Finite Group Databasebeta

← #197733

Subgroups — #197733

Order 512.

1308 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3103C211yesmaximal, characteristic
2C2×Q8×M16C211yesmaximal, characteristic, complemented
3#21200C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic, complemented
5C2×#1547C211yesmaximal, characteristic
6#9230C212yesmaximal, complemented
7#9228C212yesmaximal, complemented
8#9222C212yesmaximal, complemented
9#9220C212yesmaximal, complemented
10#8043C211yesmaximal, characteristic, complemented
11#7873C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3103—24no—
15#3103C418yescomplemented
16C2×#481—21no—
17C2×#475C2211yescharacteristic
18C2×#475C2211yescharacteristic
19C2×#460C2211yescharacteristic
20C2×#460C2211yescharacteristic
21C2×#448—21no—
22Q8×M16—24no—
23Q8×M16—24no—
24C2×#446—22no—
25C2×#446C2212yes—
26C2×C8×Q8C2212yes—
27C2×C4×SD16—21no—
28C2×C4×SD16—21no—
29C2×#431C2211yescharacteristic
30C2×#423—21no—
31C2×#423C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C2×#417—21no—
34C2×#417—21no—
35C22×#103—21no—
36C2×C4×M16—21no—
37C2×C4×M16C2211yescharacteristic
38C2×#384C2211yescharacteristic
39#1959—22no—
40#1957C2212yes—
41#1956C2212yes—
42#1803—21no—
43#1803—21no—
44#1803C2212yes—
45#1803C2212yes—
46C2×#340—21no—
47#1547—24no—
48C2×#332C2212yes—
49C2×#331C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1277C2212yes—
53#1277C2212yes—
54C22×#41—21no—
55C23×Q8D411yescharacteristic
56C22×SD16—21no—
57C22×SD16—21no—
58C22×M16—21no—
59C22×M16D411yescharacteristic
60C22×M16D411yescharacteristic
61C23×C8—21no—
62#532—22no—
63#532—22no—
64#532—22no—
65#532C2×C414yes—
66C2×C4⋊Q8C2311yescharacteristic
67C2×#127C2311yescharacteristic
68C2×#124—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—21no—
75C2×C4×Q8C2×C414yes—
76C2×C4×Q8D414yes—
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8D411yescharacteristic
79C2×C4×Q8C2311yescharacteristic
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4D411yescharacteristic
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90#481—28no—
91#475—22no—
92#475—22no—
93#475C2×C414yescomplemented
94#475C2×C414yescomplemented
95#460—22no—
96#460—22no—
97#460C2×C414yescomplemented
98#460C2×C414yescomplemented
99#448—28no—
100#446—216no—
101#446—24no—
102#446—24no—
103C8×Q8—24no—
104C8×Q8—24no—
105C4×SD16—28no—
106C4×SD16—28no—
107#431—22no—
108#431C2×C414yes—
109C2×#106—21no—
110C2×#106—21no—
111#423—28no—
112#423—22no—
113#423D414yes—
114C2×C4⋊C8—22no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2312yes—
124#417—44no—
125#417—44no—
126C2×#103—21no—
127C2×#103—21no—
128C2×#103—22no—
129C2×#103—22no—
130C2×#103—22no—
131C2×#103—22no—
132C2×#103—44no—
133C2×#102—21no—
134C2×#102—21no—
135C2×#102—22no—
136C2×#102—22no—
137C2×C22⋊C8—21no—
138C4×M16—28no—
139C4×M16—22no—
140C4×M16—22no—
141C2×#97—22no—
142C2×#97D412yes—
143C2×C4×C8—22no—
144C2×C4×C8—21no—
145C2×C4×C8C2312yes—
146#384—21no—
147#384—21no—
148#384—21no—
149#384C2312yes—
150#340—22no—
151#340—22no—
152#332—24no—
153#331—24no—
154C2×#41—24no—
155C2×#41—42no—
156C2×#41—42no—
157C2×#41—42no—
158C2×#41—42no—
159C2×#41—42no—
160C22×Q8—21no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—21no—
165C22×Q8—41no—
166C22×Q8—41no—
167C22×Q8C22⋊C414yes—
168C22×Q8C2×D411yescharacteristic
169C22×Q8C2×D411yescharacteristic
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—41no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—41no—
177C23×C4—41no—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4#4111yescharacteristic
181C2×SD16—24no—
182C2×SD16—24no—
183C2×SD16—42no—
184C2×SD16—42no—
185C2×SD16—42no—
186C2×SD16—42no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—24no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—42no—
194C2×M16—42no—
195C2×M16SD1614yescomplemented
196C22×C8—22no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—22no—
200C22×C8—42no—
201C22×C8—42no—
202C22×C8—42no—
203C22×C8C2×D412yes—
204C22×C8C2×D412yes—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8C22×C412yes—
209#127—21no—
210#127—21no—
211#127—21no—
212#127C22×C412yes—
213#124—24no—
214#124—42no—
215#121—24no—
216#121—24no—
217#121—42no—
218#121—42no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—24no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—24no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8C22⋊C414yes—
230C4×Q8C22⋊C414yes—
231C4×Q8C22×C412yes—
232C4×Q8C22×C412yes—
233C4×Q8C2×D412yes—
234C4×Q8C2×D412yes—
235C4×D4—24no—
236C4×D4—24no—
237C4×D4—42no—
238C4×D4—42no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—82no—
260C2×C4⋊C4C22⋊C414yes—
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42C22×C412yes—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D412yes—
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D412yes—
276C2×C42#4111yescharacteristic
277C2×C42#4112yes—
278C2×C42C2411yescharacteristic
279#106—24no—
280#106—24no—
281C4⋊C8—28no—
282C4⋊C8—28no—
283C4⋊C8—24no—
