Finite Group Databasebeta

← #354483

Subgroups — #354483

Order 512.

1494 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60058C211yesmaximal, characteristic
2C2×#1861C211yesmaximal, characteristic
3#20863C211yesmaximal, characteristic
4#20444C211yesmaximal, characteristic, complemented
5#20437C211yesmaximal, characteristic, complemented
6#20420C211yesmaximal, characteristic
7C2×#1764C211yesmaximal, characteristic
8#10902C211yesmaximal, characteristic, complemented
9#10900C211yesmaximal, characteristic, complemented
10#10894C211yesmaximal, characteristic, complemented
11#10893C211yesmaximal, characteristic, complemented
12#10782C211yesmaximal, characteristic
13#10779C211yesmaximal, characteristic
14#10711C211yesmaximal, characteristic, complemented
15#10706C211yesmaximal, characteristic, complemented
16C2×#568C2211yescharacteristic
17C22×#131C2211yescharacteristic
18#3428C2211yescharacteristic
19C2×#430C2211yescharacteristic
20#2911C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#421C2211yescharacteristic
23C22×#104C2211yescharacteristic
24#2896C2211yescharacteristic
25C2×#416C2211yescharacteristic
26#2875—22no—
27#2875—22no—
28C4×#131—22no—
29C4×#131—22no—
30#2873—22no—
31#2873C2211yescharacteristic
32#2873C2211yescharacteristic
33C2×#405—22no—
34C2×#405C2211yescharacteristic
35C2×#405C2211yescharacteristic
36C2×#405C2211yescharacteristic
37C2×#405C2211yescharacteristic
38C2×#405C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40#2285C2211yescharacteristic
41#1861C418yescomplemented
42#1860C414yescomplemented
43#1859C414yescomplemented
44#1773—22no—
45#1772—22no—
46#1770—22no—
47C8×C4⋊C4—22no—
48#1765—22no—
49#1764—22no—
50#1764C418yescomplemented
51C2×#344C2211yescharacteristic
52C2×#344C2211yescharacteristic
53#1742C2211yescharacteristic
54#1740C2211yescharacteristic
55#1738C2211yescharacteristic
56C2×#341C2211yescharacteristic
57C2×#341C2211yescharacteristic
58#1735C2211yescharacteristic
59#1733C2211yescharacteristic
60#1733C2211yescharacteristic
61#1733C2211yescharacteristic
62C2×#339C2211yescharacteristic
63C2×#339C2211yescharacteristic
64C2×#339C2211yescharacteristic
65#1729C2211yescharacteristic
66#1729C2211yescharacteristic
67C22×#41C2311yescharacteristic
68#568—22no—
69#568—22no—
70#568—22no—
71#568C2×C414yes—
72C2×#131—22no—
73C2×#131—21no—
74C2×#131—22no—
75C2×#131—21no—
76C2×#131—22no—
77C2×#131—22no—
78C2×#131C2×C414yes—
79C2×#131D412yes—
80C2×#131D412yes—
81C2×#131Q814yes—
82C22×M16—22no—
83C22×M16—21no—
84C22×M16C2311yescharacteristic
85C22×M16C2311yescharacteristic
86C22×M16C2311yescharacteristic
87C22×M16C2311yescharacteristic
88C22×M16C2311yescharacteristic
89C22×M16C2311yescharacteristic
90C23×C8—21no—
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93#520—22no—
94#520C2×C414yes—
95#519—21no—
96#518—21no—
97#518—22no—
98C4×#41—22no—
99C4×#41—22no—
100C2×#117—22no—
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C22×C42C2311yescharacteristic
107#430—22no—
108#430C2×C414yescomplemented
109C2×#108—22no—
110#428—22no—
111#428C2×C414yes—
112#427—22no—
113#423—22no—
114#423C2×C414yescomplemented
115C2×C4⋊C8—22no—
116#421—22no—
117#421C2×C414yes—
118C2×#104—22no—
119C2×#104C2×C412yes—
120C2×#104C2×C412yes—
121C2×#104C2×C414yes—
122C2×#104C2×C414yes—
123#419—21no—
