Finite Group Databasebeta

← #641846

Subgroups — #641846

Order 512.

4394 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2626C212yesmaximal, complemented
2C2×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
3C22×#390C211yesmaximal, characteristic
4#17077C212yesmaximal, complemented
5#17071C212yesmaximal, complemented
6#16822C214yesmaximal, complemented
7#15157C212yesmaximal, complemented
8#15148C212yesmaximal, complemented
9#15134C214yesmaximal, complemented
10#14726C214yesmaximal, complemented
11#14686C211yesmaximal, characteristic, complemented
12#14338C212yesmaximal, complemented
13#14130C212yesmaximal, complemented
14#13902C212yesmaximal
15C24×D4C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic, complemented
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2212yes—
21C23×C42C2211yescharacteristic
22#2626—216no—
23C2×#399C2214yescomplemented
24C2×#396C2218yescomplemented
25C2×#396C2218yescomplemented
26C2×#393C2214yescomplemented
27C4×C22⋊Q8—28no—
28C2×#391C2211yescharacteristic, complemented
29C2×#390—28no—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2214yes—
35C2×#390C2214yes—
36C2×#389C2211yescharacteristic, complemented
37C2×#389C2214yescomplemented
38C2×#387C2212yescomplemented
39C2×#385C2214yes—
40C2×#385C2212yes—
41C2×#384C2212yescomplemented
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#379C2214yes—
47C2×#379C2214yes—
48C2×C4×C4⋊C4C2212yescomplemented
49C2×C4×C4⋊C4C2211yescharacteristic, complemented
50C2×C4×C22⋊C4C2212yes—
51#1444C2214yescomplemented
52#1442C2214yescomplemented
53#1439C2218yescomplemented
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1439C2218yescomplemented
57#1439C2218yescomplemented
58#1439C2218yescomplemented
59#1438C2214yes—
60#1438C2214yes—
61C4⋊#95C2214yescomplemented
62C4⋊#95C2212yescomplemented
63C4⋊#95C2212yescomplemented
64C4⋊#95C2212yescomplemented
65C22⋊#95C2214yes—
66C26C2312yes—
67C23×Q8C2311yescharacteristic
68C23×D4—24no—
69C23×D4—24no—
70C23×D4—44no—
71C23×D4—44no—
72C23×D4Q818yescomplemented
73C23×D4C2312yes—
74C23×D4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8D418yescomplemented
82C2×C4×Q8—24no—
83C2×C4×Q8—44no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—44no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2318yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4—28no—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—48no—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2318yes—
110C22×C22⋊C4C2318yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2318yes—
114C22×C42—44no—
115C22×C42D418yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#399—216no—
123#396—232no—
124#396—416no—
125#393—216no—
126#391—24no—
127#390—28no—
128#390—28no—
129#390—28no—
130#390—28no—
131#390—48no—
132#390—48no—
133#389—24no—
134#389—48no—
135#387—28no—
136#385—28no—
137#385—216no—
138#384—44no—
139#380—28no—
140#380—24no—
141#380—28no—
142#380—44no—
143#379—216no—
144#379—48no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—24no—
147C4×C22⋊C4—28no—
148C2×#95C2318yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2318yes—
157C2×#95C2318yes—
158C2×#95C2314yes—
159C25—24no—
160C25—24no—
161C25—24no—
162C25—24no—
163C25—24no—
164C25—24no—
165C25—44no—
166C25—44no—
167C25—44no—
168C25—44no—
169C25C2×Q818yes—
170C25C2414yes—
171C25C2411yescharacteristic
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—42no—
175C22×Q8—42no—
176C22×Q8C2×D414yescomplemented
177C22×D4—232no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—432no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—416no—
188C22×D4—416no—
189C22×D4—432no—
190C22×D4C2×Q818yes—
191C22×D4C2×Q818yes—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4C2×D414yes—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2×Q814yes—
210C23×C4#4114yes—
211C23×C4#4114yes—
212C23×C4C2418yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C22⋊Q8—232no—
225C4×Q8—416no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—216no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—416no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4—48no—
255C2×C4⋊C4C2×D418yescomplemented
256C2×C4⋊C4C2×D414yescomplemented
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—216no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—216no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4—432no—
278C2×C22⋊C4—432no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4C2×D418yes—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—416no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42C2×D414yes—
300C2×C42C2×D414yes—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—216no—
310#95—28no—
311#95—44no—
312#95—48no—
313#95—48no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—216no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—28no—
324C24—24no—
325C24—28no—
326C24—28no—
327C24—24no—
328C24—28no—
329C24—216no—
330C24—216no—
331C24—416no—
332C24—416no—
333C24—48no—
334C24—48no—
335C24—48no—
336C24—416no—
337C24—48no—
338C24—416no—
339C24—48no—
340C24—48no—
341C24—48no—
342C24—48no—
343C24—88no—
344C24—88no—
345C24—88no—
346C24—88no—
347C24—88no—
348C24—88no—
349C24—88no—
350C24—88no—
351C24C22×D412yes—
352C24C22×D412yes—
