Finite Group Databasebeta

← C3×#32981 #10651588

Subgroups — C3×#32981 #10651588

Order 768.

1508 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#117C211yesmaximal, characteristic, complemented
2C2×C3×#482C212yesmaximal, complemented
3C3×#2847C211yesmaximal, characteristic
4C3×#1875C212yesmaximal, complemented
5C3×#1874C211yesmaximal, characteristic, complemented
6C3×#1874C211yesmaximal, characteristic, complemented
7C3×#1854C212yesmaximal, complemented
8C3×#1853C211yesmaximal, characteristic, complemented
9C3×#1852C211yesmaximal, characteristic, complemented
10C3×#1825C212yesmaximal, complemented
11C3×#1779C211yesmaximal, characteristic, complemented
12#32981C311yesmaximal, characteristic, direct product factor, Sylow
13C24×C3×C4C2211yescharacteristic
14C2×C3×#121C2211yescharacteristic
15C2×C3×#121C2211yescharacteristic
16C2×C3×#121C2211yescharacteristic
17C2×C3×#120—21no—
18C2×C3×#117—21no—
19C2×C3×#117—21no—
20C2×C3×#117—28no—
21C2×C3×#117C2212yes—
22C2×C3×#117C2212yes—
23C22×C3×C4⋊C4C2211yescharacteristic
24C22×C3×C4⋊C4C2211yescharacteristic
25C22×C3×C22⋊C4—21no—
26C22×C3×C42—21no—
27C3×#482—28no—
28C2×C3×#107C2212yescomplemented
29C2×C3×#106C2212yescomplemented
30C2×C3×#102C2212yescomplemented
31C2×C3×#102C2212yescomplemented
32C2×C3×C22⋊C8C2212yes—
33C2×C3×C22⋊C8C2211yescharacteristic
34C2×C3×C22⋊C8C2211yescharacteristic
35C2×C3×C22⋊C8C2212yes—
36C3×#386—21no—
37C3×#386—21no—
38C3×#386—21no—
39C3×#380C2211yescharacteristic
40C3×#380C2211yescharacteristic
41C3×#380C2211yescharacteristic
42C3×#340C2212yescomplemented
43C3×#340C2212yescomplemented
44C3×#340C2212yescomplemented
45C3×#340C2212yescomplemented
46C22×#117C2×C311yescharacteristic, complemented
47C2×#482C2×C312yescomplemented
48#2847C2×C311yescharacteristic
49#1875C2×C312yescomplemented
50#1874C2×C311yescharacteristic, complemented
51#1874C2×C311yescharacteristic, complemented
52#1854C2×C312yescomplemented
53#1853C2×C311yescharacteristic, complemented
54#1852C2×C311yescharacteristic, complemented
55#1825C2×C312yescomplemented
56#1779C2×C311yescharacteristic, complemented
57C25×C3D411yescharacteristic
58C22×C3×D4—21no—
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—24no—
63C23×C3×C4—24no—
64C23×C3×C4—22no—
65C23×C3×C4—22no—
66C23×C3×C4—21no—
67C23×C3×C4—21no—
68C23×C3×C4D411yescharacteristic
69C23×C3×C4D411yescharacteristic
70C23×C3×C4D411yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C23×C3×C4C2311yescharacteristic
74C22×C3×C8C2312yes—
75C22×C3×C8C2312yes—
76C3×#121—24no—
77C3×#121—22no—
78C3×#121—22no—
79C3×#121—24no—
80C3×#120—44no—
81C3×#117—28no—
82C3×#117—28no—
83C3×#117—24no—
84C3×#117—24no—
85C3×#117—24no—
86C3×#117D418yescomplemented
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—24no—
90C2×C3×C4⋊C4—24no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C22⋊C4—28no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C42—28no—
106C2×C3×C42—22no—
107C2×C3×C42—22no—
108C3×#107—24no—
109C3×#106—24no—
110C3×#102—24no—
111C3×#102—24no—
112C3×C22⋊C8—24no—
113C3×C22⋊C8—22no—
114C3×C22⋊C8—22no—
115C3×C22⋊C8—24no—
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119C24×C4C22×C311yescharacteristic
120C2×#121C22×C311yescharacteristic
121C2×#121C22×C311yescharacteristic
122C2×#121C22×C311yescharacteristic
123C2×#120—21no—
124C2×#117—21no—
125C2×#117—21no—
126C2×#117—28no—
127C2×#117C22×C312yes—
128C2×#117C22×C312yes—
129C22×C4⋊C4C22×C311yescharacteristic
130C22×C4⋊C4C22×C311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C42—21no—
133#482—28no—
134C2×#107C22×C312yescomplemented
135C2×#106C22×C312yescomplemented
136C2×#102C22×C312yescomplemented
137C2×#102C22×C312yescomplemented
138C2×C22⋊C8C22×C312yes—
139C2×C22⋊C8C22×C311yescharacteristic
140C2×C22⋊C8C22×C311yescharacteristic
141C2×C22⋊C8C22×C312yes—
142#386—21no—
143#386—21no—
144#386—21no—
145#380C22×C311yescharacteristic
146#380C22×C311yescharacteristic
147#380C22×C311yescharacteristic
