Finite Group Databasebeta

← #796921

Subgroups — #796921

Order 512.

3020 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic, complemented
2C2×#2341C211yesmaximal, characteristic
3#16691C212yesmaximal, complemented
4#15416C214yesmaximal, complemented
5#15386C214yesmaximal, complemented
6#15195C211yesmaximal, characteristic, complemented
7#15148C211yesmaximal, characteristic, complemented
8#15136C211yesmaximal, characteristic, complemented
9#15084C212yesmaximal, complemented
10#15017C212yesmaximal, complemented
11#14240C212yesmaximal, complemented
12#14022C212yesmaximal, complemented
13#14017C212yesmaximal, complemented
14#13974C212yesmaximal, complemented
15#13937C212yesmaximal
16#13931C211yesmaximal, characteristic, complemented
17C4×#399C211yesmaximal, characteristic, complemented
18C23×#41C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#399C2211yescharacteristic
22C2×#399C2212yes—
23C2×#397C2214yes—
24#2355—28no—
25#2341—28no—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#390C2211yescharacteristic
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2211yescharacteristic
33C2×#390C2212yes—
34C2×#389C2212yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#387C2214yes—
38C2×#384C2211yescharacteristic
39C2×#382C2212yes—
40C2×#380C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50#1445C2212yes—
51#1445C2214yes—
52#1445C2211yescharacteristic
53#1444C2214yes—
54#1444C2211yescharacteristic
55#1443C2214yes—
56#1442C2212yes—
57#1442C2214yes—
58#1442C2212yes—
59#1440C2214yes—
60#1440C2214yes—
61#1439C2214yes—
62#1439C2212yes—
63#1439C2212yes—
64#1439C2211yescharacteristic
65#1439C2212yes—
66#1439C2212yes—
67#1439C2212yes—
68#1439C2212yes—
69#1439C2214yes—
70#1438C2214yes—
71#1438C2214yes—
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2212yes—
81#1434C2212yes—
82C22⋊#95C2214yes—
83C22⋊#95C2214yes—
84C4×#95C2211yescharacteristic
85C22×#41—24no—
86C22×#41—28no—
87C22×#41—24no—
88C22×#41—24no—
89C22×#41—44no—
90C23×Q8C2311yescharacteristic
91C23×D4C2311yescharacteristic
92C23×D4C2312yes—
93C24×C4C2311yescharacteristic
94C24×C4C2312yes—
95C2×C4×Q8—28no—
96C2×C4×Q8—24no—
97C2×C4×Q8—44no—
98C2×C4×Q8D418yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2314yes—
133#399—24no—
134#399—28no—
135#397—48no—
136#391—28no—
137#391—28no—
138#390—28no—
139#390—24no—
140#390—28no—
141#390—28no—
142#390—24no—
143#390—44no—
144#389—44no—
145#388—48no—
146#388—48no—
147#387—216no—
148#384—24no—
149#382—28no—
150#380—28no—
151#380—24no—
152#380—28no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—44no—
157C4×C22⋊C4—28no—
158C4×C22⋊C4—24no—
159C4×C22⋊C4—44no—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C25C2411yescharacteristic
187C25C2412yes—
188C2×#41—232no—
189C2×#41—432no—
190C2×#41—432no—
191C2×#41—416no—
192C2×#41—416no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—24no—
196C22×Q8C2×D414yes—
197C22×Q8#4118yes—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—28no—
205C22×D4—24no—
206C22×D4—44no—
207C22×D4#4114yes—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—44no—
217C23×C4#4114yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C4×Q8—232no—
234C4×Q8—416no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D418yes—
266C2×C4⋊C4#4114yescomplemented
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
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—44no—
305C2×C22⋊C4—44no—
306C2×C42—28no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—28no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42C2×D418yes—
322C2×C42C2×D414yes—
323#95—22no—
324#95—24no—
325#95—22no—
326#95—28no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—22no—
334#95—24no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—22no—
339#95—44no—
340#95—42no—
341#95—44no—
342#95—42no—
343#95—42no—
344#95—42no—
345#95—42no—
346#95—44no—
347#95—44no—
348#95—44no—
349C24—22no—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24—21no—
355C24—22no—
356C24—24no—
357C24—22no—
358C24—24no—
359C24—22no—
360C24—24no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—21no—
366C24—22no—
367C24—24no—
368C24—22no—
369C24C2×#4114yes—
