Finite Group Databasebeta

← C3×#18674 #10637281

Subgroups — C3×#18674 #10637281

Order 768.

1176 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3497C211yesmaximal, characteristic, complemented
2C3×#3245C211yesmaximal, characteristic, complemented
3C3×#3169C211yesmaximal, characteristic, complemented
4C3×#3037C211yesmaximal, characteristic, complemented
5C3×#1649C212yesmaximal, complemented
6C3×#1641C212yesmaximal, complemented
7C3×#1627C212yesmaximal, complemented
8C3×#1621C212yesmaximal, complemented
9C3×#1547C211yesmaximal, characteristic, complemented
10C3×#1492C211yesmaximal, characteristic, complemented
11C2×C3×Q8⋊C8C211yesmaximal, characteristic, complemented
12#18674C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×2−1+4—21no—
14C2×C3×#137—21no—
15C3×#550—21no—
16C3×#550C2212yes—
17C3×D4×Q8—21no—
18C3×D4×Q8C2212yes—
19C3×#548—22no—
20C3×#548—22no—
21C3×#537—22no—
22C3×#537—21no—
23C3×#532—22no—
24C3×#532C2211yescharacteristic
25C3×#532C2211yescharacteristic
26C3×#532C2211yescharacteristic
27C2×C3×C4×Q8C2211yescharacteristic
28C3×#500C2211yescharacteristic
29C3×#493C2211yescharacteristic
30C3×#486C2211yescharacteristic
31C3×#486C2211yescharacteristic
32C3×#474—21no—
33C3×#474C2212yes—
34C3×#466—22no—
35C3×#466—22no—
36C3×#464—22no—
37C3×#462C2212yes—
38C3×#460C2212yes—
39C3×#459—21no—
40C3×#459C2212yes—
41C3×C4×D8—22no—
42C3×#423C2211yescharacteristic
43C2×C3×C4⋊C8C2211yescharacteristic
44C3×#417—21no—
45C2×C3×C4×C8C2211yescharacteristic
46C3×#332C2212yes—
47C3×#331C2212yes—
48C3×Q8⋊C8C2212yes—
49C3×Q8⋊C8C2212yes—
50C3×D4⋊C8C2212yes—
51C3×D4⋊C8C2212yes—
52#3497C2×C311yescharacteristic, complemented
53#3245C2×C311yescharacteristic, complemented
54#3169C2×C311yescharacteristic, complemented
55#3037C2×C311yescharacteristic, complemented
56#1649C2×C312yescomplemented
57#1641C2×C312yescomplemented
58#1627C2×C312yescomplemented
59#1621C2×C312yescomplemented
60#1547C2×C311yescharacteristic, complemented
61#1492C2×C311yescharacteristic, complemented
62C2×Q8⋊C8C2×C311yescharacteristic, complemented
63C3×2−1+4—44no—
64C3×2−1+4—44no—
65C2×C3×#41—24no—
66C2×C3×#41—22no—
67C2×C3×#41—22no—
68C2×C3×#41—21no—
69C2×C3×#41—21no—
70C2×C3×#41—21no—
71C22×C3×Q8—22no—
72C22×C3×Q8—22no—
73C22×C3×Q8D411yescharacteristic
74C3×#137—44no—
75C2×C3×Q16—22no—
76C2×C3×SD16—22no—
77C2×C3×D8—22no—
78C2×C3×D8—22no—
79C2×C3×M16—21no—
80C22×C3×C8—21no—
81C22×C3×C8—21no—
82C3×C4⋊Q8—22no—
83C3×C4⋊Q8D411yescharacteristic
84C3×C4⋊Q8C2311yescharacteristic
85C3×C4⋊Q8C2311yescharacteristic
86C3×#127—22no—
87C3×#127D411yescharacteristic
88C3×#127C2311yescharacteristic
89C3×#127C2311yescharacteristic
90C3×#124—24no—
91C3×#124—21no—
92C3×#124—21no—
93C3×#124—21no—
94C3×C22⋊Q8—22no—
95C3×C22⋊Q8—24no—
96C3×C22⋊Q8—22no—
97C3×C22⋊Q8—24no—
98C3×#121—24no—
99C3×#121—22no—
100C3×#121—24no—
101C3×#121—24no—
102C3×#121—21no—
103C3×#121—21no—
104C3×#121—22no—
105C3×#121—21no—
106C3×#121—21no—
107C3×#121D412yes—
108C3×#121D412yes—
109C3×C4×Q8—21no—
110C3×C4×Q8—21no—
111C3×C4×Q8D412yes—
112C3×C4×Q8C2312yes—
113C3×C4×D4—24no—
114C3×C4×D4—22no—
115C3×C4×D4—22no—
