Finite Group Databasebeta

← #6499938

Subgroups — #6499938

Order 512.

3274 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3538C211yesmaximal, characteristic, complemented
2C2×#3200C212yesmaximal, complemented
3#35563C211yesmaximal, characteristic
4#25560C214yesmaximal, complemented
5#25544C214yesmaximal, complemented
6#24445C214yesmaximal, complemented
7#24429C214yesmaximal, complemented
8#24072C212yesmaximal, complemented
9#24010C212yesmaximal, complemented
10#23942C212yesmaximal, complemented
11#23919C212yesmaximal, complemented
12#21744C211yesmaximal, characteristic, complemented
13#20913C212yesmaximal, complemented
14#3538—216no—
15C2×#563—22no—
16C2×#552C2214yescomplemented
17C2×#551—22no—
18C2×D4×Q8C2214yescomplemented
19C2×#544—22no—
20C2×#541C2212yes—
21C2×#539—22no—
22C2×#536C2212yes—
23C2×#535C2211yescharacteristic
24C2×#525C2211yescharacteristic
25C22×C22⋊Q8C2212yes—
26C22×#121C2211yescharacteristic
27C2×#520C2211yescharacteristic
28#3200—216no—
29C2×#485C2214yes—
30C2×#482C2214yescomplemented
31C2×#468C2214yescomplemented
32C2×#465C2214yescomplemented
33C2×#448C2214yes—
34C2×#448C2214yescomplemented
35C2×#450C2214yes—
36C2×#419C2212yescomplemented
37C22×#103C2212yescomplemented
38C2×#408C2212yes—
39#2534—22no—
40#2428—22no—
41#2426—22no—
42#2416C2214yes—
43#2414C2214yes—
44#2408—22no—
45#2316C2211yescharacteristic
46#2309C2211yescharacteristic
47#2280C2211yescharacteristic
48#2059C2214yescomplemented
49#2043C2214yescomplemented
50#2036C2214yescomplemented
51#2035C2214yescomplemented
52#2025C2212yescomplemented
53#2025C2212yescomplemented
54#2013C2212yescomplemented
55#2013C2212yescomplemented
56#2000C2214yes—
57#2000C2214yes—
58#1880C2214yescomplemented
59#1880C2214yescomplemented
60#1874C2214yes—
61#1873C2214yes—
62#1873C2214yescomplemented
63#1872C2214yes—
64#1872C2214yescomplemented
65#1870C2214yescomplemented
66#1869C2214yescomplemented
67#1865C2214yescomplemented
68#1863C2214yescomplemented
69#1853C2212yescomplemented
70#1852C2212yescomplemented
71#1818C2212yes—
72#1787C2212yescomplemented
73#1787C2212yescomplemented
74#1779C2212yes—
75#1736C2212yescomplemented
76#1735C2212yescomplemented
77C22×#41C2311yescharacteristic
78C23×Q8C2312yes—
79C23×Q8C2312yes—
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×SD16C2314yes—
84C22×SD16C2314yes—
85C22×M16C2312yes—
86C23×C8C2312yes—
87#563—232no—
88#552—216no—
89#552—216no—
90#551—232no—
91D4×Q8—216no—
92D4×Q8—216no—
93#544—232no—
94#541—216no—
95#539—232no—
96#536—28no—
97#536D4116yescomplemented
98#535—28no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8C2312yes—
102C2×#127—21no—
103C2×#126—24no—
104C2×#126—24no—
105C2×#125—24no—
106C2×#125—24no—
107C2×#125C2312yes—
108C2×#124—22no—
109C2×#124—21no—
110#525—28no—
111C2×#123—22no—
112C2×#123—22no—
113C2×#123—24no—
114C2×#123D412yes—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—216no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8D412yes—
121C2×C22⋊Q8C2314yes—
122C2×C22⋊Q8C2314yes—
123C2×C22⋊Q8C2314yes—
124C2×#121—22no—
125C2×#121—24no—
126C2×#121—44no—
127C2×#121D412yes—
128C2×#121C2314yes—
129C2×#121C2314yes—
130C2×#120—22no—
131C2×#120D412yes—
132#520—44no—
133C2×C4×Q8—21no—
134C2×C4×Q8C2314yes—
135C2×C4×D4—24no—
136C2×C4×D4—22no—
137C2×C4×D4—21no—
138C2×C4×D4C2314yes—
139C2×#117—24no—
140C2×#117C2311yescharacteristic
141C2×#117C2312yes—
142C2×#117C2312yes—
143C2×#117C2311yescharacteristic
144C2×#117C2311yescharacteristic
145C22×C4⋊C4—22no—
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2312yes—
148C22×C4⋊C4C2312yes—
149C22×C22⋊C4—22no—
150C22×C22⋊C4—21no—
151#485—216no—
152#482—216no—
153#468—216no—
154#465—216no—
155#450—216no—
156#448—216no—
157#448—48no—
158C2×#107C2314yes—
159C2×#106C2314yes—
160#419—44no—
161C2×#103—44no—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#102C2314yes—
169C2×#102C2314yes—
170#408—28no—
171C2×C22⋊C8C2314yes—
172C2×C22⋊C8C2314yes—
173C2×C22⋊C8C2312yes—
174C2×C22⋊C8C2312yes—
175C2×C22⋊C8C2314yes—
176#396—24no—
177#394—24no—
178#394—24no—
179#392—24no—
