Finite Group Databasebeta

← C3×#24183 #10642790

Subgroups — C3×#24183 #10642790

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3507C211yesmaximal, characteristic
2C3×#3078C211yesmaximal, characteristic, complemented
3C3×#3037C211yesmaximal, characteristic
4C3×#2327C211yesmaximal, characteristic, complemented
5C3×#2038C212yesmaximal, complemented
6C3×#2005C212yesmaximal, complemented
7C3×#1995C211yesmaximal, characteristic, complemented
8C3×#1963C211yesmaximal, characteristic
9C3×#1882C212yesmaximal, complemented
10C3×#1860C212yesmaximal, complemented
11C3×#1801C211yesmaximal, characteristic, complemented
12#24183C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#137C2212yes—
14C3×#559—21no—
15C3×#557—21no—
16C3×#556—21no—
17C3×#556—22no—
18C3×#554—21no—
19C3×#552—22no—
20C3×#539—22no—
21C3×#536C2212yes—
22C3×#533C2211yescharacteristic
23C3×#532C2211yescharacteristic
24C2×C3×C4⋊Q8C2211yescharacteristic
25C3×#529—22no—
26C3×#520C2212yes—
27C2×C3×C4×Q8C2211yescharacteristic
28C2×C3×C4×Q8C2211yescharacteristic
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×#101C2212yes—
43C3×#391C2211yescharacteristic
44C3×#391C2211yescharacteristic
45C3×#387—22no—
46C3×#387—22no—
47C3×#382C2211yescharacteristic
48C3×#376C2211yescharacteristic
49C3×#343C2212yes—
50C3×#339C2212yes—
51C3×#339C2212yes—
52#3507C2×C311yescharacteristic
53#3078C2×C311yescharacteristic, complemented
54#3037C2×C311yescharacteristic
55#2327C2×C311yescharacteristic, complemented
56#2038C2×C312yescomplemented
57#2005C2×C312yescomplemented
58#1995C2×C311yescharacteristic, complemented
59#1963C2×C311yescharacteristic
60#1882C2×C312yescomplemented
61#1860C2×C312yescomplemented
62#1801C2×C311yescharacteristic, complemented
63C2×C3×#41C2312yes—
64C22×C3×Q8C2311yescharacteristic
65C22×C3×Q8C2311yescharacteristic
66C3×#137—24no—
67C3×#137—42no—
68C2×C3×Q16—22no—
69C2×C3×SD16—22no—
70C2×C3×M16C2312yes—
71C2×C3×M16C2312yes—
72C3×C4⋊Q8—22no—
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8—22no—
75C3×C4⋊Q8D411yescharacteristic
76C3×C4⋊Q8D411yescharacteristic
77C3×C4⋊Q8D411yescharacteristic
78C3×#127D411yescharacteristic
79C3×#126—24no—
80C3×#126—24no—
81C3×#125—22no—
82C3×#125—24no—
83C3×#125D412yes—
84C3×#124—22no—
85C3×#124—22no—
86C3×#124D412yes—
87C3×#123—24no—
88C3×#123—24no—
89C3×C22⋊Q8—24no—
90C3×C22⋊Q8—21no—
91C3×C22⋊Q8—21no—
92C3×C22⋊Q8—22no—
93C3×C22⋊Q8—22no—
94C3×#121—21no—
95C3×#121—21no—
96C3×C4×Q8—22no—
97C3×C4×Q8—22no—
98C3×C4×Q8—21no—
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×Q8—21no—
106C3×C4×Q8—21no—
107C3×C4×Q8—21no—
108C3×C4×D4—22no—
109C3×C4×D4—22no—
110C3×C4×D4—24no—
111C3×C4×D4—21no—
112C3×C4×D4—21no—
113C3×#117—24no—
114C3×#117—22no—
115C3×#117C2312yes—
116C2×C3×C4⋊C4—22no—
117C2×C3×C4⋊C4—22no—
118C2×C3×C4⋊C4—21no—
119C2×C3×C4⋊C4D412yes—
120C2×C3×C4⋊C4C2311yescharacteristic
121C2×C3×C4⋊C4C2311yescharacteristic
122C2×C3×C42—21no—
123C2×C3×C42D412yes—
124C2×C3×C42C2311yescharacteristic
125C2×C3×C42C2311yescharacteristic
126C2×C3×C42C2311yescharacteristic
127C3×#107—22no—
128C3×#106—22no—
129C3×C4⋊C8—21no—
130C3×C4⋊C8—21no—
131C3×C4⋊C8—21no—
132C3×C4⋊C8—21no—
133C3×#104—24no—
134C3×#104—42no—
135C3×#103—22no—
136C3×#103—22no—
137C3×#103—22no—
138C3×#102—22no—
139C3×#101—42no—
140C3×#95—21no—
141C3×#95—21no—
142C3×#95—22no—
143C3×#95—22no—
144C3×#95—22no—
145C2×#137C22×C312yes—
146#559—21no—
147#557—21no—
148#556—21no—
149#556—22no—
150#554—21no—
151#552—22no—
152#539—22no—
153#536C22×C312yes—
154#533C22×C311yescharacteristic
155#532C22×C311yescharacteristic
156C2×C4⋊Q8C22×C311yescharacteristic
157#529—22no—
158#520C22×C312yes—
159C2×C4×Q8C22×C311yescharacteristic
