Finite Group Databasebeta

← C3×#24173 #10642780

Subgroups — C3×#24173 #10642780

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3511C211yesmaximal, characteristic
2C3×#3037C211yesmaximal, characteristic, complemented
3C3×C4×#137C211yesmaximal, characteristic, complemented
4C3×#2567C211yesmaximal, characteristic, complemented
5C3×#2039C212yesmaximal, complemented
6C3×#2007C212yesmaximal, complemented
7C3×#1997C211yesmaximal, characteristic, complemented
8C3×#1980C211yesmaximal, characteristic, complemented
9C3×#1882C212yesmaximal, complemented
10C3×#1859C212yesmaximal, complemented
11C3×#1806C211yesmaximal, characteristic, complemented
12#24173C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2211yescharacteristic
14C2×C3×#137C2212yes—
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×Q8C2211yescharacteristic
30C3×#464—22no—
31C3×#462—21no—
32C3×#460—21no—
33C3×#441—21no—
34C3×#440—21no—
35C3×C4×Q16—21no—
36C3×C4×SD16—21no—
37C3×#428C2211yescharacteristic
38C3×#423C2211yescharacteristic
39C2×C3×#104C2212yes—
40C2×C3×#104C2212yes—
41C2×C3×#104C2212yes—
42C3×#419C2211yescharacteristic
43C3×#417C2211yescharacteristic
44C3×#417C2212yes—
45C3×C4×M16C2211yescharacteristic
46C3×#395—22no—
47C3×#391C2212yes—
48C3×#389—22no—
49C3×#384C2212yes—
50C3×C4×C4⋊C4C2211yescharacteristic
51C3×#343C2212yes—
52C3×#343C2212yes—
53C3×#339C2212yes—
54#3511C2×C311yescharacteristic
55#3037C2×C311yescharacteristic, complemented
56C4×#137C2×C311yescharacteristic, complemented
57#2567C2×C311yescharacteristic, complemented
58#2039C2×C312yescomplemented
59#2007C2×C312yescomplemented
60#1997C2×C311yescharacteristic, complemented
61#1980C2×C311yescharacteristic, complemented
62#1882C2×C312yescomplemented
63#1859C2×C312yescomplemented
64#1806C2×C311yescharacteristic, complemented
65C2×C3×#41—22no—
66C2×C3×#41C2311yescharacteristic
67C2×C3×#41C2312yes—
68C22×C3×Q8D412yes—
69C22×C3×Q8C2311yescharacteristic
70C3×#137—22no—
71C3×#137—24no—
72C3×#137—22no—
73C3×#137—42no—
74C2×C3×Q16—21no—
75C2×C3×Q16—22no—
76C2×C3×SD16—21no—
77C2×C3×SD16—22no—
78C2×C3×M16C2312yes—
79C2×C3×M16C2311yescharacteristic
80C2×C3×M16C2311yescharacteristic
81C3×C4⋊Q8—21no—
82C3×C4⋊Q8—21no—
83C3×C4⋊Q8—21no—
84C3×C4⋊Q8—21no—
85C3×C4⋊Q8D411yescharacteristic
86C3×#127D411yescharacteristic
87C3×#126—24no—
88C3×#125—22no—
89C3×#124—22no—
90C3×#124—24no—
91C3×#124D412yes—
92C3×#123—24no—
93C3×C22⋊Q8—22no—
94C3×C22⋊Q8—24no—
95C3×C22⋊Q8—22no—
96C3×C22⋊Q8—22no—
97C3×#121—22no—
98C3×#121—22no—
99C3×#121—22no—
100C3×#121—21no—
101C3×#121—21no—
102C3×#121—24no—
103C3×C4×Q8—22no—
104C3×C4×Q8—22no—
105C3×C4×Q8—21no—
106C3×C4×Q8—21no—
107C3×C4×Q8—21no—
108C3×C4×Q8—21no—
109C3×C4×Q8D411yescharacteristic
110C3×C4×Q8D411yescharacteristic
111C3×C4×D4—22no—
112C3×C4×D4—22no—
113C3×C4×D4—21no—
114C3×C4×D4—21no—
115C3×C4×D4—24no—
116C3×C4×D4—21no—
117C3×C4×D4—21no—
118C3×C4×D4D411yescharacteristic
119C3×C4×D4D411yescharacteristic
120C3×#117—22no—
121C3×#117—22no—
122C3×#117—21no—
123C3×#117C2311yescharacteristic
124C2×C3×C4⋊C4—21no—
125C2×C3×C4⋊C4—22no—
126C2×C3×C4⋊C4—21no—
127C2×C3×C4⋊C4—24no—
128C2×C3×C4⋊C4D412yes—
129C2×C3×C4⋊C4C2311yescharacteristic
