Finite Group Databasebeta

← #608727

Subgroups — #608727

Order 512.

7608 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×D4C211yesmaximal, characteristic, complemented
2C2×C4×#123C211yesmaximal, characteristic, complemented
3C2×C4×#120C212yesmaximal, complemented
4#15203C214yesmaximal, complemented
5#15159C214yesmaximal, complemented
6#15092C211yesmaximal, characteristic, complemented
7#15091C211yesmaximal, characteristic, complemented
8#14918C212yesmaximal, complemented
9#14911C212yesmaximal, complemented
10#14724C211yesmaximal, characteristic, complemented
11#14715C211yesmaximal, characteristic
12C4×#394C214yesmaximal, complemented
13C4×#392C214yesmaximal, complemented
14C4×#381C211yesmaximal, characteristic, complemented
15C4×#379C212yesmaximal, complemented
16C24×D4C2211yescharacteristic
17C25×C4C2212yes—
18C22×#123C2211yescharacteristic
19C22×#120C2212yes—
20C22×C4×D4—28no—
21C22×C4×D4—216no—
22C22×C4×D4C2214yes—
23C22×C4×D4C2214yes—
24C22×C4×D4C2211yescharacteristic
25C22×C4×D4C2214yescomplemented
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4C2212yes—
28C23×C42C2211yescharacteristic
29C2×#394C2214yes—
30C2×#392C2214yes—
31C4×#123—28no—
32C4×#120—216no—
33C2×#388C2212yescomplemented
34C2×#387C2212yescomplemented
35C2×#386C2212yes—
36C2×#386C2211yescharacteristic
37C2×#386C2214yescomplemented
38C2×#385C2212yes—
39C2×#385C2211yescharacteristic
40C2×#385C2214yescomplemented
41C2×#381C2212yes—
42C2×#381C2211yescharacteristic
43C2×#379C2212yes—
44C2×#379C2212yes—
45C2×#379C2212yes—
46C2×C4×C4⋊C4C2212yescomplemented
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yescomplemented
50C2×C4×C22⋊C4C2214yescomplemented
51C2×C4×C22⋊C4C2218yescomplemented
52#1444C2214yescomplemented
53#1442C2214yescomplemented
54#1441C2214yescomplemented
55#1440C2218yescomplemented
56#1439C2214yescomplemented
57#1439C2218yescomplemented
58#1439C2218yescomplemented
59#1438C2214yes—
60#1438C2214yes—
61#1438C2218yescomplemented
62#1434C2214yescomplemented
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2211yescharacteristic
66C22⋊#95C2211yescharacteristic
67C22⋊#95C2218yescomplemented
68C4×#95C2214yescomplemented
69C26C2312yes—
70C23×D4—28no—
71C23×D4—216no—
72C23×D4—24no—
73C23×D4C2314yes—
74C23×D4C2311yescharacteristic
75C23×D4C2314yes—
76C24×C4—28no—
77C24×C4—28no—
78C24×C4—416no—
79C24×C4D4116yescomplemented
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2314yes—
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2314yes—
87C2×#123—24no—
88C2×#123—24no—
89C2×#123—24no—
90C2×#120—28no—
91C2×#120—28no—
92C2×#120—28no—
93C2×C4×D4—2128no—
94C2×C4×D4—232no—
95C2×C4×D4—232no—
96C2×C4×D4—28no—
97C2×C4×D4—24no—
98C2×C4×D4—216no—
99C2×C4×D4—216no—
100C2×C4×D4—216no—
101C2×C4×D4—416no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4—28no—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—28no—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2318yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2318yes—
122C22×C22⋊C4C2318yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42—28no—
128C22×C42D418yescomplemented
129C22×C42C2311yescharacteristic
130C22×C42C2314yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134#394—216no—
135#392—216no—
136#388—28no—
137#387—28no—
138#386—28no—
139#386—24no—
140#386—216no—
141#385—28no—
142#385—24no—
143#385—216no—
144#381—24no—
145#381—28no—
146#379—28no—
147#379—28no—
148#379—28no—
149C4×C4⋊C4—28no—
150C4×C22⋊C4—28no—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—216no—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—416no—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2318yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2318yes—
167C2×#95C2318yes—
168C25—24no—
169C25—24no—
170C25—216no—
