Finite Group Databasebeta

← C3×#24170 #10642777

Subgroups — C3×#24170 #10642777

Order 768.

1162 conjugacy classes of subgroups.

The subgroup lattice has 674 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×#2542C211yesmaximal, characteristic, complemented
5C3×#2039C212yesmaximal, complemented
6C3×#2006C212yesmaximal, complemented
7C3×#1997C211yesmaximal, characteristic, complemented
8C3×#1980C211yesmaximal, characteristic, complemented
9C3×#1883C212yesmaximal, complemented
10C3×#1859C212yesmaximal, complemented
11C3×#1806C211yesmaximal, characteristic, complemented
12#24170C311yesmaximal, 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
27C2×C3×C22⋊Q8—22no—
28C3×#520C2212yes—
29C3×C4×#41C2211yescharacteristic
30C2×C3×C4×D4C2211yescharacteristic
31C3×#463—22no—
32C3×#461—21no—
33C3×#459—21no—
34C3×#442—21no—
35C3×#440—21no—
36C3×C4×SD16—21no—
37C3×C4×D8—21no—
38C3×#428C2211yescharacteristic
39C3×#423C2211yescharacteristic
40C2×C3×#104C2212yes—
41C2×C3×#104C2212yes—
42C2×C3×#104C2212yes—
43C3×#418C2211yescharacteristic
44C3×#417C2211yescharacteristic
45C3×#417C2212yes—
46C3×C4×M16C2211yescharacteristic
47C3×#399—22no—
48C3×#390C2212yes—
49C3×#384C2212yes—
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#2542C2×C311yescharacteristic, complemented
58#2039C2×C312yescomplemented
59#2006C2×C312yescomplemented
60#1997C2×C311yescharacteristic, complemented
61#1980C2×C311yescharacteristic, complemented
62#1883C2×C312yescomplemented
63#1859C2×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×C22⋊Q8—48no—
99C3×#121—22no—
100C3×#121—22no—
101C3×#121—22no—
102C3×#121—21no—
103C3×#121—21no—
104C3×#121—24no—
105C3×C4×Q8—22no—
106C3×C4×Q8—22no—
107C3×C4×Q8D411yescharacteristic
108C3×C4×Q8D411yescharacteristic
109C3×C4×D4—22no—
110C3×C4×D4—22no—
111C3×C4×D4—21no—
112C3×C4×D4—21no—
113C3×C4×D4—24no—
114C3×C4×D4—21no—
115C3×C4×D4—21no—
116C3×C4×D4—21no—
117C3×C4×D4—21no—
118C3×C4×D4—21no—
119C3×C4×D4—21no—
120C3×C4×D4D411yescharacteristic
121C3×C4×D4D411yescharacteristic
122C3×#117—22no—
123C3×#117—22no—
124C3×#117—21no—
125C3×#117C2311yescharacteristic
126C2×C3×C4⋊C4—21no—
127C2×C3×C4⋊C4—22no—
128C2×C3×C4⋊C4—24no—
129C2×C3×C4⋊C4D412yes—
130C2×C3×C4⋊C4C2311yescharacteristic
131C2×C3×C4⋊C4C2312yes—
132C2×C3×C22⋊C4—24no—
133C2×C3×C22⋊C4—21no—
134C2×C3×C42—21no—
135C2×C3×C42C2312yes—
136C2×C3×C42C2311yescharacteristic
137C3×#107—21no—
138C3×#106—21no—
139C3×C4⋊C8—21no—
140C3×C4⋊C8—21no—
141C3×#104—24no—
142C3×#104—42no—
143C3×#104—42no—
144C3×#103—21no—
145C3×#103—22no—
146C3×#102—21no—
147C3×#102—22no—
148C3×#102—21no—
149C3×#102—21no—
150C3×#97—21no—
151C3×C4×C8—21no—
152C3×#95—21no—
153C2×AGL(1,ℤ/8)C22×C311yescharacteristic
154C2×AGL(1,ℤ/8)C22×C312yes—
155#556—21no—
156#555—21no—
157#554—21no—
158#553—22no—
159#548—22no—
160#547—21no—
161#542—22no—
162#537C22×C312yes—
163#533—21no—
164#532—21no—
165#532C22×C311yescharacteristic
166C2×C4⋊Q8C22×C311yescharacteristic
167C2×C22⋊Q8—22no—
168#520C22×C312yes—
169C4×#41C22×C311yescharacteristic
170C2×C4×D4C22×C311yescharacteristic
171#463—22no—
172#461—21no—
173#459—21no—
174#442—21no—
175#440—21no—
176C4×SD16—21no—
177C4×D8—21no—
178#428C22×C311yescharacteristic
179#423C22×C311yescharacteristic
