Finite Group Databasebeta

← #388474

Subgroups — #388474

Order 512.

1732 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1955C212yesmaximal, complemented
2C2×#1924C212yesmaximal, complemented
3C2×#1869C212yesmaximal, complemented
4C4×#392C211yesmaximal, characteristic
5#11042C211yesmaximal, characteristic, complemented
6#11042C212yesmaximal, complemented
7#10964C212yesmaximal, complemented
8#10877C212yesmaximal, complemented
9#10714C211yesmaximal, characteristic, complemented
10C24×C8C2211yescharacteristic
11C22×C4×D4C2212yes—
12C22×C4×D4C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×#97C2212yes—
22C2×#392C411yescharacteristic
23C2×C4×C22⋊C4C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1955—28no—
26#1924—28no—
27#1869—28no—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1444C411yescharacteristic
35#1439C412yes—
36#1439C411yescharacteristic
37#1438C411yescharacteristic
38#1438C412yes—
39C4×#95C2211yescharacteristic
40C23×D4C2×C412yes—
41C23×D4C2×C411yescharacteristic
42C24×C4C2312yes—
43C24×C4C2311yescharacteristic
44C23×C8—22no—
45C23×C8—24no—
46C23×C8—42no—
47C23×C8—41no—
48C23×C8—41no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4×D4—24no—
54C2×C4×D4—24no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C2×C4×D4—42no—
58C2×C4×D4—42no—
59C2×C4×D4—42no—
60C2×C4×D4—42no—
61C2×C4×D4—41no—
62C2×C4×D4—41no—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C412yes—
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C412yes—
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2312yes—
75C2×C4⋊C8—24no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—42no—
79C2×C4⋊C8—41no—
80C2×C22⋊C8—24no—
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—41no—
89C2×C22⋊C8—41no—
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⋊C8D414yescomplemented
98C2×#97—24no—
99C2×#97—24no—
100C2×#97—42no—
101#392—24no—
102C4×C22⋊C4—24no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—41no—
105C4×C22⋊C4—41no—
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C412yes—
108C2×#95C2×C412yes—
109C2×#95C2×C412yes—
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113#336—24no—
114#336—24no—
115#336—42no—
116#336—42no—
117#336—41no—
118#336D418yescomplemented
119C25C22×C412yes—
120C25C22×C411yescharacteristic
121C22×D4—24no—
122C22×D4—24no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—24no—
126C22×D4—24no—
127C22×D4—22no—
128C22×D4—41no—
129C22×D4—42no—
130C22×D4—42no—
131C22×D4—42no—
132C22×D4—41no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
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—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C412yes—
158C23×C4C22×C412yes—
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C412yes—
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4#4112yes—
165C23×C4C2411yescharacteristic
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
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—22no—
190C22×C8—22no—
191C22×C8—44no—
192C22×C8—48no—
193C22×C8—48no—
194C22×C8—44no—
195C22×C8—44no—
196C22×C8C2×D414yes—
197C22×C8C2×D414yes—
198C22×C8C2×D414yes—
199C4×D4—48no—
200C4×D4—48no—
201C4×D4—48no—
202C4×D4—48no—
203C4×D4—48no—
204C4×D4—44no—
205C4×D4—44no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
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—41no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4M1614yes—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42C2×D414yes—
245C2×C42#4114yes—
246C4⋊C8—48no—
247C4⋊C8—44no—
248C22⋊C8—44no—
249C22⋊C8—48no—
250C22⋊C8—48no—
251C22⋊C8—48no—
252C22⋊C8—48no—
253C22⋊C8—48no—
254#97—48no—
255#95—24no—
256#95—24no—
257#95—22no—
258#95—41no—
259#95—42no—
260#95—41no—
261#95—41no—
262C24—22no—
263C24—21no—
264C24—22no—
265C24—21no—
266C24—22no—
267C24—21no—
268C24—21no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24C4×D412yes—
285C24C2×M1612yes—
286C24#13114yes—
287C24#13112yes—
288C24C23×C411yescharacteristic
289C2×D4—216no—
290C2×D4—48no—
291C2×D4—48no—
