Finite Group Databasebeta

← #533610

Subgroups — #533610

Order 512.

2784 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2#14610C212yesmaximal, complemented
3#14489C212yesmaximal, complemented
4#14253C212yesmaximal, complemented
5#14184C214yesmaximal, complemented
6#14098C214yesmaximal
7#14053C212yesmaximal, complemented
8#13993C212yesmaximal, complemented
9#13974C212yesmaximal, complemented
10#13871C211yesmaximal, characteristic
11#13826C212yesmaximal, complemented
12#13812C212yesmaximal, complemented
13#13801C212yesmaximal, complemented
14C4×#380C212yesmaximal, complemented
15C4×#386C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2212yes—
18C22×C4×D4C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2547C4116yescomplemented
22C2×#397C2212yes—
23C2×#397C2212yes—
24C2×#391C2214yes—
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×#388C2214yes—
28C2×#386C2212yes—
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2214yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40#1446C2214yes—
41#1443C2214yes—
42#1443C2214yes—
43#1441C2214yes—
44#1440C2214yes—
45#1440C2212yes—
46#1439C2214yes—
47#1438C2212yes—
48#1437C2214yes—
49#1437C2214yes—
50#1437C2214yes—
51#1437C2211yescharacteristic
52#1437C2212yes—
53#1437C2211yescharacteristic
54#1436C2211yescharacteristic
55#1436C2214yes—
56#1436C2212yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2211yescharacteristic
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65#1434C2214yes—
66#1434C2214yes—
67#1434C2214yes—
68#1434C2214yes—
69#1434C2214yes—
70C22⋊#95C2214yes—
71C22⋊#95C2212yes—
72C4×#95C2214yes—
73C4×#95C2214yes—
74C4×#95C2214yes—
75C4×#95C2212yes—
76C23×Q8C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8—42no—
83C2×C4⋊Q8C2×C418yescomplemented
84C2×#124—48no—
85C2×#124C2×C4116yes—
86C2×C4×D4—42no—
87C2×C4×D4—42no—
88C2×C4×D4C2×C418yes—
89C2×#117—28no—
90C2×#117—42no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2314yes—
108C22×C42—28no—
109C22×C42—42no—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2314yes—
117C22×C42C2314yes—
118C22×C42C2314yes—
119C22×C42C2312yes—
120#397C2×C4116yes—
121#397C2×C4116yescomplemented
122#391—216no—
123#391C2×C4116yescomplemented
124#389—44no—
125#388—48no—
126#386—28no—
127#380—28no—
128#380—42no—
129#380C2×C4116yes—
130#380C2×C4116yescomplemented
131C4×C4⋊C4—28no—
132C4×C4⋊C4—28no—
133C4×C4⋊C4—44no—
134C4×C4⋊C4C2×C418yescomplemented
135C4×C22⋊C4—28no—
136C4×C22⋊C4—42no—
137C4×C22⋊C4—48no—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C25C2411yescharacteristic
166C25C2411yescharacteristic
167C22×Q8—28no—
168C22×Q8—44no—
169C22×Q8—42no—
170C22×Q8—42no—
171C22×Q8C22×C418yes—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4C22×C414yes—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4C22×C414yes—
193C23×C4C22×C414yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2411yescharacteristic
198C23×C4C2414yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C4⋊Q8—416no—
208#124—432no—
209C4×D4—416no—
210#117—416no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4C22×C418yes—
228C2×C4⋊C4C22×C418yes—
229C2×C4⋊C4C22×C414yes—
230C2×C4⋊C4C22×C418yes—
231C2×C4⋊C4#4118yes—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4C22×C4116yes—
257C2×C22⋊C4C22×C414yes—
258C2×C22⋊C4C22×C414yes—
259C2×C22⋊C4C22×C4116yes—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—28no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—28no—
271C2×C42—28no—
272C2×C42—28no—
273C2×C42—416no—
274C2×C42—42no—
275C2×C42—44no—
276C2×C42—41no—
277C2×C42—41no—
278C2×C42—44no—
279C2×C42—44no—
280C2×C42—44no—
281C2×C42—44no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42C22×C414yes—
285C2×C42C22×C418yes—
286C2×C42#4118yes—
287#95—24no—
288#95—24no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—24no—
298#95—42no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—41no—
304#95—41no—
305#95—44no—
306#95—42no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95C22×C418yes—
311#95C22×C4116yes—
