Finite Group Databasebeta

← #361902

Subgroups — #361902

Order 512.

1706 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×SD16C211yesmaximal, characteristic
2C2×#1834C211yesmaximal, characteristic, complemented
3C2×#1803C211yesmaximal, characteristic, complemented
4C2×#1802C211yesmaximal, characteristic
5C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
6#13926C211yesmaximal, characteristic, complemented
7#13906C211yesmaximal, characteristic
8#11378C212yesmaximal, complemented
9#11023C212yesmaximal
10#10926C212yesmaximal, complemented
11#10925C212yesmaximal
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×C4×SD16—24no—
16C2×C4×SD16—24no—
17C2×C4×SD16C418yescomplemented
18C22×#106C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2211yescharacteristic
24C22×C4×C8C2212yes—
25C22×C4×C8C2211yescharacteristic
26C2×#391C2212yes—
27C2×#390C2212yes—
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#380—22no—
31C2×C4×C4⋊C4C2211yescharacteristic
32#1834—24no—
33#1803—24no—
34#1802—24no—
35C8×C4⋊C4—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41#1441—22no—
42#1439—22no—
43C4⋊#95—22no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C22×SD16—22no—
50C22×SD16—22no—
51C22×SD16—22no—
52C22×SD16—22no—
53C22×SD16—22no—
54C22×SD16—22no—
55C22×SD16C2×C414yes—
56C23×C8C2312yes—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8C2×C414yes—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4C2×C414yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D412yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—24no—
83C22×C42—21no—
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86C4×SD16—232no—
87C2×#106—22no—
88C2×#106—22no—
89C2×#106C2×C414yes—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8D414yes—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—42no—
98C2×#103C2×C414yes—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—42no—
104C2×#102C2×C414yes—
105C2×C4×C8—24no—
106C2×C4×C8—42no—
107C2×C4×C8—42no—
108C2×C4×C8C2×C414yes—
109C2×C4×C8D414yes—
110C2×C4×C8D414yes—
111#391—24no—
112#391—42no—
113#390—24no—
114#390—42no—
115#384—22no—
116#384—22no—
117#384—22no—
118#384—22no—
119#380—48no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C2×#95—24no—
123C2×#95—24no—
124C2×#95—24no—
125C2×#95D412yes—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#336—24no—
131C25—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8C22×C412yes—
144C22×Q8#4112yes—
145C22×Q8#4112yes—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4C22×C412yes—
158C22×D4#4112yes—
159C22×D4#4112yes—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4C2×D414yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2411yescharacteristic
174C2×SD16—28no—
175C2×SD16—28no—
176C2×SD16—28no—
177C2×SD16—28no—
178C2×SD16—28no—
179C2×SD16—28no—
180C2×SD16—28no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8C22×C412yes—
196C22×C8C22×C412yes—
197C22×C8C2×D414yes—
198C22×C8C2×D412yes—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C22×C8C2×D412yes—
202C4×Q8—28no—
203C4×Q8—28no—
204C4×Q8—44no—
205C4×Q8—44no—
206C4×D4—28no—
207C4×D4—28no—
208C4×D4—44no—
209C4×D4—44no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—82no—
244C2×C4⋊C4—82no—
245C2×C4⋊C4SD1618yescomplemented
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C22×C412yes—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—48no—
252C2×C22⋊C4—48no—
253C2×C22⋊C4—84no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42C22×C412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42#4112yes—
268C2×C42#4112yes—
269#106—28no—
270C4⋊C8—28no—
271#103—28no—
272#103—48no—
273#102—28no—
274#102—48no—
275C4×C8—48no—
276C4×C8C22⋊C4116yescomplemented
277#95—24no—
278#95—48no—
279#95—48no—
280#95—84no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24C22×D411yescharacteristic
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8#11714yes—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4#11714yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—48no—
375C22×C4—416no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—88no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4C4×D412yes—
390C22×C4C4×D412yes—
391C22×C4#12112yes—
392C22×C4#12112yes—
393C22×C4#12712yes—
394C22×C4#12712yes—
395C22×C4C2×SD1618yes—
396C22×C4C2×SD1614yes—
