Finite Group Databasebeta

← #369432

Subgroups — #369432

Order 512.

1856 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2352C211yesmaximal, characteristic
3C2×#1993C211yesmaximal, characteristic, complemented
4C2×#1988C211yesmaximal, characteristic, complemented
5C2×#1955C211yesmaximal, characteristic
6#11200C212yesmaximal, complemented
7#11199C212yesmaximal, complemented
8#10850C211yesmaximal, characteristic, complemented
9#10731C212yesmaximal, complemented
10#10730C212yesmaximal, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×#97C2211yescharacteristic
20#2563—24no—
21#2563C418yescomplemented
22C2×#393—21no—
23C2×#393—21no—
24#2352—24no—
25#2352—24no—
26C2×#390—21no—
27C2×#388—21no—
28C2×#387—21no—
29C2×#386—21no—
30C2×#382—21no—
31C2×#382C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4—21no—
39#1993—24no—
40#1988—24no—
41#1955—24no—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C2×#340C2212yes—
49C2×#340C2212yes—
50C23×Q8—21no—
51C23×D4D411yescharacteristic
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C24×C4C2311yescharacteristic
55C23×C8C2312yes—
56C23×C8C2312yes—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8C2×C414yescomplemented
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4C2×C414yes—
68C2×C4×D4D414yes—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—24no—
72C2×#117—42no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—42no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—22no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4D411yescharacteristic
92C22×C22⋊C4D411yescharacteristic
93C22×C42—21no—
94C22×C42—21no—
95C22×C42—21no—
96C22×C42C2311yescharacteristic
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—42no—
105C2×C22⋊C8—42no—
106C2×#97—22no—
107C2×#97—22no—
108C2×#97—22no—
109#393—28no—
110#393—28no—
111#390—28no—
112#388—28no—
113#387—28no—
114#386—28no—
115#382—28no—
116#382—22no—
117#382D414yes—
118#380—24no—
119#380—24no—
120#380—22no—
121#380—22no—
122#380—22no—
123#380—22no—
124#380C2×C418yes—
125#380C2×C418yescomplemented
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4C2×C414yescomplemented
129C4×C4⋊C4D414yes—
130C4×C22⋊C4—28no—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139#340—24no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—24no—
144#340—24no—
145#340—42no—
146#340—42no—
147C25C2×D411yescharacteristic
148C25C2×D411yescharacteristic
149C22×Q8—24no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8—82no—
153C22×Q8—82no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—41no—
159C22×D4—41no—
160C22×D4C22⋊C414yes—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—24no—
184C23×C4—44no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—82no—
188C23×C4C22×C412yes—
189C23×C4C22×C412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2411yescharacteristic
197C22×C8—22no—
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×C8#4114yes—
211C4⋊Q8—28no—
212C4⋊Q8—28no—
213C4⋊Q8—44no—
214C4⋊Q8—44no—
215C4×D4—28no—
216C4×D4—28no—
217C4×D4—28no—
218C4×D4—28no—
219#117—48no—
220#117—44no—
221#117—44no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C22⋊C414yes—
254C2×C4⋊C4C22×C414yes—
255C2×C4⋊C4C22×C414yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4#4114yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—82no—
274C2×C22⋊C4—82no—
275C2×C22⋊C4—82no—
276C2×C22⋊C4—82no—
277C2×C22⋊C4—82no—
278C2×C22⋊C4C22⋊C414yes—
279C2×C22⋊C4C22⋊C414yes—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—44no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—82no—
287C2×C42C22×C412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42#4112yes—
292C2×C42#4112yes—
293C4⋊C8—44no—
294C22⋊C8—48no—
295C22⋊C8—48no—
296#97—44no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—24no—
305C24—21no—
306C24—21no—
307C24—22no—
308C24—22no—
309C24—21no—
310C24—21no—
311C24—22no—
312C24—22no—
313C24—21no—
314C24—21no—
315C24—22no—
316C24—22no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—22no—
322C24—22no—
323C24C2×C22⋊C412yes—
324C24C2×C22⋊C412yes—
325C24#13712yes—
326C24#13712yes—
327C24C22×D411yescharacteristic
