Finite Group Databasebeta

← #10576528

Subgroups — #10576528

Order 512.

6500 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62855C211yesmaximal, characteristic, complemented
2#62843C211yesmaximal, characteristic, complemented
3#61343C2116yesmaximal, complemented
4#61339C2116yesmaximal, complemented
5#61332C218yesmaximal, complemented
6#61325C218yesmaximal, complemented
7#60619C212yesmaximal, complemented
8#60612C211yesmaximal, characteristic, complemented
9#60490C211yesmaximal, characteristic, complemented
10#60489C212yesmaximal, complemented
11#60423C211yesmaximal, characteristic, complemented
12C2×#3041C212yesmaximal, complemented
13#60193C211yesmaximal, characteristic, complemented
14#60168C212yesmaximal, complemented
15C2×#2948C211yesmaximal, characteristic, complemented
16C22×AGL(1,ℤ/8)C2212yes—
17C23×D8C2211yescharacteristic
18#3510C2218yescomplemented
19#3508C2218yescomplemented
20#3506C2218yescomplemented
21#3501—24no—
22#3501—28no—
23#3501C2218yescomplemented
24#3501C22116yescomplemented
25#3494—28no—
26#3490—24no—
27#3460—24no—
28#3458C2212yes—
29#3458C2211yescharacteristic
30#3458C2211yescharacteristic
31C2×#543—24no—
32#3449C2211yescharacteristic
33C2×#535C2212yes—
34#3428C2212yes—
35C2×#518C2212yes—
36C2×#518C2211yescharacteristic
37C2×#518C2211yescharacteristic
38#3376C22132yescomplemented
39#3373C22132yescomplemented
40#3370C22132yescomplemented
41#3370C22132yescomplemented
42#3367C22116yescomplemented
43#3366C22116yescomplemented
44#3366C22132yescomplemented
45#3361C22132yescomplemented
46#3360C22116yescomplemented
47#3358C22116yescomplemented
48#3140C2218yescomplemented
49#3138C2218yescomplemented
50#3137C2218yescomplemented
51#3136C22116yescomplemented
52#3136C2218yescomplemented
53#3136C22116yescomplemented
54#3135C2214yescomplemented
55#3131C2214yescomplemented
56#3084C2218yescomplemented
57#3083C2218yescomplemented
58#3082C2218yescomplemented
59#3081C2218yescomplemented
60#3081C22116yescomplemented
61#3077C2218yescomplemented
62#3076C22116yescomplemented
63#3042C2218yescomplemented
64#3041C22116yes—
65#3041C2218yescomplemented
66#3041C2218yescomplemented
67#3041C2218yescomplemented
68#3036C2218yescomplemented
69C2×#463C2218yescomplemented
70C2×#459C2218yescomplemented
71#2962C22116yescomplemented
72#2961C2214yescomplemented
73#2960C2214yescomplemented
74#2949C2218yescomplemented
75#2948C22116yescomplemented
76#2948C2218yescomplemented
77#2946C22116yescomplemented
78C4×AGL(1,ℤ/8)C22116yescomplemented
79C2×#442C2214yescomplemented
80C2×C4×D8C2214yescomplemented
81#2911C2212yes—
82C2×#427C2211yescharacteristic
83#2905C2211yescharacteristic
84C2×#424C2212yes—
85#2896C2212yes—
86#2896C2212yes—
87C2×#418C2212yes—
88C2×#418C2211yescharacteristic
