Finite Group Databasebeta

← #204222

Subgroups — #204222

Order 512.

2914 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#390C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C4×#407C211yesmaximal, characteristic, complemented
4C2×#1462C214yesmaximal, complemented
5#9653C214yesmaximal, complemented
6#9649C214yesmaximal, complemented
7C24×D4C411yescharacteristic
8C23×C4⋊C4C411yescharacteristic
9C23×C22⋊C4C412yes—
10C23×C22⋊C4C2212yes—
11C23×C42C2211yescharacteristic
12C2×#423—21no—
13C2×#407C2212yes—
14C2×#407C2212yes—
15C2×C4×M16—21no—
16C2×#390—21no—
17C2×#390—21no—
18C2×#390—21no—
19C2×#390—21no—
20C2×#390—21no—
21C2×#390—21no—
22C2×#390—21no—
23C2×#390—21no—
24C2×#390C412yes—
25C2×#390C412yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28#1797—21no—
29#1797—21no—
30#1797C2212yescomplemented
31#1797C2212yescomplemented
32#1749—21no—
33#1749C2212yescomplemented
34C4×C22⋊C8—21no—
35C4×C22⋊C8C2212yescomplemented
36#1728—21no—
37#1728—21no—
38#1462—216no—
39C2×#323C22116yescomplemented
40C26C2×C412yes—
41C23×D4—28no—
42C23×D4—21no—
43C23×D4—21no—
44C23×D4—21no—
45C23×D4—21no—
46C23×D4—44no—
47C23×D4—44no—
48C23×D4C2×C412yes—
49C23×D4C2×C412yes—
50C24×C4C2×C411yescharacteristic
51C24×C4C2×C412yes—
52C24×C4C2×C411yescharacteristic
53C24×C4C2311yescharacteristic
54C22×M16—22no—
55C22×M16—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—44no—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C412yes—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—28no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—28no—
73C22×C22⋊C4—48no—
74C22×C22⋊C4C2×C414yes—
75C22×C22⋊C4C2×C414yes—
76C22×C22⋊C4C2×C414yes—
77C22×C22⋊C4C2314yes—
78C22×C42—44no—
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#423—44no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94#407—24no—
95#407—28no—
96#407D418yescomplemented
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8C2314yes—
100C2×C22⋊C8C2314yes—
101C4×M16—44no—
102C2×#97—22no—
103C2×C4×C8—22no—
104#390—22no—
105#390—22no—
106#390—22no—
107#390—22no—
108#390—22no—
109#390—22no—
110#390—22no—
111#390—22no—
112#390—22no—
113#390—22no—
114#390—22no—
115#390—22no—
116#390—22no—
117#390—22no—
118#390—22no—
119#390—22no—
120#336—22no—
121#336—22no—
122#336—22no—
123#336—22no—
124#323—232no—
125C25—22no—
126C25—22no—
127C25—28no—
128C25—28no—
129C25—44no—
130C25—44no—
131C25—44no—
132C25—44no—
133C25—44no—
134C25C22×C411yescharacteristic
135C25C22×C414yes—
136C22×D4—28no—
137C22×D4—28no—
138C22×D4—28no—
139C22×D4—28no—
140C22×D4—28no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—24no—
146C22×D4—464no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4—44no—
151C22×D4—44no—
152C22×D4—44no—
153C22×D4—44no—
154C22×D4—44no—
155C22×D4—44no—
156C22×D4—44no—
157C22×D4—44no—
158C22×D4—44no—
159C22×D4M1618yescomplemented
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—28no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C414yes—
186C23×C4C22×C411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D414yes—
192C23×C4#4114yes—
193C23×C4C2411yescharacteristic
194C2×M16—28no—
195C2×M16—44no—
196C2×M16—44no—
197C2×M16—44no—
198C2×M16—44no—
199C2×M16—44no—
200C22×C8—24no—
201C22×C8—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—216no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—48no—
224C2×C22⋊C4—48no—
225C2×C22⋊C4—48no—
226C2×C22⋊C4—48no—
227C2×C22⋊C4—48no—
228C2×C22⋊C4—48no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—48no—
234C2×C22⋊C4—48no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C4⋊C8—44no—
264C4⋊C8—44no—
265C22⋊C8—28no—
266C22⋊C8—24no—
267C22⋊C8—28no—
268C22⋊C8—44no—
269C22⋊C8C2×D418yescomplemented
270#97—44no—
271C4×C8—44no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—28no—
279C24—28no—
280C24—24no—
281C24—24no—
282C24—28no—
283C24—28no—
284C24—432no—
285C24—44no—
286C24—432no—
287C24—44no—
288C24—44no—
289C24—432no—
290C24—44no—
291C24—44no—
292C24—44no—
293C24—44no—
294C24—44no—
295C24—44no—
296C24—44no—
