Finite Group Databasebeta

← #2926252

Subgroups — #2926252

Order 512.

3440 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2804C211yesmaximal, characteristic, complemented
2C2×#2541C211yesmaximal, characteristic
3#16599C212yesmaximal, complemented
4#16504C214yesmaximal, complemented
5#15673C214yesmaximal, complemented
6#15370C214yesmaximal, complemented
7#15313C212yesmaximal, complemented
8#15261C212yesmaximal, complemented
9#15227C212yesmaximal, complemented
10#14825C212yesmaximal, complemented
11#14361C214yesmaximal, complemented
12#14241C212yesmaximal, complemented
13#14157C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#127C2211yescharacteristic
16C22×#127C2211yescharacteristic
17C22×#124C2214yes—
18C22×#123C2212yes—
19C22×#121C2214yes—
20C22×C4×D4C2211yescharacteristic
21#2804—28no—
22#2541—28no—
23C2×#400C2212yes—
24C2×#399C2212yes—
25C2×#395C2212yes—
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29C2×#393C2214yes—
30C2×#392C2214yes—
31C2×#391C2212yes—
32C2×#390C2211yescharacteristic
33C2×#390C2214yes—
34C2×#390C2211yescharacteristic
35C2×#388C2214yes—
36C2×#386C2212yes—
37C2×#386C2214yes—
38C2×#386C2212yes—
39C2×#385C2214yes—
40C2×#380C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42#1446C2212yes—
43#1444C2212yes—
44#1441C2212yes—
45#1441C2212yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2212yes—
52#1438C2214yes—
53#1438C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1438C2212yes—
57#1438C2212yes—
58#1438C2212yes—
59#1438C2214yes—
60#1438C2214yes—
61#1437C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C4×#95C2212yes—
69C23×Q8C2312yes—
70C23×D4C2311yescharacteristic
71C23×D4C2314yes—
72C23×D4C2312yes—
73C23×D4C2314yes—
74C23×D4C2311yescharacteristic
75C23×D4C2312yes—
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2312yes—
79C2×#127—24no—
80C2×#127—24no—
81C2×#127—24no—
82C2×#127—42no—
83C2×#127—42no—
84C2×#127—44no—
85C2×#127—44no—
86C2×#124—48no—
87C2×#124—48no—
88C2×#124—48no—
89C2×#123—48no—
90C2×#123—44no—
91C2×#121—216no—
92C2×#121—216no—
93C2×#121—48no—
94C2×C4×D4—24no—
95C2×C4×D4—42no—
96C2×C4×D4—42no—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2314yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#400—28no—
125#399—44no—
126#395—28no—
127#394—48no—
128#394—48no—
129#394—48no—
130#393—216no—
131#392—216no—
132#391—28no—
133#390—24no—
134#390—48no—
135#390D418yes—
136#388—48no—
137#386—28no—
138#386—44no—
139#386—48no—
140#385—216no—
141#380—44no—
142C4×C4⋊C4—24no—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C25C2412yes—
160C25C2412yes—
161C25C2412yes—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—44no—
166C22×Q8—44no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—24no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—41no—
186C22×D4—41no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—41no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—41no—
195C22×D4—42no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—42no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4C2×D414yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—24no—
210C23×C4—42no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—42no—
214C23×C4—44no—
215C23×C4—42no—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226#127—416no—
227#127—416no—
228#127—416no—
229#124—464no—
230#123—432no—
231#121—464no—
232C4×D4—416no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4#4118yes—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4C2×D4116yes—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—28no—
302C2×C42—28no—
303C2×C42—44no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42C2×D414yes—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—24no—
317#95—28no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—42no—
323C24—22no—
324C24—24no—
325C24—24no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24C22×D414yes—
351C24H(5,𝔽2)14yes—
352C24C2511yescharacteristic
353C2×Q8—416no—
