Finite Group Databasebeta

← C3×#32935 #10651542

Subgroups — C3×#32935 #10651542

Order 768.

1580 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×M16C211yesmaximal, characteristic, complemented
2C2×C3×#468C212yesmaximal, complemented
3C2×C3×#465C211yesmaximal, characteristic, complemented
4C3×#2848C211yesmaximal, characteristic, complemented
5C3×#2847C211yesmaximal, characteristic
6C3×#1895C212yesmaximal, complemented
7C3×#1894C212yesmaximal, complemented
8C3×#1893C211yesmaximal, characteristic, complemented
9C3×#1892C211yesmaximal, characteristic, complemented
10C3×#1809C212yesmaximal, complemented
11C3×#1808C211yesmaximal, characteristic, complemented
12#32935C311yesmaximal, characteristic, direct product factor, Sylow
13C24×C3×C4C2211yescharacteristic
14C22×C3×SD16C2211yescharacteristic, complemented
15C22×C3×M16—28no—
16C22×C3×M16—24no—
17C22×C3×M16C2212yes—
18C22×C3×M16C2212yes—
19C23×C3×C8C2211yescharacteristic
20C23×C3×C8C2211yescharacteristic
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×C22⋊Q8C2212yes—
23C2×C3×#121C2211yescharacteristic
24C2×C3×#121C2212yes—
25C2×C3×#120—21no—
26C3×#468—28no—
27C3×#465—24no—
28C2×C3×#107C2212yescomplemented
29C2×C3×#106C2211yescharacteristic, complemented
30C2×C3×#103C2212yescomplemented
31C2×C3×#103C2211yescharacteristic, complemented
32C2×C3×#102C2212yescomplemented
33C2×C3×#102C2211yescharacteristic, complemented
34C3×#386—22no—
35C3×#386—21no—
36C3×#385—22no—
37C3×#385—21no—
38C3×#380C2211yescharacteristic
39C3×#380C2211yescharacteristic
40C3×#380C2212yes—
41C3×#380C2212yes—
42C3×#379—21no—
43C3×#340C2212yescomplemented
44C3×#340C2212yescomplemented
45C3×#340C2212yescomplemented
46C23×M16C2×C311yescharacteristic, complemented
47C2×#468C2×C312yescomplemented
48C2×#465C2×C311yescharacteristic, complemented
49#2848C2×C311yescharacteristic, complemented
50#2847C2×C311yescharacteristic
51#1895C2×C312yescomplemented
52#1894C2×C312yescomplemented
53#1893C2×C311yescharacteristic, complemented
54#1892C2×C311yescharacteristic, complemented
55#1809C2×C312yescomplemented
56#1808C2×C311yescharacteristic, complemented
57C25×C3D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—22no—
60C22×C3×D4—21no—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—24no—
63C23×C3×C4—22no—
64C23×C3×C4—24no—
65C23×C3×C4—22no—
66C23×C3×C4D412yes—
67C23×C3×C4D411yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2312yes—
70C2×C3×SD16—22no—
71C2×C3×SD16—24no—
72C2×C3×M16—232no—
73C2×C3×M16—24no—
74C2×C3×M16—24no—
75C2×C3×M16—24no—
76C2×C3×M16—24no—
77C2×C3×M16—24no—
78C2×C3×M16D418yescomplemented
79C22×C3×C8—22no—
80C22×C3×C8—24no—
81C22×C3×C8—24no—
82C22×C3×C8D414yescomplemented
83C22×C3×C8C2312yes—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—24no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—24no—
90C3×#121—22no—
91C3×#121—24no—
92C3×#121—22no—
93C3×#121—24no—
94C3×#120—44no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—21no—
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—22no—
109C3×#107—24no—
110C3×#106—22no—
111C3×#103—22no—
112C3×#103—24no—
113C3×#102—22no—
114C3×#102—24no—
115C3×#95—21no—
116C3×#95—22no—
117C3×#95—21no—
118C3×#95—22no—
119C24×C4C22×C311yescharacteristic
120C22×SD16C22×C311yescharacteristic, complemented
121C22×M16—28no—
122C22×M16—24no—
123C22×M16C22×C312yes—
124C22×M16C22×C312yes—
125C23×C8C22×C311yescharacteristic
126C23×C8C22×C311yescharacteristic
127C2×C22⋊Q8C22×C311yescharacteristic
128C2×C22⋊Q8C22×C312yes—
129C2×#121C22×C311yescharacteristic
130C2×#121C22×C312yes—
131C2×#120—21no—
132#468—28no—
133#465—24no—
134C2×#107C22×C312yescomplemented
135C2×#106C22×C311yescharacteristic, complemented
136C2×#103C22×C312yescomplemented
137C2×#103C22×C311yescharacteristic, complemented
138C2×#102C22×C312yescomplemented
139C2×#102C22×C311yescharacteristic, complemented
140#386—22no—
141#386—21no—
142#385—22no—
143#385—21no—
144#380C22×C311yescharacteristic
145#380C22×C311yescharacteristic
146#380C22×C312yes—
