Finite Group Databasebeta

← #360346

Subgroups — #360346

Order 512.

1150 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1826C211yesmaximal, characteristic
2#20685C211yesmaximal, characteristic
3#20628C211yesmaximal, characteristic
4C2×#1753C211yesmaximal, characteristic
5#20347C211yesmaximal, characteristic
6#11317C211yesmaximal, characteristic, complemented
7#11317C211yesmaximal, characteristic, complemented
8#11291C211yesmaximal, characteristic, complemented
9#11291C211yesmaximal, characteristic, complemented
10#10883C211yesmaximal, characteristic, complemented
11#10882C211yesmaximal, characteristic, complemented
12#10819C211yesmaximal, characteristic, complemented
13#10818C211yesmaximal, characteristic, complemented
14#10733C211yesmaximal, characteristic
15#10724C211yesmaximal, characteristic
16C23×M16C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C2×#407C2211yescharacteristic
22C2×#407C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27#1826—24no—
28#1818—22no—
29#1817—22no—
30#1798—22no—
31#1797—22no—
32#1753—24no—
33C2×#341C2211yescharacteristic
34C2×#341C2211yescharacteristic
35C2×#341C2211yescharacteristic
36C2×#341C2211yescharacteristic
37#1736C412yescomplemented
38#1736C412yescomplemented
39#1736C412yescomplemented
40#1736C412yescomplemented
41#1735C2211yescharacteristic
42#1735C2211yescharacteristic
43#1735C2211yescharacteristic
44#1735C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56#1729C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C24×C4C2311yescharacteristic
62C22×M16—22no—
63C22×M16—22no—
64C22×M16—22no—
65C22×M16—22no—
66C22×M16—41no—
67C22×M16—41no—
68C22×M16C2×C412yes—
69C22×M16C2×C412yes—
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C2×#117—21no—
79C2×#117—21no—
80C2×#117—21no—
81C2×#117—21no—
82C2×#117—22no—
83C2×#117—22no—
84C2×#117—22no—
85C2×#117—22no—
86C2×#117—22no—
87C2×#117—22no—
88C2×#117—41no—
89C2×#117—41no—
90C2×#117C2×C411yescharacteristic
91C2×#117C2×C411yescharacteristic
92C2×#117C2×C412yes—
93C2×#117C2311yescharacteristic
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C22×C4⋊C4C2×C411yescharacteristic
98C22×C4⋊C4C2×C411yescharacteristic
99C22×C4⋊C4C2×C412yes—
100C22×C22⋊C4—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C42—21no—
104C22×C42—21no—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107#424—22no—
108#424—22no—
109#424—41no—
110#424—41no—
111C2×C4⋊C8—22no—
112C2×C4⋊C8—22no—
113#407—22no—
114#407—22no—
115#407—22no—
116#407—22no—
117C2×C22⋊C8—22no—
118C2×C22⋊C8—22no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8D412yes—
124C2×C22⋊C8D412yes—
125C2×#97—22no—
126C2×#97—22no—
127C2×#97—41no—
128C2×C4×C8—22no—
129C2×C4×C8—22no—
130C2×C4×C8—41no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4D414yes—
133C2×#95—21no—
134#341—22no—
135#341—22no—
136#341—41no—
137#341—41no—
138#340C2×C412yes—
139#340C2×C412yes—
140#340C2×C412yes—
141#340C2×C412yes—
142#340C2×C412yes—
143#340C2×C412yes—
144#340C2×C412yes—
145#340C2×C412yes—
146#339—22no—
147#339—22no—
148#339—22no—
149#339—22no—
150#339—22no—
151#339—22no—
152#339—41no—
153#339—41no—
154#336—22no—
155#336—22no—
156#336—22no—
157#336—22no—
158#336—22no—
159#336—22no—
160#336—22no—
161#336—22no—
162C25#4111yescharacteristic
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4#4112yes—
182C23×C4#4112yes—
183C23×C4#4111yescharacteristic
184C23×C4#4112yes—
185C23×C4C2411yescharacteristic
186C2×M16—24no—
187C2×M16—24no—
188C2×M16—24no—
189C2×M16—24no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—42no—
215C2×M16—42no—
216C2×M16—44no—
217C2×M16—44no—
218C2×M16—44no—
219C2×M16—44no—
220C22×C8—21no—
221C22×C8—21no—
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—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8C22×C412yes—
247C22×C8C22×C412yes—
248C22×C8C2×D412yes—
249C22×C8C2×D412yes—
250C22×C8#4112yes—
251C22×C8#4112yes—
252#117—21no—
253#117—21no—
254#117—22no—
255#117—22no—
256#117—22no—
257#117—22no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—21no—
270#117—24no—
271#117—42no—
272#117—42no—
273#117—42no—
274#117—42no—
275#117—42no—
276#117—42no—
277#117—42no—
278#117—42no—
279#117—42no—
280#117#4112yes—
281#117#4112yes—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4C22×C411yescharacteristic
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C4⋊C4C22×C411yescharacteristic
294C2×C4⋊C4C22×C411yescharacteristic
295C2×C4⋊C4C22×C412yes—
296C2×C4⋊C4C22×C412yes—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C4⋊C8—44no—
334C4⋊C8—44no—
335C22⋊C8—24no—
336C22⋊C8—24no—
337C22⋊C8—42no—
338C22⋊C8—42no—
339C22⋊C8—42no—
