Finite Group Databasebeta

← C3×#24180 #10642787

Subgroups — C3×#24180 #10642787

Order 768.

1146 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3506C211yesmaximal, characteristic, complemented
2C3×#3077C211yesmaximal, characteristic, complemented
3C3×#3036C211yesmaximal, characteristic, complemented
4C3×#2328C211yesmaximal, characteristic, complemented
5C3×#2037C212yesmaximal, complemented
6C3×#2002C212yesmaximal, complemented
7C3×#1995C211yesmaximal, characteristic, complemented
8C3×#1963C211yesmaximal, characteristic, complemented
9C3×#1883C212yesmaximal, complemented
10C3×#1860C212yesmaximal, complemented
11C3×#1800C211yesmaximal, characteristic, complemented
12#24180C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×AGL(1,ℤ/8)C2212yes—
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×#127C2211yescharacteristic
25C3×#525—22no—
26C3×#520C2212yes—
27C2×C3×C4×D4C2211yescharacteristic
28C2×C3×C4×D4C2211yescharacteristic
29C3×#478—22no—
30C3×#476—21no—
31C3×#474—21no—
32C3×#463—22no—
33C3×#461—21no—
34C3×#459—21no—
35C3×#428C2212yes—
36C3×#423C2211yescharacteristic
37C3×#423C2211yescharacteristic
38C2×C3×#104C2212yes—
39C2×C3×#104C2212yes—
40C3×#418C2212yes—
41C3×#417C2212yes—
42C2×C3×#100C2212yes—
43C3×#390C2211yescharacteristic
44C3×#390C2211yescharacteristic
45C3×#386—22no—
46C3×#386—22no—
47C3×#382C2211yescharacteristic
48C3×#376C2211yescharacteristic
49C3×#343C2212yes—
50C3×#339C2212yes—
51C3×#339C2212yes—
52#3506C2×C311yescharacteristic, complemented
53#3077C2×C311yescharacteristic, complemented
54#3036C2×C311yescharacteristic, complemented
55#2328C2×C311yescharacteristic, complemented
56#2037C2×C312yescomplemented
57#2002C2×C312yescomplemented
58#1995C2×C311yescharacteristic, complemented
59#1963C2×C311yescharacteristic, complemented
60#1883C2×C312yescomplemented
61#1860C2×C312yescomplemented
62#1800C2×C311yescharacteristic, complemented
63C2×C3×#41—22no—
64C2×C3×#41C2312yes—
65C22×C3×D4—22no—
66C22×C3×D4C2311yescharacteristic
67C22×C3×D4C2311yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4D412yes—
70C3×AGL(1,ℤ/8)—24no—
71C3×AGL(1,ℤ/8)—42no—
72C2×C3×SD16—22no—
73C2×C3×D8—22no—
74C2×C3×M16C2312yes—
75C2×C3×M16C2312yes—
76C3×C4⋊Q8D411yescharacteristic, complemented
77C3×C4⋊Q8D411yescharacteristic, complemented
78C3×C4⋊Q8D411yescharacteristic, complemented
79C3×#127—22no—
80C3×#127—22no—
81C3×#127—22no—
82C3×#127D411yescharacteristic, complemented
83C3×#126—24no—
84C3×#125D412yescomplemented
85C3×#124—24no—
86C3×#124—22no—
87C3×#124D412yescomplemented
88C3×#123—24no—
89C3×#123—24no—
90C3×C22⋊Q8—22no—
91C3×C22⋊Q8—21no—
92C3×C22⋊Q8—21no—
93C3×#121—22no—
94C3×#121—24no—
95C3×#121—21no—
96C3×#121—21no—
97C3×#121—22no—
98C3×#121—22no—
99C3×#120—24no—
100C3×C4×Q8—22no—
101C3×C4×Q8—21no—
102C3×C4×Q8—21no—
103C3×C4×D4—24no—
104C3×C4×D4—22no—
105C3×C4×D4—24no—
106C3×C4×D4—21no—
107C3×C4×D4—21no—
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×C4×D4—21no—
115C3×C4×D4—21no—
116C3×#117—22no—
117C3×#117—22no—
118C3×#117C2312yes—
119C2×C3×C4⋊C4—22no—
120C2×C3×C4⋊C4C2311yescharacteristic
121C2×C3×C4⋊C4C2311yescharacteristic
122C2×C3×C22⋊C4—21no—
123C2×C3×C22⋊C4—22no—
124C2×C3×C22⋊C4—22no—
125C2×C3×C22⋊C4D412yes—
