Finite Group Databasebeta

← #6514126

Subgroups — #6514126

Order 512.

1972 conjugacy classes of subgroups.

The subgroup lattice has 769 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3246C211yesmaximal, characteristic
2C2×#3104C211yesmaximal, characteristic, complemented
3#34955C211yesmaximal, characteristic
4#34214C211yesmaximal, characteristic, complemented
5#28282C212yesmaximal, complemented
6#28254C212yesmaximal, complemented
7#28204C212yesmaximal, complemented
8#28159C212yesmaximal, complemented
9#25210C214yesmaximal, complemented
10#25086C214yesmaximal, complemented
11#24818C212yesmaximal, complemented
12#24733C212yesmaximal, complemented
13#24683C212yesmaximal, complemented
14#24633C212yesmaximal, complemented
15#21972C211yesmaximal, characteristic, complemented
16#21859C211yesmaximal, characteristic
17C2×#1803C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3246—28no—
22#3104—28no—
23C2×#484C2212yes—
24C2×#477C2211yescharacteristic
25C2×#477C2211yescharacteristic
26C2×#475C2212yes—
27C2×#476C2212yes—
28C2×#465C2214yes—
29C2×#464C2214yes—
30C2×#462C2211yescharacteristic
31C2×#462C2211yescharacteristic
32C2×#460C2212yes—
33C2×#461C2212yes—
34C2×#451C2212yes—
35C2×C4×Q16C2212yes—
36C2×C4×Q16C2212yes—
37C2×#431C2211yescharacteristic
38C22×C4⋊C8C2211yescharacteristic
39C22×#103C2212yes—
40#2592—21no—
41#2592—21no—
42#2591—22no—
43#2568C2211yescharacteristic
44#2568C2211yescharacteristic
45#2548C2211yescharacteristic
46#2548C2211yescharacteristic
47#2504C2212yes—
48#2436C2212yes—
49#2398—22no—
50#2360—22no—
51#2360C2214yes—
52#2355—21no—
53#2355C2212yes—
54Q8×C4⋊C4—21no—
55Q8×C4⋊C4C2212yes—
56#2330—22no—
57#2317—21no—
58C4×C4⋊Q8C2212yes—
59C4×C4⋊Q8C2212yes—
60C4×#124C2212yes—
61C4×#124C2212yes—
62#2297—21no—
63#2297—21no—
64C2×#384C2211yescharacteristic
65#2274—21no—
66C4×#117C2211yescharacteristic
67#2085C2214yes—
68#2074C2214yes—
69#2067C2214yes—
70#2066C2214yes—
71#2056C2214yes—
72#2055C2214yes—
73#2042C2214yes—
74#2015C2214yes—
75#1990C2212yes—
76#1983C2214yes—
77#1957C2212yes—
78#1949C2214yes—
79#1939C2212yes—
80#1939C2212yes—
81#1927C2214yes—
82#1925C2214yes—
83#1880C2214yes—
84#1878C2214yes—
85#1874C2214yes—
86#1850C2214yes—
87#1842C2212yes—
88#1831C2212yes—
89#1805C2214yes—
90#1803C2212yes—
91#1803C2212yes—
92#1803C2212yes—
93#1803C2212yes—
94#1802C2212yes—
95#1802C2212yes—
96C2×#340C2212yes—
97C22×#41—21no—
98C23×Q8C2311yescharacteristic
99C22×Q16C2312yes—
100C22×Q16C2312yes—
101C22×SD16C2314yes—
102C23×C8C2312yes—
103#533—24no—
104#533D418yes—
105#532—24no—
106#532—24no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×#127C2311yescharacteristic
111C2×#125—21no—
112C2×#124—21no—
113C2×#124C2312yes—
114C2×#124C2312yes—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8C2312yes—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—22no—
123C2×C4×Q8—24no—
124C2×C4×Q8—24no—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2311yescharacteristic
127C2×C4×Q8C2311yescharacteristic
128C2×C4×Q8C2311yescharacteristic
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2314yes—
131C2×C4×D4—21no—
132C2×C4×D4C2312yes—
133C2×#117—21no—
134C2×#117D411yescharacteristic
135C2×#117D411yescharacteristic
