Finite Group Databasebeta

← #199062

Subgroups — #199062

Order 512.

1314 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60941C211yesmaximal, characteristic
2#34208C211yesmaximal, characteristic
3#21329C211yesmaximal, characteristic, complemented
4#17704C211yesmaximal, characteristic
5C2×#1518C211yesmaximal, characteristic, complemented
6#9296C214yesmaximal, complemented
7#9295C214yesmaximal, complemented
8C2×#1286C211yesmaximal, characteristic, complemented
9#7854C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C2×#532C2211yescharacteristic
12C2×#520—21no—
13#3246—22no—
14#3229C2212yes—
15#3227C412yescomplemented
16#3208C2212yes—
17#3206C412yescomplemented
18#3187—22no—
19#3167C414yescomplemented
20#3103—22no—
21#3101—22no—
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#423C2211yescharacteristic
25#2364—22no—
26#2364—22no—
27#2352—22no—
28#2352—22no—
29#2328C2212yes—
30#2327C2212yes—
31#2311—21no—
32C4×#124—22no—
33C2×#382C2211yescharacteristic
34C4×#117C2211yescharacteristic
35#1995C2214yes—
36#1994—22no—
37#1733—21no—
38#1733—21no—
39#1547—22no—
40#1546C2212yes—
41#1545C2212yes—
42#1540—22no—
43#1518—22no—
44#1518C2214yes—
45C2×#328—22no—
46#1286C2214yes—
47#1274C2214yes—
48C22×#41—21no—
49C22×M16—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533C2×C412yescomplemented
57#533C2312yes—
58#532—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532C2×C412yes—
63#532C2×C412yes—
64#532D412yes—
65#532D412yes—
66C2×C4⋊Q8C2×C411yescharacteristic, complemented
67C2×C4⋊Q8C2×C411yescharacteristic, complemented
68C2×C4⋊Q8C2311yescharacteristic
69C2×#127C2311yescharacteristic
70C2×#125—21no—
71C2×#124—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×#121—21no—
75C2×#121D412yes—
76#520—44no—
77#520—44no—
78C2×C4×Q8—22no—
79C2×C4×Q8—21no—
80C2×C4×Q8—21no—
81C2×C4×D4—24no—
82C2×C4×D4—21no—
83C2×C4×D4—22no—
84C2×C4×D4D412yes—
85C2×#117—22no—
86C2×#117—21no—
87C2×#117D411yescharacteristic
88C2×#117D411yescharacteristic
89C22×C4⋊C4—22no—
90C22×C4⋊C4—21no—
91C22×C42C2311yescharacteristic
92#498C2×C414yescomplemented
93#492C2×C414yescomplemented
94#487C2×C414yescomplemented
95#486C2×C414yescomplemented
96#483—24no—
97#481—24no—
98#476—24no—
99#475—24no—
100#465—28no—
101#461—24no—
102#460—24no—
103#450—24no—
104#448—24no—
105C4×SD16—28no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431C2×C412yescomplemented
112#431C2312yes—
113#431C2312yes—
114#423—21no—
115#423—21no—
116#423—21no—
117#423C2312yes—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8C2×C412yescomplemented
126#391—22no—
127#390—22no—
128#388—24no—
129#388—24no—
130#387—24no—
131#387—24no—
132#386—24no—
133#386—24no—
134#386—24no—
135#382—21no—
136#382—21no—
137#382D412yes—
138#382C2312yes—
139#380—22no—
140#380—22no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C22⋊C4—24no—
144C4×C22⋊C4—24no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147#376C2312yes—
148C2×#95—21no—
149C2×#95D412yes—
150C43D412yes—
151#339—24no—
152#339—24no—
153#332—24no—
154#331—24no—
155#328—28no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—22no—
160C2×#41—42no—
161C2×#41—42no—
162C2×#41—42no—
163C2×#41—42no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×D4—22no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4C2×D412yes—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C2×SD16—28no—
178C2×M16—22no—