297C24—44no—
298C24—44no—
299C24—44no—
300C24—44no—
301C24—44no—
302C24—44no—
303C24—816no—
304C24—816no—
305C24—816no—
306C24—816no—
307C24C2×C22⋊C411yescharacteristic
308C24C2×C22⋊C411yescharacteristic
309C24C4×D414yes—
310C24C2×M1618yes—
311C24C23×C411yescharacteristic
312C2×D4—28no—
313C2×D4—28no—
314C2×D4—28no—
315C2×D4—28no—
316C2×D4—432no—
317C2×D4—432no—
318C2×D4—432no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—432no—
324C2×D4—432no—
325C2×D4—416no—
326C2×D4—416no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C22×C4—232no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
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—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—28no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—41no—
380C22×C4—416no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—816no—
390C22×C4#9914yes—
391C22×C4#10014yes—
392C22×C4#10014yes—
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C4×D414yes—
404C22×C4#12114yes—
405C22×C4#12114yes—
406C22×C4#12412yes—
407C22×C4#12412yes—
408C22×C4#12712yes—
409C22×C4#12712yes—
410C22×C4C2×M1614yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4C2×#4112yes—
415M16—416no—
416M16—88no—
417C2×C8—28no—
418C2×C8—44no—
419C2×C8—44no—
420C2×C8—44no—
421C2×C8—44no—
422C2×C8—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C23—28no—
453C23—28no—
454C23—28no—
455C23—28no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—216no—
473C23—48no—
474C23—416no—
475C23—416no—
476C23—416no—
477C23—416no—
478C23—48no—
479C23—416no—
480C23—416no—
481C23—48no—
482C23—48no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—48no—
489C23—44no—
490C23—88no—
491C23—88no—
492C23—88no—
493C23—88no—
494C23—8128no—
495C23—88no—
496C23—88no—
497C23—88no—
498C23—88no—
499C23—88no—
500C23—88no—
501C23—88no—
502C23—88no—
503C23#39012yes—
504C23#39012yes—
505C23C2×#9912yes—
506C23#41012yes—
507C23C2×#10012yes—
508C23C2×#10012yes—
509C23C22×C22⋊C411yescharacteristic
510C23C2×C4×D412yes—
511C23C22×M1612yes—
512C23#56812yes—
513D4—416no—
514D4—416no—
515D4—416no—
516D4—416no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—88no—
566C2×C4—88no—
567C2×C4—88no—
568C2×C4#39012yes—
569C2×C4#39012yes—
570C2×C4#40712yes—
571C2×C4#40712yes—
572C2×C4C2×#9912yes—
573C2×C4C2×#9912yes—
574C2×C4C2×#10012yes—
575C2×C4C2×#10012yes—
576C2×C4C2×#10012yes—
577C2×C4C2×#10012yes—
578C2×C4C2×#12112yes—
579C2×C4C2×#12411yescharacteristic
580C2×C4C2×#12711yescharacteristic
581C8—48no—
582C8—84no—
583C22—232no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—48no—
603C22—48no—
604C22—48no—
605C22—816no—
606C22—816no—
607C22—832no—
608C22—832no—
609C22—84no—
610C22—84no—
611C22—84no—
612C22—84no—
613C22#196414yes—
614C22#196514yes—
615C22C2×#39011yescharacteristic
616C22C2×#40711yescharacteristic
617C22#287911yescharacteristic
618C22C22×#9911yescharacteristic
619C22C2×#41011yescharacteristic
620C22C22×#10011yescharacteristic
621C22C22×#10011yescharacteristic
622C22D4×M1614yes—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—21no—
628C4—21no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—41no—
634C4—41no—
635C4—84no—
636C4#146214yescomplemented
637C2—24no—
638C2—24no—
639C2—24no—
640C2—88no—
641C2C2×#146211yescharacteristic
642C2#1736411yescharacteristic
643C2#1739311yescharacteristic
644C2#1739611yescharacteristic
645C2#2119011yescharacteristic
646C2C2×#196411yescharacteristic
647C2C2×#196511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24×D4C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20626C2semidirect
3. C4×#407C2semidirect
4. C2×#1462C2semidirect
5. #9653C2semidirect
6. #9649C2semidirect
30. #1797C22semidirect
31. #1797C22semidirect
33. #1749C22semidirect
35. C4×C22⋊C8C22semidirect
39. C2×#323C22semidirect
96. #407D4semidirect
159. C22×D4M16semidirect
269. C22⋊C8C2×D4semidirect
636. C4#1462semidirect

Maximal subgroups

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

subgroupindex
C22×#3902
#206262
C4×#4072
C2×#14622
#96532
#96492