Finite Group Databasebeta

← #219452

Subgroups — #219452

Order 512.

763 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#496C211yesmaximal, characteristic
2#21682C211yesmaximal, characteristic, complemented
3#21633C211yesmaximal, characteristic, complemented
4#9648C211yesmaximal, characteristic, complemented
5#9646C211yesmaximal, characteristic, complemented
6#8952C211yesmaximal, characteristic, complemented
7#8952C211yesmaximal, characteristic, complemented
8#8951C211yesmaximal, characteristic, complemented
9#8949C211yesmaximal, characteristic, complemented
10#8947C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic, complemented
12#8947C211yesmaximal, characteristic, complemented
13#8945C211yesmaximal, characteristic, complemented
14#8786C211yesmaximal, characteristic, complemented
15#8136C211yesmaximal, characteristic, complemented
16C2×#496C411yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×#436C2211yescharacteristic
23C2×#435—21no—
24C2×#435—21no—
25C2×C8×D4C2211yescharacteristic
26C2×C8×D4C2211yescharacteristic
27C4×#127C2211yescharacteristic
28C4×#124C2211yescharacteristic
29#1984C2211yescharacteristic
30#1973C411yescharacteristic
31#1973C411yescharacteristic
32#1972C411yescharacteristic
33#1972C411yescharacteristic
34#1961C411yescharacteristic
35#1959C411yescharacteristic
36#1955—21no—
37#1955—21no—
38#1954C2211yescharacteristic
39#1954C2211yescharacteristic
40#1954C2211yescharacteristic
41#1924—21no—
42#1924—21no—
43#1847C411yescharacteristic
44#1842C2211yescharacteristic
45#1831C2211yescharacteristic
46#1752—21no—
47#1752—21no—
48C4×#97C2211yescharacteristic
49C4×#97C2211yescharacteristic
50C4×#97C2211yescharacteristic
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1277C2211yescharacteristic
60#1277C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C22×SD16C2×C411yescharacteristic
69C22×D8C2×C411yescharacteristic
70C22×D8C2×C411yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73C23×C8—21no—
74C2×#127C2×C411yescharacteristic
75C2×#124C2×C411yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4C2311yescharacteristic
80C2×C4×D4C2311yescharacteristic
81C2×C4×D4C2311yescharacteristic
82#496—24no—
83#446—41no—
84#446—41no—
85C4×SD16—41no—
86C4×SD16—41no—
87C4×SD16—41no—
88C4×SD16—41no—
89C4×D8—41no—
90C4×D8—41no—
91C4×D8—41no—
92C4×D8—41no—
93#436—41no—
94#436—41no—
95#435—44no—
96#435—44no—
97C8×D4—41no—
98C8×D4—41no—
99C8×D4—41no—
100C8×D4—41no—
101C2×#107C2×C411yescharacteristic
102C2×#107C2×C411yescharacteristic
103C2×#106C2×C411yescharacteristic
104C2×#106C2×C411yescharacteristic
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×#97—21no—
130C2×#97—21no—
131C2×#97—21no—
132C2×#97C2×C411yescharacteristic
133C2×#97C2×C411yescharacteristic
134C2×#97C2×C411yescharacteristic
135C2×#97C2×C411yescharacteristic
136C2×#97D411yescharacteristic
137C2×#97D411yescharacteristic
138C2×#97D411yescharacteristic
139C2×#97D411yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146#391C2×C411yescharacteristic
147#390—21no—
148#390—21no—
149#390C2×C411yescharacteristic
150#390C2×C411yescharacteristic
151#390C2×C411yescharacteristic
152#388—21no—
153#388—21no—
154#384C2×C411yescharacteristic
155#382C2×C411yescharacteristic
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C43C2311yescharacteristic
159#336—21no—
160#336—21no—
161#336—21no—
162#336—21no—
163Q8⋊C8—41no—
164Q8⋊C8—41no—
165D4⋊C8—41no—
166D4⋊C8—41no—
167D4⋊C8—41no—
168D4⋊C8—41no—
169D4⋊C8—41no—
170D4⋊C8—41no—
171C22×Q8C22×C411yescharacteristic
172C22×D4—21no—
173C22×D4—21no—
174C22×D4C22×C411yescharacteristic
175C22×D4C22×C411yescharacteristic
176C22×D4C22×C411yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C2×SD16—22no—
181C2×SD16—22no—
182C2×SD16—22no—
183C2×SD16—41no—
184C2×SD16—41no—
185C2×SD16—41no—
186C2×D8—22no—
187C2×D8—22no—
188C2×D8—41no—
189C2×D8—41no—
190C2×D8—41no—
191C2×D8M1614yescomplemented
192C2×M16—42no—
193C2×M16—42no—
194C2×M16—42no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—42no—
202C22×C8—42no—
203C22×C8—42no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8—42no—
207C22×C8C22×C411yescharacteristic
208C22×C8C22×C411yescharacteristic
209C22×C8C22×C411yescharacteristic
210C22×C8C22×C411yescharacteristic
211C22×C8C2×D411yescharacteristic
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215#127—22no—
216#127—41no—
217#124—22no—
218#124—41no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—42no—
236C4×D4—42no—
237C4×D4—42no—
