Finite Group Databasebeta

← #153885

Subgroups — #153885

Order 512.

4646 conjugacy classes of subgroups.

The subgroup lattice has 823 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊C4C211yesmaximal, characteristic
2C22×C4×C22⋊C4C211yesmaximal, characteristic
3C2×#1440C212yesmaximal
4C2×#1434C212yesmaximal, complemented
5C2×C22⋊#95C211yesmaximal, characteristic
6#7555C212yesmaximal
7#7554C212yesmaximal, complemented
8#7554C212yesmaximal, complemented
9#7554C212yesmaximal, complemented
10C25×C4C2211yescharacteristic
11C23×C22⋊C4C2212yes—
12C23×C22⋊C4C2211yescharacteristic
13C23×C22⋊C4C2211yescharacteristic
14C23×C22⋊C4C2212yes—
15C23×C42C2211yescharacteristic
16C23×C42C2212yes—
17C23×C42C2211yescharacteristic
18C2×C4×C22⋊C4—28no—
19C2×C4×C22⋊C4—24no—
20C2×C4×C22⋊C4—24no—
21C2×C4×C22⋊C4—24no—
22C2×C4×C22⋊C4C418yescomplemented
23C22×#95C2212yes—
24C22×#95C2212yes—
25C22×#95C2214yes—
26C22×#95C2214yes—
27C22×#95C2211yescharacteristic
28C22×#95C2211yescharacteristic
29C22×#95C2214yes—
30C22×#95C2212yes—
31C22×#95C2214yes—
32#1440C4116yescomplemented
33#1434—28no—
34C22⋊#95—24no—
35C26C2311yescharacteristic
36C24×C4—24no—
37C24×C4—24no—
38C24×C4—24no—
39C24×C4—24no—
40C24×C4—44no—
41C24×C4C2×C414yes—
42C24×C4D414yes—
43C24×C4D414yes—
44C24×C4C2314yes—
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C24×C4C2314yes—
48C24×C4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—24no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—24no—
58C22×C22⋊C4—24no—
59C22×C22⋊C4—28no—
60C22×C22⋊C4—24no—
61C22×C22⋊C4—24no—
62C22×C22⋊C4—28no—
63C22×C22⋊C4—24no—
64C22×C22⋊C4—24no—
65C22×C22⋊C4—44no—
66C22×C22⋊C4—44no—
67C22×C22⋊C4C2×C418yes—
68C22×C22⋊C4C2×C414yes—
69C22×C22⋊C4C2×C418yes—
70C22×C22⋊C4C2×C414yes—
71C22×C22⋊C4C2×C418yes—
72C22×C42—24no—
73C22×C42—28no—
74C22×C42—24no—
75C22×C42—22no—
76C22×C42—22no—
77C22×C42—22no—
78C22×C42—22no—
79C22×C42—24no—
80C22×C42—24no—
81C22×C42—24no—
82C22×C42—44no—
83C22×C42—44no—
84C22×C42C2×C414yes—
85C22×C42C2×C414yes—
86C4×C22⋊C4—232no—
87C4×C22⋊C4—232no—
88C2×#95—28no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95—28no—
96C2×#95—24no—
97C2×#95—24no—
98C2×#95—28no—
99C2×#95—24no—
100C2×#95—28no—
101C2×#95—28no—
102C2×#95—28no—
103C2×#95—24no—
104C2×#95—24no—
105C2×#95—24no—
106C2×#95—44no—
107C2×#95—44no—
108C2×#95—44no—
109C2×#95—44no—
110C2×#95C2×C414yes—
111C2×#95C2×C414yes—
112C2×#95C2×C4116yescomplemented
113C2×#95C2×C4116yescomplemented
114C2×#95D418yes—
115C25—22no—
116C25—22no—
117C25—22no—
118C25—24no—
119C25—22no—
120C25—22no—
121C25—22no—
122C25—22no—
123C25—24no—
124C25—22no—
125C25C22×C412yes—
126C25C22×C414yes—
127C25C2×D412yes—
128C25C2×D412yes—
129C25#4114yes—
130C25C2411yescharacteristic
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—28no—
174C23×C4—28no—
175C23×C4—216no—
176C23×C4—28no—
177C23×C4—216no—
178C23×C4—216no—
179C23×C4—216no—
180C23×C4—28no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—48no—
