Finite Group Databasebeta

← #356132

Subgroups — #356132

Order 512.

2434 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2342C211yesmaximal, characteristic
2C2×#1819C211yesmaximal, characteristic
3C22×#342C211yesmaximal, characteristic
4C22⋊C42C212yesmaximal
5#10912C214yesmaximal, complemented
6#10801C212yesmaximal, complemented
7#10735C212yesmaximal
8C4×#342C212yesmaximal, complemented
9C22×#117C411yescharacteristic, complemented
10C23×C22⋊C4C2211yescharacteristic
11C23×C22⋊C4C2212yes—
12C23×C22⋊C4C2211yescharacteristic
13#2342—22no—
14#2342—22no—
15#2342—24no—
16C2×#388C412yescomplemented
17C2×#388C412yescomplemented
18C2×#385C2212yes—
19C2×#381C411yescharacteristic, complemented
20C2×#380C411yescharacteristic, complemented
21C2×#379C2212yes—
22C2×#379C2211yescharacteristic
23C2×C4×C22⋊C4—22no—
24C2×C4×C22⋊C4C411yescharacteristic, complemented
25C2×C4×C22⋊C4C2214yes—
26C2×C4×C22⋊C4C2212yes—
27#1819C418yescomplemented
28C2×#342C418yescomplemented
29C2×#342C2214yes—
30C2×#342C2214yes—
31C2×#342C2211yescharacteristic
32C2×#342C2212yes—
33C2×#342C2211yescharacteristic
34C2×#342C2212yes—
35C2×#342C2212yes—
36#1437—24no—
37C22⋊#95—22no—
38C22⋊#95C2214yes—
39C26C2311yescharacteristic
40C24×C4—22no—
41C24×C4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2312yes—
44C2×#117—21no—
45C2×#117—22no—
46C2×#117—21no—
47C2×#117—44no—
48C22×C4⋊C4C2×C412yescomplemented
49C22×C4⋊C4C2×C412yescomplemented
50C22×C22⋊C4—22no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—24no—
53C22×C22⋊C4—28no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—24no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—44no—
59C22×C22⋊C4—44no—
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C414yes—
63C22×C22⋊C4C2×C414yes—
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C418yes—
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4D412yes—
72C22×C22⋊C4D412yes—
73C22×C22⋊C4D414yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2312yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2311yescharacteristic
78C22×C22⋊C4C2311yescharacteristic
79C22×C22⋊C4C2312yes—
80C22×C42—24no—
81C22×C42C2×C412yescomplemented
82C22×C42C2×C412yescomplemented
83#388—24no—
84#388—24no—
85#388—24no—
86#388—24no—
87#385—24no—
88#385—24no—
89#381—22no—
90#381—22no—
91#380—22no—
92#380—22no—
93#379—22no—
94#379—22no—
95#379—24no—
96#379C2×C418yes—
97C4×C22⋊C4—24no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4—21no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—21no—
102C4×C22⋊C4—28no—
103C4×C22⋊C4—44no—
104C4×C22⋊C4—48no—
105C2×#95—24no—
106C2×#95—24no—
107C2×#95—24no—
108C2×#95C2×C412yescomplemented
109C2×#95C2×C412yescomplemented
110C2×#95C2×C412yescomplemented
111C2×#95C2×C412yescomplemented
112#342—28no—
113#342—44no—
114#342C2×C414yes—
115#342C2×C418yes—
116#342C2×C414yes—
117#342C2×C418yescomplemented
118#342C2×C418yescomplemented
119C25—24no—
120C25—22no—
121C25—22no—
122C25—42no—
123C25—42no—
124C25—82no—
125C25C22×C411yescharacteristic
126C25C22×C411yescharacteristic
127C25C22×C412yes—
128C25C22×C412yes—
129C25C2×D412yes—
130C25C2×D412yes—
131C25#4114yes—
132C25C2411yescharacteristic
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—28no—
146C23×C4—44no—
147C23×C4—48no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—82no—
153C23×C4—84no—
154C23×C4—82no—
155C23×C4C4214yescomplemented
156C23×C4C4214yescomplemented
157C23×C4C22×C412yes—
158C23×C4C22×C412yes—
159C23×C4C22×C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C412yes—
165C23×C4C2×D414yes—
