Finite Group Databasebeta

← C3×#60444 #10679051

Subgroups — C3×#60444 #10679051

Order 768.

2918 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#575C211yesmaximal, characteristic, complemented
2C23×C3×M16C211yesmaximal, characteristic, complemented
3C2×C3×#525C212yesmaximal, complemented
4C3×#3056C218yesmaximal, complemented
5C3×#3055C218yesmaximal, complemented
6C3×#3054C214yesmaximal, complemented
7C3×#3050C214yesmaximal, complemented
8C3×#2911C211yesmaximal, characteristic, complemented
9C3×#2896C212yesmaximal, complemented
10#60444C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2212yes—
12C24×C3×C4C2211yescharacteristic
13C3×#575—28no—
14C2×C3×#137C2214yescomplemented
15C2×C3×AGL(1,ℤ/8)C2214yescomplemented
16C2×C3×#135C2214yescomplemented
17C22×C3×M16—28no—
18C22×C3×M16C2211yescharacteristic
19C22×C3×M16C2211yescharacteristic
20C22×C3×M16C2212yescomplemented
21C22×C3×M16C2218yescomplemented
22C23×C3×C8C2212yescomplemented
23C3×#525—22no—
24C3×#525—22no—
25C3×#525—24no—
26C3×#525C2218yes—
27C3×#525C2214yescomplemented
28C3×#525C2214yescomplemented
29C2×C3×#123—22no—
30C2×C3×C22⋊Q8C2214yescomplemented
31C2×C3×#121C2214yescomplemented
32C2×C3×#120—22no—
33C2×C3×C4×D4—24no—
34C2×C3×#117C2212yes—
35C3×#470C2218yescomplemented
36C3×#469C2218yescomplemented
37C3×#468C22116yescomplemented
38C3×#467C2218yescomplemented
39C3×#466C2218yescomplemented
40C3×#465C22116yescomplemented
41C3×#428C2218yescomplemented
42C3×#427C2214yescomplemented
43C3×#419C2214yescomplemented
44C3×#418C2214yescomplemented
45C3×#417C2218yescomplemented
46C3×#416C2218yescomplemented
47C2×#575C2×C311yescharacteristic, complemented
48C23×M16C2×C311yescharacteristic, complemented
49C2×#525C2×C312yescomplemented
50#3056C2×C318yescomplemented
51#3055C2×C318yescomplemented
52#3054C2×C314yescomplemented
53#3050C2×C314yescomplemented
54#2911C2×C311yescharacteristic, complemented
55#2896C2×C312yescomplemented
56C25×C3D411yescharacteristic
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41—28no—
60C2×C3×#41—28no—
61C2×C3×#41C2314yes—
62C2×C3×#41C2314yes—
63C22×C3×Q8C2312yes—
64C22×C3×D4—22no—
65C22×C3×D4—24no—
66C22×C3×D4C2312yes—
67C23×C3×C4—24no—
68C23×C3×C4—24no—
69C23×C3×C4—22no—
70C23×C3×C4D414yes—
71C23×C3×C4D411yescharacteristic
72C23×C3×C4D411yescharacteristic
73C23×C3×C4D411yescharacteristic
74C23×C3×C4C2311yescharacteristic
75C23×C3×C4C2314yes—
76C23×C3×C4C2311yescharacteristic
77C23×C3×C4C2311yescharacteristic
78C3×#137—216no—
79C3×AGL(1,ℤ/8)—216no—
80C3×#135—216no—
81C2×C3×Q16C2314yescomplemented
82C2×C3×SD16C2318yescomplemented
83C2×C3×D8C2314yescomplemented
84C2×C3×M16—24no—
85C2×C3×M16—232no—
86C2×C3×M16—28no—
87C2×C3×M16D418yescomplemented
88C2×C3×M16C2314yes—
89C2×C3×M16C2318yes—
90C2×C3×M16C2314yes—
91C22×C3×C8—28no—
92C22×C3×C8C2312yes—
93C22×C3×C8C2318yes—
94C22×C3×C8C2312yes—
95C3×#123—28no—
96C3×#123—24no—
97C3×#123—24no—
98C3×C22⋊Q8—24no—
99C3×C22⋊Q8—24no—
100C3×C22⋊Q8C2318yes—
