Finite Group Databasebeta

← #6686329

Subgroups — #6686329

Order 512.

1936 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3359C211yesmaximal, characteristic, complemented
2C2×#3353C211yesmaximal, characteristic
3#39360C211yesmaximal, characteristic
4#36508C211yesmaximal, characteristic, complemented
5#33196C211yesmaximal, characteristic, complemented
6#30635C214yesmaximal, complemented
7#30617C214yesmaximal, complemented
8#30279C211yesmaximal, characteristic, complemented
9#29829C211yesmaximal, characteristic, complemented
10#28675C211yesmaximal, characteristic, complemented
11#28606C211yesmaximal, characteristic, complemented
12#26552C214yesmaximal, complemented
13#26353C214yesmaximal, complemented
14#25838C211yesmaximal, characteristic, complemented
15#25705C211yesmaximal, characteristic, complemented
16#25647C211yesmaximal, characteristic, complemented
17#25571C211yesmaximal, characteristic
18#22210C211yesmaximal, characteristic, complemented
19#21120C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#556C2211yescharacteristic
22C2×D4×Q8C2211yescharacteristic
23C2×#547C2211yescharacteristic
24C2×#536—21no—
25#3359—28no—
26#3353—28no—
27C2×#500C2211yescharacteristic
28C2×#500C2211yescharacteristic
29C2×#487C2211yescharacteristic
30C2×#487C2212yes—
31C2×#487C2211yescharacteristic
32C2×#484C2212yes—
33C2×#482C2212yes—
34C2×#475C2211yescharacteristic
35C2×#475C2211yescharacteristic
36C2×#475C2212yes—
37C2×#474C2211yescharacteristic
38C2×#468C2212yes—
39C2×#461C2211yescharacteristic
40C2×#461C2212yes—
41C2×C8×Q8C2211yescharacteristic
42C2×C4×SD16C2212yes—
43C2×C4×SD16C2211yescharacteristic
44C2×#436C2211yescharacteristic
45C2×#428C2212yes—
46#2868C2211yescharacteristic
47#2862C2211yescharacteristic
48#2861—21no—
49#2680—24no—
50#2621—21no—
51#2565—21no—
52#2547C2211yescharacteristic
53#2546C2211yescharacteristic
54#2546C2211yescharacteristic
55#2545—21no—
56#2542C2211yescharacteristic
57#2519—24no—
58#2440C2214yes—
59#2434—21no—
60#2432—21no—
61#2429C2214yes—
62#2422C2214yes—
63#2362C2214yes—
64C4×C4⋊Q8—21no—
65C4×#124C2211yescharacteristic
66#2293C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2095C2214yes—
69#2095C2214yes—
70#2086C2214yes—
71#2068C2214yes—
72#2057C2214yes—
73#2055C2214yes—
74#2043C2214yes—
75#2036C2214yes—
76#2036C2214yes—
77#2000C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1988C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1954C2211yescharacteristic
84#1948C2214yes—
85#1942C2214yes—
86#1925C2212yes—
87#1923C2214yes—
88#1920C2212yes—
89#1894C2214yes—
90#1880C2212yes—
91#1872C2214yes—
92#1865C2214yes—
93#1863C2212yes—
94#1862C2214yes—
95#1854C2214yes—
96#1851C2214yes—
97#1842C2211yescharacteristic
98#1803C2214yes—
99C4×#107C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×#41—21no—
104C23×Q8—21no—
105C22×SD16C2312yes—
106C22×SD16C2311yescharacteristic
107C22×M16C2312yes—
108#557—24no—
109#557—42no—
110#556—24no—
111#556—42no—
112D4×Q8—24no—
113D4×Q8—42no—
114#547—24no—
115#547—42no—
116#536—48no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8—21no—
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2312yes—
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2311yescharacteristic
125C2×C4⋊Q8C2311yescharacteristic
126C2×C4⋊Q8C2312yes—
127C2×#126—22no—
128C2×#125—21no—
129C2×#125—22no—
130C2×#124—21no—
131C2×#124—21no—
132C2×#124D411yescharacteristic
133C2×#123—24no—
134C2×#123—22no—
135C2×C22⋊Q8—21no—
136C2×C22⋊Q8—21no—
137C2×C22⋊Q8—22no—
138C2×C22⋊Q8—24no—
139C2×C22⋊Q8C2312yes—
140C2×C22⋊Q8C2314yes—
141C2×#121C2312yes—
142C2×C4×Q8—21no—
143C2×C4×Q8—21no—
144C2×C4×Q8C2312yes—
145C2×C4×Q8C2312yes—
146C2×C4×Q8C2311yescharacteristic
147C2×C4×Q8C2311yescharacteristic
148C2×C4×D4C2312yes—
149C2×C4×D4C2311yescharacteristic
150C2×C4×D4C2311yescharacteristic
151C2×#117—21no—
152C2×#117C2312yes—
153C22×C4⋊C4C2312yes—
154#500—24no—
155#500—42no—
