Finite Group Databasebeta

← #237956

Subgroups — #237956

Order 512.

1024 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35330C211yesmaximal, characteristic, complemented
2#21209C211yesmaximal, characteristic
3#17574C211yesmaximal, characteristic
4C2×#1517C211yesmaximal, characteristic
5#10364C214yesmaximal, complemented
6#10362C214yesmaximal, complemented
7#7957C211yesmaximal, characteristic, complemented
8C2×#1286C211yesmaximal, characteristic, complemented
9#7866C211yesmaximal, characteristic, complemented
10C2×#544—21no—
11C2×#532C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#421—21no—
16C2×#421—21no—
17#2862C2212yescomplemented
18#2859C2212yescomplemented
19#2660—22no—
20#2550—22no—
21#2550—22no—
22#2549—22no—
23#2549—22no—
24#2317—21no—
25#2285C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C4×#117C2211yescharacteristic
28#1963—22no—
29#1962C2214yes—
30#1519—22no—
31#1517—22no—
32#1517C2212yes—
33#1517C2214yes—
34#1517C2212yes—
35#1502—22no—
36C2×#327—22no—
37#1286C2214yescomplemented
38#1286C2212yescomplemented
39#1286C2212yescomplemented
40#1276C2214yescomplemented
41C22×#41—21no—
42C22×M16—21no—
43#544—44no—
44#544—44no—
45#532—21no—
46#532—21no—
47#532—21no—
48#532—21no—
49#532—21no—
50#532—21no—
51#532Q812yescomplemented
52#532C2312yes—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8Q811yescharacteristic, complemented
56C2×#127Q811yescharacteristic, complemented
57C2×#125—22no—
58C2×#125—22no—
59C2×#124—21no—
60C2×C22⋊Q8—24no—
61C2×#121—21no—
62C2×#121—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×#117—21no—
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C22×C4⋊C4—21no—
69C22×C42C2311yescharacteristic
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431C2312yes—
77#431C2312yes—
78#423—21no—
79#423—21no—
80#423—21no—
81#423C2312yes—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86#421—44no—
87#421—44no—
88C2×#104—24no—
89C2×#104—24no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94#398—24no—
95#398—24no—
96#395—24no—
97#395—24no—
98#391—24no—
99#391—24no—
100#389—22no—
101#389—22no—
102#389—24no—
103#389—24no—
104#385—24no—
105#384—22no—
106#384—21no—
107#382—22no—
108#382—21no—
109#382—21no—
110#382D412yes—
111#382C2311yescharacteristic
112#382C2312yes—
113#382C2311yescharacteristic
114#380—21no—
115#380D412yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4D412yes—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122#376D412yes—
123C2×#95—21no—
124C2×#95D412yes—
125C43C2312yes—
126#327—28no—
127#327—24no—
128#327—24no—
129C2×#41—22no—
130C2×#41—22no—
131C2×#41—41no—
132C2×#41—41no—
133C2×#41—42no—
134C2×#41—42no—
135C2×#41—42no—
136C2×#41—42no—
137C22×Q8—24no—
138C22×Q8—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—41no—
143C23×C4—22no—
144C23×C4—41no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C2×M16—22no—
149C2×M16—41no—
150C2×M16—42no—
151C2×M16—42no—
152C22×C8—41no—
153C22×C8—41no—
154C22×C8—41no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—44no—
159C4⋊Q8—48no—
160C4⋊Q8—44no—
161C4⋊Q8C2×Q812yes—
162#127—21no—
163#127—21no—
164#127—21no—
165#127C2×Q812yes—
166#125—44no—
167#125—48no—
168#125—44no—
169#124—22no—
170#124—22no—
171#124—41no—
172#124—41no—
173C22⋊Q8—416no—
174#121—22no—
175#121—22no—
176#121—22no—
177#121—22no—
178#121—41no—
179#121—41no—
180#121—41no—
181#121—41no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—41no—
190#117—22no—
191#117—22no—
192#117—42no—
193#117—42no—
194#117D814yes—
195#117SD1614yes—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×Q811yescharacteristic
227C2×C42C2×Q811yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42#4111yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42C2411yescharacteristic
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—41no—
240C4⋊C8—41no—
241C4⋊C8—41no—
242C4⋊C8—41no—
243#104—48no—
244#104—48no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—41no—
248C22⋊C8—41no—
249C4×C8—22no—
250C4×C8—22no—
251C4×C8—41no—
252C4×C8—41no—
253#95—24no—
254#95—24no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—21no—
261#95—21no—
262#95C2×D414yes—
263#95C2×D412yes—
264#95C2×D412yes—
265C24—41no—
266C24#12011yescharacteristic
267#41—42no—
268#41—42no—
269#41—42no—
270#41—42no—
271#41—44no—
272#41—44no—
273#41—82no—
274#41—82no—
275#41—82no—
276#41—82no—
277C2×Q8—22no—
278C2×Q8—48no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—82no—
295C2×D4—82no—
