Finite Group Databasebeta

← #222902

Subgroups — #222902

Order 512.

1964 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2339C211yesmaximal, characteristic
2#22401C212yesmaximal, complemented
3#9878C212yesmaximal, complemented
4#9083C214yesmaximal, complemented
5#8698C212yesmaximal, complemented
6#8557C211yesmaximal, characteristic, complemented
7C4×#323C211yesmaximal, characteristic, complemented
8#7909C212yesmaximal, complemented
9C23×#41C411yescharacteristic
10C22×C4×D4C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#436—22no—
13C2×#407C2214yes—
14#2339—22no—
15#2339—24no—
16#2339—22no—
17C2×#391C411yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#386C412yes—
20C2×#386C2212yes—
21C2×#381C412yes—
22C2×#380C411yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24C22×#95C2212yes—
25#1954—22no—
26#1870—24no—
27#1797C2212yes—
28#1779C2214yes—
29C4×C22⋊C8C2212yes—
30C2×#323C2212yes—
31C2×#323C2212yes—
32#1280C2212yes—
33#1280C2214yes—
34#1280C2214yes—
35#1280C2211yescharacteristic
36#1280C2211yescharacteristic
37C22×#41—24no—
38C22×#41—22no—
39C22×#41—21no—
40C22×#41—21no—
41C22×#41—24no—
42C22×#41—44no—
43C22×#41—44no—
44C23×Q8C2×C411yescharacteristic
45C23×D4C2×C411yescharacteristic
46C23×D4C2×C412yes—
47C24×C4C2×C412yes—
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C412yes—
50C24×C4C2312yes—
51C22×M16—24no—
52C2×C4×D4—22no—
53C2×C4×D4—21no—
54C2×C4×D4—21no—
55C2×C4×D4—44no—
56C2×C4×D4—44no—
57C2×C4×D4D412yes—
58C2×C4×D4D414yes—
59C2×C4×D4D412yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2311yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2311yescharacteristic
69#436—48no—
70C2×C4⋊C8—22no—
71#407—28no—
72#407—44no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8C2314yes—
77C2×C22⋊C8C2312yes—
78C2×C22⋊C8C2312yes—
79C2×C4×C8—22no—
80#391—22no—
81#391—22no—
82#391—22no—
83#391—22no—
84#386—24no—
85#386—24no—
86#386—24no—
87#386—24no—
88#381—24no—
89#381—24no—
90#380—22no—
91#380—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C2×#95—24no—
95C2×#95—24no—
96C2×#95C2×C412yes—
97C2×#95C2×C412yes—
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C412yes—
100C2×#95C2×C412yes—
101C2×#95C2311yescharacteristic
102C2×#95C2312yes—
103#336—22no—
104#336—22no—
105#323—24no—
106#323—42no—
107C25C22×C412yes—
108C25C22×C411yescharacteristic
109C2×#41—216no—
110C2×#41—28no—
111C2×#41—28no—
112C2×#41—432no—
113C2×#41—48no—
114C2×#41—48no—
115C2×#41—432no—
116C2×#41—44no—
117C2×#41—44no—
118C2×#41—44no—
119C2×#41—44no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8M1614yescomplemented
126C22×D4—24no—
127C22×D4—24no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—24no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—24no—
134C22×D4—44no—
135C22×D4—44no—
136C22×D4C22⋊C412yes—
137C22×D4C22⋊C411yescharacteristic
138C22×D4C22⋊C411yescharacteristic
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—44no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—82no—
160C23×C4—82no—
161C23×C4M1614yescomplemented
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C412yes—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C412yes—
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C412yes—
169C23×C4C22×C411yescharacteristic
170C23×C4C2×D414yes—
171C23×C4C2×D414yes—
172C23×C4C2411yescharacteristic
173C2×M16—48no—
174C2×M16—48no—
175C2×M16—84no—
176C22×C8—22no—
177C22×C8—24no—
178C22×C8—22no—
179C4×D4—28no—
180C4×D4—44no—
181C4×D4—48no—
182C4×D4—48no—
183C4×D4—44no—
184C4×D4—44no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—82no—
204C2×C22⋊C4C22⋊C414yes—
205C2×C22⋊C4C22⋊C414yes—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C4⋊C8—44no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—22no—
218C22⋊C8—48no—
219C22⋊C8—48no—
220C22⋊C8—84no—
221C22⋊C8#4114yescomplemented
222C4×C8—44no—
223#95—24no—
224#95—24no—
225#95—22no—
226#95—24no—
227#95—24no—
228#95—22no—
229#95—24no—
230C24—22no—
231C24—22no—
232C24—24no—
233C24—24no—
234C24—21no—
235C24—22no—
236C24—22no—
237C24—21no—
238C24—22no—
239C24—22no—
240C24—21no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—42no—
