Finite Group Databasebeta

← #395259

Subgroups — #395259

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic, complemented
2C2×#2362C211yesmaximal, characteristic
3C2×#2057C211yesmaximal, characteristic, complemented
4C2×#1854C212yesmaximal
5#15125C211yesmaximal, characteristic
6#11227C211yesmaximal, characteristic, complemented
7#11124C212yesmaximal, complemented
8#11042C211yesmaximal, characteristic
9#10970C212yesmaximal
10#10908C212yesmaximal, complemented
11#10789C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#475—28no—
17C2×#475—24no—
18C22×#107C2212yes—
19C22×#106C2212yes—
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×C22⋊C8C2212yes—
24C2×#393—21no—
25#2362—24no—
26#2362—24no—
27C2×#391C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#388—21no—
30C2×#388—21no—
31C2×#387—21no—
32C2×#380C2212yes—
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36#2057—24no—
37#1854—28no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2212yes—
44#1446—21no—
45#1442—21no—
46#1439—21no—
47C4⋊#95C2211yescharacteristic
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C23×Q8C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8Q814yescomplemented
64C2×C4×Q8—22no—
65C2×C4×Q8—24no—
66C2×C4×Q8—42no—
67C2×C4×Q8D414yes—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—24no—
71C2×#117—42no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42—21no—
87C22×C42—21no—
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#475—232no—
92C2×#107—24no—
93C2×#107—24no—
94C2×#107—42no—
95C2×#106—24no—
96C2×#106—24no—
97C2×#106D418yes—
98C2×C4⋊C8—24no—
99C2×C4⋊C8—22no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—42no—
106C2×C22⋊C8—24no—
107C2×C22⋊C8—24no—
108C2×C22⋊C8—24no—
109#393—44no—
110#391—22no—
111#391—22no—
112#389—22no—
113#389D414yes—
114#388—28no—
115#388—28no—
116#387—28no—
117#380—24no—
118#380—42no—
119C4×C4⋊C4—28no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95D411yescharacteristic
134C2×#95D411yescharacteristic
135#340—24no—
136#340—24no—
137#340—24no—
138#340—24no—
139#336—24no—
140#336—22no—
141C25C2×D411yescharacteristic
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—44no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8—82no—
154C22×Q8C2×D412yes—
155C22×Q8C2×Q812yes—
156C22×Q8#4112yes—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×Q812yes—
177C23×C4#4112yes—
178C23×C4C2411yescharacteristic
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8C2×D414yes—
192C22×C8#4114yes—
193C4⋊Q8—28no—
194C4⋊Q8—48no—
195C4⋊Q8—44no—
196C22⋊Q8—28no—
197C22⋊Q8—28no—
198C22⋊Q8—28no—
199C22⋊Q8—28no—
200C4×Q8—28no—
201C4×Q8—44no—
202C4×Q8—48no—
203#117—48no—
204#117—44no—
205#117—44no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
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—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×Q812yes—
241C2×C4⋊C4#4112yes—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—82no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—44no—
266C2×C42—44no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271#107—48no—
272#106—216no—
273C4⋊C8—28no—
274#103—216no—
275#103—48no—
276C22⋊C8—216no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—22no—
282#95—22no—
283#95—44no—
284#95—42no—
285#95—44no—
286#95—42no—
287#95—82no—
288C24—21no—
289C24—21no—
290C24—22no—
291C24—22no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—22no—
297C24—22no—
298C24C22⋊Q812yes—
299C24C22×D411yescharacteristic
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—48no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—88no—
319C2×Q8—88no—
320C2×Q8—84no—
321C2×Q8—84no—
322C2×Q8C22⋊Q814yes—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4#12012yes—
397C22×C4#12012yes—
398C22×C4C22⋊Q812yes—
399C22×C4#13512yes—
400C22×C4AGL(1,ℤ/8)12yes—
401C22×C4#13712yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×Q811yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4112yes—
410C22×C4H(5,𝔽2)12yes—
411C22×C42−1+412yes—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C4⋊C4—216no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—48no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4#12014yes—
447C4⋊C4C22⋊Q814yes—
448C22⋊C4—216no—
449C22⋊C4—44no—
450C22⋊C4—48no—
451C22⋊C4—44no—
452C22⋊C4—88no—
453C42—22no—
454C42—22no—
455C42—24no—
456C42—48no—
457C42—48no—
458C42—48no—
459C42—41no—
460C42—41no—
461C42—42no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—22no—
466C23—24no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23C2×#12011yescharacteristic
486C23C2×C22⋊Q811yescharacteristic
487C23#52511yescharacteristic
488C23D4212yes—
489C23#54712yes—
490C23C2×#13511yescharacteristic
491C23C2×AGL(1,ℤ/8)11yescharacteristic
492C23C2×#13711yescharacteristic
493C23#57511yescharacteristic
494C23#57712yes—
495C23#57812yes—
496Q8—24no—
497Q8—24no—
498Q8—42no—
499Q8—42no—
500Q8—44no—
501Q8—44no—
502Q8—816no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4#44914yes—
570C2×C4C2×#12011yescharacteristic
571C2×C4C2×C22⋊Q811yescharacteristic
572C2×C4C2×C22⋊Q811yescharacteristic
573C2×C4#52511yescharacteristic
574C2×C4#52512yes—
575C2×C4#52511yescharacteristic
576C2×C4#53211yescharacteristic
577C2×C4#53212yes—
578C2×C4#53411yescharacteristic
579C2×C4#53611yescharacteristic
580C2×C4#53912yes—
581C2×C4#54212yes—
582C2×C4D4212yes—
583C2×C4#54812yes—
584C2×C4#54812yes—
585C2×C4D4×Q812yes—
586C2×C4#55112yes—
587C2×C4#55412yes—
588C8—44no—
589C8—44no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—44no—
594C22—44no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—82no—
600C22—82no—
601C22#240311yescharacteristic
602C22#243211yescharacteristic
603C22#244912yes—
604C22#246112yes—
605C22#284811yescharacteristic
606C22C2×#44911yescharacteristic
607C22#300711yescharacteristic
608C22#301011yescharacteristic
609C22#301111yescharacteristic
610C22#307611yescharacteristic
611C22#307911yescharacteristic
612C22#308212yes—
613C22#310711yescharacteristic
614C22#311511yescharacteristic
615C22#312112yes—
616C22#326512yes—
617C22#326912yes—
618C22#327612yes—
619C22#328212yes—
620C22#334212yes—
621C22#334412yes—
622C22#334812yes—
623C22#335312yes—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—42no—
628C4—42no—
629C4—44no—
630C4—44no—
631C4—82no—
632C4—84no—
633C4—81no—
634C4—162no—
635C4#238811yescharacteristic
636C4#240911yescharacteristic
637C4#245212yes—
638C4#246312yes—
639C4#256411yescharacteristic
640C4#263811yescharacteristic
641C2—82no—
642C2#1472311yescharacteristic
643C2#2385811yescharacteristic
644C2#2386511yescharacteristic
645C2#2570511yescharacteristic
646C2#2571511yescharacteristic
647C2#2633712yes—
648C2#2635312yes—
649C2#2679512yes—
650C2#2680812yes—
651C2#3295511yescharacteristic
652C2#3296411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#475C2semidirect
3. C2×#2057C2semidirect
6. #11227C2semidirect
7. #11124C2semidirect
10. #10908C2semidirect
11. #10789C2semidirect
63. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4752
C2×#23622
C2×#20572
C2×#18542
#151252
#112272
#111242
#110422
#109702
#109082
#107892