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2312yes—
141C22×C42—21no—
142C22×C42C2311yescharacteristic
143#484—28no—
144#477—24no—
145#477—24no—
146#476—28no—
147#475—28no—
148#465—216no—
149#464—48no—
150#462—24no—
151#462—24no—
152#461—28no—
153#460—28no—
154#451—28no—
155C4×Q16—28no—
156C4×Q16—28no—
157#431—24no—
158C2×#107C2312yes—
159C2×#107C2312yes—
160C2×#106C2314yes—
161C2×C4⋊C8—24no—
162C2×C4⋊C8C2312yes—
163C2×C4⋊C8C2311yescharacteristic
164C2×C4⋊C8C2311yescharacteristic
165C2×C4⋊C8C2312yes—
166C2×C4⋊C8C2314yes—
167C2×C4⋊C8C2314yes—
168C2×#103—44no—
169C2×#103C2314yes—
170C2×#103C2312yes—
171C2×#103C2312yes—
172C2×#103C2312yes—
173C2×#103C2314yes—
174C2×#103C2312yes—
175C2×#103C2314yes—
176C2×#102C2314yes—
177C2×#102C2314yes—
178C2×#102C2314yes—
179C2×C22⋊C8C2312yes—
180C2×C4×C8C2312yes—
181#400—22no—
182#400—22no—
183#397—22no—
184#397—22no—
185#394—22no—
186#394—22no—
187#393—22no—
188#393—22no—
189#391—21no—
190#391—21no—
191#391—21no—
192#391—21no—
193#391—22no—
194#391C2314yes—
195#391C2312yes—
196#391C2312yes—
197#390—21no—
198#390—21no—
199#390—21no—
200#390—21no—
201#390C2312yes—
202#390C2312yes—
203#389—22no—
204#389—22no—
205#389—22no—
206#389—24no—
207#389—24no—
208#389C2314yes—
209#388—22no—
210#388—22no—
211#388—22no—
212#388—22no—
213#387—24no—
214#387—24no—
215#386—22no—
216#386—22no—
217#386—22no—
218#386—22no—
219#385—24no—
220#384—22no—
221#384—21no—
222#384—21no—
223#384C2312yes—
224#384C2312yes—
225#382—22no—
226#382C2312yes—
227#382C2312yes—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4D411yescharacteristic
234C4×C4⋊C4D411yescharacteristic
235C4×C4⋊C4C2314yes—
236C4×C4⋊C4C2311yescharacteristic
237C4×C4⋊C4C2311yescharacteristic
238C4×C22⋊C4—24no—
239C4×C22⋊C4—22no—
240C4×C22⋊C4—22no—
241C4×C22⋊C4—22no—
242C4×C22⋊C4—22no—
243C4×C22⋊C4D412yes—
244C4×C22⋊C4C2312yes—
245#376D412yes—
246C43C2312yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#336C2314yes—
252#336C2314yes—
253C2×#41—44no—
254C2×#41—44no—
255C2×#41—44no—
256C22×Q8—21no—
257C22×Q8—21no—
258C22×Q8—21no—
259C22×Q8—22no—
260C22×Q8—22no—
261C22×Q8—22no—
262C22×Q8—22no—
263C22×Q8—41no—
264C22×Q8—41no—
265C22×Q8C2412yes—
266C22×Q8C2411yescharacteristic
267C22×Q8C2411yescharacteristic
268C22×D4—22no—
269C22×D4—21no—
270C22×D4—21no—
271C22×D4C2412yes—
272C23×C4—21no—
273C23×C4—22no—
274C23×C4—21no—
275C23×C4—22no—
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2411yescharacteristic
279C2×Q16—24no—
280C2×Q16—24no—
281C2×Q16—24no—
282C2×Q16—24no—
283C2×Q16—42no—
284C2×Q16—42no—
285C2×SD16—28no—
286C2×SD16—44no—
287C2×SD16—44no—
288C22×C8—24no—
289C22×C8—42no—
290C22×C8—42no—
291C22×C8C2414yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8C2×D414yes—
299C4⋊Q8C2×D414yes—
300#127—22no—
301#127—22no—
302#125—28no—
303#124—24no—
304#124—24no—
305#124—42no—
306#124—42no—
307#124—44no—
308C22⋊Q8—28no—
309C22⋊Q8—24no—
310C22⋊Q8C2×D418yes—
311#121—24no—
312#121—24no—
313#121—44no—
314C4×Q8—28no—
315C4×Q8—24no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×Q8—22no—
323C4×Q8—22no—
324C4×Q8—44no—
325C4×Q8—44no—
326C4×Q8—44no—
327C4×Q8—44no—
328C4×Q8—84no—
329C4×Q8C2×D418yes—
330C4×D4—24no—
331C4×D4—24no—
332C4×D4—44no—
333#117—28no—
334#117—24no—
