Finite Group Databasebeta

← #4341334

Subgroups — #4341334

Order 512.

3478 conjugacy classes of subgroups.

The subgroup lattice has 789 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2709C212yesmaximal, complemented
2C2×#2541C211yesmaximal, characteristic
3C2×#2322C211yesmaximal, characteristic, complemented
4#16660C212yesmaximal, complemented
5#16503C212yesmaximal, complemented
6#16288C212yesmaximal, complemented
7#16261C212yesmaximal, complemented
8#15458C214yesmaximal, complemented
9#15418C214yesmaximal, complemented
10#15080C212yesmaximal, complemented
11#14996C212yesmaximal, complemented
12#14747C212yesmaximal, complemented
13#14719C212yesmaximal, complemented
14#14419C212yesmaximal, complemented
15#13960C211yesmaximal, characteristic, complemented
16C22×#127C2212yes—
17C22×#127C2211yescharacteristic
18C22×#123C2211yescharacteristic
19C22×#121C2212yes—
20C22×#121C2212yes—
21C22×C4×D4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23#2709—216no—
24#2541—28no—
25C2×#396C2214yes—
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29C2×#394C2212yes—
30C2×#394C2212yes—
31C2×#392C2212yes—
32C2×#392C2212yes—
33#2322—28no—
34C2×#390C2212yes—
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#388C2214yes—
38C2×#387C2212yes—
39C2×#387C2214yes—
40C2×#386C2211yescharacteristic
41C2×#386C2214yes—
42C2×#386C2214yes—
43C2×#386C2212yes—
44C2×#386C2214yes—
45C2×#386C2214yes—
46C2×#385C2212yes—
47C2×#385C2211yescharacteristic
48C2×#381C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C22×#95C2211yescharacteristic
53#1444C2212yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2212yes—
57#1441C2212yes—
58#1441C2212yes—
59#1441C2212yes—
60#1440C2214yes—
61#1439C2214yes—
62#1439C2212yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2214yes—
67#1438C2214yes—
68#1438C2214yes—
69#1438C2214yes—
70#1438C2214yes—
71#1438C2212yes—
72#1438C2212yes—
73#1437C2214yes—
74#1434C2214yes—
75#1434C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C23×D4C2314yes—
79C23×D4C2314yes—
80C23×D4C2312yes—
81C23×D4C2312yes—
82C23×D4C2312yes—
83C23×D4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×#127—28no—
90C2×#127—28no—
91C2×#127—28no—
92C2×#127—24no—
93C2×#127—42no—
94C2×#127—42no—
95C2×#123—24no—
96C2×#123—24no—
97C2×#123D418yescomplemented
98C2×#121—28no—
99C2×#121—28no—
100C2×#121—48no—
101C2×#121—48no—
102C2×C4×D4—42no—
103C2×C4×D4—42no—
104C2×C4×D4D418yes—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—42no—
107C22×C4⋊C4—42no—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2311yescharacteristic
133C22×C22⋊C4C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137#396—48no—
138#394—216no—
139#394—216no—
140#394—216no—
141#394—44no—
142#394—44no—
143#392—28no—
144#392—28no—
145#390—28no—
146#389—28no—
147#388—28no—
148#388—48no—
149#387—216no—
150#387—28no—
151#386—24no—
152#386—28no—
153#386—216no—
154#386—216no—
155#386—48no—
156#386—48no—
157#385—28no—
158#385—24no—
159#381—24no—
160#380—42no—
161C4×C4⋊C4—24no—
162C4×C22⋊C4—24no—
163C2×#95—42no—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C25C2414yes—
184C25C2411yescharacteristic
185C25C2411yescharacteristic
186C22×D4—28no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—28no—
190C22×D4—28no—
191C22×D4—28no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—41no—
202C22×D4—41no—
203C22×D4—41no—
204C22×D4—41no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—42no—
210C22×D4—42no—
211C22×D4—42no—
212C22×D4—42no—
213C22×D4—42no—
214C22×D4—42no—
215C22×D4—44no—
216C22×D4—44no—
