Finite Group Databasebeta

← #415040

Subgroups — #415040

Order 512.

1708 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic, complemented
3C2×#1990C211yesmaximal, characteristic, complemented
4C2×#1845C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊C8C211yesmaximal, characteristic
6#14686C211yesmaximal, characteristic
7#11219C214yesmaximal, complemented
8#11096C214yesmaximal
9#10841C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×C22⋊Q8—21no—
12C23×C42C2211yescharacteristic
13C22×#107C2214yes—
14C22×C22⋊C8C2212yes—
15C22×C4×C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2857—24no—
18#2857—24no—
19C2×#391—21no—
20C2×#389—21no—
21C2×#389C2212yes—
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#380—21no—
26C2×#380—21no—
27C2×#380—21no—
28C2×#380C2212yes—
29C2×#380C2212yes—
30#1990—24no—
31#1990—24no—
32#1845—24no—
33C4×C22⋊C8—24no—
34C2×#340C2214yes—
35C2×#340C2214yes—
36C2×#340C2214yes—
37C2×#336C2212yes—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C23×Q8—21no—
42C24×C4D411yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8D414yes—
53C2×C22⋊Q8—28no—
54C2×C22⋊Q8—44no—
55C2×C22⋊Q8—44no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C42—22no—
71C22×C42—22no—
72C22×C42—22no—
73C22×C42—22no—
74C22×C42—22no—
75C22×C42—22no—
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#107—28no—
81C2×#107—28no—
82C2×#107—44no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8Q818yescomplemented
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8D414yes—
92#391—28no—
93#389—28no—
94#389—24no—
95#389—24no—
96#384—22no—
97#384—22no—
98#384—22no—
99#384—22no—
100#384—22no—
101#384D414yes—
102#380—28no—
103#380—24no—
104#380—24no—
105#380—44no—
106#380—44no—
107#380—44no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113#340—28no—
114#340—28no—
115#340—44no—
116#336—24no—
117C25C2×D411yescharacteristic
118C22×Q8—24no—
119C22×Q8—42no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—42no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4D814yes—
151C23×C4Q1614yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×Q814yes—
158C23×C4C2411yescharacteristic
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8C2×D414yes—
170C22×C8C2×Q814yes—
171C22×C8C2×Q814yes—
172C22×C8#4114yes—
173C4⋊Q8—28no—
174C4⋊Q8—28no—
175C4⋊Q8—44no—
176C4⋊Q8—44no—
177C22⋊Q8—432no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×D414yes—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—44no—
239C2×C42—44no—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247C2×C42#4112yes—
248#107—416no—
249C22⋊C8—216no—
250C4×C8—28no—
251C4×C8—44no—
252#95—28no—
253#95—28no—
254#95—44no—
255#95—44no—
256#95—44no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24C4⋊Q812yes—
270C24C2×D812yes—
271C24C2×Q1612yes—
272C24C22×D411yescharacteristic
273C2×Q8—48no—
274C2×Q8—48no—
275C2×Q8—48no—
276C2×Q8—84no—
277C2×Q8—84no—
278C2×Q8—84no—
279C2×Q8—84no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
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#12012yes—
357C22×C4#12012yes—
358C22×C4#12112yes—
359C22×C4#12112yes—
360C22×C4#12312yes—
361C22×C4#12312yes—
362C22×C4C4⋊Q812yes—
363C22×C4C2×D812yes—
364C22×C4C2×D812yes—
365C22×C4C2×Q1612yes—
366C22×C4C2×Q1612yes—
367C22×C4#13512yes—
368C22×C4#13512yes—
369C22×C4AGL(1,ℤ/8)12yes—
370C22×C4AGL(1,ℤ/8)12yes—
371C22×C4#13712yes—
372C22×C4#13712yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×Q812yes—
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4112yes—
380C22×C4C2×#4111yescharacteristic
381C22×C4H(5,𝔽2)12yes—
382C22×C42−1+412yes—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8C22⋊Q818yescomplemented
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—28no—