156#487—28no—
157#487—24no—
158#487—42no—
159#484—28no—
160#482—28no—
161#475—24no—
162#475—24no—
163#475—28no—
164#474—24no—
165#468—28no—
166#461—28no—
167#461—24no—
168C8×Q8—24no—
169C4×SD16—28no—
170C4×SD16D418yes—
171#436—24no—
172#428—28no—
173C2×#107C2312yes—
174C2×#107C2312yes—
175C2×#107C2312yes—
176C2×#107C2312yes—
177C2×#107C2314yes—
178C2×#106C2312yes—
179C2×#106C2312yes—
180C2×#106C2311yescharacteristic
181C2×C4⋊C8C2312yes—
182C2×C4⋊C8C2311yescharacteristic
183C2×C4⋊C8C2311yescharacteristic
184C2×C4⋊C8C2314yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#103C2312yes—
188C2×#103C2312yes—
189C2×#103C2312yes—
190C2×#103C2314yes—
191C2×#103C2314yes—
192C2×#103C2312yes—
193C2×#103C2311yescharacteristic
194C2×#102C2312yes—
195C2×#102C2312yes—
196C2×#102C2311yescharacteristic
197C2×C22⋊C8C2312yes—
198C2×C22⋊C8C2314yes—
199C2×C4×C8C2311yescharacteristic
200C2×C4×C8C2312yes—
201C2×C4×C8C2311yescharacteristic
202#399—22no—
203#398—22no—
204#398—22no—
205#398—24no—
206#398—24no—
207#397—22no—
208#397—24no—
209#396—24no—
210#395—24no—
211#394—24no—
212#391—21no—
213#391—21no—
214#391—21no—
215#391—21no—
216#391—21no—
217#391D411yescharacteristic
218#391C2314yes—
219#390—21no—
220#390D411yescharacteristic
221#390C2312yes—
222#389—24no—
223#389C2314yes—
224#389C2312yes—
225#389C2314yes—
226#388—24no—
227#388—24no—
228#388—24no—
229#388—22no—
230#388D414yes—
231#387—24no—
232#385—24no—
233#385—24no—
234#384—21no—
235#384—22no—
236#384C2314yes—
237#382—22no—
238#382D411yescharacteristic
239#380C2314yes—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4—24no—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2311yescharacteristic
245C4×C4⋊C4C2311yescharacteristic
246C4×C22⋊C4—21no—
247C4×C22⋊C4C2314yes—
248C43—21no—
249C43C2311yescharacteristic
250#340C2314yes—
251#340C2312yes—
252#340C2314yes—
253#340C2314yes—
254#340C2312yes—
255#340C2312yes—
256#336C2314yes—
257#336C2312yes—
258#336C2312yes—
259C2×#41—44no—
260C2×#41—44no—
261C2×#41—82no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8—21no—
265C22×Q8—22no—
266C22×Q8—44no—
267C22×Q8—44no—
268C22×Q8—82no—
269C22×Q8C2×D411yescharacteristic
270C22×Q8C2412yes—
271C22×Q8C2411yescharacteristic
272C22×D4—22no—
273C22×D4C2×D411yescharacteristic
274C22×D4C2411yescharacteristic
275C23×C4—21no—
276C23×C4—22no—
277C23×C4—22no—
278C23×C4C2412yes—
279C2×SD16—24no—
280C2×SD16—42no—
281C2×SD16—44no—
282C2×SD16C2×D414yes—
283C2×M16—24no—
284C2×M16—44no—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2411yescharacteristic
289C22×C8C2411yescharacteristic
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—42no—
301C4⋊Q8—42no—
302C4⋊Q8—44no—
303C4⋊Q8—44no—
304C4⋊Q8—44no—
305C4⋊Q8—82no—
306#126—48no—
307#125—44no—
308#125—48no—
309#124—24no—
310#124—44no—
311#124—44no—
312#123—48no—
313#123—416no—
314C22⋊Q8—24no—
315C22⋊Q8—28no—
316C22⋊Q8—48no—
317C22⋊Q8—416no—
318C22⋊Q8—44no—
319C22⋊Q8—42no—
320C22⋊Q8—44no—
321C22⋊Q8—44no—
322#121—24no—
323#121—42no—
324C4×Q8—22no—
325C4×Q8—22no—
326C4×Q8—22no—
327C4×Q8—24no—
328C4×Q8—24no—
329C4×Q8—42no—
330C4×Q8—42no—
331C4×Q8—44no—
332C4×Q8—44no—
333C4×Q8C2×D414yes—
334C4×D4—24no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—42no—
339C4×D4C2×D414yes—
340#117—24no—
341#117—44no—
342#117—42no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—42no—
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2411yescharacteristic
372C2×C4⋊C4C2411yescharacteristic
373C2×C4⋊C4C2412yes—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4C2×D412yes—
384C2×C22⋊C4C2×D414yes—
385C2×C22⋊C4C2×D414yes—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—21no—
392C2×C42—22no—
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D412yes—
396C2×C42C2412yes—
397C2×C42C2412yes—
398C2×C42C2411yescharacteristic
399#107—24no—
400#107—24no—
401#107—24no—
402#107—24no—
403#107—44no—
404#106—24no—
405#106—24no—
406#106C2×D414yes—