370C24C2×#4112yes—
371C24C2511yescharacteristic
372#41—4128no—
373C2×Q8—28no—
374C2×Q8—28no—
375C2×Q8—216no—
376C2×Q8—216no—
377C2×Q8#123116yes—
378C2×D4—216no—
379C2×D4—28no—
380C2×D4—216no—
381C2×D4—416no—
382C2×D4—416no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—48no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—216no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—28no—
423C22×C4—216no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4C22×D414yes—
470C22×C4C22×D412yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4118yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4H(5,𝔽2)14yes—
482C22×C4H(5,𝔽2)14yes—
483C22×C42−1+412yes—
484C22×C42−1+412yes—
485C4⋊C4—216no—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—84no—
497C4⋊C4—88no—
498C22⋊C4—48no—
499C22⋊C4—44no—
500C22⋊C4—48no—
501C22⋊C4—416no—
502C22⋊C4—416no—
503C22⋊C4—48no—
504C22⋊C4—44no—
505C22⋊C4—48no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C42—216no—
513C42—28no—
514C42—48no—
515C42—48no—
516C42—44no—
517C42—48no—
518C42—88no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—28no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—28no—
528C23—24no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—41no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—44no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—44no—
552C23—42no—
553C23—44no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—42no—
558C23—44no—
559C23—42no—
560C23—44no—
561C23—42no—
562C23—44no—
563C23—44no—
564C23—42no—
565C23C23×D411yescharacteristic
566C23C22×#4112yes—
567C23C22×#4111yescharacteristic
568C23C22×#4111yescharacteristic
569C23C22×#4111yescharacteristic
570C23C2×H(5,𝔽2)12yes—
571C23C2×H(5,𝔽2)12yes—
572C23C2×2−1+411yescharacteristic
573C23C2×2−1+411yescharacteristic
574C23#58512yes—
575C23#58511yescharacteristic
576Q8—232no—
577D4—432no—
578D4—416no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—432no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—416no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4C2×#12318yes—
662C2×C4C2×#12314yes—
663C2×C4#52514yes—
664C2×C4#53514yes—
665C2×C4#54814yes—
666C2×C4#55118yes—
667C2×C4#55118yes—
668C22—42no—
669C22—44no—
670C22—48no—
671C22—44no—
672C22—42no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—42no—
677C22—48no—
678C22—81no—
679C22—82no—
680C22—82no—
681C22—84no—
682C22—81no—
683C22—82no—
684C22—84no—
685C22—82no—
686C22—81no—
687C22—82no—
688C22—81no—
689C22—82no—
690C22C22×#12311yescharacteristic
691C22C2×#52511yescharacteristic
692C22C2×#53511yescharacteristic
693C22C2×#54811yescharacteristic
694C22C2×#55112yes—
695C22C2×#55112yes—
696C22#348011yescharacteristic
697C22#348012yes—
698C22#348211yescharacteristic
699C22#348711yescharacteristic
700C22#348911yescharacteristic
701C22#349811yescharacteristic
702C22#350612yes—
703C22#350911yescharacteristic
704C22#351312yes—
705C22#351511yescharacteristic
706C22#351612yes—
707C22#351712yes—
708C22#351912yes—
709C22#352212yes—
710C22#354012yes—
711C22#354112yes—
712C22#354812yes—
713C4—24no—
714C4—28no—
715C4—44no—
716C4—42no—
717C4—44no—
718C4—42no—
719C4—42no—
720C4—44no—
721C4—81no—
722C4—82no—
723C4—84no—
724C4—82no—
725C4—84no—
726C4—82no—
727C4#248218yes—
728C2—84no—
729C2—82no—
730C2C2×#248211yescharacteristic
731C2#3584411yescharacteristic
732C2#3600511yescharacteristic
733C2#3650511yescharacteristic
734C2#3657811yescharacteristic
735C2#3916911yescharacteristic
736C2#4288711yescharacteristic
737C2#4496212yes—
738C2#4506012yes—
739C2#5938112yes—
740C2#5975412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2355C2semidirect
3. #16691C2semidirect
4. #15416C2semidirect
5. #15386C2semidirect
6. #15195C2semidirect
7. #15148C2semidirect
8. #15136C2semidirect
9. #15084C2semidirect
10. #15017C2semidirect
11. #14240C2semidirect
12. #14022C2semidirect
13. #14017C2semidirect
14. #13974C2semidirect
16. #13931C2semidirect
17. C4×#399C2semidirect
266. C2×C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#23552
C2×#23412
#166912
#154162
#153862
#151952
#151482
#151362
#150842
#150172
#142402
#140222
#140172
#139742
#139372
#139312
C4×#3992