Finite Group Databasebeta

← C3×#24145 #10642752

Subgroups — C3×#24145 #10642752

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3509C211yesmaximal, characteristic, complemented
2C3×#3077C211yesmaximal, characteristic, complemented
3C3×C4×AGL(1,ℤ/8)C211yesmaximal, characteristic
4C3×#2547C211yesmaximal, characteristic, complemented
5C3×#2038C212yesmaximal, complemented
6C3×#2003C212yesmaximal
7C3×#1986C211yesmaximal, characteristic, complemented
8C3×#1966C211yesmaximal, characteristic, complemented
9C3×#1888C212yesmaximal, complemented
10C3×#1857C212yesmaximal, complemented
11C3×#1806C211yesmaximal, characteristic, complemented
12#24145C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
14C3×#557—21no—
15C3×#557—21no—
16C3×#556—21no—
17C3×#556—21no—
18C3×#552—22no—
19C3×#552—22no—
20C3×#541—22no—
21C3×#536C2212yes—
22C3×#533—21no—
23C3×#533—21no—
24C3×#533C2211yescharacteristic
25C2×C3×C4⋊Q8C2211yescharacteristic
26C2×C3×#124C2212yes—
27C3×#519C2212yes—
28C3×C4×#41C2211yescharacteristic
29C2×C3×C4×D4C2211yescharacteristic
30C3×#478—22no—
31C3×#476—21no—
32C3×#474—21no—
33C3×#442—21no—
34C3×#440—21no—
35C3×C4×SD16—21no—
36C3×C4×D8—21no—
37C3×#428C2212yes—
38C3×#428C2211yescharacteristic
39C3×#423C2211yescharacteristic
40C2×C3×#104C2212yes—
41C3×#418C2212yes—
42C3×#418C2211yescharacteristic
43C3×#417C2211yescharacteristic
44C2×C3×#101C2212yes—
45C2×C3×#101C2212yes—
46C3×C4×M16C2211yescharacteristic
47C3×#397—22no—
48C3×#391C2212yes—
49C3×#380—22no—
50C3×C4×C4⋊C4C2211yescharacteristic
51C3×#339C2212yes—
52C3×#339C2212yes—
53C3×#339C2212yes—
54#3509C2×C311yescharacteristic, complemented
55#3077C2×C311yescharacteristic, complemented
56C4×AGL(1,ℤ/8)C2×C311yescharacteristic
57#2547C2×C311yescharacteristic, complemented
58#2038C2×C312yescomplemented
59#2003C2×C312yes—
60#1986C2×C311yescharacteristic, complemented
61#1966C2×C311yescharacteristic, complemented
62#1888C2×C312yescomplemented
63#1857C2×C312yescomplemented
64#1806C2×C311yescharacteristic, complemented
65C2×C3×#41C2311yescharacteristic
66C22×C3×Q8C2312yes—
67C22×C3×D4C2311yescharacteristic
68C23×C3×C4—21no—
69C3×AGL(1,ℤ/8)—22no—
70C3×AGL(1,ℤ/8)—22no—
71C2×C3×SD16—21no—
72C2×C3×D8—21no—
73C2×C3×M16C2311yescharacteristic
74C2×C3×M16C2312yes—
75C2×C3×M16C2311yescharacteristic
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8—21no—
78C3×C4⋊Q8—21no—
79C3×C4⋊Q8—21no—
80C3×C4⋊Q8—21no—
81C3×C4⋊Q8D411yescharacteristic
82C3×C4⋊Q8D411yescharacteristic
83C3×#126—24no—
84C3×#126—24no—
85C3×#125—22no—
86C3×#125—22no—
87C3×#125—24no—
88C3×#125—24no—
89C3×#125D412yes—
90C3×#124—22no—
91C3×#124—24no—
92C3×#124—42no—
93C3×#123—24no—
94C3×C22⋊Q8—22no—
95C3×C22⋊Q8—22no—
96C3×C22⋊Q8—21no—
97C3×C22⋊Q8—21no—
98C3×C22⋊Q8—22no—
99C3×C22⋊Q8—22no—
100C3×C4×Q8—22no—
101C3×C4×Q8—22no—
102C3×C4×Q8—22no—
103C3×C4×Q8—22no—
104C3×C4×Q8—22no—
105C3×C4×Q8—22no—
106C3×C4×Q8—21no—
107C3×C4×Q8—21no—
108C3×C4×Q8D411yescharacteristic
109C3×C4×Q8D411yescharacteristic
110C3×C4×D4—21no—
111C3×C4×D4—21no—
112C3×C4×D4—21no—
113C3×C4×D4—21no—
114C3×C4×D4—21no—
115C3×C4×D4—21no—
116C3×C4×D4D411yescharacteristic
117C3×C4×D4D411yescharacteristic
118C3×#117—22no—
119C3×#117—22no—
120C3×#117—24no—
121C3×#117—21no—
122C3×#117C2311yescharacteristic
123C3×#117C2312yes—
124C2×C3×C4⋊C4—21no—
