Finite Group Databasebeta

← #6508739

Subgroups — #6508739

Order 512.

2132 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3192C211yesmaximal, characteristic
2C22×#482C211yesmaximal, characteristic, complemented
3#35262C211yesmaximal, characteristic, complemented
4#34379C211yesmaximal, characteristic, complemented
5#28126C214yesmaximal, complemented
6#27654C214yesmaximal, complemented
7#25228C214yesmaximal, complemented
8#25218C214yesmaximal, complemented
9#24583C212yesmaximal, complemented
10#24578C212yesmaximal, complemented
11#24531C212yesmaximal, complemented
12#24526C212yesmaximal, complemented
13#21933C211yesmaximal, characteristic, complemented
14#21758C211yesmaximal, characteristic, complemented
15#20702C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#3192—28no—
20C2×#481C2212yes—
21C2×#481C2212yes—
22C2×#482—28no—
23C2×#482C2214yes—
24C2×#482C2214yescomplemented
25C2×#469C2212yes—
26C2×#469C2212yes—
27C2×#447C2212yes—
28C2×#447C2212yes—
29C22×#107C2211yescharacteristic, complemented
30C22×#106C2211yescharacteristic, complemented
31C22×#102C2211yescharacteristic, complemented
32C22×#102C2211yescharacteristic, complemented
33C2×#407C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35#2854—21no—
36#2854—21no—
37#2624—22no—
38#2504C2212yes—
39#2501C2212yes—
40#2500C2214yescomplemented
41#2482—22no—
42#2405—22no—
43#2377—22no—
44#2361C2214yescomplemented
45#2345C2212yescomplemented
46D4×C22⋊C4—22no—
47D4×C22⋊C4C2212yescomplemented
48#2325—22no—
49#2324C2214yes—
50#2317C2212yescomplemented
51#2315—21no—
52#2312—21no—
53#2309C2212yescomplemented
54#2280C2211yescharacteristic
55C2×#380C2211yescharacteristic
56#2085C2214yescomplemented
57#2084C2214yescomplemented
58#2073C2214yescomplemented
59#2071C2214yescomplemented
60#2067C2214yescomplemented
61#2065C2214yescomplemented
62#2054C2214yescomplemented
63#2054C2214yescomplemented
64#1949C2214yescomplemented
65#1948C2214yescomplemented
66#1926C2214yescomplemented
67#1925C2214yescomplemented
68#1893C2212yes—
69#1893C2212yes—
70#1879C2214yescomplemented
71#1879C2214yescomplemented
72#1874C2214yescomplemented
73#1851C2212yescomplemented
74#1851C2212yescomplemented
75#1850C2212yescomplemented
76#1850C2212yescomplemented
77#1825C2214yescomplemented
78#1824C2212yescomplemented
79#1823C2212yescomplemented
80#1823C2212yescomplemented
81#1804C2214yescomplemented
82#1802C2214yescomplemented
83#1798C2214yescomplemented
84#1786C2212yescomplemented
85#1736C2212yescomplemented
86#1736C2212yescomplemented
87C23×D4—21no—
88C23×D4—21no—
89C23×D4C2311yescharacteristic
90C23×D4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×D8C2312yes—
93C22×D8C2312yes—
94C22×M16C2311yescharacteristic
95C22×M16C2311yescharacteristic
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98C2×#123—22no—
99C2×#123—22no—
100C2×#121—21no—
101C2×#121—21no—
102C2×#121—24no—
103C2×#121—21no—
104C2×#121—21no—
105C2×#121—24no—
106C2×#121—24no—
107C2×#121D418yes—
108C2×#121C2312yes—
109C2×#121C2312yes—
110C2×#121C2312yes—
111C2×#121C2312yes—
112C2×C4×D4—24no—
113C2×C4×D4C2314yes—
114C2×#117—21no—
115C2×#117—21no—
116C2×#117—24no—
117C2×#117—24no—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C2×#117C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—22no—
131C22×C42—21no—
132#482—216no—
133#482—216no—
134#481—28no—
135#481—28no—
136#469—28no—
137#469—28no—
138#447—28no—
139#447—28no—
140C2×#107—24no—
141C2×#107C2314yes—
142C2×#106—24no—
143C2×#106C2314yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148C2×#102—24no—
149C2×#102—24no—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158#407—24no—