147#380C22×C312yes—
148#379—21no—
149#340C22×C312yescomplemented
150#340C22×C312yescomplemented
151#340C22×C312yescomplemented
152C24×C3—24no—
153C24×C3—22no—
154C24×C3—22no—
155C24×C3—22no—
156C24×C3—21no—
157C24×C3—42no—
158C24×C3C2×D412yes—
159C24×C3C2×D411yescharacteristic
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×D4—21no—
165C2×C3×D4—22no—
166C2×C3×D4—21no—
167C2×C3×D4—22no—
168C2×C3×D4—44no—
169C2×C3×D4—44no—
170C2×C3×D4—44no—
171C2×C3×D4—42no—
172C2×C3×D4—42no—
173C2×C3×D4—42no—
174C22×C3×C4—28no—
175C22×C3×C4—28no—
176C22×C3×C4—216no—
177C22×C3×C4—24no—
178C22×C3×C4—22no—
179C22×C3×C4—24no—
180C22×C3×C4—24no—
181C22×C3×C4—24no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D414yes—
197C22×C3×C4#4114yes—
198C22×C3×C4#4112yes—
199C22×C3×C4C2411yescharacteristic
200C3×SD16—44no—
201C3×M16—216no—
202C3×M16—216no—
203C2×C3×C8—28no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—28no—
208C2×C3×C8C2×D414yes—
209C2×C3×C8C2×D412yes—
210C2×C3×C8C2×D412yes—
211C2×C3×C8C2×D414yes—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4—44no—
222C3×C22⋊C4—44no—
223C3×C22⋊C4—44no—
224C3×C22⋊C4—44no—
225C3×C22⋊C4—42no—
226C3×C22⋊C4—42no—
227C3×C22⋊C4—42no—
228C3×C22⋊C4—42no—
229C3×C22⋊C4—44no—
230C25C3×D411yescharacteristic
231C22×Q8C23×C311yescharacteristic
232C22×D4—22no—
233C22×D4—21no—
234C22×D4C23×C311yescharacteristic
235C23×C4—24no—
236C23×C4—22no—
237C23×C4—24no—
238C23×C4—22no—
239C23×C4C3×D412yes—
240C23×C4C3×D411yescharacteristic
241C23×C4C23×C311yescharacteristic
242C23×C4C23×C312yes—
243C2×SD16—22no—
244C2×SD16—24no—
245C2×M16—232no—
246C2×M16—24no—
247C2×M16—24no—
248C2×M16—24no—
249C2×M16—24no—
250C2×M16—24no—
251C2×M16C3×D418yescomplemented
252C22×C8—22no—
253C22×C8—24no—
254C22×C8—24no—
255C22×C8C3×D414yescomplemented
256C22×C8C23×C312yes—
257C22×C8C23×C311yescharacteristic
258C22×C8C23×C311yescharacteristic
259C22⋊Q8—22no—
260C22⋊Q8—24no—
261C22⋊Q8—22no—
262C22⋊Q8—24no—
263#121—22no—
264#121—24no—
265#121—22no—
266#121—24no—
267#120—44no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4C23×C311yescharacteristic
271C2×C4⋊C4C23×C312yes—
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C4⋊C4C23×C312yes—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—22no—
282#107—24no—
283#106—22no—
284#103—22no—
285#103—24no—
286#102—22no—
287#102—24no—
288#95—21no—
289#95—22no—
290#95—21no—
291#95—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—28no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—48no—
302C23×C3—48no—
303C23×C3—44no—
304C23×C3—44no—
305C23×C3—41no—
306C23×C3—42no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—42no—
310C23×C3#12114yes—
311C23×C3#12112yes—
312C23×C3AGL(1,ℤ/8)12yes—
313C23×C3#13712yes—
314C23×C3C22×D411yescharacteristic
315C3×Q8—41no—
316C3×Q8—41no—
317C3×Q8—42no—
318C3×D4—41no—
319C3×D4—41no—
320C3×D4—42no—
321C3×D4—88no—
322C3×D4—84no—
323C2×C3×C4—28no—
324C2×C3×C4—216no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—41no—
336C2×C3×C4—42no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—41no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—41no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4#12112yes—
349C2×C3×C4#12114yes—
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C22×D412yes—
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4112yes—
354C3×C8—24no—
355C3×C8—22no—
356C3×C8#12014yescomplemented
357C24—24no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—21no—
362C24—42no—
363C24C2×C3×D412yes—
364C24C2×C3×D411yescharacteristic
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—216no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4C2×C3×D411yescharacteristic
399C22×C4C2×C3×D412yes—
400C22×C4C2×C3×D412yes—
401C22×C4C2×C3×D414yes—
402C22×C4C3×#4114yes—
403C22×C4C3×#4112yes—
404C22×C4C24×C311yescharacteristic
405SD16—44no—
406M16—216no—
407M16—216no—
408C2×C8—28no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—28no—
413C2×C8C2×C3×D414yes—
414C2×C8C2×C3×D412yes—
415C2×C8C2×C3×D412yes—
