Finite Group Databasebeta

← #229139

Subgroups — #229139

Order 512.

770 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23084C211yesmaximal, characteristic, complemented
2#22986C211yesmaximal, characteristic, complemented
3C4×#469C211yesmaximal, characteristic
4C2×#1580C211yesmaximal, characteristic, complemented
5C2×#1569C211yesmaximal, characteristic, complemented
6#9657C211yesmaximal, characteristic, complemented
7#9596C211yesmaximal, characteristic, complemented
8#9546C211yesmaximal, characteristic, complemented
9#9545C211yesmaximal, characteristic, complemented
10#8944C211yesmaximal, characteristic, complemented
11#8943C211yesmaximal, characteristic, complemented
12#8845C211yesmaximal, characteristic, complemented
13#8843C211yesmaximal, characteristic, complemented
14#8283C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic, complemented
16C2×#469C411yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×#435—21no—
19C2×C8×D4C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×#432C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24C4×#121C2211yescharacteristic
25C4×#121C2211yescharacteristic
26#1972C411yescharacteristic
27#1955—21no—
28#1954C2211yescharacteristic
29#1954C2211yescharacteristic
30#1944C411yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1936C411yescharacteristic
34#1936C411yescharacteristic
35#1924—21no—
36#1924—21no—
37#1895C411yescharacteristic
38#1895C411yescharacteristic
39#1870—21no—
40#1869—21no—
41#1850—21no—
42#1809C411yescharacteristic
43C4×#106C2211yescharacteristic
44#1770C2211yescharacteristic
45#1770C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1752—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1580—24no—
52#1569—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×D8C2×C411yescharacteristic
70C22×M16—21no—
71C22×M16C2×C411yescharacteristic
72C22×M16C2×C411yescharacteristic
73C23×C8—21no—
74C23×C8—21no—
75C2×#121C2×C411yescharacteristic
76C2×#121C2×C411yescharacteristic
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4C2311yescharacteristic
80C2×C4×D4C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#469—24no—
83C4×D8—22no—
84C4×D8—22no—
85#435—44no—
86C8×D4—22no—
87C8×D4—22no—
88C8×D4—41no—
89C8×D4—41no—
90#432—22no—
91#432—41no—
92C2×#107C2×C411yescharacteristic
93C2×#106C2×C411yescharacteristic
94C2×#106C2×C411yescharacteristic
95#423—22no—
96#423—41no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
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—22no—
121C4×M16—22no—
122C2×#97—21no—
123C2×#97—21no—
124C2×#97—21no—
125C2×#97C2×C411yescharacteristic
126C2×#97C2×C411yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#390C2×C411yescharacteristic
134#390C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#388—21no—
138#388—21no—
139#386—21no—
140#386—21no—
141#380C2×C411yescharacteristic
142#380C2×C411yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336D411yescharacteristic
159#336D411yescharacteristic
160#335—22no—
161#335—22no—
162#334—22no—
163#334D414yescomplemented
164D4⋊C8—22no—
165D4⋊C8—22no—
166D4⋊C8—41no—
167D4⋊C8—41no—
168#322—22no—
169C82—22no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4C22×C411yescharacteristic
173C22×D4C22×C411yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C23×C4C22×C411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4#4111yescharacteristic
179C2×D8—22no—
180C2×D8—22no—
181C2×D8M1614yescomplemented
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—41no—
188C2×M16—42no—
189C2×M16—42no—
190C2×M16—81no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—42no—
198C22×C8—42no—
199C22×C8—42no—
200C22×C8—42no—
201C22×C8—81no—
202C22×C8—81no—
203C22×C8C22×C411yescharacteristic
204C22×C8C22×C411yescharacteristic
205C22×C8C22×C411yescharacteristic
206C22×C8C22×C411yescharacteristic
207C22×C8C2×D411yescharacteristic
208C22×C8C2×D411yescharacteristic
209C22×C8C2×D411yescharacteristic
210C22×C8C2×D411yescharacteristic
211#121—22no—
212#121—22no—
213#121—41no—
214#121—41no—
215C4×D4—21no—
216C4×D4—21no—
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—81no—
226C4×D4—81no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42#4111yescharacteristic
254C2×C42C2411yescharacteristic
255#107—22no—
256#106—22no—
