Finite Group Databasebeta

← #414656

Subgroups — #414656

Order 512.

1752 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2C22×C4×Q16C211yesmaximal, characteristic
3C2×#2567C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic
5C2×#1803C211yesmaximal, characteristic
6#13926C211yesmaximal, characteristic
7#11324C212yesmaximal
8#11124C212yesmaximal
9#11050C212yesmaximal
10#11024C212yesmaximal
11#10834C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C2×#477—24no—
17C2×#477—24no—
18C2×#477—24no—
19C2×C4×Q16—24no—
20C2×C4×Q16—24no—
21C2×C4×Q16—24no—
22C22×#107C2211yescharacteristic
23C22×#107C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2212yes—
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2212yes—
29C22×C4×C8C2211yescharacteristic
30#2567—24no—
31#2567—24no—
32C2×#395—22no—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#389—22no—
36C2×#384C2212yes—
37C2×#384C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39#1990—24no—
40#1803—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1441—22no—
47C4⋊#95—22no—
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C23×Q8C2311yescharacteristic
51C23×Q8C2311yescharacteristic
52C22×Q16—22no—
53C22×Q16—22no—
54C22×Q16—22no—
55C22×Q16—22no—
56C22×Q16—22no—
57C22×Q16—22no—
58C22×Q16Q814yescomplemented
59C23×C8C2311yescharacteristic
60C23×C8C2312yes—
61C23×C8C2311yescharacteristic
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8D414yes—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×Q8D414yes—
79C2×C4×Q8D414yes—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4D412yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C42—21no—
92C22×C42—21no—
93C22×C42C2312yes—
94C22×C42C2311yescharacteristic
95#477—232no—
96C4×Q16—232no—
97C2×#107—24no—
98C2×#107—22no—
99C2×#107—24no—
100C2×#107—24no—
101C2×#107—22no—
102C2×#107—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8D414yes—
106C2×#103—24no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—24no—
110C2×#103—22no—
111C2×#103—22no—
112C2×#103—24no—
113C2×#103—22no—
114C2×#103—22no—
115C2×#103—24no—
116C2×#103—24no—
117C2×#103—42no—
118C2×C4×C8—22no—
119C2×C4×C8—22no—
120C2×C4×C8D414yes—
121#395—216no—
122#391—24no—
123#391—24no—
124#391—24no—
125#391—42no—
126#389—216no—
127#384—24no—
128#384—24no—
129#384—22no—
130#384—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C2×#95—24no—
134C2×#95—24no—
135C2×#95—24no—
136C2×#95D412yes—
137#340—24no—
138#340—24no—
139#340—24no—
140#336—24no—
141#336—42no—
142C22×Q8—24no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8—82no—
168C22×Q8—82no—
169C22×Q8C2×D412yes—
170C22×Q8C2×D412yes—
171C22×Q8C2×Q812yes—
172C22×Q8C2×Q812yes—
173C22×Q8#4112yes—
174C22×Q8#4112yes—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—21no—
180C23×C4—22no—
181C23×C4C2×D414yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
185C2×Q16—28no—
186C2×Q16—28no—
187C2×Q16—28no—
188C2×Q16—28no—
189C2×Q16—28no—
190C2×Q16—28no—
191C2×Q16—28no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—22no—
208C22×C8C2×D412yes—
209C22×C8C2×D414yes—
210C22×C8C2×D412yes—
211C22×C8C2×Q812yes—
212C22×C8#4112yes—
213C4⋊Q8—28no—
214C4⋊Q8—28no—
215C4⋊Q8—44no—
216C4⋊Q8—44no—
217C4×Q8—28no—
218C4×Q8—28no—
219C4×Q8—28no—
220C4×Q8—28no—
221C4×Q8—44no—
222C4×Q8—44no—
223C4×Q8—44no—
224C4×Q8—44no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—216no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—82no—
262C2×C4⋊C4—84no—
263C2×C4⋊C4—84no—
264C2×C4⋊C4Q1618yescomplemented
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D414yes—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42#4112yes—
286#107—28no—
287#107—216no—
288C4⋊C8—28no—
289#103—28no—
290#103—28no—
291#103—216no—
292#103—48no—
293C4×C8—28no—
294#95—216no—
295#95—24no—
296#95—48no—
297#95—84no—
298C24C22×D411yescharacteristic
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—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—48no—
354C2×Q8—88no—
355C2×Q8—84no—
356C2×Q8—84no—
357C2×Q8—88no—
358C2×Q8C22⋊Q814yes—
359C2×Q8C22⋊Q814yes—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4#12012yes—
