Finite Group Databasebeta

← #369433

Subgroups — #369433

Order 512.

1932 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#2352C211yesmaximal, characteristic
3C2×#1993C211yesmaximal, characteristic, complemented
4C2×#1987C211yesmaximal, 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×#124—21no—
14C22×C4×D4C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×#97C2211yescharacteristic
21#2547—24no—
22#2547C418yescomplemented
23C2×#397—21no—
24C2×#397—21no—
25#2352—24no—
26#2352—24no—
27C2×#391—21no—
28C2×#388—21no—
29C2×#387—21no—
30C2×#386—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#1987—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×#124—28no—
64C2×#124—44no—
65C2×#124—44no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4C2×C414yes—
71C2×C4×D4D414yes—
72C2×#117—22no—
73C2×#117—22no—
74C2×#117—24no—
75C2×#117—42no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—42no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C22⋊C4—22no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4D411yescharacteristic
95C22×C22⋊C4D411yescharacteristic
96C22×C42—21no—
97C22×C42—21no—
98C22×C42—21no—
99C22×C42C2311yescharacteristic
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—24no—
107C2×C22⋊C8—42no—
108C2×C22⋊C8—42no—
109C2×#97—22no—
110C2×#97—22no—
111C2×#97—22no—
112#397—28no—
113#397—28no—
114#391—28no—
115#388—28no—
116#387—28no—
117#386—28no—
118#382—22no—
119#382D414yes—
120#380—24no—
121#380—24no—
122#380—22no—
123#380—22no—
124#380—22no—
125#380—22no—
126#380C2×C418yes—
127#380C2×C418yescomplemented
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4C2×C414yescomplemented
131C4×C4⋊C4D414yes—
132C4×C22⋊C4—28no—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141#340—24no—
142#340—24no—
143#340—24no—
144#340—24no—
145#340—24no—
146#340—24no—
147#340—42no—
148#340—42no—
149C25C2×D411yescharacteristic
150C25C2×D411yescharacteristic
151C22×Q8—24no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—41no—
163C22×D4—41no—
164C22×D4C22⋊C414yes—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—24no—
188C23×C4—44no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—82no—
192C23×C4C22×C412yes—
193C23×C4C22×C412yes—
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2411yescharacteristic
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—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8#4114yes—
215C4⋊Q8—28no—
216C4⋊Q8—28no—
217C4⋊Q8—44no—
218C4⋊Q8—44no—
219#124—432no—
220C4×D4—28no—
221C4×D4—28no—
222C4×D4—28no—
223C4×D4—28no—
224#117—48no—
225#117—44no—
226#117—44no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
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—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—48no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C22⋊C414yes—
259C2×C4⋊C4C22×C414yes—
260C2×C4⋊C4C22×C414yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4#4114yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—82no—
283C2×C22⋊C4C22⋊C414yes—
284C2×C22⋊C4C22⋊C414yes—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—44no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—82no—
292C2×C42C22×C412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42#4112yes—
297C2×C42#4112yes—
298C4⋊C8—44no—
299C22⋊C8—48no—
300C22⋊C8—48no—
301#97—44no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—24no—
310C24—21no—
311C24—21no—
312C24—22no—
313C24—22no—
314C24—21no—
315C24—21no—
316C24—22no—
317C24—22no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—22no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24C2×C22⋊C412yes—
329C24C2×C22⋊C412yes—
330C24AGL(1,ℤ/8)12yes—
331C24AGL(1,ℤ/8)12yes—
332C24C22×D411yescharacteristic
333C2×Q8—48no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×Q8—84no—
337C2×Q8—84no—
338C2×Q8—84no—
339C2×Q8—84no—
340C2×D4—28no—
