Finite Group Databasebeta

← #204142

Subgroups — #204142

Order 512.

1191 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34910C211yesmaximal, characteristic
2#20793C211yesmaximal, characteristic, complemented
3#20472C211yesmaximal, characteristic, complemented
4C2×#1495C211yesmaximal, characteristic, complemented
5C2×#1483C211yesmaximal, characteristic, complemented
6C2×#1489C211yesmaximal, characteristic, complemented
7C2×#1477C211yesmaximal, characteristic, complemented
8#9646C211yesmaximal, characteristic, complemented
9#9646C211yesmaximal, characteristic, complemented
10#9644C211yesmaximal, characteristic, complemented
11#9643C211yesmaximal, characteristic, complemented
12#9640C211yesmaximal, characteristic, complemented
13#9640C211yesmaximal, characteristic, complemented
14#9638C211yesmaximal, characteristic, complemented
15#9637C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#424—22no—
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23#2602—22no—
24#2601—22no—
25#2566C411yescharacteristic
26#2565C411yescharacteristic
27#2545C411yescharacteristic
28#2544C411yescharacteristic
29#2500C411yescharacteristic
30#2499C411yescharacteristic
31#2432C411yescharacteristic
32#2431C411yescharacteristic
33#2392—22no—
34C4×C4⋊Q8C2211yescharacteristic
35C4×#127C2211yescharacteristic
36C4×#124C2211yescharacteristic
37C4×#124C2211yescharacteristic
38#2290—22no—
39C4×#117C2211yescharacteristic
40#1843—22no—
41#1842C2211yescharacteristic
42#1842C2211yescharacteristic
43#1842C2211yescharacteristic
44#1842C2211yescharacteristic
45#1771—22no—
46#1753—22no—
47#1744C2211yescharacteristic
48#1744C2211yescharacteristic
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#424—48no—
112#423—22no—
113#423—22no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—22no—
120C2×C4⋊C8—22no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C22⋊C8—21no—
128C4×M16—22no—
129C4×M16D414yescomplemented
130C2×#97—21no—
131C2×#97—22no—
132C2×C4×C8—22no—
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137#398—22no—
138#397—22no—
139#396—22no—
140#396—22no—
141#394—22no—
142#394—22no—
143#394—22no—
144#392—22no—
145#391C2×C411yescharacteristic
146#391C2×C411yescharacteristic
147#391C2×C411yescharacteristic
148#391C2×C411yescharacteristic
149#390—22no—
150#390C2×C411yescharacteristic
151#390C2×C411yescharacteristic
152#390C2×C411yescharacteristic
153#390C2×C411yescharacteristic
154#389—22no—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#387—22no—
160#387—22no—
161#386—22no—
162#386—22no—
163#385—22no—
164#385—22no—
165#384C2×C411yescharacteristic
166#384C2×C411yescharacteristic
167#382C2×C411yescharacteristic
168#382C2×C411yescharacteristic
169C4×C4⋊C4—22no—
170C4×C4⋊C4C2×C411yescharacteristic
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4D412yes—
173C4×C22⋊C4—22no—
174C4×C22⋊C4—22no—
175C4×C22⋊C4C2×C411yescharacteristic
176C4×C22⋊C4C2×C411yescharacteristic
177C4×C22⋊C4D412yes—
178#376D412yes—
179C43C2311yescharacteristic
180C43C2311yescharacteristic
181#336—22no—
182#336—22no—
183Q8⋊C8—22no—
184Q8⋊C8—22no—
185Q8⋊C8—22no—
186Q8⋊C8—22no—
187Q8⋊C8—22no—
188Q8⋊C8—22no—
189Q8⋊C8—22no—
190Q8⋊C8—22no—
191D4⋊C8—22no—
192D4⋊C8—22no—
193D4⋊C8—22no—
194D4⋊C8—22no—
195D4⋊C8—22no—
196D4⋊C8—22no—
197D4⋊C8—22no—
198D4⋊C8—22no—
199C2×#41—22no—
200C2×#41—22no—
201C2×#41—22no—
202C2×#41—22no—
203C2×#41—41no—
204C2×#41—41no—
205C22×Q8C22×C411yescharacteristic
206C22×Q8C22×C411yescharacteristic
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—22no—
210C22×D4C22×C411yescharacteristic
211C22×D4C22×C411yescharacteristic
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4C22×C411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C2×M16—24no—
220C2×M16—44no—
221C2×M16—44no—
222C2×M16—42no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—44no—
232C22×C8—42no—
233C4⋊Q8—22no—
234C4⋊Q8—41no—
235#127—22no—
236#127—41no—
237#126—42no—
238#126—42no—
239#125—22no—
240#125—22no—
241#124—22no—
242#124—22no—
243#124—22no—
244#124—22no—
245#124—41no—
246#124—41no—
247#123—42no—
248#123—42no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251#121—22no—
252#121—22no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—42no—
274C4×D4—42no—
275C4×D4—42no—
276C4×D4—42no—
277C4×D4—42no—
278C4×D4—42no—
279C4×D4—44no—
280C4×D4—44no—
281#117—24no—
282#117—24no—
283#117—22no—
284#117—22no—
285#117—42no—
286#117—42no—
287#117—42no—
288#117—42no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4C22⋊C412yes—
299C2×C4⋊C4C22⋊C412yes—
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C4⋊C4C22×C411yescharacteristic
302C2×C4⋊C4C22×C411yescharacteristic
303C2×C4⋊C4C22×C411yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4C22⋊C412yes—
319C2×C22⋊C4C22⋊C412yes—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42C22×C411yescharacteristic