166C23×C4C2×D414yes—
167#117—44no—
168#117—48no—
169#117—44no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—82no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—216no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—48no—
214C2×C22⋊C4—48no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—416no—
218C2×C22⋊C4—48no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—84no—
224C2×C22⋊C4—82no—
225C2×C22⋊C4C4214yescomplemented
226C2×C22⋊C4C4214yescomplemented
227C2×C22⋊C4C22⋊C411yescharacteristic
228C2×C22⋊C4C22⋊C412yes—
229C2×C22⋊C4C22⋊C412yes—
230C2×C22⋊C4C22⋊C414yes—
231C2×C22⋊C4C22⋊C412yes—
232C2×C22⋊C4C22⋊C414yes—
233C2×C22⋊C4C22⋊C411yescharacteristic
234C2×C22⋊C4C22×C414yes—
235C2×C22⋊C4C22×C418yes—
236C2×C22⋊C4C22×C412yes—
237C2×C22⋊C4C22×C412yes—
238C2×C22⋊C4C22×C412yes—
239C2×C22⋊C4C22×C412yes—
240C2×C22⋊C4C22×C414yes—
241C2×C22⋊C4#4118yes—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—44no—
247C2×C42—48no—
248C2×C42—48no—
249C2×C42—84no—
250C2×C42—82no—
251#95—24no—
252#95—24no—
253#95—22no—
254#95—22no—
255#95—24no—
256#95—48no—
257#95—48no—
258#95—84no—
259C24—22no—
260C24—22no—
261C24—24no—
262C24—24no—
263C24—22no—
264C24—24no—
265C24—22no—
266C24—24no—
267C24—24no—
268C24—22no—
269C24—24no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—24no—
274C24—24no—
275C24—24no—
276C24—28no—
277C24—28no—
278C24—41no—
279C24—42no—
280C24—44no—
281C24—42no—
282C24—41no—
283C24—41no—
284C24—42no—
285C24—42no—
286C24—44no—
287C24—41no—
288C24—44no—
289C24—44no—
290C24—42no—
291C24—48no—
292C24—82no—
293C24—82no—
294C24—84no—
295C24—88no—
296C24—84no—
297C24—84no—
298C24—88no—
299C24—84no—
300C24—84no—
301C24C2×C4212yes—
302C24C2×C4212yes—
303C24C2×C22⋊C412yes—
304C24C2×C22⋊C411yescharacteristic
305C24C2×C22⋊C411yescharacteristic
306C24#11712yes—
307C24#11712yes—
308C24C4×D414yes—
309C24C4×D414yes—
310C24C4×D414yes—
311C24C4×D414yes—
312C24C23×C411yescharacteristic
313C24C23×C411yescharacteristic
314C24C23×C411yescharacteristic
315C24C22×D412yes—
316C24C2×#4112yes—
317C24C2×#4114yes—
318C24H(5,𝔽2)12yes—
319C242−1+412yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—28no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—416no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—48no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—48no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—416no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—84no—
390C22×C4—88no—
391C22×C4—84no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—84no—
395C22×C4—88no—
396C22×C4—84no—
397C22×C4—816no—
398C22×C4—816no—
399C22×C4C2×C4212yes—
400C22×C4C2×C4212yes—
401C22×C4C2×C4212yes—
402C22×C4C2×C4212yes—
403C22×C4C2×C22⋊C414yes—
404C22×C4C2×C22⋊C412yes—
405C22×C4C2×C22⋊C412yes—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—84no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—28no—
415C22⋊C4—28no—
416C22⋊C4—28no—
417C22⋊C4—24no—
418C22⋊C4—22no—
419C22⋊C4—28no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—28no—
423C22⋊C4—24no—
424C22⋊C4—28no—
425C22⋊C4—44no—
426C22⋊C4—42no—
427C22⋊C4—416no—
428C22⋊C4—416no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—84no—
439C22⋊C4—88no—
440C22⋊C4#9914yes—
441C22⋊C4#11718yescomplemented
442C42—44no—
443C42—44no—
444C42—82no—
445C42—82no—
446C42—816no—
447C23—22no—
448C23—21no—
449C23—22no—
450C23—21no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—28no—
461C23—21no—
462C23—22no—
463C23—21no—
464C23—22no—
465C23—24no—
466C23—22no—
467C23—28no—
468C23—24no—
469C23—24no—
