Finite Group Databasebeta

← #6508865

Subgroups — #6508865

Order 512.

1992 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic, complemented
2C2×#2977C211yesmaximal, characteristic
3#37442C211yesmaximal, characteristic
4#35227C211yesmaximal, characteristic
5#28108C212yesmaximal, complemented
6#28095C212yesmaximal, complemented
7#27663C212yesmaximal, complemented
8#27658C212yesmaximal, complemented
9#25254C214yesmaximal, complemented
10#25236C214yesmaximal, complemented
11#24616C214yesmaximal, complemented
12#24566C214yesmaximal, complemented
13#21941C211yesmaximal, characteristic, complemented
14#21762C211yesmaximal, characteristic, complemented
15C2×#1824C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#483—28no—
20C2×#483C2214yes—
21C2×#483C2212yes—
22C2×#483C2212yes—
23#2977—28no—
24C2×#435C2212yes—
25C2×#435C2212yes—
26C2×C8×D4C2212yes—
27C2×C8×D4C2212yes—
28C2×#433C2212yes—
29C22×#106C2212yes—
30C22×#103C2212yes—
31C2×#408C2212yes—
32C22×C22⋊C8C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34#2849—22no—
35#2814—22no—
36#2659—21no—
37#2639—21no—
38#2627—21no—
39#2625C2212yes—
40#2623—22no—
41#2618—21no—
42#2616C2212yes—
43#2504—21no—
44#2503—21no—
45#2502—21no—
46#2501—21no—
47#2487C2212yes—
48#2487C2212yes—
49#2486C2212yes—
50#2486C2214yes—
51#2417C2214yes—
52#2412—22no—
53#2409C2212yes—
54#2317C2212yes—
55#2311C2212yes—
56#2280C2211yescharacteristic
57C2×#380C2211yescharacteristic
58#2088C2214yescomplemented
59#2088C2214yescomplemented
60#2077C2214yescomplemented
61#2077C2214yescomplemented
62#2068C2214yescomplemented
63#2066C2214yescomplemented
64#2036C2214yescomplemented
65#2035C2214yescomplemented
66#1943C2214yescomplemented
67#1941C2214yescomplemented
68#1935C2214yescomplemented
69#1933C2214yescomplemented
70#1894C2214yescomplemented
71#1892C2214yes—
72#1892C2214yescomplemented
73#1873C2214yes—
74#1854C2214yes—
75#1853C2212yes—
76#1852C2212yes—
77#1824C2212yes—
78#1824C2212yes—
79#1824C2214yes—
80#1823C2211yescharacteristic
81#1823C2211yescharacteristic
82#1809C2214yescomplemented
83#1808C2212yes—
84#1807C2212yes—
85#1807C2214yescomplemented
86#1787C2211yescharacteristic
87#1787C2211yescharacteristic
88C2×#340C2214yes—
89C22×#41C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C24×C4—21no—
92C24×C4C2311yescharacteristic
93C22×M16C2312yes—
94C23×C8C2312yes—
95C23×C8C2312yes—
96C23×C8C2312yes—
97#525—24no—
98#525—24no—
99C2×#123—24no—
100C2×#123—22no—
101C2×#123D412yes—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8D411yescharacteristic
107C2×C22⋊Q8D411yescharacteristic
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2314yes—
111C2×#121D411yescharacteristic
112C2×#121D411yescharacteristic
113C2×#120D412yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×#117C2312yes—
117C2×#117C2311yescharacteristic
118C22×C4⋊C4—24no—
119C22×C4⋊C4—24no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131#483—216no—
132#483—28no—
133#483—28no—
134#435—28no—
135#435—28no—
136C8×D4—28no—
137C8×D4—28no—
138#433—28no—
139C2×#107C2314yes—
140C2×#106—28no—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#103—28no—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151#408—28no—
152C2×C22⋊C8—24no—
153C2×C22⋊C8D418yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2311yescharacteristic
157C2×C22⋊C8C2311yescharacteristic
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2311yescharacteristic
160C2×C22⋊C8C2311yescharacteristic
161C2×C22⋊C8C2312yes—
162C2×#97C2312yes—
163C2×C4×C8C2312yes—
164#400—22no—
165#400—22no—
166#400—22no—
167#399—22no—
168#399—22no—
169#398—22no—
170#398—24no—
171#398—24no—
172#398—22no—
173#397—24no—
174#397—24no—
175#397—22no—
176#396—24no—
177#395—22no—
178#395—22no—
179#394—22no—
180#394—24no—
181#393—22no—
182#391—22no—
183#391C2314yes—
184#390—22no—
185#389C2314yes—
186#389C2314yes—
187#389C2314yes—
188#386—24no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—24no—
193#385—24no—
194#385—22no—
195#385—22no—
196#385—22no—
197#384—22no—
198#382—22no—
199#381—21no—
200#381—24no—
201#381—22no—
202#380—21no—
203#380—21no—
204#380—24no—
205#380C2314yes—
206#380C2314yes—
207#380C2311yescharacteristic
208#380C2312yes—
209#380C2314yes—
210#380C2311yescharacteristic
211#380C2312yes—
212#380C2312yes—
213#379—21no—
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—
225C25C2×D411yescharacteristic
226C2×#41—22no—
227C2×#41—22no—
228C2×#41—44no—
229C22×Q8—21no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—44no—
233C22×Q8C2×D411yescharacteristic
234C22×Q8C2412yes—
235C22×D4C2×D412yes—
236C22×D4C2×D412yes—
237C22×D4C2×D412yes—
238C22×D4C2×D411yescharacteristic
239C23×C4—21no—
240C23×C4—22no—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—84no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D414yes—