171C25—24no—
172C25—24no—
173C25—24no—
174C25—24no—
175C25—48no—
176C25C2×D418yes—
177C25C2411yescharacteristic
178C25C2414yes—
179C25C2411yescharacteristic
180C22×D4—232no—
181C22×D4—2128no—
182C22×D4—264no—
183C22×D4—216no—
184C22×D4—216no—
185C22×D4—28no—
186C22×D4—24no—
187C22×D4—22no—
188C22×D4—28no—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—216no—
192C22×D4—28no—
193C22×D4—28no—
194C22×D4—28no—
195C22×D4—48no—
196C22×D4#4114yes—
197C22×D4#41116yes—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—264no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—28no—
212C23×C4—216no—
213C23×C4—216no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—28no—
217C23×C4—464no—
218C23×C4—48no—
219C23×C4—48no—
220C23×C4—48no—
221C23×C4—464no—
222C23×C4—464no—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D4116yes—
226C23×C4C2×D414yes—
227C23×C4C2×D418yes—
228C23×C4C2×D418yes—
229C23×C4C2×D4116yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242#123—232no—
243#120—264no—
244C4×D4—2128no—
245C4×D4—2128no—
246C4×D4—464no—
247C4×D4—416no—
248C2×C4⋊C4—232no—
249C2×C4⋊C4—216no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—48no—
260C2×C22⋊C4—264no—
261C2×C22⋊C4—216no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—216no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—216no—
284C2×C22⋊C4—216no—
285C2×C22⋊C4—216no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—216no—
291C2×C22⋊C4—216no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4#4118yes—
298C2×C22⋊C4#4114yes—
299C2×C42—232no—
300C2×C42—28no—
301C2×C42—28no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—216no—
307C2×C42—28no—
308C2×C42C2×D418yes—
309C2×C42C2×D418yes—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—216no—
319#95—28no—
320#95—28no—
321#95—48no—
322#95—48no—
323C24—24no—
324C24—24no—
325C24—232no—
326C24—232no—
327C24—216no—
328C24—232no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—28no—
336C24—28no—
337C24—28no—
338C24—216no—
339C24—216no—
340C24—21no—
341C24—22no—
342C24—21no—
343C24—24no—
344C24—22no—
345C24—24no—
346C24—416no—
347C24—432no—
348C24—432no—
349C24—432no—
350C24—432no—
351C24—416no—
352C24—416no—
353C24—48no—
354C24—48no—
355C24—48no—
356C24—48no—
357C24C22×D418yes—
358C24C22×D414yes—
359C24C22×D412yes—
360C24C2×#4112yes—
361C24C2×#41116yes—
362C24C2×#4112yes—
363C24C2×#4112yes—
364C24C2×#4114yes—
365C24C2511yescharacteristic
366C2×D4—232no—
367C2×D4—2256no—
368C2×D4—232no—
369C2×D4—264no—
370C2×D4—264no—
371C2×D4—232no—
372C2×D4—216no—
373C2×D4—28no—
374C2×D4—232no—
375C2×D4—432no—
376C2×D4—432no—
377C2×D4—416no—
378C2×D4—44no—
379C2×D4—416no—
380C2×D4—44no—
381C2×D4—48no—
382C22×C4—232no—
383C22×C4—28no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—216no—
388C22×C4—232no—
389C22×C4—216no—
390C22×C4—28no—
391C22×C4—216no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—232no—
415C22×C4—232no—
416C22×C4—232no—
417C22×C4—232no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—28no—
432C22×C4—24no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—432no—
437C22×C4—432no—
438C22×C4—4256no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—432no—
444C22×C4—432no—
445C22×C4—44no—
446C22×C4—416no—
447C22×C4—48no—
448C22×C4#120116yes—
449C22×C4C22×D414yes—
450C22×C4C22×D418yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C2×#4118yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4118yes—
460C22×C4H(5,𝔽2)18yes—
461C4⋊C4—232no—
462C4⋊C4—216no—
463C4⋊C4—416no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C22⋊C4—264no—
467C22⋊C4—232no—
468C22⋊C4—216no—
469C22⋊C4—216no—
470C22⋊C4—28no—
471C22⋊C4—432no—
472C22⋊C4—432no—
473C22⋊C4—416no—
474C22⋊C4—416no—
475C22⋊C4—44no—
476C22⋊C4—416no—