101C3×C22⋊Q8C2318yes—
102C3×#121—24no—
103C3×#121—24no—
104C3×#121C2318yes—
105C3×#121C2318yes—
106C3×#120—28no—
107C3×#120—24no—
108C3×#120—24no—
109C3×C4×D4—28no—
110C3×C4×D4—28no—
111C3×C4×D4—28no—
112C3×C4×D4—28no—
113C3×#117—22no—
114C3×#117—22no—
115C3×#117C2314yes—
116C3×#117C2314yes—
117C2×C3×C4⋊C4—24no—
118C2×C3×C4⋊C4C2314yes—
119C2×C3×C22⋊C4—22no—
120C2×C3×C22⋊C4—24no—
121C2×C3×C22⋊C4—24no—
122C2×C3×C42—22no—
123C3×#107C2318yescomplemented
124C3×#106C2318yescomplemented
125C3×#103C23116yescomplemented
126C3×#102C23116yescomplemented
127C22×#41C22×C312yes—
128C24×C4C22×C311yescharacteristic
129#575—28no—
130C2×#137C22×C314yescomplemented
131C2×AGL(1,ℤ/8)C22×C314yescomplemented
132C2×#135C22×C314yescomplemented
133C22×M16—28no—
134C22×M16C22×C311yescharacteristic
135C22×M16C22×C311yescharacteristic
136C22×M16C22×C312yescomplemented
137C22×M16C22×C318yescomplemented
138C23×C8C22×C312yescomplemented
139#525—22no—
140#525—22no—
141#525—24no—
142#525C22×C318yes—
143#525C22×C314yescomplemented
144#525C22×C314yescomplemented
145C2×#123—22no—
146C2×C22⋊Q8C22×C314yescomplemented
147C2×#121C22×C314yescomplemented
148C2×#120—22no—
149C2×C4×D4—24no—
150C2×#117C22×C312yes—
151#470C22×C318yescomplemented
152#469C22×C318yescomplemented
153#468C22×C3116yescomplemented
154#467C22×C318yescomplemented
155#466C22×C318yescomplemented
156#465C22×C3116yescomplemented
157#428C22×C318yescomplemented
158#427C22×C314yescomplemented
159#419C22×C314yescomplemented
160#418C22×C314yescomplemented
161#417C22×C318yescomplemented
162#416C22×C318yescomplemented
163C24×C3—24no—
164C24×C3—22no—
165C24×C3—44no—
166C24×C3C2×D411yescharacteristic
167C24×C3C2×D414yes—
168C24×C3C2×D411yescharacteristic
169C24×C3C2×D411yescharacteristic
170C3×#41—28no—
171C3×#41—28no—
172C3×#41—28no—
173C3×#41—28no—
174C3×#41—48no—
175C3×#41—48no—
176C2×C3×Q8—22no—
177C2×C3×Q8—24no—
178C2×C3×Q8—24no—
179C2×C3×Q8C2414yes—
180C2×C3×D4—28no—
181C2×C3×D4—28no—
182C2×C3×D4—24no—
183C2×C3×D4—24no—
184C2×C3×D4—24no—
185C2×C3×D4—22no—
186C2×C3×D4—24no—
187C2×C3×D4—416no—
188C2×C3×D4—48no—
189C2×C3×D4C2414yes—
190C22×C3×C4—24no—
191C22×C3×C4—216no—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4—22no—
195C22×C3×C4—28no—
196C22×C3×C4—22no—
197C22×C3×C4—28no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4—24no—
201C22×C3×C4—22no—
202C22×C3×C4—24no—
203C22×C3×C4—24no—
204C22×C3×C4—48no—
205C22×C3×C4C2×D414yes—
206C22×C3×C4C2×D414yes—
207C22×C3×C4C2×D414yes—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D412yes—
211C22×C3×C4C2×D412yes—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2×D414yes—
215C22×C3×C4#4114yes—
216C22×C3×C4C2412yes—
217C22×C3×C4C2412yes—
218C22×C3×C4C2412yes—
219C22×C3×C4C2411yescharacteristic
220C3×Q16—28no—
221C3×SD16—216no—
222C3×D8—28no—
223C3×M16—28no—
224C3×M16—216no—
225C3×M16C2×D4116yescomplemented
226C2×C3×C8—24no—
227C2×C3×C8—216no—
228C2×C3×C8C2×D418yescomplemented
229C2×C3×C8C2414yes—