284C4⋊C8—22no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—22no—
288C4⋊C8—22no—
289C4⋊C8—42no—
290C4⋊C8C22×C412yes—
291C4⋊C8C22×C412yes—
292C4⋊C8C2×D414yes—
293#103—24no—
294#103—24no—
295#103—44no—
296#103—44no—
297#103—44no—
298#103—44no—
299#102—24no—
300#102—24no—
301#102—44no—
302#102—44no—
303C22⋊C8—24no—
304#97—28no—
305#97—24no—
306C4×C8—28no—
307C4×C8—24no—
308C4×C8—22no—
309C4×C8—22no—
310C24—41no—
311C24#12111yescharacteristic
312#41—48no—
313#41—48no—
314#41—84no—
315#41—84no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—24no—
321C2×Q8—28no—
322C2×Q8—22no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8C2×C22⋊C412yes—
338C2×Q8C2×C22⋊C412yes—
339C2×Q8AGL(1,ℤ/8)12yes—
340C2×Q8#13712yes—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—44no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—81no—
357C2×D4—81no—
358C2×D4—82no—
359C2×D4—82no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4C2×C22⋊C412yes—
385C22×C4C4×D412yes—
386C22×C4C4×D412yes—
387C22×C4#12012yes—
388C22×C4#12112yes—
389C22×C4#12111yescharacteristic
390C22×C4#12111yescharacteristic
391C22×C4#12111yescharacteristic
392C22×C4#12312yes—
393C22×C4C2×SD1612yes—
394C22×C4#13512yes—
395C22×C4AGL(1,ℤ/8)12yes—
396C22×C4#13712yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4111yescharacteristic
400SD16—48no—
401SD16—48no—
402M16—28no—
403M16—44no—
404M16—48no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—24no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—84no—
423C2×C8C2×SD1614yes—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4C2×C22⋊C412yes—
446C4⋊C4C2×C22⋊C412yes—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—24no—
454C42—24no—
455C42C2×C22⋊C412yes—
456C42C4×D412yes—
457C42C4×D412yes—
458C42#12012yes—
459C42#12112yes—
460C42#12312yes—
461C42C23×C411yescharacteristic
462C42C22×D411yescharacteristic
463C42C22×D411yescharacteristic
464C42C2×#4111yescharacteristic
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—42no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23#38612yes—
477C23#46012yes—
478C23#46412yes—
479C23C2×#12111yescharacteristic
480Q8—22no—
481Q8—22no—
482Q8—22no—
483Q8—22no—
484Q8—22no—
485Q8—48no—
486Q8—41no—
487Q8—41no—
488Q8—42no—
489Q8—42no—
490Q8—81no—
491Q8—81no—
492Q8#41714yescomplemented
493D4—42no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—82no—
498D4—82no—
499D4—88no—
500D4—81no—
501D4—81no—
502D4—81no—
503D4—81no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—48no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—84no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—88no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—164no—
538C2×C4#38612yes—
539C2×C4#38612yes—
540C2×C4#38612yes—
541C2×C4C4×SD1614yes—
542C2×C4#44014yes—
543C2×C4#44814yes—
544C2×C4#44914yes—
545C2×C4#46012yes—
546C2×C4#46412yes—
547C2×C4C22×C22⋊C411yescharacteristic
548C2×C4C2×C4×D411yescharacteristic
549C2×C4C2×C4×D411yescharacteristic
550C2×C4C2×#12011yescharacteristic
551C2×C4C2×#12111yescharacteristic
552C2×C4C2×#12311yescharacteristic
553C2×C4C22×SD1611yescharacteristic
554C2×C4C2×#13511yescharacteristic
555C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
556C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
557C2×C4C2×#13711yescharacteristic
558C2×C4C2×#13711yescharacteristic
559C2×C4#57511yescharacteristic
560C2×C4#57511yescharacteristic
561C8—24no—
562C8—42no—
563C8—42no—
564C8—42no—
565C8—44no—
566C8—82no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—161no—
577C22—161no—
578C22—161no—
579C22—161no—
580C22#188014yes—
581C22C2×#38611yescharacteristic
582C22C2×#46011yescharacteristic
583C22C2×#46411yescharacteristic
584C22#305511yescharacteristic
585C22#305611yescharacteristic
586C22#306811yescharacteristic
587C22#306911yescharacteristic
588C4—21no—
589C4—21no—
590C4—22no—
591C4—22no—
592C4—24no—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—82no—
597C4—162no—
598C4#188014yes—
599C4C2×#41711yescharacteristic
600C4#289611yescharacteristic
601C4C2×C4×SD1611yescharacteristic
602C4C2×#44011yescharacteristic
603C4C2×#44811yescharacteristic
604C4C2×#44911yescharacteristic
605C4#305511yescharacteristic
606C4#305611yescharacteristic
607C2—24no—
608C2—162no—
609C2#1917011yescharacteristic
610C2#1917111yescharacteristic
611C2#1917611yescharacteristic
612C2#1917711yescharacteristic
613C2C2×#188011yescharacteristic
614C2#2097111yescharacteristic
615C2#2100411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×Q8×M16C2semidirect
4. C2×#1803C2semidirect
6. #9230C2semidirect
7. #9228C2semidirect
8. #9222C2semidirect
9. #9220C2semidirect
10. #8043C2semidirect
11. #7873C2semidirect
15. #3103C4semidirect
93. #475C2×C4semidirect
94. #475C2×C4semidirect
97. #460C2×C4semidirect
98. #460C2×C4semidirect
195. C2×M16SD16semidirect
492. Q8#417semidirect

Maximal subgroups

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

subgroupindex
C2×#31032
C2×Q8×M162
#212002
C2×#18032
C2×#15472
#92302
#92282
#92222
#92202
#80432
#78732