Finite Group Databasebeta

← #204345

Subgroups — #204345

Order 512.

1009 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3172C211yesmaximal, characteristic
2#21378C211yesmaximal, characteristic
3#21302C211yesmaximal, characteristic, complemented
4C2×#1565C211yesmaximal, characteristic, complemented
5C2×#1483C211yesmaximal, characteristic
6C2×#1485C211yesmaximal, 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
16C2×#533C2211yescharacteristic
17C2×#529C2211yescharacteristic
18#3172—24no—
19#3172—24no—
20C2×#501C411yescharacteristic
21C2×#497C411yescharacteristic
22C2×#491C411yescharacteristic
23C2×#489C411yescharacteristic
24C2×#470C411yescharacteristic
25C2×#470C411yescharacteristic
26C2×#468C411yescharacteristic
27C2×#468C411yescharacteristic
28C2×C4×Q16C2211yescharacteristic
29C2×C4×Q16C2211yescharacteristic
30C2×C4×SD16C2211yescharacteristic
31C2×C4×SD16C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
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#1750—22no—
48#1729—22no—
49#1565—24no—
50C2×#335C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1485—24no—
55#1483—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×Q16C2×C411yescharacteristic
70C22×Q16C2×C411yescharacteristic
71C22×SD16C2×C411yescharacteristic
72C22×SD16C2×C411yescharacteristic
73C22×M16—21no—
74C22×M16C2×C411yescharacteristic
75C22×M16C2×C411yescharacteristic
76C22×M16D412yes—
77C23×C8—21no—
78#533—22no—
79#533—22no—
80#533—22no—
81#533—22no—
82C2×C4⋊Q8C2×C411yescharacteristic
83C2×C4⋊Q8C2×C411yescharacteristic
84#529—22no—
85#529—22no—
86#529—22no—
87#529—22no—
88C2×#126—21no—
89C2×#125—21no—
90C2×#125C2×C411yescharacteristic
91C2×#124C2×C411yescharacteristic
92C2×#123—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8C2×C411yescharacteristic
95C2×C22⋊Q8C2×C411yescharacteristic
96C2×C22⋊Q8C2×C411yescharacteristic
97C2×#121C2×C411yescharacteristic
98C2×C4×Q8—21no—
99C2×C4×Q8C2311yescharacteristic
100C2×C4×Q8C2311yescharacteristic
101C2×C4×Q8C2311yescharacteristic
102C2×C4×D4—21no—
103C2×C4×D4C2311yescharacteristic
104C2×#117—21no—
105C2×#117—21no—
106C22×C42C2311yescharacteristic
107#501—24no—
108#497—24no—
109#491—24no—
110#489—24no—
111#470—24no—
112#470—24no—
113#468—24no—
114#468—24no—
115C4×Q16—22no—
116C4×Q16—22no—
117C4×Q16—22no—
118C4×Q16—22no—
119C4×SD16—22no—
120C4×SD16—22no—
121C4×SD16—22no—
122C4×SD16—22no—
123#431—22no—
124#431—22no—
125C2×#107C2×C411yescharacteristic
126C2×#107C2×C411yescharacteristic
127C2×#106C2×C411yescharacteristic
128C2×#106C2×C411yescharacteristic
129C2×C4⋊C8—22no—
130C2×C4⋊C8—22no—
131C2×C4⋊C8—21no—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×#103C2×C411yescharacteristic
137C2×#103C2×C411yescharacteristic
138C2×#103C2×C411yescharacteristic
139C2×#103C2×C411yescharacteristic
140C2×#103C2×C411yescharacteristic
141C2×#103C2×C411yescharacteristic
142C2×#102C2×C411yescharacteristic
143C2×#102C2×C411yescharacteristic
144C2×C22⋊C8—21no—
145C2×C22⋊C8—22no—
146C2×C22⋊C8—22no—
147C4×M16—22no—
148C4×M16—22no—
149C4×M16—22no—
150C4×M16D414yes—
151C4×M16D414yes—
152C4×M16D414yes—
153C2×#97—21no—
154C2×#97C2×C411yescharacteristic
155C2×#97C2×C411yescharacteristic
