Finite Group Databasebeta

← #7502008

Subgroups — #7502008

Order 512.

1928 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#489C211yesmaximal, characteristic
2C2×#3081C211yesmaximal, characteristic
3#35626C211yesmaximal, characteristic
4#35576C211yesmaximal, characteristic, complemented
5#30831C214yesmaximal, complemented
6#30829C214yesmaximal, complemented
7#30823C214yesmaximal, complemented
8#30821C214yesmaximal, complemented
9#23736C211yesmaximal, characteristic, complemented
10#23632C211yesmaximal, characteristic
11#23613C211yesmaximal, characteristic, complemented
12#23552C211yesmaximal, characteristic, complemented
13#23506C211yesmaximal, characteristic
14#23457C211yesmaximal, characteristic, complemented
15#21154C211yesmaximal, characteristic, complemented
16#21065C211yesmaximal, characteristic
17#21064C211yesmaximal, characteristic, complemented
18#21021C211yesmaximal, characteristic, complemented
19#20876C211yesmaximal, characteristic
20C2×#544C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C2×#518C2211yescharacteristic
24C2×#489—28no—
25C2×#489C2214yes—
26C2×#489C2214yes—
27#3081—28no—
28C2×#478C2212yes—
29C2×#478C2212yes—
30C2×#476C2212yes—
31C2×#474C2212yes—
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C2×#427C2211yescharacteristic
35C2×#427C2211yescharacteristic
36C2×#424C2211yescharacteristic
37C2×#419C2211yescharacteristic
38C2×#419C2211yescharacteristic
39C2×#418C2211yescharacteristic
40C2×#418C2211yescharacteristic
41C22×#103C2211yescharacteristic
42C22×#103C2211yescharacteristic
43C22×#102C2211yescharacteristic
44C22×#102C2211yescharacteristic
45C22×#97C2211yescharacteristic
46#2583—22no—
47#2551—22no—
48#2551C2214yescomplemented
49#2550—22no—
50#2550C2214yescomplemented
51#2483—22no—
52#2483—22no—
53#2447—22no—
54#2447—22no—
55#2447C2214yes—
56#2447C2214yes—
57#2434—22no—
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2310C2211yescharacteristic
61#2310C2211yescharacteristic
62#2309C2211yescharacteristic
63#2296C2211yescharacteristic
64#2296C2211yescharacteristic
65#2296C2211yescharacteristic
66#2274C2211yescharacteristic
67#2097C2212yescomplemented
68#2097C2212yescomplemented
69#2096C2212yescomplemented
70#2095C2212yescomplemented
71#2073C2214yes—
72#2071C2214yes—
73#2068C2214yescomplemented
74#2066C2214yescomplemented
75#2058C2214yescomplemented
76#2057C2214yescomplemented
77#2042C2214yes—
78#2042C2214yes—
79#2026C2214yescomplemented
80#2024C2214yescomplemented
81#1944C2212yescomplemented
82#1944C2212yescomplemented
83#1943C2212yescomplemented
84#1943C2212yescomplemented
85#1928C2214yescomplemented
86#1927C2214yescomplemented
87#1926C2214yes—
88#1925C2214yes—
89#1917C2214yescomplemented
90#1917C2214yescomplemented
91#1864C2214yes—
92#1862C2214yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1735C2211yescharacteristic
96#1735C2211yescharacteristic
97#1729C2211yescharacteristic
98#1729C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C23×Q8—21no—
102C23×Q8C2311yescharacteristic
103C23×D4C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#544—24no—
109#544—42no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—24no—
112C2×C4⋊Q8—24no—
113C2×C4⋊Q8—22no—
114C2×C4⋊Q8—42no—
115C2×C4⋊Q8C2314yes—
116C2×C4⋊Q8C2312yes—
117C2×#125—21no—
118C2×#125C2312yes—
119C2×#124—24no—
120C2×#124—22no—
121C2×#124—42no—
122C2×#124C2314yes—
123C2×C22⋊Q8—22no—
124#518—24no—
125#518—42no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C22×C4⋊C4—21no—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C22⋊C4—21no—
140C22×C22⋊C4—21no—
141C22×C22⋊C4—22no—
142C22×C42C2311yescharacteristic
143#489—216no—
144#489—216no—
145#478—28no—
146#478—28no—
147#476—28no—
148#474—28no—
149#428—24no—
150#428—42no—
151#427—24no—
152#427—24no—
153C2×#107C2311yescharacteristic
