Finite Group Databasebeta

← C3×#15783 #10634390

Subgroups — C3×#15783 #10634390

Order 768.

1586 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×C3×#400C211yesmaximal, characteristic
3C2×C3×#398C211yesmaximal, characteristic
4C2×C3×#391C211yesmaximal, characteristic
5C3×#1446C212yesmaximal
6C3×#1446C211yesmaximal, characteristic
7C3×#1442C212yesmaximal
8C3×#1441C212yesmaximal
9C3×#1436C211yesmaximal, characteristic
10C3×#1436C212yesmaximal
11C3×#1436C211yesmaximal, characteristic
12#15783C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C2×C3×C4⋊Q8—24no—
16C2×C3×C4⋊Q8—24no—
17C2×C3×C4⋊Q8—24no—
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C4⋊C4C2212yes—
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×C3×C4⋊C4C2211yescharacteristic
24C22×C3×C42C2211yescharacteristic
25C22×C3×C42C2211yescharacteristic
26C3×#400—24no—
27C3×#398—24no—
28C3×#391—24no—
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×C3×#95C2212yes—
39C2×C3×#95C2212yes—
40C2×C3×#95C2212yes—
41C2×C3×#95C2212yes—
42C22×C4⋊Q8C2×C311yescharacteristic
43C2×#400C2×C311yescharacteristic
44C2×#398C2×C311yescharacteristic
45C2×#391C2×C311yescharacteristic
46#1446C2×C312yes—
47#1446C2×C311yescharacteristic
48#1442C2×C312yes—
49#1441C2×C312yes—
50#1436C2×C312yes—
51#1436C2×C311yescharacteristic
52#1436C2×C311yescharacteristic
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—22no—
58C22×C3×Q8—24no—
59C22×C3×Q8—24no—
60C22×C3×Q8—42no—
61C22×C3×Q8—42no—
62C22×C3×Q8D414yes—
63C23×C3×C4C2312yes—
64C23×C3×C4C2312yes—
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2312yes—
69C23×C3×C4C2312yes—
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C23×C3×C4C2311yescharacteristic
74C3×C4⋊Q8—232no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—24no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—24no—
90C2×C3×C4⋊C4—24no—
91C2×C3×C4⋊C4D414yes—
92C2×C3×C4⋊C4D414yes—
93C2×C3×C42—22no—
94C2×C3×C42—22no—
95C2×C3×C42—22no—
96C2×C3×C42—22no—
97C2×C3×C42—42no—
98C3×#95—24no—
99C3×#95—24no—
100C3×#95—22no—
101C3×#95—24no—
102C3×#95—22no—
103C3×#95—24no—
104C3×#95—24no—
105C3×#95—24no—
106C3×#95—24no—
107C3×#95—42no—
108C3×#95—42no—
109C3×#95—42no—
110C3×#95Q818yescomplemented
111C23×Q8C22×C311yescharacteristic
112C23×Q8C22×C311yescharacteristic
113C2×C4⋊Q8—24no—
114C2×C4⋊Q8—24no—
115C2×C4⋊Q8—24no—
116C22×C4⋊C4C22×C311yescharacteristic
117C22×C4⋊C4C22×C312yes—
118C22×C4⋊C4C22×C311yescharacteristic
119C22×C4⋊C4C22×C311yescharacteristic
120C22×C4⋊C4C22×C311yescharacteristic
121C22×C4⋊C4C22×C311yescharacteristic
122C22×C42C22×C311yescharacteristic
123C22×C42C22×C311yescharacteristic
124#400—24no—
125#398—24no—
126#391—24no—
127C2×#95C22×C311yescharacteristic
128C2×#95C22×C312yes—
129C2×#95C22×C312yes—
130C2×#95C22×C311yescharacteristic
131C2×#95C22×C312yes—
132C2×#95C22×C312yes—
133C2×#95C22×C312yes—
134C2×#95C22×C312yes—
135C2×#95C22×C312yes—
136C2×#95C22×C312yes—
137C2×#95C22×C312yes—
138C2×#95C22×C312yes—
139C2×#95C22×C312yes—
140C24×C3C2411yescharacteristic
141C2×C3×Q8—28no—
142C2×C3×Q8—28no—
143C2×C3×Q8—28no—
144C2×C3×Q8—44no—
145C2×C3×Q8—48no—
146C2×C3×Q8—48no—
147C2×C3×Q8—48no—
148C2×C3×Q8—48no—
149C2×C3×Q8—44no—
150C2×C3×Q8—44no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
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—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—22no—
199C22×C3×C4—22no—
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D412yes—
206C22×C3×C4C2×D412yes—
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2×Q814yes—
210C22×C3×C4C2×Q814yes—
211C22×C3×C4C2×Q814yes—
212C22×C3×C4#4114yes—
213C22×C3×C4#4114yes—
214C3×C4⋊C4—28no—
215C3×C4⋊C4—28no—
216C3×C4⋊C4—28no—
217C3×C4⋊C4—28no—
218C3×C4⋊C4—44no—
219C3×C4⋊C4—48no—
220C3×C42—28no—
221C3×C42—44no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—24no—
228C22×Q8—24no—
229C22×Q8—42no—
230C22×Q8—42no—
231C22×Q8C3×D414yes—
232C23×C4C23×C312yes—
233C23×C4C23×C312yes—
234C23×C4C23×C311yescharacteristic
235C23×C4C23×C311yescharacteristic
236C23×C4C23×C311yescharacteristic
237C23×C4C23×C312yes—
238C23×C4C23×C312yes—
239C23×C4C23×C311yescharacteristic
240C23×C4C23×C311yescharacteristic