160C2×C4×Q8C22×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×#101C22×C312yes—
175#391C22×C311yescharacteristic
176#391C22×C311yescharacteristic
177#387—22no—
178#387—22no—
179#382C22×C311yescharacteristic
180#376C22×C311yescharacteristic
181#343C22×C312yes—
182#339C22×C312yes—
183#339C22×C312yes—
184C3×#41—22no—
185C3×#41—22no—
186C3×#41—22no—
187C3×#41—22no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
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×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8C2×D412yes—
202C2×C3×Q8#4112yes—
203C2×C3×D4—22no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4C2×D412yes—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—24no—
210C22×C3×C4—22no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2×D412yes—
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2411yescharacteristic
218C3×Q16—44no—
219C3×SD16—44no—
220C3×M16—22no—
221C3×M16—22no—
222C3×M16—22no—
223C3×M16—22no—
224C2×C3×C8—22no—
225C2×C3×C8—22no—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—28no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—41no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4C2×D412yes—
245C3×C4⋊C4C2×D412yes—
246C3×C22⋊C4—24no—
247C3×C22⋊C4—24no—
248C3×C22⋊C4—24no—
249C3×C22⋊C4—22no—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—22no—
252C3×C42—24no—
253C3×C42—22no—
254C3×C42—22no—
255C3×C42—22no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42—41no—
260C3×C42—41no—
261C3×C42C2×D411yescharacteristic
262C3×C42C2×D411yescharacteristic
263C3×C42C2×D411yescharacteristic
264C3×C42C2×D411yescharacteristic
265C3×C42#4112yes—
266C2×#41C23×C312yes—
267C22×Q8C23×C311yescharacteristic
268C22×Q8C23×C311yescharacteristic
269#137—24no—
270#137—42no—
271C2×Q16—22no—
272C2×SD16—22no—
273C2×M16C23×C312yes—
274C2×M16C23×C312yes—
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277C4⋊Q8—22no—
278C4⋊Q8C3×D411yescharacteristic
279C4⋊Q8C3×D411yescharacteristic
280C4⋊Q8C3×D411yescharacteristic
281#127C3×D411yescharacteristic
282#126—24no—
283#126—24no—
284#125—22no—
285#125—24no—
286#125C3×D412yes—
287#124—22no—
288#124—22no—
289#124C3×D412yes—
290#123—24no—
291#123—24no—
292C22⋊Q8—24no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297#121—21no—
298#121—21no—
299C4×Q8—22no—
300C4×Q8—22no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8—21no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—24no—
314C4×D4—21no—
315C4×D4—21no—
316#117—24no—
317#117—22no—
318#117C23×C312yes—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4C3×D412yes—
323C2×C4⋊C4C23×C311yescharacteristic
324C2×C4⋊C4C23×C311yescharacteristic
325C2×C42—21no—
326C2×C42C3×D412yes—
327C2×C42C23×C311yescharacteristic
328C2×C42C23×C311yescharacteristic
329C2×C42C23×C311yescharacteristic
330#107—22no—
331#106—22no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—21no—
336#104—24no—
337#104—42no—
338#103—22no—
339#103—22no—
340#103—22no—
341#102—22no—
342#101—42no—
343#95—21no—
344#95—21no—
345#95—22no—
346#95—22no—
347#95—22no—
348C23×C3—22no—
349C23×C3C22×D411yescharacteristic
350C3×Q8—22no—
351C3×Q8—22no—
352C3×Q8—22no—
353C3×Q8—22no—
354C3×Q8—42no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—81no—
362C3×D4—22no—
363C3×D4—22no—
364C3×D4—42no—
365C3×D4—42no—
366C3×D4—42no—
367C3×D4—81no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—42no—
384C2×C3×C4—42no—
385C2×C3×C4—42no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—42no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—84no—
392C2×C3×C4#12011yescharacteristic
393C2×C3×C4#12011yescharacteristic
394C2×C3×C4#12011yescharacteristic
395C2×C3×C4#12011yescharacteristic
