Finite Group Databasebeta

← #367055

Subgroups — #367055

Order 512.

1970 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#466C211yesmaximal, characteristic
2C2×#1972C211yesmaximal, characteristic, complemented
3C2×#1972C211yesmaximal, characteristic, complemented
4C2×#1943C211yesmaximal, characteristic, complemented
5C2×#1807C211yesmaximal, characteristic, complemented
6C2×#1776C211yesmaximal, characteristic, complemented
7C2×#1752C211yesmaximal, characteristic, complemented
8#13998C212yesmaximal, complemented
9#11066C212yesmaximal, complemented
10#10969C212yesmaximal, complemented
11#10931C212yesmaximal, complemented
12C23×D8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#121C2212yes—
15C2×#466—24no—
16C2×#466—24no—
17C2×#466C418yescomplemented
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C2×#390—22no—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#389C2212yes—
31C2×#380C2212yes—
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#1972—24no—
35#1972—24no—
36#1943—24no—
37#1807—24no—
38#1776—24no—
39#1752—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—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—22no—
63C23×C8—22no—
64C23×C8C2×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⋊C4D412yes—
86C22×C42—22no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#466—232no—
90C2×#107C2×C414yes—
91C2×#107D414yes—
92C2×#107D414yes—
93C2×#106—22no—
94C2×#106—22no—
95C2×#106—22no—
96C2×#106—22no—
97C2×#106—22no—
98C2×#106D414yescomplemented
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—42no—
105C2×#102—42no—
106C2×#102—42no—
107C2×#102C2×C418yes—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×#97—22no—
112C2×#97—22no—
113C2×#97—22no—
114C2×#97—22no—
115C2×#97—22no—
116C2×#97D414yescomplemented
117#390—24no—
118#390—24no—
119#390—24no—
120#390—42no—
121#390—48no—
122#389—24no—
123#389—42no—
124#380—24no—
125#380—42no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—42no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95D412yes—
134#340—24no—
135#340—24no—
136#340—24no—
137#336—22no—
138#336—22no—
139C25—22no—
140C25C2×D411yescharacteristic
141C22×D4—28no—
142C22×D4—22no—
143C22×D4—22no—
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—44no—
156C22×D4—44no—
157C22×D4—44no—
158C22×D4—44no—
159C22×D4—82no—
160C22×D4—82no—
161C22×D4C22×C414yes—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—21no—
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—22no—
176C23×C4—22no—
177C23×C4C22×C412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4#4112yes—
182C23×C4C2411yescharacteristic
183C2×D8—28no—
184C2×D8—28no—
185C2×D8—28no—
186C2×D8—44no—
187C2×D8—44no—
188C2×D8—44no—
189C2×D8—44no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—28no—
209C22×C8—28no—
210C22×C8—28no—
211C22×C8—44no—
212C22×C8—44no—
213C22×C8—44no—
214C22×C8—44no—
215C22×C8C22×C412yes—
216C22×C8C22×C412yes—
217C22×C8C2×D412yes—
218C22×C8C2×D412yes—
219C22×C8C2×D412yes—
220C22×C8C2×D412yes—
221C22×C8C2×D412yes—
222C22×C8#4112yes—
223C22×C8#4112yes—
224C22×C8#4112yes—
225#121—216no—
226#121—48no—
227#121—48no—
228#121—48no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
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—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4C22×C414yes—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2×D414yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
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—82no—
267C2×C22⋊C4—82no—
268C2×C22⋊C4—82no—
269C2×C22⋊C4—82no—
270C2×C22⋊C4—82no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42—82no—
284C2×C42C2×D412yes—
285C2×C42#4112yes—
286#107C22⋊C4116yescomplemented
287#106—28no—
288#106—44no—
289#102—216no—
290#102—48no—
291#102—48no—
292C22⋊C8—28no—
293#97—28no—
294#97—44no—
295#95—24no—
296#95—44no—
297#95—44no—
298#95—44no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24C4×D412yes—
324C24AGL(1,ℤ/8)14yes—
325C24C22×D411yescharacteristic
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—416no—
333C2×D4—416no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
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—88no—
357C2×D4—88no—
358C2×D4—88no—
359C2×D4—88no—
360C2×D4—88no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
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—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4C4×D412yes—
