Finite Group Databasebeta

← #230765

Subgroups — #230765

Order 512.

671 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23149C211yesmaximal, characteristic, complemented
2#22835C211yesmaximal, characteristic
3C4×#461C211yesmaximal, characteristic
4#9660C211yesmaximal, characteristic
5#9655C211yesmaximal, characteristic
6#9641C211yesmaximal, characteristic, complemented
7#9603C211yesmaximal, characteristic, complemented
8#9544C211yesmaximal, characteristic, complemented
9#9543C211yesmaximal, characteristic, complemented
10#8948C211yesmaximal, characteristic
11#8948C211yesmaximal, characteristic
12#8264C211yesmaximal, characteristic, complemented
13#8201C211yesmaximal, characteristic, complemented
14C8×#103C211yesmaximal, characteristic, complemented
15#8170C211yesmaximal, characteristic
16C2×#461C411yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22C42×D4C2211yescharacteristic
23#1984C2211yescharacteristic
24#1984C2211yescharacteristic
25#1981C411yescharacteristic
26#1954C2211yescharacteristic
27#1954C2211yescharacteristic
28#1948C411yescharacteristic
29#1939—21no—
30#1927C411yescharacteristic
31#1927C411yescharacteristic
32#1923—21no—
33#1923—21no—
34#1916—21no—
35#1916—21no—
36#1877C411yescharacteristic
37#1877C411yescharacteristic
38#1831C2211yescharacteristic
39#1817—21no—
40#1802C411yescharacteristic
41C8×C4⋊C4—21no—
42C8×C4⋊C4C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×C22⋊C8—21no—
50#1744C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C22×SD16C2×C411yescharacteristic
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C22×C42—21no—
76#461C818yescomplemented
77C8×Q8—22no—
78C8×Q8—22no—
79C8×Q8—41no—
80C8×Q8—41no—
81C4×SD16—41no—
82C4×SD16—41no—
83C4×SD16—41no—
84C4×SD16—41no—
85C2×#106C2×C411yescharacteristic
86C2×#106C2×C411yescharacteristic
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2×C411yescharacteristic
90C2×C4⋊C8C2×C411yescharacteristic
91C2×C4⋊C8D411yescharacteristic
92C2×C4⋊C8D411yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#103C2×C411yescharacteristic
98C2×#103C2×C411yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#391C2×C411yescharacteristic
121#391C2×C411yescharacteristic
122#389—21no—
123#389—21no—
124#384C2×C411yescharacteristic
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C43C2311yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336C2×C411yescharacteristic
150#336C2×C411yescharacteristic
151#336D411yescharacteristic
152#336D411yescharacteristic
153#335—41no—
154#334—41no—
155Q8⋊C8—22no—
156Q8⋊C8—22no—
157Q8⋊C8—41no—
158Q8⋊C8—41no—
159D4⋊C8—22no—
160D4⋊C8—22no—
161#322—41no—
162C82—41no—
163C22×Q8C22×C411yescharacteristic
164C22×Q8C22×C411yescharacteristic
165C22×D4C22×C411yescharacteristic
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C2×SD16—41no—
170C2×SD16—41no—
171C2×SD16—41no—
172C2×SD16—41no—
173C2×SD16C2×C814yes—
174C2×SD16C2×C814yes—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8C22×C411yescharacteristic
182C22×C8C22×C411yescharacteristic
183C22×C8C22×C411yescharacteristic
184C22×C8C22×C411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C4⋊Q8—22no—
190C4⋊Q8C2×C814yes—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—41no—
206C4×D4C2×C814yes—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—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×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—81no—
227C2×C42—81no—
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42C2411yescharacteristic
235#106—41no—
236#106—41no—
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—21no—
246C4⋊C8—41no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8C2×C814yes—
250C4⋊C8#4112yes—
251#103—22no—
252#103—22no—
253#103—41no—
254#103—41no—
255#103C2×C814yes—
256#103C2×C814yes—
257#102—41no—
258#102—41no—
259C22⋊C8—42no—
260C22⋊C8—42no—
261C22⋊C8—81no—
