Finite Group Databasebeta

← #378101

Subgroups — #378101

Order 512.

1990 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1954C212yesmaximal, complemented
2C2×#1869C212yesmaximal, complemented
3C2×#1869C212yesmaximal, complemented
4C2×C8×C22⋊C4C212yesmaximal, complemented
5C4×#392C211yesmaximal, characteristic
6#11147C211yesmaximal, characteristic, complemented
7#11042C212yesmaximal, complemented
8#10921C211yesmaximal, characteristic, complemented
9#10876C212yesmaximal, complemented
10C24×C8C2212yes—
11C22×C4×D4C2211yescharacteristic
12C22×C4×D4C2212yes—
13C22×C4⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C22×C4×C8C2212yes—
21C22×C4×C8C2211yescharacteristic
22C2×#392C411yescharacteristic
23C2×C4×C22⋊C4C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1954—28no—
26#1869—28no—
27#1869—28no—
28C8×C22⋊C4—28no—
29C2×#336C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35#1444C411yescharacteristic
36#1439C412yes—
37#1439C411yescharacteristic
38#1438C411yescharacteristic
39#1438C412yes—
40C4×#95C2211yescharacteristic
41C23×D4C2×C411yescharacteristic
42C23×D4C2×C412yes—
43C24×C4C2311yescharacteristic
44C24×C4C2312yes—
45C23×C8—24no—
46C23×C8—24no—
47C23×C8—24no—
48C23×C8—24no—
49C23×C8—24no—
50C23×C8—42no—
51C23×C8—42no—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4—24no—
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—42no—
63C2×C4×D4—42no—
64C2×C4×D4—41no—
65C2×C4×D4—41no—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C412yes—
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2312yes—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—42no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—41no—
93C2×C22⋊C8—41no—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8D418yescomplemented
97C2×C4×C8—24no—
98C2×C4×C8—24no—
99C2×C4×C8—24no—
100C2×C4×C8—42no—
101C2×C4×C8—41no—
102#392C818yescomplemented
103C4×C22⋊C4—24no—
104C4×C22⋊C4—24no—
105C4×C22⋊C4—41no—
106C4×C22⋊C4—41no—
107C2×#95C2×C412yes—
108C2×#95C2×C412yes—
109C2×#95C2×C412yes—
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114#336—24no—
115#336—24no—
116#336—24no—
117#336—24no—
118#336—41no—
119#336D418yescomplemented
120C25C22×C411yescharacteristic
121C25C22×C412yes—
122C22×D4—24no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—24no—
126C22×D4—24no—
127C22×D4—24no—
128C22×D4—42no—
129C22×D4—41no—
130C22×D4—42no—
131C22×D4—41no—
132C22×D4C2×C814yes—
133C22×D4C2×C818yescomplemented
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
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—21no—
153C23×C4—21no—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C412yes—
157C23×C4C22×C412yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C412yes—
162C23×C4C2×D414yes—
163C23×C4#4114yes—
164C23×C4C2411yescharacteristic
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—216no—
170C22×C8—22no—
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—48no—
190C22×C8—48no—
191C22×C8—48no—
192C22×C8—48no—
193C22×C8—48no—
194C22×C8—48no—
195C22×C8C2×D414yes—
196C22×C8C2×D414yes—
197C22×C8C2×D414yes—
198C22×C8C2×D414yes—
199C22×C8#4114yes—
200C4×D4—48no—
201C4×D4—48no—
202C4×D4—48no—
203C4×D4—48no—
204C4×D4—44no—
205C4×D4—48no—
206C4×D4—44no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—41no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4C2×C818yescomplemented
232C2×C22⋊C4C2×C814yescomplemented
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42C2×D414yes—
246C2×C42#4114yes—
247C4⋊C8—48no—
248C22⋊C8—216no—
249C22⋊C8—44no—
250C22⋊C8—44no—
251C22⋊C8—48no—
252C22⋊C8—48no—
253C22⋊C8—48no—
254C4×C8—48no—
255C4×C8—44no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95C2×C814yescomplemented
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—22no—
282C24—22no—
283C24—21no—
284C24—21no—
285C24C4×D414yes—
286C24C22×C812yes—
287C24C22×C814yes—
288C24#13114yes—
289C24C23×C411yescharacteristic
290C2×D4—216no—
