Finite Group Databasebeta

← #207020

Subgroups — #207020

Order 512.

1112 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2#20524C211yesmaximal, characteristic
3#13953C211yesmaximal, characteristic
4#9889C212yesmaximal, complemented
5#9888C212yesmaximal, complemented
6#9885C212yesmaximal, complemented
7#9884C212yesmaximal, complemented
8#8019C212yesmaximal
9#8015C212yesmaximal
10C2×#543—21no—
11C2×#407C2212yes—
12#2857C2212yes—
13#2857C2212yes—
14#2659—21no—
15#2659—21no—
16#2549—24no—
17#2440C414yescomplemented
18#2440C414yescomplemented
19#2315—21no—
20C2×#381C2211yescharacteristic
21C2×#380C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#1753—24no—
25#1749C2212yes—
26C4×C22⋊C8C2212yes—
27#1439—21no—
28#1439—21no—
29#1436—21no—
30#1436—21no—
31C4⋊#95C2211yescharacteristic
32C4⋊#95C2211yescharacteristic
33#1434C2211yescharacteristic
34#1434C2211yescharacteristic
35#1294C2214yes—
36#1294C2214yes—
37#1294C2214yes—
38#1294C2214yes—
39C24×C4C2311yescharacteristic
40C22×M16—22no—
41#543—44no—
42#543—44no—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C412yes—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—24no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C412yes—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4D411yescharacteristic
57C22×C22⋊C4D411yescharacteristic
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61#407—24no—
62#407Q818yescomplemented
63C2×C22⋊C8—22no—
64C2×C22⋊C8C2314yes—
65C2×#97—22no—
66C2×C4×C8—22no—
67#398—24no—
68#398—24no—
69#395—24no—
70#395—24no—
71#391—22no—
72#391—22no—
73#389—22no—
74#389—22no—
75#389C2×C414yes—
76#389C2×C414yes—
77#385—24no—
78#385—24no—
79#384C2×C412yes—
80#384C2×C412yes—
81#381—24no—
82#380—22no—
83#380—22no—
84#380—22no—
85#380—22no—
86#380C2×C414yes—
87#380C2314yes—
88C4×C22⋊C4—24no—
89C4×C22⋊C4C2×C414yes—
90C4×C22⋊C4D414yes—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95D411yescharacteristic
100C2×#95D411yescharacteristic
101C2×#95C2311yescharacteristic
102C2×#95C2311yescharacteristic
103C2×#95C2311yescharacteristic
104C2×#95C2311yescharacteristic
105#336—22no—
106#336—22no—
107C25C2×D411yescharacteristic
108C22×Q8—22no—
109C22×Q8—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—81no—
117C23×C4—81no—
118C23×C4C22×C412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×Q814yes—
126C23×C4#4114yes—
127C23×C4C2411yescharacteristic
128C2×M16—28no—
129C2×M16—82no—
130C2×M16—82no—
131C22×C8—24no—
132C4⋊Q8—82no—
133C4⋊Q8—82no—
134C22⋊Q8—416no—
135C22⋊Q8—416no—
136C22⋊Q8—44no—
137C22⋊Q8—44no—
138C22⋊Q8—44no—
139C22⋊Q8—44no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—81no—
157C2×C4⋊C4—81no—
158C2×C4⋊C4C22×C414yes—
159C2×C4⋊C4C22×C414yes—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—81no—
173C2×C22⋊C4—81no—
174C2×C22⋊C4C22⋊C414yes—
175C2×C22⋊C4C22⋊C414yes—
176C2×C42—42no—
177C2×C42—81no—
178C2×C42—81no—
179C2×C42—81no—
180C2×C42—81no—
181C2×C42C22×C412yes—
182C2×C42C22×C412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C22⋊C8—28no—
186C22⋊C8—24no—
187C22⋊C8C2×Q818yes—
188#97—82no—
189C4×C8—82no—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—42no—
199#95—42no—
200#95—42no—
201#95—42no—
202#95—42no—
203#95—81no—
204#95—81no—
205#95C22⋊C414yes—
206#95C22⋊C414yes—
207C24—21no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—41no—
212C24—41no—
213C24—41no—
214C24—41no—
215C24C2×C22⋊C412yes—
216C24#12412yes—
217C24C4⋊Q812yes—
218C24C22×D411yescharacteristic
219C2×Q8—44no—
220C2×Q8—44no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×Q8—82no—
224C2×Q8—82no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—84no—
278C22×C4—84no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C4×Q818yes—
289C22×C4C22⋊Q814yes—
290C22×C4C22⋊Q814yes—
291C22×C4#12412yes—
292C22×C4C4⋊Q812yes—
293C22×C4AGL(1,ℤ/8)12yes—
294C22×C4AGL(1,ℤ/8)12yes—
295C22×C4#13712yes—
296C22×C4#13712yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×Q812yes—
300C22×C4C2×#4112yes—
301M16—88no—
302C2×C8—28no—
303C2×C8—44no—
304C2×C8—44no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—48no—
309C4⋊C4—44no—
310C4⋊C4—48no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C22⋊C4—28no—
320C22⋊C4—28no—
321C22⋊C4—48no—
322C22⋊C4—48no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—84no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—84no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23#39114yes—
358C23C22×C22⋊C411yescharacteristic
359C23C2×#12411yescharacteristic
360C23C2×C4⋊Q811yescharacteristic
361C23C2×AGL(1,ℤ/8)11yescharacteristic
362C23C2×AGL(1,ℤ/8)11yescharacteristic
363C23C2×#13711yescharacteristic
364C23C2×#13711yescharacteristic
365Q8—164no—
366Q8—164no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—48no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—164no—
433C2×C4—164no—
434C2×C4#39114yes—
435C2×C4#41814yes—
436C2×C4#41914yes—
437C2×C4C2×C4×Q812yes—
438C2×C4C2×C22⋊Q812yes—
439C8—48no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—24no—
453C22—24no—
454C22—24no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22C2×#39111yescharacteristic
464C22#288311yescharacteristic
465C22#288311yescharacteristic
466C22C2×#41811yescharacteristic
467C22C2×#41911yescharacteristic
468C22#290111yescharacteristic
469C22#290111yescharacteristic
470C22#314111yescharacteristic
471C22#314111yescharacteristic
472C22#316511yescharacteristic
473C22#316511yescharacteristic
474C4—22no—
475C4—22no—
476C4—24no—
477C4—24no—
478C4—24no—
479C4—42no—
480C4—42no—
481C4—82no—
482C4—82no—
483C4—84no—
484C4—84no—
485C4—161no—
486C4—161no—
487C2—21no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—44no—
492C2#1762011yescharacteristic
493C2#1762111yescharacteristic
494C2#1762611yescharacteristic
495C2#1762711yescharacteristic
496C2#2130711yescharacteristic
497C2#2132411yescharacteristic
498C2#2133311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9889C2semidirect
5. #9888C2semidirect
6. #9885C2semidirect
7. #9884C2semidirect
17. #2440C4semidirect
18. #2440C4semidirect
62. #407Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
#205242
#139532
#98892
#98882
#98852
#98842
#80192
#80152