Finite Group Databasebeta

← #6502670

Subgroups — #6502670

Order 512.

2152 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic, complemented
2C22×#468C211yesmaximal, characteristic
3#37451C211yesmaximal, characteristic
4#34370C211yesmaximal, characteristic, complemented
5#28590C214yesmaximal, complemented
6#25217C214yesmaximal, complemented
7#25107C214yesmaximal, complemented
8#24765C214yesmaximal, complemented
9#24582C212yesmaximal, complemented
10#24532C212yesmaximal, complemented
11#24479C212yesmaximal, complemented
12#24357C212yesmaximal, complemented
13#21761C211yesmaximal, characteristic
14#20972C211yesmaximal, characteristic, complemented
15#20836C211yesmaximal, characteristic, complemented
16C23×SD16C2211yescharacteristic
17C23×M16C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21#3189—28no—
22C2×#484C2212yes—
23C2×#484C2212yescomplemented
24C2×#468—28no—
25C2×#468C2214yes—
26C2×#468C2214yes—
27C2×#467C2212yescomplemented
28C2×#467C2212yescomplemented
29C2×#464C2214yescomplemented
30C2×#451C2212yes—
31C2×#451C2212yescomplemented
32C22×#107C2211yescharacteristic
33C2×#424C2212yescomplemented
34C2×#419C2212yescomplemented
35C22×#103C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38#2816—22no—
39#2626—22no—
40#2620C2214yes—
41#2618—22no—
42#2502C2212yes—
43#2501C2212yes—
44#2436C2212yes—
45#2435C2212yes—
46#2405—21no—
47#2405—21no—
48#2378—22no—
49#2362C2214yescomplemented
50#2345—22no—
51#2345C2212yes—
52Q8×C22⋊C4C2212yes—
53#2332—22no—
54#2323C2214yescomplemented
55#2312—21no—
56#2312—21no—
57#2280C2211yescharacteristic
58#2280C2211yescharacteristic
59#2083C2214yescomplemented
60#2072C2214yescomplemented
61#2068C2214yescomplemented
62#2067C2214yescomplemented
63#2056C2214yescomplemented
64#2055C2214yescomplemented
65#2042C2214yescomplemented
66#2015C2214yescomplemented
67#1983C2214yescomplemented
68#1947C2214yescomplemented
69#1928C2214yescomplemented
70#1925C2214yescomplemented
71#1894C2214yescomplemented
72#1893C2212yes—
73#1893C2212yes—
74#1892C2212yes—
75#1892C2212yes—
76#1880C2214yescomplemented
77#1878C2214yescomplemented
78#1875C2214yes—
79#1851C2214yescomplemented
80#1850C2212yes—
81#1850C2212yes—
82#1824C2212yes—
83#1803C2214yescomplemented
84#1786C2212yes—
85#1736C2212yescomplemented
86#1729C2212yescomplemented
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×D4—21no—
90C23×D4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×Q16C2312yes—
93C22×Q16C2312yes—
94C22×SD16—24no—
95C22×SD16—44no—
96C22×SD16C2314yes—
97C22×M16—24no—
98C22×M16—44no—
99C22×M16C2312yes—
100C22×M16C2312yes—
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103C23×C8C2311yescharacteristic
104C23×C8C2311yescharacteristic
105C2×#124C2314yes—
106C2×#123—22no—
107C2×#123—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8D418yes—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120C2×#121—21no—
121C2×#121—21no—
122C2×#121—24no—
123C2×#121—24no—
124C2×#121C2312yes—
125C2×#121C2312yes—
126C2×C4×Q8C2314yes—
127C2×#117C2312yes—
128C2×#117C2312yes—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137#484—28no—
138#484—28no—
139#468—216no—
140#468—216no—
141#467—28no—
142#467—28no—
143#464—48no—
144#451—28no—
145#451—28no—
146C2×#107—24no—
147C2×#107C2312yes—
148C2×#107C2312yes—
149C2×#107C2314yes—
150#424—44no—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2314yes—
153#419—44no—