130C2×C3×C4⋊C4C2312yes—
131C2×C3×C42—21no—
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×#103—21no—
145C3×#103—21no—
146C3×#102—21no—
147C3×#102—22no—
148C3×#97—21no—
149C3×C4×C8—21no—
150C3×#95—21no—
151C3×#95—24no—
152C2×#137C22×C311yescharacteristic
153C2×#137C22×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×Q8C22×C311yescharacteristic
169#464—22no—
170#462—21no—
171#460—21no—
172#441—21no—
173#440—21no—
174C4×Q16—21no—
175C4×SD16—21no—
176#428C22×C311yescharacteristic
177#423C22×C311yescharacteristic
178C2×#104C22×C312yes—
179C2×#104C22×C312yes—
180C2×#104C22×C312yes—
181#419C22×C311yescharacteristic
182#417C22×C312yes—
183#417C22×C311yescharacteristic
184C4×M16C22×C311yescharacteristic
185#395—22no—
186#391C22×C312yes—
187#389—22no—
188#384C22×C312yes—
189C4×C4⋊C4C22×C311yescharacteristic
190#343C22×C312yes—
191#343C22×C312yes—
192#339C22×C312yes—
193C3×#41—22no—
194C3×#41—22no—
195C3×#41—21no—
196C3×#41—21no—
197C3×#41—21no—
198C3×#41—22no—
199C3×#41—22no—
200C3×#41—21no—
201C3×#41—84no—
202C2×C3×Q8—22no—
203C2×C3×Q8—22no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—82no—
210C2×C3×Q8C2×D412yes—
211C2×C3×Q8C2×D411yescharacteristic
212C2×C3×Q8#4112yescomplemented
213C2×C3×D4—21no—
214C2×C3×D4—22no—
215C2×C3×D4—24no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4C2×D412yes—
219C2×C3×D4C2×D411yescharacteristic
220C22×C3×C4—21no—
221C22×C3×C4—22no—
222C22×C3×C4—24no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—21no—
226C22×C3×C4—22no—
227C22×C3×C4—21no—
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2×D414yes—
230C22×C3×C4C2×D411yescharacteristic
231C22×C3×C4C2411yescharacteristic
232C3×Q16—24no—
233C3×Q16—44no—
234C3×SD16—24no—
235C3×SD16—44no—
236C3×M16—21no—
237C3×M16—22no—
238C3×M16—21no—
239C3×M16—22no—
240C3×M16—21no—
241C3×M16#4112yescomplemented
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—22no—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—24no—
247C3×C4⋊C4—24no—
248C3×C4⋊C4—22no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—22no—
255C3×C4⋊C4—22no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—41no—
259C3×C4⋊C4—41no—
260C3×C4⋊C4—84no—
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×C42—24no—
272C3×C42—21no—
273C3×C42—21no—
274C3×C42—21no—
275C3×C42—22no—
276C3×C42—22no—
277C3×C42—41no—
278C3×C42—41no—
279C3×C42C2×D411yescharacteristic
280C3×C42C2×D411yescharacteristic
281C3×C42C2×D411yescharacteristic
282C3×C42C2×D411yescharacteristic
283C2×#41—22no—
284C2×#41C23×C311yescharacteristic
285C2×#41C23×C312yes—
286C22×Q8C3×D412yes—
287C22×Q8C23×C311yescharacteristic
288#137—22no—
289#137—24no—
290#137—22no—
291#137—42no—
292C2×Q16—21no—
293C2×Q16—22no—
294C2×SD16—21no—
295C2×SD16—22no—
296C2×M16C23×C312yes—
297C2×M16C23×C311yescharacteristic
298C2×M16C23×C311yescharacteristic
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8C3×D411yescharacteristic
304#127C3×D411yescharacteristic
305#126—24no—
306#125—22no—
307#124—22no—
308#124—24no—
309#124C3×D412yes—
310#123—24no—
311C22⋊Q8—22no—
312C22⋊Q8—24no—
313C22⋊Q8—22no—
314C22⋊Q8—22no—
315#121—22no—
316#121—22no—
317#121—22no—
318#121—21no—
