Finite Group Databasebeta

← #7501988

Subgroups — #7501988

Order 512.

1889 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3137C211yesmaximal, characteristic
2C22×#475C211yesmaximal, characteristic, complemented
3#35576C211yesmaximal, characteristic, complemented
4#35182C211yesmaximal, characteristic
5#30828C214yesmaximal, complemented
6#30824C214yesmaximal, complemented
7#30812C214yesmaximal, complemented
8#30808C214yesmaximal, complemented
9#23738C211yesmaximal, characteristic
10#23633C211yesmaximal, characteristic, complemented
11#23609C211yesmaximal, characteristic
12#23556C211yesmaximal, characteristic, complemented
13#23510C211yesmaximal, characteristic
14#23452C211yesmaximal, characteristic
15#21155C211yesmaximal, characteristic, complemented
16#21068C211yesmaximal, characteristic
17#21066C211yesmaximal, characteristic, complemented
18#21024C211yesmaximal, characteristic, complemented
19#20874C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24#3137—28no—
25C2×#491C2212yes—
26C2×#489C2212yes—
27C2×#488C2212yes—
28C2×#487C2212yes—
29C2×#475—28no—
30C2×#475C2214yes—
31C2×#475C2214yes—
32C2×#428C2211yescharacteristic
33C2×#427C2211yescharacteristic
34C22×#106C2211yescharacteristic
35C22×#106C2211yescharacteristic
36C22×C4⋊C8C2211yescharacteristic
37C2×#417C2211yescharacteristic
38C2×#417C2211yescharacteristic
39C2×#419C2211yescharacteristic
40C2×#419C2211yescharacteristic
41C2×#416C2211yescharacteristic
42C2×#416C2211yescharacteristic
43C22×#103C2211yescharacteristic
44C22×#103C2211yescharacteristic
45C2×#405C2211yescharacteristic
46#2853—22no—
47#2623—22no—
48#2583—22no—
49#2551—22no—
50#2551C2214yes—
51#2550—22no—
52#2550C2214yes—
53#2448C2214yes—
54#2446C2214yes—
55#2434—22no—
56#2423—22no—
57#2421—22no—
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2317C2211yescharacteristic
61#2311C2211yescharacteristic
62#2310C2211yescharacteristic
63#2297C2211yescharacteristic
64#2296C2211yescharacteristic
65#2296C2211yescharacteristic
66#2274C2211yescharacteristic
67#2097C2212yes—
68#2096C2212yes—
69#2094C2212yes—
70#2092C2212yes—
71#2074C2214yes—
72#2072C2214yes—
73#2068C2212yes—
74#2067C2212yes—
75#2066C2212yes—
76#2065C2212yes—
77#2059C2214yes—
78#2058C2214yes—
79#2043C2214yes—
80#2042C2214yes—
81#2024C2214yes—
82#2023C2214yes—
83#1943C2214yes—
84#1941C2214yes—
85#1928C2214yes—
86#1927C2214yes—
87#1927C2214yes—
88#1926C2214yes—
89#1917C2214yes—
90#1916C2214yes—
91#1865C2214yes—
92#1863C2214yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1735C2211yescharacteristic
97#1729C2211yescharacteristic
98#1729C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×Q8—21no—
103C23×Q8C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#544—24no—
109#544—42no—
110#536—24no—
111#536—42no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—24no—
114C2×C4⋊Q8—24no—
115C2×C4⋊Q8—22no—
116C2×C4⋊Q8—42no—
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2314yes—
119C2×C4⋊Q8C2312yes—
120C2×#125—21no—
121C2×#125C2312yes—
122C2×#124C2312yes—
123C2×#123—22no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—22no—
127C2×C4×Q8—22no—
128C2×C4×Q8—24no—
129C2×C4×Q8—42no—
130C2×C4×Q8C2314yes—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C42—21no—
142C22×C42C2311yescharacteristic
143#491—28no—
144#489—28no—
145#488—28no—
146#487—28no—
147#475—216no—
148#475—216no—
149#428—42no—
150#427—24no—
151C2×#107C2312yes—
152C2×#107C2312yes—
153C2×#106—24no—
154C2×#106—24no—
