Finite Group Databasebeta

← #196243

Subgroups — #196243

Order 512.

1336 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic, complemented
2#34070C211yesmaximal, characteristic
3#17437C212yesmaximal, complemented
4#17349C211yesmaximal, characteristic, complemented
5#9106C212yesmaximal, complemented
6#9105C212yesmaximal, complemented
7#9104C212yesmaximal, complemented
8#9103C212yesmaximal, complemented
9C2×#1274C212yesmaximal, complemented
10C2×#532C411yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C2×C4×#41C2211yescharacteristic
13#2989—24no—
14#2983C2212yes—
15#2975C2212yes—
16#2973—22no—
17#2965—24no—
18#2920—22no—
19C2×#431C2212yes—
20C2×#431C2212yes—
21#2328C411yescharacteristic
22#2328C2211yescharacteristic
23#2327C411yescharacteristic
24#2327C2211yescharacteristic
25#2326C411yescharacteristic
26#2326C411yescharacteristic
27#2326C2211yescharacteristic
28#2326C2211yescharacteristic
29#2325—22no—
30#2325—22no—
31#2297C411yescharacteristic
32#2297C411yescharacteristic
33#2290—22no—
34#2290—22no—
35#2285C411yescharacteristic
36C2×#376C2211yescharacteristic
37#1479C2214yes—
38#1478C2214yes—
39#1477—24no—
40#1476—24no—
41#1461C2212yes—
42#1460C2212yes—
43#1459—22no—
44#1458—22no—
45#1274C2214yes—
46#1274C2214yes—
47C22×#41C2×C411yescharacteristic
48C22×#41C2×C411yescharacteristic
49#532—24no—
50#532—24no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#127C2×C411yescharacteristic
53C2×#124C2×C411yescharacteristic
54C2×#124C2×C411yescharacteristic
55C2×#121—22no—
56C4×#41—22no—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41—22no—
61C4×#41—22no—
62C4×#41—22no—
63C4×#41—22no—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×C4×Q8C2311yescharacteristic
68C2×C4×D4—22no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4D412yes—
74C2×C4×D4D412yes—
75C2×C4×D4C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×#117—21no—
78C2×#117C2×C411yescharacteristic
79C2×#117C2×C411yescharacteristic
80C2×#117D412yes—
81C22×C42—21no—
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42D412yes—
85C22×C42C2311yescharacteristic
86#446—28no—
87#436—28no—
88#435—216no—
89#432—24no—
90#432—24no—
91#431—24no—
92#431—22no—
93#431—22no—
94#431—22no—
95#431—22no—
96#431—22no—
97#431D414yescomplemented
98#431C2314yes—
99#431C2314yes—
100#423—24no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103#407—24no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C4×M16—24no—
107C2×C4×C8—22no—
108C2×C4×C8D414yescomplemented
109#391C2×C411yescharacteristic
110#391C2×C411yescharacteristic
111#391C2×C411yescharacteristic
112#391C2×C411yescharacteristic
113#390C2×C411yescharacteristic
114#390C2×C411yescharacteristic
115#390C2×C411yescharacteristic
116#390C2×C411yescharacteristic
117#388—22no—
118#388—22no—
119#388—22no—
120#388—22no—
121#387—22no—
122#387—22no—
123#387—22no—
124#387—22no—
125#386—22no—
126#386—22no—
127#386—22no—
128#386—22no—
129#386—22no—
130#386—22no—
131#385—22no—
132#385—22no—
133#385—22no—
134#385—22no—
135#384C2×C411yescharacteristic
136#384C2×C411yescharacteristic
137#382C2×C411yescharacteristic
138#382C2×C411yescharacteristic
139#380—22no—
140C4×C4⋊C4—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143#376—21no—
144#376—21no—
145#376C2×C411yescharacteristic
146#376C2×C411yescharacteristic
147#376C2311yescharacteristic
148#376C2311yescharacteristic
149C2×#95—21no—
150C2×#95—21no—
151Q8⋊C8—28no—
152D4⋊C8—28no—
153#324—24no—
154#323—24no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—22no—
160C2×#41—24no—
161C2×#41—22no—
162C2×#41—22no—
163C2×#41—24no—
164C2×#41M1614yescomplemented
165C22×Q8C22×C411yescharacteristic
166C22×Q8C22×C411yescharacteristic
167C22×D4—21no—
168C22×D4—22no—
169C22×D4C22⋊C412yes—
170C22×D4C22×C411yescharacteristic
171C22×D4C22×C411yescharacteristic
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4C22⋊C412yes—
178C23×C4C22⋊C412yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C2×M16—28no—
184C22×C8—24no—
185C22×C8—42no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188#127—22no—
189#127—22no—
190#124—22no—
191#124—22no—
192#124—22no—
193#124—22no—
194#121—216no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×D4—216no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—24no—
207C4×D4—24no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—42no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—42no—
220#117—22no—
221#117—22no—
222#117—24no—
223#117—24no—