319#121—21no—
320#121—24no—
321C4×Q8—22no—
322C4×Q8—22no—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×Q8C3×D411yescharacteristic
328C4×Q8C3×D411yescharacteristic
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4—24no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4C3×D411yescharacteristic
337C4×D4C3×D411yescharacteristic
338#117—22no—
339#117—22no—
340#117—21no—
341#117C23×C311yescharacteristic
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4C3×D412yes—
347C2×C4⋊C4C23×C311yescharacteristic
348C2×C4⋊C4C23×C312yes—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42C23×C312yes—
352C2×C42C23×C311yescharacteristic
353#107—21no—
354#106—21no—
355C4⋊C8—21no—
356C4⋊C8—21no—
357#104—24no—
358#104—42no—
359#104—42no—
360#103—21no—
361#103—22no—
362#103—21no—
363#103—21no—
364#102—21no—
365#102—22no—
366#97—21no—
367C4×C8—21no—
368#95—21no—
369#95—24no—
370C23×C3—21no—
371C23×C3—22no—
372C23×C3C22×D411yescharacteristic
373C3×Q8—22no—
374C3×Q8—21no—
375C3×Q8—21no—
376C3×Q8—22no—
377C3×Q8—22no—
378C3×Q8—22no—
379C3×Q8—44no—
380C3×Q8—41no—
381C3×Q8—41no—
382C3×Q8—82no—
383C3×D4—21no—
384C3×D4—21no—
385C3×D4—22no—
386C3×D4—22no—
387C3×D4—42no—
388C3×D4—42no—
389C3×D4—41no—
390C3×D4—42no—
391C3×D4—41no—
392C3×D4—42no—
393C3×D4—84no—
394C2×C3×C4—22no—
395C2×C3×C4—24no—
396C2×C3×C4—21no—
397C2×C3×C4—22no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—24no—
401C2×C3×C4—22no—
402C2×C3×C4—41no—
403C2×C3×C4—42no—
404C2×C3×C4—42no—
405C2×C3×C4—41no—
406C2×C3×C4—42no—
407C2×C3×C4—41no—
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—41no—
416C2×C3×C4—84no—
417C2×C3×C4#12011yescharacteristic
418C2×C3×C4#12011yescharacteristic
419C2×C3×C4#12011yescharacteristic
420C2×C3×C4#12011yescharacteristic
421C2×C3×C4C22×D411yescharacteristic
422C2×C3×C4C22×D411yescharacteristic
423C2×C3×C4C2×#4111yescharacteristic
424C2×C3×C4C2×#4111yescharacteristic
425C2×C3×C4H(5,𝔽2)11yescharacteristic
426C2×C3×C42−1+411yescharacteristic
427C3×C8—22no—
428C3×C8—41no—
429C3×C8—42no—
430#41—22no—
431#41—22no—
432#41—21no—
433#41—21no—
434#41—21no—
435#41—22no—
436#41—22no—
437#41—21no—
438#41—84no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—82no—
447C2×Q8C2×C3×D412yes—
448C2×Q8C2×C3×D411yescharacteristic
449C2×Q8C3×#4112yescomplemented
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—24no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4C2×C3×D412yes—
456C2×D4C2×C3×D411yescharacteristic
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4C2×C3×D411yescharacteristic
466C22×C4C2×C3×D414yes—
467C22×C4C2×C3×D411yescharacteristic
468C22×C4C24×C311yescharacteristic
469Q16—24no—
470Q16—44no—
471SD16—24no—
472SD16—44no—
473M16—21no—
474M16—22no—
475M16—21no—
476M16—22no—
477M16—21no—
478M16C3×#4112yescomplemented
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—84no—
498C4⋊C4C2×C3×D411yescharacteristic
499C4⋊C4C2×C3×D411yescharacteristic
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C42—24no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—22no—
513C42—22no—
514C42—41no—
515C42—41no—
516C42C2×C3×D411yescharacteristic
517C42C2×C3×D411yescharacteristic