89C2×#418C2211yescharacteristic
90C2×#416C2212yes—
91#2873C2212yes—
92C2×#405C2211yescharacteristic
93C22×#41C2312yes—
94C23×D4C2311yescharacteristic
95C23×D4C2311yescharacteristic
96C2×AGL(1,ℤ/8)—216no—
97C2×AGL(1,ℤ/8)C2318yes—
98C2×AGL(1,ℤ/8)C2318yes—
99C22×SD16C2314yes—
100C22×D8—28no—
101C22×D8C2312yes—
102C22×D8C2312yes—
103C22×D8C2314yes—
104C22×D8C2318yes—
105C22×M16C2311yescharacteristic
106C22×M16C2312yes—
107C23×C8C2311yescharacteristic
108#559C23132yescomplemented
109#556C23116yescomplemented
110#555—28no—
111#554—216no—
112#554—28no—
113#554—216no—
114#554—216no—
115#554—216no—
116#554C23116yescomplemented
117#554C23132yescomplemented
118#553—216no—
119#551—232no—
120#550C23116yescomplemented
121D4×Q8—216no—
122D4×Q8—28no—
123D4×Q8C23116yescomplemented
124#547—28no—
125#546—232no—
126#544—22no—
127#544—24no—
128#544D414yes—
129#544D418yes—
130#544C2318yes—
131#544C2318yes—
132#544C2318yes—
133#543—216no—
134#543—232no—
135#543—22no—
136#543—24no—
137#543D414yes—
138#542—216no—
139#540—216no—
140#538—216no—
141#537—22no—
142#536C2314yes—
143#535—24no—
144#535—24no—
145#535C2318yes—
146#535C2318yes—
147#535C2314yes—
148#534—22no—
149#533—24no—
150#533—24no—
151#533—24no—
152#533C2318yes—
153#533C2314yes—
154#533C2314yes—
155#533C2318yes—
156#532—216no—
157#532C2314yes—
158C2×#127C2314yes—
159#529—216no—
160C2×#124C2314yes—
161#525—216no—
162#525—24no—
163C2×C22⋊Q8—216no—
164C2×C22⋊Q8—232no—
165C2×#121—24no—
166C2×#120—24no—
167#520—24no—
168#520C2318yes—
169#519C2314yes—
170#518—24no—
171#518—24no—
172#518—24no—
173#518—24no—
174#518—24no—
175#518C2318yes—
176#518C2314yes—
177#518C2318yes—
178#518C2318yes—
179#518C2318yes—
180C4×#41—24no—
181C4×#41C2318yes—
182C2×C4×D4—22no—
183C2×C4×D4—22no—
184C2×C4×D4—24no—
185C2×C4×D4C2314yes—
186C2×C4×D4C2314yes—
187C2×C4×D4C2318yes—
188C2×#117—24no—
189C2×#117C2312yes—
190C2×#117C2311yescharacteristic
191C2×#117C2311yescharacteristic
192C2×#117C2311yescharacteristic
193C2×#117C2312yes—
194C22×C22⋊C4—22no—
195C22×C22⋊C4—24no—
196C22×C22⋊C4—22no—
197#491C2318yescomplemented
198#490C2318yescomplemented
199#490C23132yescomplemented
200#489C23132yescomplemented
201#489C23116yescomplemented
202#488C23132yescomplemented
203#488C23116yescomplemented
204#487C2318yescomplemented
205#486C2318yescomplemented
206#486C23132yescomplemented
207#479C23116yescomplemented
208#479C23116yescomplemented
209#478C23116yescomplemented
210#478C23116yescomplemented
211#478C23132yescomplemented
212#477C23116yescomplemented
213#476C23116yescomplemented
214#476C23116yescomplemented
215#475C23116yescomplemented
216#474C23116yescomplemented
217#474C23116yescomplemented
218#463C23132yes—
219#463C23132yes—
220#463C23132yescomplemented
221#461C23116yescomplemented
