Finite Group Databasebeta

← #1395699

Subgroups — #1395699

Order 512.

2888 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2422C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3#16435C214yesmaximal, complemented
4#16394C212yesmaximal, complemented
5#16360C212yesmaximal, complemented
6#16271C214yesmaximal, complemented
7#15197C211yesmaximal, characteristic, complemented
8#15183C212yesmaximal, complemented
9#15156C211yesmaximal, characteristic, complemented
10#15135C211yesmaximal, characteristic, complemented
11#15133C212yesmaximal, complemented
12#14786C212yesmaximal, complemented
13#14256C212yesmaximal, complemented
14#14018C212yesmaximal, complemented
15#13970C212yesmaximal
16#13916C211yesmaximal, characteristic, complemented
17C4×#399C211yesmaximal, characteristic, complemented
18C22×#123C2212yes—
19C22×#121C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22C23×C4⋊C4C2212yes—
23#2422—28no—
24C2×#400C2212yes—
25C2×#399C2211yescharacteristic
26C2×#399C2212yes—
27C2×#398C2214yes—
28C2×#397C2212yes—
29C2×#394C2212yes—
30D4×C4⋊C4—28no—
31C2×#390C2212yes—
32C2×#390C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2211yescharacteristic
35C2×#389C2212yes—
36C2×#386C2214yes—
37C2×#385C2212yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#384C2211yescharacteristic
41C2×#384C2211yescharacteristic
42C2×#381C2212yes—
43C2×#380C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50#1445C2211yescharacteristic
51#1445C2212yes—
52#1445C2212yes—
53#1445C2214yes—
54#1444C2212yes—
55#1444C2211yescharacteristic
56#1443C2214yes—
57#1443C2214yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2212yes—
64#1441C2212yes—
65#1440C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1439C2214yes—
69#1439C2212yes—
70#1439C2212yes—
71#1439C2211yescharacteristic
72#1439C2214yes—
73#1439C2214yes—
74#1438C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C22⋊#95C2214yes—
82C22⋊#95C2214yes—
83C22⋊#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2314yes—
90C2×#123—28no—
91C2×#123—28no—
92C2×#123—44no—
93C2×#121—24no—
94C2×#121—28no—
95C2×C4×D4—28no—
96C2×C4×D4—28no—
97C2×C4×D4—44no—
98C2×C4×D4—44no—
99C2×C4×D4D418yes—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—48no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#400—44no—
136#399—24no—
137#399—44no—
138#398—48no—
139#397—28no—
140#394—28no—
141#390—24no—
142#390—28no—
143#389—28no—
144#389—24no—
145#389—44no—
146#386—48no—
147#385—28no—
148#385—216no—
149#385—48no—
150#384—24no—
151#384—24no—
152#381—44no—
153#380—28no—
154#380—24no—
155#380—44no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—44no—
158C4×C22⋊C4—24no—
159C4×C22⋊C4—44no—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C25C2411yescharacteristic
188C25C2412yes—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—28no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—42no—
196C22×D4—44no—
197C22×D4C2×D414yes—
198C22×D4#4118yes—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—28no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C2×D418yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231#123—432no—
232#121—416no—
233C4×D4—232no—
234C4×D4—432no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—432no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4C2×D414yes—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—41no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4C2×D418yes—
315C2×C22⋊C4#4114yes—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42C2×D414yes—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—22no—
333#95—22no—
334#95—24no—
335#95—28no—
336#95—24no—
337#95—44no—
338#95—42no—
339#95—42no—
340#95—42no—
341#95—42no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—42no—
351#95—44no—
352#95—44no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—21no—
363C24—22no—
364C24—21no—
365C24—21no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—21no—
370C24—21no—
371C24—21no—
372C24C22×D414yes—
373C24C2×#4112yes—
374C24C2×#4118yes—
