Finite Group Databasebeta

← #368449

Subgroups — #368449

Order 512.

2014 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#389C211yesmaximal, characteristic
2C2×#1933C212yesmaximal, complemented
3C2×#1872C211yesmaximal, characteristic
4C2×#1872C211yesmaximal, characteristic
5C2×C4×C22⋊C8C211yesmaximal, characteristic
6#14702C211yesmaximal, characteristic
7#11293C212yesmaximal, complemented
8#11067C212yesmaximal, complemented
9#10880C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10730C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C23×C4⋊C4—21no—
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C4×C8C2212yes—
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#389—21no—
25C2×#389—21no—
26C2×#389—28no—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#380C2211yescharacteristic
30C22×#95—21no—
31#1933—28no—
32#1872—24no—
33#1872C418yescomplemented
34C4×C22⋊C8—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42#1436—21no—
43#1436—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C22⋊#95—21no—
47C22⋊#95—21no—
48C23×Q8C2311yescharacteristic
49C24×C4—21no—
50C24×C4—21no—
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4C2311yescharacteristic
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8C2×C414yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—28no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4—44no—
76C22×C4⋊C4—44no—
77C22×C4⋊C4C2×C414yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C42—22no—
86C22×C42—22no—
87C22×C42—24no—
88C22×C42—42no—
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—42no—
98C2×#103C2×C418yescomplemented
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8C2×C414yes—
104C2×C22⋊C8D414yes—
105C2×C4×C8—24no—
106C2×C4×C8—24no—
107C2×C4×C8—42no—
108#391—24no—
109#391—42no—
110#389—28no—
111#389—28no—
112#389—24no—
113#389—24no—
114#389—24no—
115#389D418yes—
116#380—22no—
117#380—22no—
118C2×#95—28no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—42no—
135#340C2×C418yescomplemented
136#336—24no—
137C25C2×D411yescharacteristic
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—22no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8C22×C412yes—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—41no—
171C23×C4—41no—
172C23×C4—41no—
173C23×C4—41no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4SD1614yes—
185C23×C4Q1614yes—
186C23×C4C22×C412yes—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4#4112yes—
194C23×C4#4112yes—
195C23×C4#4112yes—
196C23×C4#4112yes—
197C23×C4C2411yescharacteristic
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
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×C8C22×C414yes—
211C22×C8C2×D414yes—
212C22⋊Q8—28no—
213C22⋊Q8—28no—
214C22⋊Q8—44no—
215C22⋊Q8—44no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—216no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—216no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—432no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C22×C412yes—
258C2×C4⋊C4C22×C414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4#4114yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—21no—
276C2×C42—22no—
277C2×C42—21no—
278C2×C42—22no—
279C2×C42—48no—
280C2×C42—48no—
281C2×C42—41no—
282C2×C42—41no—
283C2×C42C2×D414yes—
284#103—216no—
285#103—48no—
286C22⋊C8—28no—
287C22⋊C8—28no—
288C4×C8—48no—
289#95—28no—
290#95—28no—
291#95—42no—
292#95—42no—
293#95—42no—
294#95—42no—
295#95—42no—
296#95—42no—
297#95—42no—
298#95—42no—
299#95—42no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24C4×D412yes—
309C24#12112yes—
310C24#12412yes—
311C24C2×SD1612yes—
312C24C2×Q1612yes—
313C24#13512yes—
314C24#13512yes—
315C24C22×D411yescharacteristic
316C2×Q8—22no—
317C2×Q8—24no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—41no—
321C2×Q8—42no—
322C2×Q8—41no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—216no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4#11714yes—
439C22×C4C4×D414yes—
440C22×C4C4×D412yes—
441C22×C4#12012yes—
442C22×C4#12012yes—
443C22×C4#12112yes—
444C22×C4#12314yes—
445C22×C4#12412yes—
446C22×C4#12614yes—
447C22×C4C2×SD1612yes—
448C22×C4C2×SD1612yes—
449C22×C4C2×Q1612yes—
450C22×C4C2×Q1612yes—
451C22×C4AGL(1,ℤ/8)12yes—
452C22×C4#13712yes—
453C22×C4C23×C411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C2×#4111yescharacteristic
