Finite Group Databasebeta

← #6435253

Subgroups — #6435253

Order 512.

1772 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#34060C211yesmaximal, characteristic
3#29934C214yesmaximal, complemented
4#29930C214yesmaximal, complemented
5#29926C212yesmaximal, complemented
6#29926C212yesmaximal, complemented
7#29923C212yesmaximal, complemented
8#29922C212yesmaximal, complemented
9#23736C212yesmaximal, complemented
10#23633C212yesmaximal, complemented
11#23487C212yesmaximal
12#23484C212yesmaximal
13#21259C211yesmaximal, characteristic, complemented
14C2×#1981C211yesmaximal, characteristic, complemented
15#21170C211yesmaximal, characteristic, complemented
16#21169C211yesmaximal, characteristic, complemented
17#20629C211yesmaximal, characteristic
18C2×#536C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C2×#427C2212yes—
23C2×#423C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C2×#419C2212yes—
26C2×#419C2212yes—
27C22×#103C2212yes—
28C22×#103C2212yes—
29C2×#408C2212yes—
30#2857C2212yes—
31#2857C2212yes—
32#2621C2214yes—
33#2547—21no—
34#2547—21no—
35#2546—21no—
36#2546—21no—
37#2486—22no—
38#2486—22no—
39#2320—21no—
40#2319—21no—
41#2318—22no—
42#2317C2211yescharacteristic
43#2317C2211yescharacteristic
44#2314C2212yes—
45#2314C2212yes—
46#2311C2211yescharacteristic
47#2311C2211yescharacteristic
48#2310C2212yes—
49#2295—21no—
50#2293—21no—
51#2290—22no—
52#2285C2212yes—
53#2285C2211yescharacteristic
54#2283C2212yes—
55C4×#117C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57#2097C2214yes—
58#2097C2214yes—
59#2096C2214yes—
60#2095C2214yes—
61#2074C2214yes—
62#2074C2214yes—
63#2072C2214yes—
64#2072C2214yes—
65#2036C2214yes—
66#2036C2214yes—
67#2036C2214yes—
68#2036C2214yes—
69#2035C2214yes—
70#2035C2214yes—
71#2035C2214yes—
72#2035C2214yes—
73#1981C2212yes—
74#1981C2212yes—
75#1981C2214yes—
76#1981C2212yes—
77#1981C2212yes—
78#1981C2212yes—
79#1981C2212yes—
80#1949C2212yes—
81#1949C2212yes—
82#1949C2214yes—
83#1948C2211yescharacteristic
84#1948C2211yescharacteristic
85#1948C2212yes—
86#1947C2211yescharacteristic
87#1947C2211yescharacteristic
88#1947C2212yes—
89#1798C2214yes—
90#1797C2212yes—
91#1797C2212yes—
92#1736C2212yes—
93#1735C2212yes—
94#1729C2212yes—
95C22×#41C2311yescharacteristic
96C23×Q8C2312yes—
97C22×M16C2312yes—
98C23×C8C2312yes—
99#536—44no—
100#536—44no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8—24no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#124—22no—
109C2×#123—24no—
110C2×C22⋊Q8C2314yes—
111C2×C22⋊Q8C2314yes—
112C4×#41—24no—
113C4×#41—24no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×D4D412yes—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D412yes—
119C2×C4×D4D411yescharacteristic
120C2×#117—22no—
121C2×#117C2312yes—
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2312yes—
125C2×#117C2311yescharacteristic
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C42—21no—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#428—44no—
134#427—44no—
135C2×#107C2312yes—
136C2×#107C2314yes—
137C2×#107C2312yes—
138C2×#106C2312yes—
139C2×#106C2314yes—
140C2×#106C2312yes—
141#423—24no—
142C2×C4⋊C8—24no—
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148#419—44no—
149#419—44no—
150C2×#103—44no—
151C2×#103—44no—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160#408—44no—
161C2×C22⋊C8C2314yes—
162C2×C22⋊C8C2314yes—
163#398—24no—
164#398—24no—
165#397—24no—
166#397—24no—
167#391—21no—
168#391—21no—
169#391—22no—
170#391—22no—
171#391C2314yes—
172#391C2314yes—
173#390—21no—
174#390—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#386—22no—
192#386—22no—
193#385—22no—
194#385—22no—
195#384C2312yes—
196#384C2312yes—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#383—22no—
200#382—22no—
201#382C2312yes—
202#382C2312yes—
203#382C2312yes—
204#381—22no—
205#381—24no—
206#380—22no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380—22no—
211#380—21no—
212#380—21no—
213#380C2312yes—
214#380C2312yes—
215#380C2312yes—
216#380C2312yes—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C22⋊C4—21no—
229C4×C22⋊C4—21no—
230C4×C22⋊C4—22no—
231#376—21no—
232#376—22no—
233C2×#95—21no—
234C43—21no—
235#340C2312yes—
236#340C2312yes—
237#340C2314yes—
238#340C2314yes—
239#340C2312yes—
240#340C2312yes—
241#336C2314yes—
242#336C2314yes—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C22×Q8—22no—
250C22×Q8—24no—
251C22×Q8—24no—
252C22×Q8—24no—
253C22×Q8—24no—
254C22×Q8—24no—
255C22×Q8—24no—
256C22×Q8C2×D411yescharacteristic
257C22×Q8C2414yes—
258C22×D4C2×D412yes—
259C22×D4C2×D411yescharacteristic
260C23×C4—22no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D412yes—
