Finite Group Databasebeta

← #6519258

Subgroups — #6519258

Order 512.

1526 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic
2#34953C211yesmaximal, characteristic
3#29150C212yesmaximal, complemented
4#29132C212yesmaximal, complemented
5#29132C212yesmaximal, complemented
6#28725C212yesmaximal, complemented
7#27824C212yesmaximal, complemented
8#27777C212yesmaximal, complemented
9#27757C212yesmaximal, complemented
10#27738C212yesmaximal, complemented
11#27715C214yesmaximal
12#27268C214yesmaximal
13#21885C211yesmaximal, characteristic
14#21860C211yesmaximal, characteristic
15#20734C211yesmaximal, characteristic
16C2×#1803C211yesmaximal, characteristic
17#20547C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#491C2214yes—
21C2×#431C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×#103C2212yes—
24#2859C2212yes—
25#2859C2212yes—
26#2647—22no—
27#2584—21no—
28#2584—21no—
29#2583—22no—
30#2571—21no—
31#2571C2212yes—
32#2567C2211yescharacteristic
33#2567C2211yescharacteristic
34#2552—21no—
35#2552C2212yes—
36#2546C2211yescharacteristic
37#2546C2211yescharacteristic
38#2502C2212yes—
39#2442—22no—
40#2442C2214yes—
41#2434C2212yes—
42#2399—22no—
43#2317—21no—
44#2316—21no—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2212yes—
47#2296—21no—
48#2296—21no—
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2099C2214yes—
53#2099C2214yes—
54#2074C2214yes—
55#2074C2214yes—
56#2043C2214yes—
57#2043C2214yes—
58#2026C2214yes—
59#2026C2214yes—
60#1992C2214yes—
61#1990C2212yes—
62#1987C2212yes—
63#1981C2212yes—
64#1960C2214yes—
65#1956C2212yes—
66#1935C2214yes—
67#1935C2214yes—
68#1935C2214yes—
69#1935C2214yes—
70#1892C2214yes—
71#1872C2214yes—
72#1836C2214yes—
73#1824C2212yes—
74#1803C2211yescharacteristic
75#1803C2212yes—
76#1803C2212yes—
77#1803C2212yes—
78#1803C2212yes—
79#1803C2212yes—
80#1803C2212yes—
81#1803C2212yes—
82#1803C2212yes—
83#1803C2211yescharacteristic
84#1803C2211yescharacteristic
85#1803C2211yescharacteristic
86#1787C2212yes—
87#1770C2212yes—
88#1770C2212yes—
89#1767C2212yes—
90#1767C2212yes—
91C4×#103C2212yes—
92C4×#103C2212yes—
93#1752C2214yes—
94C2×#340C2212yes—
95C23×Q8C2311yescharacteristic
96C23×C8C2312yes—
97#533—24no—
98#533—24no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125—21no—
104C2×#125—22no—
105C2×#125C2314yes—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8C2312yes—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—24no—
112C2×C4×Q8—24no—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×#117—21no—
119C2×#117—21no—
120C2×#117D411yescharacteristic
121C2×#117D411yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129#491—48no—
130#431—24no—
131C2×C4⋊C8—24no—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136C2×#103—44no—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×C22⋊C8C2312yes—
148C2×#97C2314yes—
149C2×#97C2314yes—
150C2×C4×C8C2312yes—
151#400—24no—
152#400—24no—
153#398—22no—
154#398—22no—
155#398—22no—
156#398—22no—
157#398—24no—
158#398—24no—
159#395—22no—
160#395—22no—
161#395—22no—
162#395—22no—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#391C2312yes—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—22no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389C2314yes—
188#389C2314yes—
189#385—24no—
190#385—22no—
191#385—22no—
192#385—22no—
193#385—22no—
194#384—22no—
195#384—21no—
196#384—21no—
197#384C2312yes—
198#384C2312yes—
199#384C2312yes—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2312yes—
203#382—22no—
204#382—22no—
205#382—21no—
206#382C2314yes—
207#380—21no—
208#380—24no—
209#380C2312yes—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222#376D412yes—
223C43C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#336C2314yes—
235#336C2314yes—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—41no—
243C22×Q8—41no—
244C22×Q8C2412yes—
245C22×Q8C2411yescharacteristic
