Finite Group Databasebeta

← #392819

Subgroups — #392819

Order 512.

1818 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#405C211yesmaximal, characteristic, complemented
2C2×#1936C212yesmaximal, complemented
3C2×#1895C211yesmaximal, characteristic
4C2×#1893C211yesmaximal, characteristic
5#14264C211yesmaximal, characteristic, complemented
6#14257C211yesmaximal, characteristic
7#10925C212yesmaximal, complemented
8#10880C212yesmaximal, complemented
9#10821C212yesmaximal, complemented
10#10722C211yesmaximal, characteristic, complemented
11#10721C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C2×#405—28no—
20C2×#405—24no—
21C22×#97C2212yes—
22C22×C4×C8C2212yes—
23C2×#390C2212yes—
24C2×#389C2212yes—
25C2×#380C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28#1936—28no—
29#1895C418yescomplemented
30#1893—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38#1444—21no—
39#1442—21no—
40#1439—21no—
41#1439—21no—
42#1439—22no—
43#1438—22no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C23×D4—21no—
47C23×D4C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—24no—
52C22×M16—24no—
53C22×M16C2×C414yes—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—24no—
57C23×C8—24no—
58C23×C8D414yes—
59C23×C8C2312yes—
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#121—22no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121—22no—
67C2×#121C2×C414yes—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—24no—
71C2×#117—24no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4D412yes—
87C22×C42C2312yes—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102C2×C418yes—
95#405—232no—
96C2×#97—24no—
97C2×#97—24no—
98C2×#97D418yescomplemented
99C2×C4×C8—24no—
100C2×C4×C8—24no—
101C2×C4×C8—24no—
102#390—24no—
103#390—24no—
104#389—24no—
105#389—24no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—22no—
110#380—22no—
111#380C2×C414yes—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120#340—22no—
121#340—22no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340C2×C418yes—
127C25—21no—
128C25C2×D411yescharacteristic
129C22×D4—24no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—82no—
142C22×D4—82no—
143C22×D4C22×C412yes—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4C22×C412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4#4112yes—
162C23×C4C2411yescharacteristic
163C2×M16—216no—
164C2×M16—28no—
165C2×M16—28no—
166C2×M16—28no—
167C2×M16—216no—
168C22×C8—216no—
169C22×C8—216no—
170C22×C8—28no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—28no—
182C22×C8—28no—
183C22×C8C22×C414yes—
184C22×C8C2×D414yes—
185C22×C8C2×D414yes—
186C22×C8C2×D412yes—
187C22×C8C2×D412yes—
188C22×C8#4114yes—
189C22×C8#4114yes—
190#121—28no—
191#121—28no—
192#121—44no—
193#121—44no—
194#117—216no—
195#117—28no—
196#117—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C414yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—82no—
242C2×C22⋊C4—82no—
243C2×C22⋊C4—82no—
244C2×C22⋊C4—82no—
245C2×C22⋊C4—81no—
246C2×C22⋊C4—81no—
247C2×C22⋊C4—82no—
248C2×C22⋊C4—81no—
249C2×C22⋊C4—81no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42C2×D414yes—
255C2×C42#4114yes—
256#102—216no—
257#102—44no—
258#102—44no—
259#97—216no—
260C4×C8—216no—
261#95—24no—
262#95—42no—
263#95—44no—
264#95—42no—
265#95—82no—
266#95—82no—
267#95—82no—
268#95—82no—
269C24—22no—
270C24—22no—
271C24—21no—
272C24—22no—
273C24—21no—
274C24—21no—
275C24—22no—
276C24—21no—
277C24—22no—
278C24—42no—
279C24—41no—
280C24—41no—
281C24—42no—
282C24—41no—
283C24—42no—
284C24—41no—
285C24—42no—
286C24—41no—
287C24—41no—
288C24C4×D412yes—
289C24#12112yes—
290C24#12412yes—
291C24C22×D411yescharacteristic
292C2×D4—22no—
293C2×D4—24no—
294C2×D4—22no—
295C2×D4—24no—
296C2×D4—48no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—42no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—88no—
312C2×D4—88no—
313C2×D4—84no—
314C2×D4—84no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4#11714yes—
373C22×C4C4×D414yes—
374C22×C4C4×D412yes—
375C22×C4#12114yes—
376C22×C4#12114yes—
377C22×C4#12112yes—