125C2×C3×C4⋊C4D412yes—
126C2×C3×C4⋊C4D412yes—
127C2×C3×C4⋊C4C2311yescharacteristic
128C2×C3×C22⋊C4—24no—
129C2×C3×C22⋊C4—21no—
130C2×C3×C42—21no—
131C2×C3×C42C2312yes—
132C2×C3×C42C2311yescharacteristic
133C3×#107—21no—
134C3×#107—22no—
135C3×#106—21no—
136C3×#106—22no—
137C3×C4⋊C8—21no—
138C3×C4⋊C8—21no—
139C3×#104—42no—
140C3×#103—21no—
141C3×#102—21no—
142C3×#102—21no—
143C3×#102—21no—
144C3×#102—22no—
145C3×#102—22no—
146C3×#101—24no—
147C3×#101—42no—
148C3×#97—21no—
149C3×C4×C8—21no—
150C3×#95—21no—
151C3×#95—24no—
152C2×AGL(1,ℤ/8)C22×C311yescharacteristic
153#557—21no—
154#557—21no—
155#556—21no—
156#556—21no—
157#552—22no—
158#552—22no—
159#541—22no—
160#536C22×C312yes—
161#533—21no—
162#533—21no—
163#533C22×C311yescharacteristic
164C2×C4⋊Q8C22×C311yescharacteristic
165C2×#124C22×C312yes—
166#519C22×C312yes—
167C4×#41C22×C311yescharacteristic
168C2×C4×D4C22×C311yescharacteristic
169#478—22no—
170#476—21no—
171#474—21no—
172#442—21no—
173#440—21no—
174C4×SD16—21no—
175C4×D8—21no—
176#428C22×C312yes—
177#428C22×C311yescharacteristic
178#423C22×C311yescharacteristic
179C2×#104C22×C312yes—
180#418C22×C312yes—
181#418C22×C311yescharacteristic
182#417C22×C311yescharacteristic
183C2×#101C22×C312yes—
184C2×#101C22×C312yes—
185C4×M16C22×C311yescharacteristic
186#397—22no—
187#391C22×C312yes—
188#380—22no—
189C4×C4⋊C4C22×C311yescharacteristic
190#339C22×C312yes—
191#339C22×C312yes—
192#339C22×C312yes—
193C24×C3—21no—
194C3×#41—21no—
195C3×#41—21no—
196C3×#41—21no—
197C3×#41—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—22no—
201C2×C3×Q8—22no—
202C2×C3×Q8—22no—
203C2×C3×Q8—22no—
204C2×C3×Q8—22no—
205C2×C3×Q8—22no—
206C2×C3×Q8C2×D411yescharacteristic
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4C2×D411yescharacteristic
214C2×C3×D4#4112yes—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—22no—
220C22×C3×C4—44no—
221C22×C3×C4—41no—
222C22×C3×C4—41no—
223C22×C3×C4C2×D414yes—
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2411yescharacteristic
227C3×SD16—24no—
228C3×D8—24no—
229C3×M16—21no—
230C3×M16—22no—
231C3×M16—21no—
232C3×M16—22no—
233C3×M16—21no—
234C3×M16#4112yes—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—22no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—24no—
240C3×C4⋊C4—24no—
241C3×C4⋊C4—24no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—22no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—42no—
255C3×C4⋊C4C2×D411yescharacteristic
256C3×C4⋊C4C2×D412yes—
257C3×C4⋊C4C2×D411yescharacteristic
258C3×C4⋊C4C2×D412yes—
259C3×C22⋊C4—24no—
260C3×C22⋊C4—24no—
261C3×C22⋊C4—21no—
262C3×C22⋊C4—22no—
263C3×C22⋊C4—21no—
264C3×C22⋊C4—21no—
265C3×C22⋊C4—48no—
266C3×C22⋊C4—41no—
267C3×C22⋊C4—41no—
268C3×C42—24no—
269C3×C42—24no—
270C3×C42—21no—
271C3×C42—21no—
272C3×C42—21no—
273C3×C42—22no—
274C3×C42—22no—
275C3×C42—22no—
276C3×C42C2×D411yescharacteristic
277C3×C42C2×D411yescharacteristic
278C3×C42C2×D411yescharacteristic
279C3×C42C2×D411yescharacteristic
280C2×#41C23×C311yescharacteristic
281C22×Q8C23×C312yes—
282C22×D4C23×C311yescharacteristic
283C23×C4—21no—
284AGL(1,ℤ/8)—22no—
285AGL(1,ℤ/8)—22no—
286C2×SD16—21no—
287C2×D8—21no—
288C2×M16C23×C311yescharacteristic
289C2×M16C23×C312yes—
290C2×M16C23×C311yescharacteristic