230C2×C3×C8C2414yes—
231C3×C4⋊C4—28no—
232C3×C4⋊C4—28no—
233C3×C4⋊C4—24no—
234C3×C4⋊C4C2418yes—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—24no—
237C3×C22⋊C4—28no—
238C3×C22⋊C4—28no—
239C3×C22⋊C4—28no—
240C3×C22⋊C4—28no—
241C3×C42—24no—
242C3×C42—24no—
243C25C3×D411yescharacteristic
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—28no—
247C2×#41—28no—
248C2×#41C23×C314yes—
249C2×#41C23×C314yes—
250C22×Q8C23×C312yes—
251C22×D4—24no—
252C22×D4—22no—
253C22×D4C23×C312yes—
254C23×C4—24no—
255C23×C4—24no—
256C23×C4—22no—
257C23×C4C3×D414yes—
258C23×C4C3×D411yescharacteristic
259C23×C4C3×D411yescharacteristic
260C23×C4C3×D411yescharacteristic
261C23×C4C23×C314yes—
262C23×C4C23×C311yescharacteristic
263C23×C4C23×C311yescharacteristic
264C23×C4C23×C311yescharacteristic
265#137—216no—
266AGL(1,ℤ/8)—216no—
267#135—216no—
268C2×Q16C23×C314yescomplemented
269C2×SD16C23×C318yescomplemented
270C2×D8C23×C314yescomplemented
271C2×M16—24no—
272C2×M16—232no—
273C2×M16—28no—
274C2×M16C3×D418yescomplemented
275C2×M16C23×C314yes—
276C2×M16C23×C318yes—
277C2×M16C23×C314yes—
278C22×C8—28no—
279C22×C8C23×C312yes—
280C22×C8C23×C318yes—
281C22×C8C23×C312yes—
282#123—28no—
283#123—24no—
284#123—24no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8C23×C318yes—
288C22⋊Q8C23×C318yes—
289#121—24no—
290#121—24no—
291#121C23×C318yes—
292#121C23×C318yes—
293#120—28no—
294#120—24no—
295#120—24no—
296C4×D4—28no—
297C4×D4—28no—
298C4×D4—28no—
299C4×D4—28no—
300#117—22no—
301#117—22no—
302#117C23×C314yes—
303#117C23×C314yes—
304C2×C4⋊C4—24no—
305C2×C4⋊C4C23×C314yes—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C42—22no—
310#107C23×C318yescomplemented
311#106C23×C318yescomplemented
312#103C23×C3116yescomplemented
313#102C23×C3116yescomplemented
314C23×C3—22no—
315C23×C3—24no—
316C23×C3—28no—
317C23×C3—22no—
318C23×C3—28no—
319C23×C3—24no—
320C23×C3—42no—
321C23×C3—44no—
322C23×C3—42no—
323C23×C3—416no—
324C23×C3—48no—
325C23×C3—44no—
326C23×C3#12114yes—
327C23×C3C22×D412yes—
328C23×C3C22×D411yescharacteristic
329C23×C3C22×D412yes—
330C23×C3C22×D412yes—
331C3×Q8—24no—
332C3×Q8—24no—
333C3×Q8—44no—
334C3×D4—24no—
335C3×D4—24no—
336C3×D4—416no—
337C3×D4—48no—
338C3×D4—48no—
339C3×D4—416no—
340C3×D4—44no—
341C2×C3×C4—28no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—28no—
347C2×C3×C4—28no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—28no—
353C2×C3×C4—24no—
354C2×C3×C4—48no—
355C2×C3×C4—44no—
356C2×C3×C4—48no—
357C2×C3×C4—48no—
358C2×C3×C4#12118yes—
359C2×C3×C4#12114yes—
360C2×C3×C4C22×D412yes—
361C2×C3×C4C22×D412yes—
362C2×C3×C4C22×D412yes—
363C2×C3×C4C22×D412yes—
364C2×C3×C4C22×D414yes—
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C2×#4114yes—
367C2×C3×C4C2×#4112yes—
368C2×C3×C4C2511yescharacteristic
369C3×C8—24no—
370C3×C8C22×D418yes—
371C24—24no—
372C24—22no—
373C24—44no—
374C24C2×C3×D411yescharacteristic
375C24C2×C3×D414yes—
376C24C2×C3×D411yescharacteristic