156C2×#97D412yes—
157C2×C4×C8—21no—
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164#336—22no—
165#336—22no—
166#336—22no—
167#336—22no—
168#336—22no—
169#336—22no—
170#335—22no—
171#335—22no—
172#334—22no—
173#334—22no—
174Q8⋊C8—22no—
175Q8⋊C8—22no—
176Q8⋊C8—22no—
177Q8⋊C8—22no—
178Q8⋊C8—22no—
179Q8⋊C8—22no—
180D4⋊C8—22no—
181D4⋊C8—22no—
182#322—22no—
183#322—22no—
184#322—22no—
185C82—22no—
186C22×Q8—21no—
187C22×Q8C22×C411yescharacteristic
188C22×Q8C22×C411yescharacteristic
189C22×Q8C22×C411yescharacteristic
190C22×D4—21no—
191C22×D4C22×C411yescharacteristic
192C23×C4—21no—
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4#4111yescharacteristic
196C2×Q16—22no—
197C2×Q16—22no—
198C2×Q16—22no—
199C2×Q16—22no—
200C2×Q16—41no—
201C2×Q16—41no—
202C2×SD16—22no—
203C2×SD16—22no—
204C2×SD16—22no—
205C2×SD16—22no—
206C2×SD16—41no—
207C2×SD16—41no—
208C2×M16—24no—
209C2×M16—24no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—24no—
213C2×M16—41no—
214C2×M16—41no—
215C2×M16—42no—
216C2×M16—44no—
217C2×M16C22⋊C414yes—
218C2×M16C22⋊C414yes—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—44no—
226C22×C8—42no—
227C22×C8C22×C411yescharacteristic
228C22×C8C22×C411yescharacteristic
229C22×C8C22×C411yescharacteristic
230C22×C8C22×C411yescharacteristic
231C22×C8C2×D412yes—
232C22×C8C2×D412yes—
233C4⋊Q8—22no—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237#126—42no—
238#126—42no—
239#125—22no—
240#125—22no—
241#125—42no—
242#125—42no—
243#124—22no—
244#124—22no—
245#123—42no—
246#123—42no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252C22⋊Q8—22no—
253C22⋊Q8—42no—
254C22⋊Q8—42no—
255#121—22no—
256#121—22no—
257C4×Q8—21no—
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—42no—
270C4×Q8—42no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—42no—
276C4×D4—42no—
277#117—42no—
278#117—42no—
279#117—42no—
280#117—42no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C4⋊C4C22×C411yescharacteristic
294C2×C4⋊C4C22×C411yescharacteristic
295C2×C4⋊C4C22×C411yescharacteristic
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42C22×C411yescharacteristic
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D411yescharacteristic
311C2×C42#4112yes—
312C2×C42#4111yescharacteristic
313C2×C42C2411yescharacteristic
314#107—22no—
315#107—22no—
316#106—22no—
317#106—22no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—42no—
327#103—22no—
328#103—22no—
329#103—22no—
330#103—22no—
331#103—22no—
332#103—22no—
333#102—22no—
334#102—22no—
335C22⋊C8—44no—
336C22⋊C8—42no—
337C22⋊C8—44no—
338#97—24no—
339#97—24no—
340#97C22⋊C414yes—
341#97C22⋊C414yes—
342C4×C8—21no—
343C4×C8—21no—
344C4×C8—21no—
345C4×C8—21no—
346C4×C8—21no—
347C4×C8—21no—
348C4×C8—42no—
349C4×C8C2×D412yes—
350C4×C8C2×D412yes—
351C4×C8C2×D412yes—
352C4×C8C2×D412yes—
353C4×C8C2×D412yes—
354C4×C8C2×D412yes—
355C24—21no—
356C24C4×D411yescharacteristic
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—42no—
376C2×Q8—82no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—42no—
384C2×D4—82no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4C2×C22⋊C412yes—
404C22×C4C4×D412yes—
405C22×C4C4×D411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4C4×D411yescharacteristic