180C2×#104C22×C312yes—
181C2×#104C22×C312yes—
182C2×#104C22×C312yes—
183#418C22×C311yescharacteristic
184#417C22×C312yes—
185#417C22×C311yescharacteristic
186C4×M16C22×C311yescharacteristic
187#399—22no—
188#390C22×C312yes—
189#384C22×C312yes—
190C4×C4⋊C4C22×C311yescharacteristic
191#343C22×C312yes—
192#343C22×C312yes—
193#339C22×C312yes—
194C24×C3—21no—
195C3×#41—22no—
196C3×#41—22no—
197C3×#41—21no—
198C3×#41—21no—
199C3×#41—21no—
200C3×#41—22no—
201C3×#41—22no—
202C3×#41—21no—
203C3×#41—84no—
204C2×C3×Q8—22no—
205C2×C3×Q8—22no—
206C2×C3×Q8—44no—
207C2×C3×Q8C2×D412yes—
208C2×C3×Q8C2×D411yescharacteristic
209C2×C3×D4—21no—
210C2×C3×D4—22no—
211C2×C3×D4—24no—
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×D4—21no—
219C2×C3×D4C2×D412yes—
220C2×C3×D4C2×D411yescharacteristic
221C2×C3×D4#4112yescomplemented
222C22×C3×C4—22no—
223C22×C3×C4—24no—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4—22no—
228C22×C3×C4—21no—
229C22×C3×C4—44no—
230C22×C3×C4—41no—
231C22×C3×C4—41no—
232C22×C3×C4C2×D411yescharacteristic
233C22×C3×C4C2×D414yes—
234C22×C3×C4C2×D411yescharacteristic
235C22×C3×C4C2411yescharacteristic
236C3×SD16—24no—
237C3×SD16—44no—
238C3×D8—24no—
239C3×D8—44no—
240C3×M16—21no—
241C3×M16—22no—
242C3×M16—21no—
243C3×M16—22no—
244C3×M16—21no—
245C3×M16#4112yescomplemented
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—22no—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—22no—
253C3×C4⋊C4—22no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—22no—
259C3×C4⋊C4—22no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—48no—
263C3×C4⋊C4C2×D411yescharacteristic
264C3×C4⋊C4C2×D411yescharacteristic
265C3×C22⋊C4—24no—
266C3×C22⋊C4—24no—
267C3×C22⋊C4—24no—
268C3×C22⋊C4—21no—
269C3×C22⋊C4—21no—
270C3×C22⋊C4—21no—
271C3×C22⋊C4—22no—
272C3×C22⋊C4—22no—
273C3×C22⋊C4—48no—
274C3×C22⋊C4—41no—
275C3×C22⋊C4—41no—
276C3×C42—24no—
277C3×C42—21no—
278C3×C42—21no—
279C3×C42—21no—
280C3×C42—22no—
281C3×C42—22no—
282C3×C42C2×D411yescharacteristic
283C3×C42C2×D411yescharacteristic
284C3×C42C2×D411yescharacteristic
285C3×C42C2×D411yescharacteristic
286C2×#41—22no—
287C2×#41C23×C311yescharacteristic
288C2×#41C23×C312yes—
289C22×Q8C3×D412yes—
290C22×D4C23×C311yescharacteristic
291C23×C4—21no—
292AGL(1,ℤ/8)—22no—
293AGL(1,ℤ/8)—24no—
294AGL(1,ℤ/8)—22no—
295AGL(1,ℤ/8)—42no—
296C2×SD16—21no—
297C2×SD16—22no—
298C2×D8—21no—
299C2×D8—22no—
300C2×M16C23×C312yes—
301C2×M16C23×C311yescharacteristic
302C2×M16C23×C311yescharacteristic
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8—21no—
307C4⋊Q8C3×D411yescharacteristic
308#127C3×D411yescharacteristic
309#126—24no—
310#125—22no—
311#124—22no—
312#124—24no—
313#124C3×D412yes—
314#123—24no—
315C22⋊Q8—22no—
316C22⋊Q8—24no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319C22⋊Q8—48no—
320#121—22no—
321#121—22no—
322#121—22no—
323#121—21no—
324#121—21no—
325#121—24no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×Q8C3×D411yescharacteristic
329C4×Q8C3×D411yescharacteristic
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—24no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4—21no—
340C4×D4—21no—
341C4×D4C3×D411yescharacteristic
342C4×D4C3×D411yescharacteristic
343#117—22no—
