Finite Group Databasebeta

← #368165

Subgroups — #368165

Order 512.

1714 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2362C211yesmaximal, characteristic
2C2×#1894C211yesmaximal, characteristic
3C2×#1872C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5#14130C211yesmaximal, characteristic
6#11293C212yesmaximal, complemented
7#11067C212yesmaximal, complemented
8#10884C212yesmaximal, complemented
9#10789C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20#2362—24no—
21#2362—24no—
22C2×#391C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#388—21no—
26C2×#388—21no—
27C2×#387—21no—
28C2×#385—21no—
29C2×#380C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1894C418yescomplemented
34#1872—24no—
35#1736—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1444—21no—
45#1442—21no—
46C4⋊#95C2212yes—
47C22⋊#95—21no—
48C23×Q8C2311yescharacteristic
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—42no—
54C22×M16—42no—
55C22×M16C2×C414yes—
56C23×C8C2312yes—
57C23×C8C2312yes—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8C2×C414yes—
62C2×C22⋊Q8D414yes—
63C2×C22⋊Q8Q814yes—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—24no—
67C2×C4×Q8—42no—
68C2×#117—22no—
69C2×#117—24no—
70C2×#117—42no—
71C2×#117D414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4D411yescharacteristic
90C22×C22⋊C4D411yescharacteristic
91C22×C42—21no—
92C22×C42—21no—
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—42no—
99C2×#103—42no—
100C2×#103—42no—
101C2×#103C2×C418yescomplemented
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105#391—22no—
106#391—22no—
107#389—22no—
108#389—22no—
109#389—24no—
110#389—42no—
111#388—28no—
112#388—28no—
113#387—28no—
114#385—44no—
115#380—22no—
116#380C2×C414yes—
117C4×C4⋊C4—28no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95D411yescharacteristic
131C2×#95D411yescharacteristic
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—42no—
137#340C2×C418yescomplemented
138#336—42no—
139#336—42no—
140C25C2×D411yescharacteristic
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8C22×C412yes—
149C22×Q8C2×D412yes—
150C22×Q8C2×Q812yes—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—44no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—82no—
166C23×C4C22×C412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D412yes—
174C23×C4C2×Q812yes—
175C23×C4#4112yes—
176C23×C4#4112yes—
177C23×C4C2411yescharacteristic
178C2×M16—28no—
179C2×M16—48no—
180C2×M16—44no—
181C2×M16—44no—
182C2×M16—48no—
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×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8C22×C414yes—
197C22⋊Q8—28no—
198C22⋊Q8—28no—
199C22⋊Q8—28no—
200C22⋊Q8C4⋊C4116yescomplemented
201C4×Q8—48no—
202C4×Q8—44no—
203C4×Q8—44no—
204#117—28no—
205#117—44no—
206#117—48no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—22no—
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—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—82no—
241C2×C4⋊C4C22×C414yes—
242C2×C4⋊C4C22×C412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×Q812yes—
247C2×C4⋊C4#4114yes—
248C2×C4⋊C4#4114yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42—42no—
277C2×C42—42no—
278#103—48no—
279#103—48no—
280C22⋊C8—44no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—22no—
287#95—22no—
288#95—44no—
289#95—44no—
290#95—82no—
291C24—22no—
292C24—22no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—22no—
298C24—22no—
299C24C4×D412yes—
300C24#12012yes—
301C24C22⋊Q812yes—
302C24C22×D411yescharacteristic
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—42no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8C2×C4⋊C414yes—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—48no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—41no—
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—88no—
393C22×C4—88no—
394C22×C4—84no—
395C22×C4—84no—
396C22×C4C2×C4⋊C414yes—
397C22×C4C4×D414yes—
398C22×C4C4×D412yes—
399C22×C4#12012yes—
400C22×C4C22⋊Q814yes—
401C22×C4C22⋊Q812yes—
402C22×C4#12314yes—
403C22×C4AGL(1,ℤ/8)12yes—
404C22×C4#13712yes—
405C22×C4C23×C411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×Q811yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4111yescharacteristic
413C22×C4H(5,𝔽2)12yes—
414C22×C4H(5,𝔽2)12yes—
415M16—416no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—216no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—88no—
451C4⋊C4C2×C4⋊C414yes—
452C4⋊C4#12314yes—
453C22⋊C4—216no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C42—48no—
459C42—48no—
460C42—48no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—42no—
466C42—42no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23#38514yes—
490C23C2×C4×D411yescharacteristic
491C23C2×#12011yescharacteristic
492C23C2×C22⋊Q811yescharacteristic
493C23#54612yes—
494C23#54612yes—
495C23C2×AGL(1,ℤ/8)11yescharacteristic
496C23C2×#13711yescharacteristic
497C23#57511yescharacteristic
498C23#57511yescharacteristic
499C23#57712yes—
500C23#57812yes—
501Q8—28no—
502Q8—42no—
503Q8—42no—
504Q8—44no—
505Q8—42no—
506Q8—42no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—816no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4#38514yes—
575C2×C4C22×C4⋊C411yescharacteristic
576C2×C4C2×C4×D411yescharacteristic
577C2×C4C4×#4112yes—
578C2×C4#51812yes—
579C2×C4C2×C22⋊Q811yescharacteristic
580C2×C4C2×#12311yescharacteristic
581C2×C4C2×#12311yescharacteristic
582C2×C4#52511yescharacteristic
583C2×C4#52912yes—
584C2×C4#53411yescharacteristic
585C2×C4#53511yescharacteristic
586C2×C4#53812yes—
587C2×C4#54612yes—
588C2×C4#54612yes—
589C2×C4#55312yes—
590C2×C4#55312yes—
591C2×C4#55412yes—
592C2×C4#55412yes—
593C8—44no—
594C8—82no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22C2×#38511yescharacteristic
607C22#236412yes—
608C22#240411yescharacteristic
609C22#240511yescharacteristic
610C22#245012yes—
611C22C4×AGL(1,ℤ/8)11yescharacteristic
612C22C4×#13711yescharacteristic
613C22#294712yes—
614C22#300611yescharacteristic
615C22#300711yescharacteristic
616C22#301011yescharacteristic
617C22#301111yescharacteristic
618C22#307911yescharacteristic
619C22#308011yescharacteristic
620C22#308212yes—
621C22#331012yes—
622C22#331312yes—
623C22#331512yes—
624C22#331612yes—
625C22#332312yes—
626C22#332612yes—
627C22#332812yes—
628C22#333012yes—
629C4—24no—
630C4—44no—
631C4—42no—
632C4—44no—
633C4—42no—
634C4—44no—
635C4—84no—
636C4—81no—
637C4—84no—
638C4—82no—
639C4—81no—
640C4C4×#12311yescharacteristic
641C4#232111yescharacteristic
642C4#234912yes—
643C4#244412yes—
644C4#248711yescharacteristic
645C4#257911yescharacteristic
646C2—44no—
647C2#1401611yescharacteristic
648C2#2087511yescharacteristic
649C2#2087711yescharacteristic
650C2#2313812yes—
651C2#2313912yes—
652C2#2398011yescharacteristic
653C2#2398211yescharacteristic
654C2#2405211yescharacteristic
655C2#2405411yescharacteristic
656C2#2642912yes—
657C2#2643512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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
6. #11293C2semidirect
7. #11067C2semidirect
8. #10884C2semidirect
9. #10789C2semidirect
33. #1894C4semidirect
101. C2×#103C2×C4semidirect
137. #340C2×C4semidirect
200. C22⋊Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23622
C2×#18942
C2×#18722
C2×#17362
#141302
#112932
#110672
#108842
#107892
#107312
#107152