116C3×C4×D4—21no—
117C3×C4×D4—21no—
118C3×C4×D4—21no—
119C3×C4×D4—21no—
120C3×C4×D4D412yes—
121C3×C4×D4C2312yes—
122C2×C3×C4⋊C4—21no—
123C2×C3×C4⋊C4D411yescharacteristic
124C2×C3×C42—21no—
125C2×C3×C42C2311yescharacteristic
126C3×#107—22no—
127C3×#107—22no—
128C3×C4⋊C8—21no—
129C3×C4⋊C8—21no—
130C3×C4⋊C8C2312yes—
131C3×C4⋊C8C2312yes—
132C3×#103—22no—
133C3×#102—22no—
134C3×#102—22no—
135C3×#102—22no—
136C3×#102—22no—
137C3×#102—22no—
138C3×C4×C8—21no—
139C3×C4×C8C2311yescharacteristic
140C3×C4×C8C2311yescharacteristic
141C2×2−1+4—21no—
142C2×#137—21no—
143#550—21no—
144#550C22×C312yes—
145D4×Q8—21no—
146D4×Q8C22×C312yes—
147#548—22no—
148#548—22no—
149#537—22no—
150#537—21no—
151#532—22no—
152#532C22×C311yescharacteristic
153#532C22×C311yescharacteristic
154#532C22×C311yescharacteristic
155C2×C4×Q8C22×C311yescharacteristic
156#500C22×C311yescharacteristic
157#493C22×C311yescharacteristic
158#486C22×C311yescharacteristic
159#486C22×C311yescharacteristic
160#474—21no—
161#474C22×C312yes—
162#466—22no—
163#466—22no—
164#464—22no—
165#462C22×C312yes—
166#460C22×C312yes—
167#459—21no—
168#459C22×C312yes—
169C4×D8—22no—
170#423C22×C311yescharacteristic
171C2×C4⋊C8C22×C311yescharacteristic
172#417—21no—
173C2×C4×C8C22×C311yescharacteristic
174#332C22×C312yes—
175#331C22×C312yes—
176Q8⋊C8C22×C312yes—
177Q8⋊C8C22×C312yes—
178D4⋊C8C22×C312yes—
179D4⋊C8C22×C312yes—
180C3×#41—416no—
181C3×#41—44no—
182C3×#41—48no—
183C3×#41—44no—
184C3×#41—42no—
185C3×#41—42no—
186C3×#41—42no—
187C3×#41—42no—
188C3×#41—42no—
189C3×#41—42no—
190C2×C3×Q8—24no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—22no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—44no—
197C2×C3×Q8—48no—
198C2×C3×Q8—44no—
199C2×C3×Q8D814yescomplemented
200C2×C3×Q8C2×D412yes—
201C2×C3×D4—24no—
202C2×C3×D4—22no—
203C2×C3×D4—22no—
204C2×C3×D4—24no—
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—21no—
212C2×C3×D4—41no—
213C2×C3×D4C2×D412yes—
214C2×C3×D4C2×D412yes—
215C22×C3×C4—21no—
216C22×C3×C4—24no—
217C22×C3×C4—24no—
218C22×C3×C4—21no—
219C22×C3×C4—22no—
220C22×C3×C4—22no—
221C22×C3×C4—41no—
222C22×C3×C4C2×D411yescharacteristic
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D411yescharacteristic
225C3×Q16—44no—
226C3×SD16—44no—
227C3×D8—44no—
228C3×D8—82no—
229C3×M16—42no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—22no—
233C2×C3×C8—22no—
234C2×C3×C8—42no—
235C2×C3×C8—81no—
236C3×C4⋊C4—24no—
237C3×C4⋊C4—22no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—22no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4C2×D412yes—
245C3×C4⋊C4C2×D412yes—
246C3×C4⋊C4C2×D412yes—
247C3×C22⋊C4—24no—
248C3×C22⋊C4—24no—
249C3×C22⋊C4—22no—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4—41no—
254C3×C22⋊C4—41no—
255C3×C42—22no—
256C3×C42—22no—
257C3×C42C2×D411yescharacteristic
258C3×C42C2×D411yescharacteristic
259C3×C42C2×D411yescharacteristic
260C3×C42C2411yescharacteristic
2612−1+4—44no—
2622−1+4—44no—
263C2×#41—24no—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41—21no—
267C2×#41—21no—
268C2×#41—21no—
269C22×Q8—22no—
270C22×Q8—22no—
