Finite Group Databasebeta

← #6508807

Subgroups — #6508807

Order 512.

2044 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic, complemented
2C2×#2972C211yesmaximal, characteristic
3#35084C211yesmaximal, characteristic
4#34809C211yesmaximal, characteristic
5#28098C212yesmaximal, complemented
6#28095C212yesmaximal, complemented
7#27661C212yesmaximal, complemented
8#27658C212yesmaximal, complemented
9#25239C214yesmaximal, complemented
10#25233C214yesmaximal, complemented
11#24604C214yesmaximal, complemented
12#24554C214yesmaximal, complemented
13#21939C211yesmaximal, characteristic, complemented
14#21762C211yesmaximal, characteristic, complemented
15#20699C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#485—28no—
20C2×#485C2214yes—
21C2×#485C2214yescomplemented
22#2972—28no—
23C2×C8×D4C2212yes—
24C2×C8×D4C2212yes—
25C2×C8×D4C2212yes—
26C2×C8×D4C2212yes—
27C2×#431C2212yes—
28C22×#107C2211yescharacteristic, complemented
29C22×#106C2211yescharacteristic, complemented
30C22×#103C2211yescharacteristic, complemented
31C22×#103C2211yescharacteristic, complemented
32C22×C22⋊C8C2211yescharacteristic
33C22×C22⋊C8C2212yes—
34C22×C22⋊C8C2211yescharacteristic
35#2855C2212yes—
36#2848C2212yes—
37#2638—22no—
38#2583—21no—
39#2561—21no—
40#2487C2212yes—
41#2486C2212yes—
42#2486C2212yes—
43#2485C2214yes—
44#2484—21no—
45#2483—21no—
46#2483—21no—
47#2482—21no—
48#2415C2214yes—
49#2409C2212yes—
50#2405—21no—
51#2405—21no—
52#2395—22no—
53#2322—22no—
54C4×#123—22no—
55#2296C2212yes—
56#2280C2211yescharacteristic
57#2280C2211yescharacteristic
58#2280C2212yes—
59#2088C2214yescomplemented
60#2086C2214yescomplemented
61#2077C2214yescomplemented
62#2075C2214yescomplemented
63#2068C2214yescomplemented
64#2066C2214yescomplemented
65#2036C2214yescomplemented
66#2035C2214yescomplemented
67#1942C2214yescomplemented
68#1941C2214yescomplemented
69#1933C2214yescomplemented
70#1933C2214yescomplemented
71#1892C2212yescomplemented
72#1892C2212yescomplemented
73#1892C2214yescomplemented
74#1892C2214yescomplemented
75#1873C2214yes—
76#1854C2214yes—
77#1854C2214yescomplemented
78#1825C2214yes—
79#1824C2212yescomplemented
80#1823C2212yescomplemented
81#1823C2212yescomplemented
82#1808C2214yescomplemented
83#1808C2212yescomplemented
84#1807C2214yescomplemented
85#1807C2212yescomplemented
86#1787C2212yescomplemented
87#1735C2212yescomplemented
88#1735C2212yescomplemented
89C23×Q8C2311yescharacteristic
90C23×D4D411yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C23×C8C2312yes—
94C23×C8C2312yes—
95C23×C8C2311yescharacteristic
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98C23×C8C2311yescharacteristic
99C2×#123—22no—
100C2×#123—22no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2314yes—
108C2×C4×D4—24no—
109C2×C4×D4—24no—
110C2×C4×D4D412yes—
111C2×C4×D4D412yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×#117—21no—
115C2×#117—21no—
116C2×#117—24no—
117C2×#117—24no—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C2×#117C2312yes—
121C2×#117C2312yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4D411yescharacteristic
130C22×C22⋊C4D411yescharacteristic
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133#485—216no—
134#485—216no—
135C8×D4—28no—
136C8×D4—28no—
137C8×D4—28no—
138C8×D4—28no—
139#431—28no—
140C2×#107—24no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106—24no—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×#103—24no—
149C2×#103—24no—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×C22⋊C8—24no—
155C2×C22⋊C8—28no—
156C2×C22⋊C8D418yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×C4×C8C2312yes—
164C2×C4×C8C2312yes—
165#400—22no—
166#399—22no—
167#398—22no—
168#398—22no—
169#397—22no—
170#397—24no—
171#397—22no—
172#397—24no—
173#395—22no—
174#394—24no—
175#394—24no—
176#393—22no—
177#391C2314yes—
178#389—24no—
179#389C2314yes—
180#389C2314yes—
181#389C2314yes—
182#388—24no—
183#388—24no—
184#387—24no—
185#387—24no—
186#386—22no—
187#386—22no—
188#385—24no—
189#385—22no—
190#385—22no—
191#385—22no—
192#385—22no—
193#385—22no—
194#385—22no—
195#381—22no—
196#381—21no—
197#381—21no—
198#381—22no—
199#381—22no—
200#381—22no—
201#380C2314yes—
202#380C2312yes—
203#380C2312yes—
204#380C2314yes—
205#380C2314yes—
206#380C2312yes—
207#380C2312yes—
208#379—22no—
209#379—22no—
210#379—21no—
211#379—21no—
212C4×C4⋊C4—24no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—24no—
217C4×C22⋊C4—21no—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226C25C2×D411yescharacteristic
227C25C2×D411yescharacteristic
228C22×Q8—21no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—42no—
232C22×Q8—42no—
233C22×Q8C2412yes—
234C22×D4—24no—
235C22×D4—28no—
236C22×D4C2×D412yes—
237C22×D4C2×D412yes—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4—42no—
247C23×C4—42no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D414yes—
