Finite Group Databasebeta

← #195878

Subgroups — #195878

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2C2×C4×#121C211yesmaximal, characteristic
3#20715C211yesmaximal, characteristic, complemented
4C2×#1490C212yesmaximal, complemented
5#9076C214yesmaximal, complemented
6#9074C214yesmaximal, complemented
7#7904C212yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C2211yescharacteristic
11C23×C42C2211yescharacteristic
12#2920—28no—
13C2×#432—21no—
14C2×#432—21no—
15C2×#432C2212yes—
16C2×#431C2212yescomplemented
17C2×#431C2212yescomplemented
18C2×#423—21no—
19C2×#407—21no—
20C2×C4×M16C2212yes—
21C4×#121—21no—
22C4×#121—21no—
23C4×#121—21no—
24C4×#121—21no—
25C4×#121C412yes—
26C4×#121C412yes—
27C4×#121C2212yes—
28C4×#121C2212yes—
29C2×#390C411yescharacteristic
30C2×#389C411yescharacteristic
31C2×#388—21no—
32C2×#386—21no—
33C2×#380C411yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4—21no—
36#1817—21no—
37#1797C2212yescomplemented
38#1797C2212yescomplemented
39#1779—21no—
40#1749—21no—
41#1749—21no—
42#1490—28no—
43C2×D4⋊C8C2218yescomplemented
44#1279C2218yescomplemented
45C23×D4—21no—
46C23×D4C2×C411yescharacteristic
47C24×C4—21no—
48C24×C4C2×C411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C22×M16—22no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121—28no—
55C2×#121C2×C412yes—
56C2×#121C2×C412yes—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—24no—
64C2×C4×D4—24no—
65C2×C4×D4—48no—
66C2×C4×D4C2×C412yes—
67C2×C4×D4C2312yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42—24no—
78C22×C42—24no—
79C22×C42—21no—
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#432—28no—
93#432—28no—
94#432—24no—
95#432—24no—
96#431—28no—
97#431—28no—
98#423—28no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8C2314yes—
102#407—28no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C4×M16—24no—
109C4×M16D418yescomplemented
110C2×#97—22no—
111C2×#97—22no—
112C2×C4×C8C2314yes—
113#390—21no—
114#390—21no—
115#390C2×C412yes—
116#390C2×C412yes—
117#389—21no—
118#389—21no—
119#389C2×C412yes—
120#389C2×C412yes—
121#388—22no—
122#388—22no—
123#388—22no—
124#388—22no—
125#386—22no—
126#386—22no—
127#386—22no—
128#386—22no—
129#380—21no—
130#380—21no—
131#380C2×C412yes—
132#380C2×C412yes—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2×C412yes—
136C4×C4⋊C4C2312yes—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144#336—22no—
145#336—22no—
146D4⋊C8—216no—
147C25—21no—
148C25C22⋊C411yescharacteristic
149C22×D4—28no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—24no—
153C22×D4—21no—
154C22×D4—24no—
155C22×D4—42no—
156C22×D4—44no—
157C22×D4—44no—
158C22×D4C22×C412yes—
159C23×C4—21no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C2×M16—28no—
196C2×M16—44no—
197C2×M16—44no—
198C22×C8—24no—
199C22×C8—24no—
200#121—28no—
201#121—28no—
202#121—24no—
203#121—24no—
204#121—24no—
205#121M1618yescomplemented
206C4×D4—28no—
207C4×D4—24no—
208C4×D4—24no—
209C4×D4—48no—
210C4×D4—48no—
211C4×D4—48no—
212C4×D4—48no—
213C4×D4—42no—
214C4×D4—44no—
215C4×D4—42no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—44no—
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D414yes—
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42#4114yes—
280C2×C42C2411yescharacteristic
281C4⋊C8—28no—
282C4⋊C8—28no—
283C4⋊C8—24no—
284C4⋊C8—24no—
285C22⋊C8—28no—
286C22⋊C8—28no—
287C22⋊C8—44no—
288C22⋊C8—44no—
289C22⋊C8—44no—
290#97—28no—
291#97—44no—
292C4×C8—24no—
293C4×C8C2×D418yescomplemented
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—24no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—41no—
