Finite Group Databasebeta

← #224980

Subgroups — #224980

Order 512.

899 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22639C211yesmaximal, characteristic, complemented
2#22401C211yesmaximal, characteristic
3C4×#418C211yesmaximal, characteristic
4#9560C211yesmaximal, characteristic, complemented
5#9558C211yesmaximal, characteristic, complemented
6#9542C211yesmaximal, characteristic, complemented
7#9541C211yesmaximal, characteristic, complemented
8#8849C211yesmaximal, characteristic, complemented
9#8847C211yesmaximal, characteristic, complemented
10#8708C211yesmaximal, characteristic, complemented
11#8597C211yesmaximal, characteristic, complemented
12#8310C211yesmaximal, characteristic
13#8212C211yesmaximal, characteristic, complemented
14#8210C211yesmaximal, characteristic, complemented
15#8147C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×#436C2211yescharacteristic
19C2×#418C411yescharacteristic
20C2×#418C411yescharacteristic
21C2×#407—21no—
22C2×C4×M16C2211yescharacteristic
23C4×#117C2211yescharacteristic
24#1954C2211yescharacteristic
25#1954C2211yescharacteristic
26#1939C2211yescharacteristic, complemented
27#1939C2211yescharacteristic, complemented
28#1923—21no—
29#1923—21no—
30#1870—21no—
31#1870—21no—
32#1839—21no—
33#1797C2211yescharacteristic
34#1797C2211yescharacteristic
35#1779—21no—
36C8×C4⋊C4C2211yescharacteristic, complemented
37C8×C4⋊C4C2211yescharacteristic, complemented
38C4×C4⋊C8—21no—
39#1762C411yescharacteristic
40#1762C411yescharacteristic
41#1762C411yescharacteristic
42#1762C411yescharacteristic
43C4×#102C2211yescharacteristic
44C4×#102C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×C22⋊C8C2211yescharacteristic
48C4×C22⋊C8C2211yescharacteristic
49#1736C411yescharacteristic
50#1736C411yescharacteristic
51C2×#335C2211yescharacteristic, complemented
52C2×#334C2211yescharacteristic, complemented
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×#322C2211yescharacteristic, complemented
58C2×#322C2211yescharacteristic, complemented
59#1279C2211yescharacteristic, complemented
60#1279C2211yescharacteristic, complemented
61#1279C2211yescharacteristic, complemented
62#1279C2211yescharacteristic, complemented
63#1277C2211yescharacteristic, complemented
64#1277C2211yescharacteristic, complemented
65#1273C2211yescharacteristic, complemented
66#1273C2211yescharacteristic, complemented
67C23×D4C2×C411yescharacteristic
68C24×C4—21no—
69C22×M16—21no—
70C22×M16—21no—
71C22×M16C2×C411yescharacteristic
72C22×M16C2×C411yescharacteristic
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—41no—
78C2×C4×D4—41no—
79C2×C4×D4C2×C411yescharacteristic
80C2×C4×D4C2×C411yescharacteristic
81C2×C4×D4C2311yescharacteristic
82C2×C4×D4C2311yescharacteristic
83C2×#117C2×C411yescharacteristic
84C2×#117C2×C411yescharacteristic
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C22⋊C4—21no—
87C22×C42C2311yescharacteristic
88#436—22no—
89#436—22no—
90#436—41no—
91#436—41no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98#418—24no—
99#418—24no—
100C2×#102C2×C411yescharacteristic
101C2×#102C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108#407—44no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8D411yescharacteristic
119C2×C22⋊C8D411yescharacteristic
120C4×M16—41no—
121C4×M16—41no—
122C2×#97C2×C411yescharacteristic
123C2×#97C2×C411yescharacteristic
124C2×C4×C8—21no—
125C2×C4×C8—21no—
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138#376—21no—
139C43—21no—
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336D411yescharacteristic
156#336D411yescharacteristic
157#335—41no—
158#334—41no—
159D4⋊C8—22no—
160D4⋊C8—22no—
161D4⋊C8—41no—
162D4⋊C8—41no—
163#322—41no—
164#322—41no—
165C25—21no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4—41no—
172C22×D4—41no—
173C22×D4C22×C411yescharacteristic
174C22×D4C22×C411yescharacteristic
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—81no—
185C23×C4—81no—
186C23×C4C22×C411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4#4111yescharacteristic
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—41no—
193C2×M16—41no—
194C2×M16—41no—
195C2×M16—42no—
196C2×M16—42no—
197C2×M16—42no—
198C2×M16—42no—
199C2×M16—81no—
200C2×M16—81no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8C22×C411yescharacteristic
208C22×C8C22×C411yescharacteristic
209C22×C8C22×C411yescharacteristic
210C22×C8C22×C411yescharacteristic
211C22×C8C2×D411yescharacteristic
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—42no—
224C4×D4—42no—
225C4×D4—41no—
226C4×D4—41no—
227C4×D4—42no—
228C4×D4—42no—
229C4×D4#4112yes—
230C4×D4#4112yes—
231#117—22no—
232#117—22no—
233#117—41no—
234#117M1614yescomplemented
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42C22×C411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42#4111yescharacteristic
265C2×C42C2411yescharacteristic
