Finite Group Databasebeta

← #204396

Subgroups — #204396

Order 512.

1031 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3108C211yesmaximal, characteristic
2#21378C211yesmaximal, characteristic
3#21303C211yesmaximal, characteristic, complemented
4C2×#1574C211yesmaximal, characteristic, complemented
5C2×#1495C211yesmaximal, characteristic
6C2×#1479C211yesmaximal, characteristic, complemented
7C2×#1453C211yesmaximal, 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×#529C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18#3108—24no—
19#3108—24no—
20C2×#483C411yescharacteristic
21C2×#480C411yescharacteristic
22C2×#479C411yescharacteristic
23C2×#479C411yescharacteristic
24C2×#464C411yescharacteristic
25C2×#464C411yescharacteristic
26C2×#451C411yescharacteristic
27C2×#448C411yescharacteristic
28C2×C4×Q16C2211yescharacteristic
29C2×C4×Q16C2211yescharacteristic
30C2×C4×SD16C2211yescharacteristic
31C2×C4×SD16C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C2×#431C2211yescharacteristic
34C2×#431C2211yescharacteristic
35C2×#423C2211yescharacteristic
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#1728—22no—
49#1574—24no—
50C2×#335C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1495—24no—
55#1479—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#1453—24no—
65C2×#322C2211yescharacteristic
66C2×#322C2211yescharacteristic
67C2×#322C2211yescharacteristic
68C2×C82C2211yescharacteristic
69C23×Q8C2×C411yescharacteristic
70C22×Q16C2×C411yescharacteristic
71C22×Q16C2×C411yescharacteristic
72C22×SD16C2×C411yescharacteristic
73C22×SD16C2×C411yescharacteristic
74C22×M16—21no—
75C22×M16D412yes—
76#529—22no—
77#529—22no—
78#529—22no—
79#529—22no—
80C2×#126—21no—
81C2×#125C2×C411yescharacteristic
82C2×#124C2×C411yescharacteristic
83C2×#123—21no—
84C2×C22⋊Q8C2×C411yescharacteristic
85C2×#121C2×C411yescharacteristic
86C2×C4×Q8—21no—
87C2×C4×Q8—21no—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8C2×C411yescharacteristic
93C2×C4×Q8C2×C411yescharacteristic
94C2×C4×Q8C2311yescharacteristic
95C2×C4×Q8C2311yescharacteristic
96C2×C4×Q8C2311yescharacteristic
97C2×C4×D4—21no—
98C2×C4×D4C2311yescharacteristic
99C2×#117—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4C2×C411yescharacteristic
102C22×C42—21no—
103C22×C42C2311yescharacteristic
104#483—24no—
105#480—24no—
106#479—24no—
107#479—24no—
108#464—24no—
109#464—24no—
110#451—24no—
111#448—24no—
112C4×Q16—22no—
113C4×Q16—22no—
114C4×Q16—22no—
115C4×Q16—22no—
116C4×SD16—22no—
117C4×SD16—22no—
118C4×SD16—22no—
119C4×SD16—22no—
120#431—22no—
121#431—22no—
122#431—22no—
123#431D414yes—
124#431D414yes—
125#431D414yes—
126C2×#107C2×C411yescharacteristic
127C2×#107C2×C411yescharacteristic
128C2×#106C2×C411yescharacteristic
129C2×#106C2×C411yescharacteristic
130#423—22no—
131#423—22no—
132C2×C4⋊C8—21no—
133C2×C4⋊C8—21no—
134C2×C4⋊C8C2×C411yescharacteristic
135C2×C4⋊C8C2×C411yescharacteristic
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140C2×#103C2×C411yescharacteristic
141C2×#103C2×C411yescharacteristic
142C2×#103C2×C411yescharacteristic
143C2×#103C2×C411yescharacteristic
144C2×#103C2×C411yescharacteristic
145C2×#103C2×C411yescharacteristic
146C2×#102C2×C411yescharacteristic
147C2×#102C2×C411yescharacteristic
148C2×C22⋊C8—21no—
149C2×C22⋊C8—21no—
150C2×C22⋊C8—22no—
151C2×C22⋊C8—22no—
152C2×C22⋊C8C2×C411yescharacteristic
153C2×C22⋊C8C2×C411yescharacteristic
154C4×M16—22no—
155C4×M16—22no—
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×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—41no—
192C22×Q8—41no—
193C22×Q8C22×C411yescharacteristic
194C22×Q8C22×C411yescharacteristic
195C22×Q8C22×C411yescharacteristic
196C22×D4—21no—
197C22×D4C22×C411yescharacteristic
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4C22×C411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4#4111yescharacteristic
204C2×Q16—22no—
205C2×Q16—22no—
206C2×Q16—22no—
207C2×Q16—22no—
208C2×Q16—41no—
209C2×Q16—41no—
210C2×SD16—22no—
211C2×SD16—22no—
212C2×SD16—22no—
213C2×SD16—22no—
214C2×SD16—41no—
215C2×SD16—41no—
216C2×M16—24no—
217C2×M16—24no—
218C2×M16—42no—
219C2×M16—44no—
220C2×M16—42no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8C22×C411yescharacteristic
228C22×C8C22×C411yescharacteristic
229C22×C8C22×C411yescharacteristic
230C22×C8C22×C411yescharacteristic
231C22×C8C2×D412yes—
232C22×C8C2×D412yes—
233#126—42no—
234#126—42no—
235#125—22no—
236#125—22no—
237#124—22no—
238#124—22no—
239#123—42no—
240#123—42no—
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243#121—22no—