308C24—41no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24C2×C22⋊C411yescharacteristic
316C24C2×C22⋊C411yescharacteristic
317C24C2×C22⋊C411yescharacteristic
318C2×D4—24no—
319C2×D4—28no—
320C2×D4—28no—
321C2×D4—24no—
322C2×D4—22no—
323C2×D4—24no—
324C2×D4—416no—
325C2×D4—42no—
326C2×D4—44no—
327C2×D4—432no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—42no—
332C2×D4C2×M1614yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—41no—
375C22×C4—41no—
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—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—416no—
389C22×C4—416no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—816no—
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C4×D414yes—
417C22×C4#12012yes—
418C22×C4#12114yes—
419C22×C4#12312yes—
420C22×C4C2×M1614yes—
421C22×C4AGL(1,ℤ/8)14yes—
422C22×C4C23×C411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425M16—416no—
426C2×C8—28no—
427C2×C8—44no—
428C2×C8—44no—
429C2×C8—44no—
430C2×C8—44no—
431C2×C8—44no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4C2×M1614yes—
441C22⋊C4—28no—
442C22⋊C4—28no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C42—22no—
453C42—22no—
454C42—21no—
455C42—21no—
456C42—22no—
457C42—22no—
458C42—21no—
459C42—21no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—44no—
467C42—44no—
468C42C4×D414yes—
469C42#12012yes—
470C42#12114yes—
471C42#12312yes—
472C42C22×D412yes—
473C42C2×#4112yes—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—28no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—24no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—48no—
498C23—48no—
499C23—48no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23#38612yes—
512C23#40712yes—
513C23#41812yes—
514C23#42112yes—
515C23#42112yes—
516C23C22×C22⋊C411yescharacteristic
517D4—28no—
518D4—48no—
519D4—416no—
520D4—416no—
521D4—44no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—88no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—88no—
581C2×C4#38612yes—
582C2×C4#38612yes—
583C2×C4#38612yes—
584C2×C4#40712yes—
585C2×C4#41812yes—
586C2×C4C2×C4×D412yes—
587C2×C4C2×#12011yescharacteristic
588C2×C4C2×#12112yes—
589C2×C4C2×#12311yescharacteristic
590C2×C4C22×M1612yes—
591C2×C4#56812yes—
592C2×C4C2×AGL(1,ℤ/8)12yes—
593C2×C4#57512yes—
594C8—48no—
595C8—84no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—416no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—816no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22#189714yes—
626C22C2×#38611yescharacteristic
627C22C2×#40711yescharacteristic
628C22#287911yescharacteristic
629C22C2×#41811yescharacteristic
630C22#289611yescharacteristic
631C22C2×#42111yescharacteristic
632C22C2×#42111yescharacteristic
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—21no—
638C4—21no—
639C4—22no—
640C4—44no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—82no—
646C4—82no—
647C4D4×M1614yes—
648C4#305514yes—
649C2—24no—
650C2—42no—
651C2—42no—
652C2—84no—
653C2#1744811yescharacteristic
654C2#1745211yescharacteristic
655C2#1749511yescharacteristic
656C2#1749711yescharacteristic
657C2#2089811yescharacteristic
658C2#2097311yescharacteristic
659C2C2×#189711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2920C2semidirect
3. #20715C2semidirect
4. C2×#1490C2semidirect
5. #9076C2semidirect
6. #9074C2semidirect
7. #7904C2semidirect
16. C2×#431C22semidirect
17. C2×#431C22semidirect
37. #1797C22semidirect
38. #1797C22semidirect
43. C2×D4⋊C8C22semidirect
44. #1279C22semidirect
109. C4×M16D4semidirect
205. #121M16semidirect
293. C4×C8C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C2×C4×#1212
#207152
C2×#14902
#90762
#90742
#79042