Finite Group Databasebeta

← #204369

Subgroups — #204369

Order 512.

1196 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#137C211yesmaximal, characteristic
2#21380C211yesmaximal, characteristic
3#21303C211yesmaximal, characteristic, complemented
4C2×#1569C211yesmaximal, characteristic, complemented
5C2×#1489C211yesmaximal, characteristic
6C2×#1491C211yesmaximal, characteristic, complemented
7C2×#1455C211yesmaximal, characteristic, complemented
8#9666C211yesmaximal, characteristic, complemented
9#9664C211yesmaximal, characteristic, complemented
10#9662C211yesmaximal, characteristic, complemented
11#9661C211yesmaximal, characteristic, complemented
12#9660C211yesmaximal, characteristic, complemented
13#9658C211yesmaximal, characteristic, complemented
14#9656C211yesmaximal, characteristic, complemented
15#9655C211yesmaximal, characteristic, complemented
16C22×#137C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C4×#137—24no—
20C4×#137—24no—
21C2×#441C411yescharacteristic
22C2×#441C411yescharacteristic
23C2×#440C411yescharacteristic
24C2×#440C411yescharacteristic
25C2×C4×Q16C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27C2×C4×SD16C2211yescharacteristic
28C2×C4×SD16C2211yescharacteristic
29C2×#432C2211yescharacteristic
30C2×#428C411yescharacteristic
31C2×#423C2211yescharacteristic
32C2×#417C411yescharacteristic
33C2×#419C411yescharacteristic
34C2×C4×M16C2211yescharacteristic
35C2×C4×M16C2211yescharacteristic
36C2×C4×M16C2211yescharacteristic
37#1985—22no—
38#1984C2211yescharacteristic
39#1984C2211yescharacteristic
40#1984C2211yescharacteristic
41#1984C2211yescharacteristic
42#1984C2211yescharacteristic
43#1984C2211yescharacteristic
44#1954C2211yescharacteristic
45#1954C2211yescharacteristic
46#1870—22no—
47#1749—22no—
48#1728—22no—
49#1569—24no—
50C2×#335C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1491—24no—
55#1489—24no—
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64#1455—24no—
65C2×#322C2211yescharacteristic
66C2×#322C2211yescharacteristic
67C2×#322C2211yescharacteristic
68C2×C82C2211yescharacteristic
69C22×#41C2×C411yescharacteristic
70C23×Q8C2×C411yescharacteristic
71C2×#137—24no—
72C2×#137—24no—
73C2×#137—24no—
74C22×Q16C2×C411yescharacteristic
75C22×Q16C2×C411yescharacteristic
76C22×SD16C2×C411yescharacteristic
77C22×SD16C2×C411yescharacteristic
78C22×M16—21no—
79C22×M16—21no—
80C22×M16C2×C411yescharacteristic
81C22×M16C2×C411yescharacteristic
82C22×M16D412yes—
83C4×#41—22no—
84C4×#41—22no—
85C4×#41—22no—
86C4×#41—22no—
87C2×C4×Q8—21no—
88C2×C4×Q8—21no—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—22no—
93C2×C4×Q8C2×C411yescharacteristic
94C2×C4×Q8C2×C411yescharacteristic
95C2×C4×Q8C2×C411yescharacteristic
96C2×C4×Q8C2311yescharacteristic
97C2×C4×Q8C2311yescharacteristic
98C2×C4×Q8C2311yescharacteristic
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4C2×C411yescharacteristic
102C2×C4×D4C2311yescharacteristic
103C2×#117—21no—
104C2×#117C2×C411yescharacteristic
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2×C411yescharacteristic
107C22×C42—21no—
108C22×C42—21no—
109C22×C42C2311yescharacteristic
110#441—24no—
111#441—24no—
112#440—24no—
113#440—24no—
114C4×Q16—22no—
115C4×Q16—22no—
116C4×Q16—22no—
117C4×Q16—22no—
118C4×SD16—22no—
119C4×SD16—22no—
120C4×SD16—22no—
121C4×SD16—22no—
122#432—22no—
123#432—22no—
124#428—24no—
125C2×#107C2×C411yescharacteristic
126C2×#107C2×C411yescharacteristic
127C2×#106C2×C411yescharacteristic
128C2×#106C2×C411yescharacteristic
129#423—22no—
130#423—22no—
131C2×C4⋊C8—21no—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136#419—24no—
