Finite Group Databasebeta

← #2737235

Subgroups — #2737235

Order 512.

2772 conjugacy classes of subgroups.

The subgroup lattice has 778 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic, complemented
2C2×#2492C211yesmaximal, characteristic, complemented
3#17158C214yesmaximal, complemented
4#17098C214yesmaximal, complemented
5#17080C212yesmaximal, complemented
6#16559C212yesmaximal, complemented
7#16504C212yesmaximal
8#15191C211yesmaximal, characteristic, complemented
9#15179C212yesmaximal, complemented
10#15177C211yesmaximal, characteristic, complemented
11#15177C212yesmaximal, complemented
12#15129C212yesmaximal, complemented
13#15125C211yesmaximal, characteristic, complemented
14#15125C212yesmaximal, complemented
15#14625C212yesmaximal, complemented
16#13933C211yesmaximal, characteristic, complemented
17C4×#392C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#124C2212yes—
21C22×C22⋊Q8C2211yescharacteristic
22C22×C22⋊Q8C2212yes—
23C22×C4×D4C2211yescharacteristic
24C22×C4×D4C2212yes—
25#2547—28no—
26#2492—28no—
27C2×#400C2212yes—
28C2×#397C2214yes—
29C2×#394C2214yes—
30C2×#393C2212yes—
31C2×#393C2212yes—
32C2×#393C2214yes—
33C2×#392C2211yescharacteristic
34C2×#391C2212yes—
35C2×#391C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#389C2212yes—
40C2×#388C2212yes—
41C2×#380C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49#1446C2212yes—
50#1446C2214yes—
51#1446C2212yes—
52#1446C2212yes—
53#1444C2214yes—
54#1444C2211yescharacteristic
55#1444C2212yes—
56#1442C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2212yes—
72#1439C2211yescharacteristic
73#1439C2214yes—
74#1438C2211yescharacteristic
75#1438C2212yes—
76#1438C2212yes—
77#1438C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83#1434C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C23×Q8C2312yes—
88C23×D4C2311yescharacteristic
89C23×D4C2312yes—
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—44no—
95C2×C4⋊Q8—44no—
96C2×#124—28no—
97C2×#124—48no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—44no—
100C2×C22⋊Q8—44no—
101C2×C22⋊Q8—44no—
102C2×C22⋊Q8—44no—
103C2×C4×D4—24no—
104C2×C4×D4—44no—
105C2×C4×D4—44no—
106C2×C4×D4—44no—
107C2×C4×D4—44no—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#400—28no—
139#397—216no—
140#394—48no—
141#393—28no—
142#393—28no—
143#393—48no—
144#392—24no—
145#391—24no—
146#391—28no—
147#391—24no—
148#391—28no—
149#391—44no—
150#389—44no—
151#388—28no—
152#380—24no—
153#380—28no—
154#380—28no—
155#380—44no—
156#380—44no—
157C4×C4⋊C4D418yescomplemented
158C4×C22⋊C4—24no—
159C4×C22⋊C4—44no—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C25C2411yescharacteristic
189C25C2412yes—
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—22no—
196C22×Q8—24no—
197C22×Q8—22no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—44no—
201C22×Q8—44no—
202C22×Q8—44no—
203C22×Q8—42no—
204C22×Q8—44no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—42no—
213C22×D4—42no—
214C22×D4—42no—
215C22×D4—42no—
216C22×D4—42no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—22no—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C4⋊Q8—416no—
245C4⋊Q8—416no—
246#124—432no—
247C22⋊Q8—416no—
248C22⋊Q8—432no—
249C4×D4—416no—
250C4×D4—432no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D414yes—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—41no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4#4114yes—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—44no—
328C2×C42C2×D414yes—
329C2×C42C2×D418yes—
330#95—22no—
331#95—22no—
332#95—28no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—22no—
340#95—24no—
341#95—28no—
342#95—24no—
343#95—22no—
344#95—42no—
345#95—44no—
346#95—42no—
347#95—42no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95C2×D418yes—
358C24—21no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—21no—
370C24—22no—
371C24—21no—
372C24—21no—
373C24—22no—
374C24—22no—
375C24—21no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—21no—
382C24C2×#4112yes—
383C24H(5,𝔽2)14yes—
384C24C2511yescharacteristic
