Finite Group Databasebeta

← #7502003

Subgroups — #7502003

Order 512.

1868 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
1C22×#488C211yesmaximal, characteristic
2C2×#3081C211yesmaximal, characteristic
3#35640C211yesmaximal, characteristic, complemented
4#35626C211yesmaximal, characteristic
5#30819C214yesmaximal, complemented
6#30817C214yesmaximal, complemented
7#30811C214yesmaximal, complemented
8#30809C214yesmaximal, complemented
9#23736C211yesmaximal, characteristic, complemented
10#23632C211yesmaximal, characteristic
11#23613C211yesmaximal, characteristic, complemented
12#23553C211yesmaximal, characteristic, complemented
13#23506C211yesmaximal, characteristic
14#23457C211yesmaximal, characteristic, complemented
15#21151C211yesmaximal, characteristic, complemented
16#21065C211yesmaximal, characteristic
17#21064C211yesmaximal, characteristic, complemented
18#21020C211yesmaximal, characteristic, complemented
19#20876C211yesmaximal, characteristic
20C2×#544C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C2×#518C2211yescharacteristic
24C2×#488—28no—
25C2×#488C2214yes—
26C2×#488C2214yes—
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×C4×C8C2211yescharacteristic
46#2583—22no—
47#2552—22no—
48#2552C2214yescomplemented
49#2550—22no—
50#2550C2214yescomplemented
51#2502—22no—
52#2483—22no—
53#2483—22no—
54#2447—22no—
55#2447—22no—
56#2447C2214yes—
57#2447C2214yes—
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#2059C2214yescomplemented
76#2059C2214yescomplemented
77#2042C2214yes—
78#2042C2214yes—
79#2026C2214yescomplemented
80#2024C2214yescomplemented
81#1942C2212yescomplemented
82#1942C2212yescomplemented
83#1941C2212yescomplemented
84#1941C2212yescomplemented
85#1928C2214yescomplemented
86#1927C2214yescomplemented
87#1926C2214yes—
88#1925C2214yes—
89#1916C2214yescomplemented
90#1916C2214yescomplemented
91#1864C2214yes—
92#1862C2214yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1735C2211yescharacteristic
96#1735C2211yescharacteristic
97#1729C2211yescharacteristic
98#1729C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C23×Q8C2311yescharacteristic
102C23×D4C2311yescharacteristic
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107#544—24no—
108#544—42no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8C2312yes—
111C2×#125—21no—
112C2×#125—24no—
113C2×#125—22no—
114C2×#125—42no—
115C2×#125C2312yes—
116C2×#125C2314yes—
117C2×#124—24no—
118C2×#124—22no—
119C2×#124—42no—
120C2×#124C2314yes—
121C2×C22⋊Q8—22no—
122#518—24no—
123#518—42no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C22⋊C4—21no—
139C22×C22⋊C4—21no—
140C22×C22⋊C4—22no—
141C22×C42C2311yescharacteristic
142#488—216no—
143#488—216no—
144#478—28no—
145#478—28no—
146#476—28no—
147#474—28no—
148#428—24no—
149#428—42no—
150#427—24no—
151#427—24no—
152C2×#107C2311yescharacteristic
153C2×#107C2312yes—
154C2×#107C2312yes—
155C2×#107C2311yescharacteristic
156C2×#107C2312yes—
157C2×#106C2311yescharacteristic
158C2×#106C2312yes—
159C2×#106C2312yes—
160C2×#106C2311yescharacteristic
161C2×#106C2312yes—
162#424—24no—
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2312yes—
165#419—24no—
166#419—42no—
167#418—24no—
168#418—24no—
169C2×#103—24no—
170C2×#103—24no—
171C2×#103C2312yes—
172C2×#103C2314yes—
173C2×#103C2312yes—
174C2×#103C2314yes—
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#102—24no—
178C2×#102D418yes—
179C2×#102C2312yes—
180C2×#102C2314yes—
181C2×#102C2312yes—
182C2×#102C2314yes—
183C2×#102C2312yes—
184C2×#102C2312yes—
185C2×C4×C8—24no—
186C2×C4×C8C2312yes—
187C2×C4×C8C2312yes—
188#398—24no—
189#398—24no—
190#398—24no—
191#397—24no—
192#397—24no—
193#397—24no—
194#397—24no—
195#395—24no—
196#391—22no—
197#391—21no—
198#391—21no—
199#391—24no—
200#391C2314yes—
201#391C2312yes—
202#391C2312yes—
203#390—22no—
204#390C2314yes—
205#389—22no—
206#389—24no—
207#389—24no—
208#389D412yes—
209#389D412yes—
210#389C2312yes—
211#389C2312yes—
212#389C2314yes—
213#389C2312yes—
214#389C2312yes—
215#385—21no—
216#385—21no—
217#385—21no—
218#385—21no—
219#385—24no—
220#385—24no—
221#385—22no—
222#385—22no—
223#385D412yes—
224#385D412yes—
225#384—21no—
226#384—24no—
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—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8—22no—
261C22×Q8C2412yes—
262C22×D4—21no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—22no—
266C22×D4—22no—
267C22×D4—22no—
268C22×D4C2×D414yes—
269C22×D4C2412yes—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—22no—
274C23×C4—22no—
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2411yescharacteristic
278C2×M16—22no—
279C2×M16—22no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C22×C8—22no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8C2×D414yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2411yescharacteristic
294C22×C8C2411yescharacteristic
295C4⋊Q8—24no—
296C4⋊Q8—42no—
297C4⋊Q8—44no—
298#125—28no—
299#125—24no—
300#125—44no—
301#125—42no—
302#125—48no—
303#125—44no—
304#124—28no—
305#124—44no—
306#124—48no—
307C22⋊Q8—48no—
308C4×D4—24no—
