Finite Group Databasebeta

← #231594

Subgroups — #231594

Order 512.

698 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23221C211yesmaximal, characteristic, complemented
2#23220C211yesmaximal, characteristic
3#23217C211yesmaximal, characteristic, complemented
4#23216C211yesmaximal, characteristic
5C2×#1708C211yesmaximal, characteristic
6#9746C211yesmaximal, characteristic
7#9745C211yesmaximal, characteristic, complemented
8#9738C211yesmaximal, characteristic
9#9737C211yesmaximal, characteristic, complemented
10#9064C211yesmaximal, characteristic
11#9063C211yesmaximal, characteristic
12#9059C211yesmaximal, characteristic, complemented
13#9059C211yesmaximal, characteristic, complemented
14#8282C211yesmaximal, characteristic, complemented
15C8×#106C211yesmaximal, characteristic, complemented
16C2×#498C2211yescharacteristic
17C2×#492C2211yescharacteristic
18C2×#487C2211yescharacteristic
19C2×#487C2211yescharacteristic
20C2×#486C2211yescharacteristic
21C2×#486C2211yescharacteristic
22#2369C2211yescharacteristic
23#2366C2211yescharacteristic
24#2365C2211yescharacteristic
25#2365C2211yescharacteristic
26#1988C2211yescharacteristic
27#1988C2211yescharacteristic
28#1987C2211yescharacteristic
29#1987C2211yescharacteristic
30#1982C2211yescharacteristic
31#1982C2211yescharacteristic
32#1981C2211yescharacteristic
33#1981C2211yescharacteristic
34#1936—21no—
35#1935—21no—
36#1934—21no—
37#1933—21no—
38#1928—21no—
39#1927—21no—
40#1926—21no—
41#1925—21no—
42#1865—21no—
43#1864—21no—
44#1863—21no—
45#1862—21no—
46C4×#107C2211yescharacteristic
47C4×#106C2211yescharacteristic
48#1770C2211yescharacteristic
49#1770C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1763—21no—
53#1762—21no—
54C4×#103—21no—
55C4×#102—21no—
56#1708C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#331C2211yescharacteristic
63C2×C82C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68C22×SD16—41no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×#127C2311yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77#498—41no—
78#498C2×C414yescomplemented
79#492—41no—
80#492C2×C414yes—
81#487—22no—
82#487—22no—
83#487C2×C414yescomplemented
84#487C2×C414yescomplemented
85#486—22no—
86#486—22no—
87#486C2×C414yes—
88#486C2×C414yes—
89C2×#107D411yescharacteristic
90C2×#107D411yescharacteristic
91C2×#106—41no—
92C2×#106D411yescharacteristic
93C2×#106D411yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×#102—21no—
114C2×#97—21no—
115C2×#97—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#391—21no—
122#391—21no—
123#391—41no—
124#390—21no—
125#390—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#387—41no—
131#387—41no—
132#386—41no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2311yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340D411yescharacteristic
150#340D411yescharacteristic
151#340D411yescharacteristic
152#340D411yescharacteristic
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#335—22no—
158#334—22no—
159#333—41no—
160#332—41no—
161#332—41no—
162#331—41no—
163C82C2×C414yescomplemented
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C23×C4—41no—
171C2×SD16—44no—
172C2×SD16—82no—
173C2×SD16—82no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8C22×C412yes—
193C4⋊Q8C22×C412yes—
194C4⋊Q8C22×C412yes—
195#127—21no—
196#127—21no—
197#127—21no—
198#127C22×C412yes—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—81no—
202C4×Q8—81no—
203C4×D4—42no—
204C4×D4—42no—
205C4×D4—81no—
206C4×D4—81no—
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⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—41no—
232C2×C42—41no—
233C2×C42—41no—
234C2×C42C2411yescharacteristic
235#107—41no—
236#107—41no—
237#106—22no—
238#106—22no—
239#106—44no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248#103—24no—
249#103—24no—
250#103—24no—
251#103—24no—
252#103—42no—
253#103—42no—
254#103—81no—
255#103—81no—
256#102—24no—
257#102—24no—
258#102—24no—
