Finite Group Databasebeta

← #10732202

Subgroups — #10732202

Order 768.

2446 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#888C211yesmaximal, characteristic, complemented
2C22×#878C212yesmaximal
3C2×Dic3×C22⋊C4C211yesmaximal, characteristic
4C2×#4646C212yesmaximal, complemented
5C3×#1439C211yesmaximal, characteristic
6#71568C211yesmaximal, characteristic, complemented
7#71547C212yesmaximal
8#71546C211yesmaximal, characteristic
9#69420C212yesmaximal, complemented
10#69419C211yesmaximal, characteristic, complemented
11Dic3⋊#95C211yesmaximal, characteristic, complemented
12#15178—31nomaximal, Sylow
13C24×Dic3C2211yescharacteristic
14C23×Dic6C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2212yes—
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×#217C2212yes—
19C22×#217C2211yescharacteristic
20C22×C4⋊Dic3C2212yes—
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2212yes—
24C22×Dic3⋊C4C2212yes—
25C22×C4×Dic3C2211yescharacteristic
26C22×C4×Dic3C2211yescharacteristic
27C2×#888—28no—
28C2×#888—24no—
29C2×#878—28no—
30C2×#878—28no—
31C2×#878—28no—
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2211yescharacteristic
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41Dic3×C22⋊C4—24no—
42#4646—28no—
43C22×C4⋊Q8—31no—
44C22×C22⋊Q8—32no—
45C2×#391—32no—
46C2×C4×C22⋊C4—31no—
47#1444—31no—
48#1444—32no—
49#1444—31no—
50#1439—31no—
51#1439—32no—
52#1439D311yescharacteristic
53C4⋊#95—31no—
54C25×C3C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×Dic3—22no—
60C23×Dic3—24no—
61C23×Dic3—24no—
62C23×Dic3—22no—
63C23×Dic3D414yes—
64C23×Dic3C2312yes—
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2312yes—
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C23×Dic3C2311yescharacteristic
70C22×Dic6—24no—
71C22×Dic6—24no—
72C22×Dic6—24no—
73C22×Dic6—24no—
74C22×Dic6—24no—
75C22×Dic6—24no—
76C22×Dic6D418yes—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C22⋊C4—24no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—24no—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4Q814yescomplemented
85C2×#217—24no—
86C2×#217—22no—
87C2×#217—24no—
88C2×#217—24no—
89C2×#217—22no—
90C2×#217Q814yescomplemented
91C2×C4⋊Dic3—24no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—24no—
94C2×C4⋊Dic3—24no—
95C2×C4⋊Dic3—24no—
96C2×Dic3⋊C4—24no—
97C2×Dic3⋊C4—24no—
98C2×Dic3⋊C4—24no—
99C2×Dic3⋊C4—24no—
100C2×Dic3⋊C4—24no—
101C2×Dic3⋊C4—24no—
102C2×Dic3⋊C4—24no—
103C2×Dic3⋊C4—24no—
104C2×Dic3⋊C4D418yes—
105C2×C4×Dic3—24no—
106C2×C4×Dic3—24no—
107C2×C4×Dic3—22no—
108C2×C4×Dic3D414yes—
109#888—232no—
110#878—264no—
111C3×#95—24no—
112C3×#95—22no—
113C3⋊#95—24no—
114C3⋊#95—22no—
115C3⋊#95—22no—
116C3⋊#95—24no—
117C3⋊#95—24no—
118C3⋊#95—22no—
119C3⋊#95—24no—
120C23×Q8—32no—
121C24×C4—31no—
122C2×C4⋊Q8—64no—
123C2×C4⋊Q8—68no—
124C2×C22⋊Q8—68no—
125C2×C22⋊Q8—68no—
126C2×C22⋊Q8—68no—
127C22×C4⋊C4—32no—
128C22×C4⋊C4—32no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4—32no—
131C22×C4⋊C4—31no—
132C22×C4⋊C4C2×D311yescharacteristic
133C22×C22⋊C4—31no—
134C22×C22⋊C4—32no—
135C22×C22⋊C4C2×D312yes—
136C22×C22⋊C4C2×D311yescharacteristic
137C22×C42—31no—
138C22×C42—31no—
139#391—68no—
140C4×C22⋊C4—64no—
141C2×#95—32no—
142C2×#95—31no—