245C24—42no—
246C24—41no—
247C24—41no—
248C24—41no—
249C24—41no—
250C24C2×C22⋊C412yes—
251C24C2×C22⋊C412yes—
252C24C2×M1612yes—
253C24C23×C411yescharacteristic
254#41—464no—
255#41—432no—
256#41—432no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—48no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—48no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×D4—216no—
266C2×D4—216no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—28no—
270C2×D4—28no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—416no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—416no—
280C2×D4—44no—
281C2×D4—44no—
282C2×D4#9914yes—
283C2×D4#10014yes—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—28no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—44no—
313C22×C4—41no—
314C22×C4—44no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—416no—
327C22×C4—416no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—88no—
347C22×C4—84no—
348C22×C4—84no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—88no—
352C22×C4—88no—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C414yes—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×M1612yes—
358C22×C4C2×M1612yes—
359C22×C4C2×M1612yes—
360C22×C4C2×M1614yes—
361C22×C4#13112yes—
362C22×C4C22×D412yes—
363C22×C4C2×#4112yes—
364C22×C4H(5,𝔽2)11yescharacteristic
365C22×C42−1+411yescharacteristic
366M16—816no—
367C2×C8—24no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—44no—
372C2×C8—42no—
373C2×C8—44no—
374C2×C8—44no—
375C2×C8—42no—
376C4⋊C4—24no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C22⋊C4—28no—
381C22⋊C4—44no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—84no—
388C42—24no—
389C42—42no—
390C42—42no—
391C42—42no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—28no—
397C23—28no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—81no—
446C23—81no—
447C23C2×#9914yes—
448C23C2×#10014yes—
449C23C22×C22⋊C411yescharacteristic
450C23C4×#4111yescharacteristic
451C23#52011yescharacteristic
452C23C22×M1611yescharacteristic
453C23C22×M1611yescharacteristic
454C23C2×#13111yescharacteristic
455C23#56811yescharacteristic
456Q8—48no—
457Q8—48no—
458D4—216no—
459D4—28no—
460D4—28no—
461D4—416no—
462D4—416no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—416no—
487C2×C4—416no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—48no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—84no—
506C2×C4—84no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—88no—
512C2×C4—88no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4#40712yes—
518C2×C4#40712yes—
519C2×C4C2×#9912yes—
520C2×C4C2×#10012yes—
521C2×C4#54612yes—
522C2×C4#54712yes—
523C8—44no—
524C8—82no—
525C8—84no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—24no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—48no—
545C22—48no—
546C22—48no—
547C22—48no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22#233911yescharacteristic
568C22C2×#40711yescharacteristic
569C22#288011yescharacteristic
570C22C22×#9911yescharacteristic
571C22#288311yescharacteristic
572C22C22×#10011yescharacteristic
573C22#288911yescharacteristic
574C22#296511yescharacteristic
575C22#297111yescharacteristic
576C22#298911yescharacteristic
577C22#299111yescharacteristic
578C4—22no—
579C4—22no—
580C4—22no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—44no—
585C4—44no—
586C4—82no—
587C4—81no—
588C4#146214yes—
589C2—21no—
590C2—21no—
591C2—21no—
592C2—21no—
593C2—44no—
594C2—44no—
595C2—82no—
596C2C2×#146211yescharacteristic
597C2#1737011yescharacteristic
598C2#1741611yescharacteristic
599C2#1742111yescharacteristic
600C2#2247311yescharacteristic
601C2#2247611yescharacteristic
602C2#2247911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22401C2semidirect
3. #9878C2semidirect
4. #9083C2semidirect
5. #8698C2semidirect
6. #8557C2semidirect
7. C4×#323C2semidirect
8. #7909C2semidirect
125. C22×Q8M16semidirect
161. C23×C4M16semidirect
221. C22⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
C2×#23392
#224012
#98782
#90832
#86982
#85572
C4×#3232
#79092