397C22×C4#13514yes—
398C22×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4H(5,𝔽2)11yescharacteristic
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C42−1+411yescharacteristic
411C22×C42−1+411yescharacteristic
412SD16—232no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8C2×C22⋊C414yes—
433C2×C8C2×C22⋊C414yes—
434C4⋊C4—216no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—48no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—84no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C4⋊C4—84no—
456C22⋊C4—48no—
457C22⋊C4—816no—
458C42—42no—
459C42—42no—
460C42—42no—
461C42—42no—
462C42—48no—
463C42C2×C22⋊C414yes—
464C42C4×D414yes—
465C42#12114yes—
466C42#12414yes—
467C23—24no—
468C23—44no—
469C23—44no—
470C23—42no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—82no—
481C23—82no—
482C23—81no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23C2×C4×D411yescharacteristic
491C23C2×#12111yescharacteristic
492C23C2×#12711yescharacteristic
493C23#54611yescharacteristic
494C23#54611yescharacteristic
495C23#54811yescharacteristic
496C23#54811yescharacteristic
497C23C22×SD1612yes—
498C23C2×#13512yes—
499C23#57612yes—
500C23#57812yes—
501Q8—24no—
502Q8—24no—
503Q8—42no—
504Q8—42no—
505Q8—42no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509D4—24no—
510D4—24no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—42no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—816no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4#39014yes—
557C2×C4#39014yes—
558C2×C4C4×SD1618yes—
559C2×C4#46518yes—
560C2×C4#49214yes—
561C2×C4#49514yes—
562C2×C4C22×C22⋊C411yescharacteristic
563C2×C4C2×#11711yescharacteristic
564C2×C4C2×#11711yescharacteristic
565C2×C4C2×C4×D411yescharacteristic
566C2×C4#52011yescharacteristic
567C2×C4#52011yescharacteristic
568C2×C4C2×#12111yescharacteristic
569C2×C4#52511yescharacteristic
570C2×C4#52511yescharacteristic
571C2×C4#52511yescharacteristic
572C2×C4#52511yescharacteristic
573C2×C4C2×#12411yescharacteristic
574C2×C4#53511yescharacteristic
575C2×C4#53611yescharacteristic
576C2×C4#53711yescharacteristic
577C2×C4#53711yescharacteristic
578C2×C4#54611yescharacteristic
579C2×C4#54611yescharacteristic
580C2×C4#54811yescharacteristic
581C2×C4#54811yescharacteristic
582C2×C4#55211yescharacteristic
583C2×C4#55211yescharacteristic
584C2×C4#55211yescharacteristic
585C2×C4#55211yescharacteristic
586C2×C4#55311yescharacteristic
587C2×C4#55311yescharacteristic
588C2×C4#55311yescharacteristic
589C2×C4#55311yescharacteristic
590C8—24no—
591C8—24no—
592C8—44no—
593C22—44no—
594C22—48no—
595C22—84no—
596C22—82no—
597C22—84no—
598C22—82no—
599C22—84no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22C2×#39011yescharacteristic
605C22#235211yescharacteristic
606C22#235211yescharacteristic
607C22#253511yescharacteristic
608C22#253611yescharacteristic
609C22#254011yescharacteristic
610C22#254011yescharacteristic
611C22C2×C4×SD1612yes—
612C22#295112yes—
613C22C2×#46512yes—
614C22#306112yes—
615C22C2×#49211yescharacteristic
616C22C2×#49511yescharacteristic
617C22#315511yescharacteristic
618C22#315611yescharacteristic
619C22#329512yes—
620C22#329812yes—
621C22#330212yes—
622C22#330512yes—
623C22#335912yes—
624C22#336112yes—
625C22#336212yes—
626C22#336412yes—
627C4—24no—
628C4—44no—
629C4—48no—
630C4—82no—
631C4—82no—
632C4—162no—
633C4—162no—
634C4#196818yes—
635C4#228011yescharacteristic
636C4#231111yescharacteristic
637C4#235211yescharacteristic
638C4#235211yescharacteristic
639C4#253511yescharacteristic
640C4#253611yescharacteristic
641C4#256211yescharacteristic
642C4#256211yescharacteristic
643C2—84no—
644C2#1390511yescharacteristic
645C2C2×#196811yescharacteristic
646C2#2123011yescharacteristic
647C2#2276712yes—
648C2#2277012yes—
649C2#2814112yes—
650C2#2818612yes—
651C2#2839911yescharacteristic
652C2#2840011yescharacteristic
653C2#2840311yescharacteristic
654C2#2840411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1834C2semidirect
3. C2×#1803C2semidirect
5. C2×C8×C4⋊C4C2semidirect
6. #13926C2semidirect
8. #11378C2semidirect
10. #10926C2semidirect
17. C2×C4×SD16C4semidirect
245. C2×C4⋊C4SD16semidirect
276. C4×C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×SD162
C2×#18342
C2×#18032
C2×#18022
C2×C8×C4⋊C42
#139262
#139062
#113782
#110232
#109262
#109252