291C4⋊Q8—22no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8C3×D411yescharacteristic
297C4⋊Q8C3×D411yescharacteristic
298#126—24no—
299#126—24no—
300#125—22no—
301#125—22no—
302#125—24no—
303#125—24no—
304#125C3×D412yes—
305#124—22no—
306#124—24no—
307#124—42no—
308#123—24no—
309C22⋊Q8—22no—
310C22⋊Q8—22no—
311C22⋊Q8—21no—
312C22⋊Q8—21no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8C3×D411yescharacteristic
324C4×Q8C3×D411yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C4×D4C3×D411yescharacteristic
332C4×D4C3×D411yescharacteristic
333#117—22no—
334#117—22no—
335#117—24no—
336#117—21no—
337#117C23×C311yescharacteristic
338#117C23×C312yes—
339C2×C4⋊C4—21no—
340C2×C4⋊C4C3×D412yes—
341C2×C4⋊C4C3×D412yes—
342C2×C4⋊C4C23×C311yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—24no—
345C2×C42—21no—
346C2×C42C23×C312yes—
347C2×C42C23×C311yescharacteristic
348#107—21no—
349#107—22no—
350#106—21no—
351#106—22no—
352C4⋊C8—21no—
353C4⋊C8—21no—
354#104—42no—
355#103—21no—
356#102—21no—
357#102—21no—
358#102—21no—
359#102—22no—
360#102—22no—
361#101—24no—
362#101—42no—
363#97—21no—
364C4×C8—21no—
365#95—21no—
366#95—24no—
367C23×C3—21no—
368C23×C3—22no—
369C23×C3—42no—
370C23×C3—42no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3C22×D411yescharacteristic
374C3×Q8—21no—
375C3×Q8—21no—
376C3×Q8—42no—
377C3×Q8—42no—
378C3×Q8—42no—
379C3×Q8—42no—
380C3×Q8—42no—
381C3×Q8—42no—
382C3×D4—21no—
383C3×D4—22no—
384C3×D4—21no—
385C3×D4—22no—
386C3×D4—41no—
387C3×D4—41no—
388C3×D4—41no—
389C3×D4—41no—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—24no—
396C2×C3×C4—21no—
397C2×C3×C4—24no—
398C2×C3×C4—24no—
399C2×C3×C4—41no—
400C2×C3×C4—42no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—42no—
404C2×C3×C4—42no—
405C2×C3×C4—42no—
406C2×C3×C4—42no—
407C2×C3×C4—41no—
408C2×C3×C4—42no—
409C2×C3×C4—82no—
410C2×C3×C4—82no—
411C2×C3×C4#12011yescharacteristic
412C2×C3×C4#12011yescharacteristic
413C2×C3×C4#12011yescharacteristic
414C2×C3×C4#12011yescharacteristic
415C2×C3×C4C22×D411yescharacteristic
416C2×C3×C4C22×D411yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418C2×C3×C4C2×#4111yescharacteristic
419C2×C3×C4H(5,𝔽2)11yescharacteristic
420C2×C3×C42−1+411yescharacteristic
421C3×C8—22no—
422C3×C8—41no—
423C3×C8—42no—
424C24—21no—
425#41—21no—
426#41—21no—
427#41—21no—
428#41—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8C2×C3×D411yescharacteristic
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4C2×C3×D411yescharacteristic
445C2×D4C3×#4112yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—44no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4C2×C3×D414yes—
455C22×C4C2×C3×D411yescharacteristic
456C22×C4C2×C3×D411yescharacteristic
457C22×C4C24×C311yescharacteristic
458SD16—24no—
459D8—24no—
460M16—21no—
461M16—22no—
462M16—21no—
463M16—22no—
464M16—21no—
465M16C3×#4112yes—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—22no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4C2×C3×D411yescharacteristic
487C4⋊C4C2×C3×D412yes—
488C4⋊C4C2×C3×D411yescharacteristic
489C4⋊C4C2×C3×D412yes—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—21no—
493C22⋊C4—22no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—48no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C42—24no—
