Finite Group Databasebeta

← #212124

Subgroups — #212124

Order 512.

727 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#480C211yesmaximal, characteristic
2#21488C211yesmaximal, characteristic
3C4⋊C4×M16C211yesmaximal, characteristic, complemented
4#9545C211yesmaximal, characteristic, complemented
5#9541C211yesmaximal, characteristic, complemented
6#9537C211yesmaximal, characteristic, complemented
7#9533C211yesmaximal, characteristic, complemented
8#8944C211yesmaximal, characteristic
9#8832C211yesmaximal, characteristic, complemented
10#8706C211yesmaximal, characteristic, complemented
11#8291C211yesmaximal, characteristic, complemented
12C8×#107C211yesmaximal, characteristic, complemented
13#8210C211yesmaximal, characteristic, complemented
14#8206C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic, complemented
16C2×#480C411yescharacteristic
17C2×#436—21no—
18C2×#436C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×C4×M16—21no—
21C2×C4×M16C2211yescharacteristic
22C4×#121C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1954C2211yescharacteristic
25#1939C2211yescharacteristic
26#1925C411yescharacteristic
27#1925C411yescharacteristic
28#1924—21no—
29#1918C411yescharacteristic
30#1918C411yescharacteristic
31#1874C411yescharacteristic
32#1870—21no—
33#1852C411yescharacteristic
34#1823C411yescharacteristic
35#1797C2211yescharacteristic
36C4×#107C2211yescharacteristic
37C4×#107C2211yescharacteristic
38#1771—21no—
39#1771—21no—
40#1770C2211yescharacteristic
41#1770C2211yescharacteristic
42C8×C4⋊C4C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47#1753—21no—
48C4×C22⋊C8C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50#1728—21no—
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68C22×M16—21no—
69C22×M16D411yescharacteristic
70C22×M16D411yescharacteristic
71C2×#121C2×C411yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#480—24no—
79#436—22no—
80#436—22no—
81#436—44no—
82C2×#107C2×C411yescharacteristic
83C2×#107C2×C411yescharacteristic
84C2×#107C2×C411yescharacteristic
85C2×#107C2×C411yescharacteristic
86#423—22no—
87#423—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8C2×C411yescharacteristic
104C2×C22⋊C8C2×C411yescharacteristic
105C4×M16—22no—
106C4×M16—22no—
107C4×M16—44no—
108C2×#97—21no—
109C2×#97—21no—
110C2×#97D411yescharacteristic
111C2×#97D411yescharacteristic
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#390C2×C411yescharacteristic
122#389C2×C411yescharacteristic
123#388—21no—
124#386—21no—
125#380C2×C411yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C22⋊C4—21no—
134C2×#95—21no—
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336C2×C411yescharacteristic
152#336C2×C411yescharacteristic
153#335—22no—
154#335—22no—
155D4⋊C8—22no—
156D4⋊C8—22no—
157#322—22no—
158C82—22no—
159C22×D4—21no—
160C22×D4C22×C411yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4#4111yescharacteristic
167C2×M16—24no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—41no—
173C2×M16—42no—
174C2×M16—42no—
175C2×M16—42no—
176C2×M16—42no—
177C2×M16—42no—
178C2×M16D814yes—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
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#121—22no—
194#121—41no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—42no—
200C4×D4—42no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42C2411yescharacteristic
235#107—22no—
236#107—22no—
237#107—41no—
238#107M1614yescomplemented
239C4⋊C8—24no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—42no—
247C4⋊C8#4112yes—
248C4⋊C8#4112yes—
249#102—22no—
250#102—22no—
251#102—22no—
252#102—41no—
253C22⋊C8—22no—
254C22⋊C8—41no—
255C22⋊C8—42no—
256C22⋊C8—42no—
257C22⋊C8—42no—
258C22⋊C8—42no—
259C22⋊C8—42no—
260#97—22no—
261#97—22no—
262#97—42no—
263#97—42no—
264C4×C8—21no—
265C4×C8—21no—