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—42no—
275C4⋊C8—42no—
276#102—22no—
277#102—22no—
278#102—22no—
279#102—22no—
280#102—22no—
281#102—22no—
282#102—22no—
283#102M1614yescomplemented
284C22⋊C8—22no—
285C22⋊C8—41no—
286C22⋊C8—42no—
287C22⋊C8—42no—
288C22⋊C8—42no—
289C22⋊C8—42no—
290C22⋊C8—42no—
291C22⋊C8—42no—
292C22⋊C8—81no—
293C22⋊C8—81no—
294C22⋊C8—81no—
295#97—41no—
296#97—41no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—81no—
310C4×C8—81no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315C24—21no—
316C24—21no—
317C24—42no—
318C24—42no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24—81no—
323C24—81no—
324C24C4×D411yescharacteristic
325C2×D4—28no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—44no—
341C2×D4—42no—
342C2×D4#11712yes—
343C2×D4C2×M1612yes—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
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—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4C4×D411yescharacteristic
390C22×C4C4×D411yescharacteristic
391C22×C4C4×D411yescharacteristic
392C22×C4C2×M1612yes—
393C22×C4#13112yes—
394C22×C4C23×C411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C2×#4111yescharacteristic
397M16—44no—
398M16—44no—
399M16—84no—
400M16—84no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—22no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8C2×M1612yes—
442C2×C8AGL(1,ℤ/8)12yes—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—42no—
452C4⋊C4#11712yes—
453C4⋊C4C2×M1612yes—
454C4⋊C4C2×M1612yes—
455C4⋊C4C2×M1612yes—
456C22⋊C4—24no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—24no—
471C42—42no—
472C42—42no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—81no—
478C42C2×#4111yescharacteristic
479C42C2×#4111yescharacteristic
480C42H(5,𝔽2)11yescharacteristic
481C42H(5,𝔽2)11yescharacteristic
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—22no—
490C23—48no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—84no—
504C23—84no—
505C23—84no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23#43512yes—
510C23C2×C4×D411yescharacteristic
511D4—22no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4—48no—
516D4—41no—
517D4—41no—
518D4—41no—
519D4—41no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4#43512yes—
570C2×C4C2×#11711yescharacteristic
571C2×C4#51811yescharacteristic
572C2×C4#54611yescharacteristic
573C2×C4#54611yescharacteristic
574C2×C4#55111yescharacteristic
575C2×C4#55111yescharacteristic
576C2×C4C22×M1611yescharacteristic
577C2×C4C22×M1611yescharacteristic
578C2×C4C2×#13111yescharacteristic
579C2×C4#56811yescharacteristic
580C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
581C2×C4#57511yescharacteristic
582C2×C4#57611yescharacteristic
583C2×C4#57711yescharacteristic
584C8—22no—
585C8—22no—
586C8—42no—
587C8—41no—
588C8—41no—
589C8—41no—
590C8—41no—
591C8—41no—
592C8—41no—
593C8—81no—
594C8—81no—
595C8—81no—
596C8—81no—
597C8—81no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—816no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22#234211yescharacteristic
621C22C2×#43511yescharacteristic
622C22D4×M1611yescharacteristic
623C22C4×AGL(1,ℤ/8)11yescharacteristic
624C22#294611yescharacteristic
625C22#295511yescharacteristic
626C22#295611yescharacteristic
627C4—22no—
628C4—21no—
629C4—21no—
630C4—41no—
631C4—42no—
632C4—41no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—81no—
637C4—81no—
638C4#292011yescharacteristic
639C4#292411yescharacteristic
640C4#297411yescharacteristic
641C4#297511yescharacteristic
642C4#331111yescharacteristic
643C4#331811yescharacteristic
644C4#332211yescharacteristic
645C4#332411yescharacteristic
646C2—42no—
647C2—82no—
648C2—82no—
649C2#1787111yescharacteristic
650C2#1787511yescharacteristic
651C2#1804811yescharacteristic
652C2#1805011yescharacteristic
653C2#2255311yescharacteristic
654C2#2258011yescharacteristic
655C2#2258811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22639C2semidirect
4. #9560C2semidirect
5. #9558C2semidirect
6. #9542C2semidirect
7. #9541C2semidirect
8. #8849C2semidirect
9. #8847C2semidirect
10. #8708C2semidirect
11. #8597C2semidirect
13. #8212C2semidirect
14. #8210C2semidirect
15. #8147C2semidirect
26. #1939C22semidirect
27. #1939C22semidirect
36. C8×C4⋊C4C22semidirect
37. C8×C4⋊C4C22semidirect
51. C2×#335C22semidirect
52. C2×#334C22semidirect
57. C2×#322C22semidirect
58. C2×#322C22semidirect
59. #1279C22semidirect
60. #1279C22semidirect
61. #1279C22semidirect
62. #1279C22semidirect
63. #1277C22semidirect
64. #1277C22semidirect
65. #1273C22semidirect
66. #1273C22semidirect
234. #117M16semidirect
283. #102M16semidirect

Maximal subgroups

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

subgroupindex
#226392
#224012
C4×#4182
#95602
#95582
#95422
#95412
#88492
#88472
#87082
#85972
#83102
#82122
#82102
#81472