Finite Group Databasebeta

← #212732

Subgroups — #212732

Order 512.

679 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21682C211yesmaximal, characteristic
2#21675C211yesmaximal, characteristic, complemented
3C4×#490C211yesmaximal, characteristic
4#9876C211yesmaximal, characteristic, complemented
5#9648C211yesmaximal, characteristic
6#9558C211yesmaximal, characteristic, complemented
7#9558C211yesmaximal, characteristic, complemented
8#9554C211yesmaximal, characteristic, complemented
9#9554C211yesmaximal, characteristic, complemented
10#8786C211yesmaximal, characteristic, complemented
11#8212C211yesmaximal, characteristic, complemented
12#8208C211yesmaximal, characteristic, complemented
13#8204C211yesmaximal, characteristic, complemented
14#8200C211yesmaximal, characteristic, complemented
15#7846C211yesmaximal, characteristic, complemented
16C2×#490C411yescharacteristic
17C2×#436C2211yescharacteristic
18C2×#435—21no—
19C2×#435—21no—
20C2×C8×D4C2211yescharacteristic
21C4×#127C2211yescharacteristic
22C4×#125C2211yescharacteristic
23#1992C411yescharacteristic
24#1961C411yescharacteristic
25#1955—21no—
26#1955—21no—
27#1954C2211yescharacteristic
28#1954C2211yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1936C411yescharacteristic
32#1936C411yescharacteristic
33#1936C411yescharacteristic
34#1936C411yescharacteristic
35#1917—21no—
36#1917—21no—
37#1842C2211yescharacteristic
38#1836C411yescharacteristic
39#1831C2211yescharacteristic
40#1771—21no—
41#1770—21no—
42#1770C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C4×#102C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48C4×#97C2211yescharacteristic
49C4×#97C2211yescharacteristic
50C4×#97C2211yescharacteristic
51C2×D4⋊C8C2211yescharacteristic
52C2×D4⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57C2×#322C2211yescharacteristic
58C2×C82C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69C2×#127C2×C411yescharacteristic
70C2×#125C2×C411yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C2×C4×D4C2311yescharacteristic
75#490—24no—
76#436—22no—
77#436—22no—
78#435—44no—
79#435—44no—
80C8×D4—22no—
81C8×D4—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#102C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
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⋊C8—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×#97—21no—
107C2×#97C2×C411yescharacteristic
108C2×#97C2×C411yescharacteristic
109C2×#97C2×C411yescharacteristic
110C2×#97C2×C411yescharacteristic
111C2×#97D411yescharacteristic
112C2×#97D411yescharacteristic
113C2×#97D411yescharacteristic
114C2×#97D411yescharacteristic
115C2×C4×C8—21no—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#390—21no—
123#390—21no—
124#390C2×C411yescharacteristic
125#390C2×C411yescharacteristic
126#389C2×C411yescharacteristic
127#389C2×C411yescharacteristic
128#387—21no—
129#387—21no—
130#384C2×C411yescharacteristic
131#382C2×C411yescharacteristic
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2311yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C43C2311yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151D4⋊C8—22no—
152D4⋊C8—22no—
153D4⋊C8—22no—
154D4⋊C8—22no—
155#322—22no—
156#322—22no—
157#322—22no—
158C82—22no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4C22×C411yescharacteristic
162C22×D4C22×C411yescharacteristic
163C23×C4—21no—
164C23×C4—21no—
165C2×M16—42no—
166C2×M16—42no—
167C2×M16—42no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—42no—
175C22×C8—42no—
176C22×C8—42no—
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185#127—22no—
186#127—41no—
187#125—22no—
188#125—41no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—42no—
198C4×D4—42no—
199C4×D4—42no—
200C4×D4—42no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42C2411yescharacteristic
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—42no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239C4⋊C8—42no—
240#102—22no—
241#102—22no—
