Finite Group Databasebeta

← #204156

Subgroups — #204156

Order 512.

1100 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34979C211yesmaximal, characteristic
2#21976C211yesmaximal, characteristic, complemented
3#20462C211yesmaximal, characteristic, complemented
4C2×#1497C211yesmaximal, characteristic, complemented
5C2×#1481C211yesmaximal, characteristic, complemented
6C2×#1488C211yesmaximal, characteristic
7C2×#1477C211yesmaximal, characteristic
8#9647C211yesmaximal, characteristic, complemented
9#9646C211yesmaximal, characteristic, complemented
10#9645C211yesmaximal, characteristic, complemented
11#9643C211yesmaximal, characteristic, complemented
12#9641C211yesmaximal, characteristic, complemented
13#9640C211yesmaximal, characteristic, complemented
14#9639C211yesmaximal, characteristic, complemented
15#9637C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22#2611—22no—
23#2568C411yescharacteristic
24#2547C411yescharacteristic
25#2546C411yescharacteristic
26#2542C411yescharacteristic
27#2499C411yescharacteristic
28#2499C411yescharacteristic
29#2432C411yescharacteristic
30#2432C411yescharacteristic
31#2415—22no—
32#2395—22no—
33C4×C4⋊Q8C2211yescharacteristic
34C4×C4⋊Q8C2211yescharacteristic
35C4×#124C2211yescharacteristic
36C4×#124C2211yescharacteristic
37C4×C22⋊Q8—22no—
38C4×#117C2211yescharacteristic
39#1940—22no—
40#1850—22no—
41#1842C2211yescharacteristic
42#1842C2211yescharacteristic
43#1831C2211yescharacteristic
44#1831C2211yescharacteristic
45#1768—22no—
46#1750—22no—
47#1744C2211yescharacteristic
48#1744C2211yescharacteristic
49C4×#97C2211yescharacteristic
50C42×C8C2211yescharacteristic
51#1497—24no—
52#1488—24no—
53#1481—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×Q8⋊C8C2211yescharacteristic
64C2×Q8⋊C8C2211yescharacteristic
65C2×Q8⋊C8C2211yescharacteristic
66C2×Q8⋊C8C2211yescharacteristic
67C2×D4⋊C8C2211yescharacteristic
68C2×D4⋊C8C2211yescharacteristic
69C2×D4⋊C8C2211yescharacteristic
70C2×D4⋊C8C2211yescharacteristic
71C22×#41C2×C411yescharacteristic
72C23×C8—21no—
73#533—22no—
74#533—22no—
75#533—22no—
76#533—22no—
77C2×C4⋊Q8C2×C411yescharacteristic
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×C4⋊Q8C2×C411yescharacteristic
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×#125—21no—
82C2×#124C2×C411yescharacteristic
83C2×#124C2×C411yescharacteristic
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8C2×C411yescharacteristic
87C2×C22⋊Q8C2×C411yescharacteristic
88C4×#41—22no—
89C4×#41—22no—
90C4×#41—22no—
91C4×#41—22no—
92C2×C4×Q8—21no—
93C2×C4×Q8C2×C411yescharacteristic
94C2×C4×Q8C2311yescharacteristic
95C2×C4×Q8C2311yescharacteristic
96C2×C4×Q8C2311yescharacteristic
97C2×C4×D4—21no—
98C2×C4×D4—21no—
99C2×C4×D4C2×C411yescharacteristic
100C2×C4×D4C2311yescharacteristic
101C2×#117—21no—
102C2×#117—21no—
103C2×#117C2×C411yescharacteristic
104C2×#117D412yes—
105C22×C42—21no—
106C22×C42C2311yescharacteristic
107#432—22no—
108#432—22no—
109#431—22no—
110#431—22no—
111#431D414yes—
112#431D414yes—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—22no—
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×#97—21no—
128C2×#97—22no—
129C2×C4×C8—22no—
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134#400—22no—
135#399—22no—
136#398—22no—
137#397—22no—
138#397—22no—
139#397—22no—
140#394—22no—
141#394—22no—
142#391—22no—