375C24C2511yescharacteristic
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—216no—
379C2×D4—216no—
380C2×D4—48no—
381C2×D4—416no—
382C2×D4—48no—
383C2×D4—84no—
384C2×D4—88no—
385C2×D4#123116yes—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—216no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—216no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—48no—
464C22×C4—416no—
465C22×C4—416no—
466C22×C4—416no—
467C22×C4—416no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4—88no—
476C22×C4—88no—
477C22×C4—88no—
478C22×C4C22×D412yes—
479C22×C4C22×D414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4114yes—
485C22×C4H(5,𝔽2)14yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)14yes—
489C22×C42−1+414yes—
490C22×C42−1+412yes—
491C22×C42−1+412yes—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—88no—
508C22⋊C4—216no—
509C22⋊C4—28no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—416no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—416no—
518C22⋊C4—48no—
519C22⋊C4—88no—
520C22⋊C4—88no—
521C42—28no—
522C42—44no—
523C42—48no—
524C42—48no—
525C23—24no—
526C23—24no—
527C23—28no—
528C23—28no—
529C23—24no—
530C23—28no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—22no—
538C23—24no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—44no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—42no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—41no—
566C23—42no—
567C23—41no—
568C23—42no—
569C23—42no—
570C23C2×#12318yes—
571C23C23×D411yescharacteristic
572C23C22×#4112yes—
573C23C22×#4111yescharacteristic
574C23C2×H(5,𝔽2)12yes—
575C23C2×H(5,𝔽2)11yescharacteristic
576C23C2×H(5,𝔽2)12yes—
577C23C2×H(5,𝔽2)11yescharacteristic
578C23C2×2−1+411yescharacteristic
579C23C2×2−1+411yescharacteristic
580C23C2×2−1+412yes—
581C23#58511yescharacteristic
582D4—232no—
583D4—816no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—432no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—832no—
669C2×C4—88no—
670C2×C4C2×#12314yes—
671C2×C4#53514yes—
672C2×C4#53614yes—
673C2×C4#53918yes—
674C2×C4#53918yes—
675C2×C4#54614yes—
676C22—28no—
677C22—28no—
678C22—28no—
679C22—44no—
680C22—44no—
681C22—48no—
682C22—48no—
683C22—48no—
684C22—44no—
685C22—48no—
686C22—44no—
687C22—42no—
688C22—42no—
689C22—44no—
690C22—42no—
691C22—44no—
692C22—81no—
693C22—82no—
694C22—82no—
695C22—81no—
696C22—81no—
697C22—81no—
698C22C22×#12311yescharacteristic
699C22C2×#53511yescharacteristic
700C22C2×#53611yescharacteristic
701C22C2×#53912yes—
702C22C2×#53912yes—
703C22C2×#54611yescharacteristic
704C22#348112yes—
705C22#348411yescharacteristic
706C22#348811yescharacteristic
707C22#349911yescharacteristic
708C22#350711yescharacteristic
709C22#351111yescharacteristic
710C22#351512yes—
711C22#351912yes—
712C22#352012yes—
713C22#352012yes—
714C22#353712yes—
715C22#354212yes—
716C22#354711yescharacteristic
717C22#354812yes—
718C22#354811yescharacteristic
719C22#355012yes—
720C22#355212yes—
721C4—24no—
722C4—42no—
723C4—44no—
724C4—44no—
725C4—42no—
726C4—42no—
727C4—44no—
728C4—81no—
729C4—82no—
730C4—82no—
731C4—84no—
732C4—82no—
733C4—82no—
734C4—84no—
735C4#262318yes—
736C2—48no—
737C2—82no—
738C2C2×#262311yescharacteristic
739C2#3599811yescharacteristic
740C2#3623611yescharacteristic
741C2#3895111yescharacteristic
742C2#3947011yescharacteristic
743C2#4005212yes—
744C2#4017212yes—
745C2#4242711yescharacteristic
746C2#4248011yescharacteristic
747C2#4255612yes—
748C2#4257712yes—
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
2. C2×D4×C4⋊C4C2semidirect
3. #16435C2semidirect
4. #16394C2semidirect
5. #16360C2semidirect
6. #16271C2semidirect
7. #15197C2semidirect
8. #15183C2semidirect
9. #15156C2semidirect
10. #15135C2semidirect
11. #15133C2semidirect
12. #14786C2semidirect
13. #14256C2semidirect
14. #14018C2semidirect
16. #13916C2semidirect
17. C4×#399C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24222
C2×D4×C4⋊C42
#164352
#163942
#163602
#162712
#151972
#151832
#151562
#151352
#151332
#147862
#142562
#140182
#139702
#139162
C4×#3992