Finite Group Databasebeta

← #204640

Subgroups — #204640

Order 512.

1390 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35032C211yesmaximal, characteristic
2#20524C211yesmaximal, characteristic, complemented
3C2×#1513C212yesmaximal, complemented
4C2×#1504C212yesmaximal, complemented
5#13965C211yesmaximal, characteristic, complemented
6#9682C214yesmaximal, complemented
7#9680C212yesmaximal, complemented
8#9679C212yesmaximal, complemented
9C2×#525C2212yes—
10C2×#407C2212yes—
11#2857C2211yescharacteristic
12#2856C2211yescharacteristic
13#2850C2212yes—
14#2617—22no—
15#2602—24no—
16#2500C412yescomplemented
17#2499C412yescomplemented
18#2432C412yescomplemented
19#2431C412yescomplemented
20#2325—22no—
21C2×#385—22no—
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24C22×#95C2211yescharacteristic
25#1753—24no—
26#1749C2212yes—
27C4×C22⋊C8C2212yes—
28#1513—28no—
29#1504—28no—
30C2×#328C2214yes—
31C2×#328C2214yes—
32C2×#327C2214yes—
33C2×#327C2214yes—
34#1444—22no—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C22×#41C2×C412yes—
38C24×C4—21no—
39C24×C4C2311yescharacteristic
40C22×M16—22no—
41C2×C4⋊Q8C2×C411yescharacteristic, complemented
42C2×#127C2×C411yescharacteristic, complemented
43C2×#124C2×C412yescomplemented
44#525—24no—
45#525—24no—
46#525—24no—
47#525—24no—
48C2×#123—22no—
49C2×C22⋊Q8C2×C412yes—
50C2×C22⋊Q8C2312yes—
51C2×#121—22no—
52C2×#121C2×C412yes—
53C2×#121C2312yes—
54C2×#120—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×#117C2×C412yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4D412yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65#407—24no—
66#407D418yescomplemented
67C2×C22⋊C8—22no—
68C2×C22⋊C8C2314yes—
69C2×#97—22no—
70C2×C4×C8—22no—
71#397—24no—
72#394—24no—
73#394—24no—
74#392—24no—
75#391C2×C412yescomplemented
76#391C2×C412yescomplemented
77#390—22no—
78#390C2×C412yescomplemented
79#390C2×C412yescomplemented
80#389—22no—
81#388—24no—
82#388—24no—
83#387—24no—
84#386—24no—
85#386—24no—
86#385—24no—
87#385—48no—
88#384C2×C411yescharacteristic, complemented
89#384C2×C411yescharacteristic, complemented
90#382C2×C412yescomplemented
91#380—22no—
92#380—24no—
93C4×C22⋊C4C2×C412yes—
94C4×C22⋊C4C2×C412yes—
95C4×C22⋊C4D414yes—
96C2×#95—24no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95D412yes—
101C2×#95C2312yes—
102C2×#95C2312yes—
103#336—22no—
104#336—22no—
105#328—28no—
106#328—28no—
107#327—28no—
108#327—28no—
109C25C2×D411yescharacteristic
110C2×#41—24no—
111C2×#41—24no—
112C2×#41—82no—
113C2×#41—82no—
114C22×Q8C22×C412yes—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4C22×C412yes—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—44no—
124C23×C4—42no—
125C23×C4—44no—
126C23×C4—81no—
127C23×C4—81no—
128C23×C4—81no—
129C23×C4—81no—
130C23×C4C22×C412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D414yes—
136C23×C4#4114yes—
137C23×C4C2411yescharacteristic
138C2×M16—28no—
139C2×M16—82no—
140C2×M16—82no—
141C22×C8—24no—
142C4⋊Q8—81no—
143#127—81no—
144#124—82no—
145#123—44no—
146#123—44no—
147C22⋊Q8—24no—
148C22⋊Q8—24no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153#121—24no—
154#121—24no—
155#121—22no—
156#121—22no—
157#121—22no—
158#121—22no—
159#121—44no—
160#121—44no—
161#120—44no—
162#120—44no—
163C4×D4—44no—
164C4×D4—44no—
165C4×D4—44no—
166C4×D4—44no—
167#117—24no—
168#117—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—81no—
182C2×C4⋊C4—81no—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4C22⋊C414yes—
198C2×C22⋊C4C22⋊C414yes—
199C2×C42—22no—
200C2×C42—81no—
201C2×C42—81no—
202C2×C42C22×C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C22⋊C8—28no—
208C22⋊C8—24no—
209C22⋊C8C2×D418yes—
210#97—82no—