222#459C23132yes—
223#459C23132yes—
224#459C23116yescomplemented
225#446C23116yescomplemented
226#446C23132yescomplemented
227C8×Q8C23116yescomplemented
228#442C23132yes—
229#442C23132yescomplemented
230#440C23132yescomplemented
231C4×SD16C23132yescomplemented
232C4×D8C23116yes—
233C4×D8C23116yes—
234C4×D8C23132yescomplemented
235#428C2318yes—
236#428C2318yes—
237#427C2318yes—
238#427C2318yes—
239C2×#107C2312yes—
240C2×#106C2312yes—
241#424C2318yes—
242#424C2318yes—
243#424C2314yes—
244#424C2314yes—
245#423C2314yes—
246C2×C4⋊C8C2318yes—
247#419C2314yes—
248#419C2314yes—
249#418C2318yes—
250#418C2314yes—
251#418C2318yes—
252#418C2314yes—
253#418C2318yes—
254#418C2318yes—
255#417C2318yes—
256#417C2318yes—
257#416C2318yes—
258#416C2318yes—
259#416C2314yes—
260#416C2314yes—
261#416C2318yes—
262C2×#103C2314yes—
263C2×#102C2314yes—
264C2×#102C2318yes—
265C2×#102C2314yes—
266C2×#102C2314yes—
267#405C2318yes—
268#405C2318yes—
269#405C2318yes—
270C4×M16C2318yes—
271C2×#97C2312yes—
272C2×C4×C8C2312yes—
273C25—21no—
274C25—21no—
275C2×#41—216no—
276C2×#41—22no—
277C2×#41—22no—
278C2×#41—28no—
279C2×#41—28no—
280C2×#41C2414yes—
281C2×#41C2414yes—
282C22×Q8—216no—
283C22×Q8C2412yes—
284C22×D4—22no—
285C22×D4—21no—
286C22×D4—22no—
287C22×D4—21no—
288C22×D4—22no—
289C22×D4—28no—
290C22×D4—24no—
291C22×D4—24no—
292C22×D4—24no—
293C22×D4#4118yescomplemented
294C22×D4C2412yes—
295C22×D4C2414yes—
296C22×D4C2412yes—
297C22×D4C2414yes—
298C22×D4C2412yes—
299C23×C4—22no—
300C23×C4—28no—
301C23×C4—24no—
302C23×C4—24no—
303C23×C4C2411yescharacteristic
304AGL(1,ℤ/8)—232no—
305AGL(1,ℤ/8)—232no—
306C2×SD16—216no—
307C2×SD16C24116yes—
308C2×D8—232no—
309C2×D8—28no—
310C2×D8—28no—
311C2×D8—216no—
312C2×D8C2418yes—
313C2×D8C2418yes—
314C2×D8C24116yes—
315C2×M16—28no—
316C2×M16—24no—
317C2×M16C2414yes—
318C2×M16C2414yes—
319C2×M16C2414yes—
320C22×C8—24no—
321C22×C8C2411yescharacteristic
322C22×C8C2411yescharacteristic
323C22×C8C2414yes—
324C22×C8C2414yes—
325C22×C8C2412yes—
326C4⋊Q8—28no—
327C4⋊Q8—28no—
328C4⋊Q8—28no—
329C4⋊Q8C2×D4116yes—
330C4⋊Q8C2×D418yes—
331C4⋊Q8C2414yes—
332C4⋊Q8C2418yes—
333C4⋊Q8C24116yes—
334#127—28no—
335#127C24116yes—
336#127C2414yes—
337#126—232no—
338#125—28no—
339#125—28no—
340#125—28no—
341#125C2×D4116yes—
342#125C2×D418yes—
343#125C2418yes—
344#125C24116yes—
345#124—232no—
346#124—28no—
347#124C24116yes—
348#124C2418yes—
349#123—232no—
350#123—232no—
351#123—28no—
352C22⋊Q8—216no—
353C22⋊Q8—232no—
354C22⋊Q8—216no—
355C22⋊Q8—216no—
356C22⋊Q8—28no—
357C22⋊Q8—232no—
358C22⋊Q8—232no—
359C22⋊Q8—232no—
360C22⋊Q8—216no—
361C22⋊Q8—232no—
362C22⋊Q8—28no—
363C22⋊Q8—216no—
364C22⋊Q8—216no—
365C22⋊Q8—216no—
366C22⋊Q8C2×D4116yes—
367C22⋊Q8C2×D4132yes—
368#121—28no—
369#121—216no—
370#121—216no—
