Finite Group Databasebeta

← #385940

Subgroups — #385940

Order 512.

1768 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2392C211yesmaximal, characteristic
2C2×#1954C212yesmaximal, complemented
3C2×#1916C212yesmaximal, complemented
4#11042C212yesmaximal, complemented
5#10964C212yesmaximal, complemented
6#10964C212yesmaximal, complemented
7#10921C212yesmaximal, complemented
8#10713C211yesmaximal, characteristic, complemented
9#10713C211yesmaximal, characteristic, complemented
10C22×C4×D4C2212yes—
11C22×C4×D4C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C4×C8C2212yes—
22#2392—28no—
23C2×#398C411yescharacteristic
24C2×#397C411yescharacteristic
25C2×#396C412yes—
26C2×#394C411yescharacteristic
27C2×#394C412yes—
28C2×#392C411yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2212yes—
31#1954—28no—
32#1916—28no—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C23×D4C2×C412yes—
41C23×D4C2×C411yescharacteristic
42C24×C4C2312yes—
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×C4×D4—24no—
49C2×C4×D4—24no—
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—24no—
53C2×C4×D4—41no—
54C2×C4×D4—41no—
55C2×C4×D4—42no—
56C2×C4×D4—42no—
57C2×C4×D4—42no—
58C2×#117—24no—
59C2×#117—41no—
60C2×#117—41no—
61C2×#117—42no—
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C22⋊C4C2×C412yes—
74C22×C42C2311yescharacteristic
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77C2×C4⋊C8—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—42no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—41no—
88C2×C22⋊C8—41no—
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—41no—
96C2×C22⋊C8—41no—
97C2×C22⋊C8—42no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8—42no—
100C2×C22⋊C8D414yescomplemented
101C2×C4×C8—24no—
102C2×C4×C8—24no—
103C2×C4×C8—42no—
104#398—24no—
105#397—24no—
106#396—28no—
107#394—28no—
108#394—24no—
109#392—24no—
110C4×C4⋊C4D418yes—
111C4×C22⋊C4—24no—
112C4×C22⋊C4—24no—
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115C2×#95C2×C411yescharacteristic
116C2×#95C2×C411yescharacteristic
117#336—24no—
118#336—24no—
119#336—24no—
120#336—24no—
121#336—41no—
122#336—42no—
123#336—41no—
124C25C22×C411yescharacteristic
125C25C22×C412yes—
126C22×D4—24no—
127C22×D4—24no—
128C22×D4—24no—
129C22×D4—22no—
130C22×D4—24no—
131C22×D4—24no—
132C22×D4—24no—
133C22×D4—42no—
134C22×D4—41no—
135C22×D4—41no—
136C22×D4—42no—
137C22×D4—42no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
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—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C412yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C412yes—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C412yes—
168C23×C4C2×D412yes—
169C23×C4#4112yes—
170C23×C4C2411yescharacteristic
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
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—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8C2×D414yes—
198C22×C8#4114yes—
199C4×D4—48no—
200C4×D4—48no—
201C4×D4—48no—
202C4×D4—48no—
203C4×D4—48no—
204C4×D4—48no—
205#117—48no—
206#117—48no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4C22⋊C418yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42C2×D414yes—
255C2×C42C2×D414yes—
256C2×C42C2×D414yes—
257C4⋊C8—48no—
258C22⋊C8—44no—
259C22⋊C8—48no—
260C22⋊C8—48no—
261C22⋊C8—48no—
262C22⋊C8—48no—
263C22⋊C8—44no—
264C22⋊C8—48no—
265C4×C8—48no—
266#95—24no—
267#95—24no—
268#95C22⋊C414yes—
269#95C22⋊C414yes—
270C24—21no—
271C24—22no—
272C24—22no—
273C24—21no—
274C24—22no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—21no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24C4×D412yes—
295C24#13114yes—
296C24C23×C411yescharacteristic
297C2×D4—216no—
298C2×D4—48no—
299C2×D4—48no—
300C2×D4—48no—
301C2×D4—48no—
302C2×D4—48no—
303C2×D4—44no—
