Finite Group Databasebeta

← #418661

Subgroups — #418661

Order 512.

1562 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
1C22×#501C211yesmaximal, characteristic, complemented
2C2×#2442C211yesmaximal, characteristic
3C2×#2058C211yesmaximal, characteristic, complemented
4C2×#1853C212yesmaximal
5#15125C211yesmaximal, characteristic
6#11385C211yesmaximal, characteristic, complemented
7#11325C212yesmaximal, complemented
8#11220C212yesmaximal, complemented
9#11096C212yesmaximal
10#10922C211yesmaximal, characteristic
11#10881C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#501—28no—
16C2×#501—24no—
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×#106C2212yes—
20C22×C22⋊C8C2212yes—
21C22×#97C2211yescharacteristic
22#2442—24no—
23#2442—24no—
24C2×#400—21no—
25C2×#398—21no—
26C2×#393—21no—
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#385—21no—
30C2×#382C2211yescharacteristic
31C2×#380—21no—
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×C4×C22⋊C4C2211yescharacteristic
35#2058—24no—
36#1853—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44#1446—21no—
45#1442—21no—
46#1439—21no—
47C4⋊#95C2212yes—
48C4⋊#95C2211yescharacteristic
49C23×Q8—21no—
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×#125—24no—
59C2×#125—22no—
60C2×#125—42no—
61C2×#125D414yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—42no—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D411yescharacteristic
81C22×C22⋊C4D411yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#501—232no—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—24no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—42no—
93C2×#106D418yes—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8Q818yescomplemented
97C2×#97—24no—
98C2×#97—22no—
99#400—28no—
100#398—28no—
101#393—44no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389D414yes—
106#385—28no—
107#382—22no—
108#382—22no—
109#380—28no—
110#380—22no—
111#380—22no—
112#380—24no—
113#380—42no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95D411yescharacteristic
127C2×#95D411yescharacteristic
128#340—24no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—42no—
133#336—24no—
134#336—22no—
135C25C2×D411yescharacteristic
136C22×Q8—44no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—82no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—44no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—82no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×Q814yes—
165C23×C4C2411yescharacteristic
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8C2×D414yes—
177C22×C8C2×Q814yes—
178C22×C8C2×Q814yes—
179C22×C8#4114yes—
180C4⋊Q8—28no—
181C4⋊Q8—48no—
182C4⋊Q8—44no—
183#125—28no—
184#125—44no—
185#125—48no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D412yes—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—82no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249#107—216no—
250#106—216no—
251#106—48no—
252C22⋊C8—216no—
253#97—28no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—22no—
260#95—22no—
261#95—44no—
262#95—42no—
263#95—42no—
264#95—44no—
265#95—82no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24—21no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24C4⋊Q812yes—
275C24AGL(1,ℤ/8)12yes—
276C24#13712yes—
277C24C22×D411yescharacteristic
278C2×Q8—48no—
279C2×Q8—84no—
280C2×Q8—84no—
281C2×Q8—88no—
282C2×Q8—88no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—48no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—48no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—84no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4#12012yes—
361C22×C4#12012yes—
362C22×C4C4⋊Q812yes—
363C22×C4C22×D412yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×Q812yes—
367C22×C4C2×#4112yes—
368C22×C4H(5,𝔽2)12yes—
369C22×C42−1+412yes—
370C22×C42−1+412yes—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8C22⋊Q818yes—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—84no—
409C4⋊C4—84no—
410C4⋊C4#12014yes—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—48no—
414C22⋊C4—88no—
415C42—22no—
416C42—22no—
417C42—24no—
418C42—41no—
419C42—41no—
420C42—42no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—22no—
425C23—22no—
426C23—24no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—24no—
432C23—24no—
433C23—42no—
434C23—42no—
435C23—41no—
436C23—42no—
437C23—41no—
438C23—42no—
439C23—41no—
440C23—41no—
441C23—42no—
442C23—42no—
443C23#50114yescomplemented
444C23C2×#12011yescharacteristic
445C23C2×C4⋊Q811yescharacteristic
446C23#53611yescharacteristic
447C23D4212yes—
448C23#54712yes—
449C23C2×AGL(1,ℤ/8)11yescharacteristic
450C23C2×#13711yescharacteristic
451C23#57511yescharacteristic
452C23#57511yescharacteristic
453C23#57712yes—
454C23#57712yes—
455Q8—816no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—816no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4C2×#12011yescharacteristic
521C2×C4C2×C22⋊Q812yes—
522C2×C4#53211yescharacteristic
523C2×C4#53411yescharacteristic
524C2×C4#53612yes—
525C2×C4#53611yescharacteristic
526C2×C4#54212yes—
527C2×C4#54412yes—
528C2×C4D4212yes—
529C2×C4#54812yes—
530C2×C4#54812yes—
531C2×C4D4×Q812yes—
532C2×C4D4×Q812yes—
533C2×C4D4×Q812yes—
534C2×C4#55412yes—
535C2×C4#56312yes—
536C8—28no—
537C8—44no—
538C22—28no—
539C22—24no—
540C22—42no—
541C22—42no—
542C22—44no—
543C22—44no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—44no—
549C22#240311yescharacteristic
550C22#243211yescharacteristic
551C22#243411yescharacteristic
552C22#244012yes—
553C22#267412yes—
554C22#300611yescharacteristic
555C22#300711yescharacteristic
556C22#300911yescharacteristic
557C22#301011yescharacteristic
558C22C2×#50111yescharacteristic
559C22#316312yes—
560C22#316611yescharacteristic
561C22#323411yescharacteristic
562C22#323611yescharacteristic
563C22#324412yes—
564C22#326512yes—
565C22#326612yes—
566C22#327512yes—
567C22#327512yes—
568C22#334412yes—
569C22#334512yes—
570C22#334812yes—
571C22#335012yes—
572C4—42no—
573C4—44no—
574C4—42no—
575C4—42no—
576C4—82no—
577C4—82no—
578C4—84no—
579C4—81no—
580C4—82no—
581C4—81no—
582C4—162no—
583C4#240912yes—
584C4#245212yes—
585C4#267712yes—
586C4#272912yes—
587C2—44no—
588C2#1526511yescharacteristic
589C2#2386211yescharacteristic
590C2#2386611yescharacteristic
591C2#2571411yescharacteristic
592C2#2571511yescharacteristic
593C2#2581911yescharacteristic
594C2#2582011yescharacteristic
595C2#2606812yes—
596C2#2607612yes—
597C2#3079112yes—
598C2#3079312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#501C2semidirect
3. C2×#2058C2semidirect
6. #11385C2semidirect
7. #11325C2semidirect
8. #11220C2semidirect
11. #10881C2semidirect
96. C2×C22⋊C8Q8semidirect
443. C23#501semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#24422
C2×#20582
C2×#18532
#151252
#113852
#113252
#112202
#110962
#109222
#108812