Finite Group Databasebeta

← #6470448

Subgroups — #6470448

Order 512.

2320 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2C2×Q8×C22⋊C4C211yesmaximal, characteristic
3#32960C212yesmaximal, complemented
4#32957C211yesmaximal, characteristic, complemented
5#32954C211yesmaximal, characteristic, complemented
6#29957C212yesmaximal, complemented
7#29953C211yesmaximal, characteristic, complemented
8#29951C211yesmaximal, characteristic, complemented
9#25981C214yesmaximal, complemented
10#25979C214yesmaximal, complemented
11#25973C214yesmaximal, complemented
12#25971C214yesmaximal, complemented
13#21944C212yesmaximal
14#21934C211yesmaximal, characteristic
15#20599C212yesmaximal
16C24×Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#479C2214yescomplemented
20C2×#477C2212yescomplemented
21C2×#475C2212yescomplemented
22C2×#419C2214yes—
23C2×#419C2214yes—
24C2×#407C2212yes—
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2851C2212yes—
28#2626—21no—
29#2626—21no—
30#2608—22no—
31#2607—22no—
32#2441C2212yes—
33#2441C2212yes—
34#2439C2212yes—
35#2439C2212yes—
36Q8×C22⋊C4—22no—
37Q8×C22⋊C4—22no—
38Q8×C22⋊C4C2214yes—
39Q8×C22⋊C4C2214yes—
40#2329—21no—
41#2329—21no—
42C2×#391—21no—
43C2×#391—21no—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46#2280C2212yes—
47C2×#380—21no—
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4—21no—
50C2×C4×C22⋊C4C2211yescharacteristic
51#2097C2214yescomplemented
52#2096C2212yescomplemented
53#2095C2212yescomplemented
54#2068C2214yescomplemented
55#2068C2214yescomplemented
56#2066C2214yescomplemented
57#2066C2214yescomplemented
58#2026C2214yescomplemented
59#2024C2212yescomplemented
60#2024C2212yescomplemented
61#1865C2214yescomplemented
62#1865C2214yescomplemented
63#1863C2214yescomplemented
64#1863C2214yescomplemented
65#1854C2214yes—
66#1854C2214yes—
67#1853C2212yes—
68#1853C2212yes—
69#1852C2212yes—
70#1852C2212yes—
71#1851C2214yes—
72#1851C2214yes—
73#1809C2214yes—
74#1809C2214yes—
75#1805C2214yescomplemented
76#1803C2212yescomplemented
77#1803C2212yescomplemented
78#1797C2211yescharacteristic
79#1797C2212yes—
80#1797C2211yescharacteristic
81#1787C2212yes—
82#1787C2212yes—
83#1787C2214yes—
84#1787C2214yes—
85#1787C2212yes—
86#1787C2212yes—
87C23×Q8—22no—
88C23×Q8—21no—
89C23×Q8—24no—
90C23×Q8—44no—
91C23×Q8—44no—
92C23×Q8Q818yescomplemented
93C23×Q8C2311yescharacteristic
94C23×Q8C2314yes—
95C23×Q8C2311yescharacteristic
96C24×C4—21no—
97C24×C4C2311yescharacteristic
98C22×M16C2314yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#125C2312yes—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104C2×C4×Q8—22no—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—44no—
108C2×C4×Q8—44no—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2314yes—
112C2×#117C2312yes—
113C2×#117C2312yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4D411yescharacteristic
120C22×C22⋊C4D411yescharacteristic
121C22×C42—21no—
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#479—48no—
126#477—44no—
127#475—44no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135#419—48no—
136#419—48no—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145#407—28no—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2314yes—
149#399—22no—
150#399—22no—
151#396—22no—
152#396—22no—
153#396—22no—
154#396—22no—
155#393—22no—
156#393—22no—
157#391—22no—
158#391—21no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391—22no—
163#391—22no—
164#391—24no—
165#391—24no—
166#391—24no—
167#391—24no—
168#391C2312yes—
169#391C2314yes—
170#391C2312yes—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—22no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#382C2312yes—
190#381—22no—
191#380—21no—
192#380—21no—
193#380—22no—
194#380—24no—
195#380—24no—
196#380C2314yes—
197#380C2312yes—
198#380C2312yes—
199#380C2312yes—
200#380C2312yes—
201#379—22no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—24no—
205C4×C22⋊C4—24no—
206C4×C22⋊C4D412yes—
207C4×C22⋊C4D412yes—
208C4×C22⋊C4C2312yes—
209C4×C22⋊C4C2312yes—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95—21no—
213C2×#95—21no—
214C2×#95—21no—
215C2×#95D411yescharacteristic
216C2×#95D411yescharacteristic
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#336C2312yes—
222#336C2312yes—
223#336C2314yes—
224C25C2×D411yescharacteristic
225C22×Q8—24no—
226C22×Q8—24no—
227C22×Q8—28no—
228C22×Q8—216no—
229C22×Q8—21no—
230C22×Q8—24no—
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—28no—
237C22×Q8—28no—
238C22×Q8—28no—
239C22×Q8—42no—
240C22×Q8—42no—
241C22×Q8—44no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—432no—
246C22×Q8—432no—
247C22×Q8—48no—
248C22×Q8—48no—
249C22×Q8C2×Q814yes—
250C22×Q8C2×Q8116yes—
251C22×Q8C2×Q814yes—
252C22×Q8C2412yes—
253C22×Q8C2412yes—
254C23×C4—21no—
255C23×C4—28no—
256C23×C4—21no—
257C23×C4—24no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—42no—
261C23×C4—42no—
262C23×C4—44no—
