Finite Group Databasebeta

← #644314

Subgroups — #644314

Order 512.

5844 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2847C211yesmaximal, characteristic
3C2×#2639C212yesmaximal, complemented
4#16632C214yesmaximal, complemented
5#16366C212yesmaximal, complemented
6#16253C212yesmaximal, complemented
7#14873C212yesmaximal, complemented
8#14822C212yesmaximal, complemented
9#14776C212yesmaximal, complemented
10#14759C214yesmaximal, complemented
11#14719C212yesmaximal, complemented
12#14281C212yesmaximal, complemented
13#14134C214yesmaximal, complemented
14C2×C22⋊#95C211yesmaximal, characteristic, complemented
15C24×Q8C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8—28no—
18C22×C22⋊Q8—216no—
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C22⋊Q8C2212yes—
23C22×#121C2211yescharacteristic
24C22×#121C2211yescharacteristic
25C22×#121C2211yescharacteristic
26C22×#120C2212yes—
27C23×C4⋊C4C2212yes—
28C23×C4⋊C4C2211yescharacteristic
29C23×C22⋊C4C2212yes—
30C23×C22⋊C4C2212yes—
31#2847—28no—
32#2639—216no—
33C2×#400C2214yescomplemented
34C2×#397C2214yescomplemented
35C2×#394C2214yescomplemented
36C2×#393C2214yescomplemented
37C2×#393C2214yescomplemented
38C2×#386C2218yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#386C2214yes—
43C2×#386C2214yes—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C22×#95C2212yes—
48C22×#95C2212yes—
49#1446C2214yescomplemented
50#1446C2214yescomplemented
51#1445C2214yescomplemented
52#1442C2214yescomplemented
53#1442C2218yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1440C2218yes—
57#1439C2214yescomplemented
58#1439C2218yescomplemented
59#1438C2218yes—
60C22⋊#95C2214yes—
61C22⋊#95C2212yes—
62C22⋊#95C2218yes—
63C22⋊#95C2214yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C26C2311yescharacteristic
68C23×Q8—28no—
69C23×Q8—24no—
70C23×Q8—44no—
71C23×Q8—44no—
72C23×Q8C2312yes—
73C23×Q8C2312yes—
74C23×D4C2312yes—
75C23×D4C2312yes—
76C23×D4C2311yescharacteristic
77C23×D4C2312yes—
78C24×C4—28no—
79C24×C4—44no—
80C24×C4—44no—
81C24×C4D418yes—
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2314yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C22⋊Q8—2128no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—216no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—216no—
101C2×C22⋊Q8D4116yescomplemented
102C2×#121—24no—
103C2×#121—24no—
104C2×#121—24no—
105C2×#121—28no—
106C2×#121—44no—
107C2×#121—44no—
108C2×#120—28no—
109C2×#120—216no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—28no—
112C22×C4⋊C4—216no—
113C22×C4⋊C4—28no—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—28no—
122C22×C22⋊C4—216no—
123C22×C22⋊C4—28no—
124C22×C22⋊C4—44no—
125C22×C22⋊C4—44no—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2318yes—
135C22×C22⋊C4C2318yes—
136C22×C22⋊C4C2314yes—
137C22×C22⋊C4C2312yes—
138#400—216no—
139#397—216no—
140#394—216no—
141#393—216no—
142#393—48no—
143#386—28no—
144#386—28no—
145#386—28no—
146#386—232no—
147#386—48no—
148#386—48no—
149#380—24no—
150#380—24no—
151#380—24no—
152C2×#95—28no—
153C2×#95—44no—
154C2×#95C2314yes—
155C2×#95C2318yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2318yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2318yes—
163C2×#95C2318yes—
164C2×#95C2314yes—
165C2×#95C2318yes—
166C2×#95C2318yes—
167C25—24no—
168C25—22no—
169C25—24no—
170C25—24no—
171C25—22no—
172C25—42no—
173C25—42no—
174C25C2×D414yes—
175C25#4118yes—
176C25C2412yes—
177C25C2411yescharacteristic
178C25C2411yescharacteristic
179C25C2411yescharacteristic
180C22×Q8—232no—
181C22×Q8—28no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—28no—
186C22×Q8—44no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8—464no—
190C22×Q8—416no—
191C22×Q8—416no—
192C22×Q8C2×D418yes—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—28no—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—28no—
200C22×D4—24no—
201C22×D4—28no—
202C22×D4—24no—
203C22×D4—28no—
204C22×D4—22no—
205C22×D4—24no—
206C22×D4—42no—
