Finite Group Databasebeta

← #6454259

Subgroups — #6454259

Order 512.

2106 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3106C211yesmaximal, characteristic
2C2×C4×#137C211yesmaximal, characteristic, complemented
3#35249C211yesmaximal, characteristic, complemented
4#34193C211yesmaximal, characteristic, complemented
5#24817C212yesmaximal, complemented
6#24733C212yesmaximal, complemented
7#24680C212yesmaximal, complemented
8#24631C212yesmaximal, complemented
9#24592C212yesmaximal, complemented
10#24542C212yesmaximal, complemented
11#24497C212yesmaximal, complemented
12#24375C212yesmaximal, complemented
13#23692C214yesmaximal, complemented
14#21511C214yesmaximal, complemented
15#21410C211yesmaximal, characteristic, complemented
16#21354C211yesmaximal, characteristic, complemented
17#20734C211yesmaximal, characteristic, complemented
18C22×#137C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22#3106—28no—
23C2×#482C2212yes—
24C2×#477C2211yescharacteristic
25C2×#476C2211yescharacteristic
26C2×#464C2214yes—
27C2×#460C2211yescharacteristic
28C2×#459C2211yescharacteristic
29C2×#449C2212yes—
30C4×#137—28no—
31C2×#441C2212yes—
32C2×#440C2212yes—
33C2×C4×Q16C2212yes—
34C2×C4×SD16C2212yes—
35C2×C4×SD16C2212yes—
36C2×C4×D8C2212yes—
37C2×#431C2211yescharacteristic
38C2×#428C2211yescharacteristic
39C2×#417C2211yescharacteristic
40C2×#419C2211yescharacteristic
41C2×C4×M16C2211yescharacteristic
42#2854—22no—
43#2632—22no—
44#2580—21no—
45#2580—21no—
46#2567C2212yes—
47#2548C2212yes—
48#2501C2212yes—
49#2436C2212yes—
50#2406—22no—
51#2354—22no—
52#2354C2212yes—
53Q8×C4⋊C4C2212yes—
54#2347—22no—
55#2347C2212yes—
56#2341C2212yes—
57#2326C2214yes—
58#2319—22no—
59#2319C2212yes—
60#2285—21no—
61#2285C2211yescharacteristic
62#2283—21no—
63#2283C2211yescharacteristic
64#2276C2212yes—
65#2083C2214yes—
66#2071C2214yes—
67#2068C2214yes—
68#2067C2214yes—
69#2055C2214yes—
70#2054C2214yes—
71#2042C2214yes—
72#2012C2214yes—
73#1990C2212yes—
74#1983C2214yes—
75#1957C2212yes—
76#1948C2214yes—
77#1939C2212yes—
78#1939C2212yes—
79#1927C2214yes—
80#1926C2214yes—
81#1895C2212yes—
82#1895C2212yes—
83#1880C2214yes—
84#1878C2214yes—
85#1851C2212yes—
86#1851C2212yes—
87#1824C2212yes—
88#1803C2214yes—
89#1803C2212yes—
90#1803C2212yes—
91#1787C2212yes—
92#1768C2214yes—
93#1762C2214yes—
94C4×#103C2214yes—
95C22×#41—21no—
96C22×#41C2311yescharacteristic
97C23×Q8C2311yescharacteristic
98C2×#137—24no—
99C2×#137—28no—
100C22×Q16C2312yes—
101C22×SD16C2312yes—
102C22×SD16C2312yes—
103C22×D8C2312yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106#532—24no—
107#532—24no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×#127C2311yescharacteristic
110C2×#124—21no—
111C2×#124C2314yes—
112C2×#123—24no—
113C2×#121—21no—
114C2×#121C2312yes—
115C4×#41—24no—
116C4×#41D418yes—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—24no—
120C2×C4×Q8—24no—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2314yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4C2312yes—
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2311yescharacteristic
132C2×#117—21no—
133C2×#117—22no—
134C2×#117—21no—
135C2×#117C2311yescharacteristic
136C22×C4⋊C4—21no—
137C22×C4⋊C4C2311yescharacteristic
138C22×C42—21no—
139C22×C42—21no—
140C22×C42C2311yescharacteristic
141#482—28no—
142#477—24no—
143#476—24no—
144#464—216no—
145#460—24no—
146#459—24no—
147#449—28no—
148#441—28no—
149#440—28no—
150C4×Q16—28no—
151C4×SD16—28no—
152C4×SD16—28no—
153C4×D8—28no—
154#431—24no—
155#428—24no—
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#106C2312yes—
159C2×#106C2312yes—
160C2×C4⋊C8C2311yescharacteristic
161C2×C4⋊C8C2314yes—
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2314yes—
164#419—24no—
165#417—24no—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2312yes—
