Finite Group Databasebeta

← #192383

Subgroups — #192383

Order 512.

1568 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2325C211yesmaximal, characteristic
2#21940C211yesmaximal, characteristic, complemented
3#21758C211yesmaximal, characteristic
4C2×#1519C211yesmaximal, characteristic
5#14275C211yesmaximal, characteristic, complemented
6#8869C214yesmaximal, complemented
7#8866C214yesmaximal, complemented
8#8023C211yesmaximal, characteristic, complemented
9C2×#1280C211yesmaximal, characteristic, complemented
10C22×#121C2211yescharacteristic
11C22×C4×D4—21no—
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2325—22no—
15#2325—22no—
16#2325—22no—
17#2325C2214yes—
18C2×#388—21no—
19C2×#387—21no—
20C2×#386—21no—
21C2×#386—21no—
22C2×#385—21no—
23C2×#385—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C22×#95C2211yescharacteristic
28#1893—22no—
29#1874C2212yes—
30#1874C2212yes—
31#1852—21no—
32#1852—21no—
33#1852C2212yescomplemented
34#1852C2212yescomplemented
35#1823—21no—
36#1808—22no—
37#1786—21no—
38#1519C418yescomplemented
39C2×#327C2214yes—
40#1444C2212yescomplemented
41#1442C2212yescomplemented
42#1441—22no—
43C22⋊#95—21no—
44#1294C2214yescomplemented
45#1280C2214yescomplemented
46C23×D4—21no—
47C23×D4D411yescharacteristic
48C24×C4—21no—
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C24×C4C2311yescharacteristic
52C23×C8—21no—
53C23×C8—21no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121C2×C414yes—
59C2×#121D412yes—
60C2×#121Q814yes—
61C2×#121C2312yes—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—44no—
65C2×C4×D4—44no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C42—21no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8C2×C414yescomplemented
92C2×C22⋊C8C2312yes—
93C2×C22⋊C8C2312yes—
94#388—24no—
95#388—24no—
96#387—24no—
97#387—24no—
98#386—24no—
99#386—24no—
100#386—24no—
101#386—24no—
102#385—24no—
103#385—24no—
104#385—44no—
105#380—21no—
106#380—21no—
107#380—22no—
108#380—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380C2×C414yescomplemented
113#380D412yes—
114#380C2312yes—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95D412yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#327C2×C4116yescomplemented
134C25—21no—
135C25C2×D411yescharacteristic
136C22×D4—24no—
137C22×D4—22no—
138C22×D4—24no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—81no—
147C22×D4—81no—
148C22×D4C4⋊C414yescomplemented
149C22×D4C2×D412yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—44no—
165C23×C4—81no—
166C23×C4—81no—
167C23×C4—81no—
168C23×C4—81no—
169C23×C4—81no—
170C23×C4—81no—
171C23×C4—82no—
172C23×C4—82no—
173C23×C4D814yes—
174C23×C4C22×C412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×Q812yes—
183C23×C4#4112yes—
184C23×C4#4112yes—
185C23×C4C2411yescharacteristic
186C22×C8—22no—
187C22×C8—24no—
188C22×C8—22no—
189C22×C8—42no—
190C22×C8—81no—
191C22×C8—81no—
192C22×C8—81no—
193C22×C8—81no—
194#121—24no—
195#121—24no—
196#121—24no—
197#121—24no—
198#121—24no—
199#121—22no—
200#121—22no—
201#121C4⋊C418yescomplemented
202#121C22×C414yes—
203#121C2×Q814yes—
204C4×D4—48no—
205C4×D4—416no—
206C4×D4—48no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—81no—
220C2×C4⋊C4—81no—
221C2×C4⋊C4—81no—
222C2×C4⋊C4—81no—
223C2×C4⋊C4—81no—
224C2×C4⋊C4—81no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4C4⋊C414yescomplemented
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—82no—
251C2×C22⋊C4—82no—
252C2×C42—24no—
253C2×C42—44no—
254#107—82no—
255#107—82no—
256#102—82no—
257#102—82no—
258C22⋊C8—24no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—42no—
263C22⋊C8C22×C414yescomplemented
264#95—24no—
265#95—24no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—41no—
274#95—41no—
275#95—44no—
276#95—82no—
277#95—82no—
278#95C22×C414yescomplemented
279C24—22no—
280C24—24no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24C4×D412yes—
292C24#12011yescharacteristic
293C24#12011yescharacteristic
294C24C22⋊Q812yes—
295C24C2×D812yes—
296C24#13512yes—
297C24AGL(1,ℤ/8)12yes—
298C24#13712yes—
299C24C22×D411yescharacteristic
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—28no—
303C2×D4—22no—
304C2×D4—24no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—48no—