400C4⋊C4—416no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—48no—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—84no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C22⋊C4—416no—
416C22⋊C4—88no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—44no—
422C42—44no—
423C42#12114yes—
424C42#12114yes—
425C42#12314yes—
426C42#12414yes—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23#50014yes—
462C23C2×#12011yescharacteristic
463C23C2×#12111yescharacteristic
464C23C2×#12311yescharacteristic
465C23C2×C4⋊Q811yescharacteristic
466C23#53211yescharacteristic
467C23#53511yescharacteristic
468C23#53611yescharacteristic
469C23C22×D811yescharacteristic
470C23C22×Q1611yescharacteristic
471C23C2×#13511yescharacteristic
472C23C2×#13511yescharacteristic
473C23C2×AGL(1,ℤ/8)11yescharacteristic
474C23C2×AGL(1,ℤ/8)11yescharacteristic
475C23C2×#13711yescharacteristic
476C23C2×#13711yescharacteristic
477Q8—816no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—416no—
504C2×C4—416no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4#39414yes—
534C2×C4#39414yes—
535C2×C4#44714yes—
536C2×C4#44914yes—
537C2×C4#45114yes—
538C2×C4#45214yes—
539C2×C4#46614yes—
540C2×C4#46714yes—
541C2×C4#46814yes—
542C2×C4#46814yes—
543C2×C4#48014yes—
544C2×C4#48214yes—
545C2×C4#48414yes—
546C2×C4#48514yes—
547C2×C4C2×#12111yescharacteristic
548C2×C4C2×#12111yescharacteristic
549C2×C4C2×C22⋊Q812yes—
550C2×C4C2×#12311yescharacteristic
551C2×C4C2×#12411yescharacteristic
552C2×C4#53312yes—
553C2×C4#53712yes—
554C2×C4#54212yes—
555C2×C4#54612yes—
556C2×C4D4×Q812yes—
557C2×C4D4×Q812yes—
558C2×C4#55012yes—
559C2×C4#55212yes—
560C2×C4#55412yes—
561C2×C4#55412yes—
562C2×C4#55612yes—
563C8—28no—
564C8—44no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22C2×#39411yescharacteristic
581C22#243111yescharacteristic
582C22#243411yescharacteristic
583C22#249911yescharacteristic
584C22#250211yescharacteristic
585C22#262011yescharacteristic
586C22#285711yescharacteristic
587C22C2×#44711yescharacteristic
588C22C2×#45111yescharacteristic
589C22C2×#44911yescharacteristic
590C22C2×#45211yescharacteristic
591C22C2×#46611yescharacteristic
592C22C2×#46711yescharacteristic
593C22C2×#46811yescharacteristic
594C22C2×#46811yescharacteristic
595C22C2×#48011yescharacteristic
596C22C2×#48211yescharacteristic
597C22C2×#48511yescharacteristic
598C22C2×#48411yescharacteristic
599C22C2×#50011yescharacteristic
600C22#316011yescharacteristic
601C22#316511yescharacteristic
602C22#316511yescharacteristic
603C22#316911yescharacteristic
604C22#317011yescharacteristic
605C22#318411yescharacteristic
606C22#318511yescharacteristic
607C22#320511yescharacteristic
608C22#320711yescharacteristic
609C22#320911yescharacteristic
610C22#321011yescharacteristic
611C22#322611yescharacteristic
612C22#322811yescharacteristic
613C22#323011yescharacteristic
614C22#323111yescharacteristic
615C4—42no—
616C4—42no—
617C4—42no—
618C4—84no—
619C4—84no—
620C4—162no—
621C4—162no—
622C4#203318yes—
623C4#203618yes—
624C4#241712yes—
625C4#242312yes—
626C4#249212yes—
627C4#253812yes—
628C2—44no—
629C2#1518211yescharacteristic
630C2C2×#203311yescharacteristic
631C2C2×#203611yescharacteristic
632C2#2559911yescharacteristic
633C2#2560011yescharacteristic
634C2#2581211yescharacteristic
635C2#2581311yescharacteristic
636C2#2787711yescharacteristic
637C2#2787811yescharacteristic
638C2#2794911yescharacteristic
639C2#2795011yescharacteristic
640C2#2984211yescharacteristic
641C2#2984411yescharacteristic
642C2#3320911yescharacteristic
643C2#3321011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1990C2semidirect
3. C2×#1990C2semidirect
4. C2×#1845C2semidirect
7. #11219C2semidirect
9. #10841C2semidirect
85. C2×C22⋊C8Q8semidirect
396. C2×C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#19902
C2×#18452
C2×C4×C22⋊C82
#146862
#112192
#110962
#108412