Finite Group Databasebeta

← #391257

Subgroups — #391257

Order 512.

1534 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1954C211yesmaximal, characteristic, complemented
2C2×#1923C211yesmaximal, characteristic, complemented
3C4×#392C211yesmaximal, characteristic
4#11255C211yesmaximal, characteristic, complemented
5#11147C212yesmaximal, complemented
6#11042C212yesmaximal, complemented
7#11042C211yesmaximal, characteristic, complemented
8#10964C212yesmaximal, complemented
9#10964C212yesmaximal, complemented
10#10964C211yesmaximal, characteristic, complemented
11#10797C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22C22×C22⋊C8C2212yes—
23C22×C4×C8C2211yescharacteristic
24C2×#392C411yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2212yes—
27#1954—24no—
28#1923—24no—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1444C411yescharacteristic
38#1439C411yescharacteristic
39#1439C412yes—
40#1438C412yes—
41#1438C411yescharacteristic
42C4×#95C2211yescharacteristic
43C23×D4C2×C411yescharacteristic
44C23×D4C2×C412yes—
45C24×C4C2312yes—
46C24×C4C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—42no—
56C2×C4×D4—42no—
57C2×C4×D4—42no—
58C2×C4×D4—42no—
59C2×C4×D4—42no—
60C2×C4×D4—42no—
61C2×C4×D4—42no—
62C2×C4×D4—42no—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C412yes—
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C42C2311yescharacteristic
73C22×C42C2312yes—
74C22×C42C2311yescharacteristic
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—42no—
78C2×C4⋊C8—42no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8—42no—
85C2×C22⋊C8—42no—
86C2×C22⋊C8—42no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—42no—
91C2×C22⋊C8—42no—
92C2×C22⋊C8—42no—
93C2×C22⋊C8—42no—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8—42no—
97C2×C22⋊C8—42no—
98C2×C4×C8—42no—
99C2×C4×C8D414yescomplemented
100#392—24no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—42no—
104C4×C22⋊C4—42no—
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C412yes—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C411yescharacteristic
112#336—22no—
113#336—22no—
114#336—22no—
115#336—42no—
116#336—42no—
117#336—42no—
118#336—42no—
119#336—42no—
120C25C22×C411yescharacteristic
121C25C22×C412yes—
122C22×D4—24no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—22no—
126C22×D4—24no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—42no—
130C22×D4—42no—
131C22×D4—42no—
132C22×D4—42no—
133C22×D4—42no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C412yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—21no—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—48no—
204C4×D4—48no—
205C4×D4—48no—
206C4×D4—48no—
207C4×D4—48no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42C2×D412yes—
247C2×C42#4112yes—
248C4⋊C8—44no—
249C4⋊C8—44no—
250C22⋊C8—44no—
251C22⋊C8—48no—
252C22⋊C8—48no—
253C22⋊C8—44no—
254C22⋊C8—48no—
255C22⋊C8—48no—
256C22⋊C8—48no—
257C4×C8—44no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95M1614yes—
265C24—21no—
266C24—22no—
267C24—22no—
268C24—21no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—21no—
277C24—21no—
278C24—22no—
279C24—21no—
280C24—22no—
281C24—21no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24#13112yes—
290C24#13114yes—
291C24C23×C411yescharacteristic
292C2×D4—28no—
293C2×D4—48no—
294C2×D4—48no—
295C2×D4—48no—
296C2×D4—48no—
297C2×D4—44no—
298C2×D4—48no—
299C2×D4—48no—
300C2×D4—48no—
301C2×D4—48no—
302C2×D4—48no—
303C2×D4—44no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4C4×D412yes—
391C22×C4C4×D412yes—
392C22×C4C2×M1614yes—
393C22×C4C2×M1612yes—
394C22×C4#13112yes—
