Finite Group Databasebeta

← #7505379

Subgroups — #7505379

Order 512.

2042 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35620C211yesmaximal, characteristic, complemented
2#35554C211yesmaximal, characteristic, complemented
3#31665C214yesmaximal, complemented
4#31662C214yesmaximal, complemented
5#31661C214yesmaximal, complemented
6#31658C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8#23632C211yesmaximal, characteristic, complemented
9#23544C211yesmaximal, characteristic, complemented
10#23456C211yesmaximal, characteristic, complemented
11#23456C211yesmaximal, characteristic, complemented
12#23430C211yesmaximal, characteristic
13C2×#1961C211yesmaximal, characteristic, complemented
14#21086C211yesmaximal, characteristic, complemented
15#21021C211yesmaximal, characteristic, complemented
16#20932C211yesmaximal, characteristic, complemented
17#20932C211yesmaximal, characteristic, complemented
18#20929C211yesmaximal, characteristic, complemented
19#20650C211yesmaximal, characteristic, complemented
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C22×AGL(1,ℤ/8)C2211yescharacteristic
22C2×#544C2211yescharacteristic
23C22×#127C2211yescharacteristic
24C2×#520C2211yescharacteristic
25C2×#518C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#418C2211yescharacteristic
28C2×#418C2211yescharacteristic
29C2×#416C2211yescharacteristic
30C2×#416C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#97C2211yescharacteristic
36#2568C2212yescomplemented
37#2566C2212yescomplemented
38#2565C2212yescomplemented
39#2563C2212yescomplemented
40#2450—24no—
41#2445—22no—
42#2445—22no—
43#2445C2214yes—
44#2445C2214yes—
45#2442—22no—
46#2439—22no—
47#2405—24no—
48#2317C2211yescharacteristic
49#2311C2211yescharacteristic
50#2309C2211yescharacteristic
51#2309C2211yescharacteristic
52#2309C2211yescharacteristic
53#2297C2211yescharacteristic
54#2296C2211yescharacteristic
55#2296C2211yescharacteristic
56C2×#382C2211yescharacteristic
57#2101C2214yescomplemented
58#2101C2214yescomplemented
59#2073C2214yescomplemented
60#2071C2214yescomplemented
61#2055C2214yescomplemented
62#2054C2214yescomplemented
63#2025C2214yescomplemented
64#2025C2214yescomplemented
65#2023C2214yescomplemented
66#2023C2214yescomplemented
67#2014C2212yes—
68#2014C2212yes—
69#2011C2212yes—
70#2011C2212yes—
71#1961C2214yes—
72#1961C2214yes—
73#1936C2212yescomplemented
74#1936C2212yescomplemented
75#1935C2212yescomplemented
76#1935C2212yescomplemented
77#1917C2214yescomplemented
78#1917C2214yescomplemented
79#1880C2214yescomplemented
80#1880C2214yescomplemented
81#1879C2214yescomplemented
82#1879C2214yescomplemented
83#1879C2214yescomplemented
84#1877C2214yescomplemented
85#1804C2214yescomplemented
86#1802C2214yescomplemented
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4—22no—
99C23×D4C2311yescharacteristic
100C23×D4C2311yescharacteristic
101C2×AGL(1,ℤ/8)—24no—
102C2×AGL(1,ℤ/8)—24no—
103C2×AGL(1,ℤ/8)—24no—
104C2×AGL(1,ℤ/8)—42no—
105C2×AGL(1,ℤ/8)—42no—
106C2×AGL(1,ℤ/8)—42no—
107C22×SD16C2312yes—
108C22×SD16C2312yes—
109C22×D8C2312yes—
110C22×D8C2312yes—
111C22×M16C2311yescharacteristic
112C22×M16C2311yescharacteristic
113C23×C8C2311yescharacteristic
114C23×C8C2311yescharacteristic
115#544—42no—
116#544—42no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C2312yes—
119C2×#127—24no—
120C2×#127—22no—
121C2×#127—42no—
122C2×#127C2312yes—
