Finite Group Databasebeta

← #399976

Subgroups — #399976

Order 512.

1708 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2C2×#2072C211yesmaximal, characteristic, complemented
3C2×#1928C211yesmaximal, characteristic, complemented
4C2×#1873C212yesmaximal, complemented
5#15177C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic, complemented
7#11096C212yesmaximal, complemented
8#10970C212yesmaximal, complemented
9#10964C211yesmaximal, characteristic
10#10807C211yesmaximal, characteristic, complemented
11#10731C212yesmaximal, complemented
12C22×C22⋊Q8C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×#117C2212yes—
15C22×#107C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21#2354—24no—
22#2354—24no—
23C2×#391C2211yescharacteristic
24C2×#390—21no—
25C2×#388—22no—
26C2×#384C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31#2072—24no—
32#1928—24no—
33#1873—28no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1444—21no—
43#1439—22no—
44#1438—21no—
45C4⋊#95C2211yescharacteristic
46C23×Q8C2311yescharacteristic
47C23×D4D411yescharacteristic
48C24×C4C2312yes—
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—42no—
57C2×C22⋊Q8—42no—
58C2×C4×D4—24no—
59C2×C4×D4—24no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×#117—24no—
63C2×#117—24no—
64C2×#117—24no—
65C2×#117—24no—
66C2×#117—42no—
67C2×#117—42no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D412yes—
82C22×C42—22no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85C2×#107—22no—
86C2×#107—42no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—22no—
92C2×#103—42no—
93C2×#103—42no—
94C2×#103D414yescomplemented
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8—42no—
100C2×C22⋊C8—42no—
101#391—22no—
102#391—22no—
103#390—28no—
104#388—216no—
105#384—22no—
106#384D414yes—
107#380—24no—
108#380—24no—
109#380—24no—
110#380—42no—
111C4×C4⋊C4—28no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4D414yes—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—21no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—22no—
125#340—24no—
126#340—42no—
127#336—22no—
128#336—22no—
129C25C2×D412yes—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8C2×D412yes—
140C22×D4—24no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—42no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4#4114yes—
167C23×C4#4114yes—
168C23×C4C2411yescharacteristic
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8C2×D412yes—
186C22⋊Q8—216no—
187C22⋊Q8—48no—
188C22⋊Q8—48no—
189C22⋊Q8—48no—
190C4×D4—28no—
191C4×D4—28no—
192C4×D4—216no—
193#117—48no—
194#117—48no—
195#117—48no—
196#117—48no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
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—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—81no—
227C2×C4⋊C4—81no—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4#4114yes—
233C2×C4⋊C4#4112yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—82no—
247C2×C22⋊C4—82no—
248C2×C22⋊C4—82no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—28no—
252C2×C42—24no—
253C2×C42—44no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258#107—44no—
259#103—28no—
260#103—48no—
261#103—44no—
262C22⋊C8—48no—
263C22⋊C8—48no—
264#95—28no—
265#95—24no—
266#95—44no—
267#95—42no—
268#95—42no—
269#95—42no—
270#95—81no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24#12314yes—
286C24C22×D411yescharacteristic
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—41no—
292C2×Q8—42no—
293C2×Q8—41no—
294C2×Q8—42no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×D4—28no—
306C2×D4—48no—
307C2×D4—48no—
308C2×D4—44no—
309C2×D4—44no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—42no—
342C22×C4—42no—
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—42no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4#12012yes—
385C22×C4#12012yes—
386C22×C4#12314yes—
387C22×C4#13512yes—
388C22×C4#13712yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4112yes—
394C22×C4C2×#4112yes—
395C22×C4C2×#4112yes—
396C22×C4H(5,𝔽2)11yescharacteristic
397C22×C4H(5,𝔽2)12yes—
398C22×C4H(5,𝔽2)11yescharacteristic
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—42no—
404C2×C8—41no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C4⋊C4—28no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—416no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—48no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—42no—
436C4⋊C4—84no—
437C4⋊C4#12114yes—
438C22⋊C4—216no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C42—22no—
444C42—22no—
445C42—24no—
446C42—48no—
447C42—416no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23C2×#12011yescharacteristic
484C23C2×#12312yes—
485C23D4211yescharacteristic
486C23#54612yes—
487C23#54611yescharacteristic
488C23C2×#13511yescharacteristic
489C23C2×#13711yescharacteristic
490C23#57512yes—
491C23#57611yescharacteristic
492C23#57711yescharacteristic
493C23#57812yes—
494Q8—42no—
495Q8—44no—
496Q8—42no—
497Q8—81no—
498Q8—81no—
499Q8—82no—
500D4—416no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4#45214yes—
560C2×C4C2×#12111yescharacteristic
561C2×C4#52511yescharacteristic
562C2×C4#52511yescharacteristic
563C2×C4#52512yes—
564C2×C4#52512yes—
565C2×C4#52912yes—
566C2×C4#53211yescharacteristic
567C2×C4#53511yescharacteristic
568C2×C4#53511yescharacteristic
569C2×C4#53812yes—
570C2×C4#53812yes—
571C2×C4D4211yescharacteristic
572C2×C4#54612yes—
573C2×C4#54611yescharacteristic
574C2×C4#55012yes—
575C2×C4#55112yes—
576C2×C4#55312yes—
577C2×C4#55312yes—
578C8—42no—
579C8—81no—
580C8—82no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22#240811yescharacteristic
597C22#241812yes—
598C22#243111yescharacteristic
599C22#244412yes—
600C22#284911yescharacteristic
601C22C2×#45211yescharacteristic
602C22#300511yescharacteristic
603C22#301111yescharacteristic
604C22#301211yescharacteristic
605C22#309112yes—
606C22#309512yes—
607C22#309712yes—
608C22#309812yes—
609C22#326912yes—
610C22#327811yescharacteristic
611C22#328711yescharacteristic
612C22#330212yes—
613C22#330411yescharacteristic
614C22#331512yes—
615C22#331912yes—
616C22#332112yes—
617C22#332812yes—
618C22#333311yescharacteristic
619C4—24no—
620C4—44no—
621C4—44no—
622C4—42no—
623C4—84no—
624C4—84no—
625C4—82no—
626C4—82no—
627C4—161no—
628C4#239211yescharacteristic
629C4#245612yes—
630C4#246712yes—
631C4#253511yescharacteristic
632C4#260311yescharacteristic
633C4#263111yescharacteristic
634C2—84no—
635C2#1479511yescharacteristic
636C2#2424211yescharacteristic
637C2#2425311yescharacteristic
638C2#2487212yes—
639C2#2489212yes—
640C2#2557011yescharacteristic
641C2#2557811yescharacteristic
642C2#2620712yes—
643C2#2622712yes—
644C2#3298411yescharacteristic
645C2#3298911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2072C2semidirect
3. C2×#1928C2semidirect
4. C2×#1873C2semidirect
6. #11373C2semidirect
7. #11096C2semidirect
8. #10970C2semidirect
10. #10807C2semidirect
11. #10731C2semidirect
94. C2×#103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#20722
C2×#19282
C2×#18732
#151772
#113732
#110962
#109702
#109642
#108072
#107312