Finite Group Databasebeta

← #218564

Subgroups — #218564

Order 512.

665 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22208C211yesmaximal, characteristic, complemented
2#22008C211yesmaximal, characteristic, complemented
3C4×#463C211yesmaximal, characteristic
4#9675C211yesmaximal, characteristic, complemented
5#9666C211yesmaximal, characteristic, complemented
6#9663C211yesmaximal, characteristic
7#9601C211yesmaximal, characteristic, complemented
8#9560C211yesmaximal, characteristic, complemented
9#9557C211yesmaximal, characteristic, complemented
10#8948C211yesmaximal, characteristic, complemented
11#8947C211yesmaximal, characteristic
12#8844C211yesmaximal, characteristic, complemented
13#8843C211yesmaximal, characteristic, complemented
14#8563C211yesmaximal, characteristic, complemented
15#8265C211yesmaximal, characteristic, complemented
16C2×#463C411yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C4×#124C2211yescharacteristic
22C42×D4C2211yescharacteristic
23#1984C2211yescharacteristic
24#1983C411yescharacteristic
25#1954C2211yescharacteristic
26#1954C2211yescharacteristic
27#1954C2211yescharacteristic
28#1949C411yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1928C411yescharacteristic
32#1926C411yescharacteristic
33#1923—21no—
34#1923—21no—
35#1923—21no—
36#1923—21no—
37#1879C411yescharacteristic
38#1877C411yescharacteristic
39#1850—21no—
40#1850—21no—
41#1842C2211yescharacteristic
42#1817—21no—
43#1804C411yescharacteristic
44#1797—21no—
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×#102C2211yescharacteristic
50#1744C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C22×D8C2×C411yescharacteristic
69C23×C8—21no—
70C2×#124C2×C411yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C22×C42—21no—
77#463—24no—
78C8×Q8—41no—
79C8×Q8—41no—
80C4×SD16—41no—
81C4×SD16—41no—
82C4×D8—41no—
83C4×D8—41no—
84C8×D4—41no—
85C8×D4—41no—
86C2×#107C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2×C411yescharacteristic
95C2×C4⋊C8C2×C411yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×C22⋊C8—21no—
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×C4×C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#391C2×C411yescharacteristic
124#390C2×C411yescharacteristic
125#388—21no—
126#388—21no—
127#382C2×C411yescharacteristic
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C43C2311yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336C2×C411yescharacteristic
148#336C2×C411yescharacteristic
149#336D411yescharacteristic
150#336D411yescharacteristic
151#336D411yescharacteristic
152#336D411yescharacteristic
153#335—41no—
154#335—41no—
155Q8⋊C8—41no—
156Q8⋊C8—41no—
157D4⋊C8—41no—
158D4⋊C8—41no—
159D4⋊C8—41no—
160D4⋊C8—41no—
161#322—41no—
162#322—41no—
163C22×Q8C22×C411yescharacteristic
164C22×D4C22×C411yescharacteristic
165C22×D4C22×C411yescharacteristic
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C2×SD16—22no—
171C2×SD16—22no—
172C2×SD16—41no—
173C2×D8—22no—
174C2×D8—22no—
175C2×D8—41no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—42no—
183C22×C8—81no—
184C22×C8—81no—
185C22×C8C22×C411yescharacteristic
186C22×C8C22×C411yescharacteristic
187C22×C8C22×C411yescharacteristic
188C22×C8C22×C411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193#124—22no—
194#124M1614yes—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—81no—
230C2×C42—81no—
231C2×C42C22×C411yescharacteristic
232C2×C42C22×C411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42#4111yescharacteristic
236C2×C42#4111yescharacteristic
237C2×C42C2411yescharacteristic
238#107—41no—
239#106—41no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—42no—
251C4⋊C8—42no—
252C4⋊C8—81no—
253C4⋊C8—81no—
254C4⋊C8—81no—
255C4⋊C8—81no—
256#103—22no—