123C2×#127C2312yes—
124C2×#125—21no—
125C2×#125C2312yes—
126C2×#123—24no—
127C2×#123—24no—
128C2×C22⋊Q8—22no—
129#520—42no—
130#520—42no—
131#518—24no—
132#518—42no—
133C2×C4×Q8C2312yes—
134C2×C4×D4—22no—
135C2×C4×D4—22no—
136C2×C4×D4C2314yes—
137C2×C4×D4C2312yes—
138C2×#117—21no—
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C2×#117C2311yescharacteristic
144C22×C4⋊C4—21no—
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C22⋊C4—22no—
148C22×C42C2311yescharacteristic
149#424—42no—
150C2×C4⋊C8C2314yes—
151#418—42no—
152#418—42no—
153#416—24no—
154#416—42no—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#102—42no—
158C2×#102—42no—
159C2×#102—42no—
160C2×#102—42no—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×#102C2312yes—
164C2×#102C2314yes—
165C2×#102C2312yes—
166C2×#102C2312yes—
167C2×#102C2314yes—
168C2×#102C2312yes—
169C2×#102C2314yes—
170C2×#102C2312yes—
171C2×#97—42no—
172C2×#97C2312yes—
173C2×#97C2312yes—
174#400—24no—
175#398—24no—
176#396—24no—
177#394—24no—
178#394—24no—
179#394—24no—
180#394—24no—
181#393—24no—
182#391—24no—
183#391C2311yescharacteristic
184#391C2312yes—
185#391C2311yescharacteristic
186#390—21no—
187#390—21no—
188#390—21no—
189#390—21no—
190#390—22no—
191#390C2311yescharacteristic
192#390C2312yes—
193#390C2314yes—
194#390C2314yes—
195#390C2314yes—
196#390C2311yescharacteristic
197#389—22no—
198#389D412yes—
199#389D412yes—
200#389C2312yes—
201#389C2312yes—
202#387—24no—
203#386—22no—
204#386—22no—
205#386—24no—
206#386—24no—
207#385—22no—
208#385—22no—
209#385—24no—
210#385D414yes—
211#384C2312yes—
212#382—22no—
213#382C2312yes—
214#382C2312yes—
215#382C2312yes—
216#380—24no—
217#380—22no—
218#379—22no—
219#379—22no—
220#379—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C22⋊C4—22no—
225C2×#95—22no—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#336C2314yes—
235#336C2312yes—
236#336C2312yes—
237#336C2314yes—
238#336C2314yes—
239C25—21no—
240C25—21no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41#4114yes—
247C22×Q8—22no—
248C22×Q8C2411yescharacteristic
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—24no—
253C22×D4—24no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—48no—
266C22×D4—84no—
267C22×D4—84no—
268C22×D4#4114yes—
269C22×D4C2412yes—
270C22×D4C2412yes—
271C22×D4C2411yescharacteristic
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—24no—
276C23×C4—22no—
277C23×C4C2×D412yes—
278C23×C4C2411yescharacteristic
279AGL(1,ℤ/8)—416no—
280AGL(1,ℤ/8)—416no—
281C2×SD16—24no—
282C2×SD16—24no—
283C2×SD16—24no—
284C2×SD16—42no—
285C2×SD16—42no—
286C2×SD16—42no—
287C2×D8—24no—
288C2×D8—24no—
289C2×D8—24no—
290C2×D8—42no—
291C2×D8—42no—
292C2×D8—42no—
293C2×M16—22no—
294C2×M16—22no—
295C2×M16—22no—
296C2×M16—22no—
297C2×M16—22no—
298C2×M16—22no—
299C22×C8—22no—
300C22×C8—22no—
301C22×C8—22no—
302C22×C8—22no—
303C22×C8—22no—
304C22×C8—22no—
305C22×C8C2412yes—
306C22×C8C2412yes—
307C22×C8C2412yes—
308C22×C8C2412yes—
309C4⋊Q8—42no—
310C4⋊Q8—42no—
311C4⋊Q8—44no—
312#127—24no—
313#127—24no—
314#127—42no—
315#127—42no—
316#127—48no—
317#125—42no—
318#125—42no—
319#125—44no—
320#123—88no—
321#123—88no—
322C22⋊Q8—48no—
323C4×Q8—42no—
324C4×Q8—42no—
325C4×D4—28no—
326C4×D4—44no—
327C4×D4—42no—
328C4×D4—42no—
329C4×D4—48no—
330C4×D4—48no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—22no—
338#117—22no—
339#117—22no—
340#117—22no—