354C2×Q8—48no—
355C2×Q8—84no—
356C2×Q8—84no—
357C2×Q8—88no—
358C2×D4—28no—
359C2×D4—28no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—416no—
368C2×D4—48no—
369C2×D4—416no—
370C2×D4—48no—
371C2×D4—416no—
372C2×D4—416no—
373C2×D4—48no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—416no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—416no—
380C2×D4—48no—
381C2×D4—88no—
382C2×D4—88no—
383C2×D4—88no—
384C2×D4—88no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4—84no—
388C2×D4—84no—
389C2×D4—84no—
390C2×D4—88no—
391C2×D4—84no—
392C2×D4—88no—
393C2×D4—88no—
394C2×D4—88no—
395C2×D4—88no—
396C2×D4—88no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—48no—
444C22×C4—416no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—88no—
448C22×C4—84no—
449C22×C4—84no—
450C22×C4—88no—
451C22×C4—84no—
452C22×C4—88no—
453C22×C4C22×D414yes—
454C22×C4C22×D412yes—
455C22×C4C22×D414yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4114yes—
459C22×C4H(5,𝔽2)18yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)14yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C42−1+414yes—
464C4⋊C4—216no—
465C4⋊C4—44no—
466C4⋊C4—416no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4—88no—
470C4⋊C4—84no—
471C22⋊C4—232no—
472C22⋊C4—48no—
473C22⋊C4—416no—
474C22⋊C4—48no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—416no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C22⋊C4—84no—
483C22⋊C4—88no—
484C22⋊C4—84no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C42—28no—
488C42—44no—
489C42—416no—
490C42—44no—
491C23—24no—
492C23—28no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—24no—
500C23—24no—
501C23—42no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—44no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—44no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—44no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23C23×D411yescharacteristic
557C23C22×#4112yes—
558C23C2×H(5,𝔽2)12yes—
559C23C2×H(5,𝔽2)11yescharacteristic
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×H(5,𝔽2)12yes—
562C23C2×H(5,𝔽2)14yes—
563C23C2×2−1+412yes—
564Q8—816no—
565D4—416no—
566D4—88no—
567D4—816no—
568D4—832no—
569D4—88no—
570D4—832no—
571D4—816no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—816no—
627C2×C4—816no—
628C2×C4#53614yes—
629C2×C4#54618yes—
630C2×C4#55914yes—
631C22—44no—
632C22—48no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—48no—
637C22—48no—
638C22—48no—
639C22—44no—
640C22—44no—
641C22—82no—
642C22—84no—
643C22—84no—
644C22—82no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22—82no—
649C22—82no—
650C22—84no—
651C22—82no—
652C22—84no—
653C22—84no—
654C22—82no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—82no—
659C22—84no—
660C22C2×#53611yescharacteristic
661C22C2×#54612yes—
662C22C2×#55911yescharacteristic
663C22#348411yescharacteristic
664C22#348411yescharacteristic
665C22#349812yes—
666C22#350812yes—
667C22#351914yes—
668C22#352214yes—
669C22#353214yes—
670C22#353212yes—
671C22#353612yes—
672C22#353612yes—
673C22#354311yescharacteristic
674C22#354814yes—
675C22#354912yes—
676C4—24no—
677C4—24no—
678C4—28no—
679C4—44no—
680C4—44no—
681C4—48no—
682C4—42no—
683C4—84no—
684C4—84no—
685C4—84no—
686C2—84no—
687C2—84no—
688C2—162no—
689C2#3671811yescharacteristic
690C2#3895212yes—
691C2#4389312yes—
692C2#4504712yes—
693C2#5111314yes—
694C2#5190211yescharacteristic
695C2#5485212yes—
696C2#5914311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2541C2
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×#2804C2semidirect
3. #16599C2semidirect
4. #16504C2semidirect
5. #15673C2semidirect
6. #15370C2semidirect
7. #15313C2semidirect
8. #15261C2semidirect
9. #15227C2semidirect
10. #14825C2semidirect
11. #14361C2semidirect
12. #14241C2semidirect
13. #14157C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28042
C2×#25412
#165992
#165042
#156732
#153702
#153132
#152612
#152272
#148252
#143612
#142412
#141572