291C2×D4—216no—
292C2×D4—216no—
293C2×D4—44no—
294C2×D4—48no—
295C2×D4—48no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4—48no—
299C2×D4—44no—
300C2×D4—48no—
301C2×D4—44no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
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—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4C4×D414yes—
385C22×C4C4×D414yes—
386C22×C4C4×D414yes—
387C22×C4C22×C812yes—
388C22×C4C22×C812yes—
389C22×C4C22×C814yes—
390C22×C4#13112yes—
391C22×C4#13114yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C2×#4112yes—
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4H(5,𝔽2)11yescharacteristic
398C22×C4H(5,𝔽2)12yes—
399C22×C4H(5,𝔽2)11yescharacteristic
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—432no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
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#12118yes—
429C4⋊C4—48no—
430C4⋊C4—44no—
431C22⋊C4—216no—
432C22⋊C4—216no—
433C22⋊C4—44no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—44no—
438C42—24no—
439C42—24no—
440C42—24no—
441C42—24no—
442C42—41no—
443C42—41no—
444C42—42no—
445C42—42no—
446C42—41no—
447C42—41no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—42no—
489C23—41no—
490C23—41no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23C8×D418yes—
506C23C2×C4×D412yes—
507C23C2×C4×D412yes—
508C23C4×#4111yescharacteristic
509C23#51811yescharacteristic
510C23#51811yescharacteristic
511C23C23×C811yescharacteristic
512C23C2×#13112yes—
513C23C2×#13112yes—
514C23C2×#13111yescharacteristic
515C23#56811yescharacteristic
516C23#56811yescharacteristic
517D4—432no—
518D4—416no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4C8×D418yes—
577C2×C4C2×#12112yes—
578C2×C4#52512yes—
579C2×C4#52912yes—
580C2×C4#53212yes—
581C2×C4#53812yes—
582C2×C4#54012yes—
583C2×C4D4211yescharacteristic
584C2×C4D4212yes—
585C2×C4D4211yescharacteristic
586C2×C4#54612yes—
587C2×C4#54612yes—
588C2×C4#54612yes—
589C2×C4#55012yes—
590C2×C4#55111yescharacteristic
591C2×C4#55112yes—
592C2×C4#55911yescharacteristic
593C8—28no—
594C8—44no—
595C8—44no—
596C8—82no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—42no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—44no—
617C22—81no—
618C22—82no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—82no—
623C22C4×#12112yes—
624C22#233311yescharacteristic
625C22#235812yes—
626C22#235811yescharacteristic
627C22#236611yescharacteristic
628C22C2×C8×D412yes—
629C22C2×C8×D412yes—
630C22#292912yes—
631C22#293112yes—
632C22#293312yes—
633C22#293312yes—
634C22#293312yes—
635C22#293512yes—
636C22C8×#4111yescharacteristic
637C22#296612yes—
638C22#296711yescharacteristic
639C22#297211yescharacteristic
640C22#297211yescharacteristic
641C22#297611yescharacteristic
642C22#297712yes—
643C22#297711yescharacteristic
644C22#297911yescharacteristic
645C22#298111yescharacteristic
646C4—28no—
647C4—42no—
648C4—44no—
649C4—44no—
650C4—42no—
651C4—42no—
652C4—81no—
653C4—81no—
654C4#242412yes—
655C4#242812yes—
656C4#247112yes—
657C4#255712yes—
658C2—48no—
659C2—82no—
660C2#1420711yescharacteristic
661C2C8×#12112yes—
662C2#2149612yes—
663C2#2232911yescharacteristic
664C2#2233011yescharacteristic
665C2#2284212yes—
666C2#2284211yescharacteristic
667C2#2284712yes—
668C2#2285311yescharacteristic
669C2#2317311yescharacteristic
670C2#2317711yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC2×#392C4
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
1. C2×#1954C2semidirect
2. C2×#1869C2semidirect
3. C2×#1869C2semidirect
4. C2×C8×C22⋊C4C2semidirect
6. #11147C2semidirect
7. #11042C2semidirect
8. #10921C2semidirect
9. #10876C2semidirect
96. C2×C22⋊C8D4semidirect
102. #392C8semidirect
119. #336D4semidirect
133. C22×D4C2×C8semidirect
231. C2×C22⋊C4C2×C8semidirect
232. C2×C22⋊C4C2×C8semidirect
262. #95C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#18692
C2×C8×C22⋊C42
C4×#3922
#111472
#110422
#109212
#108762