Finite Group Databasebeta

← #6986871

Subgroups — #6986871

Order 512.

1733 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36444C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic, complemented
3#32788C212yesmaximal, complemented
4#32269C214yesmaximal, complemented
5#32010C214yesmaximal, complemented
6#30298C212yesmaximal, complemented
7#30173C212yesmaximal, complemented
8#29936C211yesmaximal, characteristic, complemented
9#29163C212yesmaximal, complemented
10#28713C212yesmaximal, complemented
11#26282C212yesmaximal, complemented
12#25834C212yesmaximal, complemented
13#25811C211yesmaximal, characteristic, complemented
14#25346C212yesmaximal, complemented
15#24299C211yesmaximal, characteristic, complemented
16#23754C211yesmaximal, characteristic, complemented
17#22473C211yesmaximal, characteristic, complemented
18C2×D4×Q8C2211yescharacteristic
19C2×#548—21no—
20C22×C4⋊Q8C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#498C2212yes—
23C2×#476C2212yes—
24C2×#436C2212yes—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2857C2211yescharacteristic
29#2628—22no—
30#2568—21no—
31#2567C2212yes—
32#2563C2212yes—
33#2547C2212yes—
34#2547C2211yescharacteristic
35#2542—21no—
36#2537C2211yescharacteristic
37#2537C2212yes—
38#2487C2214yes—
39#2427—22no—
40#2427—21no—
41#2415—21no—
42#2415—22no—
43#2409—22no—
44Q8×C4⋊C4C2211yescharacteristic
45Q8×C22⋊C4—21no—
46#2317—21no—
47#2311—21no—
48C4×#121C2211yescharacteristic
49C4×#121C2212yes—
50#2285C2211yescharacteristic
51#2100C2214yes—
52#2099C2211yescharacteristic
53#2099C2212yes—
54#2093C2211yescharacteristic
55#2083C2214yes—
56#2083C2211yescharacteristic
57#2083C2212yes—
58#2077C2214yes—
59#2077C2214yes—
60#2077C2214yes—
61#2075C2214yes—
62#2075C2212yes—
63#2071C2211yescharacteristic
64#2058C2214yes—
65#2058C2214yes—
66#2058C2212yes—
67#2058C2212yes—
68#2058C2212yes—
69#2033C2214yes—
70#2033C2212yes—
71#2023C2212yes—
72#1990C2212yes—
73#1988C2212yes—
74#1987C2212yes—
75#1984C2212yes—
76#1947C2212yes—
77#1921C2214yes—
78#1921C2214yes—
79#1895C2214yes—
80#1874C2212yes—
81#1870C2214yes—
82#1864C2214yes—
83#1864C2214yes—
84#1853C2214yes—
85#1853C2212yes—
86#1817C2212yes—
87#1809C2214yes—
88#1809C2214yes—
89#1809C2214yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1728C2212yes—
95C22×#41D411yescharacteristic
96C23×Q8—21no—
97C23×Q8—21no—
98C22×M16C2314yes—
99D4×Q8—44no—
100#548—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#124—21no—
109C2×#124—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—22no—
116C2×#121—21no—
117C2×#121—22no—
118C2×#121C2311yescharacteristic
119C2×#121C2312yes—
120C4×#41—44no—
121C2×C4×Q8—21no—
122C2×C4×Q8D411yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4D412yes—
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2312yes—
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2312yes—
132C2×#117—21no—
133C2×#117D411yescharacteristic
134C2×#117D412yes—
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139#498—44no—
140#476—44no—
141#436—44no—
142C2×#107C2312yes—
143C2×#107C2312yes—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2314yes—
155C2×C4×C8C2312yes—
156#400—24no—
157#399—24no—
158#397—22no—
159#397—22no—
160#397—22no—
161#395—24no—
162#394—22no—
163#394—22no—
164#394—22no—
165#393—24no—
166#391—22no—
167#391—21no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391—21no—
172#391—22no—
173#390—22no—
174#390C2311yescharacteristic
175#389—21no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—21no—
180#389—22no—
181#389C2314yes—
182#389C2314yes—
183#389C2311yescharacteristic
184#389C2312yes—
185#388—21no—
186#388—22no—
187#386—24no—
188#386—21no—
189#386—22no—
190#385—24no—
191#384—22no—
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#382—22no—
197#382—21no—
198#380—22no—
199#380—22no—
200#380—21no—
201#380—22no—
202#380—21no—
203#380—21no—
204#380—22no—
205#380C2314yes—
206#380C2314yes—
207#380C2312yes—
208#380C2312yes—
209#380C2312yes—
210#380C2312yes—
211#380C2312yes—
212#380C2311yescharacteristic
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235C2×#41—24no—
236C2×#41—24no—
237C2×#41—42no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8—82no—
247C22×Q8—82no—
248C22×Q8C2×D411yescharacteristic
