Finite Group Databasebeta

← #7541805

Subgroups — #7541805

Order 512.

2119 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3335C212yesmaximal, complemented
2#59823C211yesmaximal, characteristic, complemented
3#36127C211yesmaximal, characteristic
4#30559C214yesmaximal, complemented
5#28687C211yesmaximal, characteristic, complemented
6#28512C211yesmaximal, characteristic, complemented
7#26234C212yesmaximal, complemented
8#26128C214yesmaximal, complemented
9#25702C212yesmaximal, complemented
10#25385C212yesmaximal, complemented
11#25341C212yesmaximal, complemented
12#25012C212yesmaximal, complemented
13#24120C212yesmaximal, complemented
14#23173C211yesmaximal, characteristic, complemented
15D4×#106C212yesmaximal, complemented
16C4×#476C211yesmaximal, characteristic, complemented
17C4×#461C211yesmaximal, characteristic, complemented
18C2×#554C2212yes—
19C2×#547C2212yes—
20C2×#532—21no—
21#3335—216no—
22C2×#498C2212yes—
23C2×#481C2214yes—
24C2×#483C2214yes—
25C2×#475C2212yes—
26C2×#475C2212yes—
27C2×#476C2211yescharacteristic
28C2×#465C2214yes—
29C2×#461C2211yescharacteristic
30C2×C4×SD16C2212yes—
31C2×C8×D4C2212yes—
32C2×C8×D4C2212yes—
33C22×#106C2214yes—
34#2868C2211yescharacteristic
35#2857—21no—
36#2732—22no—
37#2552—21no—
38#2547C2212yes—
39#2542C2212yes—
40#2538—22no—
41#2538C2212yes—
42#2537C2212yes—
43#2537C2212yes—
44#2454—22no—
45#2432—22no—
46#2422—22no—
47#2422—22no—
48#2417C2214yes—
49#2366—21no—
50D4×C4⋊C4C2212yes—
51D4×C4⋊C4C2212yes—
52D4×C4⋊C4C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×#127C2211yescharacteristic
56C42×D4C2211yescharacteristic
57#2096C2214yes—
58#2084C2214yes—
59#2077C2214yes—
60#2035C2214yes—
61#2035C2214yes—
62#2035C2214yes—
63#2034C2214yes—
64#2000C2214yes—
65#1990C2212yes—
66#1987C2211yescharacteristic
67#1981C2211yescharacteristic
68#1956C2212yes—
69#1954C2212yes—
70#1954C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2211yescharacteristic
74#1948C2211yescharacteristic
75#1927C2212yes—
76#1920C2212yes—
77#1892C2214yes—
78#1877C2212yes—
79#1872C2214yes—
80#1869C2214yes—
81#1864C2212yes—
82#1853C2214yes—
83#1853C2214yes—
84#1853C2214yes—
85#1844C2212yes—
86#1807C2214yes—
87#1802C2211yescharacteristic
88#1802C2211yescharacteristic
89C4×#106C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93C4×#102C2212yes—
94C22×#41—22no—
95C22×SD16C2312yes—
96C23×C8C2314yes—
97#554—28no—
98#554—44no—
99#547—28no—
100#547—44no—
101#532—48no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#127D411yescharacteristic
108C2×#125—22no—
109C2×#124—22no—
110C2×#123—24no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8C2314yes—
114C2×C22⋊Q8C2314yes—
115C2×#121—22no—
116C2×#121—22no—
117C2×#121C2314yes—
118C2×C4×Q8—21no—
119C2×C4×Q8C2312yes—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2312yes—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2314yes—
132C22×C42—21no—
133#498—28no—
134#483—216no—
135#481—216no—
136#476D418yescomplemented
137#475—28no—
138#475—44no—
139#465—216no—
140#461—24no—
141C4×SD16—28no—
142C8×D4—28no—
143C8×D4—28no—
144C2×#106—216no—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×#106C2312yes—
149C2×#106C2314yes—
150C2×#106C2312yes—
151C2×#106C2312yes—
152C2×C4⋊C8C2311yescharacteristic
153C2×C4⋊C8C2311yescharacteristic
154C2×C4⋊C8C2311yescharacteristic
155C2×C4⋊C8C2311yescharacteristic
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×C22⋊C8C2314yes—
165C2×C22⋊C8C2314yes—
166C2×C22⋊C8C2314yes—
167C2×C4×C8C2312yes—
168C2×C4×C8C2312yes—
169#399—24no—
170#398—24no—
171#397—24no—
172#396—24no—
173#396—24no—
174#394—24no—
175#394—24no—
176#391—22no—
177#391—21no—
178#391C2312yes—
179#390—22no—
180#390—22no—
181#390D412yes—
182#390D412yes—
183#390D412yes—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389C2312yes—
192#386—24no—
193#385—24no—
194#385—24no—
195#385—24no—
196#384—22no—
197#384—21no—
198#384—21no—
199#384D411yescharacteristic
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2312yes—
203#384C2312yes—
204#380—22no—
205#380—22no—
206#380—24no—
207#380C2314yes—
208#380C2314yes—
209#380C2314yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C43C2311yescharacteristic
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2312yes—
231C2×#41—48no—
232C2×#41—48no—
233C2×#41—84no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8C2412yes—
237C22×D4—24no—
238C22×D4C2×D412yes—
239C22×D4C2×D412yes—
240C22×D4C2×D412yes—
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—24no—
244C23×C4—24no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4C2414yes—
248C2×SD16—24no—
249C2×SD16—24no—
250C2×SD16—42no—
251C22×C8—28no—
252C22×C8—28no—
253C22×C8—44no—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8—44no—
264C4⋊Q8—82no—
265C4⋊Q8C2×D414yes—
266#127SD1618yescomplemented
267#125—48no—
268#124—48no—
269#123—416no—