341C2×D4—28no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—44no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—48no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—48no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—84no—
434C22×C4—88no—
435C22×C4—88no—
436C22×C4C2×C22⋊C412yes—
437C22×C4C2×C22⋊C412yes—
438C22×C4C2×C22⋊C412yes—
439C22×C4C2×C22⋊C412yes—
440C22×C4#12012yes—
441C22×C4#12012yes—
442C22×C4#12312yes—
443C22×C4#12312yes—
444C22×C4C23×C411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C42−1+412yes—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—28no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—48no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—48no—
490C4⋊C4#12414yescomplemented
491C22⋊C4—28no—
492C22⋊C4—28no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—416no—
496C22⋊C4—416no—
497C22⋊C4—84no—
498C42—48no—
499C42—48no—
500C42—84no—
501C42C4×D414yes—
502C42C4×D414yes—
503C42#12114yes—
504C42#12114yes—
505C23—24no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—22no—
511C23—24no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—22no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23#41814yes—
551C23C22×C22⋊C411yescharacteristic
552C23C2×#12011yescharacteristic
553C23C2×#12311yescharacteristic
554C23#54612yes—
555C23#54712yes—
556C23C2×AGL(1,ℤ/8)11yescharacteristic
557C23C2×AGL(1,ℤ/8)11yescharacteristic
558C23#57511yescharacteristic
559C23#57511yescharacteristic
560C23#57712yes—
561C23#57812yes—
562Q8—816no—
563D4—416no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—816no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4#38614yes—
632C2×C4#38614yes—
633C2×C4C2×C4×D411yescharacteristic
634C2×C4C2×C4×D411yescharacteristic
635C2×C4C4×#4112yes—
636C2×C4#52012yes—
637C2×C4C2×#12111yescharacteristic
638C2×C4C2×#12111yescharacteristic
639C2×C4C2×#12411yescharacteristic
640C2×C4#52912yes—
641C2×C4#53211yescharacteristic
642C2×C4#53511yescharacteristic
643C2×C4#53611yescharacteristic
644C2×C4#54212yes—
645C2×C4#54612yes—
646C2×C4#54612yes—
647C2×C4#54712yes—
648C2×C4#54812yes—
649C2×C4#55612yes—
650C2×C4#55912yes—
651C8—44no—
652C8—82no—
653C22—24no—
654C22—28no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—44no—
661C22—44no—
662C22—42no—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—44no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—82no—
674C22—82no—
675C22C2×#38611yescharacteristic
676C22#233912yes—
677C22#243211yescharacteristic
678C22#243511yescharacteristic
679C22#244712yes—
680C22C2×#41811yescharacteristic
681C22#289611yescharacteristic
682C22#289812yes—
683C22#300611yescharacteristic
684C22#300611yescharacteristic
685C22#300911yescharacteristic
686C22#301111yescharacteristic
687C22#309411yescharacteristic
688C22#309411yescharacteristic
689C22#309712yes—
690C22#330612yes—
691C22#330812yes—
692C22#331112yes—
693C22#331312yes—
694C22#334312yes—
695C22#334512yes—
696C22#334912yes—
697C22#335012yes—
698C4—24no—
699C4—24no—
700C4—44no—
701C4—42no—
702C4—42no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—84no—
707C4—84no—
708C4C4×#12411yescharacteristic
709C4#232611yescharacteristic
710C4#236712yes—
711C4#242712yes—
712C4#254411yescharacteristic
713C4#254711yescharacteristic
714C2—44no—
715C2—82no—
716C2#1402711yescharacteristic
717C2#2091211yescharacteristic
718C2#2091311yescharacteristic
719C2#2249412yes—
720C2#2249512yes—
721C2#2571411yescharacteristic
722C2#2571611yescharacteristic
723C2#2589611yescharacteristic
724C2#2589811yescharacteristic
725C2#2631212yes—
726C2#2631412yes—
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×#1987C2semidirect
6. #11200C2semidirect
7. #11199C2semidirect
8. #10850C2semidirect
9. #10731C2semidirect
10. #10730C2semidirect
11. #10703C2semidirect
22. #2547C4semidirect
62. C2×C4⋊Q8C2×C4semidirect
127. #380C2×C4semidirect
130. C4×C4⋊C4C2×C4semidirect
490. C4⋊C4#124semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#23522
C2×#19932
C2×#19872
C2×#19552
#112002
#111992
#108502
#107312
#107302
#107032