Finite Group Databasebeta

← #367052

Subgroups — #367052

Order 512.

1948 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#469C211yesmaximal, characteristic
2C2×#1972C211yesmaximal, characteristic, complemented
3C2×#1970C211yesmaximal, characteristic, complemented
4C2×#1944C211yesmaximal, characteristic, complemented
5C2×#1809C211yesmaximal, characteristic, complemented
6C2×C4×#106C211yesmaximal, characteristic, complemented
7C2×#1753C211yesmaximal, characteristic, complemented
8#13998C212yesmaximal, complemented
9#11066C212yesmaximal, complemented
10#10969C212yesmaximal, complemented
11#10931C212yesmaximal, complemented
12C23×D8C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×#121C2212yes—
15C2×#469—24no—
16C2×#469—24no—
17C2×#469C418yescomplemented
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C2×#390—22no—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#389C2212yes—
31C2×#380C2212yes—
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#1972—24no—
35#1970—24no—
36#1944—24no—
37#1809—24no—
38C4×#106—24no—
39#1753—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2211yescharacteristic
44C2×#336C2211yescharacteristic
45#1439—22no—
46#1438—22no—
47#1437—22no—
48C23×D4—22no—
49C23×D4C2312yes—
50C24×C4C2311yescharacteristic
51C22×D8—22no—
52C22×D8—22no—
53C22×D8—22no—
54C22×D8—22no—
55C22×D8—22no—
56C22×D8—22no—
57C22×D8C2×C414yes—
58C22×M16—22no—
59C22×M16—22no—
60C22×M16—22no—
61C22×M16—22no—
62C22×M16—22no—
63C22×M16—22no—
64C22×M16C2×C414yes—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C2×#121—24no—
70C2×#121—24no—
71C2×#121—24no—
72C2×#121—42no—
73C2×#121—42no—
74C2×#121C2×C418yes—
75C22×C4⋊C4—22no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4D411yescharacteristic
86C22×C22⋊C4D411yescharacteristic
87C22×C42—22no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#469—232no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#107—22no—
94C2×#106—22no—
95C2×#106—22no—
96C2×#106C2×C414yes—
97C2×#106D414yes—
98C2×#106D414yes—
99C2×#106D414yescomplemented
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—42no—
106C2×#102—42no—
107C2×#102—42no—
108C2×#102C2×C418yes—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×#97—22no—
113C2×#97—22no—
114C2×#97—22no—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117C2×C4×C8D414yescomplemented
118#390—24no—
119#390—24no—
120#390—24no—
121#390—42no—
122#390—48no—
123#389—24no—
124#389—42no—
125#380—24no—
126#380—42no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—42no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95D411yescharacteristic
135C2×#95D411yescharacteristic
136#340—24no—
137#340—24no—
138#340—24no—
139#336—22no—
140#336—22no—
141C25—22no—
142C25C2×D411yescharacteristic
143C22×D4—28no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—44no—
158C22×D4—44no—
159C22×D4—44no—
160C22×D4—44no—
161C22×D4—82no—
162C22×D4—82no—
163C22×D4C22×C414yes—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4C22×C412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4#4112yes—
186C23×C4C2411yescharacteristic
187C2×D8—28no—
188C2×D8—28no—
189C2×D8—28no—
190C2×D8—44no—
191C2×D8—44no—
192C2×D8—44no—
193C2×D8—44no—
194C2×M16—28no—
195C2×M16—28no—
196C2×M16—28no—
197C2×M16—44no—
198C2×M16—44no—
199C2×M16—44no—
200C2×M16—44no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8C22×C412yes—
220C22×C8C22×C412yes—
221C22×C8C2×D412yes—
222C22×C8C2×D412yes—
223C22×C8C2×D412yes—
224C22×C8C2×D412yes—
225C22×C8C2×D412yes—
226C22×C8#4112yes—
227C22×C8#4112yes—
228C22×C8#4112yes—
229#121—216no—
230#121—48no—
231#121—48no—
232#121—48no—
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—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—82no—
250C2×C4⋊C4C22×C414yes—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D414yes—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—82no—
272C2×C22⋊C4—82no—
273C2×C22⋊C4—82no—
274C2×C22⋊C4—82no—
275C2×C22⋊C4—82no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42—82no—
289C2×C42C2×D412yes—
290C2×C42#4112yes—
291#107—44no—
292#106—28no—
293#106C22⋊C4116yescomplemented
294#102—216no—
295#102—48no—
296#102—48no—
297C22⋊C8—28no—
298#97—44no—
299C4×C8—28no—
300#95—22no—
301#95—22no—
302#95—44no—
303#95—44no—
304#95—44no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—21no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24C4×D412yes—
331C24C22×D411yescharacteristic
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—416no—
339C2×D4—416no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—88no—
363C2×D4—88no—
364C2×D4—88no—
365C2×D4—88no—
366C2×D4—88no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
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—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4C4×D412yes—
440C22×C4#12112yes—
441C22×C4#12112yes—