148#340C22×C312yescomplemented
149#340C22×C312yescomplemented
150#340C22×C312yescomplemented
151#340C22×C312yescomplemented
152C24×C3—22no—
153C24×C3—22no—
154C24×C3—22no—
155C24×C3—22no—
156C24×C3—22no—
157C24×C3—21no—
158C24×C3—41no—
159C24×C3—41no—
160C24×C3C2×D411yescharacteristic
161C24×C3C2×D411yescharacteristic
162C24×C3C2×D411yescharacteristic
163C2×C3×D4—22no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—22no—
167C2×C3×D4—42no—
168C2×C3×D4—42no—
169C2×C3×D4—42no—
170C2×C3×D4—42no—
171C2×C3×D4—42no—
172C2×C3×D4—42no—
173C2×C3×D4—42no—
174C2×C3×D4—42no—
175C2×C3×D4—42no—
176C22×C3×C4—216no—
177C22×C3×C4—216no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
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—22no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—24no—
196C22×C3×C4—24no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—42no—
203C22×C3×C4—42no—
204C22×C3×C4—42no—
205C22×C3×C4—42no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D414yes—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4#4114yes—
211C22×C3×C4#4112yes—
212C22×C3×C4#4112yes—
213C22×C3×C4C2411yescharacteristic
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C3×C4⋊C4—24no—
221C3×C4⋊C4—24no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4C2×D414yes—
229C3×C4⋊C4C2×D414yes—
230C3×C22⋊C4—28no—
231C3×C22⋊C4—28no—
232C3×C22⋊C4—42no—
233C3×C22⋊C4—42no—
234C3×C22⋊C4—42no—
235C3×C22⋊C4—42no—
236C3×C22⋊C4—42no—
237C3×C22⋊C4—42no—
238C3×C42—28no—
239C3×C42—28no—
240C25C3×D411yescharacteristic
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C23×C311yescharacteristic
245C23×C4—24no—
246C23×C4—24no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4C3×D411yescharacteristic
252C23×C4C3×D411yescharacteristic
253C23×C4C3×D411yescharacteristic
254C23×C4C23×C311yescharacteristic
255C23×C4C23×C311yescharacteristic
256C23×C4C23×C311yescharacteristic
257C22×C8C23×C312yes—
258C22×C8C23×C312yes—
259#121—24no—
260#121—22no—
261#121—22no—
262#121—24no—
263#120—44no—
264#117—28no—
265#117—28no—
266#117—24no—
267#117—24no—
268#117—24no—
269#117C3×D418yescomplemented
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4C23×C311yescharacteristic
275C2×C4⋊C4C23×C311yescharacteristic
276C2×C4⋊C4C23×C311yescharacteristic
277C2×C4⋊C4C23×C312yes—
278C2×C4⋊C4C23×C312yes—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C42—28no—
289C2×C42—22no—
290C2×C42—22no—
291#107—24no—
292#106—24no—
293#102—24no—
294#102—24no—
295C22⋊C8—24no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8—24no—
299#95—21no—
300#95—21no—
301#95—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—28no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—44no—
320C23×C3—44no—
321C23×C3—44no—
322C23×C3—44no—
323C23×C3—44no—
324C23×C3—44no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3#12012yes—
333C23×C3#12312yes—
334C23×C3#12312yes—
335C23×C3#13512yes—
336C23×C3AGL(1,ℤ/8)12yes—
337C23×C3C22×D411yescharacteristic
338C3×D4—42no—
339C3×D4—42no—
340C3×D4—84no—
341C3×D4—84no—
342C3×D4—84no—
343C2×C3×C4—28no—
344C2×C3×C4—28no—
345C2×C3×C4—28no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—24no—
360C2×C3×C4—48no—
361C2×C3×C4—48no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4#12012yes—
376C2×C3×C4#12312yes—
377C2×C3×C4#12312yes—
378C2×C3×C4C22×D412yes—
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4112yes—
382C3×C8—42no—
383C3×C8—42no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—21no—
390C24—41no—
391C24—41no—
392C24C2×C3×D411yescharacteristic
393C24C2×C3×D411yescharacteristic
394C24C2×C3×D411yescharacteristic
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4C2×C3×D411yescharacteristic
439C22×C4C2×C3×D414yes—
440C22×C4C2×C3×D411yescharacteristic
441C22×C4C2×C3×D411yescharacteristic
442C22×C4C3×#4114yes—
443C22×C4C3×#4112yes—
444C22×C4C3×#4112yes—
445C22×C4C24×C311yescharacteristic
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4C2×C3×D414yes—
