Finite Group Databasebeta

← #364652

Subgroups — #364652

Order 512.

2164 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#481C211yesmaximal, characteristic
2C2×#2361C211yesmaximal, characteristic
3C2×#1893C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic, complemented
5#13924C211yesmaximal, characteristic, complemented
6#11220C212yesmaximal, complemented
7#11189C212yesmaximal, complemented
8#10908C212yesmaximal, complemented
9#10788C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic, complemented
11#10714C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C22×C4×D4—21no—
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#481—24no—
19C2×#481—24no—
20C2×#481C418yescomplemented
21C22×#106C2212yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25#2361—24no—
26#2361—24no—
27C2×#390C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#388—21no—
30C2×#386—21no—
31C2×#385—21no—
32C2×#380C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35#1893—24no—
36C2×#340—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1445—21no—
45#1445—21no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C22⋊#95—21no—
49C22⋊#95—21no—
50C23×D4—21no—
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C24×C4—21no—
54C24×C4—21no—
55C24×C4C2311yescharacteristic
56C23×C8—22no—
57C23×C8—24no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—42no—
61C23×C8C2312yes—
62C23×C8C2312yes—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121C2×C414yes—
67C2×#121D414yes—
68C2×#121D414yes—
69C2×C4×D4—28no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—24no—
73C2×C4×D4—42no—
74C2×C4×D4—44no—
75C2×C4×D4—44no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4C2×C414yescomplemented
88C22×C4⋊C4D414yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4D411yescharacteristic
100C22×C22⋊C4D411yescharacteristic
101C22×C42—21no—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104#481—232no—
105C2×#106—24no—
106C2×#106—24no—
107C2×#106C2×C418yescomplemented
108C2×#102—24no—
109C2×#102—24no—
110C2×#102—24no—
111C2×#102—24no—
112C2×#102—42no—
113C2×#102C2×C418yescomplemented
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8C2×C414yescomplemented
117#390—22no—
118#390—22no—
119#389—22no—
120#389—22no—
121#388—28no—
122#386—28no—
123#385—28no—
124#380—22no—
125#380—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95D411yescharacteristic
138C2×#95D411yescharacteristic
139#340—24no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—42no—
144#336—24no—
145#336—42no—
146C25—21no—
147C25C2×D411yescharacteristic
148C22×D4—24no—
149C22×D4—24no—
150C22×D4—24no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4C22×C412yes—
163C22×D4C2×D412yes—
164C22×D4C2×D412yes—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—82no—
186C23×C4—82no—
187C23×C4—82no—
188C23×C4—82no—
189C23×C4C22×C412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D412yes—
198C23×C4#4112yes—
199C23×C4#4112yes—
200C23×C4C2411yescharacteristic
201C22×C8—28no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—48no—
215C22×C8—48no—
216C22×C8—44no—
217C22×C8—44no—
218C22×C8C22×C414yes—
219C22×C8#4114yes—
220#121—28no—
221#121—28no—
222#121—28no—
223#121C22⋊C4116yes—
224C4×D4—432no—
225C4×D4—48no—
226C4×D4—44no—
227C4×D4—44no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—21no—
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—44no—
254C2×C4⋊C4—48no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C22×C414yes—
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4#4114yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—82no—
287C2×C22⋊C4SD1614yes—
288C2×C42—24no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—42no—
298C2×C42—42no—
299#106—216no—
300#102—216no—
301#102—48no—
302C22⋊C8—28no—
303#95—24no—
304#95—24no—
305#95—22no—
306#95—22no—
307#95—42no—
308#95—82no—
309#95—82no—
310#95—82no—
311#95—82no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—42no—
320C24—42no—
321C24—41no—
322C24—41no—
323C24—41no—
324C24—41no—
325C24—42no—
326C24—42no—
327C24C4×D412yes—
328C24#12012yes—
329C24#12112yes—
330C24C2×SD1612yes—
331C24#13512yes—
332C24AGL(1,ℤ/8)12yes—
333C24#13712yes—
334C24C22×D411yescharacteristic
335C2×D4—216no—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—42no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4C2×C22⋊C414yes—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—88no—
