Finite Group Databasebeta

← #6684271

Subgroups — #6684271

Order 512.

1744 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3387C211yesmaximal, characteristic, complemented
2C2×#3328C211yesmaximal, characteristic
3#39172C211yesmaximal, characteristic
4#36446C211yesmaximal, characteristic
5#33202C211yesmaximal, characteristic, complemented
6#30800C214yesmaximal, complemented
7#30682C214yesmaximal, complemented
8#30281C211yesmaximal, characteristic, complemented
9#29831C211yesmaximal, characteristic, complemented
10#28713C211yesmaximal, characteristic
11#28646C211yesmaximal, characteristic, complemented
12#25971C214yesmaximal, complemented
13#25936C211yesmaximal, characteristic, complemented
14#25888C211yesmaximal, characteristic
15#25840C211yesmaximal, characteristic, complemented
16#25705C211yesmaximal, characteristic
17#24922C214yesmaximal, complemented
18#22214C211yesmaximal, characteristic
19#21119C211yesmaximal, characteristic, complemented
20C2×Q82C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#556C2211yescharacteristic
23C2×D4×Q8C2211yescharacteristic
24#3387—28no—
25#3328—28no—
26C2×#501C2212yes—
27C2×#500C2211yescharacteristic
28C2×#498C2211yescharacteristic
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#485C2212yes—
32C2×#479C2212yes—
33C2×#477C2211yescharacteristic
34C2×#475C2211yescharacteristic
35C2×#475C2212yes—
36C2×#475C2211yescharacteristic
37C2×#470C2212yes—
38C2×#462C2211yescharacteristic
39C2×#451C2212yes—
40C2×#446C2211yescharacteristic
41C2×#441C2212yes—
42C2×C4×Q16C2211yescharacteristic
43C2×#436C2211yescharacteristic
44C2×#419C2212yes—
45#2862—21no—
46#2862C2211yescharacteristic
47#2862C2211yescharacteristic
48#2716—24no—
49#2640—21no—
50#2568—21no—
51#2568—21no—
52#2568C2211yescharacteristic
53#2563C2211yescharacteristic
54#2547—21no—
55#2547C2211yescharacteristic
56#2546C2211yescharacteristic
57#2545—21no—
58#2489—24no—
59#2489C2214yes—
60#2439C2214yes—
61#2421C2214yes—
62Q8×C22⋊C4C2214yes—
63#2327—21no—
64C4×#124—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2097C2214yes—
69#2097C2214yes—
70#2077C2214yes—
71#2068C2214yes—
72#2066C2214yes—
73#2059C2214yes—
74#2056C2214yes—
75#2036C2214yes—
76#2036C2214yes—
77#2024C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1954C2211yescharacteristic
84#1949C2214yes—
85#1935C2214yes—
86#1928C2212yes—
87#1924C2214yes—
88#1921C2212yes—
89#1878C2214yes—
90#1878C2212yes—
91#1863C2214yes—
92#1863C2212yes—
93#1854C2214yes—
94#1851C2214yes—
95#1842C2211yescharacteristic
96#1809C2214yes—
97#1805C2214yes—
98#1787C2214yes—
99C4×#107C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#103C2211yescharacteristic
102#1744C2211yescharacteristic
103C23×Q8C2312yes—
104C22×Q16C2311yescharacteristic
105C22×Q16C2312yes—
106C22×M16C2312yes—
107Q82—24no—
108Q82—42no—
109#557—24no—
110#557—42no—
111#556—24no—
112#556—42no—
113D4×Q8—24no—
114D4×Q8—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8—22no—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2312yes—
125C2×#126—22no—
126C2×#125—22no—
127C2×#125C2312yes—
128C2×#124—21no—
129C2×#124—21no—
130C2×#124D411yescharacteristic
131C2×C22⋊Q8—22no—
132C2×C22⋊Q8C2312yes—
133C2×C22⋊Q8C2312yes—
134C2×C4×Q8—21no—
135C2×C4×Q8—21no—
136C2×C4×Q8—21no—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2311yescharacteristic
141C2×C4×Q8C2311yescharacteristic