124#418—21no—
125#417C2×C414yes—
126#416—22no—
127#416—22no—
128#416C2×C414yes—
129C2×#103—21no—
130C2×#102—21no—
131#405—24no—
132#405—24no—
133#405—24no—
134#405—24no—
135#405—22no—
136#405—22no—
137#405—21no—
138#405—21no—
139#405—21no—
140#405—21no—
141#405—21no—
142#405—21no—
143#405—22no—
144#405—22no—
145#405—22no—
146#405—21no—
147#405—21no—
148#405—21no—
149#405—21no—
150#405—22no—
151#405—22no—
152#405C2×C414yescomplemented
153#405D412yes—
154#405D412yes—
155C4×M16—24no—
156C4×M16—24no—
157C4×M16—22no—
158C4×M16—22no—
159C2×#97—24no—
160C2×#97—21no—
161C2×#97—21no—
162C2×#97—21no—
163C2×#97—21no—
164C2×#97D412yes—
165C2×C4×C8—24no—
166C2×C4×C8—21no—
167C2×C4×C8—21no—
168C2×C4×C8—21no—
169C2×C4×C8—21no—
170C2×C4×C8—22no—
171C2×C4×C8—21no—
172C2×C4×C8—21no—
173C2×C4×C8C2×C414yescomplemented
174C2×C4×C8D412yes—
175#384C2×C412yes—
176#382C2×C412yes—
177#380—22no—
178C4×C4⋊C4—22no—
179#344—22no—
180#344C2×C414yescomplemented
181#343C2×C414yes—
182#342—22no—
183#341—22no—
184#341—22no—
185#341C2×C414yescomplemented
186#340—22no—
187#339—22no—
188#339—22no—
189#339—22no—
190#339C2×C412yes—
191#339C2×C412yes—
192#339C2×C414yescomplemented
193#339C2×C414yescomplemented
194#336—22no—
195#336—22no—
196C2×#41—21no—
197C2×#41—21no—
198C2×#41—21no—
199C2×#41—21no—
200C2×#41—21no—
201C2×#41—21no—
202C2×#41C22×C412yes—
203C2×#41C22×C412yes—
204C2×#41C22×C412yes—
205C2×#41C2×D412yes—
206C2×#41C2×Q812yes—
207C2×#41#4112yes—
208C22×Q8#4111yescharacteristic
209C22×D4—21no—
210C22×D4#4111yescharacteristic
211C23×C4—21no—
212C23×C4—21no—
213C23×C4C2411yescharacteristic
214#131—28no—
215#131—28no—
216#131—28no—
217#131—24no—
218#131—24no—
219#131—28no—
220#131—24no—
221#131—24no—
222#131—24no—
223#131—24no—
224#131C4⋊C418yescomplemented
225#131C4⋊C418yescomplemented
226C2×M16—24no—
227C2×M16—28no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—21no—
239C2×M16—21no—
240C2×M16—21no—
241C2×M16—28no—
242C2×M16—21no—
243C2×M16—21no—
244C2×M16—24no—
245C2×M16—22no—
246C2×M16—22no—
247C2×M16—24no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—21no—
255C2×M16—21no—
256C2×M16C22×C412yes—
257C2×M16C22×C412yes—
258C2×M16C22×C412yes—
259C2×M16C22×C412yes—
260C2×M16C22×C412yes—
261C2×M16C22×C412yes—
262C2×M16C22×C412yes—
263C2×M16C22×C412yes—
264C2×M16C2×D412yes—
265C2×M16C2×D412yes—
266C2×M16C2×Q812yes—
267C2×M16#4112yes—
268C2×M16#4112yes—
269C22×C8—24no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—21no—
273C22×C8—24no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8—24no—
277C22×C8—21no—
278C22×C8—21no—
279C22×C8—21no—
280C22×C8—21no—
281C22×C8—21no—
282C22×C8—21no—
283C22×C8—21no—
284C22×C8—21no—
285C22×C8C22×C412yes—
286C22×C8C22×C412yes—
287C22×C8C2×D412yes—
288C22×C8C2×D411yescharacteristic
289C22×C8C2×D411yescharacteristic
290C22×C8C2×D411yescharacteristic
291C22×C8C2×D411yescharacteristic
292C22×C8C2×D411yescharacteristic
293C22×C8C2×D411yescharacteristic
294C22×C8C2×Q812yes—
295C22×C8#4112yes—
296C22×C8#4112yes—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×D4—28no—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—28no—
303#117—24no—
304#117—24no—
305#117—21no—
306#117—21no—
307#117—21no—
308#117—21no—
309#117—21no—
310#117—21no—
311#117—24no—
312#117—21no—
313#117—21no—
314#117—21no—
315#117—21no—
