Finite Group Databasebeta

← #6856097

Subgroups — #6856097

Order 512.

1700 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36579C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3#33108C211yesmaximal, characteristic, complemented
4#32639C214yesmaximal, complemented
5#32586C214yesmaximal, complemented
6#29760C212yesmaximal
7#29336C212yesmaximal
8#29241C212yesmaximal, complemented
9#28646C212yesmaximal, complemented
10#26283C212yesmaximal, complemented
11#25902C211yesmaximal, characteristic, complemented
12#25650C212yesmaximal, complemented
13#25478C212yesmaximal, complemented
14#24298C211yesmaximal, characteristic
15#22510C211yesmaximal, characteristic, complemented
16#22084C212yesmaximal, complemented
17C4×#480C211yesmaximal, characteristic
18C2×#556C2211yescharacteristic
19C22×C4⋊Q8—21no—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C2×#487C2212yes—
23C2×#483C2212yes—
24C2×#480C2211yescharacteristic
25C2×#474C2212yes—
26C2×#446C2212yes—
27#2857C2211yescharacteristic
28#2850C2212yes—
29#2592C2212yes—
30#2581C2212yes—
31#2568C2211yescharacteristic
32#2546—21no—
33#2546—21no—
34#2546—22no—
35#2546C2212yes—
36#2546C2214yes—
37#2546C2212yes—
38#2538C2211yescharacteristic
39#2481—22no—
40#2469—22no—
41#2421—21no—
42#2409—21no—
43C2×#398—22no—
44#2362—21no—
45#2355C2211yescharacteristic
46#2352—21no—
47C4×C22⋊Q8C2212yes—
48C4×#121C2211yescharacteristic
49C2×#391—21no—
50C2×#389C2212yes—
51C2×#389C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2101C2214yes—
54#2087C2214yes—
55#2085C2214yes—
56#2068C2214yes—
57#2068C2214yes—
58#2065C2214yes—
59#2057C2212yes—
60#2042C2214yes—
61#2042C2214yes—
62#2024C2214yes—
63#2023C2212yes—
64#1990C2212yes—
65#1988C2212yes—
66#1985C2214yes—
67#1984C2212yes—
68#1981C2212yes—
69#1948C2212yes—
70#1944C2214yes—
71#1936C2214yes—
72#1935C2214yes—
73#1935C2214yes—
74#1927C2214yes—
75#1925C2212yes—
76#1921C2214yes—
77#1920C2214yes—
78#1918C2212yes—
79#1874C2212yes—
80#1874C2211yescharacteristic
81#1872C2212yes—
82#1872C2212yes—
83#1872C2212yes—
84#1863C2214yes—
85#1863C2214yes—
86#1852C2211yescharacteristic
87#1852C2212yes—
88#1823C2211yescharacteristic
89#1797C2212yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8D411yescharacteristic
97#556—44no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8—48no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#126—22no—
109C2×#124C2312yes—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8C2312yes—
113C2×#121C2311yescharacteristic
114C2×C4×Q8—44no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D412yes—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×D4—21no—
123C2×C4×D4C2311yescharacteristic
124C2×#117—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4D412yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#487—44no—
137#483—28no—
138#480—24no—
139#474—44no—
140#446—44no—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2311yescharacteristic
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2311yescharacteristic
157C2×#97C2314yes—
158C2×C4×C8C2312yes—
159#400—22no—
160#400—22no—
161#398—24no—
162#398—24no—
163#398—24no—
164#398—24no—
165#397—22no—
166#396—22no—
167#395—22no—
168#393—22no—
169#391—21no—
170#391—21no—
171#391—22no—
172#391—22no—
173#391—24no—
174#391C2312yes—
175#391C2312yes—
176#391C2312yes—
177#390C2312yes—
178#390C2311yescharacteristic
179#389—21no—
180#389—21no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389C2311yescharacteristic
186#389C2314yes—
187#389C2314yes—
188#389C2312yes—
189#389C2314yes—
190#389C2314yes—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—21no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#386—22no—
201#386—22no—
202#386—21no—
203#385—22no—
204#384C2311yescharacteristic
205#382—21no—
206#382C2312yes—
207#380—21no—
208#380—21no—
209#380—21no—
210#380C2311yescharacteristic
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C2×#95—21no—