470C23—28no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—28no—
477C23—22no—
478C23—22no—
479C23—42no—
480C23—42no—
481C23—44no—
482C23—44no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—44no—
489C23—41no—
490C23—41no—
491C23—42no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—48no—
504C23—48no—
505C23—44no—
506C23—44no—
507C23—48no—
508C23—44no—
509C23—44no—
510C23—48no—
511C23—84no—
512C23—82no—
513C23—82no—
514C23—84no—
515C23—84no—
516C23—82no—
517C23—82no—
518C23—84no—
519C23—88no—
520C23—88no—
521C23—84no—
522C23—84no—
523C23—88no—
524C23—816no—
525C23—816no—
526C23C4×C22⋊C414yes—
527C23C4×C22⋊C412yes—
528C23C4×C22⋊C412yes—
529C23C2×#9912yes—
530C23#41012yes—
531C23C22×C4211yescharacteristic
532C23C22×C22⋊C411yescharacteristic
533C23C2×#11711yescharacteristic
534C23C2×#11712yes—
535C23C2×C4×D412yes—
536C23C2×C4×D412yes—
537C23C4×#4112yes—
538C23C4×#4112yes—
539C23#51811yescharacteristic
540C23#51911yescharacteristic
541C23#52012yes—
542C23#52012yes—
543C23#54614yes—
544C23#54714yes—
545C23#55114yes—
546C23#55214yes—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—84no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—832no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4C2×#9914yes—
614C22—21no—
615C22—22no—
616C22—22no—
617C22—21no—
618C22—21no—
619C22—22no—
620C22—22no—
621C22—21no—
622C22—22no—
623C22—24no—
624C22—22no—
625C22—22no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—22no—
630C22—22no—
631C22—24no—
632C22—24no—
633C22—22no—
634C22—22no—
635C22—24no—
636C22—22no—
637C22—24no—
638C22—24no—
639C22—22no—
640C22—42no—
641C22—42no—
642C22—41no—
643C22—41no—
644C22—42no—
645C22—44no—
646C22—44no—
647C22—48no—
648C22—416no—
649C22—48no—
650C22—48no—
651C22—44no—
652C22—44no—
653C22—42no—
654C22—44no—
655C22—42no—
656C22—44no—
657C22—84no—
658C22—88no—
659C22—84no—
660C22—84no—
661C22—82no—
662C22—82no—
663C22C4×#9914yes—
664C22C2×C4×C22⋊C411yescharacteristic
665C22C4×#11711yescharacteristic
666C22#227411yescharacteristic
667C22#233912yes—
668C22#234212yes—
669C22#234312yes—
670C22#234412yes—
671C22#235914yes—
672C22C22×#9911yescharacteristic
673C22C2×#41011yescharacteristic
674C22#288312yes—
675C4—24no—
676C4—48no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—88no—
684C4—84no—
685C4—82no—
686C4—82no—
687C2—22no—
688C2—24no—
689C2—24no—
690C2—22no—
691C2—22no—
692C2—21no—
693C2—21no—
694C2—42no—
695C2—44no—
696C2#1379911yescharacteristic
697C2C2×C4×#9911yescharacteristic
698C2#2039211yescharacteristic
699C2#2247612yes—
700C2#2247712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC26C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C22⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10912C2semidirect
6. #10801C2semidirect
8. C4×#342C2semidirect
9. C22×#117C4semidirect
16. C2×#388C4semidirect
17. C2×#388C4semidirect
19. C2×#381C4semidirect
20. C2×#380C4semidirect
24. C2×C4×C22⋊C4C4semidirect
27. #1819C4semidirect
28. C2×#342C4semidirect
48. C22×C4⋊C4C2×C4semidirect
49. C22×C4⋊C4C2×C4semidirect
81. C22×C42C2×C4semidirect
82. C22×C42C2×C4semidirect
108. C2×#95C2×C4semidirect
109. C2×#95C2×C4semidirect
110. C2×#95C2×C4semidirect
111. C2×#95C2×C4semidirect
117. #342C2×C4semidirect
118. #342C2×C4semidirect
155. C23×C4C42semidirect
156. C23×C4C42semidirect
225. C2×C22⋊C4C42semidirect
226. C2×C22⋊C4C42semidirect
441. C22⋊C4#117semidirect

Maximal subgroups

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

subgroupindex
C2×#23422
C2×#18192
C22×#3422
C22⋊C422
#109122
#108012
#107352
C4×#3422