharacteristic
618C8—22no—
619C8—41no—
620C8—42no—
621C2×C3—21no—
622C2×C3—21no—
623C2×C3—81no—
624C2×C3—82no—
625C2×C3—82no—
626C2×C3#240711yescharacteristic
627C2×C3C2×#45411yescharacteristic
628C2×C3#301911yescharacteristic
629C22—21no—
630C22—21no—
631C22—44no—
632C22—42no—
633C22—41no—
634C22—42no—
635C22—41no—
636C22—42no—
637C22—41no—
638C22—42no—
639C22—82no—
640C22—81no—
641C22—81no—
642C22C2×C3×#12011yescharacteristic
643C22C3×#54811yescharacteristic
644C22C3×#55011yescharacteristic
645C4—21no—
646C4—21no—
647C4—21no—
648C4—42no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—44no—
653C4—82no—
654C4—81no—
655C4C3×#45412yes—
656C4C3×#52511yescharacteristic
657C4C3×#53711yescharacteristic
658C4C3×#54611yescharacteristic
659C4C3×#54711yescharacteristic
660C3#2417111yescharacteristic, direct product factor, Sylow
661C2—21no—
662C2—21no—
663C2—82no—
664C2—81no—
665C2—82no—
666C2C3×#240711yescharacteristic
667C2C2×C3×#45411yescharacteristic
668C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3511C2semidirect
2. C3×#3036C2semidirect
3. C3×C4×AGL(1,ℤ/8)C2semidirect
4. C3×#2563C2semidirect
5. C3×#2040C2semidirect
6. C3×#2006C2semidirect
7. C3×#1997C2semidirect
8. C3×#1980C2semidirect
9. C3×#1883C2semidirect
10. C3×#1860C2semidirect
11. C3×#1806C2semidirect
12. #24171C3direct (N × H)
54. #3511C2×C3semidirect
55. #3036C2×C3semidirect
56. C4×AGL(1,ℤ/8)C2×C3semidirect
57. #2563C2×C3semidirect
58. #2040C2×C3semidirect
59. #2006C2×C3semidirect
60. #1997C2×C3semidirect
61. #1980C2×C3semidirect
62. #1883C2×C3semidirect
63. #1860C2×C3semidirect
64. #1806C2×C3semidirect
220. C2×C3×D4#41semidirect
244. C3×M16#41semidirect
462. C2×D4C3×#41semidirect
486. M16C3×#41semidirect
660. C3#24171direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35112
C3×#30362
C3×C4×AGL(1,ℤ/8)2
C3×#25632
C3×#20402
C3×#20062
C3×#19972
C3×#19802
C3×#18832
C3×#18602
C3×#18062
#241713