154C2×#107C2312yes—
155C2×#107C2312yes—
156C2×#107C2311yescharacteristic
157C2×#107C2312yes—
158C2×#106C2311yescharacteristic
159C2×#106C2312yes—
160C2×#106C2312yes—
161C2×#106C2311yescharacteristic
162C2×#106C2312yes—
163#424—24no—
164C2×C4⋊C8C2312yes—
165C2×C4⋊C8C2312yes—
166#419—24no—
167#419—42no—
168#418—24no—
169#418—24no—
170C2×#103—24no—
171C2×#103—24no—
172C2×#103C2312yes—
173C2×#103C2314yes—
174C2×#103C2312yes—
175C2×#103C2314yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#102—24no—
179C2×#102D418yes—
180C2×#102C2312yes—
181C2×#102C2312yes—
182C2×#102C2314yes—
183C2×#102C2314yes—
184C2×#102C2312yes—
185C2×#102C2312yes—
186C2×#97—24no—
187C2×#97C2312yes—
188C2×#97C2312yes—
189#398—24no—
190#397—24no—
191#397—24no—
192#397—24no—
193#397—24no—
194#395—24no—
195#395—24no—
196#395—24no—
197#391—22no—
198#391—21no—
199#391—21no—
200#391—24no—
201#391C2314yes—
202#391C2312yes—
203#391C2312yes—
204#390—22no—
205#390C2314yes—
206#389—22no—
207#389—24no—
208#389—24no—
209#389D412yes—
210#389D412yes—
211#389C2312yes—
212#389C2312yes—
213#389C2314yes—
214#389C2312yes—
215#389C2312yes—
216#385—21no—
217#385—21no—
218#385—21no—
219#385—21no—
220#385—24no—
221#385—24no—
222#385—22no—
223#385—22no—
224#385D412yes—
225#385D412yes—
226#384—21no—
227#384C2312yes—
228#382—21no—
229#382C2312yes—
230#381—22no—
231#381—22no—
232#381—24no—
233#381—24no—
234#379—22no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4C2314yes—
239C4×C4⋊C4C2314yes—
240C4×C22⋊C4—22no—
241#340C2314yes—
242#340C2314yes—
243#340C2312yes—
244#340C2312yes—
245#340C2312yes—
246#340C2312yes—
247#336C2314yes—
248#336C2314yes—
249#336C2312yes—
250#336C2312yes—
251#336C2314yes—
252C25—21no—
253C22×Q8—21no—
254C22×Q8—24no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8—22no—
261C22×Q8—22no—
262C22×Q8—44no—
263C22×Q8—84no—
264C22×Q8C2412yes—
265C22×D4—21no—
266C22×D4—22no—
267C22×D4—22no—
268C22×D4—22no—
269C22×D4—22no—
270C22×D4—22no—
271C22×D4C2×D414yes—
272C22×D4C2412yes—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—21no—
276C23×C4—22no—
277C23×C4—22no—
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—44no—
301C4⋊Q8—42no—
302C4⋊Q8—48no—
303C4⋊Q8—44no—
304C4⋊Q8—88no—
305#125—24no—
306#125—42no—
307#125—44no—
308#124—28no—
309#124—44no—
310#124—48no—
311C22⋊Q8—48no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—42no—
315C4×D4—42no—
316C4×D4—44no—
317C4×D4—44no—
318#117—22no—
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—22no—
329#117#4114yes—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D414yes—
350C2×C4⋊C4C2×D414yes—
351C2×C4⋊C4C2×D414yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—48no—
371C2×C22⋊C4—44no—
372C2×C22⋊C4—44no—
373C2×C22⋊C4C2×D412yes—
374C2×C22⋊C4C2×D412yes—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—24no—
382C2×C42C2×D412yes—
383C2×C42C2412yes—
384C2×C42C2412yes—
385C2×C42C2412yes—
386#107—22no—
387#107—22no—
388#107—24no—
389#107—24no—
390#107—42no—
391#106—22no—
392#106—22no—
393#106—24no—
394#106—24no—
395#106—42no—
396C4⋊C8—24no—
397C4⋊C8—24no—
398#103—24no—
399#103—24no—
400#103—28no—
401#103—28no—
402#103—42no—
403#103—42no—
404#102—28no—
405#102—24no—
406#102—24no—
407#102—28no—
408#102C2×D418yes—
409#102C2×D418yes—
410#97—24no—
411#97—24no—
412#95—24no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95C2×D414yes—
424#95C2×D414yes—
425C24—22no—
426C24—24no—
427C24—42no—
428C24—42no—
429C24—82no—
430C24—82no—
431C24C22×D411yescharacteristic
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—48no—
439C2×Q8—44no—
440C2×Q8—48no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—44no—
445C2×Q8—44no—
446C2×Q8—44no—
447C2×Q8—44no—
448C2×Q8—84no—
449C2×Q8—84no—
450C2×Q8—816no—
451C2×D4—28no—
452C2×D4—24no—
453C2×D4—28no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—42no—