242#102—22no—
243#102—22no—
244#102—22no—
245#102—41no—
246#102—41no—
247#102M1614yescomplemented
248C22⋊C8—42no—
249C22⋊C8—42no—
250C22⋊C8—42no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254#97—22no—
255#97—22no—
256#97—22no—
257#97—22no—
258#97—41no—
259#97—41no—
260#97—41no—
261#97—41no—
262#97—42no—
263#97—42no—
264#97—42no—
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—42no—
274C4×C8#4112yes—
275C4×C8#4112yes—
276C4×C8#4112yes—
277C4×C8#4112yes—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284C24—21no—
285C24—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—81no—
302C2×D4—81no—
303C2×D4C2×M1612yes—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4C4×D411yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4C4×D411yescharacteristic
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C2×#4111yescharacteristic
329M16—84no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—22no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—22no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—84no—
367C2×C8#12412yes—
368C2×C8#12412yes—
369C2×C8C2×M1612yes—
370C2×C8#13112yes—
371C2×C8AGL(1,ℤ/8)12yes—
372C2×C8AGL(1,ℤ/8)12yes—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—81no—
389C4⋊C4—81no—
390C4⋊C4C2×M1612yes—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42C2×#4111yescharacteristic
414C42C2×#4111yescharacteristic
415C42C2×#4111yescharacteristic
416C42C2×#4111yescharacteristic
417C23—22no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23C2×C4×D411yescharacteristic
432D4—22no—
433D4—22no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—41no—
438D4—41no—
439D4—41no—
440D4—84no—
441D4—84no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
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—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4C4×#4111yescharacteristic
483C2×C4C4×#4111yescharacteristic
484C2×C4C2×#12411yescharacteristic
485C2×C4#52911yescharacteristic
486C2×C4#52911yescharacteristic
487C2×C4#53211yescharacteristic
488C2×C4C22×M1611yescharacteristic
489C2×C4C2×#13111yescharacteristic
490C2×C4#56811yescharacteristic
491C2×C4#56811yescharacteristic
492C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
493C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
494C2×C4#57511yescharacteristic
495C2×C4#57511yescharacteristic
496C8—24no—
497C8—22no—
498C8—22no—
499C8—22no—
500C8—22no—
501C8—22no—
502C8—22no—
503C8—42no—
504C8—41no—
505C8—41no—
506C8—81no—
507C8—81no—
508C8—81no—
509C8—81no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22C4×#12411yescharacteristic
525C22D4×M1611yescharacteristic
526C22#293511yescharacteristic
527C22C4×AGL(1,ℤ/8)11yescharacteristic
528C22C4×AGL(1,ℤ/8)11yescharacteristic
529C22#295611yescharacteristic
530C22#295611yescharacteristic
531C4—21no—
532C4—21no—
533C4—41no—
534C4—42no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—81no—
542C4—81no—
543C4#296511yescharacteristic
544C4#296711yescharacteristic
545C4#296811yescharacteristic
546C4#297011yescharacteristic
547C4#313811yescharacteristic
548C4#314211yescharacteristic
549C4#318011yescharacteristic
550C4#318011yescharacteristic
551C2—82no—
552C2—161no—
553C2#1804611yescharacteristic
554C2#1804811yescharacteristic
555C2#1807611yescharacteristic
556C2#1807811yescharacteristic
557C2#2163211yescharacteristic
558C2#2165611yescharacteristic
559C2#2166211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#127C2×C4
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. #21675C2semidirect
4. #9876C2semidirect
6. #9558C2semidirect
7. #9558C2semidirect
8. #9554C2semidirect
9. #9554C2semidirect
10. #8786C2semidirect
11. #8212C2semidirect
12. #8208C2semidirect
13. #8204C2semidirect
14. #8200C2semidirect
15. #7846C2semidirect
247. #102M16semidirect

Maximal subgroups

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

subgroupindex
#216822
#216752
C4×#4902
#98762
#96482
#95582
#95542
#87862
#82122
#82082
#82042
#82002
#78462