Finite Group Databasebeta

← #204143

Subgroups — #204143

Order 512.

1143 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34910C211yesmaximal, characteristic
2#20755C211yesmaximal, characteristic, complemented
3#20472C211yesmaximal, characteristic, complemented
4C2×#1495C211yesmaximal, characteristic, complemented
5C2×#1483C211yesmaximal, characteristic, complemented
6C2×#1489C211yesmaximal, characteristic, complemented
7C2×#1477C211yesmaximal, characteristic, complemented
8#9648C211yesmaximal, characteristic, complemented
9#9647C211yesmaximal, characteristic, complemented
10#9645C211yesmaximal, characteristic, complemented
11#9645C211yesmaximal, characteristic, complemented
12#9642C211yesmaximal, characteristic, complemented
13#9641C211yesmaximal, characteristic, complemented
14#9639C211yesmaximal, characteristic, complemented
15#9639C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2602—22no—
23#2601—22no—
24#2566C411yescharacteristic
25#2565C411yescharacteristic
26#2545C411yescharacteristic
27#2544C411yescharacteristic
28#2500C411yescharacteristic
29#2499C411yescharacteristic
30#2432C411yescharacteristic
31#2431C411yescharacteristic
32#2392—22no—
33C4×C4⋊Q8C2211yescharacteristic
34C4×#127C2211yescharacteristic
35C4×#124C2211yescharacteristic
36C4×#124C2211yescharacteristic
37#2290—22no—
38C4×#117C2211yescharacteristic
39#1832—22no—
40#1831C2211yescharacteristic
41#1831C2211yescharacteristic
42#1831C2211yescharacteristic
43#1831C2211yescharacteristic
44#1771—22no—
45#1753—22no—
46#1744C2211yescharacteristic
47#1744C2211yescharacteristic
48#1729—22no—
49C4×#97C2211yescharacteristic
50C42×C8C2211yescharacteristic
51#1495—24no—
52#1489—24no—
53#1483—24no—
54#1477—24no—
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65C2×D4⋊C8C2211yescharacteristic
66C2×D4⋊C8C2211yescharacteristic
67C2×D4⋊C8C2211yescharacteristic
68C2×D4⋊C8C2211yescharacteristic
69C2×D4⋊C8C2211yescharacteristic
70C2×D4⋊C8C2211yescharacteristic
71C22×#41C2×C411yescharacteristic
72C22×M16—21no—
73C22×M16—21no—
74C23×C8—21no—
75C2×C4⋊Q8C2×C411yescharacteristic
76C2×#127C2×C411yescharacteristic
77#529—22no—
78#529—22no—
79#529—22no—
80#529—22no—
81C2×#126—21no—
82C2×#125C2×C411yescharacteristic
83C2×#124C2×C411yescharacteristic
84C2×#124C2×C411yescharacteristic
85C2×#124C2×C411yescharacteristic
86C2×#123—21no—
87C2×C22⋊Q8C2×C411yescharacteristic
88C2×#121C2×C411yescharacteristic
89C4×#41—22no—
90C4×#41—22no—
91C4×#41—22no—
92C4×#41—22no—
93C2×C4×Q8C2×C411yescharacteristic
94C2×C4×Q8C2311yescharacteristic
95C2×C4×Q8C2311yescharacteristic
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C2×C4×D4—21no—
99C2×C4×D4—22no—
100C2×C4×D4C2×C411yescharacteristic
101C2×C4×D4C2311yescharacteristic
102C2×C4×D4C2311yescharacteristic
103C2×#117—21no—
104C2×#117—21no—
105C2×#117C2×C411yescharacteristic
106C2×#117D412yes—
107C22×C42—21no—
108C22×C42C2311yescharacteristic
109#431—22no—
110#431D414yescomplemented
111#423—22no—
112#423—22no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—22no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C22⋊C8—21no—
123C4×M16—22no—
124C4×M16D414yescomplemented
125C2×#97—21no—
126C2×#97—22no—
127C2×C4×C8—22no—
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132#398—22no—
133#397—22no—
134#396—22no—
135#396—22no—
136#394—22no—
137#394—22no—
138#394—22no—
139#392—22no—
140#391C2×C411yescharacteristic
141#391C2×C411yescharacteristic
142#391C2×C411yescharacteristic
143#391C2×C411yescharacteristic
144#390—22no—
145#390C2×C411yescharacteristic