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4H(5,𝔽2)11yescharacteristic
401C2×C3×C42−1+411yescharacteristic
402C3×C8—42no—
403C3×C8—42no—
404#41—22no—
405#41—22no—
406#41—22no—
407#41—22no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8C2×C3×D412yes—
422C2×Q8C3×#4112yes—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4C2×C3×D412yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4C2×C3×D412yes—
434C22×C4C2×C3×D412yes—
435C22×C4C2×C3×D411yescharacteristic
436C22×C4C2×C3×D411yescharacteristic
437C22×C4C24×C311yescharacteristic
438Q16—44no—
439SD16—44no—
440M16—22no—
441M16—22no—
442M16—22no—
443M16—22no—
444C2×C8—22no—
445C2×C8—22no—
446C4⋊C4—24no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4C2×C3×D412yes—
465C4⋊C4C2×C3×D412yes—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C42—24no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—41no—
480C42—41no—
481C42C2×C3×D411yescharacteristic
482C42C2×C3×D411yescharacteristic
483C42C2×C3×D411yescharacteristic
484C42C2×C3×D411yescharacteristic
485C42C3×#4112yes—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—42no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3C2×#12011yescharacteristic
492C22×C3#52511yescharacteristic
493C22×C3#53711yescharacteristic
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4—42no—
501C3×C4—42no—
502C3×C4—42no—
503C3×C4—42no—
504C3×C4—42no—
505C3×C4—81no—
506C3×C4—81no—
507C3×C4#54611yescharacteristic
508C3×C4#54711yescharacteristic
509C3×C4#54811yescharacteristic
510C3×C4#55011yescharacteristic
511C23—22no—
512C23C22×C3×D411yescharacteristic
513Q8—22no—
514Q8—22no—
515Q8—22no—
516Q8—22no—
517Q8—42no—
518Q8—41no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522Q8—41no—
523Q8—41no—
524Q8—81no—
525D4—22no—
526D4—22no—
527D4—42no—
528D4—42no—
529D4—42no—
530D4—81no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—84no—
555C2×C4C3×#12011yescharacteristic
556C2×C4C3×#12011yescharacteristic
557C2×C4C3×#12011yescharacteristic
558C2×C4C3×#12011yescharacteristic
559C2×C4C22×C3×D411yescharacteristic
560C2×C4C22×C3×D411yescharacteristic
561C2×C4C2×C3×#4111yescharacteristic
562C2×C4C2×C3×#4111yescharacteristic
563C2×C4C3×H(5,𝔽2)11yescharacteristic
564C2×C4C3×2−1+411yescharacteristic
565C8—42no—
566C8—42no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—82no—
570C2×C3#240711yescharacteristic
571C2×C3#302011yescharacteristic
572C2×C3#302211yescharacteristic
573C22—21no—
574C22—21no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22C2×C3×#12011yescharacteristic
579C22C3×#52511yescharacteristic
580C22C3×#53711yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—81no—
593C4—81no—
594C4C3×#54611yescharacteristic
595C4C3×#54711yescharacteristic
596C4C3×#54811yescharacteristic
597C4C3×#55011yescharacteristic
598C3#2418311yescharacteristic, direct product factor, Sylow
599C2—21no—
600C2—21no—
601C2—82no—
602C2C3×#240711yescharacteristic
603C2C3×#302011yescharacteristic
604C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#532C22
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
4. C3×#2327C2semidirect
5. C3×#2038C2semidirect
6. C3×#2005C2semidirect
7. C3×#1995C2semidirect
9. C3×#1882C2semidirect
10. C3×#1860C2semidirect
11. C3×#1801C2semidirect
12. #24183C3direct (N × H)
53. #3078C2×C3semidirect
55. #2327C2×C3semidirect
56. #2038C2×C3semidirect
57. #2005C2×C3semidirect
58. #1995C2×C3semidirect
60. #1882C2×C3semidirect
61. #1860C2×C3semidirect
62. #1801C2×C3semidirect
598. C3#24183direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35072
C3×#30782
C3×#30372
C3×#23272
C3×#20382
C3×#20052
C3×#19952
C3×#19632
C3×#18822
C3×#18602
C3×#18012
#241833