371#121—28no—
372#121—216no—
373#121—232no—
374#120—216no—
375#120—216no—
376#120—28no—
377C4×Q8—28no—
378C4×Q8—216no—
379C4×Q8—28no—
380C4×Q8C24116yes—
381C4×Q8C24116yes—
382C4×D4—216no—
383C4×D4—216no—
384C4×D4—216no—
385C4×D4—216no—
386C4×D4—216no—
387C4×D4—28no—
388C4×D4—232no—
389C4×D4—216no—
390C4×D4—28no—
391C4×D4—216no—
392C4×D4—28no—
393C4×D4—28no—
394C4×D4C24116yes—
395C4×D4C24132yes—
396C4×D4C24116yes—
397C4×D4C24116yes—
398#117—22no—
399#117—22no—
400#117—22no—
401#117—22no—
402#117—28no—
403#117—216no—
404#117—28no—
405#117—28no—
406#117C2×D414yes—
407#117C2×D414yes—
408#117C2×D414yes—
409#117C2414yes—
410#117C2414yes—
411#117C2414yes—
412#117C2414yes—
413#117C2414yes—
414#117C2414yes—
415#117C2414yes—
416C2×C4⋊C4—28no—
417C2×C4⋊C4—216no—
418C2×C4⋊C4—232no—
419C2×C4⋊C4C2×D412yes—
420C2×C4⋊C4C2×D414yes—
421C2×C4⋊C4C2412yes—
422C2×C4⋊C4C2414yes—
423C2×C4⋊C4C2412yes—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—24no—
428C2×C22⋊C4—28no—
429C2×C22⋊C4—24no—
430C2×C22⋊C4—28no—
431C2×C22⋊C4—216no—
432C2×C22⋊C4—216no—
433C2×C22⋊C4—216no—
434C2×C22⋊C4—216no—
435C2×C22⋊C4—28no—
436C2×C22⋊C4C2×D412yes—
437C2×C22⋊C4C2×D414yes—
438C2×C42—21no—
439C2×C42—21no—
440C2×C42—22no—
441C2×C42—28no—
442C2×C42C2412yes—
443C2×C42C2414yes—
444#107C2414yes—
445#107C2414yes—
446#107C24116yes—
447#106C2418yes—
448#106C24116yes—
449C4⋊C8C24116yes—
450C4⋊C8C24116yes—
451C4⋊C8C24116yes—
452#103C2418yes—
453#103C2418yes—
454#103C24116yes—
455#103C24116yes—
456#102C2418yes—
457#102C2418yes—
458#102C24116yes—
459#102C24116yes—
460#102C24132yes—
461#102C24116yes—
462#102C24116yes—
463#97C2418yes—
464#97C24116yes—
465C4×C8C2418yes—
466C4×C8C24116yes—
467C24—24no—
468C24—22no—
469C24—28no—
470C24—24no—
471C24—22no—
472C24—22no—
473C24—44no—
474C24—162no—
475C24—162no—
476C24C22×D411yescharacteristic
477#41—28no—
478#41—28no—
479#41—28no—
480#41—28no—
481#41—48no—
482#41—48no—
483#41—1616no—
484C2×Q8—22no—
485C2×Q8—216no—
486C2×Q8—28no—
487C2×Q8—24no—
488C2×Q8—24no—
489C2×Q8—1616no—
490C2×Q8C22×D4116yes—
491C2×Q8C2514yes—
492C2×D4—24no—
493C2×D4—24no—
494C2×D4—216no—
495C2×D4—28no—
496C2×D4—24no—
497C2×D4—22no—
498C2×D4—22no—
499C2×D4—24no—
500C2×D4—22no—
501C2×D4—28no—
502C2×D4—232no—
503C2×D4—28no—
504C2×D4—24no—
505C2×D4—24no—
506C2×D4—24no—
507C2×D4—28no—
508C2×D4—24no—
509C2×D4—28no—
510C2×D4—24no—
511C2×D4—48no—
512C2×D4—48no—
513C2×D4—44no—
514C2×D4—168no—
515C2×D4C2×#41116yes—
516C2×D4C2×#4118yes—
517C2×D4C2514yes—
518C2×D4C2514yes—
519C2×D4C2514yes—
520C22×C4—22no—
521C22×C4—22no—
522C22×C4—24no—
523C22×C4—22no—
524C22×C4—24no—
525C22×C4—24no—
526C22×C4—216no—
527C22×C4—232no—
528C22×C4—216no—
529C22×C4—216no—
530C22×C4—48no—
531C22×C4—164no—
532C22×C4—164no—
533C22×C4—168no—
534C22×C4C22×D418yes—
