Finite Group Databasebeta

← #366273

Subgroups — #366273

Order 512.

2870 conjugacy classes of subgroups.

The subgroup lattice has 781 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊C8C211yesmaximal, characteristic
2C2×#1923C212yesmaximal, complemented
3C2×#1850C211yesmaximal, characteristic, complemented
4C2×#1779C211yesmaximal, characteristic
5C2×C8×C22⋊C4C211yesmaximal, characteristic, complemented
6C4×#379C211yesmaximal, characteristic
7#11147C212yesmaximal, complemented
8#11042C212yesmaximal, complemented
9#10921C212yesmaximal, complemented
10#10714C211yesmaximal, characteristic, complemented
11#10714C211yesmaximal, characteristic, complemented
12C25×C4C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C22×C4⋊C8C2212yescomplemented
16C22×C22⋊C8—28no—
17C22×C22⋊C8—24no—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22C22×C22⋊C8C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic, complemented
25C22×C4×C8C2211yescharacteristic, complemented
26C2×#379C411yescharacteristic
27C2×#379C411yescharacteristic
28C2×C4×C22⋊C4C2212yes—
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2212yes—
32#1923—28no—
33#1850—24no—
34#1779—24no—
35C8×C22⋊C4—24no—
36C2×#336C2211yescharacteristic, complemented
37C2×#336C2212yescomplemented
38C2×#336C2212yescomplemented
39C2×#336C2212yescomplemented
40C2×#336C2211yescharacteristic, complemented
41C2×#336C2212yescomplemented
42C2×#336C2212yescomplemented
43C22⋊#95C411yescharacteristic
44C22⋊#95C412yes—
45C22⋊#95C412yes—
46C22⋊#95C411yescharacteristic
47C26C2×C411yescharacteristic
48C24×C4—24no—
49C24×C4—22no—
50C24×C4—24no—
51C24×C4—22no—
52C24×C4—22no—
53C24×C4—22no—
54C24×C4—22no—
55C24×C4—22no—
56C24×C4—42no—
57C24×C4—42no—
58C24×C4C2×C411yescharacteristic
59C24×C4C2×C412yes—
60C24×C4C2312yes—
61C24×C4C2311yescharacteristic
62C23×C8—22no—
63C23×C8—22no—
64C23×C8—22no—
65C23×C8—24no—
66C23×C8—24no—
67C23×C8—22no—
68C23×C8—22no—
69C23×C8—42no—
70C23×C8—42no—
71C23×C8D414yes—
72C23×C8C2312yes—
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2312yes—
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C22⋊C4C2×C412yes—
80C22×C22⋊C4C2×C412yes—
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C22⋊C4C2×C411yescharacteristic
83C22×C22⋊C4C2×C412yes—
84C22×C22⋊C4C2×C411yescharacteristic
85C22×C22⋊C4C2×C412yes—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2312yes—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—42no—
92C2×C22⋊C8—232no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—24no—
107C2×C22⋊C8—42no—
108C2×C22⋊C8—42no—
109C2×C22⋊C8D414yes—
110C2×C22⋊C8D418yes—
111C2×C4×C8—24no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8—42no—
115#379—24no—
116#379C818yescomplemented
117C4×C22⋊C4—22no—
118C4×C22⋊C4—24no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—24no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—42no—
124C4×C22⋊C4—42no—
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128C2×#95C2×C411yescharacteristic
129C2×#95C2×C411yescharacteristic
130C2×#95C2×C412yes—
131C2×#95C2×C412yes—
132#336—24no—
133#336—24no—
134#336—24no—
135#336—22no—
136#336—22no—
137#336—42no—
138#336—42no—
139C25—24no—
140C25—22no—
141C25—24no—
142C25—22no—
143C25—22no—
144C25—22no—
145C25—22no—
146C25—22no—
147C25—42no—
148C25—42no—
149C25C2×C814yes—
150C25C22×C412yes—
151C25C22×C411yescharacteristic
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—28no—
157C23×C4—216no—
158C23×C4—28no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—48no—
184C23×C4—48no—
185C23×C4—48no—
186C23×C4—48no—
187C23×C4—48no—
188C23×C4—48no—
189C23×C4—48no—
190C23×C4—48no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—48no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4C22×C412yes—
199C23×C4C22×C412yes—