137#417—24no—
138C2×#103C2×C411yescharacteristic
139C2×#103C2×C411yescharacteristic
140C2×#103C2×C411yescharacteristic
141C2×#103C2×C411yescharacteristic
142C2×#103C2×C411yescharacteristic
143C2×#103C2×C411yescharacteristic
144C2×#102C2×C411yescharacteristic
145C2×#102C2×C411yescharacteristic
146C2×C22⋊C8—21no—
147C2×C22⋊C8—22no—
148C2×C22⋊C8—22no—
149C4×M16—22no—
150C4×M16—22no—
151C4×M16—22no—
152C4×M16D414yes—
153C4×M16D414yes—
154C4×M16D414yes—
155C2×#97—21no—
156C2×#97—21no—
157C2×#97C2×C411yescharacteristic
158C2×#97C2×C411yescharacteristic
159C2×#97D412yes—
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166#336—22no—
167#336—22no—
168#336—22no—
169#336—22no—
170#336—22no—
171#336—22no—
172#335—22no—
173#335—22no—
174#334—22no—
175#334—22no—
176Q8⋊C8—22no—
177Q8⋊C8—22no—
178Q8⋊C8—22no—
179Q8⋊C8—22no—
180Q8⋊C8—22no—
181Q8⋊C8—22no—
182D4⋊C8—22no—
183D4⋊C8—22no—
184#322—22no—
185#322—22no—
186#322—22no—
187C82—22no—
188C2×#41—22no—
189C2×#41—22no—
190C2×#41—22no—
191C2×#41—22no—
192C2×#41—41no—
193C2×#41—41no—
194C22×Q8—21no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—41no—
200C22×Q8—41no—
201C22×Q8C22×C411yescharacteristic
202C22×Q8C22×C411yescharacteristic
203C22×Q8C22×C411yescharacteristic
204C22×D4—21no—
205C22×D4C22×C411yescharacteristic
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4C22×C411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4#4111yescharacteristic
214#137—232no—
215C2×Q16—22no—
216C2×Q16—22no—
217C2×Q16—22no—
218C2×Q16—22no—
219C2×Q16—22no—
220C2×Q16M1614yescomplemented
221C2×SD16—22no—
222C2×SD16—22no—
223C2×SD16—22no—
224C2×SD16—22no—
225C2×SD16—22no—
226C2×SD16M1614yescomplemented
227C2×M16—24no—
228C2×M16—24no—
229C2×M16—22no—
230C2×M16—22no—
231C2×M16—24no—
232C2×M16—41no—
233C2×M16—41no—
234C2×M16—42no—
235C2×M16—44no—
236C2×M16—44no—
237C2×M16—42no—
238C2×M16C22⋊C414yes—
239C2×M16C22⋊C414yes—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8C22×C411yescharacteristic
247C22×C8C22×C411yescharacteristic
248C22×C8C22×C411yescharacteristic
249C22×C8C22×C411yescharacteristic
250C22×C8C2×D412yes—
251C22×C8C2×D412yes—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—42no—
271C4×Q8—42no—
272C4×Q8—42no—
273C4×Q8—42no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—42no—
281C4×D4—42no—
282C4×D4—42no—
283C4×D4—42no—
284#117—22no—
285#117—22no—
286#117—42no—
287#117—42no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4C22×C411yescharacteristic
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C4⋊C4C22×C411yescharacteristic
302C2×C4⋊C4C22×C411yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42C22×C411yescharacteristic
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2×D411yescharacteristic
322C2×C42#4112yes—
323C2×C42#4111yescharacteristic
324C2×C42C2411yescharacteristic
325#107—22no—
326#107—22no—
327#106—22no—
328#106—22no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—21no—
336C4⋊C8—21no—
337C4⋊C8—42no—
338#103—22no—
339#103—22no—
340#103—22no—
341#103—22no—
342#103—22no—
343#103—22no—
344#102—22no—
345#102—22no—
346C22⋊C8—44no—
347C22⋊C8—42no—
348C22⋊C8—44no—
349#97—24no—
350#97—24no—
351#97—42no—
352#97C22⋊C414yes—
353#97C22⋊C414yes—
354C4×C8—21no—
355C4×C8—21no—
356C4×C8—21no—
357C4×C8—21no—
358C4×C8—21no—
359C4×C8—21no—
360C4×C8C2×D412yes—
361C4×C8C2×D412yes—
362C4×C8C2×D412yes—
363C4×C8C2×D412yes—
364C4×C8C2×D412yes—
365C4×C8C2×D412yes—
366C24—21no—
367C24C4×D411yescharacteristic
368#41—28no—
369#41—44no—
370#41—44no—
371#41—44no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—28no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—42no—
