Finite Group Databasebeta

← #2737533

Subgroups — #2737533

Order 512.

2680 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic, complemented
2C2×#2427C211yesmaximal, characteristic
3#17229C214yesmaximal, complemented
4#17150C214yesmaximal, complemented
5#17057C212yesmaximal, complemented
6#16895C212yesmaximal, complemented
7#16395C212yesmaximal, complemented
8#15243C212yesmaximal, complemented
9#15215C212yesmaximal
10#15192C211yesmaximal, characteristic, complemented
11#15190C211yesmaximal, characteristic, complemented
12#15186C212yesmaximal, complemented
13#15178C212yesmaximal, complemented
14#15178C211yesmaximal, characteristic, complemented
15#14674C212yesmaximal, complemented
16#13916C211yesmaximal, characteristic, complemented
17C4×#397C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×#124C2212yes—
21C22×#124C2211yescharacteristic
22C22×C22⋊Q8C2211yescharacteristic
23C22×C22⋊Q8C2212yes—
24C22×#121C2211yescharacteristic
25C22×C4×D4C2211yescharacteristic
26#2547—28no—
27#2427—28no—
28C2×#397C2211yescharacteristic
29C2×#397C2212yes—
30C2×#397C2214yes—
31C2×#394C2212yes—
32C2×#391C2212yes—
33C2×#391C2211yescharacteristic
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#390C2212yes—
38C2×#389C2211yescharacteristic
39C2×#386C2212yes—
40C2×#381C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47#1445C2212yes—
48#1445C2211yescharacteristic
49#1445C2214yes—
50#1444C2212yes—
51#1444C2212yes—
52#1444C2214yes—
53#1444C2214yes—
54#1444C2212yes—
55#1444C2214yes—
56#1444C2212yes—
57#1443C2214yes—
58#1443C2212yes—
59#1443C2214yes—
60#1443C2214yes—
61#1442C2211yescharacteristic
62#1442C2214yes—
63#1442C2212yes—
64#1440C2212yes—
65#1440C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2214yes—
69#1439C2211yescharacteristic
70#1439C2212yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1438C2214yes—
78#1438C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2212yes—
82C22⋊#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8—44no—
95C2×C4⋊Q8—44no—
96C2×#124—24no—
97C2×#124—28no—
98C2×#124—44no—
99C2×#124—44no—
100C2×#124—44no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—44no—
103C2×C22⋊Q8—44no—
104C2×C22⋊Q8—44no—
105C2×C22⋊Q8—44no—
106C2×#121—24no—
107C2×#121—44no—
108C2×C4×D4—44no—
109C2×C4×D4D418yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140#397—28no—
141#397—216no—
142#397—24no—
143#394—28no—
144#391—28no—
145#391—24no—
146#391—44no—
147#391—44no—
148#391—44no—
149#390—44no—
150#389—24no—
151#386—28no—
152#381—28no—
153#380—28no—
154#380—28no—
155#380—28no—
156C4×C4⋊C4—24no—
157C4×C22⋊C4—24no—
158C4×C22⋊C4—44no—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2311yescharacteristic
188C25C2412yes—
189C25C2411yescharacteristic
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—22no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—44no—
209C22×D4—42no—
210C22×D4—44no—
211C22×D4C2×D414yes—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4C2×D418yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C4⋊Q8—416no—
238C4⋊Q8—432no—
239#124—416no—
240#124—432no—
241C22⋊Q8—416no—
242C22⋊Q8—432no—
243#121—416no—
244C4×D4—416no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C2×D414yes—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4C2×D418yes—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42C2×D414yes—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—22no—
334#95—24no—
335#95—44no—
336#95—42no—
337#95—42no—
338#95—42no—
339#95—44no—
340#95—44no—
341#95—42no—
342#95—42no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95#4114yes—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—21no—
354C24—22no—
355C24—21no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—24no—
360C24—22no—
361C24—24no—
362C24—21no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—21no—
