Finite Group Databasebeta

← #6508866

Subgroups — #6508866

Order 512.

2122 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
1C22×#481C211yesmaximal, characteristic, complemented
2C2×#2977C211yesmaximal, characteristic, complemented
3#37439C211yesmaximal, characteristic
4#35227C211yesmaximal, characteristic, complemented
5#28107C212yesmaximal, complemented
6#28094C212yesmaximal, complemented
7#27663C212yesmaximal, complemented
8#27658C212yesmaximal, complemented
9#25252C214yesmaximal, complemented
10#25238C214yesmaximal, complemented
11#24618C214yesmaximal, complemented
12#24568C214yesmaximal, complemented
13#21941C211yesmaximal, characteristic, complemented
14#21761C211yesmaximal, characteristic, complemented
15C2×#1823C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#481—28no—
20C2×#481C2214yes—
21C2×#481C2212yes—
22C2×#481C2212yes—
23#2977—28no—
24C2×#435C2212yes—
25C2×#435C2212yescomplemented
26C2×C8×D4C2212yes—
27C2×C8×D4C2212yescomplemented
28C2×#433C2212yescomplemented
29C22×#106C2212yes—
30C22×#102C2212yes—
31C2×#408C2212yescomplemented
32C22×C22⋊C8C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34#2849—22no—
35#2813—22no—
36#2656—21no—
37#2639—21no—
38#2626—21no—
39#2624C2212yes—
40#2623—22no—
41#2617—21no—
42#2615C2212yes—
43#2504—21no—
44#2503—21no—
45#2502—21no—
46#2501—21no—
47#2487C2212yes—
48#2487C2214yescomplemented
49#2486C2212yes—
50#2486C2212yes—
51#2416C2214yes—
52#2410—22no—
53#2408C2212yes—
54#2317C2212yescomplemented
55#2311C2212yescomplemented
56#2280C2211yescharacteristic
57C2×#380C2211yescharacteristic
58#2088C2214yescomplemented
59#2088C2214yescomplemented
60#2077C2214yescomplemented
61#2077C2214yescomplemented
62#2067C2214yescomplemented
63#2065C2214yescomplemented
64#2034C2214yescomplemented
65#2033C2214yescomplemented
66#1943C2214yescomplemented
67#1941C2214yescomplemented
68#1936C2214yescomplemented
69#1934C2214yescomplemented
70#1895C2214yescomplemented
71#1893C2214yes—
72#1893C2214yescomplemented
73#1875C2214yes—
74#1854C2214yescomplemented
75#1853C2212yescomplemented
76#1852C2212yescomplemented
77#1824C2211yescharacteristic
78#1824C2211yescharacteristic
79#1823C2212yes—
80#1823C2212yes—
81#1823C2214yescomplemented
82#1809C2214yescomplemented
83#1808C2212yes—
84#1807C2212yes—
85#1807C2214yescomplemented
86#1786C2211yescharacteristic
87#1786C2211yescharacteristic
88C2×#340C2214yescomplemented
89C22×#41C2311yescharacteristic
90C23×D4—21no—
91C23×D4C2311yescharacteristic
92C24×C4—21no—
93C24×C4C2311yescharacteristic
94C22×M16C2312yes—
95C23×C8C2312yes—
96C23×C8C2312yes—
97C23×C8C2312yes—
98#525—24no—
99#525—24no—
100C2×#123—22no—
101C2×#123—22no—
102C2×#123D412yes—
103C2×C22⋊Q8D411yescharacteristic
104C2×C22⋊Q8D411yescharacteristic
105C2×#121—21no—
106C2×#121—21no—
107C2×#121—24no—
108C2×#121—24no—
109C2×#121—24no—
110C2×#121D411yescharacteristic
111C2×#121D411yescharacteristic
112C2×#121C2312yes—
113C2×#121C2312yes—
114C2×#121C2314yes—
115C2×#120—22no—
116C2×#120D412yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C2×#117C2312yes—
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—24no—
122C22×C4⋊C4—24no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133#481—216no—
134#481—28no—
135#481—28no—
136#435—28no—
137#435—28no—
138C8×D4—28no—
139C8×D4—28no—
140#433—28no—
141C2×#107C2314yes—
142C2×#106—28no—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×#102—28no—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153#408—28no—
154C2×C22⋊C8—24no—
155C2×C22⋊C8D418yescomplemented
