Finite Group Databasebeta

← #7501931

Subgroups — #7501931

Order 512.

1818 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3163C211yesmaximal, characteristic, complemented
2C2×#3097C211yesmaximal, characteristic
3#35620C211yesmaximal, characteristic
4#35312C211yesmaximal, characteristic
5#30800C214yesmaximal, complemented
6#30795C214yesmaximal, complemented
7#30785C214yesmaximal, complemented
8#30774C214yesmaximal, complemented
9C2×#2097C211yesmaximal, characteristic, complemented
10#23640C211yesmaximal, characteristic
11#23619C211yesmaximal, characteristic, complemented
12#23555C211yesmaximal, characteristic, complemented
13#23484C211yesmaximal, characteristic
14#23457C211yesmaximal, characteristic, complemented
15#21172C211yesmaximal, characteristic, complemented
16#21090C211yesmaximal, characteristic, complemented
17#21051C211yesmaximal, characteristic
18#21032C211yesmaximal, characteristic, complemented
19#20847C211yesmaximal, characteristic
20C2×#544C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×#520C2211yescharacteristic
24#3163—28no—
25C2×#501C2212yes—
26C2×#501C2212yes—
27C2×#499C2212yes—
28C2×#498C2212yes—
29#3097—28no—
30C2×#484C2212yes—
31C2×#485C2212yes—
32C2×#485C2212yes—
33C2×#483C2212yes—
34C2×#428C2211yescharacteristic
35C2×#428C2211yescharacteristic
36C2×#428C2211yescharacteristic
37C2×#427C2211yescharacteristic
38C22×#107C2211yescharacteristic
39C22×#106C2211yescharacteristic
40C2×#423C2211yescharacteristic
41C2×#419C2211yescharacteristic
42C2×#419C2211yescharacteristic
43C22×#103C2211yescharacteristic
44C22×#103C2211yescharacteristic
45C2×#408C2211yescharacteristic
46C2×#408C2211yescharacteristic
47C2×#405C2211yescharacteristic
48#2862—21no—
49#2861—21no—
50#2642—21no—
51#2640—21no—
52#2568—21no—
53#2566—21no—
54#2565—21no—
55#2563—21no—
56#2492C2214yes—
57#2489C2214yes—
58#2488—24no—
59#2450—24no—
60#2442C2214yes—
61#2439C2214yes—
62#2317C2211yescharacteristic
63#2311C2211yescharacteristic
64#2311C2211yescharacteristic
65#2310C2211yescharacteristic
66#2297C2211yescharacteristic
67#2296C2211yescharacteristic
68C2×#391C2211yescharacteristic
69C2×#382C2211yescharacteristic
70C4×#117C2211yescharacteristic
71#2097C2214yes—
72#2097C2214yes—
73#2077C2214yes—
74#2077C2214yes—
75#2068C2214yes—
76#2066C2214yes—
77#2059C2214yes—
78#2057C2214yes—
79#2036C2214yes—
80#2035C2214yes—
81#2026C2214yes—
82#2024C2214yes—
83#1949C2214yes—
84#1949C2214yes—
85#1935C2214yes—
86#1933C2214yes—
87#1924C2214yes—
88#1923C2214yes—
89#1921C2212yes—
90#1920C2212yes—
91#1919C2212yes—
92#1918C2212yes—
93#1854C2214yes—
94#1852C2214yes—
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1736C2211yescharacteristic
98#1735C2211yescharacteristic
99#1735C2211yescharacteristic
100#1735C2211yescharacteristic
101#1729C2211yescharacteristic
102C2×#336C2211yescharacteristic
103C22×#41C2311yescharacteristic
104C23×Q8C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2311yescharacteristic
109#544—24no—
110#544—24no—
111#544—42no—
112#544—42no—
113#536—24no—
114#536—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2312yes—
120C2×#125—21no—
121C2×#125—21no—
122C2×#125C2312yes—
123C2×#125C2312yes—
124C2×#124—21no—
125C2×#123—22no—
126C2×C22⋊Q8—22no—
127C2×C22⋊Q8—22no—
128C2×C22⋊Q8C2312yes—
129C2×C22⋊Q8C2312yes—
130#520—24no—
131#520—42no—
132C2×C4×Q8—21no—
133C2×C4×D4—21no—
134C2×C4×D4—22no—
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2312yes—
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2312yes—
145C22×C42C2311yescharacteristic
146#501—28no—
147#501—28no—
148#499—28no—
149#498—28no—
150#485—28no—
151#485—28no—
152#484—28no—
153#483—28no—
154#428—24no—
155#428—24no—
156#428—24no—
157#427D418yes—
158C2×#107—42no—