344#117—22no—
345#117—21no—
346#117C23×C311yescharacteristic
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4C3×D412yes—
351C2×C4⋊C4C23×C311yescharacteristic
352C2×C4⋊C4C23×C312yes—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—24no—
355C2×C42—21no—
356C2×C42C23×C312yes—
357C2×C42C23×C311yescharacteristic
358#107—21no—
359#106—21no—
360C4⋊C8—21no—
361C4⋊C8—21no—
362#104—24no—
363#104—42no—
364#104—42no—
365#103—21no—
366#103—22no—
367#102—21no—
368#102—22no—
369#102—21no—
370#102—21no—
371#97—21no—
372C4×C8—21no—
373#95—21no—
374C23×C3—21no—
375C23×C3—22no—
376C23×C3—22no—
377C23×C3—42no—
378C23×C3—42no—
379C23×C3—41no—
380C23×C3—41no—
381C23×C3C22×D411yescharacteristic
382C3×Q8—21no—
383C3×Q8—21no—
384C3×Q8—22no—
385C3×Q8—22no—
386C3×Q8—44no—
387C3×Q8—82no—
388C3×D4—21no—
389C3×D4—22no—
390C3×D4—21no—
391C3×D4—22no—
392C3×D4—22no—
393C3×D4—22no—
394C3×D4—42no—
395C3×D4—42no—
396C3×D4—41no—
397C3×D4—42no—
398C3×D4—41no—
399C3×D4—41no—
400C3×D4—41no—
401C3×D4—42no—
402C3×D4—84no—
403C2×C3×C4—22no—
404C2×C3×C4—24no—
405C2×C3×C4—21no—
406C2×C3×C4—22no—
407C2×C3×C4—21no—
408C2×C3×C4—21no—
409C2×C3×C4—24no—
410C2×C3×C4—41no—
411C2×C3×C4—42no—
412C2×C3×C4—42no—
413C2×C3×C4—42no—
414C2×C3×C4—42no—
415C2×C3×C4—42no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—42no—
419C2×C3×C4—44no—
420C2×C3×C4—42no—
421C2×C3×C4—41no—
422C2×C3×C4—82no—
423C2×C3×C4—84no—
424C2×C3×C4—82no—
425C2×C3×C4#12011yescharacteristic
426C2×C3×C4#12011yescharacteristic
427C2×C3×C4#12011yescharacteristic
428C2×C3×C4#12011yescharacteristic
429C2×C3×C4C22×D411yescharacteristic
430C2×C3×C4C22×D411yescharacteristic
431C2×C3×C4C2×#4111yescharacteristic
432C2×C3×C4C2×#4111yescharacteristic
433C2×C3×C4H(5,𝔽2)11yescharacteristic
434C2×C3×C42−1+411yescharacteristic
435C3×C8—22no—
436C3×C8—41no—
437C3×C8—42no—
438C24—21no—
439#41—22no—
440#41—22no—
441#41—21no—
442#41—21no—
443#41—21no—
444#41—22no—
445#41—22no—
446#41—21no—
447#41—84no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—44no—
451C2×Q8C2×C3×D412yes—
452C2×Q8C2×C3×D411yescharacteristic
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—24no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4C2×C3×D412yes—
464C2×D4C2×C3×D411yescharacteristic
465C2×D4C3×#4112yescomplemented
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—44no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4C2×C3×D411yescharacteristic
477C22×C4C2×C3×D414yes—
478C22×C4C2×C3×D411yescharacteristic
479C22×C4C24×C311yescharacteristic
480SD16—24no—
481SD16—44no—
482D8—24no—
483D8—44no—
484M16—21no—
485M16—22no—
486M16—21no—
487M16—22no—
488M16—21no—
489M16C3×#4112yescomplemented
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—48no—
507C4⋊C4C2×C3×D411yescharacteristic
508C4⋊C4C2×C3×D411yescharacteristic
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—48no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C42—24no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—22no—
525C42—22no—
526C42C2×C3×D411yescharacteristic
527C42C2×C3×D411yescharacteristic
528C42C2×C3×D411yescharacteristic
529C42C2×C3×D411yescharacteristic
530C22×C3—21no—
531C22×C3—21no—
532C22×C3—44no—
533C22×C3—42no—
534C22×C3—41no—
535C22×C3—42no—