271C22×Q8C3×D411yescharacteristic
272#137—44no—
273C2×Q16—22no—
274C2×SD16—22no—
275C2×D8—22no—
276C2×D8—22no—
277C2×M16—21no—
278C22×C8—21no—
279C22×C8—21no—
280C4⋊Q8—22no—
281C4⋊Q8C3×D411yescharacteristic
282C4⋊Q8C23×C311yescharacteristic
283C4⋊Q8C23×C311yescharacteristic
284#127—22no—
285#127C3×D411yescharacteristic
286#127C23×C311yescharacteristic
287#127C23×C311yescharacteristic
288#124—24no—
289#124—21no—
290#124—21no—
291#124—21no—
292C22⋊Q8—22no—
293C22⋊Q8—24no—
294C22⋊Q8—22no—
295C22⋊Q8—24no—
296#121—24no—
297#121—22no—
298#121—24no—
299#121—24no—
300#121—21no—
301#121—21no—
302#121—22no—
303#121—21no—
304#121—21no—
305#121C3×D412yes—
306#121C3×D412yes—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8C3×D412yes—
310C4×Q8C23×C312yes—
311C4×D4—24no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4C3×D412yes—
319C4×D4C23×C312yes—
320C2×C4⋊C4—21no—
321C2×C4⋊C4C3×D411yescharacteristic
322C2×C42—21no—
323C2×C42C23×C311yescharacteristic
324#107—22no—
325#107—22no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8C23×C312yes—
329C4⋊C8C23×C312yes—
330#103—22no—
331#102—22no—
332#102—22no—
333#102—22no—
334#102—22no—
335#102—22no—
336C4×C8—21no—
337C4×C8C23×C311yescharacteristic
338C4×C8C23×C311yescharacteristic
339C23×C3—22no—
340C23×C3—22no—
341C23×C3—41no—
342C23×C3#12011yescharacteristic
343C3×Q8—22no—
344C3×Q8—24no—
345C3×Q8—48no—
346C3×Q8—41no—
347C3×Q8—41no—
348C3×Q8—42no—
349C3×Q8—41no—
350C3×Q8—41no—
351C3×Q8C2×D814yes—
352C3×D4—22no—
353C3×D4—22no—
354C3×D4—42no—
355C3×D4—48no—
356C3×D4—42no—
357C3×D4—44no—
358C3×D4—44no—
359C3×D4—41no—
360C3×D4—41no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—41no—
364C3×D4—41no—
365C3×D4—41no—
366C3×D4—41no—
367C3×D4—42no—
368C3×D4—81no—
369C3×D4—84no—
370C3×D4—82no—
371C2×C3×C4—22no—
372C2×C3×C4—24no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—41no—
379C2×C3×C4—48no—
380C2×C3×C4—48no—
381C2×C3×C4—42no—
382C2×C3×C4—42no—
383C2×C3×C4—44no—
384C2×C3×C4—41no—
385C2×C3×C4—81no—
386C2×C3×C4—82no—
387C2×C3×C4#12011yescharacteristic
388C2×C3×C4#12011yescharacteristic
389C2×C3×C4#12011yescharacteristic
390C2×C3×C4C2×D812yes—
391C2×C3×C4AGL(1,ℤ/8)12yes—
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C22×D411yescharacteristic
395C3×C8—22no—
396C3×C8—41no—
397C3×C8—42no—
398C3×C8—42no—
399#41—416no—
400#41—44no—
401#41—48no—
402#41—44no—
403#41—42no—
404#41—42no—
405#41—42no—
406#41—42no—
407#41—42no—
408#41—42no—
409C2×Q8—24no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—44no—
416C2×Q8—48no—
417C2×Q8—44no—
418C2×Q8C3×D814yescomplemented
419C2×Q8C2×C3×D412yes—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—24no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—41no—
432C2×D4C2×C3×D412yes—
433C2×D4C2×C3×D412yes—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—41no—
441C22×C4C2×C3×D411yescharacteristic
442C22×C4C2×C3×D411yescharacteristic
443C22×C4C2×C3×D411yescharacteristic
444Q16—44no—
445SD16—44no—
446D8—44no—
447D8—82no—
448M16—42no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—42no—
454C2×C8—81no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4C2×C3×D412yes—