316#117—21no—
317#117—21no—
318#117—21no—
319#117—21no—
320#117—21no—
321#117—21no—
322#117C22×C412yes—
323#117C22×C412yes—
324#117C22×C412yes—
325#117C2×D412yes—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4C22×C412yes—
331C2×C4⋊C4C22×C412yes—
332C2×C4⋊C4#4112yes—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4#4112yes—
339C2×C42—24no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42C22×C411yescharacteristic
348C2×C42C22×C411yescharacteristic
349C2×C42C22×C412yes—
350C2×C42C22×C412yes—
351C2×C42C2×D412yes—
352#108—28no—
353#107—24no—
354#106—24no—
355C4⋊C8—28no—
356#104—28no—
357#104C4218yes—
358#104C4218yes—
359#103—24no—
360#103—24no—
361#102—24no—
362#102—24no—
363#97—24no—
364#97—24no—
365#97—24no—
366#97—22no—
367#97—22no—
368#97—22no—
369#97—22no—
370#97—41no—
371#97—41no—
372#97—42no—
373C4×C8—24no—
374C4×C8—24no—
375C4×C8—24no—
376C4×C8—24no—
377C4×C8—24no—
378C4×C8—22no—
379C4×C8—22no—
380C4×C8—22no—
381C4×C8—22no—
382C4×C8—41no—
383C4×C8—41no—
384C4×C8—42no—
385#95—24no—
386C24—21no—
387C24C2×#4111yescharacteristic
388#41—22no—
389#41—22no—
390#41—22no—
391#41—22no—
392#41—22no—
393#41—22no—
394#41—22no—
395#41—22no—
396#41—22no—
397#41—22no—
398#41—22no—
399#41—22no—
400#41C2×C4214yes—
401#41C2×C4⋊C414yes—
402#41C4×D414yes—
403#41C4×Q814yes—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8#11712yes—
408C2×Q8C4×D412yes—
409C2×Q8C4×Q812yes—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4#11712yes—
420C2×D4C4×D412yes—
421C2×D4C4×Q812yes—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—28no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4C4×D414yes—
440C22×C4C23×C411yescharacteristic
441C22×C4C23×C411yescharacteristic
442C22×C4C23×C411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×Q811yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454M16—216no—
455M16—22no—
456M16—22no—
457M16—22no—
458M16—22no—
459M16—22no—
460M16—22no—
461M16—22no—
462M16—22no—
463M16—22no—
464M16—22no—
465M16—22no—
466M16—22no—
467M16—22no—
468M16—28no—
469M16—28no—
470M16—22no—
471M16—22no—
472M16—22no—
473M16—22no—
474M16—41no—
475M16—41no—
476M16—41no—
477M16—41no—
478M16C2×C4214yes—
479M16C2×C4⋊C414yes—
480M16C4×D414yes—
481C2×C8—216no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—28no—
494C2×C8—28no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8C2×C4⋊C412yes—
509C2×C8C2×C4⋊C412yes—
510C2×C8C4×D414yes—
511C2×C8C4×D412yes—
512C2×C8C4×D412yes—
513C2×C8C4×D412yes—
514C2×C8C4×D412yes—
515C2×C8#12011yescharacteristic
516C2×C8#12011yescharacteristic
517C2×C8#12011yescharacteristic
518C2×C8#12011yescharacteristic
519C2×C8C22⋊Q812yes—
520C2×C8C22⋊Q812yes—
521C2×C8C22⋊Q812yes—
522C2×C8C22⋊Q812yes—
523C2×C8#12312yes—
524C2×C8#12312yes—
525C4⋊C4—28no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—48no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4C4×D414yes—
537C22⋊C4—28no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C42—28no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—21no—
553C42—21no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42C2×C4212yes—
564C42C2×C4212yes—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—42no—
569C23—82no—
570C23C2×C4×D411yescharacteristic
571C23C4×#4111yescharacteristic
572C23C4×#4111yescharacteristic
