Finite Group Databasebeta

← #800071

Subgroups — #800071

Order 512.

2352 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#2347C211yesmaximal, characteristic
3#17006C212yesmaximal
4#16009C214yesmaximal, complemented
5#15970C214yesmaximal
6#15242C211yesmaximal, characteristic, complemented
7#15195C211yesmaximal, characteristic
8#15176C212yesmaximal
9#15158C211yesmaximal, characteristic, complemented
10#15108C212yesmaximal, complemented
11#14174C212yesmaximal, complemented
12#14030C212yesmaximal, complemented
13#13981C212yesmaximal, complemented
14#13974C212yesmaximal
15#13936C211yesmaximal, characteristic, complemented
16#13879C212yesmaximal
17C4×#393C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×#117C2211yescharacteristic
21C2×#395C2212yes—
22C2×#393C2211yescharacteristic
23Q8×C4⋊C4—28no—
24#2347—28no—
25C2×#389C2212yes—
26C2×#389C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2214yes—
30C2×#388C2212yes—
31C2×#388C2212yes—
32C2×#388C2214yes—
33C2×#387C2214yes—
34C2×#384C2212yes—
35C2×#384C2211yescharacteristic
36C2×#383C2212yes—
37C2×#382C2212yes—
38C2×#380C2212yes—
39C2×#380C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46#1446C2211yescharacteristic
47#1445C2214yes—
48#1443C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1442C2211yescharacteristic
52#1442C2212yes—
53#1442C2214yes—
54#1442C2212yes—
55#1442C2212yes—
56#1442C2214yes—
57#1441C2212yes—
58#1441C2212yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2212yes—
62#1440C2214yes—
63#1439C2212yes—
64#1439C2211yescharacteristic
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2212yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2212yes—
81#1434C2214yes—
82#1434C2212yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×C4×Q8—24no—
88C2×C4×Q8—28no—
89C2×C4×Q8—28no—
90C2×C4×Q8—48no—
91C2×#117—24no—
92C2×#117—24no—
93C2×#117—24no—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124#395—28no—
125#393—24no—
126#389—24no—
127#389—24no—
128#389—28no—
129#389—28no—
130#389—216no—
131#388—28no—
132#388—28no—
133#388—48no—
134#387—48no—
135#384—28no—
136#384—24no—
137#383—28no—
138#382—28no—
139#380—28no—
140#380—24no—
141C4×C4⋊C4—24no—
142C4×C4⋊C4—28no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4D418yes—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—28no—
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C25C2411yescharacteristic
181C22×Q8—22no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—22no—
185C22×Q8#4114yes—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4#4114yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2411yescharacteristic
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C4×Q8—232no—
208C4×Q8—432no—
209#117—232no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4#4114yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—28no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—28no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42C2×D414yes—
288C2×C42C2×D418yes—
289C2×C42#4118yes—
290#95—22no—
291#95—22no—
292#95—24no—
293#95—24no—
294#95—28no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—22no—
304#95—24no—
305#95—28no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—22no—
310#95—44no—
311#95—42no—
312#95—44no—
313#95—42no—
314#95—42no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95C2×D418yes—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—21no—
332C24—22no—
333C24C2×#4112yes—
334C24C2511yescharacteristic
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—216no—
338C2×Q8—28no—
339C2×Q8—216no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—28no—
379C22×C4—216no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4C22×D414yes—
422C22×C4C22×D412yes—
423C22×C4C22×D414yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4114yes—
428C22×C4C2×#4114yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4114yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4112yes—
435C22×C4H(5,𝔽2)14yes—
436C22×C42−1+412yes—
437C22×C42−1+412yes—
438C22×C42−1+414yes—
439C4⋊C4—28no—
440C4⋊C4—216no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—48no—
445C4⋊C4—416no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—416no—
452C4⋊C4—88no—
453C4⋊C4—84no—
454C22⋊C4—216no—
455C22⋊C4—44no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C22⋊C4—44no—
459C22⋊C4—48no—
460C22⋊C4—84no—
461C42—216no—
462C42—28no—
463C42—216no—
464C42—48no—
465C42—416no—
466C42—44no—
467C42—48no—
468C42—84no—
469C23—22no—
470C23—24no—
471C23—22no—
472C23—24no—
473C23—22no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—42no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23C23×D411yescharacteristic
493C23C22×#4111yescharacteristic
494C23C22×#4112yes—
495C23C22×#4111yescharacteristic
496C23C22×#4111yescharacteristic
497C23C2×H(5,𝔽2)12yes—
498C23C2×2−1+412yes—
499C23C2×2−1+411yescharacteristic
500C23C2×2−1+411yescharacteristic
501C23#58511yescharacteristic
502C23#58512yes—
503Q8—232no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—416no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4#52514yes—
593C2×C4#52514yes—
594C2×C4#53614yes—
595C2×C4#54814yes—
596C2×C4#55218yes—
597C2×C4#55318yes—
598C2×C4#55618yes—
599C22—44no—
600C22—42no—
601C22—44no—
602C22—42no—
603C22—42no—
604C22—81no—
605C22—82no—
606C22—81no—
607C22—82no—
608C22—81no—
609C22—81no—
610C22C2×#52511yescharacteristic
611C22C2×#52511yescharacteristic
612C22C2×#53611yescharacteristic
613C22C2×#54811yescharacteristic
614C22C2×#55212yes—
615C22C2×#55312yes—
616C22C2×#55612yes—
617C22#348011yescharacteristic
618C22#348211yescharacteristic
619C22#348411yescharacteristic
620C22#348911yescharacteristic
621C22#349212yes—
622C22#349312yes—
623C22#350011yescharacteristic
624C22#350612yes—
625C22#350712yes—
626C22#350911yescharacteristic
627C22#351511yescharacteristic
628C22#351712yes—
629C22#351812yes—
630C22#354712yes—
631C22#355212yes—
632C22#355512yes—
633C4—24no—
634C4—28no—
635C4—42no—
636C4—44no—
637C4—44no—
638C4—42no—
639C4—44no—
640C4—44no—
641C4—42no—
642C4—81no—
643C4—82no—
644C4—84no—
645C4—84no—
646C4—82no—
647C4—82no—
648C4#259918yes—
649C2—82no—
650C2C2×#259911yescharacteristic
651C2#3570411yescharacteristic
652C2#3575911yescharacteristic
653C2#3650511yescharacteristic
654C2#3658111yescharacteristic
655C2#3917211yescharacteristic
656C2#4248511yescharacteristic
657C2#4498312yes—
658C2#4532812yes—
659C2#5921112yes—
660C2#5943412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×Q8×C4⋊C4C2semidirect
4. #16009C2semidirect
6. #15242C2semidirect
9. #15158C2semidirect
10. #15108C2semidirect
11. #14174C2semidirect
12. #14030C2semidirect
13. #13981C2semidirect
15. #13936C2semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
C2×#23472
#170062
#160092
#159702
#152422
#151952
#151762
#151582
#151082
#141742
#140302
#139812
#139742
#139362
#138792
C4×#3932