146#390C2×C411yescharacteristic
147#390C2×C411yescharacteristic
148#390C2×C411yescharacteristic
149#389—22no—
150#388—22no—
151#388—22no—
152#388—22no—
153#388—22no—
154#387—22no—
155#387—22no—
156#386—22no—
157#386—22no—
158#385—22no—
159#385—22no—
160#384C2×C411yescharacteristic
161#384C2×C411yescharacteristic
162#382C2×C411yescharacteristic
163#382C2×C411yescharacteristic
164C4×C4⋊C4—22no—
165C4×C4⋊C4C2×C411yescharacteristic
166C4×C4⋊C4C2×C411yescharacteristic
167C4×C4⋊C4D412yes—
168C4×C22⋊C4—22no—
169C4×C22⋊C4—22no—
170C4×C22⋊C4C2×C411yescharacteristic
171C4×C22⋊C4C2×C411yescharacteristic
172C4×C22⋊C4D412yes—
173#376D412yes—
174C43C2311yescharacteristic
175C43C2311yescharacteristic
176#336—22no—
177#336—22no—
178#336—22no—
179#336—22no—
180#336—22no—
181#336—22no—
182Q8⋊C8—22no—
183Q8⋊C8—22no—
184Q8⋊C8—22no—
185Q8⋊C8—22no—
186Q8⋊C8—22no—
187Q8⋊C8—22no—
188Q8⋊C8—22no—
189Q8⋊C8—22no—
190D4⋊C8—22no—
191D4⋊C8—22no—
192D4⋊C8—22no—
193D4⋊C8—22no—
194D4⋊C8—22no—
195D4⋊C8—22no—
196D4⋊C8—22no—
197D4⋊C8—22no—
198C2×#41—22no—
199C2×#41—22no—
200C2×#41—22no—
201C2×#41—22no—
202C2×#41—41no—
203C2×#41—41no—
204C22×Q8C22×C411yescharacteristic
205C22×Q8C22×C411yescharacteristic
206C22×D4—21no—
207C22×D4—21no—
208C22×D4—22no—
209C22×D4C22×C411yescharacteristic
210C22×D4C22×C411yescharacteristic
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4C22×C411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C2×M16—24no—
219C2×M16—44no—
220C2×M16—44no—
221C2×M16—42no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—44no—
231C22×C8—42no—
232C4⋊Q8—22no—
233C4⋊Q8—41no—
234#127—22no—
235#127—41no—
236#126—42no—
237#126—42no—
238#125—22no—
239#125—22no—
240#124—22no—
241#124—22no—
242#124—22no—
243#124—22no—
244#124—41no—
245#124—41no—
246#123—42no—
247#123—42no—
248C22⋊Q8—22no—
249C22⋊Q8—22no—
250#121—22no—
251#121—22no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—44no—
273C4×D4—44no—
274C4×D4—42no—
275C4×D4—42no—
276C4×D4—42no—
277C4×D4—42no—
278C4×D4—42no—
279C4×D4—42no—
280#117—22no—
281#117—22no—
282#117—42no—
283#117—42no—
284#117—42no—
285#117—42no—
286#117—42no—
287#117—42no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4C22⋊C412yes—
298C2×C4⋊C4C22⋊C412yes—
299C2×C4⋊C4C22×C411yescharacteristic
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C4⋊C4C22×C411yescharacteristic
302C2×C4⋊C4C22×C411yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4C22⋊C412yes—
318C2×C22⋊C4C22⋊C412yes—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42C22×C411yescharacteristic
326C2×C42C22×C411yescharacteristic
327C2×C42C22×C411yescharacteristic
328C2×C42C22×C411yescharacteristic
329C2×C42C22×C411yescharacteristic
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42#4112yes—
339C2×C42#4112yes—
340C2×C42C2411yescharacteristic
341C4⋊C8—24no—
342C4⋊C8—21no—
343C4⋊C8—21no—
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8—21no—
348C4⋊C8—21no—
349C4⋊C8—21no—
350C4⋊C8—42no—
351C4⋊C8—42no—
352C22⋊C8—24no—
353#97—24no—
354#97—44no—
355C4×C8—21no—
356C4×C8—21no—
357C4×C8—21no—
358C4×C8—21no—
359C4×C8—44no—
360C4×C8C2×D412yes—
361C4×C8C2×D412yes—
362C4×C8C2×D412yes—
363C4×C8C2×D412yes—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95C22⋊C412yes—
371#95C22⋊C412yes—
372#95C22⋊C412yes—
373#95C22⋊C412yes—
374C24—21no—
375C24—21no—
376C24C2×C22⋊C411yescharacteristic
377#41—28no—
378#41—44no—
379#41—44no—
380#41—44no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8#13112yes—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—82no—
411C2×D4—82no—