159C2×C22⋊C8—24no—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2312yes—
164#399—24no—
165#397—24no—
166#394—24no—
167#394—24no—
168#390—22no—
169#390—22no—
170#390—24no—
171#390C2314yes—
172#390C2314yes—
173#390C2314yes—
174#389—22no—
175#389—22no—
176#389C2314yes—
177#388—24no—
178#388—24no—
179#388—24no—
180#388—24no—
181#387—24no—
182#386—24no—
183#386—24no—
184#386—22no—
185#386—22no—
186#386—24no—
187#385—24no—
188#385—24no—
189#385—24no—
190#384C2314yes—
191#382C2314yes—
192#381D412yes—
193#380—21no—
194#380—21no—
195#380—22no—
196#380D411yescharacteristic
197#380D411yescharacteristic
198#380D412yes—
199#380C2312yes—
200#380C2312yes—
201#379—24no—
202#379—22no—
203#379D412yes—
204C4×C4⋊C4C2314yes—
205C4×C22⋊C4—22no—
206C4×C22⋊C4C2312yes—
207C4×C22⋊C4C2312yes—
208C2×#95—21no—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#336C2314yes—
216#336C2314yes—
217C25—21no—
218C25—21no—
219C25C2×D411yescharacteristic
220C22×D4—22no—
221C22×D4—28no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—21no—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—42no—
231C22×D4—42no—
232C22×D4—42no—
233C22×D4—42no—
234C22×D4—44no—
235C22×D4—82no—
236C22×D4—82no—
237C22×D4—84no—
238C22×D4C2×D414yes—
239C22×D4C2412yes—
240C22×D4C2412yes—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—24no—
247C23×C4—42no—
248C23×C4—42no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D414yes—
253C23×C4#4114yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C2×D8—24no—
258C2×D8—24no—
259C2×D8—44no—
260C2×D8—44no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—42no—
264C2×M16—42no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—42no—
268C22×C8—42no—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273#123—84no—
274#123—84no—
275#121—28no—
276#121—28no—
277#121—24no—
278#121—24no—
279#121—24no—
280#121—24no—
281#121—24no—
282#121—24no—
283#121—44no—
284#121—44no—
285#121C2×D418yes—
286#121C2×D418yes—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4—88no—
290#117—24no—
291#117—24no—
292#117—24no—
293#117—24no—
294#117—42no—
295#117—42no—
296#117—44no—
297#117—44no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—84no—
342C2×C22⋊C4—82no—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D414yes—
346C2×C22⋊C4C2×D414yes—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—24no—
350C2×C42—44no—
351C2×C42C2414yes—
352#107—28no—
353#106—28no—
354#106—24no—
355#106—24no—
356C4⋊C8—44no—
357C4⋊C8—44no—
358#102—28no—
359#102—28no—
360#102—24no—
361#102—24no—
362#102—24no—
363#102—24no—
364#102—44no—
365#102—44no—
366C22⋊C8—24no—
367C22⋊C8—24no—
368C22⋊C8—24no—
369C22⋊C8—24no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95C2×D412yes—
377#95C2×D414yes—
378#95C2×D412yes—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—24no—
383C24—22no—
384C24—42no—
385C24—42no—
386C24—44no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—82no—
391C24—82no—
392C24—81no—
393C24—81no—
394C24—82no—
395C24#12114yes—
396C24C22×D411yescharacteristic
397C24C22×D411yescharacteristic
398C24C22×D411yescharacteristic
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—48no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4—84no—
421C2×D4—84no—
422C2×D4—82no—
423C2×D4—84no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4—816no—
427C2×D4—816no—
428C2×D4C22×D414yes—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4#12114yes—
463C22×C4#12114yes—
464C22×C4#12114yes—
465C22×C4C22×D414yes—
466C22×C4C22×D414yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2511yescharacteristic
477D8—84no—
478D8—84no—
479M16—82no—
480M16—82no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—44no—