407C4⋊C8—24no—
408C4⋊C8—22no—
409C4⋊C8—22no—
410C4⋊C8—44no—
411#103—24no—
412#103—24no—
413#103—24no—
414#103—24no—
415#103—24no—
416#103—42no—
417#103—44no—
418#103—44no—
419#103C2×D414yes—
420#102—24no—
421#102—24no—
422#102C2×D414yes—
423C22⋊C8—24no—
424C22⋊C8—44no—
425C4×C8—24no—
426C4×C8—22no—
427C4×C8C2×D414yes—
428#95—22no—
429#95—22no—
430#95—21no—
431#95—24no—
432#95—21no—
433#95—24no—
434#95—24no—
435#95—24no—
436#95—24no—
437#95C2×D414yes—
438#95C2×D414yes—
439C24—21no—
440C24C22×D412yes—
441#41—816no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—21no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—24no—
449C2×Q8—22no—
450C2×Q8—24no—
451C2×Q8—44no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—44no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—82no—
463C2×Q8—816no—
464C2×Q8AGL(1,ℤ/8)14yes—
465C2×Q8C22×D412yes—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—44no—
473C2×D4—44no—
474C2×D4—82no—
475C2×D4C22×D412yes—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C2511yescharacteristic
506SD16—44no—
507SD16—48no—
508M16—48no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8C22×D412yes—
519C2×C8C22×D412yes—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—48no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—48no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—48no—
551C4⋊C4—48no—
552C4⋊C4—84no—
553C4⋊C4C22×D412yes—
554C4⋊C4C22×D412yes—
555C4⋊C42−1+414yes—
556C22⋊C4—24no—
557C22⋊C4—24no—
558C22⋊C4—28no—
559C22⋊C4—28no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—48no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—48no—
567C22⋊C4—44no—
568C42—24no—
569C42—22no—
570C42—22no—
571C42—24no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—44no—
577C42—44no—
578C42—44no—
579C42—42no—
580C42—42no—
581C42—44no—
582C42C22×D412yes—
583C42H(5,𝔽2)12yes—
584C23—22no—
585C23—22no—
586C23—24no—
587C23—22no—
588C23—22no—
589C23—24no—
590C23—42no—
591C23—41no—
592C23—41no—
593C23—42no—
594C23C23×D411yescharacteristic
595Q8—28no—
596Q8—22no—
597Q8—22no—
598Q8—42no—
599Q8—42no—
600Q8—42no—
601Q8—44no—
602Q8—88no—
603Q8—84no—
604Q8—84no—
605Q8—84no—
606D4—22no—
607D4—22no—
608D4—44no—
609D4—42no—
610D4—88no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—48no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—88no—
652C2×C4—88no—
653C2×C4D4212yes—
654C2×C4D4212yes—
655C2×C4C2×AGL(1,ℤ/8)14yes—
656C2×C4C2×AGL(1,ℤ/8)12yes—
657C2×C4#57712yes—
658C2×C4#57814yes—
659C2×C4C23×D411yescharacteristic
660C2×C4C2×H(5,𝔽2)11yescharacteristic
661C2×C4C2×2−1+412yes—
662C2×C4#58512yes—
663C8—22no—
664C8—22no—
665C8—42no—
666C8—42no—
667C8—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—44no—
675C22—44no—
676C22—82no—
677C22—82no—
678C22C2×D4211yescharacteristic
679C22#348912yes—
680C22C22×AGL(1,ℤ/8)11yescharacteristic
681C22C2×#57711yescharacteristic
682C22C2×#57812yes—
683C4—24no—
684C4—22no—
685C4—24no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—44no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4#326514yes—
699C4#344911yescharacteristic
700C4#349911yescharacteristic
701C4#355412yes—
702C2—82no—
703C2—82no—
704C2#3768011yescharacteristic
705C2C2×#326511yescharacteristic
706C2#6109111yescharacteristic
707C2#6214812yes—
708C2#6220312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3359C2semidirect
4. #36508C2semidirect
5. #33196C2semidirect
6. #30635C2semidirect
7. #30617C2semidirect
8. #30279C2semidirect
9. #29829C2semidirect
10. #28675C2semidirect
11. #28606C2semidirect
12. #26552C2semidirect
13. #26353C2semidirect
14. #25838C2semidirect
15. #25705C2semidirect
16. #25647C2semidirect
18. #22210C2semidirect
19. #21120C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33592
C2×#33532
#393602
#365082
#331962
#306352
#306172
#302792
#298292
#286752
#286062
#265522
#263532
#258382
#257052
#256472
#255712
#222102
#211202