262C22⋊C8—81no—
263C4×C8—21no—
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—81no—
278C4×C8—81no—
279C4×C8#4112yes—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287C24—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8C22×C812yes—
299C2×Q8C22×C812yes—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4C22×C812yes—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—81no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4C4×D411yescharacteristic
326C22×C4C4×D411yescharacteristic
327C22×C4C4×D411yescharacteristic
328C22×C4C4×D411yescharacteristic
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4111yescharacteristic
332SD16—44no—
333SD16—44no—
334C2×C8—21no—
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—22no—
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—21no—
356C2×C8—21no—
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—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8C22×C812yes—
374C2×C8C22×C812yes—
375C2×C8#13712yes—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—42no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—81no—
392C4⋊C4C22×C812yes—
393C4⋊C4#13112yes—
394C22⋊C4—24no—
395C22⋊C4—42no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—81no—
413C42—81no—
414C42C22×C812yes—
415C42#13112yes—
416C42C2×#4111yescharacteristic
417C42C2×#4111yescharacteristic
418C42H(5,𝔽2)11yescharacteristic
419C42H(5,𝔽2)11yescharacteristic
420C23—22no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23C2×C4×D411yescharacteristic
428Q8—22no—
429Q8—22no—
430Q8—22no—
431Q8—22no—
432Q8—41no—
433Q8—41no—
434Q8—41no—
435Q8—41no—
436D4—22no—
437D4—22no—
438D4—41no—
439D4—41no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
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—41no—
473C2×C4—84no—
474C2×C4—84no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4C8×D412yes—
480C2×C4C8×D412yes—
481C2×C4C4×#4111yescharacteristic
482C2×C4#51811yescharacteristic
483C2×C4#54611yescharacteristic
484C2×C4#55011yescharacteristic
485C2×C4#55311yescharacteristic
486C2×C4#55511yescharacteristic
487C2×C4C23×C811yescharacteristic
488C2×C4C2×#13111yescharacteristic
489C2×C4C2×#13111yescharacteristic
490C2×C4#56811yescharacteristic
491C2×C4C2×#13711yescharacteristic
492C2×C4#57511yescharacteristic
493C2×C4#57711yescharacteristic
494C2×C4#57711yescharacteristic
495C8—22no—
496C8—22no—
497C8—22no—
498C8—42no—
499C8—41no—
500C8—41no—
501C8—41no—
502C8—41no—
503C8—42no—
504C8—41no—
505C8—81no—
506C8—81no—
507C8—81no—
508C8—81no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22#236311yescharacteristic
517C22C2×C8×D411yescharacteristic
518C22#293311yescharacteristic
519C22C4×#13711yescharacteristic
520C22#294911yescharacteristic
521C22#295511yescharacteristic
522C22#295711yescharacteristic
523C4—21no—
524C4—21no—
525C4—22no—
526C4—22no—
527C4—42no—
528C4—42no—
529C4—41no—
530C4—41no—
531C4—42no—
532C4—42no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4C8×#4111yescharacteristic
537C4#296911yescharacteristic
538C4#297211yescharacteristic
539C4#297911yescharacteristic
540C4#330711yescharacteristic
541C4#332011yescharacteristic
542C4#340011yescharacteristic
543C4#340111yescharacteristic
544C2—82no—
545C2C8×#13711yescharacteristic
546C2#1785511yescharacteristic
547C2#1801311yescharacteristic
548C2#1802811yescharacteristic
549C2#2308311yescharacteristic
550C2#2311211yescharacteristic
551C2#2311511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23149C2semidirect
6. #9641C2semidirect
7. #9603C2semidirect
8. #9544C2semidirect
9. #9543C2semidirect
12. #8264C2semidirect
13. #8201C2semidirect
14. C8×#103C2semidirect
76. #461C8semidirect

Maximal subgroups

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

subgroupindex
#231492
#228352
C4×#4612
#96602
#96552
#96412
#96032
#95442
#95432
#89482
#82642
#82012
C8×#1032
#81702