143C2×#95—32no—
144C2×#95—32no—
145C2×#95—32no—
146C2×#95—31no—
147C2×#95—31no—
148C2×#95C2×D311yescharacteristic
149C2×#95C2×D312yes—
150C24×C3—22no—
151C24×C3—22no—
152C24×C3—21no—
153C24×C3—21no—
154C24×C3—21no—
155C24×C3—21no—
156C24×C3—22no—
157C24×C3—22no—
158C24×C3C2×D412yes—
159C24×C3C2×Q812yes—
160C24×C3C2×Q812yes—
161C24×C3C2411yescharacteristic
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4C2×D414yes—
181C22×C3×C4C2×D414yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×Q812yes—
184C22×C3×C4C2×Q812yes—
185C22×C3×C4#4112yes—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—21no—
203C22×Dic3—22no—
204C22×Dic3—21no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—216no—
209C22×Dic3—28no—
210C22×Dic3—44no—
211C22×Dic3—48no—
212C22×Dic3—44no—
213C22×Dic3C2×D414yes—
214C22×Dic3C2×D412yes—
215C22×Dic3C2×D414yes—
216C22×Dic3C2×D412yes—
217C22×Dic3C2×D414yes—
218C22×Dic3C2×D412yes—
219C22×Dic3C2×D412yes—
220C22×Dic3C2×D414yes—
221C22×Dic3C2×Q812yes—
222C22×Dic3C2×Q812yes—
223C22×Dic3#4112yes—
224C22×Dic3#4112yes—
225C2×Dic6—216no—
226C2×Dic6—216no—
227C2×Dic6—216no—
228C2×Dic6—48no—
229C2×Dic6—48no—
230C2×Dic6—48no—
231C2×Dic6—48no—
232C3×C4⋊C4—28no—
233C3×C22⋊C4—216no—
234C3×C22⋊C4—28no—
235#217—216no—
236#217—28no—
237C4⋊Dic3—216no—
238C4⋊Dic3—28no—
239Dic3⋊C4—216no—
240Dic3⋊C4—216no—
241Dic3⋊C4—216no—
242C4×Dic3—28no—
243C4×Dic3—44no—
244C25C22×D311yescharacteristic
245C22×Q8—38no—
246C22×Q8—64no—
247C22×Q8—64no—
248C22×Q8—64no—
249C22×Q8—64no—
250C22×Q8—64no—
251C22×Q8—64no—
252C23×C4—31no—
253C23×C4—31no—
254C23×C4—34no—
255C23×C4—32no—
256C23×C4—32no—
257C23×C4—31no—
258C23×C4—31no—
259C23×C4—64no—
260C23×C4—64no—
261C23×C4—62no—
262C23×C4—62no—
263C23×C4C22×D311yescharacteristic
264C23×C4C22×D312yes—
265C23×C4C22×D312yes—
266C23×C4C22×D311yescharacteristic
267C4⋊Q8—632no—
268C22⋊Q8—664no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—38no—
272C2×C4⋊C4—64no—
273C2×C4⋊C4—64no—
274C2×C4⋊C4—64no—
275C2×C4⋊C4—64no—
276C2×C4⋊C4—64no—
277C2×C4⋊C4—64no—
278C2×C4⋊C4—64no—
279C2×C4⋊C4—64no—
280C2×C4⋊C4—64no—
281C2×C4⋊C4—64no—
282C2×C4⋊C4—64no—
283C2×C4⋊C4—64no—
284C2×C4⋊C4—62no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—34no—
291C2×C22⋊C4—64no—
292C2×C22⋊C4—64no—
293C2×C22⋊C4—62no—
294C2×C22⋊C4—62no—
295C2×C22⋊C4—64no—
296C2×C22⋊C4Dic614yescomplemented
297C2×C42—34no—
298C2×C42—64no—
299C2×C42—62no—
300C2×C42—64no—
301#95—24no—
302#95—22no—
303#95—64no—
304#95—62no—
305#95—64no—
306#95—64no—
307#95—62no—
308#95—64no—
309#95—62no—
310C23×C3—24no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—22no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—24no—
318C23×C3—24no—
319C23×C3—24no—
320C23×C3—24no—
321C23×C3—24no—
322C23×C3—41no—
323C23×C3—42no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3—42no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3C4⋊Q814yescomplemented
333C23×C3C22×D411yescharacteristic
334C23×C3C22×D412yes—
335C23×C3C22×D411yescharacteristic
336C23×C3C22×D412yes—
337C23×C3C22×Q811yescharacteristic
338C23×C3C22×Q811yescharacteristic
339C23×C3C2×#4111yescharacteristic
340C23×C3C2×#4111yescharacteristic
341C23×C3C2×#4111yescharacteristic
342C23×C32−1+412yes—