196C23×C4—48no—
197C23×C4—48no—
198C23×C4—416no—
199C23×C4—416no—
200C23×C4—416no—
201C23×C4—416no—
202C23×C4C22⋊C4116yes—
203C23×C4C22×C418yes—
204C23×C4C22×C412yes—
205C23×C4C22×C412yes—
206C23×C4C22×C414yes—
207C23×C4C22×C414yes—
208C23×C4C22×C412yes—
209C23×C4C22×C412yes—
210C23×C4C22×C418yes—
211C23×C4C22×C412yes—
212C23×C4C22×C412yes—
213C23×C4C2×D418yes—
214C23×C4C2×D412yes—
215C23×C4C2×D412yes—
216C23×C4C2×D412yes—
217C23×C4C2×D412yes—
218C23×C4C2×D414yes—
219C23×C4#4114yes—
220C23×C4#4114yes—
221C23×C4#4118yes—
222C23×C4#4114yes—
223C23×C4#4114yes—
224C23×C4#4114yes—
225C2×C22⋊C4—232no—
226C2×C22⋊C4—216no—
227C2×C22⋊C4—216no—
228C2×C22⋊C4—216no—
229C2×C22⋊C4—216no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—216no—
234C2×C22⋊C4—216no—
235C2×C22⋊C4—216no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—232no—
239C2×C22⋊C4—216no—
240C2×C22⋊C4—416no—
241C2×C22⋊C4—48no—
242C2×C22⋊C4—48no—
243C2×C22⋊C4—416no—
244C2×C22⋊C4C42116yescomplemented
245C2×C42—216no—
246C2×C42—28no—
247C2×C42—28no—
248C2×C42—216no—
249C2×C42—48no—
250C2×C42—416no—
251C2×C42—416no—
252C2×C42—48no—
253C2×C42—48no—
254C2×C42—48no—
255C2×C42—44no—
256C2×C42—48no—
257C2×C42—48no—
258C2×C42—44no—
259#95—28no—
260#95—28no—
261#95—232no—
262#95—216no—
263#95—416no—
264#95—416no—
265#95—416no—
266#95—48no—
267#95—48no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—28no—
273C24—24no—
274C24—28no—
275C24—28no—
276C24—24no—
277C24—28no—
278C24—28no—
279C24—24no—
280C24—24no—
281C24—28no—
282C24—22no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—22no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—28no—
291C24—28no—
292C24—28no—
293C24—28no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—22no—
298C24—22no—
299C24—24no—
300C24—28no—
301C24—28no—
302C24—44no—
303C24—44no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—44no—
309C24—44no—
310C24—44no—
311C24—44no—
312C24—42no—
313C24—44no—
314C24—44no—
315C24—44no—
316C24—42no—
317C24—44no—
318C24—42no—
319C24—44no—
320C24—42no—
321C24—44no—
322C24C2×C4214yes—
323C24C2×C22⋊C414yes—
324C24#11714yes—
325C24C4×D418yes—
326C24C4×D418yes—
327C24C23×C411yescharacteristic
328C24C23×C412yes—
329C24C22×D411yescharacteristic
330C24C22×D411yescharacteristic
331C24C22×D412yes—
332C24C2×#4112yes—
333C24C2×#4112yes—
334C24C2×#4112yes—
335C24C2×#4112yes—
336C24C2×#4112yes—
337C24C2×#4112yes—
338C24C2×#4114yes—
339C24H(5,𝔽2)14yes—
340C24H(5,𝔽2)12yes—
341C242−1+412yes—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—28no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—24no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—24no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—232no—
417C22×C4—232no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—416no—
426C22×C4—432no—
427C22×C4—432no—
428C22×C4—432no—
429C22×C4—464no—
430C22×C4—432no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4C2×C4214yes—
506C22×C4C2×C4214yes—
