Finite Group Databasebeta

← #204176

Subgroups — #204176

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic, complemented
3C2×#1749C211yesmaximal, characteristic, complemented
4C2×#1459C214yesmaximal, complemented
5#9654C212yesmaximal, complemented
6#9653C212yesmaximal, complemented
7#9652C214yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4C411yescharacteristic
10C22×C4×D4C2212yes—
11C23×C42C2211yescharacteristic
12C22×C4⋊C8—21no—
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C22×#97—21no—
16C4×#121—21no—
17C4×#121—21no—
18C4×#121—21no—
19C4×#121—21no—
20C4×#121—21no—
21C4×#121—21no—
22C4×#121—21no—
23C4×#121—21no—
24C2×#390C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#388C2212yes—
27C2×#386C412yes—
28C2×#380C411yescharacteristic
29C2×C4×C4⋊C4C411yescharacteristic
30C2×C4×C22⋊C4C412yes—
31#1797—21no—
32#1797—21no—
33#1797C2212yes—
34#1797C2212yes—
35#1749—21no—
36#1749—21no—
37#1749C2212yes—
38#1749C2212yes—
39C2×#336—21no—
40C2×#336—21no—
41#1459—216no—
42C2×#324C2218yes—
43C2×#323C2218yes—
44C23×D4C2×C412yes—
45C23×D4C2×C411yescharacteristic
46C24×C4C2×C412yes—
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8—22no—
51C23×C8—22no—
52C2×#121—28no—
53C2×#121—21no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121—21no—
57C2×C4×D4—28no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—28no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—44no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C412yes—
74C22×C22⋊C4C2×C412yes—
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2312yes—
77C22×C42—44no—
78C22×C42C2×C412yes—
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—44no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—28no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8D418yescomplemented
99C2×C22⋊C8C2314yes—
100C2×C22⋊C8C2314yes—
101C2×#97—22no—
102C2×#97—22no—
103C2×#97—44no—
104#390—21no—
105#390—21no—
106#390—21no—
107#390—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#386—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134C2×#95C2×C412yes—
135#336—22no—
136#336—22no—
137#336—22no—
138#336—22no—
139#324—216no—
140#323—216no—
141C25C22×C412yes—
142C25C22×C411yescharacteristic
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—28no—
146C22×D4—24no—
147C22×D4—24no—
148C22×D4—28no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4M1618yes—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—28no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C412yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C412yes—
181C23×C4C22×C412yes—
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D414yes—
188C23×C4#4114yes—
189C23×C4C2411yescharacteristic
190C22×C8—28no—
191C22×C8—24no—
192C22×C8—24no—
193C22×C8—44no—
194C22×C8—44no—
195C22×C8—44no—
196C22×C8—44no—
197C22×C8—44no—
198#121—28no—
199#121—28no—
200#121—28no—
201#121—28no—
202C4×D4—44no—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—48no—
206C4×D4—48no—
207C4×D4—44no—
208C4×D4—44no—
209C4×D4—44no—
210C4×D4—44no—
211C4×D4—44no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—28no—
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×C42—24no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—28no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C4⋊C8—44no—
274C4⋊C8—44no—
275C22⋊C8—28no—
276C22⋊C8—24no—
277C22⋊C8—28no—
278C22⋊C8—44no—
279C22⋊C8C2×D418yes—
280#97—44no—
281#97—44no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—24no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24C2×C22⋊C411yescharacteristic
301C24C2×C22⋊C411yescharacteristic
302C24C4×D414yes—
303C24C2×M1614yes—
304C24C2×M1614yes—
305C24#13114yes—
306C24C23×C411yescharacteristic
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—28no—
310C2×D4—28no—
311C2×D4—232no—
312C2×D4—42no—
313C2×D4—416no—
314C2×D4—42no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—416no—
320C2×D4—42no—
321C2×D4—42no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
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—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—416no—
381C22×C4—416no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—816no—
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C4×D414yes—
404C22×C4#12114yes—
405C22×C4#12114yes—
406C22×C4#12412yes—
407C22×C4#12412yes—
408C22×C4#12712yes—
409C22×C4#12712yes—
410C22×C4C2×M1614yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4C2×#4112yes—
415C2×C8—28no—
416C2×C8—44no—
417C2×C8—44no—
418C2×C8—416no—
419C2×C8—44no—
420C2×C8—44no—
421C2×C8—44no—
422C2×C8—88no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C22⋊C4—28no—
433C22⋊C4—28no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—44no—
459C42—44no—
460C42#9914yes—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—24no—
474C23—28no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—84no—
493C23—84no—
494C23—84no—
495C23—84no—
496C23#39012yes—
497C23#39012yes—
498C23#40712yes—
499C23#40712yes—
500C23#40812yes—
501C23#40812yes—
502C23#435116yes—
503C23C22×C22⋊C411yescharacteristic
504C23C2×C4×D412yes—
505C23C22×M1612yes—
506C23C2×#13112yes—
507D4—48no—
508D4—48no—
509D4—416no—
510D4—416no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—28no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—88no—
570C2×C4—88no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4#39012yes—
574C2×C4#39012yes—
575C2×C4C2×#9912yes—
576C2×C4C2×#9912yes—
577C2×C4C2×#9912yes—
578C2×C4#41012yes—
579C2×C4#41312yes—
580C2×C4#41312yes—
581C2×C4C2×#12112yes—
582C2×C4C2×#12411yescharacteristic
583C2×C4C2×#12711yescharacteristic
584C8—48no—
585C8—84no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—416no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—88no—
609C22—88no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22#195514yes—
615C22#195514yes—
616C22C2×#39011yescharacteristic
617C22C2×#40711yescharacteristic
618C22C2×#40811yescharacteristic
619C22C22×#9911yescharacteristic
620C22C2×#41011yescharacteristic
621C22C2×#41311yescharacteristic
622C22C2×#41311yescharacteristic
623C22C2×#43514yes—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—21no—
628C4—22no—
629C4—21no—
630C4—22no—
631C4—22no—
632C4—44no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—82no—
638C4—82no—
639C4#196414yes—
640C2—24no—
641C2—24no—
642C2—24no—
643C2—84no—
644C2#1736211yescharacteristic
645C2#1736311yescharacteristic
646C2#1738111yescharacteristic
647C2#1738211yescharacteristic
648C2C2×#195511yescharacteristic
649C2C2×#196411yescharacteristic
650C2#2121811yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1797C2semidirect
3. C2×#1749C2semidirect
4. C2×#1459C2semidirect
5. #9654C2semidirect
6. #9653C2semidirect
7. #9652C2semidirect
98. C2×C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C2×#17972
C2×#17492
C2×#14592
#96542
#96532
#96522