159C2×#107C2312yes—
160C2×#107C2312yes—
161C2×#107C2312yes—
162C2×#107C2314yes—
163C2×#107C2311yescharacteristic
164C2×#107C2311yescharacteristic
165C2×#106—24no—
166C2×#106C2312yes—
167C2×#106C2312yes—
168C2×#106C2312yes—
169C2×#106C2312yes—
170C2×#106C2312yes—
171C2×#106C2311yescharacteristic
172C2×#106C2311yescharacteristic
173#423—24no—
174C2×C4⋊C8C2314yes—
175#419—24no—
176#419—24no—
177C2×#103—24no—
178C2×#103—42no—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185#408—24no—
186#408—24no—
187C2×C22⋊C8C2312yes—
188C2×C22⋊C8C2312yes—
189C2×C22⋊C8C2314yes—
190#405—24no—
191C2×#97C2312yes—
192C2×C4×C8C2312yes—
193#400—24no—
194#400—24no—
195#400—24no—
196#398—24no—
197#397—24no—
198#396—24no—
199#393—24no—
200#393—24no—
201#391—21no—
202#391—21no—
203#391—21no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—24no—
208#391—24no—
209#391D411yescharacteristic
210#391D411yescharacteristic
211#391C2314yes—
212#391C2314yes—
213#390—21no—
214#390—21no—
215#390D411yescharacteristic
216#390D411yescharacteristic
217#389C2314yes—
218#389C2312yes—
219#389C2312yes—
220#388—24no—
221#388—24no—
222#387—24no—
223#386—22no—
224#386—22no—
225#386D414yes—
226#385—22no—
227#385—24no—
228#385—22no—
229#384—21no—
230#382—21no—
231#382—21no—
232#382—21no—
233#382—24no—
234#382C2314yes—
235#381—22no—
236#380—24no—
237#380C2314yes—
238#380C2314yes—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2312yes—
243C4×C22⋊C4—22no—
244C4×C22⋊C4C2314yes—
245#376—21no—
246C2×#95—22no—
247C2×#95—22no—
248C43—21no—
249#340C2312yes—
250#340C2314yes—
251#340C2312yes—
252#340C2312yes—
253#340C2312yes—
254#340C2314yes—
255#340C2312yes—
256#340C2314yes—
257#340C2312yes—
258#336C2314yes—
259#336C2312yes—
260#336C2312yes—
261C2×#41—22no—
262C2×#41—22no—
263C2×#41—22no—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41—22no—
267C22×Q8—21no—
268C22×Q8—22no—
269C22×Q8—22no—
270C22×Q8—22no—
271C22×Q8—22no—
272C22×Q8—22no—
273C22×Q8—22no—
274C22×Q8—22no—
275C22×Q8—22no—
276C22×Q8C2×D411yescharacteristic
277C22×Q8C2412yes—
278C22×D4C2×D412yes—
279C22×D4C2×D411yescharacteristic
280C23×C4—21no—
281C23×C4—21no—
282C23×C4—22no—
283C23×C4—21no—
284C23×C4—21no—
285C23×C4C2×D412yes—
286C23×C4C2412yes—
287C23×C4C2411yescharacteristic
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C2×M16#4114yes—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8—22no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8C2×D414yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C22×C8C2411yescharacteristic
303C22×C8C2411yescharacteristic
304C22×C8C2412yes—
305C4⋊Q8—24no—
306C4⋊Q8—24no—
307C4⋊Q8—42no—
308C4⋊Q8—42no—
309C4⋊Q8—44no—
310C4⋊Q8—44no—
311C4⋊Q8—44no—
312#125—24no—
313#125—24no—
314#125—42no—
315#125—42no—
316#125—44no—
317#125—44no—
318#124—44no—
319#123—48no—
320C22⋊Q8—24no—
321C22⋊Q8—24no—
322C22⋊Q8—48no—
323C22⋊Q8—48no—
324C22⋊Q8—42no—
325C22⋊Q8—42no—
326C4×Q8—44no—
327C4×D4—44no—
328C4×D4—48no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—24no—
338#117—42no—
339#117C2×D414yes—
340#117C2×D414yes—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—42no—
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2411yescharacteristic
372C2×C4⋊C4C2411yescharacteristic
373C2×C4⋊C4C2411yescharacteristic
374C2×C4⋊C4C2411yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4C2×D412yes—
385C2×C22⋊C4C2×D414yes—
386C2×C22⋊C4C2×D414yes—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2412yes—
399C2×C42C2412yes—
400#107—24no—
401#107—24no—
402#107—24no—
403#107—44no—
404#107C2×D414yes—
405#107C2×D414yes—
406#106—24no—
407#106—24no—
408#106—24no—
409#106—24no—
410#106—24no—
411#106C2×D414yes—
412#106C2×D414yes—
413C4⋊C8—28no—
414#103—24no—
415#103—24no—
416#103—28no—
417#103—24no—