343C23×C32−1+412yes—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—22no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—41no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4C22⋊Q814yescomplemented
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—24no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—24no—
375C2×Dic3—24no—
376C2×Dic3—24no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—24no—
380C2×Dic3—24no—
381C2×Dic3—24no—
382C2×Dic3—42no—
383C2×Dic3—41no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—41no—
388C2×Dic3—42no—
389C2×Dic3—41no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—41no—
393C2×Dic3—42no—
394C2×Dic3—42no—
395C2×Dic3—42no—
396C2×Dic3—416no—
397C2×Dic3#12014yes—
398C2×Dic3#12018yes—
399C2×Dic3#12118yes—
400C2×Dic3#12114yes—
401C2×Dic3C22⋊Q814yescomplemented
402C2×Dic3#12714yes—
403Dic6—432no—
404C24—22no—
405C24—22no—
406C24—21no—
407C24—21no—
408C24—21no—
409C24—21no—
410C24—22no—
411C24—22no—
412C24C2×Dic612yes—
413C24D3×D412yes—
414C24D3×Q812yes—
415C24C23×D311yescharacteristic
416C2×Q8—616no—
417C2×Q8—616no—
418C2×Q8—616no—
419C2×Q8—128no—
420C2×Q8—128no—
421C2×Q8—128no—
422C2×Q8—128no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—32no—
442C22×C4—34no—
443C22×C4—34no—
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—32no—
447C22×C4—32no—
448C22×C4—32no—
449C22×C4—34no—
450C22×C4—34no—
451C22×C4—32no—
452C22×C4—32no—
453C22×C4—616no—
454C22×C4—68no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—61no—
459C22×C4—62no—
460C22×C4—61no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4—62no—
476C22×C4—62no—
477C22×C4—124no—
478C22×C4—124no—
479C22×C4—128no—
480C22×C4C2×Dic612yes—
481C22×C4C2×Dic612yes—
482C22×C4D3×D414yes—
483C22×C4D3×D414yes—
484C22×C4D3×D412yes—
485C22×C4#23712yes—
486C4⋊C4—28no—
487C4⋊C4—616no—
488C4⋊C4—616no—
489C4⋊C4—616no—
490C4⋊C4—616no—
491C4⋊C4—68no—
492C22⋊C4—216no—
493C22⋊C4—28no—
494C22⋊C4—68no—
495C22⋊C4—616no—
496C42—68no—
497C42—124no—
498C22×C3—24no—
499C22×C3—28no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—42no—
504C22×C3—44no—
505C22×C3—44no—
506C22×C3—44no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3C2×#12011yescharacteristic
510C22×C3C2×#12012yes—
511C22×C3C2×#12111yescharacteristic
512C22×C3C2×#12112yes—
513C22×C3C2×C22⋊Q811yescharacteristic
514C22×C3C2×C22⋊Q811yescharacteristic
515C22×C3C2×#12711yescharacteristic
516C22×C3C2×C4⋊Q811yescharacteristic
517C22×C3#53211yescharacteristic
518C22×C3#53311yescharacteristic
519C22×C3#53311yescharacteristic
520C22×C3#53611yescharacteristic
521C22×C3#53612yes—
522C22×C3#53611yescharacteristic
523C22×C3#54112yes—
524C22×C3#54712yes—
525C22×C3#54712yes—
526C22×C3#54812yes—
527C22×C3#54812yes—
528C22×C3D4×Q812yes—
529C22×C3D4×Q812yes—
530C22×C3D4×Q812yes—
531C22×C3D4×Q812yes—
532C3×C4—24no—
533C3×C4—44no—
534C3×C4—44no—
535C3×C4—42no—
536Dic3—24no—
537Dic3—44no—
538Dic3—42no—
539Dic3—44no—
540Dic3—42no—
541Dic3#39218yes—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—41no—
555C23—42no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—41no—
560C23—41no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23#88814yescomplemented
565C23C22×Dic611yescharacteristic
566C23C2×D3×D412yes—
