Finite Group Databasebeta

← #365732

Subgroups — #365732

Order 512.

1578 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2326C211yesmaximal, characteristic
2C2×#1959C211yesmaximal, characteristic
3C2×#1958C211yesmaximal, characteristic
4C2×#1834C211yesmaximal, characteristic, complemented
5#14277C211yesmaximal, characteristic, complemented
6#11067C212yesmaximal, complemented
7#11066C212yesmaximal
8#10843C211yesmaximal, characteristic, complemented
9#10789C212yesmaximal, complemented
10#10788C212yesmaximal
11#10694C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#97C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2326—24no—
22#2326—24no—
23C2×#391C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#389C2212yes—
26C2×#388—21no—
27C2×#388—21no—
28C2×#385—21no—
29C2×#385—21no—
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1959—24no—
35#1958C418yescomplemented
36#1834—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1444—21no—
44#1444—21no—
45#1442—21no—
46#1442—21no—
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×#124—22no—
54C2×#124—22no—
55C2×#124—22no—
56C2×#124C2×C414yes—
57C2×#124D414yes—
58C2×#124Q814yes—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—24no—
62C2×C4×Q8—42no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—24no—
66C2×C4×D4—42no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103C2×C418yescomplemented
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—42no—
95C2×#102—42no—
96C2×#102—42no—
97C2×#102C2×C418yes—
98C2×#97—22no—
99C2×#97—22no—
100C2×#97—22no—
101C2×C4×C8—22no—
102C2×C4×C8—22no—
103C2×C4×C8C2×C414yescomplemented
104#391—22no—
105#391—22no—
106#390—22no—
107#390—22no—
108#389—42no—
109#389—42no—
110#388—28no—
111#388—28no—
112#385—28no—
113#385—28no—
114#384—22no—
115#384—22no—
116#384—22no—
117#384D414yes—
118#382—22no—
119#382C2×C414yescomplemented
120#376—22no—
121#376—22no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95D411yescharacteristic
130C2×#95D411yescharacteristic
131C2×#95D411yescharacteristic
132C2×#95D411yescharacteristic
133#340—24no—
134#340—24no—
135#340—42no—
136#340—42no—
137#336—24no—
138#336—24no—
139C25—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8C22×C412yes—
148C22×Q8C2×D412yes—
149C22×Q8C2×Q812yes—
150C22×D4—21no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4C22×C412yes—
158C22×D4C2×D412yes—
159C22×D4C2×Q812yes—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—82no—
169C23×C4—82no—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2411yescharacteristic
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8C22×C414yes—
190C22×C8#4114yes—
191#124—28no—
192#124—28no—
193#124—28no—
194#124C4⋊C4116yescomplemented
195C4×Q8—48no—
196C4×Q8—48no—
197C4×D4—48no—
198C4×D4—48no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4C22×C414yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—82no—
245C2×C22⋊C4—82no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42C22×C412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×Q812yes—
260C2×C42#4114yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263#103—48no—
264#103—48no—
265#102—48no—
266#102—48no—
267#97—44no—
268C4×C8—28no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—44no—
278#95—82no—
279#95—82no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24C22×D411yescharacteristic
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8C2×C4⋊C414yes—
308C2×D4—22no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4C2×C4⋊C414yes—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—48no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—88no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4—88no—
382C22×C4C4×D412yes—
383C22×C4C4×D412yes—
384C22×C4#12012yes—
385C22×C4#12012yes—
386C22×C4C22⋊Q812yes—
387C22×C4C22⋊Q812yes—
388C22×C4#12412yes—
389C22×C4#12412yes—
390C22×C4C23×C411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×Q811yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4C2×#4111yescharacteristic
398C22×C4H(5,𝔽2)12yes—
399C22×C4H(5,𝔽2)12yes—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—48no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C4⋊C4—84no—
430C4⋊C4—84no—
431C4⋊C4—88no—
432C22⋊C4—216no—
433C22⋊C4—216no—
434C22⋊C4—48no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C42—24no—
438C42—24no—
439C42—48no—
440C42—42no—
441C42—42no—
442C42C2×C4⋊C414yes—
443C42C4×D414yes—
444C42C22⋊Q814yes—
445C42#12314yes—
446C23—24no—
447C23—42no—
448C23—42no—
449C23—44no—
450C23—44no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—44no—
455C23—44no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23C2×C4×D411yescharacteristic
470C23C2×#12011yescharacteristic
471C23C2×C22⋊Q811yescharacteristic
472C23C2×#12411yescharacteristic
473C23#53211yescharacteristic
474C23#53511yescharacteristic
475C23#53511yescharacteristic
476C23#57612yes—
477C23#57712yes—
478C23#57712yes—
479C23#57812yes—
480Q8—28no—
481Q8—44no—
482Q8—44no—
483Q8—44no—
484D4—28no—
485D4—44no—
486D4—44no—
487D4—44no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—816no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4#38514yes—
549C2×C4#38514yes—
550C2×C4C22×C4⋊C411yescharacteristic
551C2×C4C2×C4×D411yescharacteristic
552C2×C4C4×#4112yes—
553C2×C4#51812yes—
554C2×C4C2×C22⋊Q811yescharacteristic
555C2×C4C2×#12311yescharacteristic
556C2×C4#52912yes—
557C2×C4#54012yes—
558C2×C4#54612yes—
559C2×C4#54612yes—
560C2×C4#54612yes—
561C2×C4#55012yes—
562C2×C4#55312yes—
563C2×C4#55412yes—
564C2×C4#55412yes—
565C2×C4#55512yes—
566C8—44no—
567C8—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—84no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—84no—
577C22—84no—
578C22—84no—
579C22C2×#38511yescharacteristic
580C22C4×#12411yescharacteristic
581C22#232811yescharacteristic
582C22#243111yescharacteristic
583C22#243511yescharacteristic
584C22#256311yescharacteristic
585C22#256611yescharacteristic
586C22#295112yes—
587C22#295212yes—
588C22#300811yescharacteristic
589C22#300911yescharacteristic
590C22#301011yescharacteristic
591C22#301111yescharacteristic
592C22#308312yes—
593C22#308412yes—
594C22#313612yes—
595C22#313712yes—
596C22#317812yes—
597C22#317912yes—
598C22#322412yes—
599C22#322412yes—
600C22#322512yes—
601C22#322512yes—
602C4—24no—
603C4—24no—
604C4—42no—
605C4—42no—
606C4—44no—
607C4—84no—
608C4—82no—
609C4—82no—
610C4—81no—
611C4—81no—
612C4—162no—
613C4#234912yes—
614C4#236312yes—
615C4#242012yes—
616C4#245412yes—
617C2—84no—
618C2#1397911yescharacteristic
619C2#2088111yescharacteristic
620C2#2088211yescharacteristic
621C2#2164312yes—
622C2#2227412yes—
623C2#2561311yescharacteristic
624C2#2561411yescharacteristic
625C2#2586511yescharacteristic
626C2#2586611yescharacteristic
627C2#2915112yes—
628C2#2920312yes—
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
4. C2×#1834C2semidirect
5. #14277C2semidirect
6. #11067C2semidirect
8. #10843C2semidirect
9. #10789C2semidirect
11. #10694C2semidirect
35. #1958C4semidirect
91. C2×#103C2×C4semidirect
103. C2×C4×C8C2×C4semidirect
119. #382C2×C4semidirect
194. #124C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23262
C2×#19592
C2×#19582
C2×#18342
#142772
#110672
#110662
#108432
#107892
#107882
#106942