461C4⋊C4C2×C3×D414yes—
462C22⋊C4—28no—
463C22⋊C4—28no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C42—28no—
471C42—28no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—22no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—22no—
484C22×C3—416no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—81no—
494C22×C3—81no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3#48214yescomplemented
501C22×C3C2×#12011yescharacteristic
502C22×C3C2×#12311yescharacteristic
503C22×C3C2×#12311yescharacteristic
504C22×C3C2×#13511yescharacteristic
505C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
506C22×C3#57511yescharacteristic
507C22×C3#57511yescharacteristic
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—22no—
511C3×C4—44no—
512C3×C4—44no—
513C3×C4—81no—
514C3×C4—81no—
515C3×C4—81no—
516C3×C4#52512yes—
517C3×C4#52512yes—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—28no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C3×#12012yes—
549C23C3×#12312yes—
550C23C3×#12312yes—
551C23C3×#13512yes—
552C23C3×AGL(1,ℤ/8)12yes—
553C23C22×C3×D411yescharacteristic
554D4—42no—
555D4—42no—
556D4—84no—
557D4—84no—
558D4—84no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4C3×#12012yes—
592C2×C4C3×#12312yes—
593C2×C4C3×#12312yes—
594C2×C4C22×C3×D412yes—
595C2×C4C2×C3×#4111yescharacteristic
596C2×C4C2×C3×#4111yescharacteristic
597C2×C4C2×C3×#4112yes—
598C8—42no—
599C8—42no—
600C2×C3—24no—
601C2×C3—42no—
602C2×C3—42no—
603C2×C3—161no—
604C2×C3#284911yescharacteristic
605C2×C3#300211yescharacteristic
606C2×C3#300611yescharacteristic
607C2×C3C2×#48211yescharacteristic
608C2×C3#309111yescharacteristic
609C2×C3#309311yescharacteristic
610C2×C3#309411yescharacteristic
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—416no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22C3×#48214yescomplemented
640C22C2×C3×#12011yescharacteristic
641C22C2×C3×#12311yescharacteristic
642C22C2×C3×#12311yescharacteristic
643C22C2×C3×#13511yescharacteristic
644C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
645C22C3×#57511yescharacteristic
646C22C3×#57511yescharacteristic
647C4—22no—
648C4—22no—
649C4—22no—
650C4—44no—
651C4—44no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4C3×#52512yes—
656C4C3×#52512yes—
657C3#3298111yescharacteristic, direct product factor, Sylow
658C2—24no—
659C2—42no—
660C2—42no—
661C2—161no—
662C2C3×#284911yescharacteristic
663C2C3×#300211yescharacteristic
664C2C3×#300611yescharacteristic
665C2C2×C3×#48211yescharacteristic
666C2C3×#309111yescharacteristic
667C2C3×#309311yescharacteristic
668C2C3×#309411yescharacteristic
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. C22×C3×#117C2semidirect
2. C2×C3×#482C2semidirect
4. C3×#1875C2semidirect
5. C3×#1874C2semidirect
6. C3×#1874C2semidirect
7. C3×#1854C2semidirect
8. C3×#1853C2semidirect
9. C3×#1852C2semidirect
10. C3×#1825C2semidirect
11. C3×#1779C2semidirect
12. #32981C3direct (N × H)
28. C2×C3×#107C22semidirect
29. C2×C3×#106C22semidirect
30. C2×C3×#102C22semidirect
31. C2×C3×#102C22semidirect
42. C3×#340C22semidirect
43. C3×#340C22semidirect
44. C3×#340C22semidirect
45. C3×#340C22semidirect
46. C22×#117C2×C3semidirect
47. C2×#482C2×C3semidirect
49. #1875C2×C3semidirect
50. #1874C2×C3semidirect
51. #1874C2×C3semidirect
52. #1854C2×C3semidirect
53. #1853C2×C3semidirect
54. #1852C2×C3semidirect
55. #1825C2×C3semidirect
56. #1779C2×C3semidirect
86. C3×#117D4semidirect
134. C2×#107C22×C3semidirect
135. C2×#106C22×C3semidirect
136. C2×#102C22×C3semidirect
137. C2×#102C22×C3semidirect
148. #340C22×C3semidirect
149. #340C22×C3semidirect
150. #340C22×C3semidirect
151. #340C22×C3semidirect
269. #117C3×D4semidirect
500. C22×C3#482semidirect
639. C22C3×#482semidirect
657. C3#32981direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1172
C2×C3×#4822
C3×#28472
C3×#18752
C3×#18742
C3×#18542
C3×#18532
C3×#18522
C3×#18252
C3×#17792
#329813