Finite Group Databasebeta

← #369125

Subgroups — #369125

Order 512.

2622 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2541C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic, complemented
3C2×#1982C211yesmaximal, characteristic, complemented
4C2×#1957C211yesmaximal, characteristic, complemented
5C2×#1954C211yesmaximal, characteristic, complemented
6C2×#1875C214yesmaximal, complemented
7#10969C214yesmaximal, complemented
8#10849C211yesmaximal, characteristic, complemented
9C4×#340C211yesmaximal, characteristic, complemented
10C22×#127—21no—
11C22×#127C2211yescharacteristic
12C22×#121C2214yes—
13C22×C4×D4C2211yescharacteristic
14C22×#117C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2214yes—
18C22×#102C2212yes—
19C22×C22⋊C8C2214yes—
20C22×C4×C8C2211yescharacteristic
21#2541—24no—
22#2541C418yescomplemented
23C2×#392—22no—
24#2354—24no—
25#2354—24no—
26C2×#390—21no—
27C2×#390—21no—
28C2×#388—22no—
29C2×#384C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1982—24no—
35#1957—24no—
36#1954—24no—
37#1875—216no—
38C2×#340C2214yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C23×D4—24no—
42C23×D4—21no—
43C23×D4—21no—
44C23×D4D411yescharacteristic
45C23×D4C2312yes—
46C24×C4C2312yes—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#127—28no—
50C2×#127—22no—
51C2×#127—22no—
52C2×#127—22no—
53C2×#127—22no—
54C2×#127—22no—
55C2×#127—44no—
56C2×#127—44no—
57C2×#127C2×C414yes—
58C2×#121—28no—
59C2×#121—28no—
60C2×#121—28no—
61C2×#121—44no—
62C2×#121—44no—
63C2×#121C2×C4116yes—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4C2×C414yes—
69C2×C4×D4D414yes—
70C2×#117—24no—
71C2×#117—24no—
72C2×#117—28no—
73C2×#117—44no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—22no—
82C22×C22⋊C4—24no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D412yes—
85C22×C42—22no—
86C22×C42—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×#102—28no—
93C2×#102—28no—
94C2×#102—28no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—44no—
98C2×#102—44no—
99C2×C22⋊C8—28no—
100C2×C22⋊C8—28no—
101C2×C22⋊C8—44no—
102C2×C4×C8—22no—
103C2×C4×C8—22no—
104C2×C4×C8—22no—
105#392—216no—
106#390—28no—
107#390—28no—
108#388—216no—
109#384—22no—
110#384D414yescomplemented
111#380—24no—
112#380—24no—
113C4×C4⋊C4—28no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4C2×C414yes—
117C4×C4⋊C4D414yescomplemented
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—21no—
121#340—28no—
122#340—24no—
123#340D418yescomplemented
124C25—22no—
125C25C2×D412yes—
126C22×D4—216no—
127C22×D4—24no—
128C22×D4—24no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—24no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—24no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—24no—
143C22×D4—24no—
144C22×D4—48no—
145C22×D4—48no—
146C22×D4—48no—
147C22×D4—48no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—82no—
157C22×D4—82no—
158C22×D4—84no—
159C22×D4—84no—
160C22×D4C22⋊C414yes—
161C22×D4C22×C414yes—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—21no—
174C23×C4C22×C414yes—
175C23×C4C2×D412yes—
176C23×C4C2×D414yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2411yescharacteristic
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—24no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—24no—
191C22×C8C2×D414yes—
192#127—28no—
193#127—28no—
194#127—432no—
195#127—44no—
196#127—44no—
197#121—232no—
198#121—416no—
199#121—416no—
200#121—416no—
201C4×D4—28no—
202C4×D4—28no—
203C4×D4—28no—
204C4×D4—28no—
205#117—416no—
206#117—48no—
207#117—48no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4C22⋊C414yes—
227C2×C4⋊C4C22×C414yes—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4#4114yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—216no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—48no—
240C2×C22⋊C4—48no—
241C2×C22⋊C4—48no—
242C2×C22⋊C4—84no—
243C2×C22⋊C4C22⋊C418yes—
244C2×C42—28no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—48no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42C22×C412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42#4112yes—
257C2×C42#4112yes—
258C4⋊C8—44no—
259#102—416no—
260#102—48no—
261#102—48no—
262C22⋊C8—416no—
263C4×C8—44no—
264#95—28no—
265#95—24no—
266#95—24no—
267C24—22no—
268C24—22no—
269C24—24no—
270C24—24no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—24no—
276C24—24no—
277C24—24no—
278C24—42no—
279C24—42no—
280C24—44no—
281C24—44no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—44no—
287C24—44no—
288C24C2×C22⋊C414yes—
289C24C22×D411yescharacteristic
290C2×D4—28no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—28no—
294C2×D4—28no—
295C2×D4—28no—
296C2×D4—28no—
297C2×D4—432no—
298C2×D4—432no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—44no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—48no—
316C2×D4—48no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—88no—
322C2×D4—88no—
323C2×D4—84no—
