Finite Group Databasebeta

← #6684251

Subgroups — #6684251

Order 512.

1830 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3383C211yesmaximal, characteristic, complemented
2C2×#3304C211yesmaximal, characteristic
3#39744C211yesmaximal, characteristic, complemented
4#36583C211yesmaximal, characteristic
5#33196C211yesmaximal, characteristic, complemented
6#30786C214yesmaximal, complemented
7#30674C214yesmaximal, complemented
8#30276C211yesmaximal, characteristic, complemented
9#29826C211yesmaximal, characteristic, complemented
10#28704C211yesmaximal, characteristic, complemented
11#28637C211yesmaximal, characteristic, complemented
12#25961C214yesmaximal, complemented
13#25935C211yesmaximal, characteristic, complemented
14#25887C211yesmaximal, characteristic
15#25840C211yesmaximal, characteristic, complemented
16#25705C211yesmaximal, characteristic, complemented
17#24904C214yesmaximal, complemented
18#22208C211yesmaximal, characteristic, complemented
19#21119C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×#550C2211yescharacteristic
24#3383—28no—
25#3304—28no—
26C2×#500C2211yescharacteristic
27C2×#500C2211yescharacteristic
28C2×#499C2212yes—
29C2×#486C2211yescharacteristic
30C2×#486C2211yescharacteristic
31C2×#482C2212yes—
32C2×#479C2212yes—
33C2×#479C2212yes—
34C2×#475C2211yescharacteristic
35C2×#475C2211yescharacteristic
36C2×#475C2211yescharacteristic
37C2×#465C2212yes—
38C2×#460C2211yescharacteristic
39C2×#452C2212yes—
40C2×C8×Q8C2211yescharacteristic
41C2×C4×Q16C2212yes—
42C2×C4×Q16C2211yescharacteristic
43C2×C8×D4C2211yescharacteristic
44C2×#416C2212yes—
45#2868C2211yescharacteristic
46#2858—21no—
47#2858C2211yescharacteristic
48#2721—24no—
49#2658—21no—
50#2548—21no—
51#2548C2211yescharacteristic
52#2547—21no—
53#2547C2211yescharacteristic
54#2547C2211yescharacteristic
55#2546—21no—
56#2546C2211yescharacteristic
57#2545—21no—
58#2490—24no—
59#2490C2214yes—
60#2441C2214yes—
61#2423C2214yes—
62#2341C2214yes—
63#2327—21no—
64C4×#124—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2093C2214yes—
69#2093C2214yes—
70#2075C2214yes—
71#2067C2214yes—
72#2065C2214yes—
73#2059C2214yes—
74#2055C2214yes—
75#2036C2214yes—
76#2034C2214yes—
77#2025C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1988C2211yescharacteristic
81#1984C2211yescharacteristic
82#1982C2211yescharacteristic
83#1954C2211yescharacteristic
84#1948C2214yes—
85#1934C2214yes—
86#1928C2212yes—
87#1923C2214yes—
88#1921C2212yes—
89#1880C2214yes—
90#1878C2212yes—
91#1863C2214yes—
92#1863C2212yes—
93#1854C2214yes—
94#1850C2214yes—
95#1842C2211yescharacteristic
96#1808C2214yes—
97#1803C2214yes—
98#1788C2214yes—
99C4×#107C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#103C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×#41C2312yes—
104C22×Q16C2311yescharacteristic
105C22×SD16C2312yes—
106C23×C8C2312yes—
107#557—24no—
108#557—24no—
109#557—42no—
110#557—42no—
111#556—24no—
112#556—42no—
113#550—24no—
114#550—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×#127—21no—
124C2×#127C2311yescharacteristic
125C2×#126—22no—
126C2×#125—22no—
127C2×#125—22no—
128C2×#125C2312yes—
129C2×#125C2312yes—
130C2×#124—21no—
131C2×#124—21no—
132C2×#124—21no—
133C2×#124—21no—
134C2×#124—21no—
135C2×#124—21no—
136C2×#124D411yescharacteristic
137C2×C22⋊Q8C2312yes—
138C2×#121—22no—
139C2×#121C2312yes—
140C2×C4×Q8—21no—
141C2×C4×Q8—21no—
142C2×C4×Q8—21no—
143C2×C4×Q8C2312yes—
144C2×C4×Q8C2312yes—
145C2×C4×Q8C2312yes—
146C2×C4×Q8C2311yescharacteristic
147C2×C4×Q8C2311yescharacteristic
148C2×C4×Q8C2311yescharacteristic
149C2×C4×D4—21no—
150C2×C4×D4C2311yescharacteristic
151C2×#117C2312yes—
152#500—24no—