403C22×C4#12012yes—
404C22×C4#12112yes—
405C22×C4#12112yes—
406C22×C4#12712yes—
407C22×C4#12712yes—
408C22×C4C2×Q1618yes—
409C22×C4C2×Q1614yes—
410C22×C4#13714yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×Q811yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4H(5,𝔽2)11yescharacteristic
422C22×C42−1+411yescharacteristic
423C22×C42−1+411yescharacteristic
424C22×C42−1+411yescharacteristic
425Q16—232no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—42no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8C22⋊Q814yes—
447C4⋊C4—216no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—41no—
455C4⋊C4—42no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—42no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—84no—
468C4⋊C4—816no—
469C4⋊C4—84no—
470C42—42no—
471C42—42no—
472C42—42no—
473C42—42no—
474C42—48no—
475C42—48no—
476C42#12014yes—
477C42#12014yes—
478C42#12114yes—
479C42#12114yes—
480C23C2×#12011yescharacteristic
481C23C2×#12111yescharacteristic
482C23C2×#12711yescharacteristic
483C23#54611yescharacteristic
484C23#54711yescharacteristic
485C23#54811yescharacteristic
486C23D4×Q811yescharacteristic
487C23C22×Q1612yes—
488C23C2×#13712yes—
489C23#57612yes—
490C23#57712yes—
491Q8—24no—
492Q8—24no—
493Q8—24no—
494Q8—24no—
495Q8—42no—
496Q8—42no—
497Q8—42no—
498Q8—42no—
499Q8—42no—
500Q8—42no—
501Q8—42no—
502Q8—42no—
503Q8—42no—
504Q8—42no—
505Q8—42no—
506Q8—42no—
507Q8—816no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—48no—
538C2×C4—44no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4#39214yes—
554C2×C4#39214yes—
555C2×C4#45118yes—
556C2×C4#46218yes—
557C2×C4#49414yes—
558C2×C4#49714yes—
559C2×C4C2×#12011yescharacteristic
560C2×C4C2×#12011yescharacteristic
561C2×C4C2×#12111yescharacteristic
562C2×C4C2×#12111yescharacteristic
563C2×C4C2×C22⋊Q811yescharacteristic
564C2×C4C2×C22⋊Q811yescharacteristic
565C2×C4C2×C22⋊Q811yescharacteristic
566C2×C4#52511yescharacteristic
567C2×C4#52511yescharacteristic
568C2×C4#52511yescharacteristic
569C2×C4#53611yescharacteristic
570C2×C4#53611yescharacteristic
571C2×C4#53611yescharacteristic
572C2×C4#53711yescharacteristic
573C2×C4#53711yescharacteristic
574C2×C4#53711yescharacteristic
575C2×C4#54611yescharacteristic
576C2×C4#54711yescharacteristic
577C2×C4#54811yescharacteristic
578C2×C4#54811yescharacteristic
579C2×C4#54811yescharacteristic
580C2×C4D4×Q811yescharacteristic
581C2×C4D4×Q811yescharacteristic
582C2×C4#55011yescharacteristic
583C2×C4#55211yescharacteristic
584C2×C4#55211yescharacteristic
585C2×C4#55411yescharacteristic
586C2×C4#55611yescharacteristic
587C8—24no—
588C8—42no—
589C8—44no—
590C22C2×#39211yescharacteristic
591C22#240711yescharacteristic
592C22#240911yescharacteristic
593C22#241511yescharacteristic
594C22#241711yescharacteristic
595C22#254011yescharacteristic
596C22#254211yescharacteristic
597C22C2×#45112yes—
598C22#301212yes—
599C22C2×#46212yes—
600C22#304512yes—
601C22C2×#49411yescharacteristic
602C22C2×#49711yescharacteristic
603C22#315411yescharacteristic
604C22#315511yescharacteristic
605C22#329912yes—
606C22#333312yes—
607C22#333812yes—
608C22#335612yes—
609C22#336312yes—
610C22#337512yes—
611C22Q8×Q1612yes—
612C22#338712yes—
613C4—24no—
614C4—48no—
615C4—44no—
616C4—44no—
617C4—82no—
618C4—82no—
619C4—84no—
620C4—162no—
621C4#201618yes—
622C4#240711yescharacteristic
623C4#240911yescharacteristic
624C4#241511yescharacteristic
625C4#241711yescharacteristic
626C4#253611yescharacteristic
627C4#253811yescharacteristic
628C4#261611yescharacteristic
629C4#284811yescharacteristic
630C2#1517711yescharacteristic
631C2C2×#201611yescharacteristic
632C2#2342611yescharacteristic
633C2#2412212yes—
634C2#2433312yes—
635C2#2468212yes—
636C2#2481712yes—
637C2#2840111yescharacteristic
638C2#2841811yescharacteristic
639C2#2853111yescharacteristic
640C2#2854211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×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
58. C22×Q16Q8semidirect
264. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
C22×C4×Q162
C2×#25672
C2×#19902
C2×#18032
#139262
#113242
#111242
#110502
#110242
#108342