Finite Group Databasebeta

← #362764

Subgroups — #362764

Order 512.

2766 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#1991C211yesmaximal, characteristic, complemented
3C2×#1989C211yesmaximal, characteristic, complemented
4C2×#1984C211yesmaximal, characteristic, complemented
5C2×#1804C211yesmaximal, characteristic
6C2×C4×#102C211yesmaximal, characteristic
7#13925C211yesmaximal, characteristic, complemented
8#11385C212yesmaximal, complemented
9#11384C212yesmaximal, complemented
10#10788C212yesmaximal
11#10788C212yesmaximal
12C2×#554—24no—
13C2×#554—24no—
14C2×#554—24no—
15C2×#554—24no—
16C2×#554—24no—
17C2×#554C418yescomplemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C22×C22⋊Q8—21no—
22C22×C22⋊Q8—21no—
23C22×C22⋊Q8—21no—
24C22×C4×Q8C2211yescharacteristic
25C22×C4×D4C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C22×C4×D4C2211yescharacteristic
28C23×C4⋊C4—21no—
29C22×C4⋊C8C2211yescharacteristic
30C22×#102C2212yes—
31C22×#102C2212yes—
32C22×C4×C8C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1991—24no—
36#1989—24no—
37#1984—24no—
38#1804—24no—
39C4×#102—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#336C2212yes—
47C2×#336C2212yes—
48#1446—21no—
49#1445—21no—
50#1436—21no—
51#1436—21no—
52C4⋊#95C2212yes—
53C4⋊#95C2212yes—
54C23×Q8—21no—
55C23×Q8D411yescharacteristic
56C23×D4C2311yescharacteristic
57C24×C4—21no—
58C24×C4—21no—
59C24×C4—21no—
60C23×C8C2312yes—
61C23×C8C2312yes—
62#554—232no—
63#554—232no—
64#554—232no—
65#554—232no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8C2×C414yes—
72C2×#125—22no—
73C2×#125—22no—
74C2×#125—22no—
75C2×#125—22no—
76C2×#125—22no—
77C2×#125—22no—
78C2×#125—22no—
79C2×#125—22no—
80C2×#125C2×C414yes—
81C2×#125C2×C414yes—
82C2×#125D414yes—
83C2×#125D414yes—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C4×Q8—22no—
94C2×C4×Q8—22no—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×Q8C2×C414yes—
98C2×C4×Q8D414yes—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4C2×C414yes—
113C2×C4×D4C2×C414yes—
114C2×C4×D4C2×C414yes—
115C2×C4×D4D414yes—
116C2×C4×D4D414yes—
117C22×C4⋊C4—28no—
118C22×C4⋊C4—28no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—28no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C2×C4⋊C8—22no—
141C2×C4⋊C8—22no—
142C2×C4⋊C8—22no—
143C2×#102—24no—
144C2×#102—24no—
145C2×#102—24no—
146C2×#102—24no—
147C2×#102—24no—
148C2×#102Q818yescomplemented
149C2×C4×C8—22no—
150C2×C4×C8—22no—
151C2×C4×C8—22no—
152#382—22no—
153#382—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—22no—
156C2×#95—22no—
157C2×#95—22no—
158C2×#95—21no—
159C2×#95—21no—
160C2×#95—21no—
161C2×#95—22no—
162C2×#95—22no—
163#340—24no—
164#340—24no—
165#340—24no—
166#340—24no—
167#340—24no—
168#340—24no—
169#336—24no—
170#336—24no—
171C25—21no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—22no—
176C22×Q8—22no—
177C22×Q8—24no—
178C22×Q8—42no—
179C22×Q8—82no—
180C22×Q8—82no—
181C22×Q8C22⋊C414yes—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4C22×C412yes—
188C22×D4C2×D412yes—
189C22×D4C2×D412yes—
190C22×D4C2×Q814yes—
191C22×D4#4114yes—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—82no—
209C23×C4—82no—
210C23×C4—82no—
211C23×C4—82no—
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2411yescharacteristic
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8C2×Q814yes—
233C4⋊Q8—28no—
234C4⋊Q8—28no—
235C4⋊Q8—28no—
236C4⋊Q8—28no—
237#125—28no—
238#125—28no—
239#125—28no—
240#125—28no—
241#125—28no—
242#125—28no—
243#125—28no—
244#125C22⋊C4116yes—
245C22⋊Q8—264no—
246C22⋊Q8—264no—
247C22⋊Q8—264no—
248C4×Q8—28no—
249C4×Q8—28no—
250C4×Q8—28no—
251C4×Q8—28no—
252C4×D4—28no—
253C4×D4—28no—
254C4×D4—28no—
255C4×D4—28no—
256C4×D4—28no—
257C4×D4—28no—
258C4×D4—28no—
259C4×D4—28no—
260C4×D4—28no—
261C4×D4—28no—
262C4×D4—28no—
263C4×D4C22⋊C4116yes—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—264no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4C22⋊C414yes—
307C2×C4⋊C4C22⋊C414yes—
308C2×C4⋊C4C22⋊C414yes—
309C2×C4⋊C4C22×C412yes—
310C2×C4⋊C4C22×C412yes—
311C2×C4⋊C4C22×C412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×Q814yes—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—42no—
337C2×C42—82no—
338C2×C42C22×C412yes—
339C2×C42C22×C412yes—
340C2×C42C22×C412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C4⋊C8—44no—
349#102—216no—
350#102—48no—
351C4×C8—44no—
352#95—44no—
353#95—42no—
354#95—42no—
355#95—42no—
356#95—44no—
357#95—82no—