153#500—42no—
154#499—28no—
155#486—24no—
156#486—42no—
157#482—28no—
158#479—28no—
159#479—28no—
160#475—24no—
161#475—24no—
162#475—24no—
163#465—28no—
164#460—24no—
165#452—28no—
166C8×Q8—24no—
167C4×Q16—28no—
168C4×Q16D418yes—
169C8×D4—24no—
170C2×#107C2312yes—
171C2×#107C2312yes—
172C2×#107C2314yes—
173C2×#107C2312yes—
174C2×#107C2311yescharacteristic
175C2×#106C2312yes—
176C2×#106C2312yes—
177C2×#106C2312yes—
178C2×C4⋊C8C2311yescharacteristic
179C2×C4⋊C8C2311yescharacteristic
180C2×C4⋊C8C2312yes—
181C2×C4⋊C8C2314yes—
182#416—28no—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#103C2312yes—
188C2×#103C2311yescharacteristic
189C2×#103C2311yescharacteristic
190C2×#102C2312yes—
191C2×#102C2312yes—
192C2×#102C2312yes—
193C2×#102C2314yes—
194C2×#102C2314yes—
195C2×C22⋊C8C2312yes—
196C2×C22⋊C8C2314yes—
197C2×C4×C8C2311yescharacteristic
198C2×C4×C8C2312yes—
199C2×C4×C8C2311yescharacteristic
200#399—24no—
201#399—24no—
202#398—22no—
203#398—24no—
204#397—22no—
205#397—22no—
206#397—24no—
207#396—24no—
208#394—22no—
209#394—24no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391D411yescharacteristic
214#391C2312yes—
215#390—21no—
216#390—24no—
217#390—24no—
218#390—24no—
219#390—22no—
220#390D411yescharacteristic
221#390C2314yes—
222#390C2314yes—
223#389—24no—
224#389C2314yes—
225#389C2312yes—
226#388—24no—
227#388—24no—
228#388D414yes—
229#387—22no—
230#387—24no—
231#387—24no—
232#384—22no—
233#384C2314yes—
234#382D411yescharacteristic
235#380—24no—
236#380C2314yes—
237#380C2314yes—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2311yescharacteristic
244C4×C22⋊C4—24no—
245C4×C22⋊C4C2314yes—
246#376—21no—
247C43—21no—
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#340C2312yes—
254#340C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C2×#41—24no—
259C2×#41—28no—
260C2×#41—42no—
261C2×#41—44no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8—21no—
265C22×Q8C2×D411yescharacteristic
266C22×Q8C2411yescharacteristic
267C22×Q8C2411yescharacteristic
268C22×D4—22no—
269C22×D4—22no—
270C22×D4—21no—
271C22×D4—21no—
272C22×D4C2×D411yescharacteristic
273C22×D4C2412yes—
274C23×C4—22no—
275C23×C4C2412yes—
276C2×Q16—24no—
277C2×Q16C2×D414yes—
278C2×SD16—24no—
279C2×SD16—44no—
280C22×C8—24no—
281C22×C8—44no—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2411yescharacteristic
286C22×C8C2411yescharacteristic
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295C4⋊Q8—44no—
296C4⋊Q8—82no—
297C4⋊Q8—82no—
298#127—22no—
299#127—22no—
300#127—44no—
301#126—48no—
302#125—24no—
303#125—24no—
304#125—42no—
305#125—42no—
306#125—48no—
307#125—48no—
308#124—24no—
309#124—44no—
310#124—44no—
311#124—82no—
312#124—82no—
313#124—82no—
314#124—82no—
315C22⋊Q8—24no—
316C22⋊Q8—42no—
317#121—24no—
318#121—48no—
319#121—42no—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×Q8—22no—
323C4×Q8—22no—
324C4×Q8—24no—
325C4×Q8—24no—
326C4×Q8—24no—
327C4×Q8—42no—
328C4×Q8—42no—
329C4×Q8—42no—
330C4×Q8—44no—
331C4×Q8—44no—
332C4×Q8—44no—
333C4×Q8C2×D414yes—
334C4×Q8C2×D414yes—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—44no—
338#117—24no—
339#117—42no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2411yescharacteristic
366C2×C4⋊C4C2411yescharacteristic
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D414yes—
378C2×C22⋊C4C2×D414yes—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—22no—
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D412yes—
390C2×C42C2×D411yescharacteristic
391C2×C42C2412yes—
392C2×C42C2412yes—
393C2×C42C2411yescharacteristic
394#107—24no—
395#107—24no—
396#107—42no—
397#107—44no—
398#107C2×D414yes—
399#106—24no—
400#106—24no—
