Finite Group Databasebeta

← #418670

Subgroups — #418670

Order 512.

1752 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic, complemented
2C2×#2538C211yesmaximal, characteristic
3C2×#2058C211yesmaximal, characteristic, complemented
4C2×#2023C211yesmaximal, characteristic
5C2×#1864C212yesmaximal
6#11380C211yesmaximal, characteristic, complemented
7#11327C212yesmaximal, complemented
8#11207C212yesmaximal, complemented
9#11066C212yesmaximal
10#10931C211yesmaximal, characteristic
11#10884C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C22×#106C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19#2567—28no—
20#2538—24no—
21#2538—24no—
22C2×#396—21no—
23C2×#396—21no—
24C2×#395—21no—
25C2×#395—21no—
26C2×#391—21no—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#384—21no—
34C2×#384C2211yescharacteristic
35C2×#380—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37#2058—24no—
38#2023—24no—
39#1864—28no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2211yescharacteristic
46C2×#336C2212yes—
47C2×#336C2211yescharacteristic
48C2×#336C2212yes—
49C23×Q8—21no—
50C23×Q8D411yescharacteristic
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—42no—
59C2×C22⋊Q8—28no—
60C2×C22⋊Q8—44no—
61C2×C22⋊Q8—44no—
62C2×C4×Q8—24no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—42no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—24no—
69C2×C4×D4—42no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C42—21no—
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C2×#106—24no—
95C2×#106—24no—
96C2×#106—42no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—42no—
102C2×#102Q818yescomplemented
103#396—28no—
104#396—28no—
105#395—28no—
106#395—28no—
107#391—28no—
108#390—22no—
109#390—22no—
110#390—22no—
111#390D414yes—
112#389—24no—
113#389—24no—
114#389—22no—
115#389—22no—
116#389—24no—
117#389—22no—
118#389D418yes—
119#389D414yes—
120#384—28no—
121#384—22no—
122#384—22no—
123#380—28no—
124C4×C4⋊C4—24no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95—21no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—24no—
135#336—22no—
136#336—24no—
137#336—22no—
138#336—42no—
139C25—21no—
140C22×Q8—24no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—41no—
146C22×Q8—41no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—82no—
150C22×Q8—82no—
151C22×Q8SD1614yes—
152C22×D4—22no—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4C2×D412yes—
159C22×D4C2×Q814yes—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
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×C8—22no—
192C22×C8C2×Q814yes—
193C4⋊Q8—48no—
194C4⋊Q8—48no—
195C22⋊Q8—432no—
196C4×Q8—48no—
197C4×Q8—48no—
198C4×D4—48no—
199C4×D4—48no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
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—82no—
242C2×C4⋊C4—82no—
243C2×C4⋊C4—82no—
244C2×C4⋊C4—82no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×Q814yes—
252C2×C4⋊C4#4114yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—82no—
265C2×C22⋊C4—82no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—22no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42C2×D414yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281#106—48no—
282#102—216no—
283#102—48no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—24no—
288#95—28no—
289C24—22no—
290C24—24no—
291C24—41no—
292C24—41no—
293C24—42no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—42no—
299C24—42no—
300C24C22×D411yescharacteristic
301C2×Q8—28no—
302C2×Q8—28no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—48no—
309C2×Q8—84no—
310C2×Q8—84no—
311C2×Q8—88no—
312C2×Q8—88no—
313C2×D4—24no—
314C2×D4—22no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—41no—
322C2×D4—42no—
323C2×D4—41no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4C4⋊Q814yes—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4#12012yes—
395C22×C4#12012yes—
396C22×C4#12012yes—
397C22×C4#12012yes—
398C22×C4C2×SD1612yes—
399C22×C4C2×SD1612yes—
400C22×C4C2×SD1612yes—
401C22×C4AGL(1,ℤ/8)12yes—
402C22×C4#13712yes—
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×Q812yes—
407C22×C4C2×#4112yes—
408C22×C4H(5,𝔽2)12yes—
409C22×C42−1+412yes—
410C22×C42−1+412yes—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—48no—
447C4⋊C4—88no—
448C4⋊C4—84no—
449C4⋊C4—88no—
450C4⋊C4C22⋊Q818yescomplemented
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—84no—
455C22⋊C4—84no—
456C42—24no—
457C42—24no—
458C42—24no—
459C42—24no—
460C42—48no—
461C42—42no—
462C42—42no—
463C23—24no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—42no—
469C23—44no—
470C23—42no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—81no—
481C23—81no—
482C23—82no—
483C23—81no—
484C23—81no—
485C23—82no—
486C23C2×#12011yescharacteristic
487C23C2×#12011yescharacteristic
488C23#53411yescharacteristic
489C23#54812yes—
490C23D4×Q812yes—
491C23C22×SD1611yescharacteristic
492C23C2×AGL(1,ℤ/8)11yescharacteristic
493C23C2×#13711yescharacteristic
494C23#57511yescharacteristic
495C23#57612yes—
496C23#57712yes—
497Q8—416no—
498Q8—816no—
499D4—28no—
500D4—44no—
501D4—44no—
502D4—44no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—28no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—816no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4#44814yes—
570C2×C4#45014yes—
571C2×C4C2×C22⋊Q812yes—
572C2×C4C2×C4⋊Q811yescharacteristic
573C2×C4#53211yescharacteristic
574C2×C4#53611yescharacteristic
575C2×C4#53612yes—
576C2×C4#53611yescharacteristic
577C2×C4#54212yes—
578C2×C4#54412yes—
579C2×C4D4212yes—
580C2×C4D4212yes—
581C2×C4#54712yes—
582C2×C4#54812yes—
583C2×C4D4×Q812yes—
584C2×C4D4×Q812yes—
585C2×C4#55412yes—
586C2×C4#56312yes—
587C8—44no—
588C8—82no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—84no—
593C22—84no—
594C22—82no—
595C22—84no—
596C22—82no—
597C22—84no—
598C22—82no—
599C22—82no—
600C22#240311yescharacteristic
601C22#240912yes—
602C22#243211yescharacteristic
603C22#243411yescharacteristic
604C22#272912yes—
605C22C2×#45011yescharacteristic
606C22C2×#44811yescharacteristic
607C22#300611yescharacteristic
608C22#300711yescharacteristic
609C22#300811yescharacteristic
610C22#301011yescharacteristic
611C22#301212yes—
612C22#318711yescharacteristic
613C22#319311yescharacteristic
614C22#319812yes—
615C22#336212yes—
616C22#336612yes—
617C22#336912yes—
618C22#337612yes—
619C22Q8×SD1612yes—
620C22#338412yes—
621C22#338512yes—
622C22#338912yes—
623C4—24no—
624C4—24no—
625C4—48no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4—81no—
633C4—84no—
634C4#244012yes—
635C4#245212yes—
636C4#267412yes—
637C4#267712yes—
638C2—84no—
639C2#1526511yescharacteristic
640C2#2391911yescharacteristic
641C2#2392411yescharacteristic
642C2#2432912yes—
643C2#2434712yes—
644C2#2564911yescharacteristic
645C2#2566211yescharacteristic
646C2#2583411yescharacteristic
647C2#2585011yescharacteristic
648C2#3256812yes—
649C2#3259312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2567C2semidirect
3. C2×#2058C2semidirect
6. #11380C2semidirect
7. #11327C2semidirect
8. #11207C2semidirect
11. #10884C2semidirect
102. C2×#102Q8semidirect
450. C4⋊C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#25382
C2×#20582
C2×#20232
C2×#18642
#113802
#113272
#112072
#110662
#109312
#108842