460C2×D4—44no—
461C2×D4—42no—
462C2×D4—44no—
463C2×D4C22×D414yes—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—84no—
485C22×C4—84no—
486C22×C4—82no—
487C22×C4—82no—
488C22×C4C22×D414yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D412yes—
495C22×C4C2×#4112yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C42−1+412yes—
498C22×C42−1+412yes—
499C22×C42−1+412yes—
500C22×C4C2511yescharacteristic
501M16—44no—
502M16—44no—
503C2×C8—28no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—44no—
518C2×C8C22×D412yes—
519C2×C8C22×D412yes—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—48no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—42no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—84no—
550C4⋊C4—84no—
551C4⋊C4C22×D414yes—
552C4⋊C4C2×#4114yes—
553C22⋊C4—24no—
554C22⋊C4—24no—
555C22⋊C4—24no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—48no—
559C22⋊C4—48no—
560C22⋊C4—42no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—84no—
564C22⋊C4—84no—
565C42—24no—
566C42—24no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—42no—
574C42—42no—
575C42—42no—
576C42—84no—
577C23—22no—
578C23—22no—
579C23—24no—
580C23—44no—
581C23—44no—
582C23—42no—
583C23—42no—
584C23—48no—
585C23—84no—
586C23—84no—
587C23—164no—
588C23—164no—
589C23#53612yes—
590C23D4212yes—
591C23#54812yes—
592C23C23×D411yescharacteristic
593Q8—42no—
594Q8—42no—
595Q8—42no—
596Q8—42no—
597Q8—816no—
598Q8—84no—
599Q8—164no—
600D4—24no—
601D4—24no—
602D4—48no—
603D4—42no—
604D4—42no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—164no—
645C2×C4—164no—
646C2×C4#53612yes—
647C2×C4#53614yes—
648C2×C4#54214yes—
649C2×C4D4212yes—
650C2×C4#54714yes—
651C2×C4#54812yes—
652C2×C4#56314yes—
653C2×C4C23×D411yescharacteristic
654C2×C4C22×#4111yescharacteristic
655C2×C4C2×H(5,𝔽2)11yescharacteristic
656C2×C4C2×2−1+411yescharacteristic
657C2×C4C2×2−1+411yescharacteristic
658C2×C4C2×2−1+411yescharacteristic
659C8—22no—
660C8—22no—
661C8—42no—
662C8—42no—
663C8—42no—
664C22—22no—
665C22—22no—
666C22—24no—
667C22—24no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—84no—
672C22—84no—
673C22—168no—
674C22#267414yes—
675C22C2×#53611yescharacteristic
676C22C2×D4211yescharacteristic
677C22C2×#54811yescharacteristic
678C22#358512yes—
679C22#358612yes—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—42no—
684C4—42no—
685C4—44no—
686C4—44no—
687C4—82no—
688C4—82no—
689C4—81no—
690C4—81no—
691C4—81no—
692C4—81no—
693C4—82no—
694C4#267414yes—
695C4C2×#53611yescharacteristic
696C4C2×#54211yescharacteristic
697C4C2×#54711yescharacteristic
698C4C2×#56311yescharacteristic
699C2—24no—
700C2—84no—
701C2C2×#267411yescharacteristic
702C2#6088512yes—
703C2#6105811yescharacteristic
704C2#6106011yescharacteristic
705C2#6133612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
4. #35576C2semidirect
5. #30831C2semidirect
6. #30829C2semidirect
7. #30823C2semidirect
8. #30821C2semidirect
9. #23736C2semidirect
11. #23613C2semidirect
12. #23552C2semidirect
14. #23457C2semidirect
15. #21154C2semidirect
17. #21064C2semidirect
18. #21021C2semidirect
48. #2551C22semidirect
50. #2550C22semidirect
67. #2097C22semidirect
68. #2097C22semidirect
69. #2096C22semidirect
70. #2095C22semidirect
73. #2068C22semidirect
74. #2066C22semidirect
75. #2058C22semidirect
76. #2057C22semidirect
79. #2026C22semidirect
80. #2024C22semidirect
81. #1944C22semidirect
82. #1944C22semidirect
83. #1943C22semidirect
84. #1943C22semidirect
85. #1928C22semidirect
86. #1927C22semidirect
89. #1917C22semidirect
90. #1917C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4892
C2×#30812
#356262
#355762
#308312
#308292
#308232
#308212
#237362
#236322
#236132
#235522
#235062
#234572
#211542
#210652
#210642
#210212
#208762