241C23×C4C23×C311yescharacteristic
242C23×C4C23×C311yescharacteristic
243C4⋊Q8—232no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4C3×D414yes—
261C2×C4⋊C4C3×D414yes—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—42no—
267#95—24no—
268#95—24no—
269#95—22no—
270#95—24no—
271#95—22no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—42no—
277#95—42no—
278#95—42no—
279#95C3×Q818yescomplemented
280C23×C3C22×D411yescharacteristic
281C23×C3C22×D411yescharacteristic
282C23×C3C22×D411yescharacteristic
283C23×C3C22×Q812yes—
284C23×C3C2×#4112yes—
285C23×C3C2×#4112yes—
286C23×C3H(5,𝔽2)12yes—
287C23×C32−1+412yes—
288C23×C32−1+412yes—
289C3×Q8—416no—
290C3×Q8—416no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—22no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4#12014yes—
351C2×C3×C4#12314yes—
352C2×C3×C4#12414yes—
353C2×C3×C4C4⋊Q814yes—
354C24C24×C311yescharacteristic
355C2×Q8—28no—
356C2×Q8—28no—
357C2×Q8—28no—
358C2×Q8—44no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—44no—
364C2×Q8—44no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4C2×C3×D412yes—
415C22×C4C2×C3×D412yes—
416C22×C4C2×C3×D412yes—
417C22×C4C2×C3×D412yes—
418C22×C4C2×C3×D412yes—
419C22×C4C2×C3×D412yes—
420C22×C4C2×C3×D412yes—
421C22×C4C2×C3×D412yes—
422C22×C4C2×C3×D412yes—
423C22×C4C2×C3×Q814yes—
424C22×C4C2×C3×Q814yes—
425C22×C4C2×C3×Q814yes—
426C22×C4C3×#4114yes—
427C22×C4C3×#4114yes—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—28no—
431C4⋊C4—28no—
432C4⋊C4—44no—
433C4⋊C4—48no—
434C42—28no—
435C42—44no—
436C22×C3C2×#12011yescharacteristic
437C22×C3C2×#12311yescharacteristic
438C22×C3C2×#12411yescharacteristic
439C22×C3C2×C4⋊Q811yescharacteristic
440C22×C3#53312yes—
441C22×C3#53411yescharacteristic
442C22×C3#53611yescharacteristic
443C22×C3#53611yescharacteristic
444C22×C3#54112yes—
445C22×C3#54212yes—
446C22×C3#54412yes—
447C22×C3#54612yes—
448C22×C3#54612yes—
449C22×C3#54812yes—
450C22×C3#54812yes—
451C22×C3D4×Q812yes—
452C22×C3D4×Q812yes—
453C22×C3#55412yes—
454C22×C3#55612yes—
455C22×C3#55712yes—
456C22×C3#56312yes—
457C3×C4—24no—
458C3×C4—24no—
459C3×C4—24no—
460C3×C4—24no—
461C3×C4—42no—
462C3×C4—42no—
463C3×C4—42no—
464C3×C4—44no—
465C3×C4—44no—
466C3×C4—82no—
467C3×C4—82no—
468C23C22×C3×D411yescharacteristic
469C23C22×C3×D411yescharacteristic
470C23C22×C3×D411yescharacteristic
471C23C22×C3×Q812yes—
472C23C2×C3×#4112yes—
473C23C2×C3×#4112yes—
474C23C3×H(5,𝔽2)12yes—
475C23C3×2−1+412yes—
476C23C3×2−1+412yes—
477Q8—416no—
478Q8—416no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4C3×#12014yes—
539C2×C4C3×#12314yes—
540C2×C4C3×#12414yes—
541C2×C4C3×C4⋊Q814yes—
542C2×C3#240511yescharacteristic
543C2×C3#243411yescharacteristic
544C2×C3#243611yescharacteristic
545C2×C3#244612yes—
546C2×C3#249212yes—
547C2×C3#255012yes—
548C2×C3#271312yes—
549C2×C3#271612yes—
550C2×C3#272912yes—
551C22C2×C3×#12011yescharacteristic
552C22C2×C3×#12311yescharacteristic
553C22C2×C3×#12411yescharacteristic
554C22C2×C3×C4⋊Q811yescharacteristic
555C22C3×#53312yes—
556C22C3×#53411yescharacteristic
557C22C3×#53611yescharacteristic
558C22C3×#53611yescharacteristic
559C22C3×#54112yes—
560C22C3×#54212yes—
561C22C3×#54412yes—
562C22C3×#54612yes—
563C22C3×#54612yes—
564C22C3×#54812yes—
565C22C3×#54812yes—
566C22C3×D4×Q812yes—
567C22C3×D4×Q812yes—
568C22C3×#55412yes—
569C22C3×#55612yes—
570C22C3×#55712yes—
571C22C3×#56312yes—
572C4—24no—
573C4—24no—
574C4—24no—
575C4—24no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—44no—
580C4—44no—
581C4—82no—
582C4—82no—
583C3#1578311yescharacteristic, direct product factor, Sylow
584C2C3×#240511yescharacteristic
585C2C3×#243411yescharacteristic
586C2C3×#243611yescharacteristic
587C2C3×#244612yes—
588C2C3×#249212yes—
589C2C3×#255012yes—
590C2C3×#271312yes—
591C2C3×#271612yes—
592C2C3×#272912yes—
subgroupG/H
center, socle, prime radicalC24×C3C24
commutator, Frattini, cosocleC24C24×C3
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
12. #15783C3direct (N × H)
110. C3×#95Q8semidirect
279. #95C3×Q8semidirect
583. C3#15783direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C2×C3×#4002
C2×C3×#3982
C2×C3×#3912
C3×#14462
C3×#14422
C3×#14412
C3×#14362
#157833