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2311yescharacteristic
159C2×C22⋊C8C2311yescharacteristic
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2311yescharacteristic
162C2×C22⋊C8C2311yescharacteristic
163C2×C22⋊C8C2312yes—
164C2×#97C2312yes—
165C2×C4×C8C2312yes—
166#400—24no—
167#400—22no—
168#399—22no—
169#399—24no—
170#399—22no—
171#398—22no—
172#398—22no—
173#397—22no—
174#397—22no—
175#397—22no—
176#396—24no—
177#394—22no—
178#394—24no—
179#394—24no—
180#393—22no—
181#392—22no—
182#392—24no—
183#392—22no—
184#391—22no—
185#390—22no—
186#390C2314yes—
187#389C2314yes—
188#389C2314yes—
189#389C2314yes—
190#386—24no—
191#386—24no—
192#386—22no—
193#386—24no—
194#385—24no—
195#385—24no—
196#385—22no—
197#384—22no—
198#382—22no—
199#381—21no—
200#381—22no—
201#381—22no—
202#380—21no—
203#380—21no—
204#380C2314yes—
205#380C2314yes—
206#380C2311yescharacteristic
207#380C2312yes—
208#380C2314yes—
209#380C2311yescharacteristic
210#380C2312yes—
211#380C2312yes—
212#379—21no—
213#379—22no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C2×#95—21no—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225C25—21no—
226C25C2×D411yescharacteristic
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—44no—
230C22×Q8C2×D411yescharacteristic
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—24no—
234C22×D4—22no—
235C22×D4—21no—
236C22×D4—22no—
237C22×D4—44no—
238C22×D4—44no—
239C22×D4—84no—
240C22×D4C2×D412yes—
241C22×D4C2×D412yes—
242C22×D4C2×D412yes—
243C22×D4C2×D411yescharacteristic
244C22×D4C2412yes—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—44no—
251C23×C4—44no—
252C23×C4—82no—
253C23×C4—82no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D414yes—
258C23×C4#4114yes—
259C23×C4C2412yes—
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×M16—24no—
264C2×M16—44no—
265C22×C8—24no—
266C22×C8—24no—
267C22×C8—44no—
268C22×C8—44no—
269C22×C8—44no—
270C22×C8C2×D418yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275#123—28no—
276#123—84no—
277#123—84no—
278C22⋊Q8—24no—
279C22⋊Q8—24no—
280#121—24no—
281#121—24no—
282#121—24no—
283#121—24no—
284#121—24no—
285#121—24no—
286#121—44no—
287#121—44no—
288#121—44no—
289#121—44no—
290#121—88no—
291#120—28no—
292#120—84no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—24no—
297#117—22no—
298#117—22no—
299#117—44no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—82no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C42—22no—
351C2×C42C2412yes—
352#107—44no—
353#106—28no—
354#106—28no—
355#106—44no—
356C4⋊C8—24no—
357C4⋊C8—24no—
358#102—28no—
359#102—28no—
360#102—44no—
361#102—44no—
362C22⋊C8—24no—
363C22⋊C8—22no—
364C22⋊C8—22no—
365C22⋊C8—24no—
366C22⋊C8—24no—
367C22⋊C8C2×D418yes—
368C22⋊C8C2×D414yes—
369C22⋊C8C2×D414yes—
370#97—24no—
371C4×C8—24no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—21no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383C24—22no—
384C24—22no—
385C24—24no—
386C24—22no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—82no—
392C24—82no—
393C24#12114yes—
394C24C22×D412yes—
395C24C22×D411yescharacteristic
396C24C22×D411yescharacteristic
397C24C22×D411yescharacteristic
398#41—84no—
399#41—84no—
400C2×Q8—22no—
401C2×Q8—42no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—28no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—48no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—48no—
419C2×D4—42no—
420C2×D4—84no—
421C2×D4—82no—
422C2×D4—84no—
423C2×D4—84no—
424C2×D4—84no—
425C2×D4—84no—