507C22×C4C2×C22⋊C414yes—
508C22×C4C2×C22⋊C414yes—
509C22×C4#11714yes—
510C22×C4#11714yes—
511C22×C4#11714yes—
512C22×C4#11714yes—
513C22×C4C4×D4116yes—
514C22×C4C4×D418yes—
515C22×C4#12014yes—
516C22×C4#12118yes—
517C22×C4#12318yes—
518C22×C4#12318yes—
519C22×C4#12314yes—
520C22⋊C4—232no—
521C22⋊C4—264no—
522C22⋊C4—232no—
523C22⋊C4—432no—
524C42—416no—
525C42—432no—
526C42—416no—
527C23—24no—
528C23—24no—
529C23—28no—
530C23—28no—
531C23—28no—
532C23—216no—
533C23—24no—
534C23—24no—
535C23—28no—
536C23—28no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—28no—
543C23—24no—
544C23—28no—
545C23—216no—
546C23—28no—
547C23—216no—
548C23—216no—
549C23—44no—
550C23—44no—
551C23—48no—
552C23—44no—
553C23—48no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—48no—
560C23—48no—
561C23—48no—
562C23—48no—
563C23—48no—
564C23—48no—
565C23—48no—
566C23—48no—
567C23—44no—
568C23—44no—
569C23—48no—
570C23—44no—
571C23—44no—
572C23—48no—
573C23—48no—
574C23—48no—
575C23—48no—
576C23—44no—
577C23—44no—
578C23—48no—
579C23—44no—
580C23—44no—
581C23—48no—
582C23—48no—
583C23—48no—
584C23—48no—
585C23C4×C22⋊C418yes—
586C23C22×C4211yescharacteristic
587C23C22×C22⋊C411yescharacteristic
588C23C2×#11711yescharacteristic
589C23C2×#11711yescharacteristic
590C23C2×#11711yescharacteristic
591C23C2×#11711yescharacteristic
592C23C2×#11711yescharacteristic
593C23C2×C4×D412yes—
594C23C2×C4×D412yes—
595C23C2×C4×D412yes—
596C23C2×C4×D414yes—
597C23C4×#4112yes—
598C23C4×#4114yes—
599C23C4×#4114yes—
600C23C4×#4114yes—
601C23#51812yes—
602C23#51814yes—
603C23#52012yes—
604C23#52014yes—
605C23C2×#12011yescharacteristic
606C23C2×#12112yes—
607C23C2×#12312yes—
608C23C2×#12312yes—
609C23C2×#12311yescharacteristic
610C23#52512yes—
611C23#52512yes—
612C23#52512yes—
613C23#52514yes—
614C23#52914yes—
615C23#52914yes—
616C23#52914yes—
617C23#53212yes—
618C23#53412yes—
619C23#53412yes—
620C23#53511yescharacteristic
621C23#53611yescharacteristic
622C23#53712yes—
623C23#53814yes—
624C23#53914yes—
625C23#54614yes—
626C23#54614yes—
627C23#54612yes—
628C23#54712yes—
629C23#54812yes—
630C23#55012yes—
631C23#55114yes—
632C23#55114yes—
633C23#55114yes—
634C23#55214yes—
635C23#55212yes—
636C23#55314yes—
637C23#55314yes—
638C23#55314yes—
639C23#55312yes—
640C23#55514yes—
641C23#55512yes—
642C23#55614yes—
643C23#55612yes—
644C23#55914yes—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—28no—
648C2×C4—28no—
649C2×C4—28no—
650C2×C4—28no—
651C2×C4—28no—
652C2×C4—28no—
653C2×C4—216no—
654C2×C4—216no—
655C2×C4—28no—
656C2×C4—28no—
657C2×C4—24no—
658C2×C4—24no—
659C2×C4—28no—
660C2×C4—28no—
661C2×C4—216no—
662C2×C4—216no—
663C2×C4—24no—
664C2×C4—24no—
665C2×C4—28no—
666C2×C4—28no—
667C2×C4—24no—
668C2×C4—24no—
669C2×C4—216no—
670C2×C4—464no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—48no—
679C2×C4—48no—
680C2×C4—48no—
681C2×C4—48no—
682C2×C4—48no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—48no—