200C23×C4C22×C411yescharacteristic
201C23×C4C22×C412yes—
202C23×C4C22×C411yescharacteristic
203C23×C4C22×C411yescharacteristic
204C23×C4C22×C411yescharacteristic
205C23×C4C22×C411yescharacteristic
206C23×C4C2×D414yes—
207C23×C4C2×D412yes—
208C23×C4C2×D412yes—
209C23×C4#4114yes—
210C23×C4#4112yes—
211C23×C4#4112yes—
212C23×C4#4112yes—
213C23×C4#4112yes—
214C23×C4C2411yescharacteristic
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—22no—
218C22×C8—28no—
219C22×C8—28no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—22no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—22no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—28no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—44no—
247C22×C8—48no—
248C22×C8—48no—
249C22×C8—48no—
250C22×C8—48no—
251C22×C8—44no—
252C22×C8—44no—
253C22×C8C2×D414yes—
254C22×C8C2×D412yes—
255C22×C8C2×D412yes—
256C22×C8C2×D412yes—
257C22×C8C2×D414yes—
258C22×C8C2×D412yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4C2×C814yes—
279C2×C22⋊C4C2×C814yes—
280C2×C22⋊C4C2×C818yescomplemented
281C2×C22⋊C4C2×C818yescomplemented
282C2×C22⋊C4M1614yes—
283C2×C42—21no—
284C2×C42—22no—
285C2×C42—21no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—21no—
289C2×C42—22no—
290C2×C42—21no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C4⋊C8—48no—
298C22⋊C8—216no—
299C22⋊C8—28no—
300C22⋊C8—216no—
301C22⋊C8—48no—
302C22⋊C8—48no—
303C22⋊C8—44no—
304C4×C8—44no—
305C4×C8—44no—
306#95—24no—
307#95—22no—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—42no—
312#95—42no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—28no—
318C24—28no—
319C24—216no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—216no—
333C24—28no—
334C24—28no—
335C24—28no—
336C24—48no—
337C24—48no—
338C24—48no—
339C24—48no—
340C24—48no—
341C24—48no—
342C24—48no—
343C24—48no—
344C24—48no—
345C24—44no—
346C24—44no—
347C24—48no—
348C24—44no—
349C24—44no—
350C24—44no—
351C24—44no—
352C24#11712yes—
353C24C4×D414yes—
354C24C4×D412yes—
355C24C4×D412yes—
356C24C22×C814yes—
357C24C22×C812yes—
358C24C2×M1612yes—
359C24#13114yes—
360C24#13112yes—
361C24#13112yes—
362C24C23×C411yescharacteristic
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—216no—
380C22×C4—28no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—432no—
438C22×C4—432no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—48no—
460C22×C4#11712yes—
461C22×C4C4×D414yes—
462C22×C4C4×D412yes—
463C22×C4C4×D412yes—
464C22×C4#12114yes—
465C22×C4#12314yes—
466C22×C4#12414yes—
467C22×C4#12614yes—
468C22×C4C22×C812yes—
469C22×C4C22×C814yes—
470C22×C4C22×C812yes—
471C22×C4C2×M1612yes—
472C22×C4C2×M1612yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C2×#4111yescharacteristic
477C22×C4C2×#4111yescharacteristic
478C22×C4C2×#4111yescharacteristic
479C22×C4C2×#4111yescharacteristic
480C22×C4C2×#4112yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C4H(5,𝔽2)12yes—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—416no—
498C2×C8—416no—
499C2×C8—41no—
500C2×C8—42no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—42no—
505C2×C8—42no—
506C2×C8—42no—
507C2×C8—42no—
508C2×C8—42no—
509C2×C8—42no—
510C2×C8—42no—
511C2×C8—42no—
512C2×C8—42no—
513C2×C8—42no—
514C2×C8—42no—
515C2×C8—42no—
516C2×C8—42no—
517C2×C8—42no—
518C2×C8#12014yes—
519C22⋊C4—28no—
520C22⋊C4—28no—
521C22⋊C4—216no—
522C22⋊C4—28no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—44no—
527C42—41no—
528C42—42no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—42no—
533C42—42no—
534C42—42no—
535C42—42no—
536C42—42no—
537C23—22no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—22no—
543C23—24no—
544C23—22no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—22no—
553C23—232no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—432no—