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—
276C4×C8—42no—
277C4×C8#4112yes—
278C4×C8#4112yes—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286C24—21no—
287C24C4×D411yescharacteristic
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—81no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—81no—
329C22×C4—81no—
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4C4×D411yescharacteristic
333C22×C4C2×D812yes—
334C22×C4#13512yes—
335C22×C4C23×C411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C2×#4111yescharacteristic
338M16—28no—
339M16—44no—
340M16—48no—
341M16—84no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—22no—
354C2×C8—22no—
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—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
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#12312yes—
379C2×C8#12312yes—
380C2×C8C2×M1612yes—
381C2×C8C2×D812yes—
382C2×C8C2×D812yes—
383C2×C8#13712yes—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4C2×M1612yes—
399C4⋊C4C2×M1612yes—
400C4⋊C4#13112yes—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—42no—
406C22⋊C4—81no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—42no—
420C42—42no—
421C42C2×#4111yescharacteristic
422C42C2×#4111yescharacteristic
423C42C2×#4111yescharacteristic
424C42C2×#4111yescharacteristic
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23C4×D812yes—
439C23C2×C4×D411yescharacteristic
440D4—41no—
441D4—41no—
442D4—41no—
443D4—41no—
444D4—84no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
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—81no—
485C2×C4—81no—
486C2×C4—84no—
487C2×C4—84no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4C4×D812yes—
493C2×C4C4×#4111yescharacteristic
494C2×C4C4×#4111yescharacteristic
495C2×C4C2×#12311yescharacteristic
496C2×C4#52511yescharacteristic
497C2×C4#52911yescharacteristic
498C2×C4#52911yescharacteristic
499C2×C4C22×M1611yescharacteristic
500C2×C4C2×#13111yescharacteristic
501C2×C4#56811yescharacteristic
502C2×C4#56811yescharacteristic
503C2×C4C22×D811yescharacteristic
504C2×C4C2×#13511yescharacteristic
505C2×C4C2×#13711yescharacteristic
506C2×C4#57511yescharacteristic
507C8—22no—
508C8—22no—
509C8—22no—
510C8—42no—
511C8—41no—
512C8—42no—
513C8—42no—
514C8—41no—
515C8—41no—
516C8—41no—
517C8—81no—
518C8—81no—
519C8—81no—
520C8#48014yescomplemented
521C22—22no—
522C22—22no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22C4×#12311yescharacteristic
536C22#293311yescharacteristic
537C22D4×M1611yescharacteristic
538C22C2×C4×D811yescharacteristic
539C22C4×#13711yescharacteristic
540C22#295611yescharacteristic
541C22#295711yescharacteristic
542C4—22no—
543C4—21no—
544C4—21no—
545C4—41no—
546C4—42no—
547C4—41no—
548C4—42no—
549C4—42no—
550C4—81no—
551C4—81no—
552C4—81no—
553C4—81no—
554C4#296511yescharacteristic
555C4#296711yescharacteristic
556C4#296911yescharacteristic
557C4#297111yescharacteristic
558C4C2×#48011yescharacteristic
559C4#309111yescharacteristic
560C4#310011yescharacteristic
561C4#310811yescharacteristic
562C2—42no—
563C2—161no—
564C2#1799411yescharacteristic
565C2#1801211yescharacteristic
566C2M16×D811yescharacteristic
567C2#1804911yescharacteristic
568C2#2156511yescharacteristic
569C2C4×#48011yescharacteristic
570C2#2159411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4⋊C4×M16C2semidirect
4. #9545C2semidirect
5. #9541C2semidirect
6. #9537C2semidirect
7. #9533C2semidirect
9. #8832C2semidirect
10. #8706C2semidirect
11. #8291C2semidirect
12. C8×#107C2semidirect
13. #8210C2semidirect
14. #8206C2semidirect
15. #8146C2semidirect
238. #107M16semidirect
520. C8#480semidirect

Maximal subgroups

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

subgroupindex
C4×#4802
#214882
C4⋊C4×M162
#95452
#95412
#95372
#95332
#89442
#88322
#87062
#82912
C8×#1072
#82102
#82062
#81462