224#117—22no—
225#117—22no—
226#117—41no—
227#117—41no—
228#117—41no—
229#117—41no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4C22⋊C412yes—
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C2×D412yes—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4C22⋊C412yes—
251C2×C22⋊C4C22⋊C412yes—
252C2×C22⋊C4C2×D412yes—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42C22⋊C412yes—
267C2×C42C22⋊C412yes—
268C2×C42M1614yescomplemented
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42#4112yes—
280C2×C42#4112yes—
281C2×C42C2411yescharacteristic
282C4⋊C8—28no—
283C4⋊C8—24no—
284C4⋊C8—24no—
285C4⋊C8—24no—
286C4⋊C8—42no—
287C22⋊C8—28no—
288C22⋊C8—24no—
289C22⋊C8—24no—
290C22⋊C8—24no—
291C22⋊C8—42no—
292#97—28no—
293C4×C8—24no—
294C4×C8—24no—
295C4×C8—42no—
296C4×C8C2×D418yes—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311C24—21no—
312C24C2×C22⋊C412yes—
313C24C2×C22⋊C411yescharacteristic
314#41—28no—
315#41—28no—
316#41—44no—
317#41—44no—
318#41—44no—
319#41—44no—
320#41—44no—
321#41—44no—
322C2×Q8—22no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8C2×M1612yes—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—28no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—82no—
345C2×D4—82no—
346C2×D4C2×M1612yes—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—81no—
371C22×C4—81no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C4×D412yes—
387C22×C4C4×D412yes—
388C22×C4#12012yes—
389C22×C4#12112yes—
390C22×C4#12112yes—
391C22×C4#12312yes—
392C22×C4C2×M1612yes—
393C22×C4C2×M1612yes—
394C22×C4#13114yes—
395C22×C4C23×C411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398M16—88no—
399C2×C8—28no—
400C2×C8—44no—
401C2×C8—84no—
402C2×C8—84no—
403C4⋊C4—28no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C22⋊C4—28no—
417C22⋊C4—28no—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—42no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C22⋊C4—82no—
429C22⋊C4—82no—
430C42—22no—
431C42—22no—
432C42—21no—
433C42—21no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—24no—
439C42—22no—
440C42—22no—
441C42—42no—
442C42—41no—
443C42—42no—
444C42—41no—
445C42—42no—
446C42C4×D412yes—
447C42C4×D412yes—
448C42#12011yescharacteristic
449C42#12011yescharacteristic
450C42#12112yes—
451C42#12112yes—
452C42#12311yescharacteristic
453C42#12311yescharacteristic
454C42C2×M1612yes—
455C42C2×M1612yes—
456C42C22×D412yes—
457C42C2×#4112yes—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23#38612yes—
471C23#40712yes—
472C23#40812yes—
473C23C22×C22⋊C411yescharacteristic
474Q8—24no—
475Q8—42no—
476Q8—42no—
477Q8—42no—
478D4—24no—
479D4—48no—
480D4—48no—
481D4—42no—
482D4—42no—
483D4—42no—
484D4—88no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—84no—
531C2×C4#38611yescharacteristic
532C2×C4#38611yescharacteristic
533C2×C4#38612yes—
534C2×C4#38611yescharacteristic
535C2×C4#38611yescharacteristic
536C2×C4#40712yes—
537C2×C4#40812yes—
538C2×C4#43514yes—
539C2×C4#43514yes—
540C2×C4#43614yes—
541C2×C4#43614yes—
542C2×C4C2×C4×D412yes—
543C2×C4C2×#12011yescharacteristic
544C2×C4C2×#12112yes—
545C2×C4C2×#12311yescharacteristic
546C2×C4C22×M1611yescharacteristic
547C2×C4C22×M1611yescharacteristic
548C2×C4C2×#13112yes—
549C8—48no—
550C8—84no—
551C22—21no—
552C22—22no—
553C22—21no—
554C22—22no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—44no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—82no—
564C22—82no—
565C22—81no—
566C22—81no—
567C22#187014yes—
568C22C2×#38611yescharacteristic
569C22C2×#40711yescharacteristic
570C22C2×#40811yescharacteristic
571C4—22no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—22no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4#187012yes—
588C4#187012yes—
589C4C2×#43512yes—
590C4C2×#43612yes—
591C2—21no—
592C2—21no—
593C2—24no—
594C2—82no—
595C2—82no—
596C2C2×#187011yescharacteristic
597C2#2100911yescharacteristic
598C2#2101211yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC2×#532C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60277C2semidirect
3. #17437C2semidirect
4. #17349C2semidirect
5. #9106C2semidirect
6. #9105C2semidirect
7. #9104C2semidirect
8. #9103C2semidirect
9. C2×#1274C2semidirect
97. #431D4semidirect
108. C2×C4×C8D4semidirect
164. C2×#41M16semidirect
268. C2×C42M16semidirect

Maximal subgroups

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

subgroupindex
#602772
#340702
#174372
#173492
#91062
#91052
#91042
#91032
C2×#12742