358#95—82no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—24no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24C22×D411yescharacteristic
369C2×Q8—216no—
370C2×Q8—28no—
371C2×Q8—28no—
372C2×Q8—44no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—88no—
376C2×Q8—84no—
377C2×Q8—84no—
378C2×Q8—88no—
379C2×D4—22no—
380C2×D4—24no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—24no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4C2×C22⋊C414yes—
393C2×D4C4×Q818yes—
394C2×D4C22⋊Q818yes—
395C2×D4#12414yes—
396C2×D4C4⋊Q814yes—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—216no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—216no—
427C22×C4—216no—
428C22×C4—24no—
429C22×C4—48no—
430C22×C4—416no—
431C22×C4—416no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—48no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—88no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4—88no—
460C22×C4C2×C22⋊C412yes—
461C22×C4C2×C22⋊C412yes—
462C22×C4C2×C22⋊C412yes—
463C22×C4C2×C22⋊C412yes—
464C22×C4C2×C22⋊C412yes—
465C22×C4C2×C22⋊C412yes—
466C22×C4#12012yes—
467C22×C4#12012yes—
468C22×C4#12012yes—
469C22×C4#12012yes—
470C22×C4#13514yes—
471C22×C4C23×C411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×Q812yes—
478C22×C4C2×#4112yes—
479C22×C42−1+412yes—
480C22×C42−1+412yes—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—42no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—42no—
490C2×C8—42no—
491C2×C8—42no—
492C2×C8—42no—
493C2×C8—42no—
494C2×C8—42no—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—216no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—28no—
508C4⋊C4—216no—
509C4⋊C4—216no—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4C2×C22⋊C414yes—
521C4⋊C4C2×C22⋊C414yes—
522C4⋊C4C2×C22⋊C414yes—
523C22⋊C4—216no—
524C22⋊C4—216no—
525C22⋊C4—216no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—84no—
535C42C2×C22⋊C414yes—
536C42C2×C22⋊C414yes—
537C42#12014yes—
538C42#12014yes—
539C23—28no—
540C23—28no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23C22×C22⋊C411yescharacteristic
561C23C2×#12011yescharacteristic
562C23C2×#12011yescharacteristic
563C23#54812yes—
564C23D4×Q812yes—
565C23C2×#13512yes—
566C23#57512yes—
567C23#57612yes—
568C23#57712yes—
569Q8—416no—
570Q8—816no—
571D4—24no—
572D4—28no—
573D4—28no—
574D4—24no—
575D4—24no—
576D4#39118yes—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—432no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—816no—
624C2×C4—816no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4#37914yes—
637C2×C4#37914yes—
638C2×C4#41614yes—
639C2×C4C22×C22⋊C411yescharacteristic
640C2×C4C22×C22⋊C411yescharacteristic
641C2×C4C2×C4×Q812yes—
642C2×C4#51912yes—
643C2×C4C2×#12011yescharacteristic
644C2×C4C2×#12011yescharacteristic
645C2×C4C2×C22⋊Q812yes—
646C2×C4C2×#12411yescharacteristic
647C2×C4C2×C4⋊Q811yescharacteristic
648C2×C4#53612yes—
649C2×C4#53611yescharacteristic
650C2×C4#53611yescharacteristic
651C2×C4#54812yes—
652C2×C4#54812yes—
653C2×C4#54812yes—
654C2×C4D4×Q812yes—
655C2×C4D4×Q812yes—
656C2×C4D4×Q812yes—
657C8—44no—
658C8—82no—
659C22—28no—
660C22—48no—
661C22—48no—
662C22—48no—
663C22—44no—
664C22—44no—
665C22—48no—
666C22—48no—
667C22—48no—
668C22C2×#37911yescharacteristic
669C22Q8×C22⋊C412yes—
670C22#240912yes—
671C22#243411yescharacteristic
672C22#243611yescharacteristic
673C22C2×#41611yescharacteristic
674C22#289611yescharacteristic
675C22#289712yes—
676C22#300212yes—
677C22#300212yes—
678C22#300811yescharacteristic
679C22#300911yescharacteristic
680C22#301212yes—
681C22#336012yes—
682C22#336112yes—
683C22#337012yes—
684C22#337612yes—
685C22#338012yes—
686C22#338112yes—
687C22#338812yes—
688C22#338912yes—
689C4—24no—
690C4—24no—
691C4—24no—
692C4—42no—
693C4—44no—
694C4—42no—
695C4—42no—
696C4—44no—
697C4—44no—
698C4—84no—
699C4—162no—
700C4C2×#39111yescharacteristic
701C4#231011yescharacteristic
702C4Q8×C22⋊C412yes—
703C4#240912yes—
704C4#243411yescharacteristic
705C4#243611yescharacteristic
706C2—48no—
707C2#1392211yescharacteristic
708C2#2059711yescharacteristic
709C2#2060211yescharacteristic
710C2#2251912yes—
711C2#2252612yes—
712C2#2433512yes—
713C2#2433712yes—
714C2#2584312yes—
715C2#2593812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1991C2semidirect
3. C2×#1989C2semidirect
4. C2×#1984C2semidirect
7. #13925C2semidirect
8. #11385C2semidirect
9. #11384C2semidirect
17. C2×#554C4semidirect
148. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#19912
C2×#19892
C2×#19842
C2×#18042
C2×C4×#1022
#139252
#113852
#113842
#107882