Finite Group Databasebeta

← C3×#24182 #10642789

Subgroups — C3×#24182 #10642789

Order 768.

1082 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3506C211yesmaximal, characteristic
2C3×#3078C211yesmaximal, characteristic, complemented
3C3×#3037C211yesmaximal, characteristic, complemented
4C3×#2326C211yesmaximal, characteristic, complemented
5C3×#2037C212yesmaximal, complemented
6C3×#2004C212yesmaximal, complemented
7C3×#1995C211yesmaximal, characteristic, complemented
8C3×#1963C211yesmaximal, characteristic, complemented
9C3×#1882C212yesmaximal, complemented
10C3×#1859C212yesmaximal, complemented
11C3×#1800C211yesmaximal, characteristic, complemented
12#24182C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2212yes—
14C3×#559—22no—
15C3×#556—21no—
16C3×#555—21no—
17C3×#553—22no—
18C3×#550—21no—
19C3×#547—21no—
20C3×#538—22no—
21C3×#535C2212yes—
22C3×#533C2211yescharacteristic
23C3×#532C2211yescharacteristic
24C2×C3×#124C2211yescharacteristic
25C3×#525—22no—
26C3×#520C2212yes—
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×D4C2211yescharacteristic
29C3×#479—22no—
30C3×#477—21no—
31C3×#475—21no—
32C3×#464—22no—
33C3×#462—21no—
34C3×#460—21no—
35C3×#428C2212yes—
36C3×#423C2211yescharacteristic
37C3×#423C2211yescharacteristic
38C2×C3×#104C2212yes—
39C2×C3×#104C2212yes—
40C3×#419C2212yes—
41C3×#417C2212yes—
42C2×C3×#100C2212yes—
43C3×#391C2211yescharacteristic
44C3×#390C2211yescharacteristic
45C3×#388—22no—
46C3×#385—22no—
47C3×#384C2211yescharacteristic
48C3×#376C2211yescharacteristic
49C3×#343C2212yes—
50C3×#339C2212yes—
51C3×#339C2212yes—
52#3506C2×C311yescharacteristic
53#3078C2×C311yescharacteristic, complemented
54#3037C2×C311yescharacteristic, complemented
55#2326C2×C311yescharacteristic, complemented
56#2037C2×C312yescomplemented
57#2004C2×C312yescomplemented
58#1995C2×C311yescharacteristic, complemented
59#1963C2×C311yescharacteristic, complemented
60#1882C2×C312yescomplemented
61#1859C2×C312yescomplemented
62#1800C2×C311yescharacteristic, complemented
63C2×C3×#41—22no—
64C2×C3×#41C2312yes—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×D4C2311yescharacteristic
67C23×C3×C4D412yes—
68C3×#137—24no—
69C3×#137—42no—
70C2×C3×Q16—22no—
71C2×C3×SD16—22no—
72C2×C3×M16C2312yes—
73C2×C3×M16C2312yes—
74C3×C4⋊Q8D411yescharacteristic
75C3×C4⋊Q8D411yescharacteristic
76C3×C4⋊Q8D411yescharacteristic
77C3×#127—22no—
78C3×#127D411yescharacteristic
79C3×#126—24no—
80C3×#125D412yes—
81C3×#124—24no—
82C3×#124—22no—
83C3×#124—22no—
84C3×#124—22no—
85C3×#124D412yes—
86C3×#123—24no—
87C3×#123—24no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—21no—
90C3×C22⋊Q8—21no—
91C3×#121—22no—
92C3×#121—24no—
93C3×#121—21no—
94C3×#121—21no—
95C3×#121—22no—
96C3×#121—22no—
97C3×#120—24no—
98C3×C4×Q8—22no—
99C3×C4×Q8—21no—
100C3×C4×Q8—21no—
101C3×C4×Q8—21no—
102C3×C4×Q8—21no—
103C3×C4×Q8—21no—
104C3×C4×Q8—21no—
105C3×C4×D4—24no—
106C3×C4×D4—22no—
107C3×C4×D4—24no—
108C3×C4×D4—21no—
109C3×C4×D4—21no—
110C3×C4×D4—21no—
111C3×C4×D4—21no—
112C3×C4×D4—21no—
113C3×C4×D4—21no—
114C3×#117—22no—
115C3×#117—22no—
116C3×#117C2312yes—