341#117—44no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—44no—
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D412yes—
359C2×C4⋊C4C2×D414yes—
360C2×C4⋊C4C2×D414yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2411yescharacteristic
365C2×C4⋊C4C2411yescharacteristic
366C2×C4⋊C4C2412yes—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—48no—
382C2×C22⋊C4C2×D412yes—
383C2×C22⋊C4C2×D412yes—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42C2×D412yes—
391C2×C42C2412yes—
392C2×C42C2412yes—
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C4⋊C8—44no—
396#103—24no—
397#103—42no—
398#102—24no—
399#102—44no—
400#102—44no—
401#102—44no—
402#102—44no—
403#102—42no—
404#102—42no—
405#102—42no—
406#102—42no—
407#102—42no—
408#97—42no—
409#97—42no—
410#95—24no—
411#95—24no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—24no—
417#95—24no—
418#95C2×D414yes—
419#95C2×D414yes—
420C24—21no—
421C24—22no—
422C24—24no—
423C24—22no—
424C24—42no—
425C24—42no—
426C24—42no—
427C24—42no—
428C24—42no—
429C24—82no—
430C24—82no—
431C24—82no—
432C24—82no—
433C24C22×D411yescharacteristic
434#41—28no—
435#41—28no—
436#41—44no—
437#41—44no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—44no—
444C2×Q8—84no—
445C2×Q8C2×#4112yes—
446C2×D4—24no—
447C2×D4—28no—
448C2×D4—24no—
449C2×D4—28no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—48no—
467C2×D4—48no—
468C2×D4—48no—
469C2×D4—48no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—44no—
475C2×D4—44no—
476C2×D4—44no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—44no—
481C2×D4—44no—
482C2×D4—44no—
483C2×D4—84no—
484C2×D4—84no—
485C2×D4—832no—
486C2×D4C2×#4112yes—
487C2×D4C2×#4114yes—
488C22×C4—24no—
489C22×C4—21no—
490C22×C4—24no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—48no—
500C22×C4—44no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—84no—
506C22×C4—84no—
507C22×C4—84no—
508C22×C4—82no—
509C22×C4—82no—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D414yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D411yescharacteristic
516C22×C4C2×#4112yes—
517C22×C4C2×#4112yes—
518C22×C4H(5,𝔽2)12yes—
519C22×C4H(5,𝔽2)12yes—
520C22×C4H(5,𝔽2)12yes—
521C22×C42−1+412yes—
522C22×C4C2511yescharacteristic
523SD16—48no—
524SD16—48no—
525D8—48no—
526D8—48no—
527M16—44no—
528M16—44no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—44no—
542C2×C8—44no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—24no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—21no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—44no—
563C4⋊C4—44no—
564C4⋊C4—44no—
565C4⋊C4—44no—
566C4⋊C4—42no—
567C4⋊C4—42no—
568C4⋊C4—84no—
569C22⋊C4—24no—
570C22⋊C4—24no—
571C22⋊C4—48no—
572C22⋊C4—44no—
573C22⋊C4—42no—
574C22⋊C4—42no—
575C22⋊C4—42no—
576C22⋊C4—84no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C22⋊C4—84no—
580C22⋊C4—84no—
581C22⋊C4—84no—
582C22⋊C4—84no—
583C22⋊C4—84no—
584C22⋊C4—84no—
585C42—24no—
586C42—21no—
587C42—21no—
588C42—22no—
589C42—21no—
590C42—21no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—44no—
595C42—42no—
596C42—42no—
597C42—42no—
598C23—22no—
599C23—22no—
600C23—24no—
601C23—22no—
602C23—42no—
603C23—42no—
604C23—44no—
605C23—44no—
606C23—41no—
607C23—42no—
608C23—41no—
609C23—42no—
610C23—48no—
611C23—44no—
612C23—44no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—88no—
617C23—88no—
618C23—84no—
619C23—162no—