252C23×C4#4114yes—
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C2×M16—24no—
258C2×M16—44no—
259C22×C8—24no—
260C22×C8—24no—
261C22×C8—44no—
262C22×C8—44no—
263C22×C8—44no—
264C22×C8C2×D418yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269#123—28no—
270#123—84no—
271#123—88no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282#121—24no—
283#121—24no—
284#120—28no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—24no—
289#117—22no—
290#117—22no—
291#117—44no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
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—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—82no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—22no—
345C2×C42C2412yes—
346#107—44no—
347#106—28no—
348#106—28no—
349#106—44no—
350C4⋊C8—24no—
351C4⋊C8—24no—
352#103—28no—
353#103—28no—
354#103—44no—
355#103—44no—
356C22⋊C8—24no—
357C22⋊C8—22no—
358C22⋊C8—22no—
359C22⋊C8—24no—
360C22⋊C8—24no—
361C22⋊C8C2×D418yes—
362C22⋊C8C2×D414yes—
363C22⋊C8C2×D414yes—
364#97—24no—
365C4×C8—24no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—21no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379C24—22no—
380C24—24no—
381C24—22no—
382C24—42no—
383C24#12114yes—
384C24C22×D412yes—
385C24C22×D411yescharacteristic
386C24C22×D411yescharacteristic
387C24C22×D411yescharacteristic
388#41—84no—
389#41—84no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—42no—
396C2×Q8—48no—
397C2×Q8—42no—
398C2×Q8—82no—
399C2×Q8—84no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—28no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—42no—
408C2×D4—44no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—48no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—84no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4—816no—
443C22×C4#12114yes—
444C22×C4#12114yes—
445C22×C4#12114yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2511yescharacteristic
459M16—84no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—48no—
467C2×C8—84no—
468C2×C8—84no—
469C2×C8C22×D414yes—
470C2×C8C22×D414yes—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—82no—
490C4⋊C4—84no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—42no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—42no—
506C22⋊C4—82no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C42—22no—
512C42—22no—
513C42—44no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—28no—
521C23—28no—
522C23—21no—
523C23—22no—
524C23—21no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—22no—
529C23—22no—
530C23—44no—
531C23—48no—
532C23—48no—
533C23C2×#12112yes—
534C23C2×#12112yes—
535C23C2×#12112yes—
536C23#57714yes—
537C23#57712yes—
538C23#57812yes—
539C23C23×D411yescharacteristic
540Q8—44no—
541Q8—42no—
542Q8—42no—
543Q8—82no—
544Q8—84no—
545D4—48no—
546D4—48no—
547D4—82no—
548D4—84no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—88no—
589C2×C4—88no—
590C2×C4—84no—
591C2×C4—88no—
592C2×C4—84no—
593C2×C4—164no—
594C2×C4C2×#12112yes—
595C2×C4C2×#12112yes—
596C2×C4C2×#12112yes—
597C2×C4C23×D411yescharacteristic
598C2×C4C22×#4111yescharacteristic
599C2×C4#58514yes—
600C8—24no—
601C8—24no—
602C8—42no—
603C8—42no—
604C22—22no—
605C22—22no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—416no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22#305714yes—
624C22#306314yes—
625C22C22×#12111yescharacteristic
626C22#348512yes—
627C22C2×#57712yes—
628C22C2×#57711yescharacteristic
629C22C2×#57811yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—84no—
637C4—82no—
638C4—82no—
639C4—84no—
640C4—82no—
641C4—82no—
642C4#348512yes—
643C4#349112yes—
644C2—24no—
645C2—24no—
646C2—42no—
647C2—82no—
648C2#3544711yescharacteristic
649C2C2×#305711yescharacteristic
650C2C2×#306311yescharacteristic
651C2#6176512yes—
652C2#6176712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
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×#483C2semidirect
5. #28108C2semidirect
6. #28095C2semidirect
7. #27663C2semidirect
8. #27658C2semidirect
9. #25254C2semidirect
10. #25236C2semidirect
11. #24616C2semidirect
12. #24566C2semidirect
13. #21941C2semidirect
14. #21762C2semidirect
15. C2×#1824C2semidirect
58. #2088C22semidirect
59. #2088C22semidirect
60. #2077C22semidirect
61. #2077C22semidirect
62. #2068C22semidirect
63. #2066C22semidirect
64. #2036C22semidirect
65. #2035C22semidirect
66. #1943C22semidirect
67. #1941C22semidirect
68. #1935C22semidirect
69. #1933C22semidirect
70. #1894C22semidirect
72. #1892C22semidirect
82. #1809C22semidirect
85. #1807C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
C2×#29772
#374422
#352272
#281082
#280952
#276632
#276582
#252542
#252362
#246162
#245662
#219412
#217622
C2×#18242