442C22×C4AGL(1,ℤ/8)14yes—
443C22×C4C23×C411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4H(5,𝔽2)11yescharacteristic
454C22×C4H(5,𝔽2)11yescharacteristic
455C22×C4H(5,𝔽2)11yescharacteristic
456C22×C42−1+411yescharacteristic
457D8—416no—
458M16—416no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8C2×C22⋊C414yes—
481C2×C8C4×D414yes—
482C2×C8C4×D414yes—
483C2×C8#12114yes—
484C2×C8#12114yes—
485C2×C8#12414yes—
486C2×C8#12714yes—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—88no—
499C4⋊C4C2×C22⋊C418yes—
500C22⋊C4—416no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—88no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—22no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—82no—
548C23—82no—
549C23—82no—
550C23—82no—
551C23—82no—
552C23—82no—
553C23—82no—
554C23—82no—
555C23—82no—
556C23—82no—
557C23—82no—
558C23—82no—
559C23—82no—
560C23—82no—
561C23—82no—
562C23—82no—
563C23—82no—
564C23—82no—
565C23—82no—
566C23—82no—
567C23—82no—
568C23—82no—
569C23—82no—
570C23—82no—
571C23—82no—
572C23—82no—
573C23—82no—
574C23—82no—
575C23C2×C4×D411yescharacteristic
576C23C2×#12111yescharacteristic
577C23#53211yescharacteristic
578C23D4211yescharacteristic
579C23D4211yescharacteristic
580C23D4211yescharacteristic
581C23#54711yescharacteristic
582C23C2×AGL(1,ℤ/8)12yes—
583C23#57512yes—
584C23#57612yes—
585C23#57712yes—
586D4—44no—
587D4—44no—
588D4—44no—
589D4—44no—
590D4—832no—
591D4—82no—
592D4—82no—
593D4—82no—
594D4—82no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—82no—
648C2×C4—82no—
649C2×C4—82no—
650C2×C4#46914yescomplemented
651C2×C4C22×C22⋊C411yescharacteristic
652C2×C4C2×C4×D411yescharacteristic
653C2×C4C2×C4×D411yescharacteristic
654C2×C4C4×#4111yescharacteristic
655C2×C4#51811yescharacteristic
656C2×C4#52011yescharacteristic
657C2×C4C2×#12111yescharacteristic
658C2×C4C2×#12111yescharacteristic
659C2×C4#52511yescharacteristic
660C2×C4C2×#12411yescharacteristic
661C2×C4#52911yescharacteristic
662C2×C4C2×#12711yescharacteristic
663C2×C4#53511yescharacteristic
664C2×C4#53711yescharacteristic
665C2×C4#54011yescharacteristic
666C2×C4#54211yescharacteristic
667C2×C4#54611yescharacteristic
668C2×C4#54611yescharacteristic
669C2×C4#54611yescharacteristic
670C2×C4#54811yescharacteristic
671C2×C4#54811yescharacteristic
672C2×C4#54811yescharacteristic
673C2×C4#55011yescharacteristic
674C2×C4#55011yescharacteristic
675C2×C4#55011yescharacteristic
676C2×C4#55311yescharacteristic
677C2×C4#55311yescharacteristic
678C2×C4#55511yescharacteristic
679C8—42no—
680C8—42no—
681C8—42no—
682C8#39018yescomplemented
683C22—42no—
684C22—42no—
685C22—44no—
686C22—42no—
687C22—44no—
688C22—81no—
689C22—82no—
690C22—81no—
691C22—82no—
692C22—81no—
693C22—81no—
694C22—84no—
695C22—84no—
696C22—84no—
697C22—84no—
698C22—84no—
699C22—84no—
700C22—84no—
701C22—162no—
702C22—162no—
703C22—162no—
704C22—162no—
705C22—162no—
706C22—162no—
707C22—162no—
708C22—162no—
709C22C4×#12111yescharacteristic
710C22D4×C22⋊C411yescharacteristic
711C22#236111yescharacteristic
712C22#241611yescharacteristic
713C22#242811yescharacteristic
714C22#249111yescharacteristic
715C22#254111yescharacteristic
716C22C4×AGL(1,ℤ/8)12yes—
717C22#294612yes—
718C22C2×#46911yescharacteristic
719C22#305411yescharacteristic
720C22#305811yescharacteristic
721C22#306011yescharacteristic
722C22#317112yes—
723C22#317912yes—
724C22#326512yes—
725C22#326812yes—
726C22#327412yes—
727C22#327412yes—
728C22#327512yes—
729C22#327512yes—
730C22#334512yes—
731C22#334712yes—
732C4—24no—
733C4—48no—
734C4—42no—
735C4—42no—
736C4—81no—
737C4—84no—
738C4—81no—
739C4—162no—
740C4C2×#39011yescharacteristic
741C4#232411yescharacteristic
742C4#234111yescharacteristic
743C4#236311yescharacteristic
744C4#241411yescharacteristic
745C4#242611yescharacteristic
746C4#249011yescharacteristic
747C4#254011yescharacteristic
748C2—82no—
749C2—164no—
750C2#1399811yescharacteristic
751C2C4×#46911yescharacteristic
752C2#2153611yescharacteristic
753C2#2244312yes—
754C2#2303912yes—
755C2#2478211yescharacteristic
756C2#2478411yescharacteristic
757C2#2541612yes—
758C2#2779412yes—
759C2#2848711yescharacteristic
760C2#2848811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#469C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1972C2semidirect
3. C2×#1970C2semidirect
4. C2×#1944C2semidirect
5. C2×#1809C2semidirect
6. C2×C4×#106C2semidirect
7. C2×#1753C2semidirect
8. #13998C2semidirect
9. #11066C2semidirect
10. #10969C2semidirect
11. #10931C2semidirect
17. C2×#469C4semidirect
99. C2×#106D4semidirect
117. C2×C4×C8D4semidirect
293. #106C22⋊C4semidirect
650. C2×C4#469semidirect
682. C8#390semidirect

Maximal subgroups

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

subgroupindex
C22×#4692
C2×#19722
C2×#19702
C2×#19442
C2×#18092
C2×C4×#1062
C2×#17532
#139982
#110662
#109692
#109312