535C22×C4C22×D414yes—
536C22×C4C22×D411yescharacteristic
537C22×C4C22×D412yes—
538C22×C4C2×#4114yes—
539C22×C42−1+414yes—
540C22×C4C2511yescharacteristic
541C22×C4C2512yes—
542SD16—232no—
543D8—216no—
544D8—216no—
545D8—232no—
546M16—28no—
547M16—28no—
548M16—28no—
549C2×C8—28no—
550C2×C8—22no—
551C2×C8—22no—
552C2×C8—28no—
553C2×C8—24no—
554C2×C8C2514yes—
555C2×C8C2512yes—
556C2×C8C2518yes—
557C2×C8C2512yes—
558C4⋊C4—22no—
559C4⋊C4—216no—
560C4⋊C4—28no—
561C4⋊C4—216no—
562C4⋊C4—216no—
563C4⋊C4—216no—
564C4⋊C4—216no—
565C4⋊C4—24no—
566C4⋊C4—22no—
567C4⋊C4—28no—
568C4⋊C4C22×D418yes—
569C4⋊C4C22×D4116yes—
570C4⋊C4C22×D4132yes—
571C4⋊C4C22×D418yes—
572C4⋊C4C22×D418yes—
573C4⋊C4C2518yes—
574C4⋊C4C2514yes—
575C4⋊C4C2514yes—
576C22⋊C4—24no—
577C22⋊C4—24no—
578C22⋊C4—24no—
579C22⋊C4—24no—
580C22⋊C4—216no—
581C22⋊C4—232no—
582C22⋊C4—216no—
583C22⋊C4—216no—
584C22⋊C4—232no—
585C22⋊C4—216no—
586C22⋊C4C22×D418yes—
587C22⋊C4C22×D418yes—
588C22⋊C4C22×D418yes—
589C42—24no—
590C42—216no—
591C42—24no—
592C42—24no—
593C42—24no—
594C42C22×D418yes—
595C42C22×D414yes—
596C42C2514yes—
597C42C2518yes—
598C23—24no—
599C23—216no—
600C23—28no—
601C23—24no—
602C23—24no—
603C23—42no—
604C23—44no—
605C23—44no—
606C23—48no—
607C23—84no—
608C23—84no—
609C23—88no—
610C23—84no—
611C23—88no—
612C23—168no—
613C23—168no—
614C23#54814yes—
615C23C23×D412yes—
616C23C23×D411yescharacteristic
617Q8—24no—
618Q8—24no—
619Q8—416no—
620Q8—44no—
621Q8—88no—
622Q8—816no—
623D4—24no—
624D4—24no—
625D4—24no—
626D4—24no—
627D4—24no—
628D4—28no—
629D4—216no—
630D4—48no—
631D4—48no—
632D4—44no—
633D4—48no—
634D4—44no—
635D4—832no—
636D4—88no—
637D4C22×#4118yes—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—28no—
642C2×C4—44no—
643C2×C4—48no—
644C2×C4—44no—
645C2×C4—48no—
646C2×C4—48no—
647C2×C4—816no—
648C2×C4—832no—
649C2×C4—816no—
650C2×C4—816no—
651C2×C4—82no—
652C2×C4—82no—
653C2×C4—84no—
654C2×C4#54818yes—
655C2×C4#54814yes—
656C2×C4C23×D4116yes—
657C2×C4C23×D418yes—
658C2×C4C23×D412yes—
659C2×C4C23×D411yescharacteristic
660C2×C4C22×#4114yes—
661C2×C4C22×#4112yes—
662C2×C4C2×2−1+412yes—
663C2×C4C2×2−1+414yes—
664C2×C4C2611yescharacteristic
665C8—22no—
666C8—22no—
667C8—28no—
668C8—24no—
669C22—22no—
670C22—24no—
671C22—28no—
672C22—44no—
673C22—48no—
674C22—48no—
675C22—48no—
676C22—44no—
677C22—44no—
678C22—88no—
679C22—816no—
680C22—88no—
681C22—164no—
682C22—168no—
683C22C2×#54814yes—
684C22C2×#54812yes—
685C22C24×D411yescharacteristic
686C4—28no—
687C4—216no—
688C4—44no—
689C4—44no—
690C4—48no—
691C4—44no—
692C4C2×#54812yes—
693C4C2×#54814yes—
694C4C23×#4111yescharacteristic
695C4C22×2−1+411yescharacteristic
696C2—22no—
697C2—24no—
698C2—48no—
699C2—84no—
700C2—84no—
701C2C22×#54811yescharacteristic