418#103—24no—
419#103—44no—
420C22⋊C8—24no—
421C22⋊C8—24no—
422C22⋊C8—28no—
423#97—24no—
424C4×C8—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—24no—
434#95—24no—
435#95—24no—
436#95—24no—
437#95—24no—
438#95C2×D414yes—
439#95C2×D414yes—
440C24C22×D412yes—
441C24C22×D411yescharacteristic
442#41—44no—
443#41—44no—
444#41—44no—
445#41—44no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—44no—
456C2×Q8—44no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—44no—
461C2×Q8—44no—
462C2×Q8—44no—
463C2×Q8—44no—
464C2×Q8—84no—
465C2×Q8—84no—
466C2×D4—24no—
467C2×D4—24no—
468C2×D4—24no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—44no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—84no—
499C22×C4—82no—
500C22×C4—82no—
501C22×C4—82no—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D412yes—
509C22×C4C2×#4112yes—
510C22×C4H(5,𝔽2)12yes—
511C22×C42−1+412yes—
512C22×C42−1+412yes—
513C22×C42−1+412yes—
514C22×C4C2511yescharacteristic
515M16—28no—
516M16—44no—
517C2×C8—28no—
518C2×C8—22no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—44no—
531C2×C8C22×D412yes—
532C2×C8C22×D412yes—
533C2×C8C2×#4114yes—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—82no—
563C4⋊C4—84no—
564C4⋊C4—82no—
565C4⋊C4C22×D412yes—
566C4⋊C4C22×D412yes—
567C4⋊C4C22×D412yes—
568C4⋊C4C22×D412yes—
569C22⋊C4—24no—
570C22⋊C4—24no—
571C22⋊C4—24no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—42no—
578C22⋊C4—42no—
579C22⋊C4—84no—
580C22⋊C4—84no—
581C42—24no—
582C42—24no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—44no—
590C42—44no—
591C42—42no—
592C42—42no—
593C42—82no—
594C23—22no—
595C23—22no—
596C23—24no—
597C23—24no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23#53612yes—
602C23D4212yes—
603C23#54712yes—
604C23C23×D411yescharacteristic
605Q8—44no—
606Q8—42no—
607Q8—42no—
608Q8—42no—
609Q8—42no—
610Q8—84no—
611Q8—164no—
612Q8—164no—
613D4—48no—
614D4—44no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
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—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—84no—
656C2×C4—84no—
657C2×C4—164no—
658C2×C4—162no—
659C2×C4—162no—
660C2×C4—162no—
661C2×C4#53612yes—
662C2×C4#53614yes—
663C2×C4#54214yes—
664C2×C4D4212yes—
665C2×C4#54712yes—
666C2×C4#54814yes—
667C2×C4#56314yes—
668C2×C4C23×D411yescharacteristic
669C2×C4C22×#4111yescharacteristic
670C2×C4C2×H(5,𝔽2)11yescharacteristic
671C2×C4C2×2−1+411yescharacteristic
672C2×C4C2×2−1+411yescharacteristic
673C2×C4C2×2−1+411yescharacteristic
674C8—22no—
675C8—22no—
676C8—24no—
677C8—42no—
678C8—42no—
679C22—24no—
680C22—22no—
681C22—22no—
682C22—24no—
683C22—44no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—44no—
688C22#267414yes—
689C22C2×#53611yescharacteristic
690C22C2×D4211yescharacteristic
691C22C2×#54711yescharacteristic
692C22#358412yes—
693C22#358612yes—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4#267414yes—
710C4C2×#53611yescharacteristic
711C4C2×#54211yescharacteristic
712C4C2×#54811yescharacteristic
713C4C2×#56311yescharacteristic
714C2—24no—
715C2—82no—
716C2C2×#267411yescharacteristic
717C2#6089112yes—
718C2#6108311yescharacteristic
719C2#6108811yescharacteristic
720C2#6127512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3163C2semidirect
5. #30800C2semidirect
6. #30795C2semidirect
7. #30785C2semidirect
8. #30774C2semidirect
9. C2×#2097C2semidirect
11. #23619C2semidirect
12. #23555C2semidirect
14. #23457C2semidirect
15. #21172C2semidirect
16. #21090C2semidirect
18. #21032C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31632
C2×#30972
#356202
#353122
#308002
#307952
#307852
#307742
C2×#20972
#236402
#236192
#235552
#234842
#234572
#211722
#210902
#210512
#210322
#208472