154C2×#103—24no—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#102—24no—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×C22⋊C8—24no—
167C2×C22⋊C8C2312yes—
168C2×C22⋊C8C2312yes—
169#399—24no—
170#398—24no—
171#396—24no—
172#396—24no—
173#394—24no—
174#394—24no—
175#394—24no—
176#393—24no—
177#391—24no—
178#391C2314yes—
179#391C2314yes—
180#390C2314yes—
181#389—24no—
182#389C2314yes—
183#388—24no—
184#388—24no—
185#388—24no—
186#388—24no—
187#387—24no—
188#387—24no—
189#387—24no—
190#386—22no—
191#386—22no—
192#385—22no—
193#385—22no—
194#384C2314yes—
195#381—21no—
196#381D412yes—
197#380—22no—
198#380—22no—
199#380D412yes—
200#380D412yes—
201#380C2312yes—
202#380C2312yes—
203#379—21no—
204#379—22no—
205#379—22no—
206#379D411yescharacteristic
207#379D411yescharacteristic
208C4×C4⋊C4—24no—
209C4×C4⋊C4C2314yes—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4C2312yes—
215C4×C22⋊C4C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#336C2314yes—
221#336C2314yes—
222C25—21no—
223C25C2×D411yescharacteristic
224C22×Q8—21no—
225C22×Q8—21no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—42no—
230C22×Q8—42no—
231C22×Q8—42no—
232C22×Q8—42no—
233C22×Q8C2×D414yes—
234C22×Q8C2412yes—
235C22×Q8C2412yes—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—42no—
242C22×D4—42no—
243C22×D4—44no—
244C22×D4—84no—
245C22×D4C2412yes—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—42no—
252C23×C4—42no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D414yes—
257C23×C4#4114yes—
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×Q16—24no—
262C2×Q16—24no—
263C2×Q16—44no—
264C2×Q16—44no—
265C2×SD16—28no—
266C2×SD16—28no—
267C2×SD16—44no—
268C2×SD16—416no—
269C2×M16—24no—
270C2×M16—24no—
271C2×M16—24no—
272C2×M16—24no—
273C2×M16—42no—
274C2×M16—42no—
275C2×M16—416no—
276C22×C8—22no—
277C22×C8—24no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—42no—
282C22×C8—42no—
283C22×C8—42no—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2412yes—
288#124—44no—
289#124—44no—
290#123—84no—
291#123—84no—
292C22⋊Q8—28no—
293C22⋊Q8—28no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C22⋊Q8—24no—
298C22⋊Q8—24no—
299C22⋊Q8—24no—
300C22⋊Q8—44no—
301C22⋊Q8—44no—
302C22⋊Q8C2×D418yes—
303C22⋊Q8C2×D418yes—
304#121—24no—
305#121—24no—
306#121—24no—
307#121—24no—
308#121—44no—
309#121—44no—
310C4×Q8—44no—
311C4×Q8—44no—
312#117—42no—
313#117—42no—
314#117—42no—
315#117—42no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4C2×D414yes—
335C2×C4⋊C4C2×D414yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2414yes—
339C2×C4⋊C4C2412yes—
340C2×C22⋊C4—28no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—44no—
355C2×C22⋊C4—44no—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C42—22no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42C2414yes—
366#107—28no—
367#107—24no—
368#107—24no—
369C4⋊C8—28no—
370C4⋊C8—44no—
371#103—28no—
372#103—24no—
373#103—24no—
374#103—28no—
375#103—24no—
376#103—24no—
377#103—44no—
378#102—28no—
379#102—44no—
380#102—44no—
381C22⋊C8—24no—
382C22⋊C8—24no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95C2×D414yes—
392#95C2×D414yes—
393C24—24no—
394C24—22no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—44no—
399C24—82no—
400C24—82no—
401C24#12114yes—
402C24C22×D411yescharacteristic
403C24C22×D411yescharacteristic
404C24C22×D411yescharacteristic
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—48no—
416C2×Q8—42no—
417C2×Q8—82no—
418C2×Q8—82no—
419C2×Q8—82no—
420C2×Q8—82no—
421C2×Q8—84no—
