Finite Group Databasebeta

← #204109

Subgroups — #204109

Order 512.

1157 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34624C211yesmaximal, characteristic
2#21972C211yesmaximal, characteristic, complemented
3#20472C211yesmaximal, characteristic, complemented
4C2×#1483C211yesmaximal, characteristic, complemented
5C2×#1484C211yesmaximal, characteristic, complemented
6C2×#1482C211yesmaximal, characteristic, complemented
7C2×#1480C211yesmaximal, characteristic, complemented
8#9648C211yesmaximal, characteristic, complemented
9#9646C211yesmaximal, characteristic, complemented
10#9645C211yesmaximal, characteristic, complemented
11#9644C211yesmaximal, characteristic, complemented
12#9642C211yesmaximal, characteristic, complemented
13#9640C211yesmaximal, characteristic, complemented
14#9639C211yesmaximal, characteristic, complemented
15#9638C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#529C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2662—22no—
23#2661—22no—
24#2660C411yescharacteristic
25#2657C411yescharacteristic
26#2641C411yescharacteristic
27#2640C411yescharacteristic
28#2620C411yescharacteristic
29#2620C411yescharacteristic
30#2619C411yescharacteristic
31#2619C411yescharacteristic
32#2392—22no—
33#2388—22no—
34C4×#127C2211yescharacteristic
35C4×#127C2211yescharacteristic
36C4×#124C2211yescharacteristic
37C4×#124C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1939—22no—
40#1850—22no—
41#1842C2211yescharacteristic
42#1842C2211yescharacteristic
43#1831C2211yescharacteristic
44#1831C2211yescharacteristic
45#1771—22no—
46#1753—22no—
47#1744C2211yescharacteristic
48#1744C2211yescharacteristic
49C4×#97C2211yescharacteristic
50C42×C8C2211yescharacteristic
51#1484—24no—
52#1483—24no—
53#1482—24no—
54#1480—24no—
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊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×#41—21no—
72C22×M16—21no—
73C23×C8—21no—
74#532—22no—
75#532—22no—
76#532—22no—
77#532—22no—
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×#127C2×C411yescharacteristic
80C2×#127C2×C411yescharacteristic
81C2×#127C2×C411yescharacteristic
82#529—22no—
83#529—22no—
84#529—22no—
85#529—22no—
86C2×#126—21no—
87C2×#125C2×C411yescharacteristic
88C2×#124—21no—
89C2×#124C2×C411yescharacteristic
90C2×#124C2×C411yescharacteristic
91C2×#124C2×C411yescharacteristic
92C2×#123—21no—
93C2×C22⋊Q8C2×C411yescharacteristic
94C2×#121—21no—
95C2×#121C2×C411yescharacteristic
96C2×#121C2×C411yescharacteristic
97C2×#121C2×C411yescharacteristic
98C2×C4×Q8C2311yescharacteristic
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4C2311yescharacteristic
104C2×C4×D4C2311yescharacteristic
105C2×C4×D4C2311yescharacteristic
106C2×#117—21no—
107C2×#117C2×C411yescharacteristic
108C2×#117C2×C411yescharacteristic
109C22×C42C2311yescharacteristic
110#431—22no—
111#431—22no—
112#431—22no—
113#431D414yescomplemented
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—
126C4×M16—22no—
127C4×M16D414yescomplemented
128C2×#97—21no—
129C2×#97—22no—
130C2×C4×C8—21no—
131C2×C4×C8—22no—
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136#398—22no—
137#397—22no—
138#396—22no—
139#396—22no—
140#395—22no—
141#394—22no—
142#394—22no—
143#394—22no—
144#394—22no—
145#394—22no—
146#394—22no—
147#393—22no—
148#392—22no—
149#392—22no—
150#392—22no—
151#392—22no—
152#391C2×C411yescharacteristic
153#391C2×C411yescharacteristic
154#390—22no—
155#390—22no—
156#390C2×C411yescharacteristic
157#390C2×C411yescharacteristic
158#390C2×C411yescharacteristic
159#390C2×C411yescharacteristic
160#390C2×C411yescharacteristic
161#390C2×C411yescharacteristic
162#388—22no—
163#388—22no—
164#384C2×C411yescharacteristic
165#384C2×C411yescharacteristic
166#382C2×C411yescharacteristic
167#382C2×C411yescharacteristic
168C4×C4⋊C4D412yes—
169C4×C4⋊C4D412yes—
170C4×C22⋊C4—22no—
171C4×C22⋊C4—22no—
172C4×C22⋊C4D412yes—
173C4×C22⋊C4D412yes—
174#376C2×C411yescharacteristic
175#376C2×C411yescharacteristic
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—
186D4⋊C8—22no—
187D4⋊C8—22no—
188D4⋊C8—22no—
189D4⋊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—42no—
199C2×#41—42no—
200C2×#41—82no—
201C2×#41—82no—
202C22×Q8—21no—
203C22×Q8C22×C411yescharacteristic
204C22×D4—21no—
205C22×D4—21no—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—21no—
211C22×D4C22×C411yescharacteristic
212C22×D4C22×C411yescharacteristic
213C22×D4C22×C411yescharacteristic
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4C22×C411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C2×M16—24no—
221C2×M16—44no—
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—42no—
231C22×C8—82no—
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234#127—22no—
235#127—22no—
236#127—24no—
237#127—42no—
238#126—42no—
239#126—42no—
240#125—22no—
241#125—22no—
242#124—22no—
243#124—22no—
244#124—24no—
245#124—42no—
246#124—42no—
247#124—42no—
248#123—42no—
249#123—42no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252#121—22no—
253#121—22no—
254#121—22no—
255#121—22no—
256#121—22no—
257#121—22no—
258#121—42no—
259#121—42no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8—21no—
