Finite Group Databasebeta

← #2736835

Subgroups — #2736835

Order 512.

2830 conjugacy classes of subgroups.

The subgroup lattice has 788 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic
2C2×#2452C211yesmaximal, characteristic, complemented
3#17151C214yesmaximal, complemented
4#17107C214yesmaximal, complemented
5#16892C212yesmaximal, complemented
6#16259C212yesmaximal, complemented
7#16253C212yesmaximal, complemented
8#15242C211yesmaximal, characteristic, complemented
9#15223C212yesmaximal, complemented
10#15219C212yesmaximal, complemented
11#15196C211yesmaximal, characteristic, complemented
12#15129C211yesmaximal, characteristic, complemented
13#15126C212yesmaximal, complemented
14#15125C212yesmaximal, complemented
15#14660C212yesmaximal, complemented
16#13916C211yesmaximal, characteristic, complemented
17C4×#394C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×#121C2212yes—
21C22×#121C2212yes—
22C22×C4×D4C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24#2537—28no—
25#2452—28no—
26C2×#396C2212yes—
27C2×#394C2211yescharacteristic
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#394C2212yes—
31C2×#393C2212yes—
32C2×#393C2214yes—
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#390C2212yes—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#385C2212yes—
40C2×#384C2211yescharacteristic
41C2×#382C2212yes—
42C2×#381C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50#1446C2214yes—
51#1445C2214yes—
52#1445C2214yes—
53#1445C2214yes—
54#1444C2212yes—
55#1444C2212yes—
56#1442C2214yes—
57#1442C2211yescharacteristic
58#1441C2212yes—
59#1441C2211yescharacteristic
60#1441C2212yes—
61#1441C2212yes—
62#1441C2212yes—
63#1441C2212yes—
64#1440C2214yes—
65#1440C2214yes—
66#1440C2212yes—
67#1440C2214yes—
68#1440C2214yes—
69#1440C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2214yes—
75#1438C2214yes—
76#1438C2211yescharacteristic
77#1438C2214yes—
78#1438C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2211yescharacteristic
83C22⋊#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2312yes—
86C23×D4C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C23×D4C2314yes—
89C24×C4C2312yes—
90C24×C4C2312yes—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—44no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—44no—
96C2×#121—28no—
97C2×#121—28no—
98C2×#121—48no—
99C2×#121—44no—
100C2×#121—44no—
101C2×C4×D4—24no—
102C2×C4×D4—44no—
103C2×C4×D4—44no—
104C2×C4×D4D418yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#396—28no—
140#394—24no—
141#394—216no—
142#394—44no—
143#394—48no—
144#393—28no—
145#393—48no—
146#390—24no—
147#390—24no—
148#390—44no—
149#389—24no—
150#389—24no—
151#389—44no—
152#385—28no—
153#384—24no—
154#382—44no—
155#381—44no—
156#380—28no—
157#380—28no—
158#380—44no—
159#380—44no—
160C4×C4⋊C4—24no—
161C4×C22⋊C4—44no—
162C4×C22⋊C4—44no—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C25C2412yes—
189C25C2412yes—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—42no—
193C22×Q8—44no—
194C22×Q8—44no—
195C22×D4—24no—
196C22×D4—28no—
197C22×D4—28no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—42no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4C2×D414yes—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4C2×D418yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C4⋊Q8—416no—
240C22⋊Q8—432no—
241#121—432no—
242#121—432no—
243C4×D4—416no—
244C4×D4—416no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—42no—
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⋊C4#4114yes—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4C2×D418yes—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42C2×D414yes—
337#95—22no—
338#95—22no—
339#95—28no—
340#95—24no—
341#95—24no—
342#95—22no—
343#95—22no—
344#95—24no—
345#95—44no—
346#95—42no—
347#95—42no—
348#95—42no—
349#95—42no—
350#95—42no—
351#95—42no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—22no—
371C24—22no—
372C24—24no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—24no—
381C24—22no—
382C24—22no—
383C24—24no—
384C24—22no—
385C24—22no—
386C24C22×D414yes—
387C24H(5,𝔽2)14yes—
388C24C2511yescharacteristic
389C2×Q8—48no—
390C2×Q8—48no—
391C2×Q8—84no—
392C2×Q8—88no—
393C2×Q8—88no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—28no—
397C2×D4—28no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—44no—