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×#102—22no—
139C3×#102—22no—
140C3×#102—22no—
141C3×#100—42no—
142C3×#95—21no—
143C3×#95—21no—
144C3×#95—22no—
145C2×AGL(1,ℤ/8)C22×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×#127C22×C311yescharacteristic
157#525—22no—
158#520C22×C312yes—
159C2×C4×D4C22×C311yescharacteristic
160C2×C4×D4C22×C311yescharacteristic
161#478—22no—
162#476—21no—
163#474—21no—
164#463—22no—
165#461—21no—
166#459—21no—
167#428C22×C312yes—
168#423C22×C311yescharacteristic
169#423C22×C311yescharacteristic
170C2×#104C22×C312yes—
171C2×#104C22×C312yes—
172#418C22×C312yes—
173#417C22×C312yes—
174C2×#100C22×C312yes—
175#390C22×C311yescharacteristic
176#390C22×C311yescharacteristic
177#386—22no—
178#386—22no—
179#382C22×C311yescharacteristic
180#376C22×C311yescharacteristic
181#343C22×C312yes—
182#339C22×C312yes—
183#339C22×C312yes—
184C24×C3—21no—
185C24×C3C2×D412yes—
186C3×#41—22no—
187C3×#41—22no—
188C3×#41—22no—
189C3×#41—22no—
190C3×#41—84no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8C2×D412yes—
194C2×C3×D4—22no—
195C2×C3×D4—24no—
196C2×C3×D4—24no—
197C2×C3×D4—28no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
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×D4—21no—
211C2×C3×D4—82no—
212C2×C3×D4—82no—
213C2×C3×D4C2×D412yes—
214C2×C3×D4#4112yescomplemented
215C22×C3×C4—21no—
216C22×C3×C4—24no—
217C22×C3×C4—22no—
218C22×C3×C4—22no—
219C22×C3×C4—22no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—41no—
223C22×C3×C4—41no—
224C22×C3×C4—82no—
225C22×C3×C4—82no—
226C22×C3×C4C2×D412yes—
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2411yescharacteristic
230C3×SD16—44no—
231C3×D8—44no—
232C3×M16—22no—
233C3×M16—22no—
234C3×M16—22no—
235C3×M16—22no—
236C2×C3×C8—22no—
237C2×C3×C8—22no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4C2×D412yes—
250C3×C4⋊C4C2×D412yes—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—24no—
253C3×C22⋊C4—24no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—22no—
256C3×C22⋊C4—22no—
257C3×C22⋊C4—22no—
258C3×C22⋊C4—22no—
259C3×C22⋊C4—41no—
260C3×C22⋊C4—41no—
261C3×C22⋊C4—82no—
262C3×C22⋊C4—82no—
263C3×C42—24no—
264C3×C42—22no—
265C3×C42—21no—
266C3×C42—21no—
267C3×C42—21no—
268C3×C42C2×D411yescharacteristic
269C3×C42C2×D411yescharacteristic
270C3×C42C2×D411yescharacteristic
271C3×C42C2×D411yescharacteristic
272C3×C42#4112yescomplemented
273C2×#41—22no—
274C2×#41C23×C312yes—
275C22×D4—22no—
276C22×D4C23×C311yescharacteristic
277C22×D4C23×C311yescharacteristic
278C23×C4—21no—
279C23×C4C3×D412yes—
280AGL(1,ℤ/8)—24no—
281AGL(1,ℤ/8)—42no—
282C2×SD16—22no—
283C2×D8—22no—
284C2×M16C23×C312yes—
285C2×M16C23×C312yes—
286C4⋊Q8C3×D411yescharacteristic, complemented
287C4⋊Q8C3×D411yescharacteristic, complemented
288C4⋊Q8C3×D411yescharacteristic, complemented
289#127—22no—
290#127—22no—
291#127—22no—
292#127C3×D411yescharacteristic, complemented
293#126—24no—
294#125C3×D412yescomplemented
295#124—24no—
296#124—22no—
297#124C3×D412yescomplemented
298#123—24no—
299#123—24no—
300C22⋊Q8—22no—
301C22⋊Q8—21no—
302C22⋊Q8—21no—
303#121—22no—
304#121—24no—
305#121—21no—
306#121—21no—
307#121—22no—
308#121—22no—
309#120—24no—