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—48no—
393C2×Q8—48no—
394C2×Q8—48no—
395C2×Q8—88no—
396C2×Q8—88no—
397C2×Q8—84no—
398C2×Q8—88no—
399C2×Q8—88no—
400C2×D4—28no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—44no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—44no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—48no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—44no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4C22×D414yes—
496C22×C4C22×D412yes—
497C22×C4C22×D414yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4H(5,𝔽2)14yes—
503C22×C4H(5,𝔽2)14yes—
504C22×C4H(5,𝔽2)12yes—
505C22×C42−1+414yes—
506C22×C42−1+412yes—
507C22×C42−1+412yes—
508C22×C42−1+412yes—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—44no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—416no—
517C4⋊C4—48no—
518C4⋊C4—44no—
519C4⋊C4—48no—
520C4⋊C4—88no—
521C22⋊C4—28no—
522C22⋊C4—44no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—416no—
526C22⋊C4—416no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—416no—
532C22⋊C4—84no—
533C22⋊C4—88no—
534C22⋊C4—88no—
535C42—28no—
536C42—48no—
537C42—44no—
538C42—48no—
539C23—22no—
540C23—24no—
541C23—22no—
542C23—22no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—41no—
552C23—42no—
553C23—41no—
554C23—42no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—41no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—41no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—41no—
596C23—42no—
597C23C23×D411yescharacteristic
598C23C22×#4111yescharacteristic
599C23C2×H(5,𝔽2)12yes—
600C23C2×H(5,𝔽2)12yes—
601C23C2×H(5,𝔽2)11yescharacteristic
602C23C2×H(5,𝔽2)12yes—
603C23C2×2−1+411yescharacteristic
604C23C2×2−1+411yescharacteristic
605C23C2×2−1+411yescharacteristic
606C23C2×2−1+412yes—
607C23#58511yescharacteristic
608Q8—816no—
609Q8—816no—
610Q8—88no—
611D4—416no—
612D4—432no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—816no—
693C2×C4—88no—
694C2×C4#53514yes—
695C2×C4#53514yes—
696C2×C4#53714yes—
697C2×C4#54814yes—
698C2×C4H(3,𝔽4)18yes—
699C2×C4#56218yes—
700C22—44no—
701C22—44no—
702C22—42no—
703C22—44no—
704C22—44no—
705C22—44no—
706C22—44no—
707C22—44no—
708C22—42no—
709C22—44no—
710C22—44no—
711C22—42no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—81no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—81no—
721C22—81no—
722C22—82no—
723C22—82no—
724C22—81no—
725C22—82no—
726C22C2×#53511yescharacteristic
727C22C2×#53511yescharacteristic
728C22C2×#53711yescharacteristic
729C22C2×#54811yescharacteristic
730C22#348411yescharacteristic
731C22#348911yescharacteristic
732C22#349911yescharacteristic
733C22#350611yescharacteristic
734C22#350911yescharacteristic
735C22#351912yes—
736C22#352012yes—
737C22C2×H(3,𝔽4)12yes—
738C22C2×#56212yes—
739C22#353612yes—
740C22#353612yes—
741C22#353612yes—
742C22#354112yes—
743C22#354212yes—
744C22#354412yes—
745C22#355011yescharacteristic
746C22#355012yes—
747C22#355012yes—
748C22#355511yescharacteristic
749C4—24no—
750C4—44no—
751C4—42no—
752C4—44no—
753C4—44no—
754C4—42no—
755C4—42no—
756C4—84no—
757C4—82no—
758C4—81no—
759C4—82no—
760C4—84no—
761C4—82no—
762C4—82no—
763C4#281918yes—
764C2—82no—
765C2—84no—
766C2C2×#281911yescharacteristic
767C2#3912011yescharacteristic
768C2#3916711yescharacteristic
769C2#3972711yescharacteristic
770C2#4242811yescharacteristic
771C2#4247811yescharacteristic
772C2#4291111yescharacteristic
773C2#4560112yes—
774C2#4563112yes—
775C2#5295812yes—
776C2#5309712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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
2. C2×#2492C2semidirect
3. #17158C2semidirect
4. #17098C2semidirect
5. #17080C2semidirect
6. #16559C2semidirect
8. #15191C2semidirect
9. #15179C2semidirect
10. #15177C2semidirect
11. #15177C2semidirect
12. #15129C2semidirect
13. #15125C2semidirect
14. #15125C2semidirect
15. #14625C2semidirect
16. #13933C2semidirect
157. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#24922
#171582
#170982
#170802
#165592
#165042
#151912
#151792
#151772
#151292
#151252
#146252
#139332
C4×#3922