238C4×D4—42no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C42—21no—
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42#4111yescharacteristic
264C2×C42C2411yescharacteristic
265#107—41no—
266#107—41no—
267#106—41no—
268#106—41no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—42no—
278C4⋊C8—42no—
279C4⋊C8—42no—
280#103—41no—
281#103—41no—
282#102—41no—
283#102—41no—
284#102—41no—
285#102—41no—
286#102—41no—
287#102—41no—
288C22⋊C8—42no—
289C22⋊C8—42no—
290C22⋊C8—42no—
291C22⋊C8—42no—
292C22⋊C8—42no—
293C22⋊C8—42no—
294C22⋊C8—42no—
295C22⋊C8—42no—
296C22⋊C8—42no—
297#97—22no—
298#97—22no—
299#97—22no—
300#97—22no—
301#97—41no—
302#97—41no—
303#97—41no—
304#97—41no—
305#97—42no—
306#97—42no—
307#97—42no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324C24—21no—
325C24—21no—
326C24—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
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—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—81no—
354C2×D4—81no—
355C2×D4C2×M1612yes—
356C2×D4C2×M1612yes—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4C4×D411yescharacteristic
381C22×C4C4×D411yescharacteristic
382C22×C4C4×D411yescharacteristic
383C22×C4C4×D411yescharacteristic
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C2×#4111yescharacteristic
387SD16—44no—
388SD16—44no—
389D8—44no—
390D8—44no—
391M16—84no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
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—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—84no—
431C2×C8—84no—
432C2×C8C2×M1612yes—
433C2×C8#13112yes—
434C2×C8AGL(1,ℤ/8)12yes—
435C2×C8AGL(1,ℤ/8)12yes—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—81no—
461C22⋊C4—81no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—41no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42H(5,𝔽2)11yescharacteristic
475C42H(5,𝔽2)11yescharacteristic
476C42H(5,𝔽2)11yescharacteristic
477C42H(5,𝔽2)11yescharacteristic
478C23—22no—
479C23—22no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23C2×C4×D411yescharacteristic
500Q8—41no—
501Q8—41no—
502Q8—41no—
503Q8—41no—
504D4—22no—
505D4—22no—
506D4—22no—
507D4—22no—
508D4—41no—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—84no—
517D4—84no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
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—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4#51811yescharacteristic
560C2×C4#51811yescharacteristic
561C2×C4#53511yescharacteristic
562C2×C4#53511yescharacteristic
563C2×C4#54011yescharacteristic
564C2×C4#54011yescharacteristic
565C2×C4C22×M1611yescharacteristic
566C2×C4C2×#13111yescharacteristic
567C2×C4#56811yescharacteristic
568C2×C4#56811yescharacteristic
569C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
570C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
571C2×C4#57511yescharacteristic
572C2×C4#57511yescharacteristic
573C8—22no—
574C8—22no—
575C8—22no—
576C8—22no—
577C8—42no—
578C8—42no—
579C8—41no—
580C8—41no—
581C8—41no—
582C8—41no—
583C8—81no—
584C8—81no—
585C8—81no—
586C8—81no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22#232811yescharacteristic
609C22D4×M1611yescharacteristic
610C22#293511yescharacteristic
611C22C4×AGL(1,ℤ/8)11yescharacteristic
612C22C4×AGL(1,ℤ/8)11yescharacteristic
613C22#295611yescharacteristic
614C22#295611yescharacteristic
615C4—21no—
616C4—21no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4#297311yescharacteristic
627C4#297411yescharacteristic
628C4#298011yescharacteristic
629C4#298111yescharacteristic
630C4#321311yescharacteristic
631C4#321411yescharacteristic
632C4#321711yescharacteristic
633C4#321911yescharacteristic
634C2—82no—
635C2—82no—
636C2—161no—
637C2#1804611yescharacteristic
638C2#1804611yescharacteristic
639C2#1807611yescharacteristic
640C2#1807811yescharacteristic
641C2#2226611yescharacteristic
642C2#2227211yescharacteristic
643C2#2228311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#496C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21682C2semidirect
3. #21633C2semidirect
4. #9648C2semidirect
5. #9646C2semidirect
6. #8952C2semidirect
7. #8952C2semidirect
8. #8951C2semidirect
9. #8949C2semidirect
10. #8947C2semidirect
11. #8947C2semidirect
12. #8947C2semidirect
13. #8945C2semidirect
14. #8786C2semidirect
15. #8136C2semidirect
191. C2×D8M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4962
#216822
#216332
#96482
#96462
#89522
#89512
#89492
#89472
#89452
#87862
#81362