378C22×C4#12314yes—
379C22×C4#12412yes—
380C22×C4#12614yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D412yes—
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4112yes—
388C22×C4C2×#4112yes—
389C22×C4C2×#4112yes—
390C22×C4C2×#4111yescharacteristic
391M16—232no—
392C2×C8—232no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—22no—
396C2×C8—24no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8C4×D418yes—
408C2×C8#12014yes—
409C2×C8#12314yes—
410C4⋊C4—28no—
411C4⋊C4—28no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—41no—
419C4⋊C4—42no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—42no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—41no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—88no—
431C22⋊C4—216no—
432C22⋊C4—48no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C22⋊C4—84no—
438C42—24no—
439C42—24no—
440C42—24no—
441C42—24no—
442C23—22no—
443C23—24no—
444C23—22no—
445C23—22no—
446C23—24no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—44no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—44no—
469C23—81no—
470C23—82no—
471C23—81no—
472C23—82no—
473C23—81no—
474C23—81no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—81no—
480C23—81no—
481C23—81no—
482C23—81no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23#38814yes—
488C23C2×C4×D411yescharacteristic
489C23C2×#12112yes—
490C23C2×#12111yescharacteristic
491C23C2×#12411yescharacteristic
492C23#53212yes—
493C23#57612yes—
494C23#57611yescharacteristic
495C23#57712yes—
496C23#57711yescharacteristic
497C23#57711yescharacteristic
498C23#57811yescharacteristic
499D4—44no—
500D4—42no—
501D4—42no—
502D4—816no—
503D4—81no—
504D4—82no—
505D4—81no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4—81no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4#38814yes—
560C2×C4C2×#11711yescharacteristic
561C2×C4C2×C4×D412yes—
562C2×C4C2×C4×D411yescharacteristic
563C2×C4C4×#4112yes—
564C2×C4C2×#12011yescharacteristic
565C2×C4C2×#12311yescharacteristic
566C2×C4C2×#12311yescharacteristic
567C2×C4#52512yes—
568C2×C4#52512yes—
569C2×C4#52512yes—
570C2×C4#52512yes—
571C2×C4#52511yescharacteristic
572C2×C4#52511yescharacteristic
573C2×C4C2×#12611yescharacteristic
574C2×C4#52912yes—
575C2×C4#52912yes—
576C2×C4#52912yes—
577C2×C4#52912yes—
578C8—24no—
579C8—24no—
580C8—28no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—84no—
593C22—84no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—162no—
598C22—162no—
599C22—161no—
600C22—161no—
601C22—161no—
602C22—161no—
603C22C2×#38811yescharacteristic
604C22C4×#12112yes—
605C22#284711yescharacteristic
606C22#285012yes—
607C22#285411yescharacteristic
608C22#294611yescharacteristic
609C22#294612yes—
610C22#294711yescharacteristic
611C22#305711yescharacteristic
612C22#305811yescharacteristic
613C22#306011yescharacteristic
614C22#306111yescharacteristic
615C22#306212yes—
616C22#306212yes—
617C22#306212yes—
618C22#306312yes—
619C22#313612yes—
620C22#313611yescharacteristic
621C22#313711yescharacteristic
622C22#317312yes—
623C22#317412yes—
624C22#317512yes—
625C22#317612yes—
626C4—24no—
627C4—44no—
628C4—44no—
629C4—84no—
630C4—82no—
631C4—162no—
632C4—161no—
633C4—161no—
634C4#229012yes—
635C4C4×#12011yescharacteristic
636C4C4×#12311yescharacteristic
637C4#284911yescharacteristic
638C4#285212yes—
639C4#285311yescharacteristic
640C2—44no—
641C2—162no—
642C2#1469611yescharacteristic
643C2#2109111yescharacteristic
644C2#2109211yescharacteristic
645C2#2154112yes—
646C2#2154312yes—
647C2#3293811yescharacteristic
648C2#3293911yescharacteristic
649C2#3305412yes—
650C2#3305512yes—
651C2#3313711yescharacteristic
652C2#3313811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#405C2semidirect
2. C2×#1936C2semidirect
5. #14264C2semidirect
7. #10925C2semidirect
8. #10880C2semidirect
9. #10821C2semidirect
10. #10722C2semidirect
11. #10721C2semidirect
29. #1895C4semidirect
98. C2×#97D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4052
C2×#19362
C2×#18952
C2×#18932
#142642
#142572
#109252
#108802
#108212
#107222
#107212