Finite Group Databasebeta

← #204157

Subgroups — #204157

Order 512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34977C211yesmaximal, characteristic
2#21976C211yesmaximal, characteristic, complemented
3#20462C211yesmaximal, characteristic, complemented
4C2×#1496C211yesmaximal, characteristic, complemented
5C2×#1480C211yesmaximal, characteristic, complemented
6C2×#1488C211yesmaximal, characteristic, complemented
7C2×#1477C211yesmaximal, 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×C4×#41C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22#2610—22no—
23#2567C411yescharacteristic
24#2563C411yescharacteristic
25#2548C411yescharacteristic
26#2541C411yescharacteristic
27#2500C411yescharacteristic
28#2500C411yescharacteristic
29#2431C411yescharacteristic
30#2431C411yescharacteristic
31#2414—22no—
32#2388—22no—
33C4×#127C2211yescharacteristic
34C4×#127C2211yescharacteristic
35C4×#124C2211yescharacteristic
36C4×#124C2211yescharacteristic
37C4×#121—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#1496—24no—
52#1488—24no—
53#1480—24no—
54#1477—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×#41C2×C411yescharacteristic
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
82C2×#124—21no—
83C2×#124C2×C411yescharacteristic
84C2×#124C2×C411yescharacteristic
85C2×#121—21no—
86C2×#121—22no—
87C2×#121C2×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×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4C2×C411yescharacteristic
101C2×C4×D4C2311yescharacteristic
102C2×C4×D4C2311yescharacteristic
103C2×C4×D4C2311yescharacteristic
104C2×#117—21no—
105C2×#117C2×C411yescharacteristic
106C2×#117D412yes—
107C22×C42—21no—
108C22×C42C2311yescharacteristic
109#432—22no—
110#432—22no—
111#431—22no—
112#431—22no—
113#431D414yescomplemented
114#431D414yescomplemented
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—22no—
121C2×C4⋊C8—22no—
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×#97—21no—
130C2×#97—22no—
131C2×C4×C8—22no—
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136#395—22no—
137#394—22no—
138#394—22no—
139#394—22no—
140#393—22no—
141#392—22no—
142#392—22no—
143#392—22no—
144#391C2×C411yescharacteristic
145#391C2×C411yescharacteristic
146#390—22no—
147#390—22no—
148#390—22no—
149#390C2×C411yescharacteristic
150#390C2×C411yescharacteristic
151#390C2×C411yescharacteristic
152#390C2×C411yescharacteristic
153#390C2×C411yescharacteristic
154#390C2×C411yescharacteristic
155#389—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#386—22no—
161#386—22no—
162#384C2×C411yescharacteristic
163#384C2×C411yescharacteristic
164#382C2×C411yescharacteristic
165#382C2×C411yescharacteristic
166#380—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—
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—22no—
199C2×#41—22no—
200C2×#41—22no—
201C2×#41—22no—
202C2×#41—41no—
203C2×#41—41no—
204C2×#41—42no—
205C2×#41—42no—
206C2×#41—82no—
207C2×#41—82no—
208C22×Q8—21no—
209C22×Q8C22×C411yescharacteristic
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—21no—
217C22×D4C22×C411yescharacteristic
218C22×D4C22×C411yescharacteristic
219C22×D4C22×C411yescharacteristic
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4C22×C411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—42no—
237C22×C8—82no—
238C4⋊Q8—22no—
239C4⋊Q8—22no—
240#127—22no—
241#127—22no—
242#127—22no—
243#127—22no—
244#127—41no—
245#127—41no—
246#124—22no—
247#124—22no—
248#124—41no—
249#124—41no—
250#124—42no—
251#124—42no—
252#121—22no—
253#121—22no—
254#121—22no—
255#121—22no—
256#121—48no—
257#121—42no—
258#121—42no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×D4—22no—
266C4×D4—22no—
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—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—42no—
280C4×D4—42no—
281C4×D4—42no—
282C4×D4—42no—
283C4×D4—42no—
284C4×D4—42no—
285C4×D4—82no—
286C4×D4—82no—
287C4×D4—82no—
288C4×D4—82no—
289#117—22no—
290#117—22no—
291#117—44no—
292#117—42no—
293#117—42no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4C22⋊C412yes—
300C2×C4⋊C4C22⋊C412yes—
301C2×C4⋊C4C22×C411yescharacteristic
302C2×C4⋊C4C22×C411yescharacteristic
303C2×C4⋊C4C22×C411yescharacteristic
304C2×C4⋊C4C22×C411yescharacteristic
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—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4C22⋊C412yes—
321C2×C22⋊C4C22⋊C412yes—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42C22×C411yescharacteristic
328C2×C42C22×C411yescharacteristic
329C2×C42C22×C411yescharacteristic
330C2×C42C22×C411yescharacteristic
331C2×C42C22×C411yescharacteristic
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42#4112yes—
341C2×C42#4112yes—
342C2×C42C2411yescharacteristic
343C4⋊C8—24no—
