Finite Group Databasebeta

← #369383

Subgroups — #369383

Order 512.

1678 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#488C211yesmaximal, characteristic
2C2×#2326C211yesmaximal, characteristic
3C2×#1992C211yesmaximal, characteristic, complemented
4C2×#1959C211yesmaximal, characteristic
5#14283C211yesmaximal, characteristic, complemented
6#11190C212yesmaximal, complemented
7#11189C212yesmaximal
8#10841C211yesmaximal, characteristic, complemented
9#10789C212yesmaximal, complemented
10#10788C212yesmaximal
11#10694C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#488—24no—
17C2×#488—24no—
18C2×#488C418yescomplemented
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×#97C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2326—24no—
26#2326—24no—
27C2×#391C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#388—22no—
30C2×#385—22no—
31C2×#384C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1992—24no—
35#1959—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1441—22no—
43#1441—22no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125C2×C414yescomplemented
57C2×#124—22no—
58C2×#124—22no—
59C2×#124—22no—
60C2×#124C2×C414yes—
61C2×#124D414yes—
62C2×#124D414yes—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—24no—
66C2×C4×Q8—42no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—24no—
70C2×C4×D4—42no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—22no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#488—232no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—42no—
92C2×#103C2×C418yescomplemented
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—42no—
98C2×#102C2×C418yes—
99C2×#97—22no—
100C2×#97—22no—
101C2×#97—22no—
102C2×C4×C8—22no—
103C2×C4×C8—22no—
104C2×C4×C8C2×C414yescomplemented
105#391—22no—
106#391—22no—
107#390—22no—
108#390—22no—
109#388—216no—
110#385—216no—
111#384—22no—
112#384—22no—
113#384—22no—
114#384D414yes—
115#376—22no—
116#376—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95D412yes—
123C2×#95D412yes—
124#340—24no—
125#340—24no—
126#340—42no—
127#340—42no—
128#336—24no—
129#336—24no—
130C25—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8C22×C412yes—
139C22×Q8C2×D412yes—
140C22×Q8C2×D412yes—
141C22×D4—21no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C22×C412yes—
149C22×D4C2×D412yes—
150C22×D4C2×D412yes—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—82no—
160C23×C4—82no—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
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×C8—22no—
177C22×C8—22no—
178C22×C8C22×C414yes—
179C22×C8#4114yes—
180#125—28no—
181#125—28no—
182#125—44no—
183#125—44no—
184#124—28no—
185#124—28no—
186#124—28no—
187#124C22⋊C4116yes—
188C4×Q8—48no—
189C4×Q8—48no—
190C4×D4—48no—
191C4×D4—48no—
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—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
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—28no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4C22×C414yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—82no—
232C2×C22⋊C4—82no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—24no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42C22×C412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42#4114yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250#103—216no—
251#103—48no—
252#102—216no—
253#102—48no—
254#97—44no—
255C4×C8—28no—
256#95—28no—
257#95—24no—
258#95—24no—
259#95—82no—
260#95—82no—
261#95—82no—
262#95—82no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24C22×D411yescharacteristic
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8C2×C22⋊C414yes—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4C2×C22⋊C414yes—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—48no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
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—42no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4C4×D414yes—
363C22×C4#12014yes—
364C22×C4#12114yes—
365C22×C4#12412yes—
366C22×C4#12412yes—
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4112yes—
373C22×C4C2×#4112yes—
374C22×C4C2×#4111yescharacteristic
375C22×C4H(5,𝔽2)12yes—
376C22×C42−1+412yes—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—48no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—416no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—88no—
406C4⋊C4—88no—
407C22⋊C4—232no—
408C22⋊C4—48no—
409C22⋊C4—88no—
410C42—24no—
411C42—24no—
412C42—48no—
413C42—42no—
414C42—42no—
415C42C2×C22⋊C414yes—
416C42C4×D414yes—
417C42#12114yes—
418C42#12314yes—
419C23—24no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—82no—
441C23—82no—
442C23C2×C4×D411yescharacteristic
443C23C2×#12011yescharacteristic
444C23C2×#12111yescharacteristic
445C23C2×#12411yescharacteristic
446C23#53211yescharacteristic
447C23#53511yescharacteristic
448C23#53611yescharacteristic
449C23#57612yes—
450C23#57712yes—
451C23#57712yes—
452C23#57812yes—
453Q8—28no—
454Q8—44no—
455Q8—44no—
456Q8—44no—
457D4—28no—
458D4—44no—
459D4—44no—
460D4—44no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—816no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
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#38618yes—
517C2×C4C22×C22⋊C411yescharacteristic
518C2×C4C2×C4×D411yescharacteristic
519C2×C4C4×#4112yes—
520C2×C4#52012yes—
521C2×C4C2×#12111yescharacteristic
522C2×C4C2×#12311yescharacteristic
523C2×C4#52912yes—
524C2×C4#54212yes—
525C2×C4#54612yes—
526C2×C4#54612yes—
527C2×C4#54612yes—
528C2×C4#54712yes—
529C2×C4#54712yes—
530C2×C4#54812yes—
531C2×C4#55612yes—
532C2×C4#55912yes—
533C8—44no—
534C8—44no—
535C22—48no—
536C22—44no—
537C22—82no—
538C22—82no—
539C22—84no—
540C22—84no—
541C22—84no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22C2×#38611yescharacteristic
547C22C4×#12411yescharacteristic
548C22#232611yescharacteristic
549C22#243211yescharacteristic
550C22#243511yescharacteristic
551C22#254411yescharacteristic
552C22#254711yescharacteristic
553C22#295112yes—
554C22#295212yes—
555C22#301212yes—
556C22#301312yes—
557C22#304512yes—
558C22#304612yes—
559C22#313612yes—
560C22#313712yes—
561C22#317812yes—
562C22#317912yes—
563C22#322011yescharacteristic
564C22#322111yescharacteristic
565C22#322211yescharacteristic
566C22#322311yescharacteristic
567C22#324312yes—
568C22#324412yes—
569C4—28no—
570C4—42no—
571C4—42no—
572C4—44no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4—84no—
577C4—162no—
578C4#233912yes—
579C4#236712yes—
580C4#242712yes—
581C4#244712yes—
582C2—84no—
583C2#1402711yescharacteristic
584C2#2093712yes—
585C2#2164312yes—
586C2#2222711yescharacteristic
587C2#2222811yescharacteristic
588C2#2568412yes—
589C2#2586711yescharacteristic
590C2#2586811yescharacteristic
591C2#2862911yescharacteristic
592C2#2863011yescharacteristic
593C2#2873012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1992C2semidirect
5. #14283C2semidirect
6. #11190C2semidirect
8. #10841C2semidirect
9. #10789C2semidirect
11. #10694C2semidirect
18. C2×#488C4semidirect
56. C2×#125C2×C4semidirect
92. C2×#103C2×C4semidirect
104. C2×C4×C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4882
C2×#23262
C2×#19922
C2×#19592
#142832
#111902
#111892
#108412
#107892
#107882
#106942