249C22×D4—22no—
250C22×D4—21no—
251C22×D4C2×D411yescharacteristic
252C22×D4C2×D412yes—
253C22×D4C2411yescharacteristic
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2411yescharacteristic
262C23×C4C2412yes—
263C2×M16—28no—
264C2×M16—44no—
265C2×M16—44no—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C4⋊Q8—24no—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276#124—48no—
277#124—44no—
278C22⋊Q8—44no—
279C22⋊Q8—48no—
280C22⋊Q8—48no—
281C22⋊Q8—44no—
282C22⋊Q8—84no—
283C22⋊Q8—84no—
284#121—22no—
285#121—24no—
286#121—41no—
287#121—42no—
288#121—44no—
289#121—48no—
290C4×Q8—24no—
291C4×Q8—42no—
292C4×Q8—44no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—42no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—48no—
300C4×D4—44no—
301#117—44no—
302#117—42no—
303#117—44no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—44no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—24no—
365#107—42no—
366#107—44no—
367#106—44no—
368#106—44no—
369#106—44no—
370C4⋊C8—24no—
371#102—24no—
372#102—24no—
373#102—44no—
374C22⋊C8—24no—
375C22⋊C8—42no—
376C22⋊C8—44no—
377C4×C8—24no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393C24—21no—
394C24C22×D412yes—
395C24C22×D411yescharacteristic
396#41—416no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8—816no—
412C2×Q8—816no—
413C2×Q8#13714yes—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—81no—
428C2×D4—82no—
429C2×D42−1+412yes—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—42no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4AGL(1,ℤ/8)14yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2511yescharacteristic
476M16—88no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C82−1+414yes—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—81no—
517C4⋊C4—82no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C22⋊C4—24no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—42no—
531C22⋊C4—44no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—24no—
541C42—42no—
542C42—42no—
543C42—44no—
544C42—44no—
545C42—42no—
546C42H(5,𝔽2)14yes—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23C2×AGL(1,ℤ/8)12yes—
568C23C23×D411yescharacteristic
569Q8—28no—
570Q8—88no—
571Q8—88no—
572Q8—88no—
573Q8—84no—
574D4—24no—
575D4—48no—
576D4—44no—
577D4—42no—
578D4—84no—
579D4—88no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—28no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—84no—
629C2×C4—88no—
630C2×C4—88no—
631C2×C4—84no—
632C2×C4#53412yes—
633C2×C4#53412yes—
634C2×C4C2×AGL(1,ℤ/8)12yes—
635C2×C4C2×AGL(1,ℤ/8)12yes—
636C2×C4C2×#13714yes—
637C2×C4C2×#13712yes—
638C2×C4#57712yes—
639C2×C4C2×H(5,𝔽2)12yes—
640C2×C4C2×2−1+411yescharacteristic
641C2×C4C2×2−1+412yes—
642C2×C4#58511yescharacteristic
643C8—24no—
644C8—42no—
645C8—44no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22C2×#53411yescharacteristic
668C22#348911yescharacteristic
669C22#349911yescharacteristic
670C22C22×AGL(1,ℤ/8)11yescharacteristic
671C22C22×#13711yescharacteristic
672C22C2×#57711yescharacteristic
673C22#358511yescharacteristic
674C4—22no—
675C4—24no—
676C4—22no—
677C4—22no—
678C4—42no—
679C4—41no—
680C4—42no—
681C4—81no—
682C4—82no—
683C4—82no—
684C4—81no—
685C4—84no—
686C4—82no—
687C4—81no—
688C4—82no—
689C4#319314yes—
690C4#355212yes—
691C4#356112yes—
692C2—42no—
693C2—82no—
694C2—82no—
695C2#4291211yescharacteristic
696C2C2×#319311yescharacteristic
697C2#6075711yescharacteristic
698C2#6211011yescharacteristic
699C2#6214711yescharacteristic
700C2#6247711yescharacteristic
701C2#6251011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36444C2
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
2. #35279C2semidirect
3. #32788C2semidirect
4. #32269C2semidirect
5. #32010C2semidirect
6. #30298C2semidirect
7. #30173C2semidirect
8. #29936C2semidirect
9. #29163C2semidirect
10. #28713C2semidirect
11. #26282C2semidirect
12. #25834C2semidirect
13. #25811C2semidirect
14. #25346C2semidirect
15. #24299C2semidirect
16. #23754C2semidirect
17. #22473C2semidirect

Maximal subgroups

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

subgroupindex
#364442
#352792
#327882
#322692
#320102
#302982
#301732
#299362
#291632
#287132
#262822
#258342
#258112
#253462
#242992
#237542
#224732