702C2#6309112yes—
subgroupG/H
center, socleC22C24×D4
commutator, Frattini, cosocleC2×C4C26
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #3458 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62855C2semidirect
2. #62843C2semidirect
3. #61343C2semidirect
4. #61339C2semidirect
5. #61332C2semidirect
6. #61325C2semidirect
7. #60619C2semidirect
8. #60612C2semidirect
9. #60490C2semidirect
10. #60489C2semidirect
11. #60423C2semidirect
12. C2×#3041C2semidirect
13. #60193C2semidirect
14. #60168C2semidirect
15. C2×#2948C2semidirect
18. #3510C22semidirect
19. #3508C22semidirect
20. #3506C22semidirect
23. #3501C22semidirect
24. #3501C22semidirect
38. #3376C22semidirect
39. #3373C22semidirect
40. #3370C22semidirect
41. #3370C22semidirect
42. #3367C22semidirect
43. #3366C22semidirect
44. #3366C22semidirect
45. #3361C22semidirect
46. #3360C22semidirect
47. #3358C22semidirect
48. #3140C22semidirect
49. #3138C22semidirect
50. #3137C22semidirect
51. #3136C22semidirect
52. #3136C22semidirect
53. #3136C22semidirect
54. #3135C22semidirect
55. #3131C22semidirect
56. #3084C22semidirect
57. #3083C22semidirect
58. #3082C22semidirect
59. #3081C22semidirect
60. #3081C22semidirect
61. #3077C22semidirect
62. #3076C22semidirect
63. #3042C22semidirect
65. #3041C22semidirect
66. #3041C22semidirect
67. #3041C22semidirect
68. #3036C22semidirect
69. C2×#463C22semidirect
70. C2×#459C22semidirect
71. #2962C22semidirect
72. #2961C22semidirect
73. #2960C22semidirect
74. #2949C22semidirect
75. #2948C22semidirect
76. #2948C22semidirect
77. #2946C22semidirect
78. C4×AGL(1,ℤ/8)C22semidirect
79. C2×#442C22semidirect
80. C2×C4×D8C22semidirect
108. #559C23semidirect
109. #556C23semidirect
116. #554C23semidirect
117. #554C23semidirect
120. #550C23semidirect
123. D4×Q8C23semidirect
197. #491C23semidirect
198. #490C23semidirect
199. #490C23semidirect
200. #489C23semidirect
201. #489C23semidirect
202. #488C23semidirect
203. #488C23semidirect
204. #487C23semidirect
205. #486C23semidirect
206. #486C23semidirect
207. #479C23semidirect
208. #479C23semidirect
209. #478C23semidirect
210. #478C23semidirect
211. #478C23semidirect
212. #477C23semidirect
213. #476C23semidirect
214. #476C23semidirect
215. #475C23semidirect
216. #474C23semidirect
217. #474C23semidirect
220. #463C23semidirect
221. #461C23semidirect
224. #459C23semidirect
225. #446C23semidirect
226. #446C23semidirect
227. C8×Q8C23semidirect
229. #442C23semidirect
230. #440C23semidirect
231. C4×SD16C23semidirect
234. C4×D8C23semidirect
293. C22×D4#41semidirect

Maximal subgroups

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

subgroupindex
#628552
#628432
#613432
#613392
#613322
#613252
#606192
#606122
#604902
#604892
#604232
C2×#30412
#601932
#601682
C2×#29482