179C2×M16—41no—
180C2×M16—42no—
181C2×M16—42no—
182C22×C8—22no—
183C22×C8—41no—
184C22×C8—41no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8C22×C412yes—
194C4⋊Q8C22×C412yes—
195C4⋊Q8C22×C412yes—
196C4⋊Q8C2×D412yes—
197#127—21no—
198#127—21no—
199#127C22×C412yes—
200#127C2×D412yes—
201#125—22no—
202#125—22no—
203#125—41no—
204#125—41no—
205#124—22no—
206#124—22no—
207#124—22no—
208#124—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—41no—
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216C22⋊Q8—41no—
217#121—22no—
218#121—22no—
219#121—22no—
220#121—22no—
221#121—22no—
222#121—22no—
223#121C22⋊C414yes—
224#121C22⋊C414yes—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—44no—
230C4×Q8—41no—
231C4×Q8—41no—
232C4×Q8—41no—
233C4×Q8—41no—
234C4×Q8—44no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—44no—
242C4×D4—416no—
243C4×D4—44no—
244C4×D4C22⋊C414yes—
245C4×D4C22⋊C414yes—
246#117—22no—
247#117—22no—
248#117—22no—
249#117—22no—
250#117—22no—
251#117—48no—
252#117—41no—
253#117—41no—
254#117SD1614yescomplemented
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4C2×D412yes—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—41no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—41no—
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42#4111yescharacteristic
293C2×C42#4111yescharacteristic
294C2×C42#4111yescharacteristic
295C2×C42#4111yescharacteristic
296C2×C42C2411yescharacteristic
297#106—28no—
298C4⋊C8—22no—
299C4⋊C8—22no—
300C4⋊C8—22no—
301C4⋊C8—22no—
302C4⋊C8—22no—
303C4⋊C8—41no—
304C4⋊C8—41no—
305C4⋊C8—41no—
306#103—28no—
307#102—28no—
308C22⋊C8—22no—
309C22⋊C8—22no—
310C22⋊C8—22no—
311C22⋊C8—41no—
312C4×C8—22no—
313C4×C8—22no—
314C4×C8—41no—
315C4×C8C22×C414yes—
316#95—24no—
317#95—24no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95C2×D414yes—
324C24—22no—
325C24#12011yescharacteristic
326#41—44no—
327#41—44no—
328#41—44no—
329#41—44no—
330#41—44no—
331#41—44no—
332#41—44no—
333#41—44no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—82no—
345C2×Q8—82no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—82no—
360C2×D4—82no—
361C2×D4C2×C22⋊C414yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—48no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—44no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4—84no—
377C22×C4—84no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C4×D411yescharacteristic
383C22×C4C4×D411yescharacteristic
384C22×C4C4×D411yescharacteristic
385C22×C4C4×D411yescharacteristic
386C22×C4#12011yescharacteristic
387C22×C4#12011yescharacteristic
388C22×C4#12011yescharacteristic
389C22×C4#12111yescharacteristic
390C22×C4#12111yescharacteristic
391C22×C4#12111yescharacteristic
392C22×C4#12111yescharacteristic
393C22×C4#12311yescharacteristic
394C22×C4#12311yescharacteristic
395C22×C4C2×SD1612yes—
396C22×C4#13512yes—
397C22×C4AGL(1,ℤ/8)12yes—
398C22×C4#13712yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402SD16—88no—
403M16—42no—
404M16—42no—
405M16—82no—
406C2×C8—24no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—82no—
410C2×C8—82no—
411C2×C8—82no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—48no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4C2×C22⋊C414yes—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—48no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—84no—