324C2×D4—84no—
325C2×D4—816no—
326C2×D4—816no—
327C2×D4—816no—
328C2×D4—816no—
329C2×D4—816no—
330C2×D4—84no—
331C2×D4—84no—
332C2×D4—84no—
333C2×D4—84no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—28no—
348C22×C4—28no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4C2×C22⋊C414yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4#12012yes—
384C22×C4#12012yes—
385C22×C4#12312yes—
386C22×C4#12312yes—
387C22×C4#13512yes—
388C22×C4#13512yes—
389C22×C4AGL(1,ℤ/8)12yes—
390C22×C4AGL(1,ℤ/8)12yes—
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D412yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4112yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)12yes—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—44no—
412C2×C8—44no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—28no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—28no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—48no—
426C4⋊C4—84no—
427C4⋊C4C4×D418yes—
428C4⋊C4#12714yescomplemented
429C22⋊C4—216no—
430C22⋊C4—432no—
431C22⋊C4—416no—
432C22⋊C4—88no—
433C42—416no—
434C42—48no—
435C42—48no—
436C42C4×D414yes—
437C42C4×D414yes—
438C42#12114yes—
439C42#12114yes—
440C23—28no—
441C23—24no—
442C23—24no—
443C23—28no—
444C23—24no—
445C23—42no—
446C23—42no—
447C23—44no—
448C23—42no—
449C23—42no—
450C23—44no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—48no—
464C23—44no—
465C23—44no—
466C23—48no—
467C23—44no—
468C23—82no—
469C23—82no—
470C23—84no—
471C23—82no—
472C23—82no—
473C23—84no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—84no—
479C23—84no—
480C23—84no—
481C23—84no—
482C23—84no—
483C23—84no—
484C23—84no—
485C23—84no—
486C23C22×C22⋊C411yescharacteristic
487C23C2×#12011yescharacteristic
488C23C2×#12311yescharacteristic
489C23D4212yes—
490C23#54612yes—
491C23C2×#13511yescharacteristic
492C23C2×#13511yescharacteristic
493C23C2×AGL(1,ℤ/8)11yescharacteristic
494C23C2×AGL(1,ℤ/8)11yescharacteristic
495C23#57612yes—
496C23#57712yes—
497D4—44no—
498D4—44no—
499D4—48no—
500D4—416no—
501D4—816no—
502D4—864no—
503D4—816no—
504D4—84no—
505D4—82no—
506D4—82no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—48no—
531C2×C4—48no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4#38614yes—
545C2×C4#38614yes—
546C2×C4#41614yes—
547C2×C4#41814yes—
548C2×C4#44914yes—
549C2×C4#44914yes—
550C2×C4#48214yes—
551C2×C4#48214yes—
552C2×C4C2×C4×D411yescharacteristic
553C2×C4C2×C4×D412yes—
554C2×C4C2×C4×D411yescharacteristic
555C2×C4#51812yes—
556C2×C4C2×#12111yescharacteristic
557C2×C4C2×#12111yescharacteristic
558C2×C4C2×#12711yescharacteristic
559C2×C4#53212yes—
560C2×C4#53211yescharacteristic
561C2×C4#53511yescharacteristic
562C2×C4#53511yescharacteristic
563C2×C4#54012yes—
564C2×C4D4212yes—
565C2×C4D4212yes—
566C2×C4#54612yes—
567C2×C4#55012yes—
568C2×C4#55012yes—
569C2×C4#55912yes—
570C8—44no—
571C8—82no—
572C22—48no—
573C22—48no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—84no—
578C22—84no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—88no—
584C22—88no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22—164no—
589C22—164no—
590C22—162no—
591C22—162no—
592C22—162no—
593C22—162no—
594C22C2×#38611yescharacteristic
595C22D4×C22⋊C412yes—
596C22#243111yescharacteristic
597C22#243311yescharacteristic
598C22#244512yes—
599C22C2×#41611yescharacteristic
600C22C2×#41811yescharacteristic
601C22#289712yes—
602C22C2×#44911yescharacteristic
603C22C2×#44911yescharacteristic
604C22#300811yescharacteristic
605C22#301011yescharacteristic
606C22C2×#48211yescharacteristic
607C22C2×#48211yescharacteristic
608C22#309612yes—
609C22#327212yes—
610C22#327412yes—
611C22#328412yes—
612C22#328712yes—
613C22#330412yes—
614C22#330512yes—
615C22#332412yes—
616C22#332512yes—
617C4—24no—
618C4—44no—
619C4—48no—
620C4—44no—
621C4—84no—
622C4—82no—
623C4#187518yescomplemented
624C4C4×#12711yescharacteristic
625C4#232811yescharacteristic
626C4#236612yes—
627C4#242812yes—
628C4#254111yescharacteristic
629C4#254411yescharacteristic
630C2—84no—
631C2—164no—
632C2#1402511yescharacteristic
633C2C2×#187511yescharacteristic
634C2#2091411yescharacteristic
635C2#2241812yes—
636C2#2242812yes—
637C2#2557211yescharacteristic
638C2#2557411yescharacteristic
639C2#2575411yescharacteristic
640C2#2575911yescharacteristic
641C2#2623512yes—
642C2#2623712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2541C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2354C2semidirect
3. C2×#1982C2semidirect
4. C2×#1957C2semidirect
5. C2×#1954C2semidirect
6. C2×#1875C2semidirect
7. #10969C2semidirect
8. #10849C2semidirect
9. C4×#340C2semidirect
22. #2541C4semidirect
110. #384D4semidirect
117. C4×C4⋊C4D4semidirect
123. #340D4semidirect
428. C4⋊C4#127semidirect
623. C4#1875semidirect

Maximal subgroups

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

subgroupindex
C2×#25412
C2×#23542
C2×#19822
C2×#19572
C2×#19542
C2×#18752
#109692
#108492
C4×#3402