310C4×Q8—22no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×D4—24no—
314C4×D4—22no—
315C4×D4—24no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326#117—22no—
327#117—22no—
328#117C23×C312yes—
329C2×C4⋊C4—22no—
330C2×C4⋊C4C23×C311yescharacteristic
331C2×C4⋊C4C23×C311yescharacteristic
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4C3×D412yes—
336C2×C42C23×C311yescharacteristic
337C2×C42C23×C311yescharacteristic
338C2×C42C23×C311yescharacteristic
339#107—22no—
340#106—22no—
341C4⋊C8—21no—
342C4⋊C8—21no—
343C4⋊C8—21no—
344C4⋊C8—21no—
345#104—24no—
346#104—42no—
347#103—22no—
348#102—22no—
349#102—22no—
350#102—22no—
351#100—42no—
352#95—21no—
353#95—21no—
354#95—22no—
355C23×C3—22no—
356C23×C3—22no—
357C23×C3—22no—
358C23×C3—22no—
359C23×C3—22no—
360C23×C3—42no—
361C23×C3—42no—
362C23×C3—42no—
363C23×C3—42no—
364C23×C3—41no—
365C23×C3—41no—
366C23×C3C22×D411yescharacteristic
367C3×Q8—22no—
368C3×Q8—22no—
369C3×Q8—42no—
370C3×Q8—81no—
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—22no—
374C3×D4—22no—
375C3×D4—42no—
376C3×D4—42no—
377C3×D4—42no—
378C3×D4—41no—
379C3×D4—41no—
380C3×D4—41no—
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—81no—
385C3×D4—84no—
386C3×D4—88no—
387C3×D4—84no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—44no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—42no—
399C2×C3×C4—42no—
400C2×C3×C4—42no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—42no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—82no—
407C2×C3×C4—82no—
408C2×C3×C4—82no—
409C2×C3×C4—84no—
410C2×C3×C4#12011yescharacteristic
411C2×C3×C4#12011yescharacteristic
412C2×C3×C4#12011yescharacteristic
413C2×C3×C4#12011yescharacteristic
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C22×D411yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418C2×C3×C4H(5,𝔽2)11yescharacteristic
419C2×C3×C42−1+411yescharacteristic
420C3×C8—42no—
421C3×C8—42no—
422C24—21no—
423C24C2×C3×D412yes—
424#41—22no—
425#41—22no—
426#41—22no—
427#41—22no—
428#41—84no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8C2×C3×D412yes—
432C2×D4—22no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—28no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—82no—
450C2×D4—82no—
451C2×D4C2×C3×D412yes—
452C2×D4C3×#4112yescomplemented
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4C2×C3×D412yes—
465C22×C4C2×C3×D411yescharacteristic
466C22×C4C2×C3×D411yescharacteristic
467C22×C4C24×C311yescharacteristic
468SD16—44no—
469D8—44no—
470M16—22no—
471M16—22no—
472M16—22no—
473M16—22no—
474C2×C8—22no—
475C2×C8—22no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4C2×C3×D412yes—
488C4⋊C4C2×C3×D412yes—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C42—24no—
502C42—22no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42C2×C3×D411yescharacteristic
507C42C2×C3×D411yescharacteristic
508C42C2×C3×D411yescharacteristic
509C42C2×C3×D411yescharacteristic
510C42C3×#4112yescomplemented
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—44no—
514C22×C3—44no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—82no—
523C22×C3—82no—
524C22×C3—81no—
525C22×C3—81no—
526C22×C3C2×#12011yescharacteristic
527C22×C3#52511yescharacteristic
528C22×C3#53711yescharacteristic