500C42—24no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42C2×C3×D411yescharacteristic
508C42C2×C3×D411yescharacteristic
509C42C2×C3×D411yescharacteristic
510C42C2×C3×D411yescharacteristic
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—44no—
514C22×C3—41no—
515C22×C3—42no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—82no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3C2×#12011yescharacteristic
522C22×C3#54611yescharacteristic
523C22×C3#54711yescharacteristic
524C3×C4—21no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—44no—
528C3×C4—41no—
529C3×C4—41no—
530C3×C4—42no—
531C3×C4—41no—
532C3×C4—42no—
533C3×C4—82no—
534C3×C4—81no—
535C3×C4#45412yes—
536C3×C4#52511yescharacteristic
537C3×C4#53711yescharacteristic
538C3×C4#54811yescharacteristic
539C3×C4#55011yescharacteristic
540C23—21no—
541C23—22no—
542C23—42no—
543C23—42no—
544C23—41no—
545C23—41no—
546C23C22×C3×D411yescharacteristic
547Q8—21no—
548Q8—21no—
549Q8—42no—
550Q8—42no—
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555D4—21no—
556D4—22no—
557D4—21no—
558D4—22no—
559D4—41no—
560D4—41no—
561D4—41no—
562D4—41no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—21no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4C3×#12011yescharacteristic
585C2×C4C3×#12011yescharacteristic
586C2×C4C3×#12011yescharacteristic
587C2×C4C3×#12011yescharacteristic
588C2×C4C22×C3×D411yescharacteristic
589C2×C4C22×C3×D411yescharacteristic
590C2×C4C2×C3×#4111yescharacteristic
591C2×C4C2×C3×#4111yescharacteristic
592C2×C4C3×H(5,𝔽2)11yescharacteristic
593C2×C4C3×2−1+411yescharacteristic
594C8—22no—
595C8—41no—
596C8—42no—
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—81no—
600C2×C3—82no—
601C2×C3#240711yescharacteristic
602C2×C3C2×#45411yescharacteristic
603C2×C3#302011yescharacteristic
604C22—21no—
605C22—21no—
606C22—44no—
607C22—41no—
608C22—42no—
609C22—41no—
610C22—41no—
611C22—82no—
612C22—81no—
613C22—81no—
614C22C2×C3×#12011yescharacteristic
615C22C3×#54611yescharacteristic
616C22C3×#54711yescharacteristic
617C4—21no—
618C4—21no—
619C4—21no—
620C4—44no—
621C4—41no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—42no—
626C4—82no—
627C4—81no—
628C4C3×#45412yes—
629C4C3×#52511yescharacteristic
630C4C3×#53711yescharacteristic
631C4C3×#54811yescharacteristic
632C4C3×#55011yescharacteristic
633C3#2414511yescharacteristic, direct product factor, Sylow
634C2—21no—
635C2—21no—
636C2—82no—
637C2—81no—
638C2C3×#240711yescharacteristic
639C2C2×C3×#45411yescharacteristic
640C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×AGL(1,ℤ/8)C22
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×#3509C2semidirect
2. C3×#3077C2semidirect
4. C3×#2547C2semidirect
5. C3×#2038C2semidirect
7. C3×#1986C2semidirect
8. C3×#1966C2semidirect
9. C3×#1888C2semidirect
10. C3×#1857C2semidirect
11. C3×#1806C2semidirect
12. #24145C3direct (N × H)
54. #3509C2×C3semidirect
55. #3077C2×C3semidirect
57. #2547C2×C3semidirect
58. #2038C2×C3semidirect
60. #1986C2×C3semidirect
61. #1966C2×C3semidirect
62. #1888C2×C3semidirect
63. #1857C2×C3semidirect
64. #1806C2×C3semidirect
633. C3#24145direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35092
C3×#30772
C3×C4×AGL(1,ℤ/8)2
C3×#25472
C3×#20382
C3×#20032
C3×#19862
C3×#19662
C3×#18882
C3×#18572
C3×#18062
#241453