416C2×C8C2×C3×D414yes—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—44no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—24no—
438C22×C3—24no—
439C22×C3—24no—
440C22×C3—24no—
441C22×C3—24no—
442C22×C3—416no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—81no—
448C22×C3—81no—
449C22×C3—81no—
450C22×C3—82no—
451C22×C3—82no—
452C22×C3#46814yescomplemented
453C22×C3C2×#12111yescharacteristic
454C22×C3C2×#12112yes—
455C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
456C22×C3C2×#13711yescharacteristic
457C22×C3#57512yes—
458C3×C4—22no—
459C3×C4—24no—
460C3×C4—81no—
461C3×C4—81no—
462C3×C4—81no—
463C3×C4—82no—
464C3×C4—82no—
465C3×C4C2×#12011yescharacteristic
466C3×C4#52511yescharacteristic
467C3×C4#52512yes—
468C23—22no—
469C23—22no—
470C23—28no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—48no—
478C23—48no—
479C23—44no—
480C23—44no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—41no—
485C23—42no—
486C23C3×#12114yes—
487C23C3×#12112yes—
488C23C3×AGL(1,ℤ/8)12yes—
489C23C3×#13712yes—
490C23C22×C3×D411yescharacteristic
491Q8—41no—
492Q8—41no—
493Q8—42no—
494D4—41no—
495D4—41no—
496D4—42no—
497D4—88no—
498D4—84no—
499C2×C4—28no—
500C2×C4—216no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4C3×#12112yes—
525C2×C4C3×#12114yes—
526C2×C4C22×C3×D411yescharacteristic
527C2×C4C22×C3×D412yes—
528C2×C4C2×C3×#4111yescharacteristic
529C2×C4C2×C3×#4112yes—
530C8—24no—
531C8—22no—
532C8C3×#12014yescomplemented
533C2×C3—24no—
534C2×C3—44no—
535C2×C3—161no—
536C2×C3#284711yescharacteristic
537C2×C3C2×#46811yescharacteristic
538C2×C3#305411yescharacteristic
539C2×C3#305512yes—
540C2×C3#305612yes—
541C22—22no—
542C22—24no—
543C22—24no—
544C22—22no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—416no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—82no—
556C22—82no—
557C22—81no—
558C22C3×#46814yescomplemented
559C22C2×C3×#12111yescharacteristic
560C22C2×C3×#12112yes—
561C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
562C22C2×C3×#13711yescharacteristic
563C22C3×#57512yes—
564C4—22no—
565C4—24no—
566C4—81no—
567C4—81no—
568C4—82no—
569C4—82no—
570C4—81no—
571C4C2×C3×#12011yescharacteristic
572C4C3×#52511yescharacteristic
573C4C3×#52512yes—
574C3#3293511yescharacteristic, direct product factor, Sylow
575C2—24no—
576C2—44no—
577C2—161no—
578C2C3×#284711yescharacteristic
579C2C2×C3×#46811yescharacteristic
580C2C3×#305411yescharacteristic
581C2C3×#305512yes—
582C2C3×#305612yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×C3×M16C2semidirect
2. C2×C3×#468C2semidirect
3. C2×C3×#465C2semidirect
4. C3×#2848C2semidirect
6. C3×#1895C2semidirect
7. C3×#1894C2semidirect
8. C3×#1893C2semidirect
9. C3×#1892C2semidirect
10. C3×#1809C2semidirect
11. C3×#1808C2semidirect
12. #32935C3direct (N × H)
14. C22×C3×SD16C22semidirect
28. C2×C3×#107C22semidirect
29. C2×C3×#106C22semidirect
30. C2×C3×#103C22semidirect
31. C2×C3×#103C22semidirect
32. C2×C3×#102C22semidirect
33. C2×C3×#102C22semidirect
43. C3×#340C22semidirect
44. C3×#340C22semidirect
45. C3×#340C22semidirect
46. C23×M16C2×C3semidirect
47. C2×#468C2×C3semidirect
48. C2×#465C2×C3semidirect
49. #2848C2×C3semidirect
51. #1895C2×C3semidirect
52. #1894C2×C3semidirect
53. #1893C2×C3semidirect
54. #1892C2×C3semidirect
55. #1809C2×C3semidirect
56. #1808C2×C3semidirect
78. C2×C3×M16D4semidirect
82. C22×C3×C8D4semidirect
120. C22×SD16C22×C3semidirect
134. C2×#107C22×C3semidirect
135. C2×#106C22×C3semidirect
136. C2×#103C22×C3semidirect
137. C2×#103C22×C3semidirect
138. C2×#102C22×C3semidirect
139. C2×#102C22×C3semidirect
149. #340C22×C3semidirect
150. #340C22×C3semidirect
151. #340C22×C3semidirect
251. C2×M16C3×D4semidirect
255. C22×C8C3×D4semidirect
356. C3×C8#120semidirect
452. C22×C3#468semidirect
532. C8C3×#120semidirect
558. C22C3×#468semidirect
574. C3#32935direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×M162
C2×C3×#4682
C2×C3×#4652
C3×#28482
C3×#28472
C3×#18952
C3×#18942
C3×#18932
C3×#18922
C3×#18092
C3×#18082
#329353