567C23C2×D3×D412yes—
568C23C2×D3×D411yescharacteristic
569C23C2×D3×D411yescharacteristic
570C23C2×#23711yescharacteristic
571C23C2×D3×Q811yescharacteristic
572C23D3×#4111yescharacteristic
573C23D3×#4111yescharacteristic
574C23#101012yes—
575C23#101012yes—
576Q8—1232no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—34no—
590C2×C4—34no—
591C2×C4—38no—
592C2×C4—34no—
593C2×C4—38no—
594C2×C4—34no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—64no—
607C2×C4—62no—
608C2×C4—64no—
609C2×C4—62no—
610C2×C4—64no—
611C2×C4—64no—
612C2×C4—64no—
613C2×C4—64no—
614C2×C4—64no—
615C2×C4—64no—
616C2×C4—62no—
617C2×C4—64no—
618C2×C4—64no—
619C2×C4—62no—
620C2×C4—1216no—
621C2×C4—122no—
622C2×C4—121no—
623C2×C4—122no—
624C2×C4—122no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—122no—
628C2×C4—121no—
629C2×C4—122no—
630C2×C4—122no—
631C2×C4—122no—
632C2×C4—122no—
633C2×C4—122no—
634C2×C4—122no—
635C2×C4#87814yescomplemented
636C2×C3—44no—
637C2×C3C2×#39211yescharacteristic
638C2×C3#240911yescharacteristic
639C2×C3#240911yescharacteristic
640C2×C3#242912yes—
641C2×C3#242912yes—
642C2×C3#243212yes—
643C2×C3#243412yes—
644C2×C3#254211yescharacteristic
645C2×C3#254211yescharacteristic
646C2×C3#262111yescharacteristic
647C2×C3#285711yescharacteristic
648C22—24no—
649C22—28no—
650C22—42no—
651C22—44no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—44no—
658C22—42no—
659C22C2×#87811yescharacteristic
660C22C2×#88811yescharacteristic
661C22#551711yescharacteristic
662C22D4×Dic612yes—
663C22D4×Dic612yes—
664C22D3×#12012yes—
665C22D3×#12011yescharacteristic
666C22#559611yescharacteristic
667C22#559812yes—
668C22D3×#12111yescharacteristic
669C22D3×#12112yes—
670C22#561112yes—
671C22#562211yescharacteristic
672C22D3×C22⋊Q811yescharacteristic
673C22#563212yes—
674C22#563212yes—
675C22#564812yes—
676C22#564811yescharacteristic
677C22#565512yes—
678C22#567511yescharacteristic
679C22#567712yes—
680C22#570112yes—
681C22D3×#12711yescharacteristic
682C4—24no—
683C4—38no—
684C4—42no—
685C4—44no—
686C4—44no—
687C4—64no—
688C4—124no—
689C4—122no—
690C4—124no—
691C4—122no—
692C3#1517811yescharacteristic, complemented, Sylow
693C2—44no—
694C2#7400111yescharacteristic
695C2#7441212yes—
696C2#7468911yescharacteristic
697C2#7550012yes—
698C2#7634911yescharacteristic
699C2D3×#39211yescharacteristic
700C2#7678011yescharacteristic
701C2#7690412yes—
702C2#7728712yes—
703C2#7753811yescharacteristic
704C2#7803711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1439C2
prime radicalC25×C3C23
hypercenter#1439D3
hypocenterC3#15178
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
1. C22×#888C2semidirect
4. C2×#4646C2semidirect
6. #71568C2semidirect
9. #69420C2semidirect
10. #69419C2semidirect
11. Dic3⋊#95C2semidirect
84. C2×C3×C22⋊C4Q8semidirect
90. C2×#217Q8semidirect
296. C2×C22⋊C4Dic6semidirect
332. C23×C3C4⋊Q8semidirect
367. C2×C3×C4C22⋊Q8semidirect
401. C2×Dic3C22⋊Q8semidirect
564. C23#888semidirect
635. C2×C4#878semidirect
692. C3#15178semidirect

Maximal subgroups

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

subgroupindex
C22×#8882
C22×#8782
C2×Dic3×C22⋊C42
C2×#46462
C3×#14392
#715682
#715472
#715462
#694202
#694192
Dic3⋊#952
#151783