117C2×C3×C4⋊C4—22no—
118C2×C3×C4⋊C4—21no—
119C2×C3×C4⋊C4—22no—
120C2×C3×C4⋊C4C2311yescharacteristic
121C2×C3×C4⋊C4C2311yescharacteristic
122C2×C3×C22⋊C4—22no—
123C2×C3×C22⋊C4—22no—
124C2×C3×C22⋊C4D412yes—
125C2×C3×C42—21no—
126C2×C3×C42C2311yescharacteristic
127C2×C3×C42C2311yescharacteristic
128C2×C3×C42C2311yescharacteristic
129C3×#107—22no—
130C3×#106—22no—
131C3×C4⋊C8—21no—
132C3×C4⋊C8—21no—
133C3×C4⋊C8—21no—
134C3×C4⋊C8—21no—
135C3×#104—24no—
136C3×#104—42no—
137C3×#103—22no—
138C3×#103—22no—
139C3×#103—22no—
140C3×#102—22no—
141C3×#100—42no—
142C3×#95—21no—
143C3×#95—21no—
144C3×#95—22no—
145C2×#137C22×C312yes—
146#559—22no—
147#556—21no—
148#555—21no—
149#553—22no—
150#550—21no—
151#547—21no—
152#538—22no—
153#535C22×C312yes—
154#533C22×C311yescharacteristic
155#532C22×C311yescharacteristic
156C2×#124C22×C311yescharacteristic
157#525—22no—
158#520C22×C312yes—
159C2×C4×Q8C22×C311yescharacteristic
160C2×C4×D4C22×C311yescharacteristic
161#479—22no—
162#477—21no—
163#475—21no—
164#464—22no—
165#462—21no—
166#460—21no—
167#428C22×C312yes—
168#423C22×C311yescharacteristic
169#423C22×C311yescharacteristic
170C2×#104C22×C312yes—
171C2×#104C22×C312yes—
172#419C22×C312yes—
173#417C22×C312yes—
174C2×#100C22×C312yes—
175#391C22×C311yescharacteristic
176#390C22×C311yescharacteristic
177#388—22no—
178#385—22no—
179#384C22×C311yescharacteristic
180#376C22×C311yescharacteristic
181#343C22×C312yes—
182#339C22×C312yes—
183#339C22×C312yes—
184C24×C3C2×D412yes—
185C3×#41—22no—
186C3×#41—22no—
187C3×#41—22no—
188C3×#41—22no—
189C3×#41—84no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8C2×D412yes—
198C2×C3×Q8#4112yescomplemented
199C2×C3×D4—22no—
200C2×C3×D4—24no—
201C2×C3×D4—24no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4C2×D412yes—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—24no—
214C22×C3×C4—22no—
215C22×C3×C4—22no—
216C22×C3×C4—22no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—82no—
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C3×Q16—44no—
225C3×SD16—44no—
226C3×M16—22no—
227C3×M16—22no—
228C3×M16—22no—
229C3×M16—22no—
230C2×C3×C8—22no—
231C2×C3×C8—22no—
232C3×C4⋊C4—24no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—22no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4—41no—
245C3×C4⋊C4—82no—
246C3×C4⋊C4C2×D412yes—
247C3×C4⋊C4C2×D412yes—
248C3×C22⋊C4—28no—
249C3×C22⋊C4—24no—
250C3×C22⋊C4—24no—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—22no—
256C3×C22⋊C4—22no—
257C3×C22⋊C4—82no—
258C3×C42—24no—
259C3×C42—22no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—41no—
264C3×C42—41no—
265C3×C42C2×D411yescharacteristic
266C3×C42C2×D411yescharacteristic
267C3×C42C2×D411yescharacteristic
268C3×C42C2×D411yescharacteristic
269C3×C42#4112yescomplemented
270C2×#41—22no—
271C2×#41C23×C312yes—
272C22×Q8C23×C311yescharacteristic
273C22×D4C23×C311yescharacteristic
274C23×C4C3×D412yes—
275#137—24no—
276#137—42no—
277C2×Q16—22no—
278C2×SD16—22no—
279C2×M16C23×C312yes—
280C2×M16C23×C312yes—
281C4⋊Q8C3×D411yescharacteristic
282C4⋊Q8C3×D411yescharacteristic
283C4⋊Q8C3×D411yescharacteristic