573C23#52511yescharacteristic
574C23#52511yescharacteristic
575C23#52911yescharacteristic
576C23#53311yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—22no—
580Q8C4×C4⋊C414yes—
581D4—24no—
582D4—24no—
583D4—24no—
584D4—24no—
585D4—22no—
586D4—22no—
587D4—22no—
588D4C4×C4⋊C414yes—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—82no—
610C2×C4—88no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4C22×C4211yescharacteristic
614C2×C4C22×C4⋊C411yescharacteristic
615C2×C4C2×#11711yescharacteristic
616C2×C4C2×C4×D411yescharacteristic
617C2×C4C2×C4×D411yescharacteristic
618C2×C4C2×C4×D411yescharacteristic
619C2×C4C2×C4×D411yescharacteristic
620C2×C4C2×C4×D411yescharacteristic
621C2×C4C2×C4×D411yescharacteristic
622C2×C4C2×C4×D411yescharacteristic
623C2×C4C2×C4×Q811yescharacteristic
624C2×C4C2×C4×Q811yescharacteristic
625C2×C4C4×#4111yescharacteristic
626C2×C4C4×#4111yescharacteristic
627C2×C4C4×#4111yescharacteristic
628C2×C4C4×#4111yescharacteristic
629C2×C4C2×#12011yescharacteristic
630C2×C4C2×C22⋊Q811yescharacteristic
631C2×C4C2×C22⋊Q811yescharacteristic
632C2×C4C2×#12311yescharacteristic
633C2×C4#52511yescharacteristic
634C2×C4#52511yescharacteristic
635C2×C4#52511yescharacteristic
636C2×C4#52511yescharacteristic
637C2×C4#52911yescharacteristic
638C2×C4#52911yescharacteristic
639C2×C4#52911yescharacteristic
640C2×C4#53311yescharacteristic
641C8—24no—
642C8—24no—
643C8—24no—
644C8—24no—
645C8—42no—
646C8—42no—
647C8—42no—
648C8#38512yes—
649C8#38512yes—
650C8#38512yes—
651C8#38512yes—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—24no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—44no—
661C22—82no—
662C22—82no—
663C22C42×D411yescharacteristic
664C22#229011yescharacteristic
665C22#229611yescharacteristic
666C22C4×#12011yescharacteristic
667C22C4×#12311yescharacteristic
668C22C4×C22⋊Q811yescharacteristic
669C22C4×C22⋊Q811yescharacteristic
670C4—24no—
671C4—24no—
672C4—42no—
673C4—44no—
674C4—44no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4C2×C4×C4⋊C411yescharacteristic
679C4C42×D411yescharacteristic
680C4C2×#38511yescharacteristic
681C4#229011yescharacteristic
682C4C4×#12011yescharacteristic
683C4C4×#12311yescharacteristic
684C4C4×C22⋊Q811yescharacteristic
685C4C4×C22⋊Q811yescharacteristic
686C2—21no—
687C2—21no—
688C2—21no—
689C2—21no—
690C2—21no—
691C2—21no—
692C2—44no—
693C2C4×#38511yescharacteristic
694C2#2147712yes—
subgroupG/H
centerC2×C8#120
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC22C4×#120
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C23×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20444C2semidirect
5. #20437C2semidirect
8. #10902C2semidirect
9. #10900C2semidirect
10. #10894C2semidirect
11. #10893C2semidirect
14. #10711C2semidirect
15. #10706C2semidirect
41. #1861C4semidirect
42. #1860C4semidirect
43. #1859C4semidirect
50. #1764C4semidirect
108. #430C2×C4semidirect
114. #423C2×C4semidirect
152. #405C2×C4semidirect
173. C2×C4×C8C2×C4semidirect
180. #344C2×C4semidirect
185. #341C2×C4semidirect
192. #339C2×C4semidirect
193. #339C2×C4semidirect
224. #131C4⋊C4semidirect
225. #131C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#600582
C2×#18612
#208632
#204442
#204372
#204202
C2×#17642
#109022
#109002
#108942
#108932
#107822
#107792
#107112
#107062