Finite Group Databasebeta

← #395920

Subgroups — #395920

Order 512.

1788 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic, complemented
2C2×#2067C212yesmaximal, complemented
3C2×#1895C211yesmaximal, characteristic
4C2×#1875C211yesmaximal, characteristic
5C2×#1851C211yesmaximal, characteristic, complemented
6#14137C211yesmaximal, characteristic
7#13951C211yesmaximal, characteristic, complemented
8#11231C212yesmaximal, complemented
9#11219C212yesmaximal, complemented
10#11121C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#428—28no—
17C2×#428—24no—
18C22×#107C2212yes—
19C22×#106C2212yes—
20C22×C4⋊C8C2212yes—
21C22×#102C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C2×#390C2212yes—
26C2×#389C2212yes—
27C2×#386—22no—
28C2×#380C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#2067—28no—
31#1895—24no—
32#1875—24no—
33#1851—24no—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40#1439—21no—
41#1439—21no—
42#1439—22no—
43C4⋊#95—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46#1434—21no—
47C23×D4—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—24no—
54C22×M16—24no—
55C22×M16D414yes—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2312yes—
59C2×#121—22no—
60C2×#121—22no—
61C2×#121—22no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121Q814yes—
65C2×#117—22no—
66C2×#117—24no—
67C2×#117—42no—
68C2×#117D414yes—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4D412yes—
86C22×C42—21no—
87C22×C42C2312yes—
88#428—232no—
89C2×#107—24no—
90C2×#107—24no—
91C2×#107D418yescomplemented
92C2×#106—24no—
93C2×#106—24no—
94C2×#106—24no—
95C2×C4⋊C8—24no—
96C2×C4⋊C8—24no—
97C2×C4⋊C8—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—42no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108#390—24no—
109#390—42no—
110#389—24no—
111#389—42no—
112#386—48no—
113#380—22no—
114#380—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95D411yescharacteristic
124C2×#95D411yescharacteristic
125#340—24no—
126#340—24no—
127#340—22no—
128#340—22no—
129#340—42no—
130#336—24no—
131C25—21no—
132C25C2×D411yescharacteristic
133C22×D4—24no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—44no—
144C22×D4—44no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4C2×Q812yes—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—44no—
158C23×C4—42no—
159C23×C4—44no—
160C23×C4—81no—
161C23×C4—81no—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×Q812yes—
168C23×C4#4112yes—
169C23×C4#4112yes—
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C2×M16—216no—
173C2×M16—28no—
174C2×M16—28no—
175C2×M16—28no—
176C2×M16—216no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8C2×D414yes—
192C22×C8C2×D414yes—
193C22×C8#4114yes—
194#121—28no—
195#121—28no—
196#121—44no—
197#121—44no—
198#117—28no—
199#117—44no—
200#117—48no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
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—48no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×Q812yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—82no—
249C2×C22⋊C4—82no—
250C2×C22⋊C4—81no—
251C2×C22⋊C4—81no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—44no—
260#107—216no—
261#106—216no—
262C4⋊C8—48no—
263#102—44no—
264#102—48no—
265#102—44no—
266C22⋊C8—44no—
267#95—22no—
268#95—22no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—42no—
273#95—44no—
274#95—82no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—22no—
284C24—41no—
285C24—41no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24#12112yes—
295C24#12112yes—
296C24C22⋊Q812yes—
297C24C22×D411yescharacteristic
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—416no—
303C2×D4—48no—
304C2×D4—41no—
305C2×D4—42no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—42no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—84no—
319C2×D4—84no—
320C2×D4—88no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
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—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—48no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