207C22×D4—42no—
208C22×D4—44no—
209C22×D4—44no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—232no—
218C23×C4—28no—
219C23×C4—28no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—28no—
226C23×C4—28no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—28no—
230C23×C4—28no—
231C23×C4—416no—
232C23×C4—416no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—44no—
243C23×C4—44no—
244C23×C4—44no—
245C23×C4—44no—
246C23×C4—464no—
247C23×C4C2×D414yes—
248C23×C4C2×D414yes—
249C23×C4C2×D414yes—
250C23×C4C2×D414yes—
251C23×C4C2×D414yes—
252C23×C4C2×D414yes—
253C23×C4C2×D418yes—
254C23×C4C2418yes—
255C23×C4C2412yes—
256C23×C4C2412yes—
257C23×C4C2412yes—
258C23×C4C2414yes—
259C23×C4C2414yes—
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2411yescharacteristic
264C22⋊Q8—264no—
265C22⋊Q8—264no—
266C22⋊Q8—264no—
267C22⋊Q8—264no—
268#121—416no—
269#121—416no—
270#121—416no—
271#120—432no—
272C2×C4⋊C4—232no—
273C2×C4⋊C4—264no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4C2×D4116yes—
290C2×C4⋊C4C2×D418yes—
291C2×C22⋊C4—264no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—216no—
313C2×C22⋊C4—216no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—432no—
318C2×C22⋊C4—48no—
319C2×C22⋊C4—48no—
320C2×C22⋊C4—48no—
321C2×C22⋊C4—48no—
322C2×C22⋊C4C2×D4116yes—
323#95—28no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—216no—
329#95—216no—
330#95—28no—
331#95—44no—
332#95—48no—
333#95—48no—
334#95—48no—
335#95—48no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—21no—
341C24—22no—
342C24—21no—
343C24—22no—
344C24—22no—
345C24—28no—
346C24—28no—
347C24—28no—
348C24—28no—
349C24—216no—
350C24—216no—
351C24—232no—
352C24—22no—
353C24—21no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—21no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—24no—
362C24—24no—
363C24—22no—
364C24—24no—
365C24—24no—
366C24—22no—
367C24—24no—
368C24—44no—
369C24—44no—
370C24—416no—
371C24—416no—
372C24—48no—
373C24—48no—
374C24—416no—
375C24—48no—
376C24—48no—
377C24#124116yes—
378C24C22×D412yes—
379C24C22×D412yes—
380C24C22×D414yes—
381C24C22×D412yes—
382C24C2×#4114yes—
383C24C2×#4114yes—
384C24C2×#4114yes—
385C24H(5,𝔽2)14yes—
386C24H(5,𝔽2)12yes—
387C242−1+414yes—
388C242−1+412yes—
389C24C2511yescharacteristic
390C2×Q8—216no—
391C2×Q8—216no—
392C2×Q8—416no—
393C2×Q8—432no—
394C2×Q8—4128no—
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8—48no—
398C2×Q8—416no—
399C2×Q8—48no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—416no—
412C2×D4—416no—
413C2×D4—432no—
414C2×D4—44no—
415C2×D4—88no—
416C2×D4—816no—
417C22×C4—232no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—216no—
428C22×C4—216no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—28no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—28no—
448C22×C4—28no—
449C22×C4—28no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—28no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—28no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—28no—
463C22×C4—28no—
464C22×C4—28no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—28no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—416no—
477C22×C4—416no—
478C22×C4—416no—
479C22×C4—416no—
480C22×C4—416no—
481C22×C4—416no—
482C22×C4—432no—
483C22×C4—4128no—
484C22×C4—416no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—416no—
490C22×C4—416no—
491C22×C4—416no—
492C22×C4—88no—
493C22×C4—88no—
494C22×C4—88no—
495C22×C4—88no—
496C22×C4#120116yes—
497C22×C4C22×D418yes—
498C22×C4C22×D418yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D414yes—
502C22×C4C22×D414yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4H(5,𝔽2)14yes—
506C4⋊C4—216no—
507C4⋊C4—232no—
508C4⋊C4—216no—
509C4⋊C4—232no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C22⋊C4—232no—
514C22⋊C4—232no—
515C22⋊C4—48no—
516C22⋊C4—416no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—416no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—816no—