536C22×C3—41no—
537C22×C3—42no—
538C22×C3—41no—
539C22×C3—42no—
540C22×C3—82no—
541C22×C3—81no—
542C22×C3—81no—
543C22×C3C2×#12011yescharacteristic
544C22×C3#54811yescharacteristic
545C22×C3#55011yescharacteristic
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—21no—
549C3×C4—44no—
550C3×C4—42no—
551C3×C4—41no—
552C3×C4—41no—
553C3×C4—44no—
554C3×C4—41no—
555C3×C4—81no—
556C3×C4#45312yes—
557C3×C4#52511yescharacteristic
558C3×C4#53711yescharacteristic
559C3×C4#54611yescharacteristic
560C3×C4#54711yescharacteristic
561C23—21no—
562C23—22no—
563C23—22no—
564C23—42no—
565C23—42no—
566C23—41no—
567C23—41no—
568C23C22×C3×D411yescharacteristic
569Q8—21no—
570Q8—21no—
571Q8—22no—
572Q8—22no—
573Q8—44no—
574Q8—82no—
575D4—21no—
576D4—22no—
577D4—21no—
578D4—22no—
579D4—22no—
580D4—22no—
581D4—42no—
582D4—42no—
583D4—41no—
584D4—42no—
585D4—41no—
586D4—42no—
587D4—41no—
588D4—41no—
589D4—84no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—24no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—82no—
610C2×C4—84no—
611C2×C4—82no—
612C2×C4C3×#12011yescharacteristic
613C2×C4C3×#12011yescharacteristic
614C2×C4C3×#12011yescharacteristic
615C2×C4C3×#12011yescharacteristic
616C2×C4C22×C3×D411yescharacteristic
617C2×C4C22×C3×D411yescharacteristic
618C2×C4C2×C3×#4111yescharacteristic
619C2×C4C2×C3×#4111yescharacteristic
620C2×C4C3×H(5,𝔽2)11yescharacteristic
621C2×C4C3×2−1+411yescharacteristic
622C8—22no—
623C8—41no—
624C8—42no—
625C2×C3—21no—
626C2×C3—21no—
627C2×C3—81no—
628C2×C3—82no—
629C2×C3—82no—
630C2×C3#240711yescharacteristic
631C2×C3C2×#45311yescharacteristic
632C2×C3#302011yescharacteristic
633C22—21no—
634C22—21no—
635C22—44no—
636C22—42no—
637C22—41no—
638C22—42no—
639C22—41no—
640C22—42no—
641C22—41no—
642C22—42no—
643C22—82no—
644C22—81no—
645C22—81no—
646C22C2×C3×#12011yescharacteristic
647C22C3×#54811yescharacteristic
648C22C3×#55011yescharacteristic
649C4—21no—
650C4—21no—
651C4—21no—
652C4—44no—
653C4—42no—
654C4—41no—
655C4—41no—
656C4—44no—
657C4—41no—
658C4—81no—
659C4C3×#45312yes—
660C4C3×#52511yescharacteristic
661C4C3×#53711yescharacteristic
662C4C3×#54611yescharacteristic
663C4C3×#54711yescharacteristic
664C3#2417011yescharacteristic, direct product factor, Sylow
665C2—21no—
666C2—21no—
667C2—82no—
668C2—81no—
669C2—82no—
670C2C3×#240711yescharacteristic
671C2C2×C3×#45311yescharacteristic
672C2C3×#302011yescharacteristic
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×#2542C2semidirect
5. C3×#2039C2semidirect
6. C3×#2006C2semidirect
7. C3×#1997C2semidirect
8. C3×#1980C2semidirect
9. C3×#1883C2semidirect
10. C3×#1859C2semidirect
11. C3×#1806C2semidirect
12. #24170C3direct (N × H)
54. #3511C2×C3semidirect
55. #3036C2×C3semidirect
56. C4×AGL(1,ℤ/8)C2×C3semidirect
57. #2542C2×C3semidirect
58. #2039C2×C3semidirect
59. #2006C2×C3semidirect
60. #1997C2×C3semidirect
61. #1980C2×C3semidirect
62. #1883C2×C3semidirect
63. #1859C2×C3semidirect
64. #1806C2×C3semidirect
221. C2×C3×D4#41semidirect
245. C3×M16#41semidirect
465. C2×D4C3×#41semidirect
489. M16C3×#41semidirect
664. C3#24170direct (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×#25422
C3×#20392
C3×#20062
C3×#19972
C3×#19802
C3×#18832
C3×#18592
C3×#18062
#241703