171C2×#103C2314yes—
172C2×#102C2312yes—
173C2×#102C2312yes—
174C2×#102C2312yes—
175C2×#102C2312yes—
176C2×#102C2314yes—
177C2×#102C2314yes—
178C2×C22⋊C8C2312yes—
179C4×M16—24no—
180C2×#97C2312yes—
181C2×C4×C8C2312yes—
182C2×C4×C8C2312yes—
183#399—22no—
184#399—22no—
185#395—24no—
186#394—24no—
187#393—22no—
188#393—22no—
189#391C2314yes—
190#390—22no—
191#390—22no—
192#390—24no—
193#390C2314yes—
194#390C2314yes—
195#389—22no—
196#389—22no—
197#389—24no—
198#389C2314yes—
199#388—24no—
200#388—24no—
201#388—24no—
202#388—24no—
203#388—24no—
204#386—24no—
205#385—24no—
206#385—24no—
207#384—22no—
208#384C2314yes—
209#384C2312yes—
210#383—22no—
211#383D412yes—
212#382—21no—
213#382D411yescharacteristic
214#382D411yescharacteristic
215#381—22no—
216#381D412yes—
217#380—21no—
218#380—22no—
219#380—22no—
220#380—22no—
221#380D412yes—
222#380C2312yes—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—24no—
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2314yes—
230C4×C4⋊C4C2312yes—
231C4×C22⋊C4—22no—
232C4×C22⋊C4C2312yes—
233#376—22no—
234#376C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#336C2314yes—
240#336C2314yes—
241C2×#41—22no—
242C2×#41—24no—
243C2×#41—42no—
244C2×#41—44no—
245C2×#41—84no—
246C2×#41C2×D414yes—
247C22×Q8—21no—
248C22×Q8—21no—
249C22×Q8—24no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—42no—
253C22×Q8C2412yes—
254C22×Q8C2411yescharacteristic
255C22×D4—21no—
256C22×D4—22no—
257C22×D4—21no—
258C22×D4—21no—
259C22×D4C2412yes—
260C22×D4C2411yescharacteristic
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—22no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2411yescharacteristic
270#137—232no—
271C2×Q16—24no—
272C2×Q16—24no—
273C2×Q16#4118yescomplemented
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—28no—
278C2×SD16—24no—
279C2×D8—24no—
280C2×D8—44no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—24no—
284C2×M16—42no—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291#127—22no—
292#127—22no—
293#124—28no—
294#124—44no—
295#124—44no—
296#123—88no—
297#121—24no—
298#121—24no—
299#121—44no—
300C4×Q8—22no—
301C4×Q8—22no—
302C4×Q8—28no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×Q8—44no—
310C4×Q8C2×D414yes—
311C4×Q8C2×D414yes—
312C4×D4—28no—
313C4×D4—28no—
314C4×D4—24no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—24no—
318C4×D4—44no—
319C4×D4C2×D414yes—
320C4×D4C2×D414yes—
321#117—22no—
322#117—28no—
323#117—84no—
324#117—82no—
325#117C2×D414yes—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—44no—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2414yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4C2×D414yes—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—28no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—24no—
368C2×C42—24no—
369C2×C42—22no—
370C2×C42—44no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D414yes—
374C2×C42#4114yes—
375C2×C42C2414yes—
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378#107—24no—
379#107—24no—
380#106—24no—
381#106—24no—
382C4⋊C8—28no—
383C4⋊C8—22no—
384C4⋊C8—22no—
385C4⋊C8—44no—
386#103—24no—
387#103—24no—
388#103—28no—
389#103—24no—
390#103—24no—
391#103—44no—
392#102—24no—
393#102—24no—
394#102—28no—
395#102—24no—
396#102—24no—
397#102—44no—
398C22⋊C8—24no—
399#97—24no—
400C4×C8—24no—
401C4×C8—24no—
402#95—22no—
403#95—24no—
404#95—22no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95C2×D412yes—
409#95C2×D412yes—
410#95C2×D414yes—
411#95C2×D412yes—
412#95C2×D412yes—
413#95C2×D414yes—
414C24—21no—
415C24—22no—
416C24C22×D411yescharacteristic
417#41—28no—
418#41—44no—
419#41—48no—
420#41—816no—
421C2×Q8—21no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—28no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—44no—