401#106—24no—
402C4⋊C8—24no—
403C4⋊C8—22no—
404C4⋊C8—22no—
405C4⋊C8—44no—
406#103—24no—
407#103—24no—
408#103—24no—
409#103—24no—
410#103—42no—
411#103C2×D414yes—
412#103C2×D414yes—
413#102—24no—
414#102—24no—
415#102—24no—
416#102—44no—
417#102—44no—
418C22⋊C8—24no—
419C22⋊C8—44no—
420C4×C8—24no—
421C4×C8—22no—
422C4×C8C2×D414yes—
423#95—22no—
424#95—22no—
425#95—24no—
426#95—22no—
427#95—21no—
428#95—22no—
429#95—21no—
430#95—24no—
431#95—24no—
432#95C2×D414yes—
433#95C2×D414yes—
434C24—22no—
435C24C22×D412yes—
436#41—48no—
437#41—84no—
438#41—88no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8C22×D412yes—
457C2×Q8C22×D412yes—
458C2×Q82−1+414yes—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—44no—
467C2×D4—48no—
468C2×D4—44no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—82no—
474C2×D4—84no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—44no—
489C22×C4—48no—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C2511yescharacteristic
498Q16—28no—
499SD16—84no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—24no—
508C2×C8—24no—
509C2×C8—84no—
510C2×C8C22×D412yes—
511C2×C8C22×D412yes—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—48no—
542C4⋊C4—84no—
543C4⋊C4—84no—
544C4⋊C4C22×D412yes—
545C4⋊C4C22×D412yes—
546C22⋊C4—24no—
547C22⋊C4—28no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—48no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C22⋊C4—84no—
557C22⋊C4—84no—
558C42—24no—
559C42—24no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—42no—
566C42—44no—
567C42—44no—
568C42—44no—
569C42—44no—
570C42—42no—
571C42—42no—
572C42—82no—
573C42#13514yes—
574C42C22×D412yes—
575C42H(5,𝔽2)12yes—
576C23—24no—
577C23—22no—
578C23—24no—
579C23—22no—
580C23—22no—
581C23—44no—
582C23—42no—
583C23—44no—
584C23—42no—
585C23—42no—
586C23C23×D411yescharacteristic
587Q8—22no—
588Q8—24no—
589Q8—22no—
590Q8—22no—
591Q8—44no—
592Q8—42no—
593Q8—84no—
594Q8—84no—
595Q8—84no—
596Q8#53614yes—
597D4—42no—
598D4—42no—
599D4—44no—
600D4—44no—
601D4—88no—
602D4—84no—
603D4—162no—
604D4—164no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—88no—
645C2×C4D4212yes—
646C2×C4D4212yes—
647C2×C4C2×#13514yes—
648C2×C4C2×#13512yes—
649C2×C4#57612yes—
650C2×C4C23×D411yescharacteristic
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C2×C4C2×2−1+412yes—
653C2×C4#58512yes—
654C8—22no—
655C8—22no—
656C8—42no—
657C8—42no—
658C8—42no—
659C22—42no—
660C22—42no—
661C22—44no—
662C22—42no—
663C22—44no—
664C22—82no—
665C22—82no—
666C22—84no—
667C22—82no—
668C22—84no—
669C22C2×D4211yescharacteristic
670C22#348912yes—
671C22C22×#13511yescharacteristic
672C22C2×#57611yescharacteristic
673C22#358612yes—
674C4—24no—
675C4—24no—
676C4—22no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4#328714yes—
690C4C2×#53611yescharacteristic
691C4#350011yescharacteristic
692C4#356112yes—
693C2—82no—
694C2—162no—
695C2#3767611yescharacteristic
696C2C2×#328711yescharacteristic
697C2#6108311yescharacteristic
698C2#6210112yes—
699C2#6218612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
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×#3383C2semidirect
3. #39744C2semidirect
5. #33196C2semidirect
6. #30786C2semidirect
7. #30674C2semidirect
8. #30276C2semidirect
9. #29826C2semidirect
10. #28704C2semidirect
11. #28637C2semidirect
12. #25961C2semidirect
13. #25935C2semidirect
15. #25840C2semidirect
16. #25705C2semidirect
17. #24904C2semidirect
18. #22208C2semidirect
19. #21119C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33832
C2×#33042
#397442
#365832
#331962
#307862
#306742
#302762
#298262
#287042
#286372
#259612
#259352
#258872
#258402
#257052
#249042
#222082
#211192