395C22×C4#13112yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4H(5,𝔽2)12yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)11yescharacteristic
401C22×C4H(5,𝔽2)12yes—
402C22×C4H(5,𝔽2)12yes—
403C22×C4H(5,𝔽2)11yescharacteristic
404C22×C4H(5,𝔽2)11yescharacteristic
405C22×C42−1+411yescharacteristic
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—41no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—41no—
420C2×C8—42no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C22⋊C4—28no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—44no—
449C42—22no—
450C42—22no—
451C42—24no—
452C42—41no—
453C42—41no—
454C42—42no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42—42no—
459C23—22no—
460C23—24no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—41no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23C2×C4×D411yescharacteristic
518C23#51812yes—
519C23#51811yescharacteristic
520C23#51812yes—
521C23#52011yescharacteristic
522C23C22×M1611yescharacteristic
523C23C2×#13111yescharacteristic
524C23C2×#13111yescharacteristic
525C23C2×#13112yes—
526C23C2×#13111yescharacteristic
527C23#56812yes—
528D4—416no—
529D4—432no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—82no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—81no—
586C2×C4—82no—
587C2×C4—81no—
588C2×C4—82no—
589C2×C4#43614yes—
590C2×C4#53411yescharacteristic
591C2×C4#53511yescharacteristic
592C2×C4#53511yescharacteristic
593C2×C4#53511yescharacteristic
594C2×C4#53511yescharacteristic
595C2×C4#53711yescharacteristic
596C2×C4#53812yes—
597C2×C4#54011yescharacteristic
598C2×C4#54012yes—
599C2×C4#54211yescharacteristic
600C2×C4#56012yes—
601C2×C4#56012yes—
602C2×C4#56012yes—
603C2×C4#56012yes—
604C2×C4#56012yes—
605C2×C4H(3,𝔽4)12yes—
606C2×C4#56212yes—
607C2×C4#56212yes—
608C8—42no—
609C8—42no—
610C8—82no—
611C8—81no—
612C8—82no—
613C8—81no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—42no—
624C22—42no—
625C22—44no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—81no—
631C22—82no—
632C22—81no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—81no—
637C22—82no—
638C22—81no—
639C22—82no—
640C22#232111yescharacteristic
641C22#232411yescharacteristic
642C22#232811yescharacteristic
643C22#237512yes—
644C22#237712yes—
645C22C2×#43611yescharacteristic
646C22#292911yescharacteristic
647C22#292911yescharacteristic
648C22#293211yescharacteristic
649C22#297411yescharacteristic
650C22#297412yes—
651C22#297512yes—
652C22#297612yes—
653C22#297612yes—
654C22#297712yes—
655C22#297712yes—
656C22#297712yes—
657C22#297911yescharacteristic
658C22#298112yes—
659C22#298112yes—
660C22#299111yescharacteristic
661C22#299212yes—
662C22#299411yescharacteristic
663C4—24no—
664C4—44no—
665C4—42no—
666C4—42no—
667C4—82no—
668C4—82no—
669C4—81no—
670C4—81no—
671C4#281311yescharacteristic
672C4#281511yescharacteristic
673C4#281811yescharacteristic
674C4#281911yescharacteristic
675C4#282812yes—
676C4#282912yes—
677C2—82no—
678C2—84no—
679C2#1462311yescharacteristic
680C2#2203111yescharacteristic
681C2#2203211yescharacteristic
682C2#2213611yescharacteristic
683C2#2213711yescharacteristic
684C2#2226111yescharacteristic
685C2#2226411yescharacteristic
686C2#2330812yes—
687C2#2331012yes—
688C2#2332712yes—
689C2#2333012yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC2×#392C4
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
1. C2×#1954C2semidirect
2. C2×#1923C2semidirect
4. #11255C2semidirect
5. #11147C2semidirect
6. #11042C2semidirect
7. #11042C2semidirect
8. #10964C2semidirect
9. #10964C2semidirect
10. #10964C2semidirect
11. #10797C2semidirect
99. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#19232
C4×#3922
#112552
#111472
#110422
#109642
#107972