143#391C2×C411yescharacteristic
144#391C2×C411yescharacteristic
145#391C2×C411yescharacteristic
146#391C2×C411yescharacteristic
147#391C2×C411yescharacteristic
148#391C2×C411yescharacteristic
149#390C2×C411yescharacteristic
150#390C2×C411yescharacteristic
151#389—22no—
152#389—22no—
153#389—22no—
154#388—22no—
155#388—22no—
156#387—22no—
157#387—22no—
158#385—22no—
159#385—22no—
160#384C2×C411yescharacteristic
161#384C2×C411yescharacteristic
162#382C2×C411yescharacteristic
163#382C2×C411yescharacteristic
164#380—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4C2×C411yescharacteristic
168C4×C4⋊C4C2×C411yescharacteristic
169C4×C4⋊C4D412yes—
170C4×C22⋊C4—22no—
171C4×C22⋊C4—22no—
172C4×C22⋊C4C2×C411yescharacteristic
173C4×C22⋊C4C2×C411yescharacteristic
174C4×C22⋊C4D412yes—
175#376D412yes—
176C43C2311yescharacteristic
177C43C2311yescharacteristic
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—
190Q8⋊C8—22no—
191Q8⋊C8—22no—
192Q8⋊C8—22no—
193Q8⋊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×Q8—21no—
205C22×Q8C22×C411yescharacteristic
206C22×Q8C22×C411yescharacteristic
207C22×Q8C22×C411yescharacteristic
208C22×D4—21no—
209C22×D4C22×C411yescharacteristic
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4C22×C411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—42no—
225C22×C8—82no—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—41no—
233C4⋊Q8—41no—
234#125—42no—
235#125—42no—
236#124—22no—
237#124—22no—
238#124—41no—
239#124—41no—
240C22⋊Q8—22no—
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244C22⋊Q8—48no—
245C22⋊Q8—42no—
246C22⋊Q8—42no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—42no—
262C4×Q8—42no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—42no—
270C4×D4—42no—
271C4×D4—42no—
272C4×D4—42no—
273#117—22no—
274#117—22no—
275#117—44no—
276#117—42no—
277#117—42no—
278#117—42no—
279#117—42no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4C22⋊C412yes—
294C2×C4⋊C4C22⋊C412yes—
295C2×C4⋊C4C22×C411yescharacteristic
296C2×C4⋊C4C22×C411yescharacteristic
297C2×C4⋊C4C22×C411yescharacteristic
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4C22⋊C412yes—
309C2×C22⋊C4C22⋊C412yes—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42C22×C411yescharacteristic
318C2×C42C22×C411yescharacteristic
319C2×C42C22×C411yescharacteristic
320C2×C42C22×C411yescharacteristic
321C2×C42C22×C411yescharacteristic
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42#4112yes—
331C2×C42#4112yes—
332C2×C42C2411yescharacteristic
333C4⋊C8—24no—
334C4⋊C8—24no—
335C4⋊C8—21no—
336C4⋊C8—21no—
337C4⋊C8—21no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—21no—
341C4⋊C8—21no—
342C4⋊C8—21no—
343C4⋊C8—42no—
344C4⋊C8—82no—
345C4⋊C8—82no—
346C22⋊C8—24no—
347C22⋊C8—24no—
348C22⋊C8—42no—
349#97—44no—
350#97—42no—
351C4×C8—21no—
352C4×C8—21no—
353C4×C8—21no—
354C4×C8—21no—
355C4×C8—44no—
356C4×C8C2×D412yes—
357C4×C8C2×D412yes—
358C4×C8C2×D412yes—
359C4×C8C2×D412yes—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95C22⋊C412yes—
369#95C22⋊C412yes—
370#95C22⋊C412yes—
371#95C22⋊C412yes—
372C24—21no—
373C24C2×C22⋊C411yescharacteristic
374#41—28no—
375#41—44no—
376#41—44no—
377#41—44no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