335#117—24no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—42no—
357C2×C4⋊C4—42no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4—44no—
360C2×C4⋊C4—82no—
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2411yescharacteristic
366C2×C4⋊C4C2411yescharacteristic
367C2×C4⋊C4C2414yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4C2×D412yes—
382C2×C22⋊C4C2×D412yes—
383C2×C42—22no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—24no—
389C2×C42—24no—
390C2×C42—44no—
391C2×C42C2×D412yes—
392C2×C42C2×D412yes—
393C2×C42C2×D412yes—
394C2×C42C2×D412yes—
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D414yes—
398C2×C42#4114yes—
399C2×C42C2412yes—
400C2×C42C2412yes—
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403#107—24no—
404#107—24no—
405#106—28no—
406C4⋊C8—24no—
407C4⋊C8—24no—
408C4⋊C8—22no—
409C4⋊C8—22no—
410C4⋊C8—44no—
411C4⋊C8—44no—
412#103—24no—
413#103—24no—
414#103—28no—
415#103—28no—
416#103—24no—
417#103—24no—
418#103—44no—
419#102—28no—
420#102—44no—
421#102—44no—
422C22⋊C8—24no—
423C4×C8—24no—
424#95—24no—
425#95—24no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—24no—
435#95—24no—
436#95C2×D412yes—
437#95C2×D412yes—
438#95C2×D412yes—
439#95C2×D412yes—
440C24—22no—
441C24C22×D411yescharacteristic
442#41—816no—
443C2×Q8—24no—
444C2×Q8—22no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—28no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—21no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—44no—
463C2×Q8—44no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—44no—
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8C22×D414yes—
472C2×Q8H(5,𝔽2)12yes—
473C2×Q82−1+412yes—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—44no—
481C2×D4—44no—
482C2×D4—44no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—81no—
488C2×D4—81no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—24no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4—44no—
510C22×C4—44no—
511C22×C4—81no—
512C22×C4—81no—
513C22×C4—82no—
514C22×C4—82no—
515C22×C4#12114yes—
516C22×C4#12114yes—
517C22×C4#13514yes—
518C22×C4#13714yes—
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D411yescharacteristic
526C22×C4C22×D412yes—
527C22×C4C22×D411yescharacteristic
528C22×C4C22×D411yescharacteristic
529C22×C4C2×#4112yes—
530C22×C4C2511yescharacteristic
531Q16—48no—
532Q16—48no—
533SD16—88no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—24no—
537C2×C8—22no—
538C2×C8—24no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—24no—
543C2×C8—84no—
544C4⋊C4—24no—
545C4⋊C4—24no—
546C4⋊C4—24no—
547C4⋊C4—28no—
548C4⋊C4—28no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—22no—
560C4⋊C4—24no—
561C4⋊C4—24no—
562C4⋊C4—44no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—42no—
566C4⋊C4—42no—
567C4⋊C4—42no—
568C4⋊C4—42no—
569C4⋊C4—44no—
570C4⋊C4—44no—
571C4⋊C4—82no—
572C4⋊C4—82no—
573C4⋊C4—84no—
574C4⋊C4—84no—
575C4⋊C4C22×D414yes—
576C22⋊C4—28no—
577C22⋊C4—24no—
578C22⋊C4—24no—
579C22⋊C4—44no—
580C22⋊C4—44no—
581C22⋊C4—44no—
582C22⋊C4—44no—
583C22⋊C4—44no—
584C22⋊C4—44no—
585C22⋊C4—82no—
586C22⋊C4—82no—
587C22⋊C4—82no—
588C22⋊C4—82no—
589C42—28no—
590C42—24no—
591C42—24no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—42no—