222C2×#95—22no—
223C2×#95—22no—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C22×Q8—24no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2×D412yes—
248C22×Q8C2412yes—
249C22×D4—21no—
250C22×D4C2411yescharacteristic
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2411yescharacteristic
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2414yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—416no—
270C4⋊Q8—48no—
271C4⋊Q8—48no—
272#126—48no—
273#124—44no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278#121—22no—
279#121—22no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×Q8—42no—
285C4×Q8—42no—
286C4×Q8—44no—
287C4×D4—24no—
288C4×D4—44no—
289#117—44no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4#4114yes—
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—44no—
347C2×C42—44no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356#107—24no—
357#107—24no—
358#107—44no—
359#106—28no—
360C4⋊C8—24no—
361C4⋊C8—44no—
362#103—28no—
363#103—28no—
364#103—44no—
365#103—44no—
366#102—24no—
367#102—24no—
368#102—44no—
369C22⋊C8—22no—
370C22⋊C8—24no—
371C22⋊C8—22no—
372#97—44no—
373C4×C8—24no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393C24—21no—
394C24C22×D411yescharacteristic
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—416no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—816no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4AGL(1,ℤ/8)14yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2×#4112yes—
467C22×C42−1+412yes—
468C22×C4C2511yescharacteristic
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—81no—
512C4⋊C4C2×#4114yes—
513C4⋊C42−1+414yes—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—81no—
523C22⋊C4—82no—
524C42—22no—
525C42—22no—
526C42—24no—
527C42—22no—
528C42—24no—
529C42—42no—
530C42—42no—
531C42—44no—
532C42—44no—
533C42—44no—
534C42—42no—
535C42—82no—
536C42H(5,𝔽2)14yes—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23#54812yes—
549C23C2×AGL(1,ℤ/8)12yes—
550C23#57812yes—
551C23C23×D411yescharacteristic
552Q8—48no—
553Q8—44no—
554Q8—44no—
555Q8—44no—
556Q8—44no—
557Q8—88no—
558Q8—88no—
559D4—42no—
560D4—42no—
561D4—84no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—84no—
615C2×C4—88no—
616C2×C4—84no—
617C2×C4#53412yes—
618C2×C4#53412yes—
619C2×C4#54812yes—
620C2×C4C2×AGL(1,ℤ/8)12yes—
621C2×C4C2×AGL(1,ℤ/8)12yes—
622C2×C4#57512yes—
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)12yes—
625C2×C4C2×2−1+412yes—
626C2×C4C2×2−1+411yescharacteristic
627C8—42no—
628C8—44no—
629C8—42no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22#336714yes—
644C22C2×#53411yescharacteristic
645C22C2×#54811yescharacteristic
646C22#348611yescharacteristic
647C22C22×AGL(1,ℤ/8)11yescharacteristic
648C22C2×#57511yescharacteristic
649C22C2×#57811yescharacteristic
650C22#358611yescharacteristic
651C4—22no—
652C4—22no—
653C4—22no—
654C4—42no—
655C4—42no—
656C4—41no—
657C4—42no—
658C4—42no—
659C4—44no—
660C4—44no—
661C4—82no—
662C4—81no—
663C4—84no—
664C4—81no—
665C4—81no—
666C4—82no—
667C4#319414yes—
668C4#351112yes—
669C4#355212yes—
670C2—24no—
671C2—161no—
672C2#3972111yescharacteristic
673C2C2×#319411yescharacteristic
674C2#6075911yescharacteristic
675C2C2×#336711yescharacteristic
676C2#6132011yescharacteristic
677C2#6182711yescharacteristic
678C2#6184211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2546C2semidirect
3. #33108C2semidirect
4. #32639C2semidirect
5. #32586C2semidirect
8. #29241C2semidirect
9. #28646C2semidirect
10. #26283C2semidirect
11. #25902C2semidirect
12. #25650C2semidirect
13. #25478C2semidirect
15. #22510C2semidirect
16. #22084C2semidirect

Maximal subgroups

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

subgroupindex
#365792
C2×#25462
#331082
#326392
#325862
#297602
#293362
#292412
#286462
#262832
#259022
#256502
#254782
#242982
#225102
#220842
C4×#4802