Finite Group Databasebeta

← #154108

Subgroups — #154108

Order 512.

4766 conjugacy classes of subgroups.

The subgroup lattice has 825 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#380C211yesmaximal, characteristic
2C22×C4×C22⋊C4C211yesmaximal, characteristic
3C2×#1444C212yesmaximal, complemented
4C2×#1439C214yesmaximal
5C2×C4⋊#95C212yesmaximal, complemented
6C2×C22⋊#95C211yesmaximal, characteristic
7#7557C212yesmaximal, complemented
8#7554C212yesmaximal, complemented
9C25×C4C2211yescharacteristic
10C23×C4⋊C4C2214yes—
11C23×C4⋊C4C2212yes—
12C23×C22⋊C4C2212yes—
13C23×C22⋊C4C2212yes—
14C23×C22⋊C4C2212yes—
15C23×C42C2212yes—
16C2×#380—28no—
17C2×#380—24no—
18C2×C4×C22⋊C4—28no—
19C2×C4×C22⋊C4C418yescomplemented
20C22×#95C2214yes—
21C22×#95C2212yes—
22C22×#95C2211yescharacteristic
23C22×#95C2211yescharacteristic
24C22×#95C2214yes—
25C22×#95C2214yes—
26C22×#95C2211yescharacteristic
27C22×#95C2212yes—
28C22×#95C2211yescharacteristic
29#1444—28no—
30#1439—216no—
31C4⋊#95—28no—
32C22⋊#95—24no—
33C26C2311yescharacteristic
34C24×C4—24no—
35C24×C4—24no—
36C24×C4—24no—
37C24×C4—24no—
38C24×C4—44no—
39C24×C4C2×C414yes—
40C24×C4D414yes—
41C24×C4Q814yes—
42C24×C4C2314yes—
43C24×C4C2312yes—
44C24×C4C2312yes—
45C24×C4C2312yes—
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C24×C4C2312yes—
49C22×C4⋊C4—28no—
50C22×C4⋊C4—28no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—28no—
53C22×C4⋊C4—28no—
54C22×C4⋊C4—28no—
55C22×C4⋊C4—28no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—44no—
59C22×C4⋊C4—44no—
60C22×C4⋊C4—44no—
61C22×C22⋊C4—28no—
62C22×C22⋊C4—28no—
63C22×C22⋊C4—24no—
64C22×C22⋊C4—24no—
65C22×C22⋊C4—28no—
66C22×C22⋊C4—24no—
67C22×C22⋊C4—24no—
68C22×C22⋊C4—28no—
69C22×C22⋊C4—28no—
70C22×C22⋊C4—42no—
71C22×C22⋊C4—44no—
72C22×C22⋊C4—42no—
73C22×C22⋊C4C2×C418yes—
74C22×C22⋊C4D418yes—
75C22×C22⋊C4Q818yes—
76C22×C42—24no—
77C22×C42—28no—
78C22×C42—24no—
79C22×C42—44no—
80C22×C42C2×C418yes—
81#380—232no—
82C4×C22⋊C4—232no—
83C2×#95—24no—
84C2×#95—24no—
85C2×#95—24no—
86C2×#95—22no—
87C2×#95—24no—
88C2×#95—22no—
89C2×#95—28no—
90C2×#95—24no—
91C2×#95—28no—
92C2×#95—28no—
93C2×#95—28no—
94C2×#95—28no—
95C2×#95—24no—
96C2×#95—28no—
97C2×#95—28no—
98C2×#95—24no—
99C2×#95—24no—
100C2×#95—44no—
101C2×#95—44no—
102C2×#95C2×C414yes—
103C2×#95C2×C414yes—
104C2×#95D418yes—
105C2×#95D418yes—
106C25—22no—
107C25—22no—
108C25—24no—
109C25—22no—
110C25—22no—
111C25—22no—
112C25—22no—
113C25—21no—
114C25—21no—
115C25—24no—
116C25—21no—
117C25—21no—
118C25C22×C412yes—
119C25C2×D414yes—
120C25C2×D412yes—
121C25C2×Q814yes—
122C25C2×Q812yes—
123C25C2411yescharacteristic
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—28no—
145C23×C4—28no—
146C23×C4—216no—
147C23×C4—28no—
148C23×C4—216no—
149C23×C4—216no—
150C23×C4—216no—
151C23×C4—28no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—48no—
187C23×C4—48no—
188C23×C4—48no—
189C23×C4—416no—
190C23×C4—416no—
191C23×C4—416no—
192C23×C4—416no—
193C23×C4C4⋊C4116yescomplemented
194C23×C4C22×C414yes—
195C23×C4C22×C414yes—
196C23×C4C22×C412yes—
197C23×C4C22×C412yes—
198C23×C4C2×D418yes—
199C23×C4C2×D414yes—