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—42no—
534C3×C4—42no—
535C3×C4—42no—
536C3×C4—42no—
537C3×C4—42no—
538C3×C4—81no—
539C3×C4—81no—
540C3×C4#54611yescharacteristic
541C3×C4#54711yescharacteristic
542C3×C4#54811yescharacteristic
543C3×C4#55011yescharacteristic
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—41no—
554C23—41no—
555C23C22×C3×D411yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—42no—
559Q8—81no—
560D4—22no—
561D4—22no—
562D4—22no—
563D4—22no—
564D4—42no—
565D4—42no—
566D4—42no—
567D4—41no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—41no—
573D4—81no—
574D4—88no—
575D4—84no—
576D4—84no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—84no—
599C2×C4C3×#12011yescharacteristic
600C2×C4C3×#12011yescharacteristic
601C2×C4C3×#12011yescharacteristic
602C2×C4C3×#12011yescharacteristic
603C2×C4C22×C3×D411yescharacteristic
604C2×C4C22×C3×D411yescharacteristic
605C2×C4C2×C3×#4111yescharacteristic
606C2×C4C2×C3×#4111yescharacteristic
607C2×C4C3×H(5,𝔽2)11yescharacteristic
608C2×C4C3×2−1+411yescharacteristic
609C8—42no—
610C8—42no—
611C2×C3—21no—
612C2×C3—21no—
613C2×C3—82no—
614C2×C3—82no—
615C2×C3—82no—
616C2×C3#240711yescharacteristic
617C2×C3#301911yescharacteristic
618C2×C3#302111yescharacteristic
619C22—21no—
620C22—21no—
621C22—44no—
622C22—44no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—82no—
631C22—82no—
632C22—81no—
633C22—81no—
634C22C2×C3×#12011yescharacteristic
635C22C3×#52511yescharacteristic
636C22C3×#53711yescharacteristic
637C4—21no—
638C4—21no—
639C4—21no—
640C4—21no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—81no—
647C4—81no—
648C4C3×#54611yescharacteristic
649C4C3×#54711yescharacteristic
650C4C3×#54811yescharacteristic
651C4C3×#55011yescharacteristic
652C3#2418011yescharacteristic, direct product factor, Sylow
653C2—21no—
654C2—21no—
655C2—82no—
656C2—82no—
657C2—82no—
658C2C3×#240711yescharacteristic
659C2C3×#301911yescharacteristic
660C2C3×#302111yescharacteristic
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×#3506C2semidirect
2. C3×#3077C2semidirect
3. C3×#3036C2semidirect
4. C3×#2328C2semidirect
5. C3×#2037C2semidirect
6. C3×#2002C2semidirect
7. C3×#1995C2semidirect
8. C3×#1963C2semidirect
9. C3×#1883C2semidirect
10. C3×#1860C2semidirect
11. C3×#1800C2semidirect
12. #24180C3direct (N × H)
52. #3506C2×C3semidirect
53. #3077C2×C3semidirect
54. #3036C2×C3semidirect
55. #2328C2×C3semidirect
56. #2037C2×C3semidirect
57. #2002C2×C3semidirect
58. #1995C2×C3semidirect
59. #1963C2×C3semidirect
60. #1883C2×C3semidirect
61. #1860C2×C3semidirect
62. #1800C2×C3semidirect
76. C3×C4⋊Q8D4semidirect
77. C3×C4⋊Q8D4semidirect
78. C3×C4⋊Q8D4semidirect
82. C3×#127D4semidirect
84. C3×#125D4semidirect
87. C3×#124D4semidirect
214. C2×C3×D4#41semidirect
272. C3×C42#41semidirect
286. C4⋊Q8C3×D4semidirect
287. C4⋊Q8C3×D4semidirect
288. C4⋊Q8C3×D4semidirect
292. #127C3×D4semidirect
294. #125C3×D4semidirect
297. #124C3×D4semidirect
452. C2×D4C3×#41semidirect
510. C42C3×#41semidirect
652. C3#24180direct (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×#30772
C3×#30362
C3×#23282
C3×#20372
C3×#20022
C3×#19952
C3×#19632
C3×#18832
C3×#18602
C3×#18002
#241803