180#390—22no—
181#390—24no—
182#390C2312yes—
183#390C2312yes—
184#389—22no—
185#389C2314yes—
186#388—24no—
187#386—24no—
188#381—21no—
189#380—22no—
190#380C2312yes—
191#380C2312yes—
192#379—21no—
193#379—22no—
194C4×C22⋊C4—22no—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2312yes—
201#336C2312yes—
202C25—21no—
203C25C2×D411yescharacteristic
204C2×#41—42no—
205C2×#41—41no—
206C2×#41—41no—
207C2×#41C2×D418yes—
208C22×Q8—28no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—24no—
212C22×Q8—42no—
213C22×Q8—42no—
214C22×Q8—42no—
215C22×Q8—42no—
216C22×Q8—42no—
217C22×Q8—42no—
218C22×Q8—44no—
219C22×Q8C2×D411yescharacteristic
220C22×Q8C2×D418yescomplemented
221C22×Q8C2414yes—
222C22×D4—24no—
223C22×D4—24no—
224C22×D4—24no—
225C22×D4—21no—
226C22×D4—42no—
227C22×D4—41no—
228C22×D4—41no—
229C22×D4—44no—
230C22×D4—84no—
231C22×D4C2×D414yes—
232C22×D4C2×D412yes—
233C22×D4C2×D411yescharacteristic
234C22×D4C2412yes—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—24no—
240C23×C4—22no—
241C23×C4—42no—
242C23×C4—41no—
243C23×C4—41no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D412yes—
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C23×C4C2412yes—
249C2×SD16—28no—
250C2×SD16—28no—
251C2×SD16—44no—
252C2×SD16—44no—
253C2×SD16—44no—
254C2×SD16—44no—
255C2×M16—24no—
256C2×M16—42no—
257C2×M16—42no—
258C22×C8—24no—
259C22×C8—42no—
260C22×C8—42no—
261C22×C8C2414yes—
262C22×C8C2414yes—
263C4⋊Q8—232no—
264C4⋊Q8—216no—
265C4⋊Q8—28no—
266#127—44no—
267#126—232no—
268#126—232no—
269#125—232no—
270#125—232no—
271#125—28no—
272#124—216no—
273#124—44no—
274#123—232no—
275#123—216no—
276#123—216no—
277#123—28no—
278C22⋊Q8—216no—
279C22⋊Q8—216no—
280C22⋊Q8—232no—
281C22⋊Q8—232no—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C22⋊Q8—28no—
286C22⋊Q8—28no—
287C22⋊Q8C2×D4116yes—
288C22⋊Q8C2×D4116yescomplemented
289#121—28no—
290#121—28no—
291#121—28no—
292#121—28no—
293#121—44no—
294#121—48no—
295#121—88no—
296#120—28no—
297#120—48no—
298C4×Q8—28no—
299C4×Q8—28no—
300C4×Q8—44no—
301C4×D4—232no—
302C4×D4—28no—
303C4×D4—28no—
304C4×D4—44no—
305C4×D4—48no—
306#117—232no—
307#117—24no—
308#117—24no—
309#117—24no—
310#117—24no—
311#117—42no—
312#117—42no—
313#117C2×D418yescomplemented
314C2×C4⋊C4—216no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D414yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2414yes—
335C2×C4⋊C4C2412yes—
336C2×C22⋊C4—216no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4C2×D414yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D414yes—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42C2412yes—
360#107—28no—
361#106—28no—
362#103—28no—
363#103—28no—
364#103—28no—
365#103—28no—
366#103—44no—
367#103—44no—
368#102—28no—
369#102—28no—
370C22⋊C8—28no—
371C22⋊C8—28no—
372C22⋊C8—42no—
373C22⋊C8—42no—
374C22⋊C8—44no—
375#95—22no—
376#95—22no—
377#95—24no—
378C24—22no—
379C24—24no—
380C24—22no—
381C24—41no—
382C24—42no—
383C24—41no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—82no—
388C24—82no—
389C24C22×D411yescharacteristic
390C24C22×D412yes—
391C24C22×D412yes—
392#41—48no—
393#41—48no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—41no—
402C2×Q8—41no—
403C2×Q8—48no—
404C2×Q8—48no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—82no—
409C2×Q8—84no—
410C2×Q8—84no—
411C2×Q8—88no—
412C2×Q8—88no—
413C2×Q8#12014yes—
414C2×Q8C22×D418yes—
415C2×D4—28no—
416C2×D4—28no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—41no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4—41no—
425C2×D4—42no—
426C2×D4—48no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—42no—
432C2×D4—84no—
433C2×D4—84no—
434C2×D4—84no—
435C2×D4—84no—
436C2×D4—84no—
437C2×D4—816no—
438C2×D4#12018yes—
439C2×D4#12014yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—28no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—28no—
452C22×C4—24no—
453C22×C4—28no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—41no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—41no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—84no—