256C23×C4#4114yes—
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—24no—
264C22×C8—24no—
265C22×C8—42no—
266C22×C8—42no—
267C22×C8—44no—
268C22×C8—44no—
269C22×C8—42no—
270C22×C8—42no—
271C22×C8C2×D414yes—
272C22×C8C2×D414yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277#123—48no—
278#123—48no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8—44no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286C22⋊Q8—44no—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—24no—
293#117—24no—
294#117—24no—
295#117—24no—
296#117—24no—
297#117—42no—
298#117—42no—
299#117—44no—
300#117—44no—
301#117—42no—
302#117—42no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2414yes—
326C2×C4⋊C4C2414yes—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—44no—
356C2×C42C2×D412yes—
357C2×C42C2412yes—
358#107—28no—
359#107—44no—
360#106—28no—
361#106—44no—
362C4⋊C8—24no—
363C4⋊C8—24no—
364#103—28no—
365#103—28no—
366#103—44no—
367#103—44no—
368C22⋊C8—24no—
369C22⋊C8—24no—
370C22⋊C8—24no—
371C22⋊C8—24no—
372C22⋊C8C2×D418yes—
373C22⋊C8C2×D418yes—
374C4×C8—24no—
375C4×C8—24no—
376#95—24no—
377#95—22no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388C24—22no—
389C24—24no—
390C24—24no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24#12114yes—
396C24C22×D411yescharacteristic
397C24C22×D411yescharacteristic
398C24C22×D412yes—
399C24C22×D411yescharacteristic
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8—82no—
408C2×Q8—84no—
409C2×D4—232no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4#13518yes—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—84no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4#12114yes—
458C22×C4#12114yes—
459C22×C4#12114yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2511yescharacteristic
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—44no—
480C2×C8—44no—
481C2×C8—84no—
482C2×C8—84no—
483C2×C8—82no—
484C2×C8—82no—
485C2×C8C22×D414yes—
486C2×C8C22×D414yes—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—82no—
523C22⋊C4—84no—
524C22⋊C4—82no—
525C22⋊C4—84no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—44no—
530C42—44no—
531C42—44no—
532C42—82no—
533C23—22no—
534C23—22no—
535C23—28no—
536C23—22no—
537C23—24no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—28no—
547C23—44no—
548C23—44no—
549C23—82no—
550C23—88no—
551C23—88no—
552C23—82no—
553C23C2×#12112yes—
554C23C2×#12112yes—
555C23C2×#12112yes—
556C23C2×#13514yes—
557C23C2×#13514yes—
558C23C23×D411yescharacteristic
559Q8—42no—
560Q8—42no—
561Q8—44no—
562Q8—84no—
563D4—216no—
564D4—216no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—84no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4C2×#12112yes—
612C2×C4C2×#12112yes—
613C2×C4C2×#12112yes—
614C2×C4C2×#13514yes—
615C2×C4C23×D411yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4#58512yes—
618C2×C4#58512yes—
619C8—24no—
620C8—24no—
621C8—42no—
622C8—42no—
623C22—28no—
624C22—22no—
625C22—22no—
626C22—28no—
627C22—22no—
628C22—22no—
629C22—24no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—42no—
638C22—42no—
639C22—88no—
640C22—88no—
641C22#305014yes—
642C22C22×#12111yescharacteristic
643C22#348512yes—
644C22C22×#13512yes—
645C22#358612yes—
646C4—24no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—42no—
651C4—42no—
652C4—84no—
653C4—84no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4#348511yescharacteristic
659C4#348511yescharacteristic
660C4#349112yes—
661C2—24no—
662C2—44no—
663C2—42no—
664C2—42no—
665C2#3544711yescharacteristic
666C2C2×#305011yescharacteristic
667C2#6045511yescharacteristic
668C2#6170314yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#485C2semidirect
5. #28098C2semidirect
6. #28095C2semidirect
7. #27661C2semidirect
8. #27658C2semidirect
9. #25239C2semidirect
10. #25233C2semidirect
11. #24604C2semidirect
12. #24554C2semidirect
13. #21939C2semidirect
14. #21762C2semidirect
15. #20699C2semidirect
21. C2×#485C22semidirect
28. C22×#107C22semidirect
29. C22×#106C22semidirect
30. C22×#103C22semidirect
31. C22×#103C22semidirect
59. #2088C22semidirect
60. #2086C22semidirect
61. #2077C22semidirect
62. #2075C22semidirect
63. #2068C22semidirect
64. #2066C22semidirect
65. #2036C22semidirect
66. #2035C22semidirect
67. #1942C22semidirect
68. #1941C22semidirect
69. #1933C22semidirect
70. #1933C22semidirect
71. #1892C22semidirect
72. #1892C22semidirect
73. #1892C22semidirect
74. #1892C22semidirect
77. #1854C22semidirect
79. #1824C22semidirect
80. #1823C22semidirect
81. #1823C22semidirect
82. #1808C22semidirect
83. #1808C22semidirect
84. #1807C22semidirect
85. #1807C22semidirect
86. #1787C22semidirect
87. #1735C22semidirect
88. #1735C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4852
C2×#29722
#350842
#348092
#280982
#280952
#276612
#276582
#252392
#252332
#246042
#245542
#219392
#217622
#206992