401C2×D4—416no—
402C2×D4—416no—
403C2×D4—44no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—88no—
407C2×D4—88no—
408C2×D4—88no—
409C2×D4—88no—
410C2×D4—88no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—216no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—88no—
498C22×C4—88no—
499C22×C4C22×D412yes—
500C22×C4C22×D414yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4112yes—
505C22×C4C2×#4112yes—
506C22×C4H(5,𝔽2)12yes—
507C22×C4H(5,𝔽2)14yes—
508C22×C4H(5,𝔽2)12yes—
509C22×C4H(5,𝔽2)12yes—
510C22×C42−1+412yes—
511C22×C42−1+414yes—
512C4⋊C4—28no—
513C4⋊C4—28no—
514C4⋊C4—44no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—84no—
525C4⋊C4—88no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—416no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—416no—
537C22⋊C4—84no—
538C22⋊C4—88no—
539C22⋊C4—88no—
540C22⋊C4—88no—
541C22⋊C4—88no—
542C42—28no—
543C42—44no—
544C42—44no—
545C42—48no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—44no—
571C23—42no—
572C23—42no—
573C23—44no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—44no—
586C23—44no—
587C23—42no—
588C23—44no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—44no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—44no—
598C23—42no—
599C23—42no—
600C23—44no—
601C23—42no—
602C23—44no—
603C23—42no—
604C23C23×D411yescharacteristic
605C23C22×#4111yescharacteristic
606C23C2×H(5,𝔽2)12yes—
607C23C2×H(5,𝔽2)11yescharacteristic
608C23C2×H(5,𝔽2)11yescharacteristic
609C23C2×H(5,𝔽2)11yescharacteristic
610C23C2×H(5,𝔽2)12yes—
611C23C2×2−1+412yes—
612C23C2×2−1+411yescharacteristic
613C23#58511yescharacteristic
614C23#58512yes—
615Q8—816no—
616D4—416no—
617D4—416no—
618D4—832no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—41no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
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—42no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—44no—
690C2×C4—42no—
691C2×C4—44no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—44no—
696C2×C4—42no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—42no—
701C2×C4—44no—
702C2×C4—816no—
703C2×C4—816no—
704C2×C4#53414yes—
705C2×C4#53514yes—
706C2×C4#53614yes—
707C2×C4#54614yes—
708C2×C4#56218yes—
709C2×C4#56218yes—
710C22—44no—
711C22—44no—
712C22—44no—
713C22—44no—
714C22—44no—
715C22—44no—
716C22—44no—
717C22—44no—
718C22—44no—
719C22—44no—
720C22—44no—
721C22—44no—
722C22—44no—
723C22—44no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—84no—
728C22—82no—
729C22—82no—
730C22—84no—
731C22—82no—
732C22—84no—
733C22—82no—
734C22—82no—
735C22—84no—
736C22C2×#53411yescharacteristic
737C22C2×#53511yescharacteristic
738C22C2×#53611yescharacteristic
739C22C2×#54611yescharacteristic
740C22#348411yescharacteristic
741C22#348911yescharacteristic
742C22#350011yescharacteristic
743C22#350611yescharacteristic
744C22#351111yescharacteristic
745C22#351812yes—
746C22#351912yes—
747C22C2×#56212yes—
748C22C2×#56212yes—
749C22#353712yes—
750C22#354112yes—
751C22#354212yes—
752C22#354412yes—
753C22#354611yescharacteristic
754C22#354712yes—
755C22#354811yescharacteristic
756C22#355112yes—
757C22#355812yes—
758C22#355912yes—
759C4—24no—
760C4—42no—
761C4—42no—
762C4—42no—
763C4—44no—
764C4—42no—
765C4—82no—
766C4—82no—
767C4—82no—
768C4—82no—
769C4—81no—
770C4—84no—
771C4—84no—
772C4—82no—
773C4#281618yes—
774C2—84no—
775C2—84no—
776C2C2×#281611yescharacteristic
777C2#3890411yescharacteristic
778C2#3895811yescharacteristic
779C2#3947111yescharacteristic
780C2#4242611yescharacteristic
781C2#4248211yescharacteristic
782C2#4290811yescharacteristic
783C2#4565012yes—
784C2#4572012yes—
785C2#5297012yes—
786C2#5310112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2537C2
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×#2452C2semidirect
3. #17151C2semidirect
4. #17107C2semidirect
5. #16892C2semidirect
6. #16259C2semidirect
7. #16253C2semidirect
8. #15242C2semidirect
9. #15223C2semidirect
10. #15219C2semidirect
11. #15196C2semidirect
12. #15129C2semidirect
13. #15126C2semidirect
14. #15125C2semidirect
15. #14660C2semidirect
16. #13916C2semidirect
17. C4×#394C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
C2×#24522
#171512
#171072
#168922
#162592
#162532
#152422
#152232
#152192
#151962
#151292
#151262
#151252
#146602
#139162
C4×#3942