309C4×D4—24no—
310C4×D4—42no—
311C4×D4—42no—
312C4×D4—44no—
313C4×D4—44no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117#4114yes—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—44no—
343C2×C4⋊C4—44no—
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2×D414yes—
348C2×C4⋊C4C2×D414yes—
349C2×C4⋊C4C2×D414yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2411yescharacteristic
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—48no—
369C2×C22⋊C4—44no—
370C2×C22⋊C4—44no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D412yes—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—24no—
380C2×C42C2×D412yes—
381C2×C42C2412yes—
382C2×C42C2412yes—
383C2×C42C2412yes—
384#107—22no—
385#107—22no—
386#107—24no—
387#107—24no—
388#107—42no—
389#106—22no—
390#106—22no—
391#106—24no—
392#106—24no—
393#106—42no—
394C4⋊C8—24no—
395C4⋊C8—24no—
396#103—24no—
397#103—24no—
398#103—28no—
399#103—28no—
400#103—42no—
401#103—42no—
402#102—28no—
403#102—24no—
404#102—24no—
405#102—28no—
406#102C2×D418yes—
407#102C2×D418yes—
408C4×C8—24no—
409C4×C8—24no—
410#95—24no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—24no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95C2×D414yes—
422#95C2×D414yes—
423C24—22no—
424C24—24no—
425C24—42no—
426C24—42no—
427C24—82no—
428C24—82no—
429C24C22×D411yescharacteristic
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—44no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—84no—
445C2×D4—28no—
446C2×D4—24no—
447C2×D4—28no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—42no—
454C2×D4—44no—
455C2×D4—42no—
456C2×D4—44no—
457C2×D4C22×D414yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4C22×D414yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C42−1+412yes—
492C22×C42−1+412yes—
493C22×C42−1+412yes—
494C22×C4C2511yescharacteristic
495M16—44no—
496M16—44no—
497C2×C8—28no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—44no—
512C2×C8C22×D412yes—
513C2×C8C22×D412yes—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—22no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—48no—
535C4⋊C4—48no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4C22×D414yes—
547C4⋊C4C2×#4114yes—
548C22⋊C4—24no—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—48no—
554C22⋊C4—48no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C22⋊C4—84no—
559C22⋊C4—84no—
560C42—24no—
561C42—24no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—42no—
569C42—42no—
570C42—42no—
571C42—84no—
572C23—22no—
573C23—22no—
574C23—24no—
575C23—44no—
576C23—44no—
577C23—42no—
578C23—42no—
579C23—48no—
580C23—84no—
581C23—84no—
582C23—164no—
583C23—164no—
584C23#53612yes—
585C23D4212yes—
586C23#54812yes—
587C23C23×D411yescharacteristic
588Q8—42no—
589Q8—42no—
590Q8—42no—
591Q8—42no—
592Q8—84no—
593Q8—164no—
594D4—24no—
595D4—24no—
596D4—48no—
597D4—42no—
598D4—42no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—84no—
638C2×C4—164no—
639C2×C4—164no—
640C2×C4#53612yes—
641C2×C4#53614yes—
642C2×C4#54214yes—
643C2×C4D4212yes—
644C2×C4#54714yes—
645C2×C4#54812yes—
646C2×C4#56314yes—
647C2×C4C23×D411yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4C2×2−1+411yescharacteristic
651C2×C4C2×2−1+411yescharacteristic
652C2×C4C2×2−1+411yescharacteristic
653C8—22no—
654C8—22no—
655C8—42no—
656C8—42no—
657C8—42no—
658C22—22no—
659C22—22no—
660C22—24no—
661C22—24no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—84no—
666C22—84no—
667C22—168no—
668C22#267414yes—
669C22C2×#53611yescharacteristic
670C22C2×D4211yescharacteristic
671C22C2×#54811yescharacteristic
672C22#358412yes—
673C22#358412yes—
674C4—42no—
675C4—42no—
676C4—44no—
677C4—42no—
678C4—42no—
679C4—44no—
680C4—44no—
681C4—81no—
682C4—81no—
683C4—82no—
684C4—81no—
685C4—82no—
686C4—81no—
687C4—82no—
688C4#267414yes—
689C4C2×#53611yescharacteristic
690C4C2×#54211yescharacteristic
691C4C2×#54711yescharacteristic
692C4C2×#56311yescharacteristic
693C2—24no—
694C2—84no—
695C2C2×#267411yescharacteristic
696C2#6087912yes—
697C2#6105111yescharacteristic
698C2#6105211yescharacteristic
699C2#6133112yes—
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
3. #35640C2semidirect
5. #30819C2semidirect
6. #30817C2semidirect
7. #30811C2semidirect
8. #30809C2semidirect
9. #23736C2semidirect
11. #23613C2semidirect
12. #23553C2semidirect
14. #23457C2semidirect
15. #21151C2semidirect
17. #21064C2semidirect
18. #21020C2semidirect
48. #2552C22semidirect
50. #2550C22semidirect
67. #2097C22semidirect
68. #2097C22semidirect
69. #2096C22semidirect
70. #2095C22semidirect
73. #2068C22semidirect
74. #2066C22semidirect
75. #2059C22semidirect
76. #2059C22semidirect
79. #2026C22semidirect
80. #2024C22semidirect
81. #1942C22semidirect
82. #1942C22semidirect
83. #1941C22semidirect
84. #1941C22semidirect
85. #1928C22semidirect
86. #1927C22semidirect
89. #1916C22semidirect
90. #1916C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4882
C2×#30812
#356402
#356262
#308192
#308172
#308112
#308092
#237362
#236322
#236132
#235532
#235062
#234572
#211512
#210652
#210642
#210202
#208762