Finite Group Databasebeta

← #195336

Subgroups — #195336

Order 512.

1686 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3187C211yesmaximal, characteristic
2#34205C211yesmaximal, characteristic
3#21759C211yesmaximal, characteristic, complemented
4C22×#327C211yesmaximal, characteristic
5#14159C211yesmaximal, characteristic, complemented
6#9045C214yesmaximal, complemented
7#9039C214yesmaximal, complemented
8C2×#1294C211yesmaximal, characteristic, complemented
9#7907C211yesmaximal, characteristic, complemented
10C22×C22⋊Q8C2211yescharacteristic
11C22×#121C2211yescharacteristic
12C2×#518—21no—
13#3187—24no—
14#3187C418yescomplemented
15C2×#481C2212yes—
16C2×#483—21no—
17C2×#465—22no—
18C2×#448C2212yes—
19C2×#450—21no—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22#2364—22no—
23#2361—22no—
24#2344—21no—
25#2342—22no—
26D4×C22⋊C4—21no—
27#2325C2214yes—
28#2308—21no—
29C4×#121—22no—
30C2×#380C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32C22×#95C2211yescharacteristic
33#1892—22no—
34#1872—22no—
35#1824C2212yescomplemented
36#1787C2212yescomplemented
37C2×#327—24no—
38C2×#327C2214yes—
39#1446C2212yescomplemented
40#1445C2212yescomplemented
41#1436—22no—
42C22⋊#95—21no—
43C22⋊#95—21no—
44#1294C2214yescomplemented
45#1280C2214yescomplemented
46C23×Q8C2311yescharacteristic
47C23×D4—21no—
48C23×D4D411yescharacteristic
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C22×SD16—22no—
52C23×C8—21no—
53C23×C8—21no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8C2×C414yescomplemented
62C2×#121—21no—
63C2×#121—21no—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121C2×C414yes—
67C2×#121D414yes—
68C2×#121D412yes—
69C2×#121C2312yes—
70#518—44no—
71#518—44no—
72C2×C4×D4—24no—
73C2×C4×D4—24no—
74C2×C4×D4—22no—
75C2×#117—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D412yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42D412yes—
87#483—28no—
88#481—24no—
89#481C2×C418yescomplemented
90#465—216no—
91#450—28no—
92#448—24no—
93#448C2×C418yescomplemented
94C2×#106—22no—
95C2×#103—22no—
96C2×#103—24no—
97C2×#102—22no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8C2×C414yescomplemented
104C2×C22⋊C8C2312yes—
105C2×C22⋊C8C2312yes—
106#390—22no—
107#389—22no—
108#388—24no—
109#388—24no—
110#387—24no—
111#386—24no—
112#386—24no—
113#386—24no—
114#385—24no—
115#381—22no—
116#380—21no—
117#380—21no—
118#380D412yes—
119#380C2312yes—
120#379—22no—
121C4×C4⋊C4—22no—
122C4×C22⋊C4—24no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95C2312yes—
136C2×#95C2312yes—
137#340—24no—
138#327—28no—
139C25—21no—
140C25C2×D411yescharacteristic
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—82no—
147C22×Q8C22×C412yes—
148C22×D4—24no—
149C22×D4—22no—
150C22×D4—24no—
151C22×D4—22no—
152C22×D4—21no—
153C22×D4—22no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4—82no—
159C22×D4C22⋊C414yes—
160C22×D4C2×D412yes—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—44no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—82no—
172C23×C4—82no—
173C23×C4C22×C412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4#4112yes—
183C23×C4#4112yes—
184C23×C4C2411yescharacteristic
185C2×SD16—28no—
186C2×SD16—84no—
187C22×C8—22no—
188C22×C8—24no—
189C22×C8—22no—
190C22×C8—42no—
191C22×C8—82no—
192C22×C8—82no—
193C22⋊Q8—24no—
194C22⋊Q8—24no—
195C22⋊Q8—24no—
196C22⋊Q8—24no—
197C22⋊Q8—42no—
198C22⋊Q8—42no—
199C22⋊Q8—42no—
200C22⋊Q8—42no—
201#121—24no—
202#121—24no—
203#121—24no—
204#121—24no—
205#121—24no—
206#121—22no—
207#121—22no—
208#121C22⋊C418yes—
209#121C22×C414yes—
210#121C2×D414yes—
211C4×D4—416no—
212C4×D4—48no—
213C4×D4—48no—
214C4×D4—84no—
215#117—48no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4C22⋊C414yes—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D414yes—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—48no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4SD1614yescomplemented
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—82no—
262#106—28no—
263#103—28no—
264#103—84no—
265#102—28no—
266C22⋊C8—24no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—42no—
271C22⋊C8C22×C414yes—
272#95—24no—
273#95—24no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—44no—
281#95—44no—
282#95—42no—
283#95—82no—
284#95—82no—
285C24—22no—
286C24—24no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—82no—