200C23×C4C2×D418yes—
201C23×C4C2×D414yes—
202C23×C4C2×D412yes—
203C23×C4C2×D414yes—
204C23×C4C2×D412yes—
205C23×C4C2×D414yes—
206C23×C4C2×Q814yes—
207C23×C4C2×Q814yes—
208C23×C4C2×Q812yes—
209C23×C4C2×Q812yes—
210C23×C4#4114yes—
211C23×C4#4114yes—
212C23×C4#4118yes—
213C23×C4#4114yes—
214C23×C4#4114yes—
215C2×C4⋊C4—232no—
216C2×C4⋊C4—216no—
217C2×C4⋊C4—416no—
218C2×C4⋊C4—416no—
219C2×C4⋊C4—416no—
220C2×C4⋊C4—416no—
221C2×C4⋊C4—416no—
222C2×C4⋊C4—416no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—416no—
225C2×C4⋊C4—48no—
226C2×C4⋊C4—416no—
227C2×C22⋊C4—216no—
228C2×C22⋊C4—216no—
229C2×C22⋊C4—232no—
230C2×C22⋊C4—232no—
231C2×C22⋊C4—216no—
232C2×C22⋊C4—216no—
233C2×C22⋊C4—416no—
234C2×C22⋊C4—416no—
235C2×C22⋊C4—416no—
236C2×C22⋊C4—416no—
237C2×C22⋊C4—48no—
238C2×C22⋊C4—48no—
239C2×C22⋊C4—48no—
240C2×C22⋊C4—48no—
241C2×C22⋊C4C4⋊C4132yescomplemented
242C2×C42—216no—
243C2×C42—216no—
244C2×C42—48no—
245C2×C42—416no—
246C2×C42—416no—
247#95—28no—
248#95—216no—
249#95—416no—
250#95—416no—
251#95—44no—
252#95—48no—
253#95—416no—
254#95—44no—
255#95C22⋊C4116yescomplemented
256C24—22no—
257C24—24no—
258C24—22no—
259C24—24no—
260C24—22no—
261C24—22no—
262C24—24no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—28no—
267C24—24no—
268C24—28no—
269C24—24no—
270C24—24no—
271C24—22no—
272C24—24no—
273C24—24no—
274C24—28no—
275C24—28no—
276C24—24no—
277C24—24no—
278C24—24no—
279C24—22no—
280C24—22no—
281C24—24no—
282C24—22no—
283C24—22no—
284C24—24no—
285C24—24no—
286C24—28no—
287C24—28no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—28no—
293C24—28no—
294C24—24no—
295C24—24no—
296C24—44no—
297C24—44no—
298C24—44no—
299C24—44no—
300C24—44no—
301C24—44no—
302C24—44no—
303C24—44no—
304C24—44no—
305C24—44no—
306C24—44no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—44no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24C2×C4⋊C418yes—
317C24C2×C4⋊C414yes—
318C24#12714yes—
319C24C4⋊Q814yes—
320C24C4⋊Q818yes—
321C24C23×C411yescharacteristic
322C24C22×D412yes—
323C24C22×D411yescharacteristic
324C24C22×D412yes—
325C24C22×D412yes—
326C24C22×Q811yescharacteristic
327C24C22×Q812yes—
328C24C2×#4114yes—
329C24C2×#4112yes—
330C24C2×#4112yes—
331C24C2×#4112yes—
332C24C2×#4112yes—
333C24H(5,𝔽2)14yes—
334C24H(5,𝔽2)11yescharacteristic
335C24H(5,𝔽2)11yescharacteristic
336C242−1+411yescharacteristic
337C242−1+411yescharacteristic
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
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—28no—
353C22×C4—28no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—24no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—24no—
377C22×C4—28no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—24no—
387C22×C4—28no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—232no—
393C22×C4—232no—
394C22×C4—28no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—416no—
412C22×C4—432no—
413C22×C4—432no—
414C22×C4—432no—
415C22×C4—432no—
416C22×C4—464no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4C2×C22⋊C414yes—
490C22×C4C2×C22⋊C418yes—
491C22×C4C2×C4⋊C418yes—
492C22×C4C2×C4⋊C414yes—
493C22×C4C2×C4⋊C418yes—
494C22×C4C2×C4⋊C414yes—