270C22⋊Q8—28no—
271C22⋊Q8—416no—
272C22⋊Q8—44no—
273C22⋊Q8—48no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276#121—28no—
277#121—44no—
278#121—48no—
279#121—48no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—82no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—22no—
290C4×D4—41no—
291C4×D4—41no—
292C4×D4—42no—
293C4×D4—42no—
294C4×D4—48no—
295C4×D4—48no—
296C4×D4—48no—
297C4×D4—84no—
298C4×D4C2×D414yes—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4C2×D414yes—
341C2×C22⋊C4C2×D414yes—
342C2×C22⋊C4C2×D414yes—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—44no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#106—28no—
357#106—24no—
358#106—28no—
359#106—28no—
360#106—42no—
361#106—44no—
362#106C2×D418yes—
363C4⋊C8—22no—
364C4⋊C8—22no—
365C4⋊C8C2×D414yes—
366C4⋊C8#4114yescomplemented
367#103—28no—
368#103—24no—
369#103—24no—
370#103—42no—
371#103—44no—
372#102—24no—
373#102—42no—
374#102C2×D418yes—
375C22⋊C8—28no—
376C22⋊C8—28no—
377C22⋊C8—44no—
378C4×C8—24no—
379C4×C8—24no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—24no—
390C24—21no—
391C24C22×D414yes—
392#41—832no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—81no—
404C2×Q8—81no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—84no—
419C2×D4C2×SD1618yes—
420C2×D4C2×SD1618yes—
421C2×D4C2×SD1618yes—
422C2×D4C22×D412yes—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—48no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—48no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—48no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D414yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2511yescharacteristic
460SD16—48no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—416no—
472C2×C8C22×D414yes—
473C2×C8C2×#4114yes—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—42no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4—82no—
504C4⋊C4—84no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4C22×D414yes—
508C4⋊C4C22×D412yes—
509C22⋊C4—24no—
510C22⋊C4—28no—
511C22⋊C4—28no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—42no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—42no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C42—22no—
522C42—22no—
523C42—21no—
524C42—21no—
525C42—41no—
526C42—44no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42—44no—
531C42C2×SD1614yes—
532C42C22×D412yes—
533C42C2×#4112yes—
534C42H(5,𝔽2)12yes—
535C422−1+412yes—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C22×SD1618yes—
550C23C23×D411yescharacteristic
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—84no—
556Q8—88no—
557D4—216no—
558D4—22no—
559D4—22no—
560D4—216no—
561D4—48no—
562D4—41no—
563D4—41no—
564D4—816no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—84no—
607C2×C4—816no—
608C2×C4—84no—
609C2×C4—168no—
610C2×C4D4212yes—
611C2×C4D4212yes—
612C2×C4#54712yes—
613C2×C4#54712yes—
614C2×C4C22×SD1612yes—
615C2×C4C22×SD1612yes—
616C2×C4C22×SD1612yes—
617C2×C4#57712yes—
618C2×C4#57712yes—
619C2×C4C23×D411yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C2×C4C2×2−1+411yescharacteristic
623C2×C4#58511yescharacteristic
624C2×C4#58511yescharacteristic
625C8—24no—
626C8—24no—
627C8—42no—
628C8—42no—
629C22—28no—
630C22—28no—
631C22—28no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22C2×D4211yescharacteristic
644C22C2×#54711yescharacteristic
645C22#348511yescharacteristic
646C22C23×SD1611yescharacteristic
647C22C2×#57711yescharacteristic
648C22C2×#57711yescharacteristic
649C22#358611yescharacteristic
650C4—22no—
651C4—22no—
652C4—22no—
653C4—41no—
654C4—42no—
655C4—44no—
656C4—44no—
657C4—81no—
658C4—82no—
659C4—84no—
660C4—82no—
661C4—82no—
662C4—81no—
663C4D4×SD1614yes—
664C4#333514yescomplemented
665C4D4×#4111yescharacteristic
666C4#349711yescharacteristic
667C4#351711yescharacteristic
668C4#355411yescharacteristic
669C2—48no—
670C2—82no—
671C2#5906611yescharacteristic
672C2C2×D4×SD1611yescharacteristic
673C2#6109411yescharacteristic
674C2C2×#333511yescharacteristic
675C2#6128111yescharacteristic
676C2#6169511yescharacteristic
677C2#6179211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3335C2semidirect
2. #59823C2semidirect
4. #30559C2semidirect
5. #28687C2semidirect
6. #28512C2semidirect
7. #26234C2semidirect
8. #26128C2semidirect
9. #25702C2semidirect
10. #25385C2semidirect
11. #25341C2semidirect
12. #25012C2semidirect
13. #24120C2semidirect
14. #23173C2semidirect
15. D4×#106C2semidirect
16. C4×#476C2semidirect
17. C4×#461C2semidirect
136. #476D4semidirect
266. #127SD16semidirect
366. C4⋊C8#41semidirect
664. C4#3335semidirect

Maximal subgroups

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

subgroupindex
C2×#33352
#598232
#361272
#305592
#286872
#285122
#262342
#261282
#257022
#253852
#253412
#250122
#241202
#231732
D4×#1062
C4×#4762
C4×#4612