412C2×D4#13112yes—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—82no—
439C22×C4C2×C22⋊C412yes—
440C22×C4C2×C22⋊C412yes—
441C22×C4C2×C22⋊C412yes—
442C22×C4C2×C22⋊C412yes—
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C4×D412yes—
447C22×C4C4×D412yes—
448C22×C4#12112yes—
449C22×C4#12112yes—
450C22×C4#12412yes—
451C22×C4#12712yes—
452C22×C4C23×C411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455M16—48no—
456M16—84no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—84no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—82no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—42no—
526C42—42no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—42no—
532C42—42no—
533C42C4×D412yes—
534C42C4×D412yes—
535C42#12112yes—
536C42#12112yes—
537C42#12412yes—
538C42#12712yes—
539C42C22×D411yescharacteristic
540C42C22×D411yescharacteristic
541C42C2×#4111yescharacteristic
542C42C2×#4111yescharacteristic
543C23—21no—
544C23—21no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—42no—
555C23—42no—
556C23—41no—
557C23#39012yes—
558C23C22×C22⋊C411yescharacteristic
559Q8—22no—
560Q8—22no—
561Q8—41no—
562Q8—41no—
563Q8—41no—
564Q8—41no—
565Q8—41no—
566Q8—41no—
567D4—22no—
568D4—22no—
569D4—48no—
570D4—41no—
571D4—41no—
572D4—41no—
573D4—41no—
574D4—41no—
575D4—41no—
576D4—88no—
577D4—84no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—48no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—84no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4#39012yes—
627C2×C4#39012yes—
628C2×C4#39012yes—
629C2×C4C2×C4×D411yescharacteristic
630C2×C4C2×C4×D411yescharacteristic
631C2×C4C2×#12111yescharacteristic
632C2×C4C2×#12111yescharacteristic
633C2×C4C2×#12411yescharacteristic
634C2×C4C2×#12711yescharacteristic
635C2×C4C2×#13111yescharacteristic
636C2×C4C2×#13111yescharacteristic
637C2×C4#56811yescharacteristic
638C2×C4#56811yescharacteristic
639C2×C4#57512yes—
640C2×C4#57512yes—
641C8—24no—
642C8—42no—
643C8—42no—
644C8—42no—
645C8—81no—
646C8—81no—
647C8—81no—
648C8—81no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—42no—
653C22—42no—
654C22—44no—
655C22—42no—
656C22—82no—
657C22—81no—
658C22—81no—
659C22—82no—
660C22—82no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22C2×#39011yescharacteristic
665C22#288011yescharacteristic
666C22#288011yescharacteristic
667C22#289612yes—
668C22#290112yes—
669C4—22no—
670C4—22no—
671C4—21no—
672C4—21no—
673C4—22no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—81no—
680C4—81no—
681C4—81no—
682C4—81no—
683C4—81no—
684C4#292911yescharacteristic
685C4#293011yescharacteristic
686C4#293311yescharacteristic
687C4#293311yescharacteristic
688C4#314212yes—
689C4#315311yescharacteristic
690C4#315311yescharacteristic
691C2—42no—
692C2—82no—
693C2—161no—
694C2#1751312yes—
695C2#1751612yes—
696C2#2119111yescharacteristic
697C2#2120111yescharacteristic
698C2#2121011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2431C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20755C2semidirect
3. #20472C2semidirect
4. C2×#1495C2semidirect
5. C2×#1483C2semidirect
6. C2×#1489C2semidirect
7. C2×#1477C2semidirect
8. #9648C2semidirect
9. #9647C2semidirect
10. #9645C2semidirect
11. #9645C2semidirect
12. #9642C2semidirect
13. #9641C2semidirect
14. #9639C2semidirect
15. #9639C2semidirect
110. #431D4semidirect
124. C4×M16D4semidirect

Maximal subgroups

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

subgroupindex
#349102
#207552
#204722
C2×#14952
C2×#14832
C2×#14892
C2×#14772
#96482
#96472
#96452
#96422
#96412
#96392