Finite Group Databasebeta

← #210599

Subgroups — #210599

Order 512.

683 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23681C211yesmaximal, characteristic
2#22208C211yesmaximal, characteristic
3#20574C211yesmaximal, characteristic, complemented
4C2×SD16⋊C8C211yesmaximal, characteristic
5C2×C8×Q16C211yesmaximal, characteristic, complemented
6#9676C211yesmaximal, characteristic
7#9604C211yesmaximal, characteristic, complemented
8#9548C211yesmaximal, characteristic, complemented
9#9545C211yesmaximal, characteristic
10#8952C211yesmaximal, characteristic
11#8950C211yesmaximal, characteristic, complemented
12#8568C211yesmaximal, characteristic, complemented
13#8264C211yesmaximal, characteristic, complemented
14#8201C211yesmaximal, characteristic, complemented
15C8×#102C211yesmaximal, characteristic
16C2×#446C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C4×#124C2211yescharacteristic
23#2295C2211yescharacteristic
24#2085C411yescharacteristic
25#2074C411yescharacteristic
26#2067C411yescharacteristic
27#2066C411yescharacteristic
28#2056C411yescharacteristic
29#2055C411yescharacteristic
30#2042C411yescharacteristic
31#2015C411yescharacteristic
32#1984C2211yescharacteristic
33#1954C2211yescharacteristic
34#1923—21no—
35#1923—21no—
36#1850—21no—
37#1850—21no—
38#1842C2211yescharacteristic
39#1771—21no—
40#1770—21no—
41#1770C2211yescharacteristic
42C8×C4⋊C4C2211yescharacteristic
43#1768—21no—
44#1767C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1746—21no—
50#1744C2211yescharacteristic
51SD16⋊C8—24no—
52C8×Q16—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×Q16C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C23×C8—21no—
72C2×#124C2×C411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76#446—22no—
77#446—22no—
78C8×Q8—22no—
79C8×Q8—22no—
80C8×Q8—22no—
81C8×Q8D414yes—
82C4×Q16—22no—
83C4×Q16—22no—
84C4×SD16—22no—
85C4×SD16—22no—
86C8×D4—22no—
87C8×D4—22no—
88C2×#107C2×C411yescharacteristic
89C2×#106C2×C411yescharacteristic
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2×C411yescharacteristic
97C2×C4⋊C8C2×C411yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#103C2×C411yescharacteristic
106C2×#103C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×#97—21no—
114C2×#97—21no—
115C2×#97—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125#391C2×C411yescharacteristic
126#391C2×C411yescharacteristic
127#391C2×C411yescharacteristic
128#390C2×C411yescharacteristic
129#389C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#388—21no—
132#388—21no—
133#387—21no—
134#387—21no—
135#382C2×C411yescharacteristic
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2311yescharacteristic
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140#376—21no—
141C43C2311yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336C2×C411yescharacteristic
153#336C2×C411yescharacteristic
154#336D411yescharacteristic
155#336D411yescharacteristic
156#336D411yescharacteristic
157#336D411yescharacteristic
158#335—22no—
159#335—22no—
160Q8⋊C8—22no—
161Q8⋊C8—22no—
162Q8⋊C8—22no—
163Q8⋊C8—22no—
164D4⋊C8—22no—
165D4⋊C8—22no—
166#322—22no—
167C82—22no—
168C22×Q8C22×C411yescharacteristic
169C22×Q8C22×C411yescharacteristic
170C22×D4C22×C411yescharacteristic
171C23×C4—21no—
172C2×Q16—22no—
173C2×Q16—22no—
174C2×Q16—22no—
175C2×SD16—22no—
176C2×SD16—22no—
177C2×SD16—22no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—42no—
185C22×C8—81no—
186C22×C8—81no—
187C22×C8C22×C411yescharacteristic
188C22×C8C22×C411yescharacteristic
189C22×C8C22×C411yescharacteristic
190C22×C8C22×C411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195#124—22no—
196#124—22no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8C2×D412yes—
204C4×Q8#4112yes—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42C2411yescharacteristic
233#107—22no—
234#106—22no—
235C4⋊C8—24no—
236C4⋊C8—24no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8—81no—
248C4⋊C8—81no—
249C4⋊C8C2×D412yes—
250C4⋊C8#4112yes—
251#103—22no—
252#103—22no—
253#103—22no—
