Finite Group Databasebeta

← #6458585

Subgroups — #6458585

Order 512.

1790 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#35181C211yesmaximal, characteristic, complemented
3#27764C214yesmaximal, complemented
4#27762C214yesmaximal, complemented
5#27757C212yesmaximal, complemented
6#27756C212yesmaximal, complemented
7#27749C212yesmaximal, complemented
8#27748C212yesmaximal, complemented
9#23633C212yesmaximal, complemented
10#23510C212yesmaximal
11C2×#1981C211yesmaximal, characteristic, complemented
12#21205C212yesmaximal
13#21202C211yesmaximal, characteristic, complemented
14#21090C212yesmaximal, complemented
15#20645C211yesmaximal, characteristic
16#20441C211yesmaximal, characteristic, complemented
17C4×#417C211yesmaximal, characteristic
18C2×#536C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#417C2211yescharacteristic
23C2×#417C2211yescharacteristic
24C2×#419C2212yes—
25C2×#419C2212yes—
26C2×#416C2212yes—
27C22×#103C2212yes—
28C22×#103C2212yes—
29C2×#405C2212yes—
30C2×C4×M16C2211yescharacteristic
31#2857—21no—
32#2857—21no—
33#2851—22no—
34#2646—22no—
35#2621—22no—
36#2571—21no—
37#2571C2212yes—
38#2570—22no—
39#2570C2214yes—
40#2552—21no—
41#2552C2212yes—
42#2547C2212yes—
43#2547C2212yes—
44#2546C2212yes—
45#2546C2212yes—
46#2486—22no—
47#2486—22no—
48#2317C2212yes—
49#2317C2212yes—
50#2310C2212yes—
51#2285C2211yescharacteristic
52#2285C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2074C2214yes—
55#2074C2214yes—
56#2072C2214yes—
57#2072C2214yes—
58#2043C2214yes—
59#2043C2214yes—
60#2042C2214yes—
61#2042C2214yes—
62#1981C2212yes—
63#1981C2212yes—
64#1981C2214yes—
65#1960C2214yes—
66#1960C2214yes—
67#1959C2214yes—
68#1958C2214yes—
69#1956C2212yes—
70#1956C2212yes—
71#1956C2214yes—
72#1935C2214yes—
73#1935C2214yes—
74#1933C2214yes—
75#1933C2214yes—
76#1805C2212yes—
77#1804C2212yes—
78#1803C2211yescharacteristic
79#1803C2211yescharacteristic
80#1802C2211yescharacteristic
81#1802C2211yescharacteristic
82#1771C2214yes—
83#1770C2212yes—
84C8×C4⋊C4C2212yes—
85#1763C2212yes—
86#1762C2212yes—
87C4×#103C2212yes—
88C4×#102C2212yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2212yes—
92#1729C2212yes—
93C22×#41C2311yescharacteristic
94C23×Q8C2312yes—
95C22×M16C2311yescharacteristic
96C22×M16C2311yescharacteristic
97C23×C8C2312yes—
98#536—28no—
99#536—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2314yes—
107C2×#125—22no—
108C2×#124C2314yes—
109C2×#123—24no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—22no—
112C4×#41—24no—
113C4×#41—24no—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—21no—
121C2×#117C2311yescharacteristic
122C2×#117C2312yes—
123C2×#117C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C42—21no—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C2×C4⋊C8—24no—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135#419—28no—
136#419—44no—
137#417—24no—
138#417—24no—
139#416—44no—
140C2×#103—44no—
141C2×#103—44no—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158#405—44no—
159C4×M16—24no—
160C2×#97C2312yes—
161C2×#97C2314yes—
162C2×C4×C8C2312yes—
163C2×C4×C8C2314yes—
164#400—24no—
165#400—24no—
166#398—24no—
167#398—24no—
168#398—24no—
169#398—24no—
170#397—24no—
171#397—24no—
172#391—22no—
173#391—22no—
174#391C2314yes—
175#391C2314yes—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389C2314yes—
185#389C2314yes—
186#387—24no—
187#387—24no—
188#385—24no—
189#385—24no—
190#384—22no—
191#384D411yescharacteristic
192#384D411yescharacteristic
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2314yes—
196#382—22no—
197#382—22no—
198#382D412yes—
199#382C2312yes—
200#382C2314yes—
201#381—24no—
202#380—21no—
203#380—21no—
204#380—22no—
205#380—22no—
206#380—22no—
207#380—22no—
208#380D412yes—
209#380D412yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C2×#95—21no—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#336C2314yes—
228#336C2314yes—
229C2×#41—22no—
230C2×#41—22no—
231C2×#41—22no—
232C2×#41—22no—
233C2×#41—22no—
234C2×#41#4114yes—
235C22×Q8—22no—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—24no—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—24no—
242C22×Q8C2414yes—
243C22×Q8C2411yescharacteristic
244C22×D4—21no—
245C22×D4C2411yescharacteristic
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×M16—22no—