353C24C22×Q818yes—
354C24C22×Q812yes—
355C24C2×#4112yes—
356C24C2×#4112yes—
357C24H(5,𝔽2)12yes—
358C242−1+412yes—
359C24C2511yescharacteristic
360C2×Q8—28no—
361C2×Q8—48no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—48no—
365C2×D4—216no—
366C2×D4—216no—
367C2×D4—232no—
368C2×D4—416no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—416no—
372C2×D4—4128no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—416no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—216no—
388C22×C4—216no—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—28no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4#121116yes—
438C22×C4C22⋊Q8116yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×Q812yes—
450C22×C4C22×Q812yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C4H(5,𝔽2)14yes—
457C22×C4H(5,𝔽2)14yes—
458C4⋊C4—28no—
459C4⋊C4—216no—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—416no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C4⋊C4—816no—
470C22⋊C4—216no—
471C22⋊C4—416no—
472C22⋊C4—48no—
473C22⋊C4—416no—
474C22⋊C4—416no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—816no—
478C22⋊C4—816no—
479C22⋊C4—816no—
480C22⋊C4—816no—
481C42—28no—
482C42—28no—
483C42—44no—
484C42—44no—
485C42—48no—
486C42—48no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—216no—
505C23—28no—
506C23—28no—
507C23—216no—
508C23—432no—
509C23—432no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—48no—
517C23—44no—
518C23—44no—
519C23—48no—
520C23—48no—
521C23—48no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—48no—
526C23—48no—
527C23—48no—
528C23—432no—
529C23—832no—
530C23—832no—
531C23—816no—
532C23—832no—
533C23—816no—
534C23—816no—
535C23—816no—
536C23—832no—
537C23C2×#12114yes—
538C23C2×C22⋊Q814yes—
539C23#53714yes—
540C23D4×Q814yes—
541C23#55014yes—
542C23#55414yes—
543C23#55614yes—
544C23C23×D411yescharacteristic
545C23C23×D411yescharacteristic
546C23C23×Q811yescharacteristic
547C23C22×#4111yescharacteristic
548C23C22×#4111yescharacteristic
549C23C2×H(5,𝔽2)12yes—
550C23C2×H(5,𝔽2)12yes—
551C23C2×H(5,𝔽2)12yes—
552C23C2×H(5,𝔽2)11yescharacteristic
553C23C2×2−1+411yescharacteristic
554C23#58512yes—
555Q8—416no—
556D4—432no—
557D4—432no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—232no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—416no—
583C2×C4—416no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—48no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—44no—
614C2×C4—816no—
615C2×C4C2×#12114yes—
616C2×C4C2×#12114yes—
617C2×C4C2×C22⋊Q814yes—
618C2×C4C2×C22⋊Q814yes—
619C2×C4#53414yes—
620C2×C4#53514yes—
621C2×C4#53514yes—
622C2×C4#53818yes—
623C2×C4#54318yes—
624C22—24no—
625C22—28no—
626C22—28no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—28no—
631C22—24no—
632C22—28no—
633C22—24no—
634C22—416no—
635C22—48no—
636C22—48no—
637C22—416no—
638C22—48no—
639C22—864no—
640C22—84no—
641C22—88no—
642C22—84no—
643C22—84no—
644C22—84no—
645C22—88no—
646C22#253818yes—
647C22C22×#12111yescharacteristic
648C22C22×C22⋊Q811yescharacteristic
649C22C2×#53411yescharacteristic
650C22C2×#53511yescharacteristic
651C22C2×#53511yescharacteristic
652C22C2×#53711yescharacteristic
653C22#344911yescharacteristic
654C22C2×#53812yes—
655C22C2×#54312yes—
656C22C2×D4×Q811yescharacteristic
657C22C2×#55011yescharacteristic
658C22C2×#55411yescharacteristic
659C22C2×#55611yescharacteristic
660C22#349612yes—
661C22#349812yes—
662C22#350112yes—
663C22#350512yes—
664C22#350612yes—
665C22#350812yes—
666C22#353612yes—
667C22#353712yes—
668C22#355812yes—
669C22#356112yes—
670C4—24no—
671C4—24no—
672C4—24no—
673C4—24no—
674C4—42no—
675C4—44no—
676C4—44no—
677C4—82no—
678C4—82no—
679C4—84no—
680C4—88no—
681C4#262618yescomplemented
682C2—28no—
683C2—88no—
684C2C2×#262611yescharacteristic
685C2C2×#253811yescharacteristic
686C2#3500011yescharacteristic
687C2#3517211yescharacteristic
688C2#3527211yescharacteristic
689C2#3743911yescharacteristic
690C2#3754611yescharacteristic
691C2#3936712yes—
692C2#3938312yes—
693C2#3949512yes—
694C2#3951112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2626C2semidirect
2. C2×C4×C22⋊Q8C2semidirect
4. #17077C2semidirect
5. #17071C2semidirect
6. #16822C2semidirect
7. #15157C2semidirect
8. #15148C2semidirect
9. #15134C2semidirect
10. #14726C2semidirect
11. #14686C2semidirect
12. #14338C2semidirect
13. #14130C2semidirect
17. C22×C4×Q8C22semidirect
23. C2×#399C22semidirect
24. C2×#396C22semidirect
25. C2×#396C22semidirect
26. C2×#393C22semidirect
28. C2×#391C22semidirect
36. C2×#389C22semidirect
37. C2×#389C22semidirect
38. C2×#387C22semidirect
41. C2×#384C22semidirect
48. C2×C4×C4⋊C4C22semidirect
49. C2×C4×C4⋊C4C22semidirect
51. #1444C22semidirect
52. #1442C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
61. C4⋊#95C22semidirect
62. C4⋊#95C22semidirect
63. C4⋊#95C22semidirect
64. C4⋊#95C22semidirect
72. C23×D4Q8semidirect
81. C2×C22⋊Q8D4semidirect
176. C22×Q8C2×D4semidirect
255. C2×C4⋊C4C2×D4semidirect
256. C2×C4⋊C4C2×D4semidirect
681. C4#2626semidirect

Maximal subgroups

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

subgroupindex
C2×#26262
C2×C4×C22⋊Q82
C22×#3902
#170772
#170712
#168222
#151572
#151482
#151342
#147262
#146862
#143382
#141302
#139022