296C2×D4—82no—
297C2×D4—82no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—81no—
308C22×C4—81no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4#12011yescharacteristic
314C22×C4#12011yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4C22⋊Q811yescharacteristic
317C22×C4C22⋊Q811yescharacteristic
318C22×C4C22⋊Q811yescharacteristic
319C22×C4C22⋊Q811yescharacteristic
320C22×C4C22⋊Q811yescharacteristic
321C22×C4C22⋊Q811yescharacteristic
322C22×C4#12411yescharacteristic
323C22×C4#12411yescharacteristic
324C22×C4#12411yescharacteristic
325C22×C4#12411yescharacteristic
326C22×C4#12411yescharacteristic
327C22×C4#12411yescharacteristic
328C22×C4C2×D812yes—
329C22×C4C2×SD1612yes—
330C22×C4AGL(1,ℤ/8)12yes—
331C22×C4#13712yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335M16—42no—
336M16—42no—
337M16—82no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—82no—
342C2×C8—82no—
343C2×C8—82no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—48no—
349C4⋊C4—48no—
350C4⋊C4—48no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—44no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—84no—
363C4⋊C4—84no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—44no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C42—22no—
372C42—22no—
373C42—44no—
374C42—82no—
375C42—82no—
376C42C22⋊Q811yescharacteristic
377C42C22⋊Q811yescharacteristic
378C42C22⋊Q811yescharacteristic
379C42C22⋊Q811yescharacteristic
380C42C22⋊Q811yescharacteristic
381C42C22⋊Q811yescharacteristic
382C42#12411yescharacteristic
383C42#12411yescharacteristic
384C42#12411yescharacteristic
385C42#12411yescharacteristic
386C42#12411yescharacteristic
387C42#12411yescharacteristic
388C42C2×D814yes—
389C42C2×SD1614yes—
390C42C22×Q811yescharacteristic
391C42C2×#4111yescharacteristic
392C42C2×#4111yescharacteristic
393C42C2×#4111yescharacteristic
394C23—41no—
395C23—41no—
396C23—42no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23#39311yescharacteristic
401C23#39311yescharacteristic
402C23#44712yes—
403C23#44812yes—
404C23C2×#12011yescharacteristic
405Q8—42no—
406Q8—42no—
407Q8—82no—
408Q8—168no—
409D4—42no—
410D4—42no—
411D4—82no—
412D4—82no—
413D4—82no—
414D4—82no—
415D4—82no—
416D4—82no—
417D4—84no—
418D4—82no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—84no—
451C2×C4—164no—
452C2×C4#39311yescharacteristic
453C2×C4#39311yescharacteristic
454C2×C4#39311yescharacteristic
455C2×C4#39311yescharacteristic
456C2×C4#39311yescharacteristic
457C2×C4#39311yescharacteristic
458C2×C4#44712yes—
459C2×C4#44812yes—
460C2×C4#47412yes—
461C2×C4#47412yes—
462C2×C4#47612yes—
463C2×C4#47612yes—
464C2×C4#48612yes—
465C2×C4#48612yes—
466C2×C4#48912yes—
467C2×C4#48912yes—
468C2×C4C2×C22⋊Q811yescharacteristic
469C2×C4C2×C22⋊Q811yescharacteristic
470C2×C4C2×C22⋊Q811yescharacteristic
471C2×C4C2×#12411yescharacteristic
472C2×C4C2×#12411yescharacteristic
473C2×C4C2×#12411yescharacteristic
474C2×C4C22×D811yescharacteristic
475C2×C4C22×SD1611yescharacteristic
476C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
477C2×C4C2×#13711yescharacteristic
478C8—82no—
479C8—82no—
480C8—82no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—42no—
488C22—42no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—164no—
493C22#202312yes—
494C22#202312yes—
495C22C2×#39311yescharacteristic
496C22C2×#44711yescharacteristic
497C22C2×#44811yescharacteristic
498C4—21no—
499C4—21no—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—44no—
507C4—44no—
508C4—82no—
509C4—84no—
510C4—82no—
511C4—82no—
512C4#202312yes—
513C4#202312yes—
514C4C2×#47411yescharacteristic
515C4C2×#47611yescharacteristic
516C4C2×#48611yescharacteristic
517C4C2×#48911yescharacteristic
518C2—21no—
519C2—21no—
520C2—22no—
521C2—22no—
522C2—162no—
523C2C2×#202311yescharacteristic
524C2#2347211yescharacteristic
525C2#2347211yescharacteristic
subgroupG/H
center, socleC22C2×#393
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35330C2semidirect
5. #10364C2semidirect
6. #10362C2semidirect
7. #7957C2semidirect
8. C2×#1286C2semidirect
9. #7866C2semidirect
17. #2862C22semidirect
18. #2859C22semidirect
37. #1286C22semidirect
38. #1286C22semidirect
39. #1286C22semidirect
40. #1276C22semidirect
51. #532Q8semidirect
55. C2×C4⋊Q8Q8semidirect
56. C2×#127Q8semidirect

Maximal subgroups

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

subgroupindex
#353302
#212092
#175742
C2×#15172
#103642
#103622
#79572
C2×#12862
#78662