284#127—22no—
285#127C3×D411yescharacteristic
286#126—24no—
287#125C3×D412yes—
288#124—24no—
289#124—22no—
290#124—22no—
291#124—22no—
292#124C3×D412yes—
293#123—24no—
294#123—24no—
295C22⋊Q8—22no—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298#121—22no—
299#121—24no—
300#121—21no—
301#121—21no—
302#121—22no—
303#121—22no—
304#120—24no—
305C4×Q8—22no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×D4—24no—
313C4×D4—22no—
314C4×D4—24no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—21no—
321#117—22no—
322#117—22no—
323#117C23×C312yes—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4C23×C311yescharacteristic
328C2×C4⋊C4C23×C311yescharacteristic
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4C3×D412yes—
332C2×C42—21no—
333C2×C42C23×C311yescharacteristic
334C2×C42C23×C311yescharacteristic
335C2×C42C23×C311yescharacteristic
336#107—22no—
337#106—22no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—21no—
341C4⋊C8—21no—
342#104—24no—
343#104—42no—
344#103—22no—
345#103—22no—
346#103—22no—
347#102—22no—
348#100—42no—
349#95—21no—
350#95—21no—
351#95—22no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—42no—
357C23×C3—42no—
358C23×C3C22×D411yescharacteristic
359C3×Q8—22no—
360C3×Q8—22no—
361C3×Q8—22no—
362C3×Q8—22no—
363C3×Q8—42no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×Q8—81no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—42no—
370C3×D4—42no—
371C3×D4—42no—
372C3×D4—41no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—41no—
376C3×D4—81no—
377C3×D4—84no—
378C3×D4—84no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—44no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—42no—
393C2×C3×C4—42no—
394C2×C3×C4—42no—
395C2×C3×C4—41no—
396C2×C3×C4—42no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—82no—
401C2×C3×C4—84no—
402C2×C3×C4#12011yescharacteristic
403C2×C3×C4#12011yescharacteristic
404C2×C3×C4#12011yescharacteristic
405C2×C3×C4#12011yescharacteristic
406C2×C3×C4C22×D411yescharacteristic
407C2×C3×C4C22×D411yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4H(5,𝔽2)11yescharacteristic
411C2×C3×C42−1+411yescharacteristic
412C3×C8—42no—
413C3×C8—42no—
414C24C2×C3×D412yes—
415#41—22no—
416#41—22no—
417#41—22no—
418#41—22no—
419#41—84no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8C2×C3×D412yes—
428C2×Q8C3×#4112yescomplemented
429C2×D4—22no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4C2×C3×D412yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—82no—
450C22×C4C2×C3×D412yes—
451C22×C4C2×C3×D411yescharacteristic
452C22×C4C2×C3×D411yescharacteristic
453C22×C4C24×C311yescharacteristic
454Q16—44no—
455SD16—44no—
456M16—22no—
457M16—22no—
458M16—22no—
459M16—22no—
460C2×C8—22no—
461C2×C8—22no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—82no—
476C4⋊C4C2×C3×D412yes—
477C4⋊C4C2×C3×D412yes—
478C22⋊C4—28no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—82no—
488C42—24no—
489C42—22no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—41no—
494C42—41no—
495C42C2×C3×D411yescharacteristic
496C42C2×C3×D411yescharacteristic