422C2×Q8C22×D414yes—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—42no—
434C2×D4—84no—
435C2×D4—82no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4—816no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4#12114yes—
468C22×C4#12114yes—
469C22×C4#12114yes—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2511yescharacteristic
482Q16—84no—
483Q16—84no—
484SD16—416no—
485M16—48no—
486M16—48no—
487C2×C8—28no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—44no—
498C2×C8—82no—
499C2×C8—82no—
500C2×C8H(5,𝔽2)14yes—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4C22×D414yes—
521C22⋊C4—28no—
522C22⋊C4—28no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C42—24no—
541C42—24no—
542C42—84no—
543C42—82no—
544C42—84no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—28no—
554C23—44no—
555C23—44no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23—88no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—168no—
566C23C2×#12112yes—
567C23C2×#12112yes—
568C23C2×#12112yes—
569C23#57814yes—
570C23C23×D411yescharacteristic
571Q8—24no—
572Q8—24no—
573Q8—44no—
574Q8—42no—
575Q8—42no—
576Q8—44no—
577Q8—84no—
578Q8—84no—
579D4—42no—
580D4—42no—
581D4—44no—
582D4—84no—
583D4—88no—
584D4—88no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
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—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4C2×#12112yes—
626C2×C4C2×#12112yes—
627C2×C4C2×#12112yes—
628C2×C4C23×D411yescharacteristic
629C2×C4C22×#4111yescharacteristic
630C2×C4C2×H(5,𝔽2)12yes—
631C2×C4#58512yes—
632C8—24no—
633C8—42no—
634C8—42no—
635C8—42no—
636C22—22no—
637C22—22no—
638C22—28no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—44no—
646C22—44no—
647C22—88no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—84no—
652C22—84no—
653C22—168no—
654C22#304414yes—
655C22C22×#12111yescharacteristic
656C22#349612yes—
657C22C2×#57812yes—
658C22#358512yes—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—44no—
663C4—44no—
664C4—44no—
665C4—44no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—84no—
670C4#348411yescharacteristic
671C4#348511yescharacteristic
672C4#350512yes—
673C2—24no—
674C2—42no—
675C2—42no—
676C2—162no—
677C2#3535811yescharacteristic
678C2C2×#304411yescharacteristic
679C2#6042611yescharacteristic
680C2#6231214yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3189C2semidirect
4. #34370C2semidirect
5. #28590C2semidirect
6. #25217C2semidirect
7. #25107C2semidirect
8. #24765C2semidirect
9. #24582C2semidirect
10. #24532C2semidirect
11. #24479C2semidirect
12. #24357C2semidirect
14. #20972C2semidirect
15. #20836C2semidirect
23. C2×#484C22semidirect
27. C2×#467C22semidirect
28. C2×#467C22semidirect
29. C2×#464C22semidirect
31. C2×#451C22semidirect
33. C2×#424C22semidirect
34. C2×#419C22semidirect
49. #2362C22semidirect
54. #2323C22semidirect
59. #2083C22semidirect
60. #2072C22semidirect
61. #2068C22semidirect
62. #2067C22semidirect
63. #2056C22semidirect
64. #2055C22semidirect
65. #2042C22semidirect
66. #2015C22semidirect
67. #1983C22semidirect
68. #1947C22semidirect
69. #1928C22semidirect
70. #1925C22semidirect
71. #1894C22semidirect
76. #1880C22semidirect
77. #1878C22semidirect
79. #1851C22semidirect
83. #1803C22semidirect
85. #1736C22semidirect
86. #1729C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
C22×#4682
#374512
#343702
#285902
#252172
#251072
#247652
#245822
#245322
#244792
#243572
#217612
#209722
#208362