327C2×C42C22×C411yescharacteristic
328C2×C42C22×C411yescharacteristic
329C2×C42C22×C411yescharacteristic
330C2×C42C22×C411yescharacteristic
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42#4112yes—
340C2×C42#4112yes—
341C2×C42C2411yescharacteristic
342C4⋊C8—24no—
343C4⋊C8—21no—
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8—21no—
348C4⋊C8—21no—
349C4⋊C8—21no—
350C4⋊C8—21no—
351C4⋊C8—44no—
352C4⋊C8—44no—
353C4⋊C8—42no—
354C4⋊C8—42no—
355C4⋊C8—44no—
356C4⋊C8—44no—
357C22⋊C8—24no—
358#97—24no—
359#97—44no—
360C4×C8—21no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8—21no—
364C4×C8—44no—
365C4×C8C2×D412yes—
366C4×C8C2×D412yes—
367C4×C8C2×D412yes—
368C4×C8C2×D412yes—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95C22⋊C412yes—
376#95C22⋊C412yes—
377#95C22⋊C412yes—
378#95C22⋊C412yes—
379C24—21no—
380C24—21no—
381C24C2×C22⋊C411yescharacteristic
382#41—28no—
383#41—44no—
384#41—44no—
385#41—44no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8#13112yes—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4#13112yes—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—82no—
444C22×C4C2×C22⋊C412yes—
445C22×C4C2×C22⋊C412yes—
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C412yes—
450C22×C4C2×C22⋊C412yes—
451C22×C4C4×D412yes—
452C22×C4C4×D412yes—
453C22×C4#12112yes—
454C22×C4#12112yes—
455C22×C4#12412yes—
456C22×C4#12712yes—
457C22×C4C23×C411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460M16—48no—
461M16—84no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—84no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—44no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—82no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—22no—
527C42—22no—
528C42—42no—
529C42—42no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42—42no—
535C42—42no—
536C42C4×D412yes—
537C42C4×D412yes—
538C42#12112yes—
539C42#12112yes—
540C42#12412yes—
541C42#12712yes—
542C42AGL(1,ℤ/8)14yes—
543C42#13714yes—
544C42C22×D411yescharacteristic
545C42C22×D411yescharacteristic
546C42C2×#4111yescharacteristic
547C42C2×#4111yescharacteristic
548C23—21no—
549C23—21no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—42no—
555C23—42no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23#39012yes—
563C23C22×C22⋊C411yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—41no—
567Q8—41no—
568Q8—41no—
569Q8—41no—
570Q8—41no—
571Q8—41no—
572D4—22no—
573D4—22no—
574D4—48no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—88no—
582D4—84no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
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—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—48no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—84no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4#39012yes—
630C2×C4#39012yes—
631C2×C4#39012yes—
632C2×C4#41714yes—
633C2×C4#41714yes—
634C2×C4C2×C4×D411yescharacteristic
635C2×C4C2×C4×D411yescharacteristic
636C2×C4C2×#12111yescharacteristic
637C2×C4C2×#12111yescharacteristic
638C2×C4C2×#12411yescharacteristic
639C2×C4C2×#12711yescharacteristic
640C2×C4C2×#13111yescharacteristic
641C2×C4C2×#13111yescharacteristic
642C2×C4#56811yescharacteristic
643C2×C4#56811yescharacteristic
644C2×C4C2×AGL(1,ℤ/8)12yes—
645C2×C4C2×#13712yes—
646C8—24no—
647C8—42no—
648C8—42no—
649C8—42no—
650C8—81no—
651C8—81no—
652C8—81no—
653C8—81no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—42no—
658C22—42no—
659C22—44no—
660C22—42no—
661C22—82no—
662C22—81no—
663C22—81no—
664C22—82no—
665C22—82no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22C2×#39011yescharacteristic
670C22#288011yescharacteristic
671C22#288011yescharacteristic
672C22C2×#41712yes—
673C22#290112yes—
674C4—24no—
675C4—24no—
676C4—22no—
677C4—22no—
678C4—21no—
679C4—21no—
680C4—22no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—81no—
688C4—81no—
689C4#292911yescharacteristic
690C4#293011yescharacteristic
691C4#293311yescharacteristic
692C4#293311yescharacteristic
693C4#314112yes—
694C4#315111yescharacteristic
695C4#315211yescharacteristic
696C2—42no—
697C2—82no—
698C2—161no—
699C2#1751212yes—
700C2#1751512yes—
701C2#2119111yescharacteristic
702C2#2120011yescharacteristic
703C2#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. #20793C2semidirect
3. #20472C2semidirect
4. C2×#1495C2semidirect
5. C2×#1483C2semidirect
6. C2×#1489C2semidirect
7. C2×#1477C2semidirect
8. #9646C2semidirect
9. #9646C2semidirect
10. #9644C2semidirect
11. #9643C2semidirect
12. #9640C2semidirect
13. #9640C2semidirect
14. #9638C2semidirect
15. #9637C2semidirect
110. #431D4semidirect
129. C4×M16D4semidirect

Maximal subgroups

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

subgroupindex
#349102
#207932
#204722
C2×#14952
C2×#14832
C2×#14892
C2×#14772
#96462
#96442
#96432
#96402
#96382
#96372