497C42C2×C3×D411yescharacteristic
498C42C2×C3×D411yescharacteristic
499C42C3×#4112yescomplemented
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—44no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—82no—
510C22×C3C2×#12011yescharacteristic
511C22×C3#52511yescharacteristic
512C22×C3#53711yescharacteristic
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—42no—
518C3×C4—42no—
519C3×C4—42no—
520C3×C4—42no—
521C3×C4—42no—
522C3×C4—42no—
523C3×C4—81no—
524C3×C4—81no—
525C3×C4#54611yescharacteristic
526C3×C4#54711yescharacteristic
527C3×C4#54811yescharacteristic
528C3×C4#55011yescharacteristic
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—42no—
534C23—42no—
535C23C22×C3×D411yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—22no—
539Q8—22no—
540Q8—42no—
541Q8—41no—
542Q8—41no—
543Q8—81no—
544D4—22no—
545D4—22no—
546D4—42no—
547D4—42no—
548D4—42no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—81no—
554D4—84no—
555D4—84no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—82no—
578C2×C4—84no—
579C2×C4C3×#12011yescharacteristic
580C2×C4C3×#12011yescharacteristic
581C2×C4C3×#12011yescharacteristic
582C2×C4C3×#12011yescharacteristic
583C2×C4C22×C3×D411yescharacteristic
584C2×C4C22×C3×D411yescharacteristic
585C2×C4C2×C3×#4111yescharacteristic
586C2×C4C2×C3×#4111yescharacteristic
587C2×C4C3×H(5,𝔽2)11yescharacteristic
588C2×C4C3×2−1+411yescharacteristic
589C8—42no—
590C8—42no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—82no—
594C2×C3—82no—
595C2×C3#240711yescharacteristic
596C2×C3#301911yescharacteristic
597C2×C3#302211yescharacteristic
598C22—21no—
599C22—21no—
600C22—44no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—82no—
608C22C2×C3×#12011yescharacteristic
609C22C3×#52511yescharacteristic
610C22C3×#53711yescharacteristic
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—81no—
622C4—81no—
623C4C3×#54611yescharacteristic
624C4C3×#54711yescharacteristic
625C4C3×#54811yescharacteristic
626C4C3×#55011yescharacteristic
627C3#2418211yescharacteristic, direct product factor, Sylow
628C2—21no—
629C2—21no—
630C2—82no—
631C2—82no—
632C2C3×#240711yescharacteristic
633C2C3×#301911yescharacteristic
634C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3506C2
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
2. C3×#3078C2semidirect
3. C3×#3037C2semidirect
4. C3×#2326C2semidirect
5. C3×#2037C2semidirect
6. C3×#2004C2semidirect
7. C3×#1995C2semidirect
8. C3×#1963C2semidirect
9. C3×#1882C2semidirect
10. C3×#1859C2semidirect
11. C3×#1800C2semidirect
12. #24182C3direct (N × H)
53. #3078C2×C3semidirect
54. #3037C2×C3semidirect
55. #2326C2×C3semidirect
56. #2037C2×C3semidirect
57. #2004C2×C3semidirect
58. #1995C2×C3semidirect
59. #1963C2×C3semidirect
60. #1882C2×C3semidirect
61. #1859C2×C3semidirect
62. #1800C2×C3semidirect
198. C2×C3×Q8#41semidirect
269. C3×C42#41semidirect
428. C2×Q8C3×#41semidirect
499. C42C3×#41semidirect
627. C3#24182direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35062
C3×#30782
C3×#30372
C3×#23262
C3×#20372
C3×#20042
C3×#19952
C3×#19632
C3×#18822
C3×#18592
C3×#18002
#241823