292C24—82no—
293C24C4×D412yes—
294C24#12011yescharacteristic
295C24#12011yescharacteristic
296C24#12112yes—
297C24C2×SD1612yes—
298C24#13512yes—
299C24AGL(1,ℤ/8)12yes—
300C24#13712yes—
301C24C22×D411yescharacteristic
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—82no—
309C2×Q8—82no—
310C2×Q8—82no—
311C2×Q8—82no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—28no—
315C2×D4—22no—
316C2×D4—24no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—48no—
324C2×D4—48no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—84no—
330C2×D4—84no—
331C2×D4C2×C22⋊C414yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—48no—
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—48no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4—88no—
381C22×C4—88no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C412yes—
386C22×C4C4×D412yes—
387C22×C4C4×D412yes—
388C22×C4C4×D412yes—
389C22×C4#12011yescharacteristic
390C22×C4#12011yescharacteristic
391C22×C4#12112yes—
392C22×C4#12112yes—
393C22×C4#12112yes—
394C22×C4#12312yes—
395C22×C4#12312yes—
396C22×C4C2×SD1614yes—
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403SD16—88no—
404C2×C8—24no—
405C2×C8—44no—
406C2×C8—44no—
407C2×C8—42no—
408C2×C8—84no—
409C2×C8—84no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—48no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4C2×C22⋊C414yes—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—28no—
427C22⋊C4—48no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C42—44no—
438C42—48no—
439C42—82no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—24no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—44no—
451C23—44no—
452C23—42no—
453C23—44no—
454C23—44no—
455C23—42no—
456C23—42no—
457C23—44no—
458C23—44no—
459C23—48no—
460C23—84no—
461C23—84no—
462C23—164no—
463C23—164no—
464C23#38612yes—
465C23#38612yes—
466C23C4×SD1614yes—
467C23#44014yes—
468C23#45012yes—
469C23#45012yes—
470C23#45212yes—
471C23#45212yes—
472C23#46114yes—
473C23#46314yes—
474C23C2×C4×D411yescharacteristic
475C23C2×#12011yescharacteristic
476C23C2×#12111yescharacteristic
477C23C22×SD1611yescharacteristic
478C23C2×#13511yescharacteristic
479C23C2×AGL(1,ℤ/8)11yescharacteristic
480C23C2×#13711yescharacteristic
481Q8—44no—
482Q8—44no—
483Q8—82no—
484Q8—82no—
485D4—44no—
486D4—48no—
487D4—416no—
488D4—44no—
489D4—44no—
490D4—44no—
491D4—44no—
492D4—82no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—816no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—168no—
547C2×C4—164no—
548C2×C4#38612yes—
549C2×C4#38612yes—
550C2×C4C22×C22⋊C411yescharacteristic
551C2×C4C2×C4×D411yescharacteristic
552C2×C4C2×#12111yescharacteristic
553C2×C4C2×#12311yescharacteristic
554C8—44no—
555C8—82no—
556C22—22no—
557C22—22no—
558C22—24no—
559C22—24no—
560C22—21no—
561C22—21no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—24no—
567C22—24no—
568C22—21no—
569C22—21no—
570C22—22no—
571C22—22no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—84no—
580C22—84no—
581C22—168no—
582C22#187714yes—
583C22#187714yes—
584C22C2×#38611yescharacteristic
585C22C2×C4×SD1611yescharacteristic
586C22C2×#44011yescharacteristic
587C22C2×#45011yescharacteristic
588C22C2×#45211yescharacteristic
589C22#301812yes—
590C22#302612yes—
591C22C2×#46111yescharacteristic
592C22C2×#46311yescharacteristic
593C4—21no—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—22no—
598C4—22no—
599C4—44no—
600C4—44no—
601C4—84no—
602C4—84no—
603C4—82no—
604C4—82no—
605C4—82no—
606C2—21no—
607C2—24no—
608C2—24no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—84no—
613C2#1838112yes—
614C2#1838512yes—
615C2C2×#187711yescharacteristic
616C2#2094211yescharacteristic
617C2#2096011yescharacteristic
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
3. #21759C2semidirect
5. #14159C2semidirect
6. #9045C2semidirect
7. #9039C2semidirect
8. C2×#1294C2semidirect
9. #7907C2semidirect
14. #3187C4semidirect
35. #1824C22semidirect
36. #1787C22semidirect
39. #1446C22semidirect
40. #1445C22semidirect
44. #1294C22semidirect
45. #1280C22semidirect
61. C2×C22⋊Q8C2×C4semidirect
89. #481C2×C4semidirect
93. #448C2×C4semidirect
103. C2×C22⋊C8C2×C4semidirect
257. C2×C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#31872
#342052
#217592
C22×#3272
#141592
#90452
#90392
C2×#12942
#79072