Finite Group Databasebeta

← #388073

Subgroups — #388073

Order 512.

1171 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21190C211yesmaximal, characteristic
2#20898C211yesmaximal, characteristic
3C2×#1870C211yesmaximal, characteristic
4#11269C212yesmaximal, complemented
5#11097C212yesmaximal, complemented
6#11097C212yesmaximal, complemented
7#10911C212yesmaximal, complemented
8#10734C212yesmaximal, complemented
9#10724C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C23×M16C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C22×#108C2212yes—
14C2×#423C2211yescharacteristic
15C2×#407C2212yes—
16C2×#407C2212yes—
17C2×#407C2212yes—
18C2×#407C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C2×C4×M16C2211yescharacteristic
22#1955—21no—
23#1955—21no—
24#1954—21no—
25#1954—21no—
26#1870—21no—
27#1870—21no—
28#1870C412yes—
29#1870C412yes—
30#1870C412yes—
31#1870C412yes—
32#1869—21no—
33#1869—21no—
34C2×#341C2212yes—
35C2×#341C2212yes—
36C2×#341C2212yes—
37C2×#341C2212yes—
38C2×#341C2212yes—
39C2×#341C2212yes—
40C2×#341C2212yes—
41#1728C2212yes—
42C2×#336C2212yes—
43C23×D4Q811yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—21no—
47C22×M16—24no—
48C22×M16—41no—
49C22×M16—41no—
50C22×M16—42no—
51C22×M16—42no—
52C22×M16—42no—
53C22×M16—41no—
54C22×M16—41no—
55C22×M16C2×C412yes—
56C22×M16C2×C412yes—
57C22×M16C2312yes—
58C22×M16C2312yes—
59C22×M16C2312yes—
60C22×M16C2312yes—
61C23×C8—21no—
62C23×C8C2312yes—
63C23×C8C2312yes—
64C2×C4×D4—41no—
65C2×C4×D4—41no—
66C2×C4×D4—41no—
67C2×C4×D4—41no—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C22×C4⋊C4Q811yescharacteristic
77C22×C22⋊C4Q811yescharacteristic
78C22×C22⋊C4Q811yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#108—24no—
81C2×#108—42no—
82C2×#108—42no—
83#423—41no—
84#423—41no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87#407—24no—
88#407—24no—
89#407—24no—
90#407—42no—
91#407—42no—
92#407—42no—
93#407—41no—
94#407—41no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—42no—
104C2×C22⋊C8—42no—
105C2×C22⋊C8—41no—
106C2×C22⋊C8—41no—
107C2×C22⋊C8C2×C414yes—
108C2×C22⋊C8C2×C414yes—
109C2×C22⋊C8C2×C412yes—
110C2×C22⋊C8C2×C412yes—
111C4×M16—41no—
112C4×M16—41no—
113C2×#97—21no—
114C2×C4×C8—21no—
115#341—24no—
116#341—42no—
117#341—42no—
118#341—42no—
119#341—42no—
120#341—42no—
121#341—42no—
122#336—22no—
123#336—22no—
124#336C2×C414yes—
125#336C2×C414yes—
126C25C2×Q811yescharacteristic
127C25C2×Q811yescharacteristic
128C22×D4—42no—
129C22×D4—41no—
130C22×D4—41no—
131C22×D4—42no—
132C22×D4C4⋊C411yescharacteristic
133C22×D4C4⋊C411yescharacteristic
134C22×D4C4⋊C411yescharacteristic
135C22×D4C4⋊C411yescharacteristic
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×Q811yescharacteristic
161C23×C4C2×Q811yescharacteristic
162C23×C4C2×Q811yescharacteristic
163C23×C4C2×Q811yescharacteristic
164C23×C4C2411yescharacteristic
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—44no—
188C2×M16—44no—
189C2×M16—42no—
190C2×M16—42no—
191C2×M16—44no—
192C2×M16—44no—
193C2×M16—48no—
194C2×M16—42no—
195C2×M16—44no—
196C2×M16—48no—
197C2×M16—44no—
198C2×M16—44no—
199C2×M16#4114yes—
200C2×M16#4114yes—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—44no—
218C22×C8—42no—
219C22×C8C22×C414yes—
220C22×C8C22×C414yes—
221C4×D4—41no—
222C4×D4—41no—
223C4×D4—41no—
224C4×D4—41no—
225C4×D4—41no—
226C4×D4—41no—
227C4×D4—41no—
228C4×D4—41no—
229C4×D4—41no—
230C4×D4—41no—
231C4×D4—41no—
232C4×D4—41no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4C4⋊C411yescharacteristic
240C2×C4⋊C4C4⋊C411yescharacteristic
241C2×C4⋊C4C4⋊C411yescharacteristic
242C2×C4⋊C4C4⋊C411yescharacteristic
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4C4⋊C411yescharacteristic
248C2×C22⋊C4C4⋊C411yescharacteristic
249C2×C22⋊C4C4⋊C411yescharacteristic
250C2×C22⋊C4C4⋊C411yescharacteristic
251C2×C22⋊C4C4⋊C411yescharacteristic
252C2×C22⋊C4C4⋊C411yescharacteristic
253C2×C22⋊C4C4⋊C411yescharacteristic
254C2×C22⋊C4C4⋊C411yescharacteristic
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263#108—48no—