244#121—22no—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—42no—
262C4×Q8—42no—
263C4×Q8—42no—
264C4×Q8—42no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—42no—
270C4×D4—42no—
271#117—42no—
272#117—42no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42C22×C411yescharacteristic
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D411yescharacteristic
307C2×C42#4112yes—
308C2×C42#4111yescharacteristic
309C2×C42C2411yescharacteristic
310#107—22no—
311#107—22no—
312#106—22no—
313#106—22no—
314C4⋊C8—24no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—42no—
324C4⋊C8C22⋊C414yes—
325C4⋊C8C22⋊C414yes—
326#103—22no—
327#103—22no—
328#103—22no—
329#103—22no—
330#103—22no—
331#103—22no—
332#102—22no—
333#102—22no—
334C22⋊C8—24no—
335C22⋊C8—44no—
336C22⋊C8—42no—
337C22⋊C8—44no—
338C22⋊C8C22⋊C414yes—
339C22⋊C8C22⋊C414yes—
340#97—24no—
341C4×C8—21no—
342C4×C8—21no—
343C4×C8—21no—
344C4×C8—21no—
345C4×C8—21no—
346C4×C8—21no—
347C4×C8—42no—
348C4×C8C2×D412yes—
349C4×C8C2×D412yes—
350C4×C8C2×D412yes—
351C4×C8C2×D412yes—
352C4×C8C2×D412yes—
353C4×C8C2×D412yes—
354C24—21no—
355C24C4×D411yescharacteristic
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—28no—
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—42no—
374C2×Q8—42no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—42no—
379C2×Q8#13112yes—
380C2×Q8#13112yes—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—42no—
388C2×D4—82no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—82no—
412C22×C4C2×C22⋊C412yes—
413C22×C4C4×D412yes—
414C22×C4C4×D411yescharacteristic
415C22×C4C4×D411yescharacteristic
416C22×C4C4×D411yescharacteristic
417C22×C4#12112yes—
418C22×C4#12112yes—
419C22×C4#12412yes—
420C22×C4#12712yes—
421C22×C4C23×C411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2×#4111yescharacteristic
424Q16—44no—
425Q16—44no—
426SD16—44no—
427SD16—44no—
428M16—48no—
429M16—84no—
430C2×C8—22no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—42no—
455C2×C8C2×C22⋊C412yes—
456C2×C8C2×C22⋊C412yes—
457C2×C8C2×C22⋊C412yes—
458C2×C8C2×C22⋊C412yes—
459C2×C8#13714yes—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—42no—
493C42—42no—
494C42C2×C22⋊C412yes—
495C42C4×D412yes—
496C42#12112yes—
497C42#12112yes—
498C42#12412yes—
499C42#12712yes—
500C42C22×D411yescharacteristic
501C42C22×D411yescharacteristic
502C42C22×D411yescharacteristic
503C42C2×#4111yescharacteristic
504C23—21no—
505C23—21no—
506C23—22no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23#39012yes—
514C23C2×C4×D411yescharacteristic
515Q8—22no—
516Q8—22no—
517Q8—22no—
518Q8—22no—
519Q8—48no—
520Q8—41no—
521Q8—41no—
522Q8—41no—
523Q8—41no—
524Q8—41no—
525Q8—41no—
526Q8—41no—
527Q8—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—84no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—48no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—84no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—82no—
569C2×C4#39012yes—
570C2×C4#39012yes—
571C2×C4#39012yes—
572C2×C4C22×C22⋊C411yescharacteristic
573C2×C4C2×C4×D411yescharacteristic
574C2×C4C2×#12111yescharacteristic
575C2×C4C2×#12111yescharacteristic
576C2×C4C2×#12411yescharacteristic
577C2×C4C2×#12711yescharacteristic
578C2×C4C2×#13111yescharacteristic
579C2×C4C2×#13111yescharacteristic
580C2×C4#56811yescharacteristic
581C2×C4#56811yescharacteristic
582C2×C4C2×#13712yes—
583C2×C4#57512yes—
584C8—22no—
585C8—22no—
586C8—22no—
587C8—22no—
588C8—24no—
589C8—42no—
590C8—42no—
591C8—42no—
592C8—81no—
593C8—81no—
594C8—81no—
595C8—81no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—42no—
600C22—42no—
601C22—82no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22C2×#39011yescharacteristic
607C22#293311yescharacteristic
608C22#293311yescharacteristic
609C22C4×#13712yes—
610C22#295712yes—
611C4—22no—
612C4—22no—
613C4—21no—
614C4—21no—
615C4—22no—
616C4—42no—
617C4—42no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4#288011yescharacteristic
624C4#288011yescharacteristic
625C4#292911yescharacteristic
626C4#293011yescharacteristic
627C4#305612yes—
628C4#315211yescharacteristic
629C4#315311yescharacteristic
630C2—42no—
631C2—161no—
632C2#1801012yes—
633C2#1801212yes—
634C2#2119111yescharacteristic
635C2#2123711yescharacteristic
636C2#2124011yescharacteristic
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. #21303C2semidirect
4. C2×#1574C2semidirect
6. C2×#1479C2semidirect
7. C2×#1453C2semidirect
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×#31082
#213782
#213032
C2×#15742
C2×#14952
C2×#14792
C2×#14532
#96662
#96642
#96622
#96612
#96602
#96582
#96562
#96552