246C22×Q8C2411yescharacteristic
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—21no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C22×C8—24no—
254C22×C8—42no—
255C22×C8—42no—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—42no—
266C4⋊Q8—42no—
267#125—44no—
268#125—44no—
269#125—44no—
270#125—84no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8—44no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—44no—
285C4×Q8—44no—
286C4×Q8—44no—
287#117—24no—
288#117—24no—
289#117—44no—
290#117—44no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—42no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—82no—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—44no—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42#4114yes—
357C2×C42#4114yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C4⋊C8—24no—
363C4⋊C8—24no—
364C4⋊C8—22no—
365C4⋊C8—22no—
366#103—28no—
367#103—24no—
368#103—24no—
369#103—44no—
370#103—42no—
371#103—44no—
372#103—44no—
373#103—42no—
374#103—44no—
375#103—44no—
376C22⋊C8—24no—
377#97—44no—
378#97—44no—
379C4×C8—24no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24C22×D411yescharacteristic
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—28no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—81no—
424C2×Q8—81no—
425C2×Q8H(5,𝔽2)12yes—
426C2×Q82−1+412yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4#12414yes—
452C22×C4#12414yes—
453C22×C4#13714yes—
454C22×C4#13714yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4C2511yescharacteristic
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—84no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—42no—
535C42—42no—
536C42—42no—
537C42—42no—
538C42—42no—
539C42—42no—
540C42—44no—
541C42—44no—
542C42#12414yes—
543C42#12414yes—
544C42C2×#4112yes—
545C42C2×#4112yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23C2×#12412yes—
553C23C2×#13712yes—
554C23C2×#13712yes—
555C23C23×D411yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—22no—
559Q8—22no—
560Q8—48no—
561Q8—41no—
562Q8—41no—
563Q8—41no—
564Q8—41no—
565Q8—42no—
566Q8—42no—
567Q8—42no—
568Q8—42no—
569Q8—84no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—84no—
622C2×C4—164no—
623C2×C4#491116yescomplemented
624C2×C4C2×#12412yes—
625C2×C4C2×#12412yes—
626C2×C4C2×#12412yes—
627C2×C4C2×#12412yes—
628C2×C4C2×#12412yes—
629C2×C4C2×#13712yes—
630C2×C4C2×#13712yes—
631C2×C4C2×#13712yes—
632C2×C4C2×#13712yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C2×C4#58511yescharacteristic
638C2×C4#58511yescharacteristic
639C8—42no—
640C8—42no—
641C8—44no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22C2×#49114yes—
649C22C22×#12411yescharacteristic
650C22#348811yescharacteristic
651C22#348911yescharacteristic
652C22C22×#13711yescharacteristic
653C22C22×#13711yescharacteristic
654C22#358511yescharacteristic
655C22#358611yescharacteristic
656C4—22no—
657C4—22no—
658C4—22no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—84no—
666C4—82no—
667C4—82no—
668C4—81no—
669C4—81no—
670C4—82no—
671C4—82no—
672C4C2×#49114yes—
673C4C2×#49114yes—
674C4#349211yescharacteristic
675C4#349311yescharacteristic
676C4#349411yescharacteristic
677C4#349511yescharacteristic
678C2—24no—
679C2#3554211yescharacteristic
680C2C22×#49111yescharacteristic
681C2#6062111yescharacteristic
682C2#6196511yescharacteristic
683C2#6196611yescharacteristic
684C2#6211311yescharacteristic
685C2#6211411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #29150C2semidirect
4. #29132C2semidirect
5. #29132C2semidirect
6. #28725C2semidirect
7. #27824C2semidirect
8. #27777C2semidirect
9. #27757C2semidirect
10. #27738C2semidirect
623. C2×C4#491semidirect

Maximal subgroups

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

subgroupindex
#352862
#349532
#291502
#291322
#287252
#278242
#277772
#277572
#277382
#277152
#272682
#218852
#218602
#207342
C2×#18032
#205472