Finite Group Databasebeta

← #2737123

Subgroups — #2737123

Order 512.

2180 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic, complemented
2C2×#2494C211yesmaximal, characteristic
3#17155C214yesmaximal, complemented
4#17106C214yesmaximal
5#17039C212yesmaximal
6#16603C212yesmaximal, complemented
7#16506C212yesmaximal
8#15221C212yesmaximal, complemented
9#15197C211yesmaximal, characteristic, complemented
10#15156C211yesmaximal, characteristic, complemented
11#15135C211yesmaximal, characteristic
12#15132C212yesmaximal
13#15128C212yesmaximal, complemented
14#15125C212yesmaximal
15#14629C212yesmaximal, complemented
16#13936C211yesmaximal, characteristic, complemented
17C4×#399C211yesmaximal, characteristic
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2212yes—
20#2551—28no—
21#2494—28no—
22C2×#401C2212yes—
23C2×#400C2212yes—
24C2×#399C2211yescharacteristic
25C2×#399C2212yes—
26C2×#397C2212yes—
27C2×#397C2214yes—
28C2×#395C2214yes—
29C2×#395C2214yes—
30C2×#393C2214yes—
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2211yescharacteristic
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#387C2212yes—
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48#1446C2214yes—
49#1446C2214yes—
50#1445C2212yes—
51#1445C2212yes—
52#1445C2214yes—
53#1445C2211yescharacteristic
54#1444C2214yes—
55#1444C2212yes—
56#1444C2211yescharacteristic
57#1444C2214yes—
58#1443C2214yes—
59#1443C2214yes—
60#1443C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2214yes—
67#1441C2212yes—
68#1441C2212yes—
69#1441C2212yes—
70#1440C2214yes—
71#1439C2214yes—
72#1439C2212yes—
73#1439C2211yescharacteristic
74#1439C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2211yescharacteristic
82#1434C2212yes—
83#1434C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2314yes—
86C24×C4C2311yescharacteristic
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—44no—
90C2×C4⋊Q8—44no—
91C2×C4⋊Q8—48no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
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×C42C2311yescharacteristic
120C22×C42C2312yes—
121#401—28no—
122#400—28no—
123#399—24no—
124#399—44no—
125#397—28no—
126#397—48no—
127#395—216no—
128#395—48no—
129#393—48no—
130#389—24no—
131#389—28no—
132#389—24no—
133#389—28no—
134#389—44no—
135#388—28no—
136#387—28no—
137#384—24no—
138#382—24no—
139#380—24no—
140#380—24no—
141#380—44no—
142C4×C4⋊C4—24no—
143C4×C4⋊C4—44no—
144C4×C4⋊C4D418yes—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—44no—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2311yescharacteristic
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C25C2411yescharacteristic
182C22×Q8—28no—
183C22×Q8—28no—
184C22×Q8—44no—
185C22×Q8—44no—
186C22×Q8—44no—
187C22×Q8—44no—
188C22×Q8—44no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4C2411yescharacteristic
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C4⋊Q8—432no—
211C4⋊Q8—432no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4#4114yes—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—44no—
288C2×C42C2×D414yes—
289C2×C42C2×D418yes—
290#95—22no—
291#95—22no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—28no—
299#95—24no—
300#95—22no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—28no—
307#95—22no—
308#95—44no—
309#95—42no—
310#95—42no—
311#95—42no—
312#95—42no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—42no—
323#95C2×D418yes—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—21no—
329C24—22no—
330C24—21no—
331C24—22no—
332C24—21no—
333C24—21no—
334C24H(5,𝔽2)12yes—
335C24C2511yescharacteristic
336C2×Q8—416no—
337C2×Q8—416no—
338C2×Q8—88no—
339C2×Q8—88no—
340C2×Q8—88no—
341C2×Q8—88no—
342C2×Q8—88no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4C22×D412yes—
429C22×C4C22×D414yes—
430C22×C4C22×D414yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4112yes—
435C22×C4H(5,𝔽2)14yes—
436C22×C4H(5,𝔽2)14yes—
437C22×C42−1+414yes—
438C22×C42−1+412yes—
439C22×C42−1+414yes—
440C22×C42−1+412yes—
441C22×C42−1+412yes—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—48no—
447C4⋊C4—416no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—416no—
451C4⋊C4—48no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4—88no—
459C22⋊C4—48no—
460C22⋊C4—44no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—44no—
464C22⋊C4—84no—
465C22⋊C4—88no—
466C42—28no—
467C42—48no—
468C42—44no—
469C42—48no—
470C42—48no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—22no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23—42no—
494C23C23×D411yescharacteristic
495C23C22×#4111yescharacteristic
496C23C2×H(5,𝔽2)12yes—
497C23C2×H(5,𝔽2)12yes—
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×2−1+412yes—
500C23C2×2−1+411yescharacteristic
501C23C2×2−1+411yescharacteristic
502C23C2×2−1+411yescharacteristic
503C23C2×2−1+412yes—
504C23#58511yescharacteristic
505Q8—832no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—88no—
595C2×C4#53514yes—
596C2×C4#53614yes—
597C2×C4#53614yes—
598C2×C4#54814yes—
599C2×C4#56318yes—
600C2×C4#56318yes—
601C22—44no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—42no—
606C22—81no—
607C22—82no—
608C22—81no—
609C22—82no—
610C22—81no—
611C22—81no—
612C22C2×#53511yescharacteristic
613C22C2×#53611yescharacteristic
614C22C2×#53611yescharacteristic
615C22C2×#54811yescharacteristic
616C22#348411yescharacteristic
617C22#348811yescharacteristic
618C22#349911yescharacteristic
619C22#351112yes—
620C22#351811yescharacteristic
621C22#352012yes—
622C22#352211yescharacteristic
623C22C2×#56312yes—
624C22C2×#56312yes—
625C22#353712yes—
626C22#354212yes—
627C22#354412yes—
628C22#354411yescharacteristic
629C22#355012yes—
630C22#355012yes—
631C22#355212yes—
632C22#355212yes—
633C22#355511yescharacteristic
634C22#356312yes—
635C4—24no—
636C4—44no—
637C4—44no—
638C4—42no—
639C4—44no—
640C4—44no—
641C4—44no—
642C4—82no—
643C4—84no—
644C4—81no—
645C4—82no—
646C4—82no—
647C4—81no—
648C4—84no—
649C4—81no—
650C4#282118yes—
651C2—82no—
652C2C2×#282111yescharacteristic
653C2#3912511yescharacteristic
654C2#3916711yescharacteristic
655C2#3972011yescharacteristic
656C2#4242711yescharacteristic
657C2#4248011yescharacteristic
658C2#4296411yescharacteristic
659C2#4562712yes—
660C2#4563312yes—
661C2#5296812yes—
662C2#5312412yes—
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×#2551C2semidirect
3. #17155C2semidirect
6. #16603C2semidirect
8. #15221C2semidirect
9. #15197C2semidirect
10. #15156C2semidirect
13. #15128C2semidirect
15. #14629C2semidirect
16. #13936C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#24942
#171552
#171062
#170392
#166032
#165062
#152212
#151972
#151562
#151352
#151322
#151282
#151252
#146292
#139362
C4×#3992