142C2×C4×Q8C2311yescharacteristic
143C2×C4×D4—21no—
144C2×C4×D4C2311yescharacteristic
145C2×#117C2312yes—
146#501—28no—
147#500—42no—
148#498—24no—
149#487—24no—
150#487—42no—
151#485—28no—
152#479—28no—
153#477—24no—
154#475—24no—
155#475—24no—
156#475—28no—
157#470—28no—
158#462—24no—
159#451—28no—
160#446—24no—
161#441—28no—
162C4×Q16D418yes—
163#436—24no—
164C2×#107C2312yes—
165C2×#107C2314yes—
166C2×#107C2312yes—
167C2×#107C2311yescharacteristic
168C2×#106C2312yes—
169C2×#106C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×C4⋊C8C2311yescharacteristic
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2314yes—
176#419—28no—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2314yes—
185C2×#103C2314yes—
186C2×#103C2312yes—
187C2×#103C2311yescharacteristic
188C2×#103C2311yescharacteristic
189C2×C22⋊C8C2312yes—
190C2×C22⋊C8C2314yes—
191C2×#97C2312yes—
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2311yescharacteristic
194#400—22no—
195#400—24no—
196#400—24no—
197#398—22no—
198#397—24no—
199#397—22no—
200#397—24no—
201#396—24no—
202#393—22no—
203#393—24no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—22no—
212#391—24no—
213#391—24no—
214#391—24no—
215#391D411yescharacteristic
216#391C2312yes—
217#391C2314yes—
218#391C2314yes—
219#390—21no—
220#390D411yescharacteristic
221#389C2314yes—
222#389C2312yes—
223#388—24no—
224#388—24no—
225#388D414yes—
226#387—22no—
227#387—24no—
228#387—24no—
229#382—21no—
230#382—21no—
231#382—22no—
232#382—24no—
233#382D411yescharacteristic
234#382C2314yes—
235#380—24no—
236#380C2314yes—
237#380C2314yes—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4C2311yescharacteristic
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2312yes—
244C4×C22⋊C4—24no—
245C4×C22⋊C4C2314yes—
246#376—21no—
247C43—21no—
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2312yes—
251#340C2314yes—
252#340C2314yes—
253#340C2312yes—
254#340C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—22no—
261C22×Q8—22no—
262C22×Q8—21no—
263C22×Q8—24no—
264C22×Q8—28no—
265C22×Q8—42no—
266C22×Q8—44no—
267C22×Q8C2×D411yescharacteristic
268C22×Q8C2412yes—
269C22×Q8C2411yescharacteristic
270C22×Q8C2411yescharacteristic
271C22×D4C2×D411yescharacteristic
272C23×C4—22no—
273C23×C4C2412yes—
274C2×Q16—24no—
275C2×Q16—24no—
276C2×Q16—44no—
277C2×Q16C2×D414yes—
278C2×M16—24no—
279C2×M16—44no—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2411yescharacteristic
284C22×C8C2411yescharacteristic
285C4⋊Q8—24no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—42no—
295C4⋊Q8—44no—
296C4⋊Q8—44no—
297C4⋊Q8—44no—
298C4⋊Q8—48no—
299C4⋊Q8—44no—
300#126—48no—
301#125—24no—
302#125—42no—
303#125—48no—
304#124—42no—
305#124—44no—
306#124—44no—
307C22⋊Q8—24no—
308C22⋊Q8—24no—
309C22⋊Q8—48no—
310C22⋊Q8—42no—
311C22⋊Q8—42no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—24no—
317C4×Q8—24no—
318C4×Q8—24no—
319C4×Q8—42no—
320C4×Q8—42no—
321C4×Q8—42no—
322C4×Q8—44no—
323C4×Q8—44no—
324C4×Q8—44no—
325C4×Q8C2×D414yes—
326C4×Q8C2×D414yes—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—44no—
330#117—24no—
331#117—42no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2411yescharacteristic
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4C2×D412yes—
365C2×C22⋊C4C2×D414yes—
366C2×C22⋊C4C2×D414yes—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D412yes—
379C2×C42C2412yes—