431C22×C4—84no—
432C22×C4—88no—
433C22×C4—84no—
434C22×C4—88no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4—88no—
440C22×C4C2×C22⋊C414yes—
441C22×C4C4×D414yes—
442C22×C4C4×D412yes—
443C22×C4#12012yes—
444C22×C4#12114yes—
445C22×C4#12112yes—
446C22×C4#12314yes—
447C22×C4C2×SD1612yes—
448C22×C4C2×SD1612yes—
449C22×C4C23×C411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4111yescharacteristic
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+412yes—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—416no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—48no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4C2×C22⋊C414yes—
494C4⋊C4#12314yes—
495C22⋊C4—28no—
496C22⋊C4—216no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—44no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C42—48no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—42no—
508C42—42no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23—82no—
544C23—82no—
545C23—82no—
546C23—82no—
547C23—82no—
548C23#38614yes—
549C23C4×SD1614yes—
550C23#44014yes—
551C23#45014yes—
552C23#45214yes—
553C23#46114yes—
554C23#46314yes—
555C23C2×C4×D411yescharacteristic
556C23C2×#12011yescharacteristic
557C23C2×#12111yescharacteristic
558C23#54612yes—
559C23#54712yes—
560C23C22×SD1611yescharacteristic
561C23C2×#13511yescharacteristic
562C23C2×AGL(1,ℤ/8)11yescharacteristic
563C23C2×#13711yescharacteristic
564C23#57612yes—
565C23#57712yes—
566D4—28no—
567D4—432no—
568D4—42no—
569D4—44no—
570D4—42no—
571D4—42no—
572D4—42no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
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—24no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—816no—
622C2×C4—816no—
623C2×C4—816no—
624C2×C4—82no—
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—82no—
637C2×C4#38614yes—
638C2×C4C22×C22⋊C411yescharacteristic
639C2×C4C2×C4×D411yescharacteristic
640C2×C4C4×#4112yes—
641C2×C4#52012yes—
642C2×C4C2×#12111yescharacteristic
643C2×C4C2×#12311yescharacteristic
644C2×C4C2×#12311yescharacteristic
645C2×C4#52511yescharacteristic
646C2×C4#52912yes—
647C2×C4#53411yescharacteristic
648C2×C4#53611yescharacteristic
649C2×C4#53912yes—
650C2×C4#54612yes—
651C2×C4#54612yes—
652C2×C4#54712yes—
653C2×C4#54712yes—
654C2×C4#55112yes—
655C2×C4#55212yes—
656C8—44no—
657C8—44no—
658C22—24no—
659C22—24no—
660C22—28no—
661C22—28no—
662C22—28no—
663C22—24no—
664C22—28no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—82no—
672C22—82no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—82no—
678C22—82no—
679C22#187718yes—
680C22C2×#38611yescharacteristic
681C22#235912yes—
682C22#240511yescharacteristic
683C22#240611yescharacteristic
684C22#242512yes—
685C22C2×C4×SD1611yescharacteristic
686C22C2×#44011yescharacteristic
687C22#294612yes—
688C22C2×#45011yescharacteristic
689C22C2×#45211yescharacteristic
690C22#300811yescharacteristic
691C22#301011yescharacteristic
692C22C2×#46111yescharacteristic
693C22C2×#46311yescharacteristic
694C22#304512yes—
695C22#329512yes—
696C22#330112yes—
697C22#331612yes—
698C22#331812yes—
699C22#333512yes—
700C22#333912yes—
701C22#334512yes—
702C22#334612yes—
703C4—24no—
704C4—42no—
705C4—44no—
706C4—42no—
707C4—44no—
708C4—81no—
709C4—84no—
710C4—82no—
711C4—81no—
712C4—162no—
713C4C4×#12311yescharacteristic
714C4#232211yescharacteristic
715C4#233912yes—
716C4#241912yes—
717C4#248611yescharacteristic
718C4#253411yescharacteristic
719C2—28no—
720C2—84no—
721C2#1396811yescharacteristic
722C2C2×#187711yescharacteristic
723C2#2093211yescharacteristic
724C2#2291812yes—
725C2#2293612yes—
726C2#2403311yescharacteristic
727C2#2403611yescharacteristic
728C2#2408911yescharacteristic
729C2#2409211yescharacteristic
730C2#2520212yes—
731C2#2521412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C22×#340C2semidirect
5. #13924C2semidirect
6. #11220C2semidirect
7. #11189C2semidirect
8. #10908C2semidirect
9. #10788C2semidirect
10. #10730C2semidirect
11. #10714C2semidirect
20. C2×#481C4semidirect
87. C22×C4⋊C4C2×C4semidirect
107. C2×#106C2×C4semidirect
113. C2×#102C2×C4semidirect
116. C2×C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4812
C2×#23612
C2×#18932
C22×#3402
#139242
#112202
#111892
#109082
#107882
#107302
#107142