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—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8#13112yes—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4#13112yes—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4C2×C22⋊C412yes—
431C22×C4C2×C22⋊C412yes—
432C22×C4C2×C22⋊C412yes—
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C22⋊C412yes—
437C22×C4C4×D412yes—
438C22×C4C4×D412yes—
439C22×C4#12112yes—
440C22×C4#12112yes—
441C22×C4#12412yes—
442C22×C4#12712yes—
443C22×C4C23×C411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—42no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—84no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C22⋊C4—24no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—24no—
514C42—42no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—42no—
522C42—42no—
523C42C4×D412yes—
524C42C4×D412yes—
525C42#12112yes—
526C42#12112yes—
527C42#12412yes—
528C42#12712yes—
529C42C22×D411yescharacteristic
530C42C22×D411yescharacteristic
531C42C2×#4111yescharacteristic
532C42C2×#4111yescharacteristic
533C23—21no—
534C23—21no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23#39012yes—
543C23C22×C22⋊C411yescharacteristic
544Q8—22no—
545Q8—22no—
546Q8—41no—
547Q8—41no—
548Q8—41no—
549Q8—41no—
550Q8—41no—
551Q8—41no—
552Q8—41no—
553Q8—41no—
554Q8—41no—
555Q8—41no—
556Q8—84no—
557D4—22no—
558D4—22no—
559D4—48no—
560D4—41no—
561D4—41no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—48no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—84no—
606C2×C4—82no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4#39012yes—
610C2×C4#39012yes—
611C2×C4#39012yes—
612C2×C4#42114yes—
613C2×C4C2×C4×D411yescharacteristic
614C2×C4C2×C4×D411yescharacteristic
615C2×C4C2×#12111yescharacteristic
616C2×C4C2×#12111yescharacteristic
617C2×C4C2×#12411yescharacteristic
618C2×C4C2×#12711yescharacteristic
619C2×C4C2×#13111yescharacteristic
620C2×C4C2×#13111yescharacteristic
621C2×C4#56811yescharacteristic
622C2×C4#56811yescharacteristic
623C2×C4#57712yes—
624C2×C4#57812yes—
625C8—42no—
626C8—42no—
627C8—42no—
628C8—42no—
629C8—81no—
630C8—81no—
631C8—81no—
632C8—81no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—42no—
637C22—42no—
638C22—44no—
639C22—42no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22C2×#39011yescharacteristic
644C22#288011yescharacteristic
645C22#288011yescharacteristic
646C22#289812yes—
647C22C2×#42112yes—
648C4—24no—
649C4—22no—
650C4—22no—
651C4—21no—
652C4—21no—
653C4—22no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4#292911yescharacteristic
665C4#293011yescharacteristic
666C4#293311yescharacteristic
667C4#293311yescharacteristic
668C4#313712yes—
669C4#315411yescharacteristic
670C4#315611yescharacteristic
671C2—42no—
672C2—82no—
673C2#1752312yes—
674C2#1752412yes—
675C2#2119111yescharacteristic
676C2#2120511yescharacteristic
677C2#2120911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41C2×C4
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. #21976C2semidirect
3. #20462C2semidirect
4. C2×#1497C2semidirect
5. C2×#1481C2semidirect
8. #9647C2semidirect
9. #9646C2semidirect
10. #9645C2semidirect
11. #9643C2semidirect
12. #9641C2semidirect
13. #9640C2semidirect
14. #9639C2semidirect
15. #9637C2semidirect

Maximal subgroups

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

subgroupindex
#349792
#219762
#204622
C2×#14972
C2×#14812
C2×#14882
C2×#14772
#96472
#96462
#96452
#96432
#96412
#96402
#96392
#96372