518C42C2×C3×D411yescharacteristic
519C42C2×C3×D411yescharacteristic
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—42no—
523C22×C3—41no—
524C22×C3—41no—
525C22×C3—42no—
526C22×C3—41no—
527C22×C3—42no—
528C22×C3C2×#12011yescharacteristic
529C22×C3#54811yescharacteristic
530C22×C3#55011yescharacteristic
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—44no—
535C3×C4—42no—
536C3×C4—41no—
537C3×C4—42no—
538C3×C4—41no—
539C3×C4—41no—
540C3×C4—44no—
541C3×C4—81no—
542C3×C4#45612yes—
543C3×C4#52511yescharacteristic
544C3×C4#53711yescharacteristic
545C3×C4#54611yescharacteristic
546C3×C4#54711yescharacteristic
547C23—21no—
548C23—22no—
549C23C22×C3×D411yescharacteristic
550Q8—22no—
551Q8—21no—
552Q8—21no—
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—44no—
557Q8—41no—
558Q8—41no—
559Q8—82no—
560D4—21no—
561D4—21no—
562D4—22no—
563D4—22no—
564D4—42no—
565D4—42no—
566D4—41no—
567D4—42no—
568D4—41no—
569D4—42no—
570D4—84no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—84no—
594C2×C4C3×#12011yescharacteristic
595C2×C4C3×#12011yescharacteristic
596C2×C4C3×#12011yescharacteristic
597C2×C4C3×#12011yescharacteristic
598C2×C4C22×C3×D411yescharacteristic
599C2×C4C22×C3×D411yescharacteristic
600C2×C4C2×C3×#4111yescharacteristic
601C2×C4C2×C3×#4111yescharacteristic
602C2×C4C3×H(5,𝔽2)11yescharacteristic
603C2×C4C3×2−1+411yescharacteristic
604C8—22no—
605C8—41no—
606C8—42no—
607C2×C3—21no—
608C2×C3—21no—
609C2×C3—81no—
610C2×C3—82no—
611C2×C3#240711yescharacteristic
612C2×C3C2×#45611yescharacteristic
613C2×C3#301911yescharacteristic
614C22—21no—
615C22—21no—
616C22—42no—
617C22—41no—
618C22—41no—
619C22—42no—
620C22—41no—
621C22—42no—
622C22C2×C3×#12011yescharacteristic
623C22C3×#54811yescharacteristic
624C22C3×#55011yescharacteristic
625C4—21no—
626C4—21no—
627C4—21no—
628C4—44no—
629C4—42no—
630C4—41no—
631C4—42no—
632C4—41no—
633C4—41no—
634C4—44no—
635C4—81no—
636C4C3×#45612yes—
637C4C3×#52511yescharacteristic
638C4C3×#53711yescharacteristic
639C4C3×#54611yescharacteristic
640C4C3×#54711yescharacteristic
641C3#2417311yescharacteristic, direct product factor, Sylow
642C2—21no—
643C2—21no—
644C2—81no—
645C2—82no—
646C2C3×#240711yescharacteristic
647C2C2×C3×#45611yescharacteristic
648C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3511C2
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×#3037C2semidirect
3. C3×C4×#137C2semidirect
4. C3×#2567C2semidirect
5. C3×#2039C2semidirect
6. C3×#2007C2semidirect
7. C3×#1997C2semidirect
8. C3×#1980C2semidirect
9. C3×#1882C2semidirect
10. C3×#1859C2semidirect
11. C3×#1806C2semidirect
12. #24173C3direct (N × H)
55. #3037C2×C3semidirect
56. C4×#137C2×C3semidirect
57. #2567C2×C3semidirect
58. #2039C2×C3semidirect
59. #2007C2×C3semidirect
60. #1997C2×C3semidirect
61. #1980C2×C3semidirect
62. #1882C2×C3semidirect
63. #1859C2×C3semidirect
64. #1806C2×C3semidirect
212. C2×C3×Q8#41semidirect
241. C3×M16#41semidirect
449. C2×Q8C3×#41semidirect
478. M16C3×#41semidirect
641. C3#24173direct (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×#30372
C3×C4×#1372
C3×#25672
C3×#20392
C3×#20072
C3×#19972
C3×#19802
C3×#18822
C3×#18592
C3×#18062
#241733