155C2×#106C2314yes—
156C2×#106C2312yes—
157C2×#106C2312yes—
158C2×#106C2312yes—
159C2×#106C2312yes—
160C2×C4⋊C8—24no—
161C2×C4⋊C8C2312yes—
162C2×C4⋊C8C2312yes—
163#419—24no—
164#419—24no—
165#417—24no—
166#417—42no—
167#416—24no—
168#416—24no—
169C2×#103—24no—
170C2×#103D418yes—
171C2×#103C2314yes—
172C2×#103C2311yescharacteristic
173C2×#103C2312yes—
174C2×#103C2311yescharacteristic
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2314yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#102C2311yescharacteristic
183C2×#102C2312yes—
184C2×#102C2311yescharacteristic
185C2×#102C2312yes—
186C2×#102C2312yes—
187#405—24no—
188C2×#97C2312yes—
189C2×C4×C8C2312yes—
190#400—24no—
191#399—24no—
192#398—24no—
193#398—24no—
194#397—24no—
195#395—24no—
196#395—24no—
197#395—24no—
198#391—22no—
199#391—24no—
200#391C2314yes—
201#391C2312yes—
202#391C2312yes—
203#390C2312yes—
204#390C2312yes—
205#389—22no—
206#389—24no—
207#389—24no—
208#389D412yes—
209#389D412yes—
210#389C2314yes—
211#389C2314yes—
212#389C2312yes—
213#389C2312yes—
214#388—24no—
215#388—24no—
216#387—24no—
217#387—24no—
218#386—21no—
219#386—21no—
220#386—22no—
221#385—21no—
222#385—21no—
223#385—22no—
224#385—22no—
225#385—22no—
226#385D412yes—
227#385D412yes—
228#384—21no—
229#384—21no—
230#384C2312yes—
231#382—21no—
232#382—21no—
233#382C2312yes—
234#381—22no—
235#380—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4C2314yes—
239C4×C4⋊C4C2314yes—
240C4×C22⋊C4—22no—
241C4×C22⋊C4—22no—
242#340C2312yes—
243#340C2314yes—
244#340C2312yes—
245#340C2314yes—
246#340C2314yes—
247#336C2314yes—
248#336C2314yes—
249#336C2312yes—
250#336C2312yes—
251#336C2314yes—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—22no—
258C22×Q8—21no—
259C22×Q8—24no—
260C22×Q8—22no—
261C22×Q8—22no—
262C22×Q8—22no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8—22no—
266C22×Q8—44no—
267C22×Q8—84no—
268C22×Q8C2×D414yes—
269C22×Q8C2411yescharacteristic
270C22×Q8C2412yes—
271C22×D4—21no—
272C22×D4C2411yescharacteristic
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—21no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2411yescharacteristic
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8C2×D414yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2411yescharacteristic
297C22×C8C2411yescharacteristic
298C4⋊Q8—28no—
299C4⋊Q8—24no—
300C4⋊Q8—24no—
301C4⋊Q8—44no—
302C4⋊Q8—42no—
303C4⋊Q8—48no—
304C4⋊Q8—42no—
305C4⋊Q8—44no—
306C4⋊Q8—88no—
307#125—24no—
308#125—42no—
309#125—44no—
310#124—24no—
311#124—42no—
312#123—48no—
313C22⋊Q8—48no—
314C22⋊Q8—44no—
315C22⋊Q8—44no—
316C4×Q8—28no—
317C4×Q8—44no—
318C4×Q8—48no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—22no—
328#117#4114yes—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4C2×D412yes—
373C2×C22⋊C4C2×D412yes—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—24no—
380C2×C42—24no—
381C2×C42—44no—
382C2×C42C2×D412yes—
383C2×C42C2412yes—
384C2×C42C2412yes—
385C2×C42C2412yes—
386#107—24no—
387#107—42no—
388#106—24no—
389#106—24no—
390#106—28no—
391#106—24no—
392#106—42no—
393C4⋊C8—24no—
394C4⋊C8—24no—
395#103—28no—
396#103—24no—
397#103—24no—
398#103—22no—
399#103—22no—
400#103—24no—
401#103—24no—
402#103—28no—
403#103—42no—
404#103C2×D418yes—
405#103C2×D418yes—
406#102—22no—
407#102—22no—
408#102—24no—
409#102—24no—
410#102—42no—
411#97—24no—
412C4×C8—24no—
413#95—24no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95C2×D414yes—
425#95C2×D414yes—
426C24—21no—