266C23×C4C2×D411yescharacteristic
267C23×C4C2412yes—
268C23×C4C2411yescharacteristic
269C2×M16—24no—
270C2×M16—24no—
271C2×M16—24no—
272C22×C8—24no—
273C22×C8—24no—
274C22×C8—24no—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2414yes—
278C4⋊Q8—24no—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—48no—
283C4⋊Q8—44no—
284C4⋊Q8—44no—
285#124—48no—
286#123—416no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290C22⋊Q8—44no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×D4—28no—
294C4×D4—28no—
295C4×D4—24no—
296C4×D4—24no—
297#117—22no—
298#117—22no—
299#117—24no—
300#117—24no—
301#117—24no—
302#117—24no—
303#117—22no—
304#117—22no—
305#117—22no—
306#117—22no—
307#117—84no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—22no—
365C2×C42—82no—
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42#4114yes—
371C2×C42#4114yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—42no—
376#107—42no—
377#107—44no—
378#106—42no—
379#106—42no—
380#106—44no—
381C4⋊C8—22no—
382C4⋊C8—22no—
383C4⋊C8—24no—
384C4⋊C8—24no—
385C4⋊C8—24no—
386#103—44no—
387#103—44no—
388#103—44no—
389#103—44no—
390#103—44no—
391#103—44no—
392#103—44no—
393#103—44no—
394C22⋊C8—44no—
395C22⋊C8—44no—
396#95—22no—
397#95—22no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—24no—
416C24C22×D412yes—
417C24C22×D411yescharacteristic
418#41—44no—
419#41—44no—
420#41—44no—
421#41—44no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—24no—
426C2×Q8—24no—
427C2×Q8—24no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—48no—
435C2×Q8—48no—
436C2×Q8—48no—
437C2×Q8—48no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—24no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4—81no—
478C22×C4#12314yes—
479C22×C4#12314yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C42−1+412yes—
494C22×C4C2511yescharacteristic
495M16—48no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—48no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—48no—
529C4⋊C4—48no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—42no—
548C22⋊C4—44no—
549C22⋊C4—42no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C42—24no—
555C42—24no—
556C42—22no—
557C42—22no—
558C42—42no—
559C42—44no—
560C42—42no—
561C42—44no—
562C42—42no—
563C42—42no—
564C42—81no—
565C42—81no—
566C42—82no—
567C42#12314yes—
568C42#12314yes—
569C42C2×#4112yes—
570C42C2×#4112yes—
571C42C2×#4112yes—
572C42C2×#4112yes—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23C2×#12312yes—
585C23#53512yes—
586C23#53712yes—
587C23C23×D411yescharacteristic
588Q8—44no—
589Q8—44no—
590Q8—44no—
591Q8—44no—
592Q8—44no—
593Q8—88no—
594D4—48no—
595D4—44no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—44no—
613C2×C4—48no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—84no—
648C2×C4—88no—
649C2×C4C2×#12312yes—
650C2×C4C2×#12312yes—
651C2×C4C2×#12312yes—
652C2×C4C2×#12312yes—
653C2×C4C2×#12312yes—
654C2×C4#53512yes—
655C2×C4#53712yes—
656C2×C4#54014yes—
657C2×C4#54014yes—
658C2×C4#54214yes—
659C2×C4#54214yes—
660C2×C4#57712yes—
661C2×C4#57812yes—
662C2×C4C22×#4111yescharacteristic
663C2×C4C22×#4111yescharacteristic
664C2×C4C2×H(5,𝔽2)11yescharacteristic
665C2×C4C2×H(5,𝔽2)11yescharacteristic
666C2×C4C2×H(5,𝔽2)11yescharacteristic
667C2×C4C2×2−1+411yescharacteristic
668C8—42no—
669C8—42no—
670C8—44no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22#262414yes—
685C22C22×#12311yescharacteristic
686C22C2×#53511yescharacteristic
687C22C2×#53711yescharacteristic
688C22C2×#57711yescharacteristic
689C22C2×#57811yescharacteristic
690C22#358611yescharacteristic
691C22#358611yescharacteristic
692C4—22no—
693C4—22no—
694C4—22no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—84no—
699C4—81no—
700C4—82no—
701C4—82no—
702C4—81no—
703C4—82no—
704C4—81no—
705C4—82no—
706C4—81no—
707C4#262414yes—
708C4#309714yes—
709C4C2×#54011yescharacteristic
710C4C2×#54011yescharacteristic
711C4C2×#54211yescharacteristic
712C4C2×#54211yescharacteristic
713C2—24no—
714C2—82no—
715C2C2×#262411yescharacteristic
716C2C2×#309711yescharacteristic
717C2#6051911yescharacteristic
718C2#6083611yescharacteristic
719C2#6083711yescharacteristic
720C2#6092311yescharacteristic
721C2#6092511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29934C2semidirect
4. #29930C2semidirect
5. #29926C2semidirect
6. #29926C2semidirect
7. #29923C2semidirect
8. #29922C2semidirect
9. #23736C2semidirect
10. #23633C2semidirect
13. #21259C2semidirect
14. C2×#1981C2semidirect
15. #21170C2semidirect
16. #21169C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#340602
#299342
#299302
#299262
#299232
#299222
#237362
#236332
#234872
#234842
#212592
C2×#19812
#211702
#211692
#206292