292C2×D4—48no—
293C2×D4—44no—
294C2×D4—48no—
295C2×D4—48no—
296C2×D4—44no—
297C2×D4—48no—
298C2×D4—48no—
299C2×D4—44no—
300C2×D4—48no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
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×C4C4×D414yes—
389C22×C4C4×D414yes—
390C22×C4C4×D412yes—
391C22×C4C2×M1612yes—
392C22×C4C2×M1612yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D412yes—
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4112yes—
397C22×C4H(5,𝔽2)11yescharacteristic
398C22×C4H(5,𝔽2)11yescharacteristic
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)11yescharacteristic
401C22×C4H(5,𝔽2)12yes—
402C22×C42−1+411yescharacteristic
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—416no—
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—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C4⋊C4—48no—
433C4⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—44no—
440C22⋊C4—48no—
441C42—24no—
442C42—24no—
443C42—24no—
444C42—24no—
445C42—41no—
446C42—42no—
447C42—41no—
448C42—42no—
449C42—41no—
450C42—41no—
451C23—22no—
452C23—24no—
453C23—24no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—22no—
458C23—24no—
459C23—22no—
460C23—24no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—42no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
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—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23#43614yes—
509C23C2×C4×D412yes—
510C23C2×C4×D411yescharacteristic
511C23#51812yes—
512C23#51811yescharacteristic
513C23#52011yescharacteristic
514C23C22×M1611yescharacteristic
515C23C2×#13111yescharacteristic
516C23C2×#13112yes—
517C23#56811yescharacteristic
518C23#56811yescharacteristic
519C23#56812yes—
520D4—432no—
521D4—416no—
522C2×C4—22no—
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—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4#53411yescharacteristic
583C2×C4#53512yes—
584C2×C4#53712yes—
585C2×C4#53711yescharacteristic
586C2×C4#53811yescharacteristic
587C2×C4#53911yescharacteristic
588C2×C4#54012yes—
589C2×C4#54212yes—
590C2×C4D4211yescharacteristic
591C2×C4D4212yes—
592C2×C4#54612yes—
593C2×C4#54612yes—
594C2×C4#55012yes—
595C2×C4#55112yes—
596C2×C4#55911yescharacteristic
597C2×C4#56012yes—
598C2×C4#56011yescharacteristic
599C2×C4H(3,𝔽4)11yescharacteristic
600C8—44no—
601C8—44no—
602C8—44no—
603C8—82no—
604C22—28no—
605C22—24no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22#232412yes—
631C22#232511yescharacteristic
632C22#235812yes—
633C22#236611yescharacteristic
634C22#237511yescharacteristic
635C22C2×#43611yescharacteristic
636C22#292912yes—
637C22#293211yescharacteristic
638C22#293212yes—
639C22#293312yes—
640C22#293312yes—
641C22D4×M1612yes—
642C22#297311yescharacteristic
643C22#297412yes—
644C22#297612yes—
645C22#297612yes—
646C22#297611yescharacteristic
647C22#297711yescharacteristic
648C22#298012yes—
649C22#298011yescharacteristic
650C22#298911yescharacteristic
651C22#299211yescharacteristic
652C22#299212yes—
653C4—42no—
654C4—44no—
655C4—44no—
656C4—42no—
657C4—42no—
658C4—81no—
659C4—81no—
660C4—82no—
661C4#267011yescharacteristic
662C4#267312yes—
663C4#267511yescharacteristic
664C4#270512yes—
665C4#271211yescharacteristic
666C4#275211yescharacteristic
667C2—44no—
668C2—84no—
669C2#1452111yescharacteristic
670C2#2213612yes—
671C2#2214112yes—
672C2#2217111yescharacteristic
673C2#2217611yescharacteristic
674C2#2284812yes—
675C2#2285112yes—
676C2#2317511yescharacteristic
677C2#2317711yescharacteristic
678C2#2330711yescharacteristic
679C2#2331111yescharacteristic
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×#1955C2semidirect
2. C2×#1924C2semidirect
3. C2×#1869C2semidirect
5. #11042C2semidirect
6. #11042C2semidirect
7. #10964C2semidirect
8. #10877C2semidirect
9. #10714C2semidirect
97. C2×C22⋊C8D4semidirect
118. #336D4semidirect

Maximal subgroups

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

subgroupindex
C2×#19552
C2×#19242
C2×#18692
C4×#3922
#110422
#109642
#108772
#107142