377C24C2×C3×D411yescharacteristic
378#41—28no—
379#41—28no—
380#41—28no—
381#41—28no—
382#41—48no—
383#41—48no—
384C2×Q8—22no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8C24×C314yes—
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—24no—
395C2×D4—416no—
396C2×D4—48no—
397C2×D4C24×C314yes—
398C22×C4—24no—
399C22×C4—216no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—48no—
413C22×C4C2×C3×D414yes—
414C22×C4C2×C3×D412yes—
415C22×C4C2×C3×D412yes—
416C22×C4C2×C3×D414yes—
417C22×C4C2×C3×D414yes—
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C2×C3×D412yes—
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C2×C3×D414yes—
423C22×C4C3×#4114yes—
424C22×C4C24×C312yes—
425C22×C4C24×C312yes—
426C22×C4C24×C312yes—
427C22×C4C24×C311yescharacteristic
428Q16—28no—
429SD16—216no—
430D8—28no—
431M16—28no—
432M16—216no—
433M16C2×C3×D4116yescomplemented
434C2×C8—24no—
435C2×C8—216no—
436C2×C8C2×C3×D418yescomplemented
437C2×C8C24×C314yes—
438C2×C8C24×C314yes—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—24no—
442C4⋊C4C24×C318yes—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—28no—
446C22⋊C4—28no—
447C22⋊C4—28no—
448C22⋊C4—28no—
449C42—24no—
450C42—24no—
451C22×C3—28no—
452C22×C3—22no—
453C22×C3—24no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—24no—
458C22×C3—44no—
459C22×C3—416no—
460C22×C3—44no—
461C22×C3—44no—
462C22×C3—48no—
463C22×C3—88no—
464C22×C3C2×#12114yes—
465C22×C3C2×#12112yes—
466C22×C3#57514yescomplemented
467C22×C3C23×D411yescharacteristic
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—44no—
471C3×C4—48no—
472C3×C4C2×#12112yes—
473C3×C4C2×#12114yes—
474C3×C4C23×D411yescharacteristic
475C3×C4C22×#4111yescharacteristic
476C23—24no—
477C23—22no—
478C23—28no—
479C23—28no—
480C23—22no—
481C23—24no—
482C23—42no—
483C23—44no—
484C23—42no—
485C23—416no—
486C23—48no—
487C23—44no—
488C23C3×#12114yes—
489C23C22×C3×D412yes—
490C23C22×C3×D412yes—
491C23C22×C3×D411yescharacteristic
492C23C22×C3×D412yes—
493Q8—24no—
494Q8—24no—
495Q8—44no—
496D4—24no—
497D4—24no—
498D4—416no—
499D4—416no—
500D4—48no—
501D4—48no—
502D4—44no—
503C2×C4—28no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—48no—
517C2×C4—44no—
518C2×C4—48no—
519C2×C4—48no—
520C2×C4C3×#12118yes—
521C2×C4C3×#12114yes—
522C2×C4C22×C3×D412yes—
523C2×C4C22×C3×D412yes—
524C2×C4C22×C3×D412yes—
525C2×C4C22×C3×D414yes—
526C2×C4C22×C3×D411yescharacteristic
527C2×C4C22×C3×D412yes—
528C2×C4C2×C3×#4114yes—
529C2×C4C2×C3×#4112yes—
530C2×C4C25×C311yescharacteristic
531C8—24no—
532C8C22×C3×D418yes—
533C2×C3—22no—
534C2×C3—24no—
535C2×C3—24no—
536C2×C3—42no—
537C2×C3—84no—
538C2×C3C22×#12111yescharacteristic
539C2×C3C2×#57512yes—
540C22—28no—
541C22—22no—
542C22—24no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—24no—
547C22—44no—
548C22—416no—
549C22—44no—
550C22—44no—
551C22—48no—
552C22—88no—
553C22C2×C3×#12114yes—
554C22C2×C3×#12112yes—
555C22C3×#57514yescomplemented