217C22×D4—44no—
218C22×D4—44no—
219C22×D4—44no—
220C22×D4—44no—
221C22×D4C2×D414yes—
222C22×D4C2×D414yes—
223C22×D4#4114yes—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—24no—
232C23×C4—41no—
233C23×C4—41no—
234C23×C4—41no—
235C23×C4—41no—
236C23×C4—41no—
237C23×C4—41no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—44no—
244C23×C4—44no—
245C23×C4C2×D414yes—
246C23×C4C2×D414yes—
247C23×C4C2×D414yes—
248C23×C4C2414yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2414yes—
252C23×C4C2412yes—
253C23×C4C2411yescharacteristic
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2414yes—
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260#127—416no—
261#127—432no—
262#123—232no—
263#121—432no—
264#121—432no—
265C4×D4—416no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—41no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—416no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4C2×D418yes—
287C2×C4⋊C4C2×D414yes—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—41no—
305C2×C22⋊C4—41no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—41no—
309C2×C22⋊C4—41no—
310C2×C22⋊C4—41no—
311C2×C22⋊C4—41no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
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—42no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4C2×D418yes—
338C2×C22⋊C4C2×D414yes—
339C2×C22⋊C4C2×D414yes—
340C2×C22⋊C4C2×D414yes—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—44no—
348C2×C42C2×D414yes—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—28no—
355#95—24no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—24no—
360#95—24no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—42no—
365#95—44no—
366#95—44no—
367#95—44no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—21no—
372C24—24no—
373C24—24no—
374C24—21no—
375C24—21no—
376C24—21no—
377C24—24no—
378C24—21no—
379C24—24no—
380C24—24no—
381C24—24no—
382C24—24no—
383C24—24no—
384C24—24no—
385C24—24no—
386C24—24no—
387C24—24no—
388C24—24no—
389C24—24no—
390C24—24no—
391C24—24no—
392C24—24no—
393C24—24no—
394C24—24no—
395C24C22×D412yes—
396C24C22×D412yes—
397C24C22×D412yes—
398C24C22×D412yes—
399C24C2×#4112yes—
400C24C2×#4112yes—
401C24C2511yescharacteristic
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—416no—
409C2×D4—416no—
410C2×D4—416no—
411C2×D4—416no—
412C2×D4—416no—
413C2×D4—416no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—416no—
420C2×D4—416no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—84no—
424C2×D4—84no—
425C2×D4—84no—
426C2×D4—84no—
427C2×D4—88no—
428C2×D4—88no—
429C2×D4—88no—
430C2×D4—88no—
431C2×D4—88no—
432C2×D4—88no—
433C2×D4—88no—
434C2×D4—88no—
435C2×D4—88no—
436C2×D4—88no—
437C2×D4—84no—
438C2×D4—84no—
439C2×D4#121116yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—28no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—28no—
472C22×C4—28no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—48no—
507C22×C4—84no—
508C22×C4—84no—
509C22×C4—84no—
510C22×C4—84no—
511C22×C4—84no—
512C22×C4—88no—
513C22×C4—84no—
514C22×C4—84no—
515C22×C4—84no—
516C22×C4—84no—
517C22×C4—88no—
518C22×C4C22×D412yes—
519C22×C4C22×D414yes—
520C22×C4C22×D414yes—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C22×D412yes—
524C22×C4C22×D412yes—
525C22×C4C22×D412yes—
526C22×C4C2×#4112yes—
527C22×C4H(5,𝔽2)18yes—
528C22×C4H(5,𝔽2)12yes—
529C22×C4H(5,𝔽2)12yes—
530C4⋊C4—216no—
531C4⋊C4—216no—
532C4⋊C4—44no—
533C4⋊C4—416no—
534C4⋊C4—48no—
535C4⋊C4—48no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C22⋊C4—28no—