257#106—22no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—42no—
265C4⋊C8—42no—
266C4⋊C8—81no—
267C4⋊C8C2×D412yes—
268C4⋊C8C2×D412yes—
269#102—22no—
270#102—22no—
271#102—22no—
272#102—22no—
273#102—22no—
274#102—22no—
275C22⋊C8—22no—
276C22⋊C8—41no—
277C22⋊C8—42no—
278C22⋊C8—42no—
279C22⋊C8—42no—
280C22⋊C8—42no—
281C22⋊C8—42no—
282C22⋊C8—42no—
283C22⋊C8—81no—
284C22⋊C8—81no—
285#97—22no—
286#97—22no—
287#97—42no—
288#97—42no—
289#97—81no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8C2×D412yes—
301C4×C8#4112yes—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308C24—21no—
309C24—21no—
310C24C4×D411yescharacteristic
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—81no—
326C2×D4—81no—
327C2×D4C2×M1612yes—
328C2×D4C2×M1612yes—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4C4×D411yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4C4×D411yescharacteristic
366C22×C4C23×C411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4111yescharacteristic
369D8—28no—
370M16—44no—
371M16—44no—
372M16—84no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—22no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—84no—
410C2×C8—84no—
411C2×C8C4×D412yes—
412C2×C8C4×D412yes—
413C2×C8C2×M1612yes—
414C2×C8#13112yes—
415C2×C8#13112yes—
416C2×C8AGL(1,ℤ/8)12yes—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—42no—
430C4⋊C4—81no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—81no—
441C22⋊C4—81no—
442C22⋊C4—81no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42C22×D411yescharacteristic
454C42C2×#4111yescharacteristic
455C42H(5,𝔽2)11yescharacteristic
456C422−1+411yescharacteristic
457C23—21no—
458C23—21no—
459C23—22no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23C2×C4×D411yescharacteristic
479D4—22no—
480D4—22no—
481D4—22no—
482D4—22no—
483D4—41no—
484D4—41no—
485D4—41no—
486D4—41no—
487D4—84no—
488D4—84no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4C2×C4×D411yescharacteristic
537C2×C4#52011yescharacteristic
538C2×C4D4211yescharacteristic
539C2×C4#54711yescharacteristic
540C2×C4#54811yescharacteristic
541C2×C4#55311yescharacteristic
542C2×C4C22×M1611yescharacteristic
543C2×C4C2×#13111yescharacteristic
544C2×C4C2×#13111yescharacteristic
545C2×C4#56811yescharacteristic
546C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
547C2×C4#57511yescharacteristic
548C2×C4#57611yescharacteristic
549C2×C4#57711yescharacteristic
550C8—22no—
551C8—22no—
552C8—22no—
553C8—42no—
554C8—41no—
555C8—41no—
556C8—41no—
557C8—42no—
558C8—41no—
559C8—42no—
560C8—81no—
561C8—81no—
562C8—81no—
563C8#43514yescomplemented
564C22—41no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—41no—
570C22—42no—
571C22—42no—
572C22—41no—
573C22—42no—
574C22—42no—
575C22—41no—
576C22—41no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22#236111yescharacteristic
586C22#293111yescharacteristic
587C22D4×M1611yescharacteristic
588C22C4×AGL(1,ℤ/8)11yescharacteristic
589C22#294611yescharacteristic
590C22#295511yescharacteristic
591C22#295611yescharacteristic
592C4—21no—
593C4—21no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4C2×#43511yescharacteristic
604C4#293311yescharacteristic
605C4#299011yescharacteristic
606C4#299311yescharacteristic
607C4#327411yescharacteristic
608C4#327511yescharacteristic
609C4#334511yescharacteristic
610C4#334711yescharacteristic
611C2—81no—
612C2—82no—
613C2—82no—
614C2#1796611yescharacteristic
615C2#1796811yescharacteristic
616C2#1804611yescharacteristic
617C2#1805011yescharacteristic
618C2#2298611yescharacteristic
619C2#2302111yescharacteristic
620C2#2303911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#469C4
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. #23084C2semidirect
2. #22986C2semidirect
4. C2×#1580C2semidirect
5. C2×#1569C2semidirect
6. #9657C2semidirect
7. #9596C2semidirect
8. #9546C2semidirect
9. #9545C2semidirect
10. #8944C2semidirect
11. #8943C2semidirect
12. #8845C2semidirect
13. #8843C2semidirect
14. #8283C2semidirect
15. #8146C2semidirect
163. #334D4semidirect
181. C2×D8M16semidirect
563. C8#435semidirect

Maximal subgroups

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

subgroupindex
#230842
#229862
C4×#4692
C2×#15802
C2×#15692
#96572
#95962
#95462
#95452
#89442
#89432
#88452
#88432
#82832
#81462