304C2×D4—48no—
305C2×D4—44no—
306C2×D4—48no—
307C2×D4—44no—
308C2×D4—48no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
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×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4C2×C22⋊C414yes—
395C22×C4C2×C22⋊C412yes—
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C414yes—
398C22×C4C4×D412yes—
399C22×C4#13114yes—
400C22×C4#13114yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D412yes—
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4112yes—
405C22×C4H(5,𝔽2)11yescharacteristic
406C22×C4H(5,𝔽2)11yescharacteristic
407C22×C4H(5,𝔽2)11yescharacteristic
408C22×C4H(5,𝔽2)12yes—
409C22×C4H(5,𝔽2)12yes—
410C22×C42−1+411yescharacteristic
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
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—
439C4⋊C4—216no—
440C4⋊C4—48no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C42—24no—
452C42—24no—
453C42—24no—
454C42—24no—
455C42—24no—
456C42—24no—
457C42—42no—
458C42—42no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—41no—
487C23—42no—
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—41no—
503C23—42no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—42no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23C22×C22⋊C411yescharacteristic
518C23C2×C4×D411yescharacteristic
519C23C4×#4111yescharacteristic
520C23#51811yescharacteristic
521C23#51811yescharacteristic
522C23#51811yescharacteristic
523C23#52011yescharacteristic
524C23C2×#13112yes—
525C23C2×#13112yes—
526C23C2×#13112yes—
527C23#56812yes—
528D4—432no—
529D4—416no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
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—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4#40818yes—
590C2×C4#53411yescharacteristic
591C2×C4#53512yes—
592C2×C4#53512yes—
593C2×C4#53711yescharacteristic
594C2×C4#53812yes—
595C2×C4#53811yescharacteristic
596C2×C4#53911yescharacteristic
597C2×C4#54012yes—
598C2×C4D4212yes—
599C2×C4#54612yes—
600C2×C4#54612yes—
601C2×C4#54812yes—
602C2×C4#55012yes—
603C2×C4#55312yes—
604C2×C4#56012yes—
605C2×C4#56212yes—
606C8—44no—
607C8—44no—
608C8—82no—
609C8—82no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—44no—
621C22—42no—
622C22—82no—
623C22—82no—
624C22—81no—
625C22—82no—
626C22—81no—
627C22—82no—
628C22—82no—
629C22—81no—
630C22—81no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22#230911yescharacteristic
637C22#232511yescharacteristic
638C22#233311yescharacteristic
639C22D4×C22⋊C411yescharacteristic
640C22#234111yescharacteristic
641C22#236411yescharacteristic
642C22#237711yescharacteristic
643C22C2×#40812yes—
644C22#288012yes—
645C22#292912yes—
646C22#293312yes—
647C22#296612yes—
648C22#296912yes—
649C22#297612yes—
650C22#297612yes—
651C22#297711yescharacteristic
652C22#297712yes—
653C22#297811yescharacteristic
654C22#297912yes—
655C22#297911yescharacteristic
656C22#297911yescharacteristic
657C22#299312yes—
658C22#299412yes—
659C4—28no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—81no—
665C4—81no—
666C4—82no—
667C4#267312yes—
668C4#267612yes—
669C4#271112yes—
670C4#271412yes—
671C2—84no—
672C2—82no—
673C2#1441811yescharacteristic
674C2#2178711yescharacteristic
675C2#2179211yescharacteristic
676C2#2216911yescharacteristic
677C2#2217911yescharacteristic
678C2#2233011yescharacteristic
679C2#2233211yescharacteristic
680C2#2239812yes—
681C2#2253112yes—
682C2#2312212yes—
683C2#2332812yes—
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
2. C2×#1954C2semidirect
3. C2×#1916C2semidirect
4. #11042C2semidirect
5. #10964C2semidirect
6. #10964C2semidirect
7. #10921C2semidirect
8. #10713C2semidirect
9. #10713C2semidirect
100. C2×C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23922
C2×#19542
C2×#19162
#110422
#109642
#109212
#107132