380C2×C42C2412yes—
381C2×C42C2411yescharacteristic
382#107—24no—
383#107—42no—
384#107—44no—
385#107C2×D414yes—
386#106—24no—
387#106—24no—
388#106—24no—
389#106—24no—
390C4⋊C8—24no—
391C4⋊C8—22no—
392C4⋊C8—22no—
393C4⋊C8—44no—
394#103—24no—
395#103—24no—
396#103—24no—
397#103—24no—
398#103—24no—
399#103—24no—
400#103—24no—
401#103—44no—
402#103—42no—
403#103—44no—
404#103C2×D414yes—
405#103C2×D414yes—
406C22⋊C8—24no—
407C22⋊C8—44no—
408#97—24no—
409C4×C8—22no—
410C4×C8C2×D414yes—
411#95—22no—
412#95—22no—
413#95—24no—
414#95—24no—
415#95—22no—
416#95—21no—
417#95—22no—
418#95—21no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95C2×D414yes—
427#95C2×D414yes—
428C24C22×D412yes—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—24no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—44no—
442C2×Q8—48no—
443C2×Q8—44no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—48no—
452C2×Q8—82no—
453C2×Q8—84no—
454C2×Q8—84no—
455C2×Q8—88no—
456C2×Q8C22×D412yes—
457C2×Q8C22×D412yes—
458C2×Q82−1+414yes—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—44no—
477C22×C4—48no—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C2511yescharacteristic
486Q16—28no—
487Q16—84no—
488M16—84no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8C22×D412yes—
499C2×C8C22×D412yes—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—82no—
532C4⋊C4C22×D412yes—
533C4⋊C4C22×D412yes—
534C22⋊C4—24no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—48no—
541C22⋊C4—84no—
542C42—24no—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—42no—
551C42—44no—
552C42—44no—
553C42—44no—
554C42—44no—
555C42—44no—
556C42—42no—
557C42—42no—
558C42C22×D412yes—
559C42H(5,𝔽2)12yes—
560C23—24no—
561C23—22no—
562C23—24no—
563C23—22no—
564C23—22no—
565C23C23×D411yescharacteristic
566Q8—22no—
567Q8—24no—
568Q8—22no—
569Q8—22no—
570Q8—44no—
571Q8—42no—
572Q8—42no—
573Q8—42no—
574Q8—44no—
575Q8—88no—
576Q8—84no—
577Q8—84no—
578Q8—162no—
579Q8—164no—
580Q8#53614yes—
581D4—44no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—88no—
624C2×C4D4212yes—
625C2×C4D4212yes—
626C2×C4#57512yes—
627C2×C4#57812yes—
628C2×C4C23×D411yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×2−1+412yes—
631C2×C4#58512yes—
632C8—22no—
633C8—22no—
634C8—42no—
635C8—42no—
636C8—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—44no—
641C22—44no—
642C22C2×D4211yescharacteristic
643C22#348912yes—
644C22C2×#57511yescharacteristic
645C22C2×#57811yescharacteristic
646C22#358612yes—
647C4—24no—
648C4—22no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4#327714yes—
664C4C2×#53611yescharacteristic
665C4#350011yescharacteristic
666C4#356112yes—
667C2—82no—
668C2#3767611yescharacteristic
669C2C2×#327711yescharacteristic
670C2#6108311yescharacteristic
671C2#6212412yes—
672C2#6219212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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×#3387C2semidirect
5. #33202C2semidirect
6. #30800C2semidirect
7. #30682C2semidirect
8. #30281C2semidirect
9. #29831C2semidirect
11. #28646C2semidirect
12. #25971C2semidirect
13. #25936C2semidirect
15. #25840C2semidirect
17. #24922C2semidirect
19. #21119C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33872
C2×#33282
#391722
#364462
#332022
#308002
#306822
#302812
#298312
#287132
#286462
#259712
#259362
#258882
#258402
#257052
#249222
#222142
#211192