Finite Group Databasebeta

← #366274

Subgroups — #366274

Order 512.

2246 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×#408C211yesmaximal, characteristic
2C2×#1923C212yesmaximal, complemented
3C2×#1851C211yesmaximal, characteristic, complemented
4C2×#1780C211yesmaximal, characteristic
5C2×C8×C22⋊C4C211yesmaximal, characteristic, complemented
6#13851C211yesmaximal, characteristic
7#11147C212yesmaximal, complemented
8#11042C212yesmaximal, complemented
9#10921C212yesmaximal, complemented
10#10715C211yesmaximal, characteristic, complemented
11#10714C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C2×#408—28no—
17C2×#408—24no—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22C22×C22⋊C8C2212yes—
23C22×C22⋊C8C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×C4×C22⋊C4C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2212yes—
30#1923—28no—
31#1851—24no—
32#1780—24no—
33C8×C22⋊C4—24no—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1446C411yescharacteristic
42#1446C411yescharacteristic
43#1438C411yescharacteristic
44#1438C412yes—
45#1438C412yes—
46#1438C411yescharacteristic
47C22×#41—24no—
48C22×#41—22no—
49C22×#41—22no—
50C22×#41—24no—
51C22×#41—22no—
52C22×#41—22no—
53C22×#41—22no—
54C22×#41—22no—
55C22×#41—42no—
56C22×#41—42no—
57C23×Q8C2×C411yescharacteristic
58C23×D4C2×C412yes—
59C23×D4C2×C411yescharacteristic
60C24×C4C2312yes—
61C24×C4C2311yescharacteristic
62C22×M16—22no—
63C22×M16—22no—
64C22×M16—22no—
65C22×M16—24no—
66C22×M16—42no—
67C23×C8—22no—
68C23×C8—22no—
69C23×C8—24no—
70C23×C8—42no—
71C23×C8D414yes—
72C23×C8C2312yes—
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2312yes—
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4C2×C412yes—
82C22×C22⋊C4C2×C412yes—
83C22×C22⋊C4C2×C412yes—
84C22×C22⋊C4C2×C412yes—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2312yes—
88C2×C4⋊C8—24no—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—42no—
91#408—232no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—24no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—42no—
107C2×C22⋊C8—42no—
108C2×C22⋊C8D414yes—
109C2×C22⋊C8D418yes—
110C2×C4×C8—24no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8—42no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—24no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—24no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—42no—
121C4×C22⋊C4—42no—
122C2×#95C2×C411yescharacteristic
123C2×#95C2×C412yes—
124C2×#95C2×C412yes—
125C2×#95C2×C411yescharacteristic
126C2×#95C2×C411yescharacteristic
127C2×#95C2×C411yescharacteristic
128C2×#95C2×C412yes—
129C2×#95C2×C412yes—
130#336—24no—
131#336—24no—
132#336—24no—
133#336—22no—
134#336—22no—
135#336—42no—
136#336—42no—
137C25C22×C412yes—
138C25C22×C411yescharacteristic
139C2×#41—28no—
140C2×#41—216no—
141C2×#41—28no—
142C2×#41—48no—
143C2×#41—48no—
144C2×#41—48no—
145C2×#41—48no—
146C2×#41—48no—
147C2×#41—48no—
148C2×#41—48no—
149C2×#41—48no—
150C2×#41—44no—
151C2×#41—44no—
152C2×#41—48no—
153C2×#41—44no—
154C2×#41—44no—
155C2×#41—44no—
156C2×#41—44no—
157C22×Q8—22no—
158C22×Q8—24no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—42no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—22no—
168C22×D4—24no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—42no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4C22×C412yes—
191C23×C4C22×C412yes—
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C412yes—
195C23×C4C22×C411yescharacteristic
196C23×C4C22×C411yescharacteristic
197C23×C4C22×C411yescharacteristic
198C23×C4C2×D414yes—
199C23×C4C2×D412yes—
200C23×C4C2×D412yes—
201C23×C4#4114yes—
202C23×C4#4112yes—
203C23×C4#4112yes—
204C23×C4#4112yes—
205C23×C4#4112yes—
206C23×C4C2411yescharacteristic
207C2×M16—28no—
208C2×M16—48no—
209C2×M16—48no—
210C2×M16—44no—
211C2×M16—44no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—22no—
215C22×C8—28no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—22no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—22no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—28no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—44no—
243C22×C8—48no—
244C22×C8—48no—
245C22×C8C2×D412yes—
246C22×C8C2×D414yes—
247C22×C8C2×D412yes—
248C22×C8C2×D412yes—
249C22×C8C2×D414yes—
250C22×C8C2×D412yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—21no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—21no—
278C2×C42—22no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C4⋊C8—48no—
287C22⋊C8—216no—