495C22×C4#11714yes—
496C22×C4#11718yes—
497C22×C4C4×D418yes—
498C22×C4#12014yes—
499C22×C4#12118yes—
500C22×C4#121116yes—
501C22×C4C22⋊Q818yes—
502C22×C4C22⋊Q8116yes—
503C22×C4C22⋊Q818yes—
504C22×C4#12314yes—
505C4⋊C4—432no—
506C4⋊C4—464no—
507C22⋊C4—264no—
508C22⋊C4—432no—
509C22⋊C4—432no—
510C42—432no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—216no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—24no—
522C23—24no—
523C23—28no—
524C23—28no—
525C23—24no—
526C23—24no—
527C23—28no—
528C23—24no—
529C23—28no—
530C23—24no—
531C23—28no—
532C23—24no—
533C23—216no—
534C23—24no—
535C23—44no—
536C23—48no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—44no—
546C23—48no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—48no—
556C23—48no—
557C23—48no—
558C23—48no—
559C23—48no—
560C23—44no—
561C23—44no—
562C23—48no—
563C23—48no—
564C23—48no—
565C23—48no—
566C23—48no—
567C23—48no—
568C23—48no—
569C23—48no—
570C23—44no—
571C23—44no—
572C23—48no—
573C23—48no—
574C23—48no—
575C23#38418yes—
576C23C22×C22⋊C411yescharacteristic
577C23C22×C4⋊C411yescharacteristic
578C23C22×C4⋊C412yes—
579C23C2×#11712yes—
580C23C2×#11711yescharacteristic
581C23C2×C4×D412yes—
582C23C4×#4111yescharacteristic
583C23C4×#4111yescharacteristic
584C23#51812yes—
585C23#52011yescharacteristic
586C23#52011yescharacteristic
587C23C2×#12011yescharacteristic
588C23C2×#12112yes—
589C23C2×#12114yes—
590C23C2×C22⋊Q812yes—
591C23C2×C22⋊Q814yes—
592C23C2×C22⋊Q812yes—
593C23C2×#12311yescharacteristic
594C23#52512yes—
595C23#52514yes—
596C23#52914yes—
597C23#52914yes—
598C23C2×#12711yescharacteristic
599C23C2×C4⋊Q811yescharacteristic
600C23C2×C4⋊Q812yes—
601C23#53212yes—
602C23#53214yes—
603C23#53214yes—
604C23#53314yes—
605C23#53312yes—
606C23#53312yes—
607C23#53412yes—
608C23#53511yescharacteristic
609C23#53611yescharacteristic
610C23#53814yes—
611C23#53914yes—
612C23#54014yes—
613C23#54214yes—
614C23#54312yes—
615C23#54411yescharacteristic
616C23#54411yescharacteristic
617C23D4214yes—
618C23D4212yes—
619C23D4212yes—
620C23#54614yes—
621C23#54612yes—
622C23#54614yes—
623C23#54614yes—
624C23#54614yes—
625C23#54712yes—
626C23#54712yes—
627C23#54712yes—
628C23#54812yes—
629C23D4×Q812yes—
630C23D4×Q812yes—
631C23#55012yes—
632C23#55114yes—
633C23#55212yes—
634C23#55312yes—
635C23#55412yes—
636C23#55414yes—
637C23#55414yes—
638C23#55412yes—
639C23#55512yes—
640C23#55612yes—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—28no—
644C2×C4—28no—
645C2×C4—28no—
646C2×C4—28no—
647C2×C4—28no—
648C2×C4—28no—
649C2×C4—28no—
650C2×C4—28no—
651C2×C4—28no—
652C2×C4—28no—
653C2×C4—28no—
654C2×C4—28no—
655C2×C4—28no—
656C2×C4—216no—
657C2×C4—216no—
658C2×C4—216no—
659C2×C4—216no—
660C2×C4—216no—
661C2×C4—216no—
662C2×C4—24no—
663C2×C4—24no—
664C2×C4—28no—
665C2×C4—28no—
666C2×C4—28no—
667C2×C4—28no—
668C2×C4—464no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—48no—
673C2×C4—48no—
674C2×C4—48no—
675C2×C4—48no—
676C2×C4—44no—
677C2×C4—48no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—48no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—48no—
684C2×C4—48no—
685C2×C4—48no—