477C22⋊C4—44no—
478C22⋊C4—48no—
479C22⋊C4—44no—
480C22⋊C4—48no—
481C42—216no—
482C42—216no—
483C42—44no—
484C42—416no—
485C42—48no—
486C23—28no—
487C23—24no—
488C23—28no—
489C23—24no—
490C23—28no—
491C23—216no—
492C23—28no—
493C23—264no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—216no—
498C23—216no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—216no—
503C23—216no—
504C23—216no—
505C23—216no—
506C23—22no—
507C23—24no—
508C23—28no—
509C23—464no—
510C23—464no—
511C23—432no—
512C23—464no—
513C23—464no—
514C23—464no—
515C23—464no—
516C23—432no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—48no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—48no—
526C23—48no—
527C23—41no—
528C23—42no—
529C23—44no—
530C23—42no—
531C23—41no—
532C23—44no—
533C23—42no—
534C23—48no—
535C23—44no—
536C23C2×#12014yes—
537C23#525116yes—
538C23#52514yes—
539C23#52518yes—
540C23#53214yes—
541C23#53218yes—
542C23C23×D412yes—
543C23C23×D411yescharacteristic
544C23C22×#4114yes—
545C23C22×#4111yescharacteristic
546C23C22×#4112yes—
547C23C22×#4111yescharacteristic
548C23C2×H(5,𝔽2)14yes—
549D4—2128no—
550D4—464no—
551D4—416no—
552C2×C4—232no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—216no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—232no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—216no—
571C2×C4—464no—
572C2×C4—432no—
573C2×C4—416no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—464no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—88no—
604C2×C4C2×#12014yes—
605C2×C4C2×#12014yes—
606C2×C4#52518yes—
607C2×C4#534116yes—
608C2×C4#53518yes—
609C22—24no—
610C22—216no—
611C22—216no—
612C22—28no—
613C22—28no—
614C22—24no—
615C22—216no—
616C22—216no—
617C22—216no—
618C22—4128no—
619C22—48no—
620C22—416no—
621C22—48no—
622C22—416no—
623C22—416no—
624C22—416no—
625C22—48no—
626C22—48no—
627C22—48no—
628C22—44no—
629C22—42no—
630C22—81no—
631C22—84no—
632C22—82no—
633C22—81no—
634C22#238818yes—
635C22C22×#12011yescharacteristic
636C22C2×#52514yes—
637C22C2×#52512yes—
638C22C2×#52512yes—
639C22C2×#52511yescharacteristic
640C22C2×#53212yes—
641C22C2×#53211yescharacteristic
642C22C2×#53414yes—
643C22C2×#53512yes—
644C22#348114yes—
645C22#348118yes—
646C22#348114yes—
647C4—28no—
648C4—24no—
649C4—48no—
650C4—44no—
651C4—44no—
652C4—42no—
653C4—48no—
654C4—81no—
655C4—84no—
656C4—82no—
657C4—81no—
658C4#261318yes—
659C2—28no—
660C2—416no—
661C2—82no—
662C2C2×#238811yescharacteristic
663C2C2×#261311yescharacteristic
664C2#3495614yes—
665C2#3495714yes—
666C2#3496312yes—
667C2#3496512yes—
668C2#3661411yescharacteristic
subgroupG/H
centerC23×C4C24
commutator, Frattini, socle, cosocleC24C25
prime radical#14715C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C23×C4×D4C2semidirect
2. C2×C4×#123C2semidirect
3. C2×C4×#120C2semidirect
4. #15203C2semidirect
5. #15159C2semidirect
6. #15092C2semidirect
7. #15091C2semidirect
8. #14918C2semidirect
9. #14911C2semidirect
10. #14724C2semidirect
12. C4×#394C2semidirect
13. C4×#392C2semidirect
14. C4×#381C2semidirect
15. C4×#379C2semidirect
25. C22×C4×D4C22semidirect
33. C2×#388C22semidirect
34. C2×#387C22semidirect
37. C2×#386C22semidirect
40. C2×#385C22semidirect
46. C2×C4×C4⋊C4C22semidirect
49. C2×C4×C22⋊C4C22semidirect
50. C2×C4×C22⋊C4C22semidirect
51. C2×C4×C22⋊C4C22semidirect
52. #1444C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
55. #1440C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
61. #1438C22semidirect
62. #1434C22semidirect
67. C22⋊#95C22semidirect
68. C4×#95C22semidirect
79. C24×C4D4semidirect
128. C22×C42D4semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D42
C2×C4×#1232
C2×C4×#1202
#152032
#151592
#150922
#150912
#149182
#149112
#147242
#147152
C4×#3942
C4×#3922
C4×#3812
C4×#3792