257#103—22no—
258#103—41no—
259#102—22no—
260#102—22no—
261#102—41no—
262#102—41no—
263#102—41no—
264C22⋊C8—42no—
265C22⋊C8—81no—
266C22⋊C8—81no—
267C22⋊C8—81no—
268C22⋊C8—81no—
269C22⋊C8—81no—
270C22⋊C8—81no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—42no—
284C4×C8—81no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292C24—21no—
293C24—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8C2×M1612yes—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4C2×M1612yes—
311C2×D4#13112yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4—81no—
335C22×C4—81no—
336C22×C4C4×D411yescharacteristic
337C22×C4C4×D411yescharacteristic
338C22×C4C4×D411yescharacteristic
339C22×C4C4×D411yescharacteristic
340C22×C4C23×C411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4111yescharacteristic
343SD16—44no—
344D8—44no—
345C2×C8—22no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—84no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4#13112yes—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—42no—
408C22⋊C4—81no—
409C22⋊C4—81no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—81no—
423C42—81no—
424C42C2×M1612yes—
425C42#13112yes—
426C42H(5,𝔽2)11yescharacteristic
427C42H(5,𝔽2)11yescharacteristic
428C42H(5,𝔽2)11yescharacteristic
429C422−1+411yescharacteristic
430C23—22no—
431C23—22no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23C2×C4×D411yescharacteristic
445Q8—22no—
446Q8—22no—
447Q8—41no—
448Q8—41no—
449D4—22no—
450D4—22no—
451D4—22no—
452D4—22no—
453D4—41no—
454D4—41no—
455D4—41no—
456D4—41no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—84no—
488C2×C4—84no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4#43512yes—
498C2×C4#43512yes—
499C2×C4#51811yescharacteristic
500C2×C4#52011yescharacteristic
501C2×C4#53511yescharacteristic
502C2×C4#53711yescharacteristic
503C2×C4#54011yescharacteristic
504C2×C4#54211yescharacteristic
505C2×C4C22×M1611yescharacteristic
506C2×C4C2×#13111yescharacteristic
507C2×C4C2×#13111yescharacteristic
508C2×C4C2×#13111yescharacteristic
509C2×C4#57611yescharacteristic
510C2×C4#57611yescharacteristic
511C2×C4#57711yescharacteristic
512C2×C4#57811yescharacteristic
513C8—42no—
514C8—41no—
515C8—41no—
516C8—41no—
517C8—41no—
518C8—41no—
519C8—41no—
520C8—41no—
521C8—41no—
522C8—81no—
523C8—81no—
524C8—81no—
525C8—81no—
526C8—81no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22#232411yescharacteristic
542C22C2×#43511yescharacteristic
543C22#292911yescharacteristic
544C22#294611yescharacteristic
545C22#295111yescharacteristic
546C22#295611yescharacteristic
547C22#295711yescharacteristic
548C4—21no—
549C4—21no—
550C4—22no—
551C4—22no—
552C4—41no—
553C4—42no—
554C4—41no—
555C4—42no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4#297411yescharacteristic
561C4#297911yescharacteristic
562C4#299011yescharacteristic
563C4#299411yescharacteristic
564C4#322011yescharacteristic
565C4#322511yescharacteristic
566C4#325711yescharacteristic
567C4#326211yescharacteristic
568C2—82no—
569C2—82no—
570C2#1787511yescharacteristic
571C2#1788111yescharacteristic
572C2#1791811yescharacteristic
573C2#1792611yescharacteristic
574C2#2213511yescharacteristic
575C2#2215711yescharacteristic
576C2#2216711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22208C2semidirect
2. #22008C2semidirect
4. #9675C2semidirect
5. #9666C2semidirect
7. #9601C2semidirect
8. #9560C2semidirect
9. #9557C2semidirect
10. #8948C2semidirect
12. #8844C2semidirect
13. #8843C2semidirect
14. #8563C2semidirect
15. #8265C2semidirect

Maximal subgroups

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

subgroupindex
#222082
#220082
C4×#4632
#96752
#96662
#96632
#96012
#95602
#95572
#89482
#89472
#88442
#88432
#85632
#82652