524C22⋊C4—816no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—28no—
543C23—28no—
544C23—28no—
545C23—28no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—232no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—24no—
558C23—24no—
559C23—232no—
560C23—28no—
561C23—28no—
562C23—416no—
563C23—416no—
564C23—416no—
565C23—464no—
566C23—416no—
567C23—464no—
568C23—416no—
569C23—44no—
570C23—42no—
571C23—42no—
572C23—44no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—44no—
583C23—44no—
584C23—48no—
585C23—48no—
586C23—44no—
587C23—44no—
588C23—48no—
589C23—48no—
590C23—44no—
591C23—44no—
592C23C2×#12014yes—
593C23C2×#12414yes—
594C23C2×#12414yes—
595C23C2×#12414yes—
596C23#53414yes—
597C23#53614yes—
598C23#53714yes—
599C23#53818yes—
600C23#53918yes—
601C23#548116yes—
602C23C23×D411yescharacteristic
603C23C23×D412yes—
604C23C22×#4112yes—
605C23C2×H(5,𝔽2)12yes—
606C23C2×H(5,𝔽2)12yes—
607C23C2×H(5,𝔽2)11yescharacteristic
608C23C2×2−1+412yes—
609C23C2×2−1+411yescharacteristic
610C23#58512yes—
611Q8—432no—
612Q8—432no—
613D4—88no—
614D4—816no—
615D4—816no—
616D4—816no—
617C2×C4—232no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—28no—
625C2×C4—28no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—28no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—216no—
633C2×C4—28no—
634C2×C4—28no—
635C2×C4—416no—
636C2×C4—416no—
637C2×C4—416no—
638C2×C4—416no—
639C2×C4—416no—
640C2×C4—432no—
641C2×C4—432no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—48no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—48no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—48no—
663C2×C4—48no—
664C2×C4—48no—
665C2×C4—48no—
666C2×C4—48no—
667C2×C4—48no—
668C2×C4—48no—
669C2×C4—48no—
670C2×C4—832no—
671C2×C4C2×#12014yes—
672C2×C4C2×#12014yes—
673C2×C4#53414yes—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—216no—
678C22—216no—
679C22—24no—
680C22—24no—
681C22—216no—
682C22—216no—
683C22—24no—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—24no—
688C22—24no—
689C22—24no—
690C22—24no—
691C22—24no—
692C22—24no—
693C22—464no—
694C22—42no—
695C22—48no—
696C22—42no—
697C22—48no—
698C22—42no—
699C22—42no—
700C22—82no—
701C22—82no—
702C22—88no—
703C22—82no—
704C22—88no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22#243618yes—
710C22#263918yescomplemented
711C22C22×#12011yescharacteristic
712C22C22×#12411yescharacteristic
713C22C2×#53411yescharacteristic
714C22C2×#53411yescharacteristic
715C22C2×#53611yescharacteristic
716C22C2×#53711yescharacteristic
717C22#344911yescharacteristic
718C22C2×#53812yes—
719C22C2×#53912yes—
720C22C2×#54814yes—
721C22#348414yes—
722C22#348812yes—
723C22#348912yes—
724C22#350514yes—
725C22#353514yes—
726C22#353612yes—
727C22#353812yes—
728C4—24no—
729C4—24no—
730C4—24no—
731C4—42no—
732C4—48no—
733C4—48no—
734C4—82no—
735C4—88no—
736C4—82no—
737C4—82no—
738C2—28no—
739C2—28no—
740C2—44no—
741C2—162no—
742C2C2×#263911yescharacteristic
743C2C2×#243611yescharacteristic
744C2#3498011yescharacteristic
745C2#3501911yescharacteristic
746C2#3525111yescharacteristic
747C2#3744211yescharacteristic
748C2#3744711yescharacteristic
749C2#3770214yes—
750C2#3770314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2847C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊Q8C2semidirect
3. C2×#2639C2semidirect
4. #16632C2semidirect
5. #16366C2semidirect
6. #16253C2semidirect
7. #14873C2semidirect
8. #14822C2semidirect
9. #14776C2semidirect
10. #14759C2semidirect
11. #14719C2semidirect
12. #14281C2semidirect
13. #14134C2semidirect
14. C2×C22⋊#95C2semidirect
33. C2×#400C22semidirect
34. C2×#397C22semidirect
35. C2×#394C22semidirect
36. C2×#393C22semidirect
37. C2×#393C22semidirect
49. #1446C22semidirect
50. #1446C22semidirect
51. #1445C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
101. C2×C22⋊Q8D4semidirect
710. C22#2639semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#28472
C2×#26392
#166322
#163662
#162532
#148732
#148222
#147762
#147592
#147192
#142812
#141342
C2×C22⋊#952