Finite Group Databasebeta

← #416126

Subgroups — #416126

Order 512.

1538 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2545C211yesmaximal, characteristic
2C2×#1983C211yesmaximal, characteristic
3C2×#1958C211yesmaximal, characteristic
4C2×#1805C211yesmaximal, characteristic, complemented
5C2×C4×#103C211yesmaximal, characteristic, complemented
6#13955C211yesmaximal, characteristic, complemented
7#11190C212yesmaximal, complemented
8#11189C212yesmaximal
9#11010C212yesmaximal, complemented
10#11009C212yesmaximal
11#10834C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×#124C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×C4×C8C2211yescharacteristic
24#2545—24no—
25#2545—24no—
26C2×#396—22no—
27C2×#391C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#388—22no—
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1983—24no—
34#1958—24no—
35#1805—24no—
36C4×#103—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1442—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44#1434—22no—
45C23×Q8C2311yescharacteristic
46C23×Q8C2311yescharacteristic
47C23×D4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—42no—
54C2×#124—22no—
55C2×#124—22no—
56C2×#124—24no—
57C2×#124—22no—
58C2×#124—22no—
59C2×#124—22no—
60C2×#124—22no—
61C2×#124—22no—
62C2×#124—42no—
63C2×#124D414yes—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8D414yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4D412yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—21no—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—42no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103—42no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—42no—
104C2×#102—42no—
105C2×#102—42no—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109#396—216no—
110#391—22no—
111#391—22no—
112#390—22no—
113#390—22no—
114#388—216no—
115#382—22no—
116#382—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4D414yes—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95D412yes—
130#340—24no—
131#340—24no—
132#336—42no—
133#336—42no—
134C25—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8C2×D412yes—
153C22×Q8C2×D412yes—
154C22×Q8#4114yes—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4C2×D412yes—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2411yescharacteristic
175C22×C8—22no—
176C22×C8—22no—
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#4114yes—
189C4⋊Q8—48no—
190C4⋊Q8—44no—
191C4⋊Q8—44no—
192#124—28no—
193#124—28no—
194#124—48no—
195#124—44no—
196#124—44no—
197#124—44no—
198#124—44no—
199C4×Q8—28no—
200C4×Q8—28no—
201C4×Q8—44no—
202C4×Q8—44no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D414yes—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—44no—
242C2×C42—24no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—82no—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42#4112yes—
259C2×C42#4112yes—
260C2×C42#4112yes—
261C4⋊C8—44no—
262#103—216no—
263#103—48no—
264#103—48no—
265#103—48no—
266#102—48no—
267#102—48no—
268C4×C8—44no—
269#95—28no—
270#95—24no—
271#95—24no—
272#95—44no—
273#95—42no—
274#95—42no—
275#95—44no—
276#95—82no—
277#95—82no—
278C24—22no—
279C24—41no—
280C24—41no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24C22×D411yescharacteristic
288C2×Q8—22no—
289C2×Q8—24no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—42no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8#12414yes—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—42no—
331C2×D4—41no—
332C2×D4—42no—
333C2×D4—41no—
334C2×D4—42no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4#12012yes—
389C22×C4#12012yes—
390C22×C4#12112yes—
391C22×C4#12112yes—
392C22×C4#12412yes—
393C22×C4#12412yes—
394C22×C4#13514yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4112yes—
402C22×C4C2×#4112yes—
403C22×C4H(5,𝔽2)12yes—
404C22×C4H(5,𝔽2)12yes—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—48no—
424C4⋊C4—416no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—48no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C22⋊C4—416no—
435C22⋊C4—416no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—42no—
443C42—42no—
444C42—48no—
445C42—84no—
446C42#12114yes—
447C42#12114yes—
448C42#12314yes—
449C42#12314yes—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—44no—
454C23—44no—
455C23—81no—
456C23—81no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—81no—
466C23—81no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23C2×#12011yescharacteristic
472C23C2×#12111yescharacteristic
473C23C2×#12411yescharacteristic
474C23#54612yes—
475C23#55012yes—
476C23C2×#13512yes—
477C23#57512yes—
478C23#57712yes—
479C23#57812yes—
480Q8—24no—
481Q8—24no—
482Q8—42no—
483Q8—44no—
484Q8—42no—
485Q8—44no—
486Q8—44no—
487Q8—44no—
488Q8—81no—
489Q8—81no—
490Q8—81no—
491Q8—81no—
492D4—44no—
493D4—44no—
494D4—81no—
495D4—81no—
496D4—81no—
497D4—81no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—81no—
534C2×C4—81no—
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—82no—
549C2×C4#39414yes—
550C2×C4#39414yes—
551C2×C4C2×#12111yescharacteristic
552C2×C4C2×#12111yescharacteristic
553C2×C4C2×#12311yescharacteristic
554C2×C4C2×#12311yescharacteristic
555C2×C4#52512yes—
556C2×C4C2×#12411yescharacteristic
557C2×C4#52912yes—
558C2×C4#53211yescharacteristic
559C2×C4#53511yescharacteristic
560C2×C4#53511yescharacteristic
561C2×C4#53512yes—
562C2×C4#54012yes—
563C2×C4#54612yes—
564C2×C4#54612yes—
565C2×C4#55312yes—
566C2×C4#55312yes—
567C2×C4#55312yes—
568C2×C4#55512yes—
569C8—44no—
570C8—82no—
571C22—84no—
572C22—84no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—162no—
577C22—162no—
578C22—162no—
579C22—162no—
580C22C2×#39411yescharacteristic
581C22#241412yes—
582C22#243111yescharacteristic
583C22#243511yescharacteristic
584C22#249012yes—
585C22#300212yes—
586C22#301312yes—
587C22#303512yes—
588C22#304211yescharacteristic
589C22#304411yescharacteristic
590C22#313111yescharacteristic
591C22#313511yescharacteristic
592C22#313712yes—
593C22#330012yes—
594C22#330112yes—
595C22#332612yes—
596C22#332712yes—
597C22#339312yes—
598C22#339412yes—
599C22#340112yes—
600C22#340212yes—
601C4—24no—
602C4—24no—
603C4—44no—
604C4—44no—
605C4—82no—
606C4—84no—
607C4—82no—
608C4—82no—
609C4—162no—
610C4#242012yes—
611C4#250011yescharacteristic
612C4#250411yescharacteristic
613C4#253512yes—
614C4#261911yescharacteristic
615C4#285011yescharacteristic
616C2—162no—
617C2#1521911yescharacteristic
618C2#2350411yescharacteristic
619C2#2351211yescharacteristic
620C2#2463612yes—
621C2#2463712yes—
622C2#2563612yes—
623C2#2588912yes—
624C2#2775212yes—
625C2#2775412yes—
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
4. C2×#1805C2semidirect
5. C2×C4×#103C2semidirect
6. #13955C2semidirect
7. #11190C2semidirect
9. #11010C2semidirect
11. #10834C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25452
C2×#19832
C2×#19582
C2×#18052
C2×C4×#1032
#139552
#111902
#111892
#110102
#110092
#108342