556C22C23×C3×D411yescharacteristic
557C4—22no—
558C4—22no—
559C4—44no—
560C4—48no—
561C4C2×C3×#12112yes—
562C4C2×C3×#12114yes—
563C4C23×C3×D411yescharacteristic
564C4C22×C3×#4111yescharacteristic
565C3#6044411yescharacteristic, direct product factor, Sylow
566C2—24no—
567C2—22no—
568C2—24no—
569C2—42no—
570C2—84no—
571C2C22×C3×#12111yescharacteristic
572C2C2×C3×#57512yes—
subgroupG/H
centerC2×C3×C4C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
socleC22×C3C23×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#575C2semidirect
2. C23×C3×M16C2semidirect
3. C2×C3×#525C2semidirect
4. C3×#3056C2semidirect
5. C3×#3055C2semidirect
6. C3×#3054C2semidirect
7. C3×#3050C2semidirect
8. C3×#2911C2semidirect
9. C3×#2896C2semidirect
10. #60444C3direct (N × H)
14. C2×C3×#137C22semidirect
15. C2×C3×AGL(1,ℤ/8)C22semidirect
16. C2×C3×#135C22semidirect
20. C22×C3×M16C22semidirect
21. C22×C3×M16C22semidirect
22. C23×C3×C8C22semidirect
27. C3×#525C22semidirect
28. C3×#525C22semidirect
30. C2×C3×C22⋊Q8C22semidirect
31. C2×C3×#121C22semidirect
35. C3×#470C22semidirect
36. C3×#469C22semidirect
37. C3×#468C22semidirect
38. C3×#467C22semidirect
39. C3×#466C22semidirect
40. C3×#465C22semidirect
41. C3×#428C22semidirect
42. C3×#427C22semidirect
43. C3×#419C22semidirect
44. C3×#418C22semidirect
45. C3×#417C22semidirect
46. C3×#416C22semidirect
47. C2×#575C2×C3semidirect
48. C23×M16C2×C3semidirect
49. C2×#525C2×C3semidirect
50. #3056C2×C3semidirect
51. #3055C2×C3semidirect
52. #3054C2×C3semidirect
53. #3050C2×C3semidirect
54. #2911C2×C3semidirect
55. #2896C2×C3semidirect
81. C2×C3×Q16C23semidirect
82. C2×C3×SD16C23semidirect
83. C2×C3×D8C23semidirect
87. C2×C3×M16D4semidirect
123. C3×#107C23semidirect
124. C3×#106C23semidirect
125. C3×#103C23semidirect
126. C3×#102C23semidirect
130. C2×#137C22×C3semidirect
131. C2×AGL(1,ℤ/8)C22×C3semidirect
132. C2×#135C22×C3semidirect
136. C22×M16C22×C3semidirect
137. C22×M16C22×C3semidirect
138. C23×C8C22×C3semidirect
143. #525C22×C3semidirect
144. #525C22×C3semidirect
146. C2×C22⋊Q8C22×C3semidirect
147. C2×#121C22×C3semidirect
151. #470C22×C3semidirect
152. #469C22×C3semidirect
153. #468C22×C3semidirect
154. #467C22×C3semidirect
155. #466C22×C3semidirect
156. #465C22×C3semidirect
157. #428C22×C3semidirect
158. #427C22×C3semidirect
159. #419C22×C3semidirect
160. #418C22×C3semidirect
161. #417C22×C3semidirect
162. #416C22×C3semidirect
225. C3×M16C2×D4semidirect
228. C2×C3×C8C2×D4semidirect
268. C2×Q16C23×C3semidirect
269. C2×SD16C23×C3semidirect
270. C2×D8C23×C3semidirect
274. C2×M16C3×D4semidirect
310. #107C23×C3semidirect
311. #106C23×C3semidirect
312. #103C23×C3semidirect
313. #102C23×C3semidirect
433. M16C2×C3×D4semidirect
436. C2×C8C2×C3×D4semidirect
466. C22×C3#575semidirect
555. C22C3×#575semidirect
565. C3#60444direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5752
C23×C3×M162
C2×C3×#5252
C3×#30562
C3×#30552
C3×#30542
C3×#30502
C3×#29112
C3×#28962
#604443