Finite Group Databasebeta

← #403450

Subgroups — #403450

Order 512.

1374 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic, complemented
2C2×#2023C211yesmaximal, characteristic
3C2×#1926C211yesmaximal, characteristic
4#15148C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11346C211yesmaximal, characteristic, complemented
7#11327C212yesmaximal, complemented
8#11066C212yesmaximal
9#10931C212yesmaximal
10#10884C212yesmaximal, complemented
11#10807C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C2×#399—21no—
18#2369—24no—
19#2369—24no—
20C2×#391—22no—
21C2×#390C2211yescharacteristic
22C2×#390C2212yes—
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2023—24no—
31#1926—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1439—22no—
43#1439—21no—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×D4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8D414yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—21no—
68C22×C42—21no—
69C22×C42—22no—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72C2×#106—22no—
73C2×#106—42no—
74C2×#102—22no—
75C2×#102—22no—
76C2×#102—22no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—42no—
81C2×#102Q814yescomplemented
82#399—44no—
83#391—216no—
84#390—24no—
85#390—22no—
86#390—22no—
87#390—42no—
88#389—28no—
89#389—24no—
90#389—24no—
91#389—24no—
92#389—42no—
93C4×C4⋊C4—28no—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4D414yes—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95D412yes—
107#340—22no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113#336—22no—
114#336—22no—
115#336—42no—
116#336—42no—
117C25—21no—
118C22×Q8—28no—
119C22×Q8—24no—
120C22×Q8—81no—
121C22×Q8—82no—
122C22×Q8—81no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4C2×Q812yes—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8C2×Q812yes—
160C4⋊Q8—28no—
161C4⋊Q8—28no—
162C4⋊Q8—216no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4SD1614yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D414yes—
197C2×C4⋊C4C2×Q812yes—
198C2×C4⋊C4#4114yes—
199C2×C4⋊C4#4112yes—
200C2×C4⋊C4#4114yes—
201C2×C4⋊C4#4114yes—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—82no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—28no—
223C2×C42—44no—
224C2×C42—82no—
225C2×C42—82no—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228#106—44no—
229#102—28no—
230#102—44no—
231#102—48no—
232#95—28no—
233#95—28no—
234#95—24no—
235#95—24no—
236#95—44no—
237#95—42no—
238#95—81no—
239C24—22no—
240C24—41no—
241C24—42no—
242C24—42no—
243C24—41no—
244C24—41no—
245C24—41no—
246C24—42no—
247C24—41no—
248C24—42no—
249C24—41no—
250C24C22×D411yescharacteristic
251C2×Q8—216no—
252C2×Q8—84no—
253C2×Q8—84no—
254C2×Q8—88no—
255C2×Q8—88no—
256C2×D4—22no—
257C2×D4—24no—
258C2×D4—22no—
259C2×D4—24no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—42no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4#12012yes—
323C22×C4#12012yes—
324C22×C4C2×SD1614yes—
325C22×C4C2×SD1612yes—
326C22×C4#13712yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×Q811yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4112yes—
332C22×C4C2×#4112yes—
333C22×C4C2×#4112yes—
334C22×C4H(5,𝔽2)12yes—
335C22×C42−1+411yescharacteristic
336C22×C42−1+411yescharacteristic
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C4⋊C4—216no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—28no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—44no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—88no—
374C4⋊C4—88no—
375C4⋊C4C22⋊Q814yes—
376C4⋊C4#12318yes—
377C22⋊C4—88no—
378C22⋊C4—84no—
379C22⋊C4—88no—
380C22⋊C4—84no—
381C42—24no—
382C42—22no—
383C42—22no—
384C42—48no—
385C42—42no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—88no—
390C23—42no—
391C23—42no—
392C23—44no—
393C23—42no—
394C23—44no—
395C23—82no—
396C23—82no—
397C23—81no—
398C23—81no—
399C23—81no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—82no—
404C23—81no—
405C23—82no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23C2×#12011yescharacteristic
414C23#52512yes—
415C23#54612yes—
416C23#54811yescharacteristic
417C23D4×Q811yescharacteristic
418C23C22×SD1611yescharacteristic
419C23C2×#13711yescharacteristic
420C23#57512yes—
421C23#57612yes—
422C23#57611yescharacteristic
423C23#57711yescharacteristic
424Q8—816no—
425D4—42no—
426D4—42no—
427D4—44no—
428D4—81no—
429D4—81no—
430D4—82no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—28no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4#44814yes—
484C2×C4C2×C22⋊Q811yescharacteristic
485C2×C4C2×#12312yes—
486C2×C4#52511yescharacteristic
487C2×C4#52511yescharacteristic
488C2×C4#52912yes—
489C2×C4#53211yescharacteristic
490C2×C4#53312yes—
491C2×C4#53511yescharacteristic
492C2×C4#53611yescharacteristic
493C2×C4#54212yes—
494C2×C4#54212yes—
495C2×C4#54811yescharacteristic
496C2×C4D4×Q811yescharacteristic
497C2×C4#55012yes—
498C2×C4#55212yes—
499C2×C4#55412yes—
500C2×C4#55612yes—
501C2×C4#55912yes—
502C8—42no—
503C8—81no—
504C8—82no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—84no—
509C22—84no—
510C22—162no—
511C22—162no—
512C22—161no—
513C22—161no—
514C22—161no—
515C22—161no—
516C22#240911yescharacteristic
517C22#243111yescharacteristic
518C22#245812yes—
519C22#246512yes—
520C22#284911yescharacteristic
521C22C2×#44811yescharacteristic
522C22#300511yescharacteristic
523C22#300811yescharacteristic
524C22#301211yescharacteristic
525C22#310312yes—
526C22#311712yes—
527C22#311812yes—
528C22#312212yes—
529C22#329512yes—
530C22#331912yes—
531C22#332112yes—
532C22#333312yes—
533C22#336211yescharacteristic
534C22#336911yescharacteristic
535C22#337012yes—
536C22Q8×SD1611yescharacteristic
537C22#338511yescharacteristic
538C22#338912yes—
539C4—24no—
540C4—44no—
541C4—44no—
542C4—42no—
543C4—48no—
544C4—84no—
545C4—82no—
546C4—82no—
547C4—161no—
548C4#239611yescharacteristic
549C4#242312yes—
550C4#244812yes—
551C4#258011yescharacteristic
552C4#260411yescharacteristic
553C4#263211yescharacteristic
554C2—162no—
555C2#1487511yescharacteristic
556C2#2432911yescharacteristic
557C2#2433911yescharacteristic
558C2#2556911yescharacteristic
559C2#2557811yescharacteristic
560C2#2669212yes—
561C2#2671012yes—
562C2#2700212yes—
563C2#2701812yes—
564C2#3300511yescharacteristic
565C2#3301411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2369C2semidirect
5. #11384C2semidirect
6. #11346C2semidirect
7. #11327C2semidirect
10. #10884C2semidirect
11. #10807C2semidirect
81. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
C2×#20232
C2×#19262
#151482
#113842
#113462
#113272
#110662
#109312
#108842
#108072