490C2×C8—44no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—82no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4C22×D414yes—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—28no—
514C22⋊C4—28no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C22⋊C4—82no—
530C42—24no—
531C42—24no—
532C42—44no—
533C42—44no—
534C42—84no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—28no—
546C23—24no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—42no—
552C23—42no—
553C23—48no—
554C23—44no—
555C23—42no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—164no—
567C23—164no—
568C23—164no—
569C23—164no—
570C23C2×#12112yes—
571C23C2×#12112yes—
572C23C2×#12112yes—
573C23#57612yes—
574C23#57712yes—
575C23C23×D411yescharacteristic
576D4—24no—
577D4—24no—
578D4—416no—
579D4—42no—
580D4—42no—
581D4—44no—
582D4—44no—
583D4—84no—
584D4—84no—
585D4—88no—
586D4—88no—
587D4—88no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
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—24no—
606C2×C4—24no—
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—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—168no—
628C2×C4—164no—
629C2×C4C2×#12112yes—
630C2×C4C2×#12112yes—
631C2×C4C2×#12112yes—
632C2×C4C23×D411yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4#58512yes—
635C2×C4#58512yes—
636C8—42no—
637C8—42no—
638C8—42no—
639C8—42no—
640C22—22no—
641C22—24no—
642C22—24no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—88no—
658C22—84no—
659C22—84no—
660C22—82no—
661C22—82no—
662C22—84no—
663C22—84no—
664C22—82no—
665C22—168no—
666C22—168no—
667C22#304514yes—
668C22C22×#12111yescharacteristic
669C22#348512yes—
670C22C2×#57611yescharacteristic
671C22C2×#57711yescharacteristic
672C22#358512yes—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—44no—
677C4—44no—
678C4—44no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—84no—
683C4#348511yescharacteristic
684C4#348511yescharacteristic
685C4#349112yes—
686C2—24no—
687C2—42no—
688C2—42no—
689C2—84no—
690C2—162no—
691C2#3544711yescharacteristic
692C2C2×#304511yescharacteristic
693C2#6042311yescharacteristic
694C2#6175212yes—
695C2#6175812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3192C2
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
2. C22×#482C2semidirect
3. #35262C2semidirect
4. #34379C2semidirect
5. #28126C2semidirect
6. #27654C2semidirect
7. #25228C2semidirect
8. #25218C2semidirect
9. #24583C2semidirect
10. #24578C2semidirect
11. #24531C2semidirect
12. #24526C2semidirect
13. #21933C2semidirect
14. #21758C2semidirect
15. #20702C2semidirect
24. C2×#482C22semidirect
29. C22×#107C22semidirect
30. C22×#106C22semidirect
31. C22×#102C22semidirect
32. C22×#102C22semidirect
40. #2500C22semidirect
44. #2361C22semidirect
45. #2345C22semidirect
47. D4×C22⋊C4C22semidirect
50. #2317C22semidirect
53. #2309C22semidirect
56. #2085C22semidirect
57. #2084C22semidirect
58. #2073C22semidirect
59. #2071C22semidirect
60. #2067C22semidirect
61. #2065C22semidirect
62. #2054C22semidirect
63. #2054C22semidirect
64. #1949C22semidirect
65. #1948C22semidirect
66. #1926C22semidirect
67. #1925C22semidirect
70. #1879C22semidirect
71. #1879C22semidirect
72. #1874C22semidirect
73. #1851C22semidirect
74. #1851C22semidirect
75. #1850C22semidirect
76. #1850C22semidirect
77. #1825C22semidirect
78. #1824C22semidirect
79. #1823C22semidirect
80. #1823C22semidirect
81. #1804C22semidirect
82. #1802C22semidirect
83. #1798C22semidirect
84. #1786C22semidirect
85. #1736C22semidirect
86. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31922
C22×#4822
#352622
#343792
#281262
#276542
#252282
#252182
#245832
#245782
#245312
#245262
#219332
#217582
#207022