426C2×D4—816no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—816no—
459C22×C4#12114yes—
460C22×C4#12114yes—
461C22×C4#12114yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2511yescharacteristic
475M16—84no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—48no—
483C2×C8—84no—
484C2×C8—84no—
485C2×C8C22×D414yes—
486C2×C8C22×D414yes—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—82no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—82no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C42—22no—
527C42—22no—
528C42—44no—
529C23—22no—
530C23—22no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—22no—
535C23—28no—
536C23—28no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—44no—
546C23—48no—
547C23—48no—
548C23—42no—
549C23—44no—
550C23—42no—
551C23—44no—
552C23—42no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—164no—
557C23—164no—
558C23C2×#12112yes—
559C23C2×#12112yes—
560C23C2×#12112yes—
561C23#57612yes—
562C23#57714yes—
563C23#57712yes—
564C23C23×D411yescharacteristic
565Q8—82no—
566D4—48no—
567D4—48no—
568D4—44no—
569D4—42no—
570D4—42no—
571D4—82no—
572D4—84no—
573D4—84no—
574D4—88no—
575D4—88no—
576D4—168no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—88no—
614C2×C4—88no—
615C2×C4—84no—
616C2×C4—88no—
617C2×C4—164no—
618C2×C4C2×#12112yes—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4C23×D411yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4#58514yes—
624C8—24no—
625C8—24no—
626C8—42no—
627C8—42no—
628C22—22no—
629C22—22no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—416no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—44no—
645C22—44no—
646C22—42no—
647C22—82no—
648C22—82no—
649C22—84no—
650C22—84no—
651C22—82no—
652C22—168no—
653C22#305714yes—
654C22#306214yes—
655C22C22×#12111yescharacteristic
656C22#348512yes—
657C22C2×#57611yescharacteristic
658C22C2×#57712yes—
659C22C2×#57711yescharacteristic
660C4—22no—
661C4—22no—
662C4—22no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—82no—
667C4—84no—
668C4—82no—
669C4—82no—
670C4—84no—
671C4#348512yes—
672C4#349112yes—
673C2—24no—
674C2—24no—
675C2—42no—
676C2—82no—
677C2—162no—
678C2#3544711yescharacteristic
679C2C2×#305711yescharacteristic
680C2C2×#306211yescharacteristic
681C2#6176512yes—
682C2#6176612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#37439C2
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. C22×#481C2semidirect
2. C2×#2977C2semidirect
4. #35227C2semidirect
5. #28107C2semidirect
6. #28094C2semidirect
7. #27663C2semidirect
8. #27658C2semidirect
9. #25252C2semidirect
10. #25238C2semidirect
11. #24618C2semidirect
12. #24568C2semidirect
13. #21941C2semidirect
14. #21761C2semidirect
15. C2×#1823C2semidirect
25. C2×#435C22semidirect
27. C2×C8×D4C22semidirect
28. C2×#433C22semidirect
31. C2×#408C22semidirect
48. #2487C22semidirect
54. #2317C22semidirect
55. #2311C22semidirect
58. #2088C22semidirect
59. #2088C22semidirect
60. #2077C22semidirect
61. #2077C22semidirect
62. #2067C22semidirect
63. #2065C22semidirect
64. #2034C22semidirect
65. #2033C22semidirect
66. #1943C22semidirect
67. #1941C22semidirect
68. #1936C22semidirect
69. #1934C22semidirect
70. #1895C22semidirect
72. #1893C22semidirect
74. #1854C22semidirect
75. #1853C22semidirect
76. #1852C22semidirect
81. #1823C22semidirect
82. #1809C22semidirect
85. #1807C22semidirect
88. C2×#340C22semidirect
155. C2×C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4812
C2×#29772
#374392
#352272
#281072
#280942
#276632
#276582
#252522
#252382
#246182
#245682
#219412
#217612
C2×#18232