288C22⋊C8—28no—
289C22⋊C8—216no—
290C22⋊C8—48no—
291C22⋊C8—48no—
292C22⋊C8—44no—
293C4×C8—44no—
294C4×C8—44no—
295#95—24no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—24no—
300#95—22no—
301#95—42no—
302#95—42no—
303C24—22no—
304C24—22no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24#11712yes—
326C24C4×D414yes—
327C24C4×D412yes—
328C24C4×D412yes—
329C24C23×C411yescharacteristic
330#41—416no—
331#41—416no—
332#41—432no—
333#41—432no—
334#41—416no—
335#41—416no—
336C2×Q8—28no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×D4—28no—
342C2×D4—216no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—44no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—44no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
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—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
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×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4#11712yes—
433C22×C4C4×D414yes—
434C22×C4C4×D412yes—
435C22×C4C4×D412yes—
436C22×C4#12114yes—
437C22×C4#12314yes—
438C22×C4#12414yes—
439C22×C4#12614yes—
440C22×C4#13112yes—
441C22×C4#13112yes—
442C22×C4#13114yes—
443C22×C4#13112yes—
444C22×C4#13112yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4112yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C4H(5,𝔽2)12yes—
455M16—416no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—416no—
471C2×C8—41no—
472C2×C8—42no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—42no—
487C2×C8—42no—
488C2×C8—42no—
489C2×C8—42no—
490C2×C8#12014yes—
491C22⋊C4—28no—
492C22⋊C4—28no—
493C22⋊C4—216no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C42—41no—
499C42—42no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42—42no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23#38814yes—
566C23C2×#11711yescharacteristic
567C23C2×C4×D411yescharacteristic
568C23C2×C4×D412yes—
569C23C2×C4×D411yescharacteristic
570C23#51812yes—
571C23C2×#13111yescharacteristic
572C23C2×#13111yescharacteristic
573C23C2×#13111yescharacteristic
574C23C2×#13111yescharacteristic
575C23C2×#13112yes—
576C23#56812yes—
577Q8—416no—
578D4—416no—
579D4—432no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4#38814yes—
640C2×C4C2×#12011yescharacteristic
641C2×C4C2×#12111yescharacteristic
642C2×C4C2×#12311yescharacteristic
643C2×C4#52512yes—
644C2×C4#52511yescharacteristic
645C2×C4C2×#12411yescharacteristic
646C2×C4C2×#12611yescharacteristic
647C2×C4#53411yescharacteristic
648C2×C4#53511yescharacteristic
649C2×C4#53812yes—
650C2×C4D4212yes—
651C2×C4#54612yes—
652C2×C4#54612yes—
653C2×C4#54612yes—
654C2×C4#54612yes—
655C2×C4#55112yes—
656C2×C4#55112yes—
657C2×C4#55112yes—
658C8—24no—
659C8—42no—
660C8—42no—
661C8—44no—
662C8—42no—
663C8—82no—
664C22—24no—
665C22—24no—
666C22—24no—
667C22—42no—
668C22—42no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—42no—
674C22—42no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22C2×#38811yescharacteristic
691C22C4×#12011yescharacteristic
692C22#232111yescharacteristic
693C22#234212yes—
694C22#235812yes—
695C22#292211yescharacteristic
696C22#292211yescharacteristic
697C22#292512yes—
698C22#292911yescharacteristic
699C22#292911yescharacteristic
700C22#292912yes—
701C22#293111yescharacteristic
702C22#293112yes—
703C22#293111yescharacteristic
704C22#293111yescharacteristic
705C22#293211yescharacteristic
706C22#293312yes—
707C22#293312yes—
708C22#293512yes—
709C22#297612yes—
710C22#297712yes—
711C22#297912yes—
712C22#298112yes—
713C4—24no—
714C4—42no—
715C4—42no—
716C4—44no—
717C4—44no—
718C4—42no—
719C4—42no—
720C4—82no—
721C4#240511yescharacteristic
722C4#240811yescharacteristic
723C4#247212yes—
724C4#248012yes—
725C4#248211yescharacteristic
726C4#253411yescharacteristic
727C2—44no—
728C2—84no—
729C2#1399111yescharacteristic
730C2#2104611yescharacteristic
731C2#2105111yescharacteristic
732C2#2144211yescharacteristic
733C2#2144311yescharacteristic
734C2#2202911yescharacteristic
735C2#2203111yescharacteristic
736C2#2255612yes—
737C2#2256012yes—
738C2#2284712yes—
739C2#2285312yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×#41C22
LCSG ▷ C24 ▷ 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×#1851C2semidirect
5. C2×C8×C22⋊C4C2semidirect
7. #11147C2semidirect
8. #11042C2semidirect
9. #10921C2semidirect
10. #10715C2semidirect
11. #10714C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4082
C2×#19232
C2×#18512
C2×#17802
C2×C8×C22⋊C42
#138512
#111472
#110422
#109212
#107152
#107142