256C2×M16—22no—
257C2×M16—22no—
258C2×M16—22no—
259C2×M16—22no—
260C2×M16#4114yes—
261C22×C8—24no—
262C22×C8—24no—
263C22×C8—24no—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C4⋊Q8—28no—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276#125—84no—
277#124—28no—
278#124—44no—
279#123—416no—
280C22⋊Q8—48no—
281C22⋊Q8—48no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—44no—
291C4×D4—44no—
292#117—24no—
293#117—24no—
294#117—22no—
295#117—22no—
296#117—24no—
297#117—24no—
298#117—44no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4C2×D414yes—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—44no—
349C2×C42—82no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42#4114yes—
353C2×C42#4114yes—
354C2×C42C2414yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C4⋊C8—22no—
358C4⋊C8—22no—
359C4⋊C8—24no—
360#103—24no—
361#103—24no—
362#103—28no—
363#103—28no—
364#103—44no—
365#103—44no—
366#103—44no—
367#103—44no—
368#103—42no—
369#103—42no—
370#103—44no—
371#103—44no—
372#102—24no—
373#102—24no—
374#102—42no—
375#102—42no—
376#97—24no—
377#97—44no—
378C4×C8—24no—
379C4×C8—44no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95C2×D414yes—
393#95C2×D414yes—
394C24—21no—
395C24C22×D411yescharacteristic
396#41—28no—
397#41—28no—
398#41—44no—
399#41—44no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—21no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—48no—
415C2×Q8—48no—
416C2×Q8—48no—
417C2×Q8—48no—
418C2×Q8C2×#4112yes—
419C2×D4—24no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C2×#4112yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—81no—
458C22×C4#12414yes—
459C22×C4#12414yes—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C4C2511yescharacteristic
474M16—28no—
475M16—44no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—48no—
488C2×C8C2×#4114yes—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4—84no—
525C22⋊C4—28no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C42—24no—
533C42—24no—
534C42—42no—
535C42—42no—
536C42—42no—
537C42—44no—
538C42—44no—
539C42—82no—
540C42#12414yes—
541C42#12414yes—
542C42C2×#4112yes—
543C42C2×#4112yes—
544C42C2×#4112yes—
545C42C2×#4112yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C2×#12412yes—
558C23#54612yes—
559C23#55012yes—
560C23C23×D411yescharacteristic
561Q8—22no—
562Q8—22no—
563Q8—41no—
564Q8—41no—
565Q8—44no—
566Q8—44no—
567Q8—44no—
568Q8—44no—
569Q8—88no—
570D4—22no—
571D4—22no—
572D4—48no—
573D4—41no—
574D4—41no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—88no—
618C2×C4—84no—
619C2×C4C2×#12412yes—
620C2×C4C2×#12412yes—
621C2×C4C2×#12412yes—
622C2×C4C2×#12412yes—
623C2×C4C2×#12412yes—
624C2×C4#52914yes—
625C2×C4#54014yes—
626C2×C4#54612yes—
627C2×C4#55012yes—
628C2×C4#55314yes—
629C2×C4#55314yes—
630C2×C4#57712yes—
631C2×C4#57812yes—
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C8—24no—
639C8—42no—
640C8—42no—
641C8—42no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22#249014yes—
656C22C22×#12411yescharacteristic
657C22C2×#54611yescharacteristic
658C22C2×#55011yescharacteristic
659C22C2×#57711yescharacteristic
660C22C2×#57811yescharacteristic
661C22#358611yescharacteristic
662C22#358611yescharacteristic
663C4—22no—
664C4—22no—
665C4—22no—
666C4—44no—
667C4—44no—
668C4—42no—
669C4—82no—
670C4—81no—
671C4—81no—
672C4—84no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4#249014yes—
677C4#313714yes—
678C4C2×#52911yescharacteristic
679C4C2×#54011yescharacteristic
680C4C2×#55311yescharacteristic
681C4C2×#55311yescharacteristic
682C2—24no—
683C2—82no—
684C2C2×#249011yescharacteristic
685C2C2×#313711yescharacteristic
686C2#6062011yescharacteristic
687C2#6120911yescharacteristic
688C2#6121011yescharacteristic
689C2#6142011yescharacteristic
690C2#6142211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35181C2semidirect
3. #27764C2semidirect
4. #27762C2semidirect
5. #27757C2semidirect
6. #27756C2semidirect
7. #27749C2semidirect
8. #27748C2semidirect
9. #23633C2semidirect
11. C2×#1981C2semidirect
13. #21202C2semidirect
14. #21090C2semidirect
16. #20441C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#351812
#277642
#277622
#277572
#277562
#277492
#277482
#236332
#235102
C2×#19812
#212052
#212022
#210902
#206452
#204412
C4×#4172