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—41no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—88no—
381C22×C4—88no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4#12114yes—
385C22×C4#12112yes—
386C22×C4#12112yes—
387C22×C4C22⋊Q814yes—
388C22×C4C22⋊Q814yes—
389C22×C4C22⋊Q812yes—
390C22×C4#12614yes—
391C22×C4AGL(1,ℤ/8)14yes—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×Q811yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4112yes—
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4H(5,𝔽2)12yes—
401C22×C42−1+412yes—
402M16—232no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—42no—
413C2×C8—41no—
414C2×C8—42no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8#12118yes—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—48no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—88no—
439C4⋊C4#12014yes—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C22⋊C4—84no—
446C42—48no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C23—24no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—44no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—81no—
483C23—81no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—81no—
490C23—81no—
491C23—81no—
492C23—81no—
493C23—82no—
494C23—81no—
495C23—81no—
496C23#39614yes—
497C23C2×#12111yescharacteristic
498C23C2×#12111yescharacteristic
499C23C2×C22⋊Q811yescharacteristic
500C23D4212yes—
501C23#54712yes—
502C23C2×AGL(1,ℤ/8)12yes—
503C23#57512yes—
504C23#57611yescharacteristic
505C23#57711yescharacteristic
506C23#57711yescharacteristic
507C23#57811yescharacteristic
508D4—42no—
509D4—42no—
510D4—44no—
511D4—816no—
512D4—81no—
513D4—81no—
514D4—82no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—816no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4#39614yes—
570C2×C4C2×#12011yescharacteristic
571C2×C4C2×#12111yescharacteristic
572C2×C4C2×#12112yes—
573C2×C4C2×C22⋊Q811yescharacteristic
574C2×C4C2×C22⋊Q811yescharacteristic
575C2×C4#52511yescharacteristic
576C2×C4#52512yes—
577C2×C4C2×#12611yescharacteristic
578C2×C4#53411yescharacteristic
579C2×C4#53611yescharacteristic
580C2×C4#53712yes—
581C2×C4#53912yes—
582C2×C4#54612yes—
583C2×C4#54812yes—
584C2×C4D4×Q812yes—
585C2×C4#55212yes—
586C2×C4#55312yes—
587C2×C4#55412yes—
588C8—28no—
589C8—81no—
590C8—81no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—84no—
606C22—84no—
607C22—161no—
608C22—161no—
609C22—161no—
610C22—161no—
611C22—162no—
612C22—162no—
613C22C2×#39611yescharacteristic
614C22#241011yescharacteristic
615C22#241311yescharacteristic
616C22#241612yes—
617C22#244912yes—
618C22#303811yescharacteristic
619C22#304011yescharacteristic
620C22#304511yescharacteristic
621C22#304611yescharacteristic
622C22#305512yes—
623C22#306211yescharacteristic
624C22#306311yescharacteristic
625C22#307712yes—
626C22#308311yescharacteristic
627C22#308411yescharacteristic
628C22#326412yes—
629C22#327112yes—
630C22#327212yes—
631C22#327712yes—
632C22#334512yes—
633C22#334712yes—
634C22#335112yes—
635C22#335312yes—
636C4—24no—
637C4—44no—
638C4—44no—
639C4—84no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—162no—
644C4#240911yescharacteristic
645C4#241712yes—
646C4#248112yes—
647C4#257911yescharacteristic
648C4#261811yescharacteristic
649C4#284711yescharacteristic
650C2—44no—
651C2—162no—
652C2#1472911yescharacteristic
653C2#2361611yescharacteristic
654C2#2361811yescharacteristic
655C2#2437211yescharacteristic
656C2#2437511yescharacteristic
657C2#2459211yescharacteristic
658C2#2459911yescharacteristic
659C2#2479012yes—
660C2#2479412yes—
661C2#2635212yes—
662C2#2636612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#428C2semidirect
2. C2×#2067C2semidirect
5. C2×#1851C2semidirect
7. #13951C2semidirect
8. #11231C2semidirect
9. #11219C2semidirect
10. #11121C2semidirect
11. #10730C2semidirect
91. C2×#107D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#20672
C2×#18952
C2×#18752
C2×#18512
#141372
#139512
#112312
#112192
#111212
#107302