312C2×D4—48no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—48no—
316C2×D4—48no—
317C2×D4—81no—
318C2×D4—81no—
319C2×D4—84no—
320C2×D4—84no—
321C2×D4C2×C4⋊C414yes—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—48no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—84no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4—816no—
395C22×C4—84no—
396C22×C4C2×C4⋊C412yes—
397C22×C4C2×C4⋊C412yes—
398C22×C4C4×D412yes—
399C22×C4C4×D412yes—
400C22×C4C4×D412yes—
401C22×C4#12011yescharacteristic
402C22×C4#12011yescharacteristic
403C22×C4C22⋊Q812yes—
404C22×C4C22⋊Q812yes—
405C22×C4C22⋊Q812yes—
406C22×C4#12312yes—
407C22×C4#12312yes—
408C22×C4C2×D814yes—
409C22×C4C23×C411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×Q811yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C2×C8—24no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—84no—
422C2×C8—84no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—48no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C4⋊C4—84no—
435C4⋊C4—88no—
436C4⋊C4C2×C4⋊C414yes—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—28no—
440C22⋊C4—48no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—48no—
444C22⋊C4—44no—
445C22⋊C4—84no—
446C22⋊C4—84no—
447C22⋊C4—82no—
448C22⋊C4—82no—
449C42—48no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—24no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—48no—
476C23—48no—
477C23—48no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—84no—
488C23#38512yes—
489C23#38512yes—
490C23C4×D814yes—
491C23#44014yes—
492C23#44712yes—
493C23#44712yes—
494C23#45212yes—
495C23#45212yes—
496C23#47414yes—
497C23#47814yes—
498C23C2×C4×D411yescharacteristic
499C23C2×#12011yescharacteristic
500C23C2×C22⋊Q811yescharacteristic
501C23C22×D811yescharacteristic
502C23C2×#13511yescharacteristic
503C23C2×AGL(1,ℤ/8)11yescharacteristic
504C23C2×#13711yescharacteristic
505D4—44no—
506D4—44no—
507D4—416no—
508D4—44no—
509D4—44no—
510D4—44no—
511D4—82no—
512D4—84no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—82no—
558C2×C4—84no—
559C2×C4—816no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4#38512yes—
580C2×C4#38512yes—
581C2×C4C22×C4⋊C411yescharacteristic
582C2×C4C2×C4×D411yescharacteristic
583C2×C4C2×C22⋊Q811yescharacteristic
584C2×C4C2×#12311yescharacteristic
585C8—44no—
586C8—82no—
587C22—21no—
588C22—22no—
589C22—21no—
590C22—22no—
591C22—21no—
592C22—22no—
593C22—21no—
594C22—22no—
595C22—21no—
596C22—22no—
597C22—21no—
598C22—22no—
599C22—21no—
600C22—21no—
601C22—22no—
602C22—22no—
603C22—21no—
604C22—22no—
605C22—21no—
606C22—22no—
607C22—21no—
608C22—22no—
609C22—21no—
610C22—22no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—816no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22#186214yes—
624C22#186214yes—
625C22C2×#38511yescharacteristic
626C22C2×C4×D811yescharacteristic
627C22C2×#44011yescharacteristic
628C22C2×#44711yescharacteristic
629C22C2×#45211yescharacteristic
630C22#301911yescharacteristic
631C22#302011yescharacteristic
632C22#302611yescharacteristic
633C22#302611yescharacteristic
634C22C2×#47411yescharacteristic
635C22C2×#47811yescharacteristic
636C4—21no—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—22no—
641C4—22no—
642C4—44no—
643C4—44no—
644C4—84no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—161no—
649C4—161no—
650C2—21no—
651C2—24no—
652C2—21no—
653C2—24no—
654C2—21no—
655C2—21no—
656C2—84no—
657C2#1828411yescharacteristic
658C2#1828911yescharacteristic
659C2#1829211yescharacteristic
660C2#1829711yescharacteristic
661C2#2086411yescharacteristic
662C2C2×#186211yescharacteristic
663C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21940C2semidirect
5. #14275C2semidirect
6. #8869C2semidirect
7. #8866C2semidirect
8. #8023C2semidirect
9. C2×#1280C2semidirect
33. #1852C22semidirect
34. #1852C22semidirect
38. #1519C4semidirect
40. #1444C22semidirect
41. #1442C22semidirect
44. #1294C22semidirect
45. #1280C22semidirect
91. C2×C22⋊C8C2×C4semidirect
112. #380C2×C4semidirect
133. #327C2×C4semidirect
148. C22×D4C4⋊C4semidirect
201. #121C4⋊C4semidirect
227. C2×C4⋊C4C4⋊C4semidirect
263. C22⋊C8C22×C4semidirect
278. #95C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23252
#219402
#217582
C2×#15192
#142752
#88692
#88662
#80232
C2×#12802