254#103—22no—
255#103—22no—
256#102—22no—
257#102—22no—
258#102—22no—
259C22⋊C8—42no—
260C22⋊C8—81no—
261C22⋊C8—81no—
262#97—24no—
263#97—81no—
264#97—81no—
265C4×C8—24no—
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—21no—
275C4×C8—21no—
276C4×C8—42no—
277C4×C8—81no—
278C4×C8C2×D412yes—
279C4×C8#4112yes—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291C24—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8C4×D412yes—
303C2×Q8#13112yes—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4#13112yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4C4×D411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4C4×D411yescharacteristic
326C22×C4C4×D411yescharacteristic
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C2×#4111yescharacteristic
330Q16—28no—
331SD16—28no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—22no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—22no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—22no—
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—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—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—84no—
371C2×C8#12112yes—
372C2×C8#12112yes—
373C2×C8#13112yes—
374C2×C8#13112yes—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—81no—
389C4⋊C4—81no—
390C4⋊C4C4×D412yes—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—81no—
396C22⋊C4—81no—
397C42—22no—
398C42—22no—
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—
413C42C22×D411yescharacteristic
414C42C2×#4111yescharacteristic
415C42C2×#4111yescharacteristic
416C42C2×#4111yescharacteristic
417C23—22no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23C2×C4×D411yescharacteristic
425Q8—22no—
426Q8—22no—
427Q8—22no—
428Q8—22no—
429Q8—41no—
430Q8—41no—
431Q8—41no—
432Q8—41no—
433D4—22no—
434D4—22no—
435D4—41no—
436D4—41no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
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—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4C2×C4×D411yescharacteristic
477C2×C4C4×#4111yescharacteristic
478C2×C4C2×#12111yescharacteristic
479C2×C4#52511yescharacteristic
480C2×C4#52911yescharacteristic
481C2×C4#53211yescharacteristic
482C2×C4C2×#13111yescharacteristic
483C2×C4C2×#13111yescharacteristic
484C2×C4C2×#13111yescharacteristic
485C2×C4C2×#13111yescharacteristic
486C2×C4#57611yescharacteristic
487C2×C4#57711yescharacteristic
488C2×C4#57811yescharacteristic
489C2×C4#57811yescharacteristic
490C8—22no—
491C8—22no—
492C8—22no—
493C8—22no—
494C8—42no—
495C8—41no—
496C8—41no—
497C8—42no—
498C8—41no—
499C8—41no—
500C8—42no—
501C8—81no—
502C8—81no—
503C8—81no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22C4×#12111yescharacteristic
512C22#292911yescharacteristic
513C22#293111yescharacteristic
514C22#294711yescharacteristic
515C22#295111yescharacteristic
516C22#295611yescharacteristic
517C22#295711yescharacteristic
518C4—21no—
519C4—21no—
520C4—41no—
521C4—41no—
522C4—42no—
523C4—41no—
524C4—41no—
525C4—42no—
526C4—42no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—81no—
531C4#292911yescharacteristic
532C4#292911yescharacteristic
533C4#296611yescharacteristic
534C4#296611yescharacteristic
535C4#304411yescharacteristic
536C4#304511yescharacteristic
537C4#312111yescharacteristic
538C4#312411yescharacteristic
539C2—82no—
540C2#1792111yescharacteristic
541C2#1792611yescharacteristic
542C2#1796911yescharacteristic
543C2#1797311yescharacteristic
544C2#2149111yescharacteristic
545C2#2151211yescharacteristic
546C2#2151911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×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
3. #20574C2semidirect
5. C2×C8×Q16C2semidirect
7. #9604C2semidirect
8. #9548C2semidirect
11. #8950C2semidirect
12. #8568C2semidirect
13. #8264C2semidirect
14. #8201C2semidirect

Maximal subgroups

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

subgroupindex
#236812
#222082
#205742
C2×SD16⋊C82
C2×C8×Q162
#96762
#96042
#95482
#95452
#89522
#89502
#85682
#82642
#82012
C8×#1022