369C24—22no—
370C24C22×D414yes—
371C242−1+412yes—
372C24C2511yescharacteristic
373C2×Q8—44no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—28no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—84no—
393C2×D4—88no—
394C2×D4—88no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—216no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—88no—
479C22×C4—88no—
480C22×C4C22×D412yes—
481C22×C4C22×D414yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C2×#4114yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)14yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)14yes—
490C22×C42−1+414yes—
491C22×C42−1+412yes—
492C4⋊C4—28no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—416no—
498C4⋊C4—416no—
499C4⋊C4—416no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—84no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—416no—
511C22⋊C4—416no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—416no—
515C22⋊C4—44no—
516C22⋊C4—88no—
517C22⋊C4—88no—
518C42—28no—
519C42—48no—
520C42—44no—
521C42—48no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—44no—
541C23—42no—
542C23—44no—
543C23—42no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—42no—
552C23—44no—
553C23—41no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—44no—
559C23—42no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23C23×D411yescharacteristic
569C23C22×#4111yescharacteristic
570C23C2×H(5,𝔽2)12yes—
571C23C2×H(5,𝔽2)12yes—
572C23C2×H(5,𝔽2)11yescharacteristic
573C23C2×H(5,𝔽2)11yescharacteristic
574C23C2×2−1+411yescharacteristic
575C23C2×2−1+411yescharacteristic
576C23C2×2−1+412yes—
577C23#58512yes—
578C23#58511yescharacteristic
579Q8—88no—
580Q8—816no—
581D4—416no—
582D4—816no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—816no—
665C2×C4—88no—
666C2×C4#53414yes—
667C2×C4#53514yes—
668C2×C4#53514yes—
669C2×C4#54714yes—
670C2×C4#56018yes—
671C2×C4#56018yes—
672C22—44no—
673C22—44no—
674C22—42no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—42no—
682C22—42no—
683C22—44no—
684C22—81no—
685C22—82no—
686C22—84no—
687C22—81no—
688C22—84no—
689C22—81no—
690C22—84no—
691C22—81no—
692C22—84no—
693C22—82no—
694C22C2×#53411yescharacteristic
695C22C2×#53511yescharacteristic
696C22C2×#53511yescharacteristic
697C22C2×#54711yescharacteristic
698C22#348511yescharacteristic
699C22#348911yescharacteristic
700C22#349911yescharacteristic
701C22#351112yes—
702C22#351412yes—
703C22#351711yescharacteristic
704C22#352011yescharacteristic
705C22C2×#56012yes—
706C22C2×#56012yes—
707C22#354112yes—
708C22#354711yescharacteristic
709C22#355211yescharacteristic
710C22#355212yes—
711C22#355212yes—
712C22#355412yes—
713C22#356112yes—
714C22#356112yes—
715C22#356112yes—
716C22#356212yes—
717C4—24no—
718C4—44no—
719C4—44no—
720C4—44no—
721C4—42no—
722C4—82no—
723C4—81no—
724C4—82no—
725C4—84no—
726C4—81no—
727C4—82no—
728C4—81no—
729C4—82no—
730C4—84no—
731C4#281518yes—
732C2—84no—
733C2—82no—
734C2C2×#281511yescharacteristic
735C2#3891811yescharacteristic
736C2#3892011yescharacteristic
737C2#3948811yescharacteristic
738C2#4264811yescharacteristic
739C2#4268011yescharacteristic
740C2#4291211yescharacteristic
741C2#4566912yes—
742C2#4606312yes—
743C2#5302212yes—
744C2#5304112yes—
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
1. C2×#2547C2semidirect
3. #17229C2semidirect
4. #17150C2semidirect
5. #17057C2semidirect
6. #16895C2semidirect
7. #16395C2semidirect
8. #15243C2semidirect
10. #15192C2semidirect
11. #15190C2semidirect
12. #15186C2semidirect
13. #15178C2semidirect
14. #15178C2semidirect
15. #14674C2semidirect
16. #13916C2semidirect
17. C4×#397C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#24272
#172292
#171502
#170572
#168952
#163952
#152432
#152152
#151922
#151902
#151862
#151782
#146742
#139162
C4×#3972