344C4⋊C8—24no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8—21no—
348C4⋊C8—21no—
349C4⋊C8—21no—
350C4⋊C8—21no—
351C4⋊C8—21no—
352C4⋊C8—21no—
353C4⋊C8—42no—
354C4⋊C8—82no—
355C4⋊C8—82no—
356C22⋊C8—24no—
357C22⋊C8—24no—
358C22⋊C8—42no—
359#97—44no—
360#97—42no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8—21no—
364C4×C8—21no—
365C4×C8—44no—
366C4×C8C2×D412yes—
367C4×C8C2×D412yes—
368C4×C8C2×D412yes—
369C4×C8C2×D412yes—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95C22⋊C412yes—
375#95C22⋊C412yes—
376#95C22⋊C412yes—
377#95C22⋊C412yes—
378C24—21no—
379C24—21no—
380C24—21no—
381C24C2×C22⋊C411yescharacteristic
382#41—28no—
383#41—44no—
384#41—44no—
385#41—44no—
386#41—88no—
387#41—84no—
388#41—84no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—42no—
395C2×Q8—82no—
396C2×Q8#13112yes—
397C2×D4—21no—
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—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—82no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4#13112yes—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4C2×C22⋊C412yes—
462C22×C4C2×C22⋊C412yes—
463C22×C4C2×C22⋊C412yes—
464C22×C4C2×C22⋊C411yescharacteristic
465C22×C4C2×C22⋊C411yescharacteristic
466C22×C4C2×C22⋊C411yescharacteristic
467C22×C4C2×C22⋊C412yes—
468C22×C4C4×D412yes—
469C22×C4C4×D412yes—
470C22×C4#12112yes—
471C22×C4#12112yes—
472C22×C4#12412yes—
473C22×C4#12712yes—
474C22×C4C23×C411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—42no—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8—42no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—42no—
490C2×C8—42no—
491C2×C8—42no—
492C2×C8—42no—
493C2×C8—84no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—42no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—44no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—24no—
543C42—42no—
544C42—41no—
545C42—41no—
546C42—42no—
547C42—42no—
548C42C4×D412yes—
549C42C4×D412yes—
550C42#12112yes—
551C42#12112yes—
552C42#12412yes—
553C42#12712yes—
554C42C22×D411yescharacteristic
555C42C22×D411yescharacteristic
556C42C2×#4111yescharacteristic
557C42C2×#4111yescharacteristic
558C23—21no—
559C23—21no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—41no—
576C23—41no—
577C23#39012yes—
578C23C22×C22⋊C411yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—41no—
582Q8—41no—
583Q8—84no—
584D4—22no—
585D4—22no—
586D4—48no—
587D4—41no—
588D4—41no—
589D4—41no—
590D4—41no—
591D4—41no—
592D4—41no—
593D4—41no—
594D4—41no—
595D4—41no—
596D4—41no—
597D4—88no—
598D4—84no—
599D4—84no—
600D4—88no—
601D4—84no—
602D4—84no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—48no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—84no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—81no—
650C2×C4#39012yes—
651C2×C4#39012yes—
652C2×C4#39012yes—
653C2×C4#42114yes—
654C2×C4C2×C4×D411yescharacteristic
655C2×C4C2×C4×D411yescharacteristic
656C2×C4C2×#12111yescharacteristic
657C2×C4C2×#12111yescharacteristic
658C2×C4C2×#12411yescharacteristic
659C2×C4C2×#12711yescharacteristic
660C2×C4C2×#13111yescharacteristic
661C2×C4C2×#13111yescharacteristic
662C2×C4#56811yescharacteristic
663C2×C4#56811yescharacteristic
664C2×C4#57612yes—
665C2×C4#57712yes—
666C8—42no—
667C8—42no—
668C8—42no—
669C8—42no—
670C8—81no—
671C8—81no—
672C8—81no—
673C8—81no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—42no—
678C22—42no—
679C22—44no—
680C22—42no—
681C22—82no—
682C22—81no—
683C22—81no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22C2×#39011yescharacteristic
695C22#288011yescharacteristic
696C22#288011yescharacteristic
697C22#289712yes—
698C22C2×#42112yes—
699C4—24no—
700C4—22no—
701C4—22no—
702C4—21no—
703C4—21no—
704C4—22no—
705C4—42no—
706C4—42no—
707C4—42no—
708C4—42no—
709C4—81no—
710C4—81no—
711C4—81no—
712C4—81no—
713C4#292911yescharacteristic
714C4#293011yescharacteristic
715C4#293311yescharacteristic
716C4#293311yescharacteristic
717C4#313612yes—
718C4#315411yescharacteristic
719C4#315511yescharacteristic
720C2—42no—
721C2—82no—
722C2—161no—
723C2—161no—
724C2#1752112yes—
725C2#1752612yes—
726C2#2119111yescharacteristic
727C2#2120611yescharacteristic
728C2#2120911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#34977C2
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×#1496C2semidirect
5. C2×#1480C2semidirect
6. C2×#1488C2semidirect
7. C2×#1477C2semidirect
8. #9648C2semidirect
9. #9646C2semidirect
10. #9645C2semidirect
11. #9644C2semidirect
12. #9642C2semidirect
13. #9640C2semidirect
14. #9639C2semidirect
15. #9638C2semidirect
113. #431D4semidirect
114. #431D4semidirect

Maximal subgroups

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

subgroupindex
#349772
#219762
#204622
C2×#14962
C2×#14802
C2×#14882
C2×#14772
#96482
#96462
#96452
#96442
#96422
#96402
#96392
#96382