430C2×Q8—48no—
431C2×Q8—44no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—82no—
435C2×Q8C22×D412yes—
436C2×Q8C2×#4114yes—
437C2×Q8C2×#4114yes—
438C2×D4—24no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—24no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—24no—
446C2×D4—44no—
447C2×D4—48no—
448C2×D4—44no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—82no—
454C2×D4C22×D412yes—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—82no—
479C22×C4—84no—
480C22×C4#12114yes—
481C22×C4#12114yes—
482C22×C4C22×D414yes—
483C22×C4C22×D414yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C2×#4112yes—
491C22×C4C2511yescharacteristic
492Q16—216no—
493SD16—216no—
494SD16—48no—
495D8—48no—
496M16—28no—
497M16—44no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8—22no—
507C2×C8C2×#4114yes—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—48no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—82no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4C22×D412yes—
531C4⋊C4C22×D412yes—
532C22⋊C4—24no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—44no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—84no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C22⋊C4—82no—
545C42—24no—
546C42—24no—
547C42—24no—
548C42—24no—
549C42—24no—
550C42—48no—
551C42—44no—
552C42—44no—
553C42—82no—
554C42—84no—
555C42#12114yes—
556C42#12114yes—
557C42C22×D412yes—
558C42C22×D412yes—
559C42C2×#4112yes—
560C42C2×#4112yes—
561C23—22no—
562C23—22no—
563C23—21no—
564C23—21no—
565C23—22no—
566C23—42no—
567C23—42no—
568C23—41no—
569C23—41no—
570C23—44no—
571C23—42no—
572C23—44no—
573C23—42no—
574C23—42no—
575C23C2×#12112yes—
576C23C23×D411yescharacteristic
577Q8—22no—
578Q8—24no—
579Q8—24no—
580Q8—24no—
581Q8—48no—
582Q8—42no—
583Q8—42no—
584Q8—84no—
585Q8#52514yes—
586D4—22no—
587D4—22no—
588D4—48no—
589D4—42no—
590D4—42no—
591D4—42no—
592D4—44no—
593D4—84no—
594D4—88no—
595D4—84no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—44no—
617C2×C4—48no—
618C2×C4—48no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—84no—
634C2×C4—88no—
635C2×C4—84no—
636C2×C4C2×#12112yes—
637C2×C4C2×#12112yes—
638C2×C4C2×#12112yes—
639C2×C4C2×#12112yes—
640C2×C4C2×#12112yes—
641C2×C4#57712yes—
642C2×C4#57812yes—
643C2×C4C23×D411yescharacteristic
644C2×C4C22×#4112yes—
645C2×C4C22×#4111yescharacteristic
646C2×C4#58512yes—
647C8—24no—
648C8—42no—
649C8—42no—
650C8—42no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—84no—
664C22—84no—
665C22—82no—
666C22C22×#12111yescharacteristic
667C22D4×#4112yes—
668C22C2×#57711yescharacteristic
669C22C2×#57811yescharacteristic
670C22#358512yes—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—24no—
676C4—42no—
677C4—42no—
678C4—44no—
679C4—44no—
680C4—44no—
681C4—82no—
682C4—82no—
683C4—84no—
684C4#304614yes—
685C4C2×#52511yescharacteristic
686C4#344911yescharacteristic
687C4#347512yes—
688C2—42no—
689C2—82no—
690C2—162no—
691C2#3473711yescharacteristic
692C2C2×#304611yescharacteristic
693C2#6042611yescharacteristic
694C2#6154612yes—
695C2#6155812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3106C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×#137C2semidirect
3. #35249C2semidirect
4. #34193C2semidirect
5. #24817C2semidirect
6. #24733C2semidirect
7. #24680C2semidirect
8. #24631C2semidirect
9. #24592C2semidirect
10. #24542C2semidirect
11. #24497C2semidirect
12. #24375C2semidirect
13. #23692C2semidirect
14. #21511C2semidirect
15. #21410C2semidirect
16. #21354C2semidirect
17. #20734C2semidirect
273. C2×Q16#41semidirect

Maximal subgroups

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

subgroupindex
C2×#31062
C2×C4×#1372
#352492
#341932
#248172
#247332
#246802
#246312
#245922
#245422
#244972
#243752
#236922
#215112
#214102
#213542
#207342