394C2×Q8C2×M1612yes—
395C2×Q8C2×M1612yes—
396C2×Q8C2×M1612yes—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4C2×M1612yes—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4C2×C22⋊C412yes—
435C22×C4C4×D412yes—
436C22×C4C4×D411yescharacteristic
437C22×C4C4×D411yescharacteristic
438C22×C4C4×D411yescharacteristic
439C22×C4#12112yes—
440C22×C4#12112yes—
441C22×C4#12412yes—
442C22×C4#12712yes—
443C22×C4C23×C411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C2×#4111yescharacteristic
446Q16—28no—
447Q16—28no—
448SD16—28no—
449SD16—28no—
450M16—28no—
451M16—44no—
452M16—48no—
453M16—48no—
454M16—84no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—42no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8C2×C22⋊C412yes—
479C2×C8C2×C22⋊C412yes—
480C2×C8C2×C22⋊C412yes—
481C2×C8C2×C22⋊C412yes—
482C2×C8C2×M1612yes—
483C2×C8C2×M1612yes—
484C2×C8#13714yescomplemented
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—42no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42C2×C22⋊C412yes—
518C42C4×D412yes—
519C42#12112yes—
520C42#12112yes—
521C42#12412yes—
522C42#12712yes—
523C42C22×D411yescharacteristic
524C42C22×D411yescharacteristic
525C42C22×D411yescharacteristic
526C42C2×#4111yescharacteristic
527C23—21no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23#39012yes—
537C23C2×C4×D411yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—22no—
541Q8—22no—
542Q8—22no—
543Q8—22no—
544Q8—48no—
545Q8—41no—
546Q8—41no—
547Q8—41no—
548Q8—41no—
549Q8—41no—
550Q8—41no—
551D4—22no—
552D4—22no—
553D4—48no—
554D4—41no—
555D4—41no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—82no—
594C2×C4#39012yes—
595C2×C4#39012yes—
596C2×C4#39012yes—
597C2×C4C22×C22⋊C411yescharacteristic
598C2×C4C2×C4×D411yescharacteristic
599C2×C4C2×#12111yescharacteristic
600C2×C4C2×#12111yescharacteristic
601C2×C4C2×#12411yescharacteristic
602C2×C4C2×#12711yescharacteristic
603C2×C4C22×M1611yescharacteristic
604C2×C4C22×M1611yescharacteristic
605C2×C4#56811yescharacteristic
606C2×C4#56811yescharacteristic
607C2×C4C2×#13712yes—
608C2×C4#57512yes—
609C8—24no—
610C8—22no—
611C8—22no—
612C8—22no—
613C8—24no—
614C8—42no—
615C8—42no—
616C8—81no—
617C8—81no—
618C8—81no—
619C8—81no—
620C8#40714yes—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—42no—
625C22—42no—
626C22—44no—
627C22—42no—
628C22—82no—
629C22—81no—
630C22—81no—
631C22C2×#39011yescharacteristic
632C22D4×M1611yescharacteristic
633C22D4×M1611yescharacteristic
634C22C4×#13712yes—
635C22#295612yes—
636C4—22no—
637C4—22no—
638C4—21no—
639C4—22no—
640C4—21no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—81no—
645C4—81no—
646C4—81no—
647C4—81no—
648C4C2×#40711yescharacteristic
649C4#287911yescharacteristic
650C4D4×M1611yescharacteristic
651C4D4×M1611yescharacteristic
652C4#305612yes—
653C4#315211yescharacteristic
654C4#315311yescharacteristic
655C2—42no—
656C2—82no—
657C2#1804712yes—
658C2#1804912yes—
659C2#2119011yescharacteristic
660C2#2123711yescharacteristic
661C2#2124111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21303C2semidirect
4. C2×#1569C2semidirect
6. C2×#1491C2semidirect
7. C2×#1455C2semidirect
8. #9666C2semidirect
9. #9664C2semidirect
10. #9662C2semidirect
11. #9661C2semidirect
12. #9660C2semidirect
13. #9658C2semidirect
14. #9656C2semidirect
15. #9655C2semidirect
220. C2×Q16M16semidirect
226. C2×SD16M16semidirect
484. C2×C8#137semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1372
#213802
#213032
C2×#15692
C2×#14892
C2×#14912
C2×#14552
#96662
#96642
#96622
#96612
#96602
#96582
#96562
#96552