263C23×C4—44no—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D412yes—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×Q814yes—
271C23×C4#4114yes—
272C23×C4C2411yescharacteristic
273C23×C4C2412yes—
274C2×M16—28no—
275C2×M16—44no—
276C2×M16—44no—
277C22×C8C2414yes—
278C22×C8C2414yes—
279C4⋊Q8—42no—
280C4⋊Q8—42no—
281#125—44no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—28no—
287C4×Q8—44no—
288C4×Q8—42no—
289C4×Q8—42no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×Q8—48no—
293#117—44no—
294#117—44no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—42no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—42no—
342C2×C42—42no—
343C2×C42—44no—
344C2×C42—44no—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—44no—
352#107—44no—
353#106—44no—
354#106—44no—
355C4⋊C8—42no—
356C4⋊C8—42no—
357C4⋊C8—44no—
358#103—28no—
359#103—28no—
360#103—28no—
361#103—28no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366C22⋊C8—24no—
367C22⋊C8—24no—
368C22⋊C8—28no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24—22no—
391C24—22no—
392C24—41no—
393C24—41no—
394C24—41no—
395C24—41no—
396C24C22⋊Q814yes—
397C24C22×D412yes—
398C24C22×D411yescharacteristic
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—216no—
402C2×Q8—216no—
403C2×Q8—24no—
404C2×Q8—216no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—232no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×Q8—416no—
416C2×Q8—48no—
417C2×Q8—416no—
418C2×Q8—416no—
419C2×Q8—432no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—48no—
423C2×Q8—42no—
424C2×Q8—464no—
425C2×Q8—42no—
426C2×Q8—81no—
427C2×Q8—81no—
428C2×Q8—81no—
429C2×Q8—81no—
430C2×Q8C22×Q814yes—
431C2×Q8C22×Q814yes—
432C2×Q8H(5,𝔽2)14yes—
433C2×Q8H(5,𝔽2)14yes—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—216no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—432no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4C22⋊Q818yes—
480C22×C4C22⋊Q814yes—
481C22×C4#13714yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×Q814yes—
491C22×C4C22×Q812yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4114yes—
494C22×C4C2511yescharacteristic
495M16—88no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—81no—
525C4⋊C4—81no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—41no—
541C42—41no—
542C42—42no—
543C42—44no—
544C42—44no—
545C42—48no—
546C42—82no—
547C42—82no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—22no—
554C23—28no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23C2×C22⋊Q814yes—
566C23C2×C22⋊Q812yes—
567C23C2×#13712yes—
568C23#57512yes—
569C23C23×D411yescharacteristic
570Q8—216no—
571Q8—24no—
572Q8—24no—
573Q8—24no—
574Q8—28no—
575Q8—432no—
576Q8—48no—
577Q8—44no—
578Q8—162no—
579Q8—162no—
580Q8#54318yes—
581C2×C4—216no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—24no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—48no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—432no—
614C2×C4—44no—
615C2×C4—48no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4C2×C22⋊Q812yes—
641C2×C4C2×C22⋊Q814yes—
642C2×C4C2×#13712yes—
643C2×C4C2×#13712yes—
644C2×C4C23×Q811yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4C2×H(5,𝔽2)12yes—
647C2×C4C2×H(5,𝔽2)12yes—
648C8—44no—
649C8—44no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—416no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22#307814yes—
665C22C22×C22⋊Q811yescharacteristic
666C22#348411yescharacteristic
667C22#348411yescharacteristic
668C22C22×#13711yescharacteristic
669C22C2×#57511yescharacteristic
670C22#358611yescharacteristic
671C22#358611yescharacteristic
672C4—22no—
673C4—22no—
674C4—28no—
675C4—22no—
676C4—24no—
677C4—42no—
678C4—41no—
679C4—41no—
680C4—42no—
681C4—44no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4C2×#54312yes—
689C4#349012yes—
690C2—24no—
691C2—44no—
692C2#3498211yescharacteristic
693C2C2×#307811yescharacteristic
694C2#6049111yescharacteristic
695C2#6157411yescharacteristic
696C2#6157811yescharacteristic
697C2#6158311yescharacteristic
698C2#6158411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32960C2semidirect
4. #32957C2semidirect
5. #32954C2semidirect
6. #29957C2semidirect
7. #29953C2semidirect
8. #29951C2semidirect
9. #25981C2semidirect
10. #25979C2semidirect
11. #25973C2semidirect
12. #25971C2semidirect
19. C2×#479C22semidirect
20. C2×#477C22semidirect
21. C2×#475C22semidirect
51. #2097C22semidirect
52. #2096C22semidirect
53. #2095C22semidirect
54. #2068C22semidirect
55. #2068C22semidirect
56. #2066C22semidirect
57. #2066C22semidirect
58. #2026C22semidirect
59. #2024C22semidirect
60. #2024C22semidirect
61. #1865C22semidirect
62. #1865C22semidirect
63. #1863C22semidirect
64. #1863C22semidirect
75. #1805C22semidirect
76. #1803C22semidirect
77. #1803C22semidirect
92. C23×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×Q8×C22⋊C42
#329602
#329572
#329542
#299572
#299532
#299512
#259812
#259792
#259732
#259712
#219442
#219342
#205992