312#95C22×C4116yes—
313#95#4118yes—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—21no—
318C24—21no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—21no—
326C24—21no—
327C24—22no—
328C24—22no—
329C24C23×C412yes—
330C24C23×C412yes—
331C24H(5,𝔽2)14yes—
332C24C2511yescharacteristic
333C2×Q8—216no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×Q8—88no—
337C2×Q8—88no—
338C2×D4—28no—
339C2×D4—28no—
340C2×D4—44no—
341C2×D4—48no—
342C2×D4—48no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—216no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—216no—
394C22×C4—216no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4C23×C412yes—
424C22×C4C23×C412yes—
425C22×C4C23×C418yes—
426C22×C4C23×C412yes—
427C22×C4C23×C418yes—
428C22×C4C23×C412yes—
429C22×C4C23×C412yes—
430C22×C4C2×#4114yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4118yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4114yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4114yes—
437C22×C4C2×#4114yes—
438C22×C4H(5,𝔽2)12yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C42−1+414yes—
441C4⋊C4—216no—
442C4⋊C4—216no—
443C4⋊C4—216no—
444C4⋊C4—44no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—88no—
448C22⋊C4—232no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—416no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—416no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—84no—
458C42—28no—
459C42—216no—
460C42—48no—
461C42—44no—
462C42—44no—
463C42—48no—
464C42—48no—
465C42—88no—
466C42—88no—
467C42—88no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
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—41no—
508C23—41no—
509C23#51814yes—
510C23C24×C411yescharacteristic
511C23C22×#4112yes—
512C23C22×#4112yes—
513C23C22×#4112yes—
514C23C2×H(5,𝔽2)11yescharacteristic
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×H(5,𝔽2)12yes—
517C23C2×2−1+412yes—
518C23#58512yes—
519Q8—816no—
520D4—416no—
521C2×C4—22no—
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—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—432no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—416no—
583C2×C4—416no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4C4×#4114yes—
599C2×C4C4×#4118yes—
600C2×C4#51814yes—
601C2×C4#55114yes—
602C2×C4#55114yes—
603C2×C4#55518yes—
604C2×C4#55618yes—
605C22—24no—
606C22—28no—
607C22—44no—
608C22—44no—
609C22—42no—
610C22—44no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—44no—
617C22—42no—
618C22—44no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22C2×C4×#4112yes—
624C22C2×C4×#4111yescharacteristic
625C22C2×#51811yescharacteristic
626C22C2×#51811yescharacteristic
627C22C4×2−1+412yes—
628C22C2×#55111yescharacteristic
629C22C2×#55111yescharacteristic
630C22C2×#55512yes—
631C22C2×#55612yes—
632C22#347512yes—
633C22#348012yes—
634C22#348212yes—
635C22#349212yes—
636C22#349412yes—
637C22#351312yes—
638C22#351912yes—
639C22#351912yes—
640C22#352212yes—
641C22#354112yes—
642C22#355011yescharacteristic
643C22#355111yescharacteristic
644C4—28no—
645C4—24no—
646C4—44no—
647C4—42no—
648C4—42no—
649C4—44no—
650C4—42no—
651C4—48no—
652C4—48no—
653C4—48no—
654C4—82no—
655C4—84no—
656C4#237218yes—
657C2—44no—
658C2—82no—
659C2C2×#237211yescharacteristic
660C2C4×#55612yes—
661C2#3424611yescharacteristic
662C2#3429812yes—
663C2#3441311yescharacteristic
664C2#5920412yes—
665C2#5923412yes—
666C2#5923612yes—
667C2#5938112yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14610C2semidirect
3. #14489C2semidirect
4. #14253C2semidirect
5. #14184C2semidirect
7. #14053C2semidirect
8. #13993C2semidirect
9. #13974C2semidirect
11. #13826C2semidirect
12. #13812C2semidirect
13. #13801C2semidirect
14. C4×#380C2semidirect
21. #2547C4semidirect
83. C2×C4⋊Q8C2×C4semidirect
121. #397C2×C4semidirect
123. #391C2×C4semidirect
130. #380C2×C4semidirect
134. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
#146102
#144892
#142532
#141842
#140982
#140532
#139932
#139742
#138712
#138262
#138122
#138012
C4×#3802
C4×#3862