686C2×C4—48no—
687C2×C4—48no—
688C2×C4—44no—
689C2×C4—48no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—48no—
693C2×C4—48no—
694C2×C4—48no—
695C2×C4—48no—
696C2×C4—44no—
697C2×C4—48no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—44no—
701C2×C4—42no—
702C2×C4—42no—
703C2×C4—44no—
704C2×C4—48no—
705C2×C4—48no—
706C2×C4—44no—
707C2×C4—48no—
708C2×C4—48no—
709C2×C4—48no—
710C2×C4—48no—
711C2×C4—48no—
712C2×C4—48no—
713C2×C4—48no—
714C2×C4—84no—
715C2×C4—84no—
716C2×C4—84no—
717C2×C4—84no—
718C2×C4—84no—
719C2×C4—84no—
720C2×C4—84no—
721C2×C4—84no—
722C2×C4#38118yes—
723C2×C4#38618yes—
724C22—28no—
725C22—28no—
726C22—28no—
727C22—44no—
728C22—44no—
729C22—48no—
730C22—48no—
731C22—416no—
732C22—44no—
733C22—44no—
734C22—416no—
735C22—416no—
736C22—416no—
737C22—48no—
738C22—48no—
739C22C2×C4×C22⋊C411yescharacteristic
740C22C4×#11711yescharacteristic
741C22C4×#11711yescharacteristic
742C22C42×D412yes—
743C22#227512yes—
744C22C2×#38111yescharacteristic
745C22#228011yescharacteristic
746C22#228311yescharacteristic
747C22#228311yescharacteristic
748C22C2×#38611yescharacteristic
749C22#229012yes—
750C22#229711yescharacteristic
751C22C4×#12012yes—
752C22C4×#12114yes—
753C22C4×#12314yes—
754C22C4×#12314yes—
755C22C4×#12312yes—
756C22#230811yescharacteristic
757C22#231111yescharacteristic
758C22#231812yes—
759C22#232112yes—
760C22#232212yes—
761C22#232514yes—
762C22#234212yes—
763C22#234212yes—
764C22#234612yes—
765C22#234712yes—
766C22#235212yes—
767C22#235312yes—
768C22#235412yes—
769C22#235712yes—
770C22#235914yes—
771C22#236314yes—
772C22#236414yes—
773C22#236714yes—
774C22#237214yes—
775C22#237314yes—
776C22#237314yes—
777C22#237414yes—
778C22#240512yes—
779C22#240712yes—
780C22#241214yes—
781C22#241412yes—
782C22#241512yes—
783C22#245914yes—
784C22#246014yes—
785C22#246614yes—
786C22#246714yes—
787C22#248312yes—
788C22#248512yes—
789C22#248514yes—
790C22#249912yes—
791C22#250012yes—
792C22#251714yes—
793C22#251814yes—
794C22#251814yes—
795C22#252314yes—
796C22#260014yes—
797C22#260214yes—
798C4—28no—
799C4—28no—
800C4—44no—
801C4—416no—
802C4—44no—
803C4—44no—
804C4—44no—
805C4—88no—
806C2—48no—
807C2C4×#38611yescharacteristic
808C2C4×#38111yescharacteristic
809C2#1378411yescharacteristic
810C2#1381612yes—
811C2#1382212yes—
812C2#1386911yescharacteristic
813C2#1387111yescharacteristic
814C2#1388812yes—
815C2#1389212yes—
816C2#1393911yescharacteristic
817C2#1394011yescharacteristic
818C2#1417314yes—
819C2#1419914yes—
820C2#1420214yes—
821C2#1424814yes—
subgroupG/H
center, Frattini, socle, cosocleC25C24
commutatorC23C22×C42
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1434C2semidirect
7. #7554C2semidirect
8. #7554C2semidirect
9. #7554C2semidirect
22. C2×C4×C22⋊C4C4semidirect
32. #1440C4semidirect
112. C2×#95C2×C4semidirect
113. C2×#95C2×C4semidirect
244. C2×C22⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊C42
C2×#14402
C2×#14342
C2×C22⋊#952
#75552
#75542