Finite Group Databasebeta

← #762843

Subgroups — #762843

Order 512.

3038 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#17046C214yesmaximal, complemented
4#15233C212yesmaximal, complemented
5#15130C212yesmaximal
6#15129C214yesmaximal, complemented
7#14888C214yesmaximal, complemented
8#14048C212yesmaximal, complemented
9#14002C212yesmaximal, complemented
10#13993C214yesmaximal, complemented
11#13911C211yesmaximal, characteristic, complemented
12#13910C211yesmaximal, characteristic, complemented
13#13906C211yesmaximal, characteristic, complemented
14#13906C211yesmaximal, characteristic, complemented
15C4×#380C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2212yes—
19C22×#117C2212yes—
20C23×C42C2211yescharacteristic
21C2×#394C2218yes—
22C4×#121—28no—
23C2×#390C2212yes—
24C2×#390C2211yescharacteristic
25C2×#390C2212yes—
26C2×#389C2211yescharacteristic
27C2×#389C2214yes—
28C2×#388C2212yes—
29C2×#386C2212yes—
30#2285—28no—
31C2×#384C2212yes—
32C2×#384C2211yescharacteristic
33C2×#384C2211yescharacteristic
34C2×#382C2212yes—
35C2×#382C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2214yes—
39C2×#380C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2214yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49#1444C2214yes—
50#1443C2218yes—
51#1442C2214yes—
52#1441C2218yes—
53#1440C2218yes—
54#1440C2218yes—
55#1439C2218yes—
56#1439C2214yes—
57#1439C2214yes—
58#1437C2218yes—
59#1436C2218yes—
60C4⋊#95C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66#1434C2212yes—
67C22⋊#95C2212yes—
68C4×#95C2212yes—
69C23×D4C2311yescharacteristic
70C23×D4C2312yes—
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C2×#121—24no—
76C2×#121—24no—
77C2×#121—24no—
78C2×C4×D4—28no—
79C2×C4×D4—24no—
80C2×C4×D4—24no—
81C2×C4×D4—24no—
82C2×C4×D4—48no—
83C2×#117—28no—
84C2×#117—48no—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2318yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2318yes—
104C22×C22⋊C4C2318yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C42—24no—
108C22×C42—24no—
109C22×C42D418yes—
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2314yes—
117C22×C42C2311yescharacteristic
118#394—416no—
119#390—28no—
120#390—24no—
121#390—44no—
122#389—24no—
123#389—48no—
124#388—28no—
125#386—28no—
126#384—28no—
127#384—24no—
128#384—24no—
129#382—28no—
130#382—44no—
131#380—28no—
132#380—28no—
133#380—24no—
134#380—28no—
135#380—216no—
136#380—48no—
137C4×C4⋊C4—28no—
138C4×C4⋊C4—28no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4—24no—
142C4×C22⋊C4—28no—
143C4×C22⋊C4—44no—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2318yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2318yes—
155C2×#95C2314yes—
156C2×#95C2318yes—
157C2×#95C2318yes—
158C2×#95C2318yes—
159C25C2412yes—
160C25C2411yescharacteristic
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—28no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—24no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—44no—
172C22×D4#4114yes—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4C2×D414yes—
193C23×C4C2×D414yes—
194C23×C4C2×D414yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208#121—232no—
209C4×D4—432no—
210C4×D4—416no—
211#117—432no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—216no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4#4114yes—
245C2×C4⋊C4#4118yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—216no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—216no—
261C2×C22⋊C4—216no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—48no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—48no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—232no—
276C2×C42—28no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—48no—
286C2×C42C2×D414yes—
287C2×C42C2×D414yes—
288C2×C42C2×D414yes—
289C2×C42C2×D414yes—
290C2×C42#4114yes—
291C2×C42#4114yes—
292#95—24no—
293#95—24no—
294#95—28no—
295#95—28no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—216no—
301#95—28no—
302#95—44no—
303#95—48no—
304#95—48no—
305#95—48no—
306#95—48no—
307C24—22no—
308C24—21no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—22no—