478C22×C4C22×D412yes—
479C22×C4C22×D414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D414yes—
483C22×C4C22×D414yes—
484C22×C4C22×D414yes—
485C22×C4C2511yescharacteristic
486SD16—88no—
487SD16—88no—
488M16—84no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—84no—
496C4⋊C4—24no—
497C4⋊C4—216no—
498C4⋊C4—28no—
499C4⋊C4—216no—
500C4⋊C4—216no—
501C4⋊C4—216no—
502C4⋊C4—216no—
503C4⋊C4—216no—
504C4⋊C4—24no—
505C4⋊C4—28no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—44no—
512C4⋊C4—84no—
513C4⋊C4C22×D418yes—
514C4⋊C4H(5,𝔽2)14yes—
515C22⋊C4—216no—
516C22⋊C4—216no—
517C22⋊C4—28no—
518C22⋊C4—216no—
519C22⋊C4—24no—
520C22⋊C4—28no—
521C22⋊C4—28no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—48no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—82no—
531C42—216no—
532C42—28no—
533C42—24no—
534C42—44no—
535C42—42no—
536C42—42no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—22no—
541C23—22no—
542C23—21no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—21no—
547C23—22no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—44no—
552C23—48no—
553C23—44no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—82no—
559C23—84no—
560C23—84no—
561C23—82no—
562C23—84no—
563C23—84no—
564C23—164no—
565C23—164no—
566C23C2×#12012yes—
567C23C2×#12014yes—
568C23#57714yes—
569C23C23×D411yescharacteristic
570Q8—28no—
571Q8—28no—
572Q8—44no—
573Q8—44no—
574Q8—44no—
575Q8—88no—
576Q8—88no—
577D4—44no—
578D4—48no—
579D4—416no—
580D4—44no—
581D4—44no—
582D4—84no—
583D4—816no—
584D4—168no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—416no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—84no—
620C2×C4—82no—
621C2×C4—88no—
622C2×C4—84no—
623C2×C4—88no—
624C2×C4—164no—
625C2×C4C2×#12014yes—
626C2×C4C2×#12012yes—
627C2×C4C23×D412yes—
628C2×C4C2×H(5,𝔽2)12yes—
629C2×C4#58511yescharacteristic
630C2×C4#58511yescharacteristic
631C8—44no—
632C8—44no—
633C22—24no—
634C22—22no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—28no—
639C22—24no—
640C22—42no—
641C22—44no—
642C22—42no—
643C22—42no—
644C22—41no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—41no—
649C22—42no—
650C22—88no—
651C22—82no—
652C22—84no—
653C22—82no—
654C22—82no—
655C22—84no—
656C22—168no—
657C22C22×#12011yescharacteristic
658C22#348411yescharacteristic
659C22#348511yescharacteristic
660C22C2×#57712yes—
661C22#358612yes—
662C4—22no—
663C4—24no—
664C4—42no—
665C4—48no—
666C4—44no—
667C4—44no—
668C4—48no—
669C4—81no—
670C4—81no—
671C4—82no—
672C4#349612yes—
673C4#349612yes—
674C2—44no—
675C2—44no—
676C2—42no—
677C2—162no—
678C2#3534711yescharacteristic
679C2#6037412yes—
680C2#6163111yescharacteristic
681C2#6165211yescharacteristic
682C2#6175312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35563C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3538C2semidirect
2. C2×#3200C2semidirect
4. #25560C2semidirect
5. #25544C2semidirect
6. #24445C2semidirect
7. #24429C2semidirect
8. #24072C2semidirect
9. #24010C2semidirect
10. #23942C2semidirect
11. #23919C2semidirect
12. #21744C2semidirect
13. #20913C2semidirect
16. C2×#552C22semidirect
18. C2×D4×Q8C22semidirect
30. C2×#482C22semidirect
31. C2×#468C22semidirect
32. C2×#465C22semidirect
34. C2×#448C22semidirect
36. C2×#419C22semidirect
37. C22×#103C22semidirect
48. #2059C22semidirect
49. #2043C22semidirect
50. #2036C22semidirect
51. #2035C22semidirect
52. #2025C22semidirect
53. #2025C22semidirect
54. #2013C22semidirect
55. #2013C22semidirect
58. #1880C22semidirect
59. #1880C22semidirect
62. #1873C22semidirect
64. #1872C22semidirect
65. #1870C22semidirect
66. #1869C22semidirect
67. #1865C22semidirect
68. #1863C22semidirect
69. #1853C22semidirect
70. #1852C22semidirect
72. #1787C22semidirect
73. #1787C22semidirect
75. #1736C22semidirect
76. #1735C22semidirect
97. #536D4semidirect
220. C22×Q8C2×D4semidirect
288. C22⋊Q8C2×D4semidirect
313. #117C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#35382
C2×#32002
#355632
#255602
#255442
#244452
#244292
#240722
#240102
#239422
#239192
#217442
#209132