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—21no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—42no—
277C4×D4—42no—
278C4×D4—42no—
279C4×D4—42no—
280C4×D4—82no—
281C4×D4—82no—
282C4×D4—82no—
283C4×D4—82no—
284#117—24no—
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—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4C22⋊C412yes—
296C2×C4⋊C4C22⋊C412yes—
297C2×C4⋊C4C22×C411yescharacteristic
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C4⋊C4C22×C411yescharacteristic
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4C22⋊C412yes—
319C2×C22⋊C4C22⋊C412yes—
320C2×C42—21no—
321C2×C42C22×C411yescharacteristic
322C2×C42C22×C411yescharacteristic
323C2×C42C22×C411yescharacteristic
324C2×C42C22×C411yescharacteristic
325C2×C42C22×C411yescharacteristic
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42#4112yes—
335C2×C42#4112yes—
336C2×C42C2411yescharacteristic
337C4⋊C8—24no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—21no—
341C4⋊C8—21no—
342C4⋊C8—21no—
343C4⋊C8—21no—
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—42no—
347C4⋊C8—82no—
348C4⋊C8—82no—
349C22⋊C8—24no—
350C22⋊C8—42no—
351#97—24no—
352#97—44no—
353C4×C8—21no—
354C4×C8—21no—
355C4×C8—21no—
356C4×C8—21no—
357C4×C8—44no—
358C4×C8—42no—
359C4×C8C2×D412yes—
360C4×C8C2×D412yes—
361C4×C8C2×D412yes—
362C4×C8C2×D412yes—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95C22⋊C412yes—
368#95C22⋊C412yes—
369#95C22⋊C412yes—
370#95C22⋊C412yes—
371C24—21no—
372C24—21no—
373C24—21no—
374C24C2×C22⋊C411yescharacteristic
375#41—88no—
376#41—84no—
377#41—84no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—42no—
383C2×Q8—82no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—44no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4—82no—
404C2×D4—82no—
405C2×D4—82no—
406C2×D4—82no—
407C2×D4—82no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4—82no—
411C2×D4—82no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—42no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4C2×C22⋊C412yes—
435C22×C4C2×C22⋊C412yes—
436C22×C4C2×C22⋊C412yes—
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C412yes—
441C22×C4C4×D412yes—
442C22×C4C4×D412yes—
443C22×C4#12112yes—
444C22×C4#12112yes—
445C22×C4#12412yes—
446C22×C4#12712yes—
447C22×C4C23×C411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450M16—48no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—42no—
457C2×C8—42no—
458C2×C8—42no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—84no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C22⋊C4—24no—
487C22⋊C4—42no—
488C22⋊C4—44no—
489C22⋊C4—42no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—42no—
516C42—42no—
517C42—41no—
518C42—41no—
519C42C4×D412yes—
520C42C4×D412yes—
521C42#12112yes—
522C42#12112yes—
523C42#12412yes—
524C42#12712yes—
525C42#13114yes—
526C42C22×D411yescharacteristic
527C42C22×D411yescharacteristic
528C42C2×#4111yescharacteristic
529C42C2×#4111yescharacteristic
530C23—21no—
531C23—21no—
532C23—22no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23#39012yes—
550C23C22×C22⋊C411yescharacteristic
551Q8—42no—
552Q8—42no—
553Q8—84no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—88no—
561D4—84no—
562D4—84no—
563D4—88no—
564D4—84no—
565D4—84no—
566D4—84no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4#39012yes—
611C2×C4#39012yes—
612C2×C4#39012yes—
613C2×C4#40814yes—
614C2×C4C2×C4×D411yescharacteristic
615C2×C4C2×C4×D411yescharacteristic
616C2×C4C2×#12111yescharacteristic
617C2×C4C2×#12111yescharacteristic
618C2×C4C2×#12411yescharacteristic
619C2×C4C2×#12711yescharacteristic
620C2×C4C2×#13112yes—
621C2×C4#56812yes—
622C2×C4#57612yes—
623C2×C4#57712yes—
624C8—24no—
625C8—42no—
626C8—42no—
627C8—42no—
628C8—81no—
629C8—81no—
630C8—81no—
631C8—81no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—42no—
636C22—42no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22C2×#39011yescharacteristic
653C22C2×#40811yescharacteristic
654C22#287911yescharacteristic
655C22#289712yes—
656C22#290112yes—
657C4—24no—
658C4—22no—
659C4—22no—
660C4—21no—
661C4—21no—
662C4—22no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—81no—
667C4—81no—
668C4—81no—
669C4—81no—
670C4—81no—
671C4#293512yes—
672C4#293512yes—
673C4#313612yes—
674C4#315411yescharacteristic
675C4#315511yescharacteristic
676C2—42no—
677C2—161no—
678C2—161no—
679C2—161no—
680C2#1746812yes—
681C2#1750512yes—
682C2#2118911yescharacteristic
683C2#2120611yescharacteristic
684C2#2121011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#34624C2
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. #21972C2semidirect
3. #20472C2semidirect
4. C2×#1483C2semidirect
5. C2×#1484C2semidirect
6. C2×#1482C2semidirect
7. C2×#1480C2semidirect
8. #9648C2semidirect
9. #9646C2semidirect
10. #9645C2semidirect
11. #9644C2semidirect
12. #9642C2semidirect
13. #9640C2semidirect
14. #9639C2semidirect
15. #9638C2semidirect
113. #431D4semidirect
127. C4×M16D4semidirect

Maximal subgroups

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

subgroupindex
#346242
#219722
#204722
C2×#14832
C2×#14842
C2×#14822
C2×#14802
#96482
#96462
#96452
#96442
#96422
#96402
#96392
#96382