597C42—42no—
598C42—42no—
599C42—42no—
600C42—42no—
601C42—42no—
602C42—84no—
603C42—84no—
604C42#12114yes—
605C42#12114yes—
606C42C22×D412yes—
607C42C22×D412yes—
608C42C2×#4112yes—
609C42C2×#4112yes—
610C23—21no—
611C23—21no—
612C23—21no—
613C23—21no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23C2×#12112yes—
622C23C2×#13512yes—
623C23C2×#13712yes—
624C23C23×D411yescharacteristic
625Q8—24no—
626Q8—22no—
627Q8—22no—
628Q8—24no—
629Q8—22no—
630Q8—22no—
631Q8—48no—
632Q8—48no—
633Q8—41no—
634Q8—41no—
635Q8—42no—
636Q8—42no—
637Q8—41no—
638Q8—41no—
639Q8—84no—
640D4—42no—
641D4—42no—
642D4—42no—
643D4—42no—
644D4—84no—
645D4—88no—
646D4—84no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—24no—
662C2×C4—24no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—22no—
673C2×C4—44no—
674C2×C4—48no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—42no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—42no—
683C2×C4—44no—
684C2×C4—41no—
685C2×C4—41no—
686C2×C4—41no—
687C2×C4—41no—
688C2×C4—41no—
689C2×C4—41no—
690C2×C4—42no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—84no—
696C2×C4—88no—
697C2×C4—164no—
698C2×C4#464116yescomplemented
699C2×C4C2×#12112yes—
700C2×C4C2×#12112yes—
701C2×C4C2×#12112yes—
702C2×C4C2×#12112yes—
703C2×C4C2×#12112yes—
704C2×C4C2×#13512yes—
705C2×C4C2×#13512yes—
706C2×C4C2×#13712yes—
707C2×C4C2×#13712yes—
708C2×C4C23×D411yescharacteristic
709C2×C4C22×#4111yescharacteristic
710C2×C4C2×H(5,𝔽2)11yescharacteristic
711C2×C4C2×2−1+411yescharacteristic
712C2×C4#58511yescharacteristic
713C2×C4#58511yescharacteristic
714C8—44no—
715C8—42no—
716C8—42no—
717C22—22no—
718C22—22no—
719C22—22no—
720C22—22no—
721C22—22no—
722C22—22no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22C2×#46414yes—
731C22C22×#12111yescharacteristic
732C22#348811yescharacteristic
733C22#348911yescharacteristic
734C22C22×#13511yescharacteristic
735C22C22×#13711yescharacteristic
736C22#358511yescharacteristic
737C22#358611yescharacteristic
738C4—22no—
739C4—22no—
740C4—22no—
741C4—42no—
742C4—44no—
743C4—42no—
744C4—42no—
745C4—42no—
746C4—42no—
747C4—42no—
748C4—44no—
749C4—84no—
750C4—81no—
751C4—81no—
752C4—82no—
753C4C2×#46414yes—
754C4C2×#46414yes—
755C4#348811yescharacteristic
756C4#348911yescharacteristic
757C4#349211yescharacteristic
758C4#349311yescharacteristic
759C2—24no—
760C2—162no—
761C2#3551811yescharacteristic
762C2C22×#46411yescharacteristic
763C2#6042511yescharacteristic
764C2#6195311yescharacteristic
765C2#6196411yescharacteristic
766C2#6210111yescharacteristic
767C2#6211211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3104C2semidirect
4. #34214C2semidirect
5. #28282C2semidirect
6. #28254C2semidirect
7. #28204C2semidirect
8. #28159C2semidirect
9. #25210C2semidirect
10. #25086C2semidirect
11. #24818C2semidirect
12. #24733C2semidirect
13. #24683C2semidirect
14. #24633C2semidirect
15. #21972C2semidirect
17. C2×#1803C2semidirect
698. C2×C4#464semidirect

Maximal subgroups

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

subgroupindex
C2×#32462
C2×#31042
#349552
#342142
#282822
#282542
#282042
#281592
#252102
#250862
#248182
#247332
#246832
#246332
#219722
#218592
C2×#18032