259#102—24no—
260#102—42no—
261#102—42no—
262#102—81no—
263#102—81no—
264#97—81no—
265#97—81no—
266C4×C8—21no—
267C4×C8—81no—
268C4×C8—81no—
269C4×C8C22×C412yes—
270C4×C8C22×C412yes—
271C4×C8C22×C412yes—
272C4×C8#4112yes—
273C4×C8#4112yes—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—41no—
279#95—41no—
280#95—41no—
281#95—41no—
282#95—41no—
283#95—41no—
284C24—41no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—81no—
300C2×Q8—81no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—81no—
316C2×D4—81no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—161no—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4H(5,𝔽2)11yescharacteristic
334SD16—88no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
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—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8C4×D412yes—
359C2×C8C4×D412yes—
360C2×C8C4×D412yes—
361C2×C8C4×D412yes—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
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—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C4⋊C4—161no—
391C4⋊C4#13512yes—
392C4⋊C4#13512yes—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—161no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—82no—
405C42—82no—
406C42—161no—
407C42C23×C411yescharacteristic
408C42C2×#4111yescharacteristic
409C42C2×#4111yescharacteristic
410C42H(5,𝔽2)11yescharacteristic
411C23—42no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23D4211yescharacteristic
419Q8—42no—
420Q8—42no—
421Q8—42no—
422Q8—42no—
423Q8—82no—
424Q8—81no—
425Q8—81no—
426Q8—81no—
427Q8—81no—
428Q8—161no—
429D4—42no—
430D4—42no—
431D4—42no—
432D4—42no—
433D4—82no—
434D4—81no—
435D4—81no—
436D4—81no—
437D4—81no—
438D4—161no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
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—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—164no—
475C2×C4C2×C4×D411yescharacteristic
476C2×C4C2×C4×D411yescharacteristic
477C2×C4#51811yescharacteristic
478C2×C4#55011yescharacteristic
479C2×C4#55011yescharacteristic
480C2×C4#55911yescharacteristic
481C2×C4C2×#13511yescharacteristic
482C2×C4C2×#13511yescharacteristic
483C2×C4#57511yescharacteristic
484C2×C4#57511yescharacteristic
485C2×C4#57611yescharacteristic
486C2×C4#57711yescharacteristic
487C2×C4#57711yescharacteristic
488C2×C4#57811yescharacteristic
489C8—24no—
490C8—22no—
491C8—22no—
492C8—22no—
493C8—22no—
494C8—81no—
495C8—81no—
496C8—81no—
497C8—81no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—161no—
502C22—161no—
503C22—161no—
504C22—161no—
505C22#236611yescharacteristic
506C22#327511yescharacteristic
507C22#327511yescharacteristic
508C22#328011yescharacteristic
509C22#328511yescharacteristic
510C22#329111yescharacteristic
511C22#329111yescharacteristic
512C4—21no—
513C4—21no—
514C4—42no—
515C4—42no—
516C4—82no—
517C4—82no—
518C4—82no—
519C4—81no—
520C4—81no—
521C4—81no—
522C4—81no—
523C4—81no—
524C4—81no—
525C4C4×#13511yescharacteristic
526C4#294311yescharacteristic
527C4#295111yescharacteristic
528C4#295211yescharacteristic
529C4#339311yescharacteristic
530C4#339611yescharacteristic
531C4#340111yescharacteristic
532C4#340111yescharacteristic
533C2—162no—
534C2#1990111yescharacteristic
535C2#1990411yescharacteristic
536C2#1993611yescharacteristic
537C2#1993911yescharacteristic
538C2#2318611yescharacteristic
539C2#2318911yescharacteristic
540C2#2319811yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23221C2semidirect
3. #23217C2semidirect
7. #9745C2semidirect
9. #9737C2semidirect
12. #9059C2semidirect
13. #9059C2semidirect
14. #8282C2semidirect
15. C8×#106C2semidirect
56. #1708C4semidirect
78. #498C2×C4semidirect
83. #487C2×C4semidirect
84. #487C2×C4semidirect
163. C82C2×C4semidirect

Maximal subgroups

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

subgroupindex
#232212
#232202
#232172
#232162
C2×#17082
#97462
#97452
#97382
#97372
#90642
#90632
#90592
#82822
C8×#1062