211C4×C8—82no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—44no—
218#95—42no—
219#95—44no—
220#95C22⋊C414yes—
221#95C22⋊C414yes—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—42no—
226C24—41no—
227C24—41no—
228C24C2×C22⋊C412yes—
229C24#12412yes—
230C24#12712yes—
231C24C22×D411yescharacteristic
232#41—88no—
233#41—88no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—82no—
250C2×D4—82no—
251C2×D4—84no—
252C2×D4—82no—
253C2×D4—82no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—48no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4—88no—
292C22×C4—88no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4C2×C22⋊C414yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C4×D414yes—
300C22×C4C4×D414yes—
301C22×C4#12114yes—
302C22×C4#12114yes—
303C22×C4#12412yes—
304C22×C4#12712yes—
305C22×C4AGL(1,ℤ/8)14yes—
306C22×C4#13714yes—
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D412yes—
310C22×C4C2×#4112yes—
311M16—88no—
312C2×C8—28no—
313C2×C8—44no—
314C2×C8—44no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—88no—
326C4⋊C4—88no—
327C4⋊C4—84no—
328C22⋊C4—28no—
329C22⋊C4—28no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—44no—
337C22⋊C4—48no—
338C22⋊C4—84no—
339C22⋊C4—84no—
340C22⋊C4—84no—
341C42—42no—
342C42—42no—
343C42—42no—
344C42—41no—
345C42—41no—
346C42—44no—
347C23—21no—
348C23—21no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—24no—
353C23—24no—
354C23—22no—
355C23—22no—
356C23—24no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—44no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23—42no—
366C23—42no—
367C23—44no—
368C23—44no—
369C23—42no—
370C23#39014yes—
371C23C22×C22⋊C411yescharacteristic
372C23C2×#12411yescharacteristic
373C23C2×#12711yescharacteristic
374C23C2×AGL(1,ℤ/8)12yes—
375C23C2×#13712yes—
376Q8—82no—
377Q8—82no—
378D4—82no—
379D4—82no—
380D4—88no—
381D4—88no—
382D4—84no—
383D4—82no—
384D4—82no—
385D4—164no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—88no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—816no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4#39014yes—
435C2×C4#41714yes—
436C2×C4#41714yes—
437C2×C4C2×C4×D412yes—
438C2×C4C2×#12112yes—
439C8—48no—
440C22—21no—
441C22—21no—
442C22—22no—
443C22—22no—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—24no—
449C22—24no—
450C22—24no—
451C22—44no—
452C22—48no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22—82no—
457C22—82no—
458C22—84no—
459C22—82no—
460C22—84no—
461C22C2×#39011yescharacteristic
462C22#288311yescharacteristic
463C22#288311yescharacteristic
464C22C2×#41712yes—
465C22#290112yes—
466C22#314112yes—
467C22#315111yescharacteristic
468C22#315211yescharacteristic
469C4—24no—
470C4—24no—
471C4—22no—
472C4—24no—
473C4—22no—
474C4—44no—
475C4—84no—
476C4—82no—
477C4—82no—
478C4—82no—
479C4—162no—
480C2—21no—
481C2—22no—
482C2—21no—
483C2—44no—
484C2—162no—
485C2#1762212yes—
486C2#1763012yes—
487C2#2120011yescharacteristic
488C2#2121311yescharacteristic
489C2#2125311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35032C2
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
2. #20524C2semidirect
3. C2×#1513C2semidirect
4. C2×#1504C2semidirect
5. #13965C2semidirect
6. #9682C2semidirect
7. #9680C2semidirect
8. #9679C2semidirect
16. #2500C4semidirect
17. #2499C4semidirect
18. #2432C4semidirect
19. #2431C4semidirect
41. C2×C4⋊Q8C2×C4semidirect
42. C2×#127C2×C4semidirect
43. C2×#124C2×C4semidirect
66. #407D4semidirect
75. #391C2×C4semidirect
76. #391C2×C4semidirect
78. #390C2×C4semidirect
79. #390C2×C4semidirect
88. #384C2×C4semidirect
89. #384C2×C4semidirect
90. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350322
#205242
C2×#15132
C2×#15042
#139652
#96822
#96802
#96792