Finite Group Databasebeta

← #196024

Subgroups — #196024

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 666 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×#1466C212yesmaximal, complemented
5#9083C214yesmaximal, complemented
6#9080C214yesmaximal, complemented
7#7912C212yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4C412yes—
10C22×C4×D4C2211yescharacteristic
11C23×C42C2211yescharacteristic
12#2920—28no—
13C2×#432—21no—
14C2×#432C2212yes—
15C2×#432C2212yes—
16C2×#431—21no—
17C2×#431C2212yes—
18C2×#423—21no—
19C2×#407C2212yes—
20C2×C4×M16—21no—
21C4×#121—21no—
22C4×#121—21no—
23C4×#121—21no—
24C4×#121—21no—
25C4×#121—21no—
26C4×#121—21no—
27C4×#121—21no—
28C4×#121—21no—
29C2×#390C411yescharacteristic
30C2×#389C411yescharacteristic
31C2×#388C2212yes—
32C2×#386C2212yes—
33C2×#380C411yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C412yes—
36#1817C2212yes—
37#1797—21no—
38#1797—21no—
39#1779C2212yes—
40#1749—21no—
41#1749—21no—
42#1466—28no—
43C2×#323C2218yes—
44#1280C2218yes—
45C23×D4C2×C411yescharacteristic
46C23×D4C2×C412yes—
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2×C412yes—
50C24×C4C2311yescharacteristic
51C22×M16—22no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121—28no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—24no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—24no—
67C2×C4×D4—48no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C412yes—
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2312yes—
77C22×C42—24no—
78C22×C42—24no—
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C412yes—
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#432—28no—
93#432—24no—
94#432—24no—
95#432—24no—
96#432D418yescomplemented
97#431—28no—
98#431—28no—
99#423—28no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103#407—28no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8C2314yes—
108C2×C22⋊C8C2314yes—
109C4×M16—28no—
110C2×#97—22no—
111C2×#97—22no—
112C2×C4×C8—22no—
113#390—21no—
114#390—21no—
115#390—21no—
116#390—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
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#380—21no—
132#380—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141C2×#95C2×C412yes—
142C2×#95C2×C412yes—
143C2×#95C2312yes—
144#336—22no—
145#336—22no—
146#323—216no—
147C25C22×C411yescharacteristic
148C25C22×C412yes—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—42no—
158C22×D4—44no—
159C22×D4—44no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—28no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
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×C412yes—
189C23×C4C22×C412yes—
190C23×C4C22×C412yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2411yescharacteristic
196C2×M16—28no—
197C2×M16—44no—
198C2×M16—44no—
199C22×C8—24no—
200C22×C8—24no—
201#121—28no—
202#121—28no—
203#121—28no—
204#121—28no—
205C4×D4—48no—
206C4×D4—48no—
207C4×D4—48no—
208C4×D4—48no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—44no—
212C4×D4—44no—
213C4×D4—42no—
214C4×D4—42no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
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—24no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4M1618yescomplemented
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
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×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D414yes—
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42#4114yes—
281C4⋊C8—28no—
282C4⋊C8—28no—
283C4⋊C8—44no—
284C22⋊C8—28no—
285C22⋊C8—24no—
286C22⋊C8—24no—
287C22⋊C8—24no—
288C22⋊C8—44no—
289C22⋊C8—44no—
290C22⋊C8C2×D418yes—
291#97—28no—
292#97—44no—
293C4×C8—28no—
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—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—24no—
308C24—24no—
309C24—41no—
310C24—41no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24C2×C22⋊C411yescharacteristic
316C24C2×C22⋊C411yescharacteristic
317C24C2×M1614yes—
318C24C23×C411yescharacteristic
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—28no—
322C2×D4—28no—
323C2×D4—416no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—44no—
327C2×D4—416no—
328C2×D4—44no—
329C2×D4—416no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—42no—
333C2×D4—42no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
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—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
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—22no—
375C22×C4—24no—
376C22×C4—24no—
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—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—416no—
392C22×C4—416no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—816no—
408C22×C4#10014yes—
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×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C4×D414yes—
420C22×C4C4×D414yes—
421C22×C4C2×M1614yes—
422C22×C4C2×M1614yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D412yes—
426C22×C4C2×#4112yes—
427M16—416no—
428C2×C8—28no—
429C2×C8—44no—
430C2×C8—44no—
431C2×C8—44no—
432C2×C8—44no—
433C2×C8—44no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C22⋊C4—28no—
443C22⋊C4—28no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—42no—
449C22⋊C4—44no—
450C22⋊C4—42no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C42—21no—
454C42—21no—
455C42—22no—
456C42—22no—
457C42—21no—
458C42—21no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—44no—
468C42—44no—
469C42#12012yes—
470C42#12012yes—
471C42#12114yes—
472C42#12114yes—
473C42#12312yes—
474C42#12312yes—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—28no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—24no—
493C23—28no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—42no—
511C23—84no—
512C23—84no—
513C23—84no—
514C23—84no—
515C23#41012yes—
516C23C2×#10012yes—
517C23C22×C22⋊C411yescharacteristic
518C23C2×C4×D412yes—
519C23C22×M1612yes—
520C23#56812yes—
521D4—48no—
522D4—48no—
523D4—416no—
524D4—416no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—88no—
582C2×C4—88no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4#38612yes—
586C2×C4#38612yes—
587C2×C4#38612yes—
588C2×C4#38612yes—
589C2×C4#40712yes—
590C2×C4#40712yes—
591C2×C4#41012yes—
592C2×C4C2×#10012yes—
593C2×C4C2×#10012yes—
594C2×C4#41312yes—
595C2×C4C2×#12011yescharacteristic
596C2×C4C2×#12112yes—
597C2×C4C2×#12311yescharacteristic
598C8—48no—
599C8—84no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—416no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—88no—
625C22—88no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22C2×#38611yescharacteristic
631C22C2×#40711yescharacteristic
632C22#287911yescharacteristic
633C22C2×#41011yescharacteristic
634C22C2×#41011yescharacteristic
635C22C22×#10011yescharacteristic
636C22C2×#41311yescharacteristic
637C22D4×M1614yes—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—21no—
643C4—22no—
644C4—21no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—44no—
649C4—41no—
650C4—41no—
651C4—82no—
652C4—82no—
653C4#190014yes—
654C2—24no—
655C2—42no—
656C2—42no—
657C2—84no—
658C2#1736111yescharacteristic
659C2#1736411yescharacteristic
660C2#1740211yescharacteristic
661C2#1740311yescharacteristic
662C2#2089811yescharacteristic
663C2#2098311yescharacteristic
664C2C2×#190011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2920C2semidirect
3. #20715C2semidirect
4. C2×#1466C2semidirect
5. #9083C2semidirect
6. #9080C2semidirect
7. #7912C2semidirect
96. #432D4semidirect
243. C2×C22⋊C4M16semidirect

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×#14662
#90832
#90802
#79122