Finite Group Databasebeta

← #231806

Subgroups — #231806

Order 512.

859 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23798C211yesmaximal, characteristic
2#23174C211yesmaximal, characteristic, complemented
3#23173C211yesmaximal, characteristic, complemented
4C2×#1586C211yesmaximal, characteristic, complemented
5C2×#1580C211yesmaximal, characteristic, complemented
6#9678C211yesmaximal, characteristic, complemented
7#9674C211yesmaximal, characteristic, complemented
8#9665C211yesmaximal, characteristic, complemented
9#9659C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8947C211yesmaximal, characteristic, complemented
13#8947C211yesmaximal, characteristic, complemented
14#8277C211yesmaximal, characteristic, complemented
15#8265C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×#436C2211yescharacteristic
21C2×#435—21no—
22C2×C8×D4—21no—
23C2×C8×D4C2211yescharacteristic
24C2×C8×D4C2211yescharacteristic
25C2×C8×D4C2211yescharacteristic
26C4×#127C2211yescharacteristic
27C4×#127C2211yescharacteristic
28#2093C411yescharacteristic
29#2093C411yescharacteristic
30#2093C411yescharacteristic
31#2093C411yescharacteristic
32#2011C411yescharacteristic
33#2011C411yescharacteristic
34#2011C411yescharacteristic
35#2011C411yescharacteristic
36#1955—21no—
37#1954—21no—
38#1954C2211yescharacteristic
39#1954C2211yescharacteristic
40#1954C2211yescharacteristic
41#1954C2211yescharacteristic
42#1870—21no—
43#1869—21no—
44#1869—21no—
45#1869—21no—
46#1767C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×C4⋊C8C2211yescharacteristic
49C4×C4⋊C8C2211yescharacteristic
50#1744C2211yescharacteristic
51#1586—24no—
52#1580—24no—
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×#322C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×D8C2×C411yescharacteristic
70C22×D8C2×C411yescharacteristic
71C22×D8C2×C411yescharacteristic
72C22×D8C2×C411yescharacteristic
73C22×M16—21no—
74C23×C8—21no—
75C23×C8—21no—
76C23×C8—21no—
77C2×#127C2×C411yescharacteristic
78C2×#127C2×C411yescharacteristic
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4C2311yescharacteristic
84C2×C4×D4C2311yescharacteristic
85C2×C4×D4C2311yescharacteristic
86C2×C4×D4C2311yescharacteristic
87C4×D8—22no—
88C4×D8—22no—
89C4×D8—22no—
90C4×D8—22no—
91C4×D8—41no—
92C4×D8—41no—
93C4×D8—41no—
94C4×D8—41no—
95#436—22no—
96#436—41no—
97#435—44no—
98C8×D4—22no—
99C8×D4—22no—
100C8×D4—22no—
101C8×D4—41no—
102C8×D4—41no—
103C8×D4—41no—
104C8×D4—44no—
105C2×#107C2×C411yescharacteristic
106C2×#107C2×C411yescharacteristic
107C2×#107C2×C411yescharacteristic
108C2×#107C2×C411yescharacteristic
109C2×C4⋊C8—21no—
110C2×C4⋊C8D411yescharacteristic
111C2×C4⋊C8D411yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123C2×#102C2×C411yescharacteristic
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×#97—21no—
137C2×C4×C8—21no—
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144#390—21no—
145#390—21no—
146#390—21no—
147#390—21no—
148#390C2×C411yescharacteristic
149#390C2×C411yescharacteristic
150#390C2×C411yescharacteristic
151#390C2×C411yescharacteristic
152#384C2×C411yescharacteristic
153#384C2×C411yescharacteristic
154C43C2311yescharacteristic
155#336—21no—
156#336—21no—
157#336—21no—
158#336C2×C411yescharacteristic
159#336C2×C411yescharacteristic
160#336C2×C411yescharacteristic
161#336C2×C411yescharacteristic
162#336D411yescharacteristic
163#336D411yescharacteristic
164#334—22no—
165#334D414yescomplemented
166D4⋊C8—22no—
167D4⋊C8—22no—
168D4⋊C8—22no—
169D4⋊C8—22no—
170D4⋊C8—41no—
171D4⋊C8—41no—
172D4⋊C8—41no—
173D4⋊C8—41no—
174#322—22no—
175#322—22no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4C22×C411yescharacteristic
181C22×D4C22×C411yescharacteristic
182C22×D4C22×C411yescharacteristic
183C22×D4C22×C411yescharacteristic
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C2×D8—22no—
189C2×D8—22no—
190C2×D8—22no—
191C2×D8—22no—
192C2×D8—22no—
193C2×D8—22no—
194C2×D8—22no—
195C2×D8—22no—
196C2×D8—41no—
197C2×D8—41no—
198C2×D8—41no—
199C2×D8—41no—
200C2×M16—42no—
201C2×M16—42no—
202C2×M16—81no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—42no—
210C22×C8—42no—
211C22×C8—42no—
212C22×C8—42no—
213C22×C8—42no—
214C22×C8—42no—
215C22×C8—81no—
216C22×C8—81no—
217C22×C8—81no—
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C22×C411yescharacteristic
221C22×C8C22×C411yescharacteristic
222C22×C8C2×D411yescharacteristic
223C22×C8C2×D411yescharacteristic
224C22×C8C2×D411yescharacteristic
225C22×C8C2×D411yescharacteristic
226#127—22no—
227#127—22no—
228#127—22no—