444C22⋊C4—84no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—48no—
449C42—42no—
450C42—42no—
451C42—82no—
452C42—82no—
453C42C2×C22⋊C411yescharacteristic
454C42C2×C22⋊C411yescharacteristic
455C42C4×D411yescharacteristic
456C42C4×D411yescharacteristic
457C42C4×D411yescharacteristic
458C42C4×D411yescharacteristic
459C42#12111yescharacteristic
460C42#12111yescharacteristic
461C42#12111yescharacteristic
462C42#12111yescharacteristic
463C42#12311yescharacteristic
464C42#12311yescharacteristic
465C42C2×SD1614yes—
466C42C23×C411yescharacteristic
467C42C22×D411yescharacteristic
468C42C2×#4111yescharacteristic
469C42C2×#4111yescharacteristic
470C23—24no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23#38611yescharacteristic
480C23#38611yescharacteristic
481C23#45012yes—
482C23#45212yes—
483C23C2×#12011yescharacteristic
484Q8—44no—
485Q8—42no—
486Q8—42no—
487Q8—42no—
488Q8—42no—
489Q8—82no—
490Q8—82no—
491Q8—82no—
492D4—44no—
493D4—48no—
494D4—48no—
495D4—42no—
496D4—42no—
497D4—42no—
498D4—42no—
499D4—82no—
500D4—82no—
501D4—82no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—48no—
521C2×C4—48no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—168no—
533C2×C4—164no—
534C2×C4#38611yescharacteristic
535C2×C4#38611yescharacteristic
536C2×C4#38611yescharacteristic
537C2×C4#38611yescharacteristic
538C2×C4#38611yescharacteristic
539C2×C4#38611yescharacteristic
540C2×C4C4×SD1612yes—
541C2×C4C4×SD1612yes—
542C2×C4#44012yes—
543C2×C4#44012yes—
544C2×C4#45012yes—
545C2×C4#45212yes—
546C2×C4#46112yes—
547C2×C4#46112yes—
548C2×C4#46312yes—
549C2×C4#46312yes—
550C2×C4C22×C22⋊C411yescharacteristic
551C2×C4C2×C4×D411yescharacteristic
552C2×C4C2×C4×D411yescharacteristic
553C2×C4C2×#12111yescharacteristic
554C2×C4C2×#12111yescharacteristic
555C2×C4C2×#12311yescharacteristic
556C2×C4C22×SD1611yescharacteristic
557C2×C4C2×#13511yescharacteristic
558C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
559C2×C4C2×#13711yescharacteristic
560C8—44no—
561C8—82no—
562C8—82no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—42no—
570C22—42no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22#187712yes—
579C22#187712yes—
580C22C2×#38611yescharacteristic
581C22C2×#45011yescharacteristic
582C22C2×#45211yescharacteristic
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—44no—
592C4—44no—
593C4—82no—
594C4—84no—
595C4—82no—
596C4—82no—
597C4#187712yes—
598C4#187712yes—
599C4C2×C4×SD1611yescharacteristic
600C4C2×#44011yescharacteristic
601C4C2×#46111yescharacteristic
602C4C2×#46311yescharacteristic
603C2—21no—
604C2—21no—
605C2—22no—
606C2—22no—
607C2—84no—
608C2C2×#187711yescharacteristic
609C2#2099811yescharacteristic
610C2#2101211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21329C2semidirect
5. C2×#1518C2semidirect
6. #9296C2semidirect
7. #9295C2semidirect
8. C2×#1286C2semidirect
9. #7854C2semidirect
15. #3227C4semidirect
17. #3206C4semidirect
19. #3167C4semidirect
56. #533C2×C4semidirect
66. C2×C4⋊Q8C2×C4semidirect
67. C2×C4⋊Q8C2×C4semidirect
92. #498C2×C4semidirect
93. #492C2×C4semidirect
94. #487C2×C4semidirect
95. #486C2×C4semidirect
111. #431C2×C4semidirect
125. C2×C4×C8C2×C4semidirect
254. #117SD16semidirect

Maximal subgroups

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

subgroupindex
#609412
#342082
#213292
#177042
C2×#15182
#92962
#92952
C2×#12862
#78542