340C22⋊C8—42no—
341#97—44no—
342C4×C8—44no—
343#95—24no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—21no—
348C24—42no—
349C24—42no—
350C24C4×D411yescharacteristic
351C24C4×D411yescharacteristic
352C24#12312yes—
353C24#12612yes—
354C24C2×#4111yescharacteristic
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4#11712yes—
415C22×C4#11712yes—
416C22×C4C4×D412yes—
417C22×C4C4×D412yes—
418C22×C4C4×D411yescharacteristic
419C22×C4C4×D411yescharacteristic
420C22×C4#12112yes—
421C22×C4#12112yes—
422C22×C4#12312yes—
423C22×C4#12412yes—
424C22×C4#12412yes—
425C22×C4#12612yes—
426C22×C4C23×C411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4H(5,𝔽2)11yescharacteristic
437C22×C4H(5,𝔽2)11yescharacteristic
438C22×C42−1+411yescharacteristic
439C22×C42−1+411yescharacteristic
440M16—48no—
441M16—48no—
442M16—41no—
443M16—41no—
444M16—41no—
445M16—41no—
446M16—41no—
447M16—41no—
448M16—41no—
449M16—41no—
450M16—41no—
451M16—41no—
452M16—41no—
453M16—41no—
454M16—41no—
455M16—41no—
456M16—41no—
457M16—41no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8#11714yes—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4#11714yes—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—44no—
516C22⋊C4—48no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—44no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—41no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—42no—
543C42—42no—
544C23—22no—
545C23—22no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—82no—
563C23—82no—
564C23—82no—
565C23—82no—
566C23#38812yes—
567C23#38812yes—
568C23C2×C4×D411yescharacteristic
569C23C2×#12311yescharacteristic
570C23C2×#12611yescharacteristic
571C23#55111yescharacteristic
572C23#55111yescharacteristic
573C23#55211yescharacteristic
574C23#55211yescharacteristic
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
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—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—24no—
600C2×C4—24no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
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—88no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4#38812yes—
627C2×C4#38812yes—
628C2×C4C2×#11711yescharacteristic
629C2×C4C2×#11711yescharacteristic
630C2×C4C2×#11711yescharacteristic
631C2×C4C2×C4×D411yescharacteristic
632C2×C4#51811yescharacteristic
633C2×C4#51911yescharacteristic
634C2×C4C2×#12111yescharacteristic
635C2×C4C2×#12411yescharacteristic
636C2×C4#52911yescharacteristic
637C2×C4#52911yescharacteristic
638C2×C4#52911yescharacteristic
639C2×C4#52911yescharacteristic
640C2×C4#53911yescharacteristic
641C2×C4#53911yescharacteristic
642C2×C4#53911yescharacteristic
643C2×C4#53911yescharacteristic
644C2×C4#54611yescharacteristic
645C2×C4#54611yescharacteristic
646C2×C4#54611yescharacteristic
647C2×C4#54611yescharacteristic
648C2×C4#54711yescharacteristic
649C2×C4#54711yescharacteristic
650C2×C4#54711yescharacteristic
651C2×C4#54711yescharacteristic
652C2×C4#55111yescharacteristic
653C2×C4#55111yescharacteristic
654C2×C4#55211yescharacteristic
655C2×C4#55211yescharacteristic
656C8—24no—
657C8—42no—
658C8—42no—
659C8—42no—
660C8—81no—
661C8—81no—
662C8—81no—
663C8—81no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—21no—
669C22—21no—
670C22—21no—
671C22—21no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22C2×#38811yescharacteristic
687C22#234211yescharacteristic
688C22#234411yescharacteristic
689C22#241911yescharacteristic
690C22#241911yescharacteristic
691C22#249311yescharacteristic
692C22#249311yescharacteristic
693C22#295512yes—
694C22#295612yes—
695C4—22no—
696C4—22no—
697C4—44no—
698C4—44no—
699C4—82no—
700C4—82no—
701C4—81no—
702C4—81no—
703C4—81no—
704C4—81no—
705C4#228311yescharacteristic
706C4#231311yescharacteristic
707C4#234211yescharacteristic
708C4#234411yescharacteristic
709C4#242511yescharacteristic
710C4#242511yescharacteristic
711C4#248811yescharacteristic
712C4#248811yescharacteristic
713C2—21no—
714C2—21no—
715C2—21no—
716C2—21no—
717C2—42no—
718C2—42no—
719C2#1387411yescharacteristic
720C2#2109911yescharacteristic
721C2#2110011yescharacteristic
722C2#2258812yes—
723C2#2260812yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11317C2semidirect
7. #11317C2semidirect
8. #11291C2semidirect
9. #11291C2semidirect
10. #10883C2semidirect
11. #10882C2semidirect
12. #10819C2semidirect
13. #10818C2semidirect
37. #1736C4semidirect
38. #1736C4semidirect
39. #1736C4semidirect
40. #1736C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18262
#206852
#206282
C2×#17532
#203472
#113172
#112912
#108832
#108822
#108192
#108182
#107332
#107242