620C23—162no—
621C23—162no—
622C23—162no—
623C23—164no—
624C23—164no—
625C23#53512yes—
626C23#54612yes—
627C23#54812yes—
628C23C23×D411yescharacteristic
629Q8—22no—
630Q8—22no—
631Q8—41no—
632Q8—41no—
633Q8—164no—
634D4—22no—
635D4—24no—
636D4—22no—
637D4—24no—
638D4—48no—
639D4—42no—
640D4—42no—
641D4—41no—
642D4—42no—
643D4—48no—
644D4—42no—
645D4—42no—
646D4—41no—
647D4—42no—
648D4—816no—
649D4—816no—
650D4—84no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—24no—
664C2×C4—22no—
665C2×C4—48no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—48no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—41no—
678C2×C4—42no—
679C2×C4—41no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—168no—
684C2×C4—164no—
685C2×C4—162no—
686C2×C4—162no—
687C2×C4#53512yes—
688C2×C4#53814yes—
689C2×C4#53914yes—
690C2×C4#54612yes—
691C2×C4#54812yes—
692C2×C4#55114yes—
693C2×C4#56214yes—
694C2×C4C22×#4111yescharacteristic
695C2×C4C22×#4111yescharacteristic
696C2×C4C2×H(5,𝔽2)11yescharacteristic
697C2×C4C2×H(5,𝔽2)11yescharacteristic
698C2×C4C2×H(5,𝔽2)11yescharacteristic
699C2×C4C2×2−1+411yescharacteristic
700C8—42no—
701C8—42no—
702C8—42no—
703C8—42no—
704C22—22no—
705C22—22no—
706C22—24no—
707C22—24no—
708C22—42no—
709C22—44no—
710C22—44no—
711C22—48no—
712C22—81no—
713C22—81no—
714C22—82no—
715C22—82no—
716C22—84no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—84no—
722C22—84no—
723C22—168no—
724C22—168no—
725C22#269614yes—
726C22C2×#53511yescharacteristic
727C22C2×#54611yescharacteristic
728C22C2×#54811yescharacteristic
729C22#358512yes—
730C22#358512yes—
731C4—42no—
732C4—42no—
733C4—42no—
734C4—42no—
735C4—42no—
736C4—44no—
737C4—81no—
738C4—82no—
739C4—82no—
740C4—81no—
741C4—82no—
742C4—82no—
743C4#269614yes—
744C4C2×#53811yescharacteristic
745C4C2×#53911yescharacteristic
746C4C2×#55111yescharacteristic
747C4C2×#56211yescharacteristic
748C2—24no—
749C2—82no—
750C2—84no—
751C2—162no—
752C2C2×#269611yescharacteristic
753C2#6082511yescharacteristic
754C2#6082511yescharacteristic
755C2#6117512yes—
756C2#6133712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23430C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35620C2semidirect
2. #35554C2semidirect
3. #31665C2semidirect
4. #31662C2semidirect
5. #31661C2semidirect
6. #31658C2semidirect
7. #23746C2semidirect
8. #23632C2semidirect
9. #23544C2semidirect
10. #23456C2semidirect
11. #23456C2semidirect
13. C2×#1961C2semidirect
14. #21086C2semidirect
15. #21021C2semidirect
16. #20932C2semidirect
17. #20932C2semidirect
18. #20929C2semidirect
19. #20650C2semidirect
36. #2568C22semidirect
37. #2566C22semidirect
38. #2565C22semidirect
39. #2563C22semidirect
57. #2101C22semidirect
58. #2101C22semidirect
59. #2073C22semidirect
60. #2071C22semidirect
61. #2055C22semidirect
62. #2054C22semidirect
63. #2025C22semidirect
64. #2025C22semidirect
65. #2023C22semidirect
66. #2023C22semidirect
73. #1936C22semidirect
74. #1936C22semidirect
75. #1935C22semidirect
76. #1935C22semidirect
77. #1917C22semidirect
78. #1917C22semidirect
79. #1880C22semidirect
80. #1880C22semidirect
81. #1879C22semidirect
82. #1879C22semidirect
83. #1879C22semidirect
84. #1877C22semidirect
85. #1804C22semidirect
86. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#356202
#355542
#316652
#316622
#316612
#316582
#237462
#236322
#235442
#234562
#234302
C2×#19612
#210862
#210212
#209322
#209292
#206502