464C4⋊C4C2×C3×D412yes—
465C4⋊C4C2×C3×D412yes—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C42—22no—
475C42—22no—
476C42C2×C3×D411yescharacteristic
477C42C2×C3×D411yescharacteristic
478C42C2×C3×D411yescharacteristic
479C42C24×C311yescharacteristic
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3#44712yes—
492C22×C3C2×#12011yescharacteristic
493C3×C4—24no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—42no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4—81no—
501C3×C4#44712yes—
502C3×C4C22×D811yescharacteristic
503C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
504C3×C4#57711yescharacteristic
505C3×C4#57811yescharacteristic
506C23—22no—
507C23—22no—
508C23—41no—
509C23C3×#12011yescharacteristic
510Q8—22no—
511Q8—24no—
512Q8—48no—
513Q8—41no—
514Q8—41no—
515Q8—42no—
516Q8—41no—
517Q8—41no—
518Q8C2×C3×D814yes—
519D4—22no—
520D4—22no—
521D4—42no—
522D4—48no—
523D4—42no—
524D4—44no—
525D4—44no—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—41no—
533D4—41no—
534D4—42no—
535D4—81no—
536D4—84no—
537D4—82no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—48no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4C3×#12011yescharacteristic
555C2×C4C3×#12011yescharacteristic
556C2×C4C3×#12011yescharacteristic
557C2×C4C2×C3×D812yes—
558C2×C4C3×AGL(1,ℤ/8)12yes—
559C2×C4C22×C3×D411yescharacteristic
560C2×C4C22×C3×D411yescharacteristic
561C2×C4C22×C3×D411yescharacteristic
562C8—22no—
563C8—41no—
564C8—42no—
565C8—42no—
566C2×C3—22no—
567C2×C3—82no—
568C2×C3—82no—
569C2×C3—161no—
570C2×C3C2×#44711yescharacteristic
571C2×C3#301311yescharacteristic
572C2×C3#301811yescharacteristic
573C22—21no—
574C22—21no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22C3×#44712yes—
585C22C2×C3×#12011yescharacteristic
586C4—24no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—81no—
594C4C3×#44712yes—
595C4C22×C3×D811yescharacteristic
596C4C2×C3×AGL(1,ℤ/8)11yescharacteristic
597C4C3×#57711yescharacteristic
598C4C3×#57811yescharacteristic
599C3#1867411yescharacteristic, direct product factor, Sylow
600C2—22no—
601C2—82no—
602C2—82no—
603C2—161no—
604C2C2×C3×#44711yescharacteristic
605C2C3×#301311yescharacteristic
606C2C3×#301811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#3497C2semidirect
2. C3×#3245C2semidirect
3. C3×#3169C2semidirect
4. C3×#3037C2semidirect
5. C3×#1649C2semidirect
6. C3×#1641C2semidirect
7. C3×#1627C2semidirect
8. C3×#1621C2semidirect
9. C3×#1547C2semidirect
10. C3×#1492C2semidirect
11. C2×C3×Q8⋊C8C2semidirect
12. #18674C3direct (N × H)
52. #3497C2×C3semidirect
53. #3245C2×C3semidirect
54. #3169C2×C3semidirect
55. #3037C2×C3semidirect
56. #1649C2×C3semidirect
57. #1641C2×C3semidirect
58. #1627C2×C3semidirect
59. #1621C2×C3semidirect
60. #1547C2×C3semidirect
61. #1492C2×C3semidirect
62. C2×Q8⋊C8C2×C3semidirect
199. C2×C3×Q8D8semidirect
418. C2×Q8C3×D8semidirect
599. C3#18674direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34972
C3×#32452
C3×#31692
C3×#30372
C3×#16492
C3×#16412
C3×#16272
C3×#16212
C3×#15472
C3×#14922
C2×C3×Q8⋊C82
#186743