427C24C22×D411yescharacteristic
428#41—44no—
429#41—44no—
430#41—44no—
431#41—44no—
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—28no—
435C2×Q8—28no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—48no—
447C2×Q8—44no—
448C2×Q8—42no—
449C2×Q8—44no—
450C2×Q8—42no—
451C2×Q8—44no—
452C2×Q8—48no—
453C2×Q8—84no—
454C2×Q8—84no—
455C2×Q8—816no—
456C2×Q8C22×D414yes—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—82no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4C22×D414yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D412yes—
498C22×C4C2×#4112yes—
499C22×C4H(5,𝔽2)12yes—
500C22×C42−1+412yes—
501C22×C42−1+412yes—
502C22×C42−1+412yes—
503C22×C4C2511yescharacteristic
504M16—44no—
505M16—44no—
506C2×C8—28no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—44no—
521C2×C8C22×D412yes—
522C2×C8C22×D412yes—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—28no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—22no—
540C4⋊C4—44no—
541C4⋊C4—48no—
542C4⋊C4—48no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4—84no—
553C4⋊C4—84no—
554C4⋊C4—84no—
555C4⋊C4C22×D414yes—
556C4⋊C4C2×#4114yes—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—44no—
565C22⋊C4—84no—
566C22⋊C4—82no—
567C42—24no—
568C42—24no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—48no—
576C42—42no—
577C42—42no—
578C42—84no—
579C23—22no—
580C23—22no—
581C23—42no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—42no—
586C23#53612yes—
587C23D4212yes—
588C23#54812yes—
589C23C23×D411yescharacteristic
590Q8—24no—
591Q8—24no—
592Q8—48no—
593Q8—42no—
594Q8—42no—
595Q8—41no—
596Q8—41no—
597Q8—41no—
598Q8—41no—
599Q8—816no—
600Q8—164no—
601D4—41no—
602D4—41no—
603D4—41no—
604D4—41no—
605D4—84no—
606C2×C4—28no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4#53612yes—
648C2×C4#53614yes—
649C2×C4#54214yes—
650C2×C4D4212yes—
651C2×C4#54714yes—
652C2×C4#54812yes—
653C2×C4#56314yes—
654C2×C4#57714yes—
655C2×C4C23×D411yescharacteristic
656C2×C4C22×#4111yescharacteristic
657C2×C4C2×H(5,𝔽2)11yescharacteristic
658C2×C4C2×2−1+411yescharacteristic
659C2×C4C2×2−1+411yescharacteristic
660C2×C4C2×2−1+411yescharacteristic
661C8—22no—
662C8—22no—
663C8—42no—
664C8—42no—
665C8—42no—
666C22—22no—
667C22—22no—
668C22—24no—
669C22—24no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—82no—
674C22—82no—
675C22#267414yes—
676C22C2×#53611yescharacteristic
677C22C2×D4211yescharacteristic
678C22C2×#54811yescharacteristic
679C22C2×#57712yes—
680C22#358612yes—
681C4—24no—
682C4—42no—
683C4—44no—
684C4—44no—
685C4—42no—
686C4—42no—
687C4—44no—
688C4—44no—
689C4—82no—
690C4—81no—
691C4—81no—
692C4—81no—
693C4—82no—
694C4—82no—
695C4#267414yes—
696C4C2×#53611yescharacteristic
697C4C2×#54211yescharacteristic
698C4C2×#54711yescharacteristic
699C4C2×#56311yescharacteristic
700C2—24no—
701C2—161no—
702C2C2×#267411yescharacteristic
703C2#6088912yes—
704C2#6105011yescharacteristic
705C2#6106111yescharacteristic
706C2#6134612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#475C2semidirect
3. #35576C2semidirect
5. #30828C2semidirect
6. #30824C2semidirect
7. #30812C2semidirect
8. #30808C2semidirect
10. #23633C2semidirect
12. #23556C2semidirect
15. #21155C2semidirect
17. #21066C2semidirect
18. #21024C2semidirect
19. #20874C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31372
C22×#4752
#355762
#351822
#308282
#308242
#308122
#308082
#237382
#236332
#236092
#235562
#235102
#234522
#211552
#210682
#210662
#210242
#208742