328C2×Q8—48no—
329C2×Q8—88no—
330C2×Q8—88no—
331C2×Q8—84no—
332C2×Q8—84no—
333C2×D4—28no—
334C2×D4—28no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—44no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—48no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—48no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—84no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4C2×C22⋊C412yes—
430C22×C4C2×C22⋊C412yes—
431C22×C4C2×C22⋊C412yes—
432C22×C4C2×C22⋊C412yes—
433C22×C4#12012yes—
434C22×C4#12012yes—
435C22×C4#12312yes—
436C22×C4#12312yes—
437C22×C4C23×C411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C42−1+412yes—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8—42no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—28no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—48no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—48no—
483C4⋊C4#12414yes—
484C22⋊C4—28no—
485C22⋊C4—28no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C22⋊C4—84no—
491C42—48no—
492C42—48no—
493C42—84no—
494C42C4×D414yes—
495C42C4×D414yes—
496C42#12114yes—
497C42#12114yes—
498C23—24no—
499C23—24no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—22no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23#41914yes—
544C23C22×C22⋊C411yescharacteristic
545C23C2×#12011yescharacteristic
546C23C2×#12311yescharacteristic
547C23#54612yes—
548C23#54712yes—
549C23C2×#13711yescharacteristic
550C23C2×#13711yescharacteristic
551C23#57511yescharacteristic
552C23#57511yescharacteristic
553C23#57612yes—
554C23#57712yes—
555Q8—816no—
556D4—416no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—816no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4#38614yes—
625C2×C4#38614yes—
626C2×C4C2×C4×D411yescharacteristic
627C2×C4C2×C4×D411yescharacteristic
628C2×C4C4×#4112yes—
629C2×C4#52012yes—
630C2×C4C2×#12111yescharacteristic
631C2×C4C2×#12111yescharacteristic
632C2×C4C2×#12411yescharacteristic
633C2×C4#52912yes—
634C2×C4#53211yescharacteristic
635C2×C4#53511yescharacteristic
636C2×C4#53611yescharacteristic
637C2×C4#54212yes—
638C2×C4#54612yes—
639C2×C4#54612yes—
640C2×C4#54712yes—
641C2×C4#54812yes—
642C2×C4#55612yes—
643C2×C4#55912yes—
644C8—44no—
645C8—82no—
646C22—24no—
647C22—28no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—44no—
654C22—42no—
655C22—44no—
656C22—44no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—44no—
661C22—44no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—82no—
667C22—82no—
668C22C2×#38611yescharacteristic
669C22#233912yes—
670C22#243211yescharacteristic
671C22#243511yescharacteristic
672C22#244712yes—
673C22C2×#41911yescharacteristic
674C22#289611yescharacteristic
675C22#289712yes—
676C22#300711yescharacteristic
677C22#300711yescharacteristic
678C22#300811yescharacteristic
679C22#301011yescharacteristic
680C22#309511yescharacteristic
681C22#309511yescharacteristic
682C22#309612yes—
683C22#331412yes—
684C22#331612yes—
685C22#331912yes—
686C22#332112yes—
687C22#334412yes—
688C22#334612yes—
689C22#334712yes—
690C22#334812yes—
691C4—24no—
692C4—24no—
693C4—44no—
694C4—42no—
695C4—42no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—84no—
700C4—84no—
701C4C4×#12411yescharacteristic
702C4#232611yescharacteristic
703C4#236712yes—
704C4#242712yes—
705C4#254411yescharacteristic
706C4#254711yescharacteristic
707C2—44no—
708C2—82no—
709C2#1402711yescharacteristic
710C2#2091111yescharacteristic
711C2#2091411yescharacteristic
712C2#2249212yes—
713C2#2249712yes—
714C2#2571311yescharacteristic
715C2#2571511yescharacteristic
716C2#2589511yescharacteristic
717C2#2589711yescharacteristic
718C2#2631312yes—
719C2#2631512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1993C2semidirect
4. C2×#1988C2semidirect
6. #11200C2semidirect
7. #11199C2semidirect
8. #10850C2semidirect
9. #10731C2semidirect
10. #10730C2semidirect
11. #10703C2semidirect
21. #2563C4semidirect
62. C2×C4⋊Q8C2×C4semidirect
125. #380C2×C4semidirect
128. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#23522
C2×#19932
C2×#19882
C2×#19552
#112002
#111992
#108502
#107312
#107302
#107032