315C24—21no—
316C24—22no—
317C24—24no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—22no—
326C24C22×D412yes—
327C24C2×#4114yes—
328C24H(5,𝔽2)14yes—
329C24C2511yescharacteristic
330C2×D4—216no—
331C2×D4—28no—
332C2×D4—28no—
333C2×D4—28no—
334C2×D4—416no—
335C2×D4—416no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C22×C4—216no—
343C22×C4—216no—
344C22×C4—28no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—28no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—28no—
398C22×C4—24no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4—44no—
409C22×C4—48no—
410C22×C4—44no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—416no—
415C22×C4—416no—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C2×#4118yes—
423C22×C4C2×#4114yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4114yes—
433C22×C4H(5,𝔽2)14yes—
434C22×C4H(5,𝔽2)12yes—
435C22×C42−1+414yes—
436C22×C42−1+412yes—
437C4⋊C4—28no—
438C4⋊C4—216no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—816no—
450C22⋊C4—216no—
451C22⋊C4—416no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—416no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—816no—
458C22⋊C4—816no—
459C42—28no—
460C42—28no—
461C42—28no—
462C42—416no—
463C42—48no—
464C42—48no—
465C42—48no—
466C42#123116yes—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—28no—
483C23—28no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—41no—
488C23—42no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23#53514yes—
513C23C23×D411yescharacteristic
514C23C22×#4111yescharacteristic
515C23C22×#4111yescharacteristic
516C23C22×#4111yescharacteristic
517C23C22×#4111yescharacteristic
518C23C22×#4112yes—
519C23C2×H(5,𝔽2)12yes—
520C23C2×H(5,𝔽2)12yes—
521C23C2×H(5,𝔽2)11yescharacteristic
522C23C2×2−1+412yes—
523C23C2×2−1+411yescharacteristic
524D4—432no—
525D4—416no—
526C2×C4—232no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—416no—
560C2×C4—416no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—816no—
589C2×C4C2×#12314yes—
590C2×C4C2×#12314yes—
591C2×C4C2×#12314yes—
592C2×C4#52514yes—
593C2×C4#53214yes—
594C2×C4#53514yes—
595C2×C4#53614yes—
596C2×C4#53714yes—
597C2×C4#55214yes—
598C2×C4#55314yes—
599C2×C4#55318yes—
600C2×C4#55514yes—
601C2×C4#55518yes—
602C2×C4#55614yes—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—42no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—48no—
615C22—48no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—82no—
620C22—84no—
621C22—84no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22C22×#12311yescharacteristic
626C22C2×#52511yescharacteristic
627C22C2×#53211yescharacteristic
628C22C2×#53511yescharacteristic
629C22C2×#53511yescharacteristic
630C22C2×#53611yescharacteristic
631C22C2×#53711yescharacteristic
632C22C2×#55211yescharacteristic
633C22C2×#55311yescharacteristic
634C22C2×#55312yes—
635C22C2×#55511yescharacteristic
636C22C2×#55512yes—
637C22C2×#55611yescharacteristic
638C22#348212yes—
639C22#348212yes—
640C22#349212yes—
641C22#349212yes—
642C22#351912yes—
643C22#351912yes—
644C22#351912yes—
645C22#352212yes—
646C22#354812yes—
647C22#355012yes—
648C4—24no—
649C4—24no—
650C4—48no—
651C4—44no—
652C4—42no—
653C4—42no—
654C4—82no—
655C4—84no—
656C4—84no—
657C4—84no—
658C4—82no—
659C4#248518yes—
660C4#250018yes—
661C2—44no—
662C2—84no—
663C2C2×#248511yescharacteristic
664C2C2×#250011yescharacteristic
665C2#3497911yescharacteristic
666C2#3522711yescharacteristic
667C2#3523711yescharacteristic
668C2#3523811yescharacteristic
669C2#3764011yescharacteristic
670C2#5921712yes—
671C2#5921712yes—
672C2#5921912yes—
673C2#5923412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2285C2semidirect
3. #17046C2semidirect
4. #15233C2semidirect
6. #15129C2semidirect
7. #14888C2semidirect
8. #14048C2semidirect
9. #14002C2semidirect
10. #13993C2semidirect
11. #13911C2semidirect
12. #13910C2semidirect
13. #13906C2semidirect
14. #13906C2semidirect
15. C4×#380C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C2×#22852
#170462
#152332
#151302
#151292
#148882
#140482
#140022
#139932
#139112
#139102
#139062
C4×#3802