264C4⋊C8—42no—
265C4⋊C8—42no—
266C22⋊C8—42no—
267C22⋊C8—42no—
268C22⋊C8—44no—
269C22⋊C8—44no—
270C22⋊C8—44no—
271C22⋊C8—44no—
272C22⋊C8—44no—
273C22⋊C8—44no—
274C22⋊C8—44no—
275C22⋊C8—44no—
276C22⋊C8—42no—
277C22⋊C8—42no—
278#97—42no—
279C4×C8—42no—
280C24—21no—
281C24—21no—
282C24—22no—
283C24—21no—
284C24—21no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24—42no—
293C24—42no—
294C24C2×C4⋊C411yescharacteristic
295C24C2×C4⋊C411yescharacteristic
296C24C2×C4⋊C411yescharacteristic
297C24C2×C4⋊C411yescharacteristic
298C24C22×Q811yescharacteristic
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—48no—
312C2×D4—48no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—41no—
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—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4C2×C4⋊C411yescharacteristic
381C22×C4C2×C4⋊C411yescharacteristic
382C22×C4C2×C4⋊C411yescharacteristic
383C22×C4C2×C4⋊C411yescharacteristic
384C22×C4C2×C4⋊C411yescharacteristic
385C22×C4C2×C4⋊C411yescharacteristic
386C22×C4C2×C4⋊C411yescharacteristic
387C22×C4C2×C4⋊C411yescharacteristic
388C22×C4C23×C411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C2×#4112yes—
391C22×C4C2×#4112yes—
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C4H(5,𝔽2)11yescharacteristic
394C22×C4H(5,𝔽2)11yescharacteristic
395C22×C4H(5,𝔽2)11yescharacteristic
396C22×C4H(5,𝔽2)12yes—
397C22×C4H(5,𝔽2)12yes—
398M16—24no—
399M16—24no—
400M16—24no—
401M16—24no—
402M16—48no—
403M16—48no—
404M16—48no—
405M16—42no—
406M16—42no—
407M16—42no—
408M16—42no—
409M16—42no—
410M16—42no—
411M16—42no—
412M16—42no—
413M16—42no—
414M16—42no—
415M16—42no—
416M16—42no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—48no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C23—21no—
465C23—21no—
466C23—22no—
467C23—21no—
468C23—22no—
469C23—21no—
470C23—22no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—81no—
501C23—81no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—82no—
508C23#43012yes—
509C23C22×C4⋊C411yescharacteristic
510C23#54311yescharacteristic
511C23#54311yescharacteristic
512C23#55412yes—
513C23#55412yes—
514D4—44no—
515D4—44no—
516D4—44no—
517D4—44no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4#43012yes—
570C2×C4C4×#4112yes—
571C2×C4#51811yescharacteristic
572C2×C4#51812yes—
573C2×C4#51811yescharacteristic
574C2×C4#53411yescharacteristic
575C2×C4#53411yescharacteristic
576C2×C4#53812yes—
577C2×C4#53812yes—
578C2×C4#54012yes—
579C2×C4#54012yes—
580C2×C4#54612yes—
581C2×C4#54612yes—
582C2×C4#55111yescharacteristic
583C2×C4#55312yes—
584C2×C4#55911yescharacteristic
585C2×C4#56012yes—
586C2×C4#56011yescharacteristic
587C2×C4H(3,𝔽4)11yescharacteristic
588C8—44no—
589C8—44no—
590C8—44no—
591C8—44no—
592C8—82no—
593C8—82no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—24no—
604C22—41no—
605C22—41no—
606C22—42no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—82no—
623C22—84no—
624C22—82no—
625C22—82no—
626C22—84no—
627C22—82no—
628C22#231511yescharacteristic
629C22#234912yes—
630C22#274811yescharacteristic
631C22#278012yes—
632C22#280411yescharacteristic
633C22C2×#43011yescharacteristic
634C22C2×#43011yescharacteristic
635C22#291612yes—
636C4—22no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4#233312yes—
645C4#237411yescharacteristic
646C4#237511yescharacteristic
647C4#268411yescharacteristic
648C4#268812yes—
649C4#270911yescharacteristic
650C2—21no—
651C2—22no—
652C2—21no—
653C2—42no—
654C2—81no—
655C2—81no—
656C2—81no—
657C2#1451311yescharacteristic
658C2#2194911yescharacteristic
659C2#2195011yescharacteristic
660C2#2271812yes—
661C2#2271912yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C4⋊C4
prime radicalC23×D4Q8
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11269C2semidirect
5. #11097C2semidirect
6. #11097C2semidirect
7. #10911C2semidirect
8. #10734C2semidirect
9. #10724C2semidirect

Maximal subgroups

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

subgroupindex
#211902
#208982
C2×#18702
#112692
#110972
#109112
#107342
#107242