539C22⋊C4—216no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—416no—
543C22⋊C4—416no—
544C22⋊C4—44no—
545C22⋊C4—416no—
546C22⋊C4—88no—
547C22⋊C4—88no—
548C22⋊C4—88no—
549C22⋊C4—88no—
550C22⋊C4—84no—
551C22⋊C4—88no—
552C22⋊C4—84no—
553C22⋊C4—88no—
554C22⋊C4—88no—
555C22⋊C4—88no—
556C42—28no—
557C42—48no—
558C42—84no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—24no—
564C23—24no—
565C23—28no—
566C23—28no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—24no—
572C23—24no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—28no—
578C23—28no—
579C23—44no—
580C23—44no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23—44no—
613C23—44no—
614C23—44no—
615C23—44no—
616C23—44no—
617C23—44no—
618C23—44no—
619C23—44no—
620C23—44no—
621C23—44no—
622C23—44no—
623C23—44no—
624C23—44no—
625C23—44no—
626C23C2×#12114yes—
627C23C2×#12114yes—
628C23C23×D411yescharacteristic
629C23C23×D411yescharacteristic
630C23C22×#4111yescharacteristic
631C23C2×H(5,𝔽2)11yescharacteristic
632C23C2×H(5,𝔽2)11yescharacteristic
633C23C2×H(5,𝔽2)14yes—
634C23#58511yescharacteristic
635C23#58511yescharacteristic
636C23#58514yes—
637D4—416no—
638D4—816no—
639D4—816no—
640D4—832no—
641D4—832no—
642D4—816no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—24no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—24no—
658C2×C4—28no—
659C2×C4—24no—
660C2×C4—28no—
661C2×C4—28no—
662C2×C4—24no—
663C2×C4—24no—
664C2×C4—24no—
665C2×C4—28no—
666C2×C4—416no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—42no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—44no—
706C2×C4—44no—
707C2×C4—44no—
708C2×C4—88no—
709C2×C4—88no—
710C2×C4—88no—
711C2×C4—816no—
712C2×C4C2×#12114yes—
713C2×C4#53414yes—
714C22—28no—
715C22—28no—
716C22—24no—
717C22—24no—
718C22—42no—
719C22—42no—
720C22—44no—
721C22—44no—
722C22—44no—
723C22—44no—
724C22—44no—
725C22—44no—
726C22—42no—
727C22—42no—
728C22—44no—
729C22—44no—
730C22—44no—
731C22—44no—
732C22—84no—
733C22—84no—
734C22—84no—
735C22—84no—
736C22—84no—
737C22—84no—
738C22—84no—
739C22—84no—
740C22—84no—
741C22—84no—
742C22—84no—
743C22—84no—
744C22—84no—
745C22—84no—
746C22—84no—
747C22C22×#12111yescharacteristic
748C22C2×#53411yescharacteristic
749C22#344911yescharacteristic
750C22#348412yes—
751C22#348814yes—
752C22#349611yescharacteristic
753C22#349611yescharacteristic
754C22#350012yes—
755C22#350511yescharacteristic
756C22#350511yescharacteristic
757C22#351314yes—
758C22#353914yes—
759C22#354012yes—
760C22#354312yes—
761C22#354312yes—
762C22#355112yes—
763C22#356214yes—
764C4—24no—
765C4—24no—
766C4—42no—
767C4—42no—
768C4—42no—
769C4—44no—
770C4—48no—
771C4—44no—
772C4—82no—
773C4—84no—
774C4—82no—
775C4—84no—
776C2—44no—
777C2—44no—
778C2—164no—
779C2#3500011yescharacteristic
780C2#3572812yes—
781C2#3723211yescharacteristic
782C2#3741911yescharacteristic
783C2#4050512yes—
784C2#4654812yes—
785C2#4745712yes—
786C2#4818412yes—
787C2#5068412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2541C2
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×#2709C2semidirect
3. C2×#2322C2semidirect
4. #16660C2semidirect
5. #16503C2semidirect
6. #16288C2semidirect
7. #16261C2semidirect
8. #15458C2semidirect
9. #15418C2semidirect
10. #15080C2semidirect
11. #14996C2semidirect
12. #14747C2semidirect
13. #14719C2semidirect
14. #14419C2semidirect
15. #13960C2semidirect
97. C2×#123D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27092
C2×#25412
C2×#23222
#166602
#165032
#162882
#162612
#154582
#154182
#150802
#149962
#147472
#147192
#144192
#139602