430C22×C4#12114yes—
431C22×C4C23×C411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4H(5,𝔽2)11yescharacteristic
442C22×C4H(5,𝔽2)11yescharacteristic
443C22×C4H(5,𝔽2)11yescharacteristic
444C22×C42−1+411yescharacteristic
445D8—416no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—416no—
468C2×C8C2×C22⋊C414yes—
469C2×C8C4×D414yes—
470C2×C8C4×D414yes—
471C2×C8#12114yes—
472C2×C8#12114yes—
473C2×C8#12414yes—
474C2×C8#12714yes—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—88no—
487C4⋊C4C2×C22⋊C418yes—
488C22⋊C4—416no—
489C22⋊C4—48no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—88no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23—82no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23—82no—
544C23—82no—
545C23—82no—
546C23—82no—
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#44214yes—
562C23C2×C4×D411yescharacteristic
563C23C2×#12111yescharacteristic
564C23#53211yescharacteristic
565C23D4211yescharacteristic
566C23D4211yescharacteristic
567C23D4211yescharacteristic
568C23#54711yescharacteristic
569C23C2×AGL(1,ℤ/8)12yes—
570C23#57512yes—
571C23#57612yes—
572C23#57712yes—
573D4—44no—
574D4—44no—
575D4—44no—
576D4—44no—
577D4—832no—
578D4—82no—
579D4—82no—
580D4—82no—
581D4—82no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4C22×C22⋊C411yescharacteristic
633C2×C4C2×C4×D411yescharacteristic
634C2×C4C2×C4×D411yescharacteristic
635C2×C4C4×#4111yescharacteristic
636C2×C4#51811yescharacteristic
637C2×C4#52011yescharacteristic
638C2×C4C2×#12111yescharacteristic
639C2×C4C2×#12111yescharacteristic
640C2×C4#52511yescharacteristic
641C2×C4C2×#12411yescharacteristic
642C2×C4#52911yescharacteristic
643C2×C4C2×#12711yescharacteristic
644C2×C4#53511yescharacteristic
645C2×C4#53711yescharacteristic
646C2×C4#54011yescharacteristic
647C2×C4#54211yescharacteristic
648C2×C4#54611yescharacteristic
649C2×C4#54611yescharacteristic
650C2×C4#54611yescharacteristic
651C2×C4#54811yescharacteristic
652C2×C4#54811yescharacteristic
653C2×C4#54811yescharacteristic
654C2×C4#55011yescharacteristic
655C2×C4#55011yescharacteristic
656C2×C4#55011yescharacteristic
657C2×C4#55311yescharacteristic
658C2×C4#55311yescharacteristic
659C2×C4#55511yescharacteristic
660C8—42no—
661C8—42no—
662C8—42no—
663C8#39018yescomplemented
664C22—24no—
665C22—24no—
666C22—24no—
667C22—42no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—42no—
672C22—42no—
673C22—44no—
674C22—42no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—84no—
682C22—162no—
683C22—162no—
684C22—162no—
685C22—162no—
686C22—162no—
687C22—162no—
688C22—162no—
689C22—162no—
690C22C4×#12111yescharacteristic
691C22D4×C22⋊C411yescharacteristic
692C22#236111yescharacteristic
693C22#241611yescharacteristic
694C22#242811yescharacteristic
695C22#249111yescharacteristic
696C22#254111yescharacteristic
697C22C2×#44211yescharacteristic
698C22#294311yescharacteristic
699C22#294811yescharacteristic
700C22#295211yescharacteristic
701C22#305512yes—
702C22#306212yes—
703C22#317912yes—
704C22#318012yes—
705C22#326412yes—
706C22#326412yes—
707C22#326512yes—
708C22#326812yes—
709C22#327112yes—
710C22#327112yes—
711C22#334512yes—
712C22#334712yes—
713C4—48no—
714C4—42no—
715C4—44no—
716C4—82no—
717C4—84no—
718C4—162no—
719C4C2×#39011yescharacteristic
720C4#232411yescharacteristic
721C4#234111yescharacteristic
722C4#236311yescharacteristic
723C4#241411yescharacteristic
724C4#242611yescharacteristic
725C4#249011yescharacteristic
726C4#254011yescharacteristic
727C2—44no—
728C2—164no—
729C2#1399811yescharacteristic
730C2#2152611yescharacteristic
731C2#2152911yescharacteristic
732C2#2243611yescharacteristic
733C2#2244111yescharacteristic
734C2#2302911yescharacteristic
735C2#2303111yescharacteristic
736C2#2479012yes—
737C2#2542212yes—
738C2#2779812yes—
739C2#2849312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#466C2
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×#1972C2semidirect
4. C2×#1943C2semidirect
5. C2×#1807C2semidirect
6. C2×#1776C2semidirect
7. C2×#1752C2semidirect
8. #13998C2semidirect
9. #11066C2semidirect
10. #10969C2semidirect
11. #10931C2semidirect
17. C2×#466C4semidirect
98. C2×#106D4semidirect
116. C2×#97D4semidirect
286. #107C22⋊C4semidirect
663. C8#390semidirect

Maximal subgroups

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

subgroupindex
C22×#4662
C2×#19722
C2×#19432
C2×#18072
C2×#17762
C2×#17522
#139982
#110662
#109692
#109312