457C22×C4C2×#4111yescharacteristic
458C22×C4C2×#4112yes—
459C22×C4C2×#4111yescharacteristic
460C22×C4C2×#4111yescharacteristic
461C22×C4H(5,𝔽2)12yes—
462C22×C42−1+412yes—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C4⋊C4—216no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—48no—
486C4⋊C4—416no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—48no—
492C4⋊C4—416no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4C4×D418yes—
498C22⋊C4—48no—
499C22⋊C4—84no—
500C42—24no—
501C42—24no—
502C42—44no—
503C42—44no—
504C42—42no—
505C42—42no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23#38814yes—
535C23C4×SD1614yes—
536C23C4×Q1614yes—
537C23#46514yes—
538C23#46714yes—
539C23#48714yes—
540C23#48814yes—
541C23C2×C4×D411yescharacteristic
542C23C2×#12011yescharacteristic
543C23C2×#12111yescharacteristic
544C23#52511yescharacteristic
545C23C2×#12411yescharacteristic
546C23#53411yescharacteristic
547C23#53611yescharacteristic
548C23C22×SD1611yescharacteristic
549C23C22×Q1611yescharacteristic
550C23C2×#13511yescharacteristic
551C23C2×#13511yescharacteristic
552C23C2×AGL(1,ℤ/8)11yescharacteristic
553C23C2×#13711yescharacteristic
554C23#57511yescharacteristic
555C23#57511yescharacteristic
556Q8—44no—
557Q8—44no—
558Q8—82no—
559Q8—82no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—28no—
589C2×C4—416no—
590C2×C4—416no—
591C2×C4—432no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—816no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—81no—
624C2×C4—82no—
625C2×C4—81no—
626C2×C4—82no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4#38814yes—
643C2×C4#45014yes—
644C2×C4#45114yes—
645C2×C4C2×#11711yescharacteristic
646C2×C4C2×C4×D412yes—
647C2×C4C2×C4×D411yescharacteristic
648C2×C4#52012yes—
649C2×C4C2×#12311yescharacteristic
650C2×C4#52512yes—
651C2×C4C2×#12611yescharacteristic
652C2×C4#53912yes—
653C2×C4D4212yes—
654C2×C4#54612yes—
655C2×C4#54712yes—
656C2×C4#54812yes—
657C2×C4#54812yes—
658C2×C4#55212yes—
659C2×C4#55312yes—
660C2×C4#55312yes—
661C8—44no—
662C8—44no—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22#193318yescomplemented
678C22C2×#38811yescharacteristic
679C22C4×#12011yescharacteristic
680C22#232211yescharacteristic
681C22#241011yescharacteristic
682C22#241311yescharacteristic
683C22#243611yescharacteristic
684C22#259111yescharacteristic
685C22C2×C4×SD1611yescharacteristic
686C22C2×C4×Q1611yescharacteristic
687C22C4×AGL(1,ℤ/8)11yescharacteristic
688C22C4×#13711yescharacteristic
689C22C2×#45011yescharacteristic
690C22C2×#45111yescharacteristic
691C22#300211yescharacteristic
692C22#300211yescharacteristic
693C22C2×#46511yescharacteristic
694C22C2×#46711yescharacteristic
695C22#305511yescharacteristic
696C22#305611yescharacteristic
697C22#310111yescharacteristic
698C22#310411yescharacteristic
699C22#310511yescharacteristic
700C22#310811yescharacteristic
701C22C2×#48711yescharacteristic
702C22C2×#48811yescharacteristic
703C22#314111yescharacteristic
704C22#314211yescharacteristic
705C22#318711yescharacteristic
706C22#318811yescharacteristic
707C22#318911yescharacteristic
708C22#319011yescharacteristic
709C22#322911yescharacteristic
710C22#323011yescharacteristic
711C22#323211yescharacteristic
712C22#323211yescharacteristic
713C4—24no—
714C4—24no—
715C4—24no—
716C4—42no—
717C4—42no—
718C4—48no—
719C4—84no—
720C4—82no—
721C4—82no—
722C4—161no—
723C4—161no—
724C4#235212yes—
725C4#236112yes—
726C4#244312yes—
727C4#248112yes—
728C2—28no—
729C2#1401911yescharacteristic
730C2C2×#193311yescharacteristic
731C2#2109311yescharacteristic
732C2C4×#45011yescharacteristic
733C2C4×#45111yescharacteristic
734C2#2207811yescharacteristic
735C2#2207911yescharacteristic
736C2#2437911yescharacteristic
737C2#2438011yescharacteristic
738C2#2460111yescharacteristic
739C2#2460211yescharacteristic
740C2#2590211yescharacteristic
741C2#2590711yescharacteristic
742C2#2932911yescharacteristic
743C2#2933011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1933C2semidirect
7. #11293C2semidirect
8. #11067C2semidirect
9. #10880C2semidirect
33. #1872C4semidirect
98. C2×#103C2×C4semidirect
135. #340C2×C4semidirect
677. C22#1933semidirect

Maximal subgroups

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

subgroupindex
C22×#3892
C2×#19332
C2×#18722
C2×C4×C22⋊C82
#147022
#112932
#110672
#108802
#107302