229#127—41no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—42no—
247C4×D4—42no—
248C4×D4—42no—
249C4×D4—42no—
250C4×D4—81no—
251C4×D4—81no—
252C4×D4—81no—
253C4×D4—81no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42#4111yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42C2411yescharacteristic
281#107—22no—
282#107—22no—
283#107—41no—
284#107—41no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—42no—
294C4⋊C8C2×D412yes—
295C4⋊C8C2×D412yes—
296#102—22no—
297#102—22no—
298#102—22no—
299#102—22no—
300#102—41no—
301#102—41no—
302#102—41no—
303#102—41no—
304C22⋊C8—42no—
305C22⋊C8—42no—
306C22⋊C8—42no—
307C22⋊C8—42no—
308C22⋊C8—42no—
309C22⋊C8—42no—
310C22⋊C8—42no—
311C22⋊C8—42no—
312C22⋊C8—81no—
313C22⋊C8—81no—
314C22⋊C8—81no—
315C22⋊C8—81no—
316#97—42no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—42no—
328C4×C8C2×D412yes—
329C4×C8#4112yes—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—21no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—81no—
364C2×D4—81no—
365C2×D4—81no—
366C2×D4—81no—
367C2×D4#13112yes—
368C2×D4#13112yes—
369C2×D4#13112yes—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4C4×D411yescharacteristic
399C22×C4C4×D411yescharacteristic
400C22×C4C4×D411yescharacteristic
401C22×C4C4×D411yescharacteristic
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C2×#4111yescharacteristic
405D8—28no—
406D8—44no—
407D8—44no—
408D8—44no—
409M16—84no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—22no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—84no—
451C2×C8—84no—
452C2×C8—84no—
453C2×C8C4×D412yes—
454C2×C8C4×D412yes—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C22⋊C4—81no—
482C22⋊C4—81no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—41no—
490C42—41no—
491C42AGL(1,ℤ/8)12yes—
492C42AGL(1,ℤ/8)12yes—
493C42C22×D411yescharacteristic
494C42C2×#4111yescharacteristic
495C42H(5,𝔽2)11yescharacteristic
496C42H(5,𝔽2)11yescharacteristic
497C23—22no—
498C23—22no—
499C23—22no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C2×C4×D411yescharacteristic
526D4—22no—
527D4—22no—
528D4—22no—
529D4—22no—
530D4—22no—
531D4—22no—
532D4—48no—
533D4—41no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—41no—
542D4—41no—
543D4—84no—
544D4—84no—
545D4—84no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4#44212yes—
587C2×C4#44212yes—
588C2×C4C2×C4×D411yescharacteristic
589C2×C4#51811yescharacteristic
590C2×C4D4211yescharacteristic
591C2×C4#55011yescharacteristic
592C2×C4#55011yescharacteristic
593C2×C4#55911yescharacteristic
594C2×C4C2×#13111yescharacteristic
595C2×C4C2×#13111yescharacteristic
596C2×C4C2×#13111yescharacteristic
597C2×C4#56811yescharacteristic
598C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
599C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
600C2×C4#57611yescharacteristic
601C2×C4#57611yescharacteristic
602C8—22no—
603C8—22no—
604C8—42no—
605C8—41no—
606C8—41no—
607C8—41no—
608C8—41no—
609C8—41no—
610C8—41no—
611C8—42no—
612C8—42no—
613C8—81no—
614C8—81no—
615C8—81no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22#236611yescharacteristic
645C22#292911yescharacteristic
646C22#293311yescharacteristic
647C22C2×#44211yescharacteristic
648C22#294811yescharacteristic
649C22#295511yescharacteristic
650C22#295611yescharacteristic
651C4—22no—
652C4—21no—
653C4—21no—
654C4—22no—
655C4—41no—
656C4—41no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4#292911yescharacteristic
662C4#293311yescharacteristic
663C4#297611yescharacteristic
664C4#297711yescharacteristic
665C4#327411yescharacteristic
666C4#327411yescharacteristic
667C4#339911yescharacteristic
668C4#339911yescharacteristic
669C2—82no—
670C2—82no—
671C2—82no—
672C2—161no—
673C2#1790711yescharacteristic
674C2#1792211yescharacteristic
675C2#1800111yescharacteristic
676C2#1802611yescharacteristic
677C2#2317711yescharacteristic
678C2#2318111yescharacteristic
679C2#2318811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radical#23798C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23174C2semidirect
3. #23173C2semidirect
4. C2×#1586C2semidirect
5. C2×#1580C2semidirect
6. #9678C2semidirect
7. #9674C2semidirect
8. #9665C2semidirect
9. #9659C2semidirect
10. #8951C2semidirect
11. #8947C2semidirect
12. #8947C2semidirect
13. #8947C2semidirect
14. #8277C2semidirect
15. #8265C2semidirect
165. #334D4semidirect

Maximal subgroups

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

subgroupindex
#237982
#231742
#231732
C2×#15862
C2×#15802
#96782
#96742
#96652
#96592
#89512
#89472
#82772
#82652