573C23—432no—
574C23—416no—
575C23—416no—
576C23—416no—
577C23—432no—
578C23—416no—
579C23—432no—
580C23—416no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—416no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23#38814yes—
599C23#43114yes—
600C23#43314yes—
601C23C8×D418yes—
602C23C8×D414yes—
603C23C8×D414yes—
604C23#43514yes—
605C23#43614yes—
606C23C2×#11711yescharacteristic
607C23C2×C4×D411yescharacteristic
608C23C2×C4×D412yes—
609C23C2×C4×D411yescharacteristic
610C23#51812yes—
611C23C23×C811yescharacteristic
612C23C22×M1611yescharacteristic
613C23C2×#13111yescharacteristic
614C23C2×#13111yescharacteristic
615C23C2×#13112yes—
616C23#56812yes—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—28no—
624C2×C4—28no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—432no—
642C2×C4—416no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4#38814yes—
679C2×C4C2×#12011yescharacteristic
680C2×C4C2×#12111yescharacteristic
681C2×C4C2×#12311yescharacteristic
682C2×C4#52512yes—
683C2×C4#52511yescharacteristic
684C2×C4C2×#12411yescharacteristic
685C2×C4C2×#12611yescharacteristic
686C2×C4#53411yescharacteristic
687C2×C4#53511yescharacteristic
688C2×C4#53812yes—
689C2×C4D4212yes—
690C2×C4#54612yes—
691C2×C4#54612yes—
692C2×C4#54612yes—
693C2×C4#54612yes—
694C2×C4#55112yes—
695C2×C4#55112yes—
696C2×C4#55112yes—
697C8—24no—
698C8—42no—
699C8—42no—
700C8—44no—
701C8—42no—
702C8—82no—
703C22—24no—
704C22—24no—
705C22—28no—
706C22—28no—
707C22—24no—
708C22—28no—
709C22—28no—
710C22—24no—
711C22—24no—
712C22—24no—
713C22—28no—
714C22—28no—
715C22—28no—
716C22—464no—
717C22—44no—
718C22—44no—
719C22—44no—
720C22—44no—
721C22—44no—
722C22—44no—
723C22—44no—
724C22—44no—
725C22—44no—
726C22—44no—
727C22—44no—
728C22—44no—
729C22#192318yescomplemented
730C22C2×#38811yescharacteristic
731C22C4×#12011yescharacteristic
732C22#232111yescharacteristic
733C22#234212yes—
734C22#235812yes—
735C22C2×#43111yescharacteristic
736C22C2×#43311yescharacteristic
737C22#292412yes—
738C22C2×C8×D411yescharacteristic
739C22C2×C8×D412yes—
740C22C2×C8×D411yescharacteristic
741C22C2×#43511yescharacteristic
742C22C2×#43611yescharacteristic
743C22#292911yescharacteristic
744C22#292912yes—
745C22#293011yescharacteristic
746C22#293312yes—
747C22#293312yes—
748C22D4×M1612yes—
749C22#297212yes—
750C22#297412yes—
751C22#297612yes—
752C22#297712yes—
753C4—24no—
754C4—42no—
755C4—42no—
756C4—44no—
757C4—44no—
758C4—42no—
759C4—42no—
760C4—82no—
761C4#240511yescharacteristic
762C4#240811yescharacteristic
763C4#247212yes—
764C4#248012yes—
765C4#248211yescharacteristic
766C4#253411yescharacteristic
767C2—28no—
768C2—48no—
769C2#1399111yescharacteristic
770C2C2×#192311yescharacteristic
771C2#2105011yescharacteristic
772C2C8×#12011yescharacteristic
773C2#2143911yescharacteristic
774C2#2202811yescharacteristic
775C2#2203011yescharacteristic
776C2#2255212yes—
777C2#2255812yes—
778C2#2284212yes—
779C2#2284812yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC26C2×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1923C2semidirect
3. C2×#1850C2semidirect
5. C2×C8×C22⋊C4C2semidirect
7. #11147C2semidirect
8. #11042C2semidirect
9. #10921C2semidirect
10. #10714C2semidirect
11. #10714C2semidirect
15. C22×C4⋊C8C22semidirect
24. C22×C4×C8C22semidirect
25. C22×C4×C8C22semidirect
36. C2×#336C22semidirect
37. C2×#336C22semidirect
38. C2×#336C22semidirect
39. C2×#336C22semidirect
40. C2×#336C22semidirect
41. C2×#336C22semidirect
42. C2×#336C22semidirect
116. #379C8semidirect
280. C2×C22⋊C4C2×C8semidirect
281. C2×C22⋊C4C2×C8semidirect
729. C22#1923semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊C82
C2×#19232
C2×#18502
C2×#17792
C2×C8×C22⋊C42
C4×#3792
#111472
#110422
#109212
#107142