408C22×C4#12112yes—
409C22×C4#12112yes—
410C22×C4#12412yes—
411C22×C4#12712yes—
412C22×C4C23×C411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C2×#4111yescharacteristic
415Q16—44no—
416Q16—44no—
417SD16—44no—
418SD16—44no—
419M16—28no—
420M16—44no—
421M16—48no—
422M16—48no—
423C2×C8—24no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—42no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—84no—
447C2×C8C2×C22⋊C412yes—
448C2×C8C2×C22⋊C412yes—
449C2×C8C2×C22⋊C412yes—
450C2×C8C2×C22⋊C412yes—
451C2×C8#13112yes—
452C2×C8#13112yes—
453C2×C8#13714yes—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42C2×C22⋊C412yes—
491C42C4×D412yes—
492C42#12112yes—
493C42#12112yes—
494C42#12412yes—
495C42#12712yes—
496C42C22×D411yescharacteristic
497C42C22×D411yescharacteristic
498C42C22×D411yescharacteristic
499C42C2×#4111yescharacteristic
500C23—21no—
501C23—21no—
502C23—22no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23#39012yes—
510C23C2×C4×D411yescharacteristic
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—41no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522Q8—41no—
523Q8—84no—
524D4—41no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—84no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—82no—
563C2×C4#39012yes—
564C2×C4#39012yes—
565C2×C4#39012yes—
566C2×C4C22×C22⋊C411yescharacteristic
567C2×C4C2×C4×D411yescharacteristic
568C2×C4C2×#12111yescharacteristic
569C2×C4C2×#12111yescharacteristic
570C2×C4C2×#12411yescharacteristic
571C2×C4C2×#12711yescharacteristic
572C2×C4C2×#13111yescharacteristic
573C2×C4C2×#13111yescharacteristic
574C2×C4#56811yescharacteristic
575C2×C4#56811yescharacteristic
576C2×C4C2×#13712yes—
577C2×C4#57512yes—
578C8—24no—
579C8—22no—
580C8—22no—
581C8—22no—
582C8—24no—
583C8—42no—
584C8—42no—
585C8—81no—
586C8—81no—
587C8—81no—
588C8—81no—
589C8#40814yes—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—42no—
594C22—42no—
595C22—82no—
596C22—82no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22C2×#39011yescharacteristic
601C22#293511yescharacteristic
602C22#293511yescharacteristic
603C22C4×#13712yes—
604C22#295612yes—
605C4—22no—
606C4—22no—
607C4—21no—
608C4—21no—
609C4—22no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—81no—
616C4—81no—
617C4C2×#40811yescharacteristic
618C4#287911yescharacteristic
619C4#293511yescharacteristic
620C4#293511yescharacteristic
621C4#305612yes—
622C4#315211yescharacteristic
623C4#315311yescharacteristic
624C2—42no—
625C2—161no—
626C2#1807712yes—
627C2#1807912yes—
628C2#2118911yescharacteristic
629C2#2123711yescharacteristic
630C2#2124111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×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. #21302C2semidirect
4. C2×#1565C2semidirect
6. C2×#1485C2semidirect
7. C2×#1455C2semidirect
8. #9666C2semidirect
9. #9664C2semidirect
10. #9662C2semidirect
11. #9661C2semidirect
12. #9660C2semidirect
13. #9658C2semidirect
14. #9656C2semidirect
15. #9655C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31722
#213782
#213022
C2×#15652
C2×#14832
C2×#14852
C2×#14552
#96662
#96642
#96622
#96612
#96602
#96582
#96562
#96552