686C2×C4—48no—
687C2×C4—48no—
688C2×C4—48no—
689C2×C4—48no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—48no—
693C2×C4—48no—
694C2×C4—84no—
695C2×C4—82no—
696C2×C4—82no—
697C2×C4—82no—
698C2×C4—84no—
699C2×C4—82no—
700C2×C4—82no—
701C2×C4—82no—
702C2×C4—84no—
703C2×C4—84no—
704C2×C4—84no—
705C2×C4—84no—
706C2×C4—84no—
707C2×C4—82no—
708C2×C4—84no—
709C2×C4—82no—
710C2×C4—84no—
711C2×C4—84no—
712C2×C4—84no—
713C2×C4—84no—
714C2×C4#38018yes—
715C2×C4#380116yescomplemented
716C2×C4#38518yes—
717C22—28no—
718C22—28no—
719C22—28no—
720C22—44no—
721C22—44no—
722C22—44no—
723C22—48no—
724C22—48no—
725C22—48no—
726C22—48no—
727C22—44no—
728C22—48no—
729C22—48no—
730C22—416no—
731C22—416no—
732C22—48no—
733C22—48no—
734C22C2×#38011yescharacteristic
735C22C2×#38012yes—
736C22C2×#38411yescharacteristic
737C22#228511yescharacteristic
738C22#228512yes—
739C22C2×#38511yescharacteristic
740C22#229012yes—
741C22#229312yes—
742C22#229611yescharacteristic
743C22#231511yescharacteristic
744C22#231711yescharacteristic
745C22#231812yes—
746C22#231912yes—
747C22D4×C22⋊C412yes—
748C22#233911yescharacteristic
749C22#234111yescharacteristic
750C22#234212yes—
751C22#234611yescharacteristic
752C22#234711yescharacteristic
753C22D4×C4⋊C412yes—
754C22D4×C4⋊C412yes—
755C22#234912yes—
756C22#234912yes—
757C22#238812yes—
758C22#239512yes—
759C22#239614yes—
760C22#241012yes—
761C22#241212yes—
762C22#241412yes—
763C22#241512yes—
764C22#242414yes—
765C22#242514yes—
766C22#242714yes—
767C22#242814yes—
768C22#243311yescharacteristic
769C22#243411yescharacteristic
770C22#243714yes—
771C22#243912yes—
772C22#244012yes—
773C22#244012yes—
774C22#244112yes—
775C22#244914yes—
776C22#245014yes—
777C22#245214yes—
778C22#245414yes—
779C22#246414yes—
780C22#246514yes—
781C22#250111yescharacteristic
782C22#250211yescharacteristic
783C22#252114yes—
784C22#252214yes—
785C22#253912yes—
786C22#254914yes—
787C22#255314yes—
788C22#255514yes—
789C22#255614yes—
790C22#257414yes—
791C22#258312yes—
792C22#266112yes—
793C22#266312yes—
794C22#266512yes—
795C22#266712yes—
796C22#285612yes—
797C22#285712yes—
798C22#285714yes—
799C4—28no—
800C4—28no—
801C4—216no—
802C4—48no—
803C4—88no—
804C4—84no—
805C4—84no—
806C2—48no—
807C2#1384411yescharacteristic
808C2#1387711yescharacteristic
809C2#1388011yescharacteristic
810C2#1389112yes—
811C2#1389912yes—
812C2#1391512yes—
813C2#1392012yes—
814C2#1395111yescharacteristic
815C2#1396511yescharacteristic
816C2#1435511yescharacteristic
817C2#1468611yescharacteristic
818C2#1476212yes—
819C2#1476312yes—
820C2#1477014yes—
821C2#1477114yes—
822C2#1483912yes—
823C2#1484012yes—
subgroupG/H
center, Frattini, socle, cosocleC25C24
commutatorC24C23×C4
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1444C2semidirect
5. C2×C4⋊#95C2semidirect
7. #7557C2semidirect
8. #7554C2semidirect
19. C2×C4×C22⋊C4C4semidirect
193. C23×C4C4⋊C4semidirect
241. C2×C22⋊C4C4⋊C4semidirect
255. #95C22⋊C4semidirect
715. C2×C4#380semidirect

Maximal subgroups

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

subgroupindex
C22×#3802
C22×C4×C22⋊C42
C2×#14442
C2×#14392
C2×C4⋊#952
C2×C22⋊#952
#75572
#75542