Finite Group Databasebeta

← #374153

Subgroups — #374153

Order 512.

1702 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic
2#34178C211yesmaximal, characteristic
3#20972C212yesmaximal
4C22×#343C211yesmaximal, characteristic
5#11210C212yesmaximal, complemented
6#11055C212yesmaximal, complemented
7#10957C212yesmaximal, complemented
8#10891C212yesmaximal, complemented
9#10732C212yesmaximal
10C23×M16C2212yes—
11C22×C22⋊Q8C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C2×#419C2214yes—
14#2502—24no—
15#2502C418yescomplemented
16C2×#398—22no—
17#2345—24no—
18#2344—24no—
19#2316C2212yes—
20#2313—22no—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#385—22no—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26#2280—22no—
27#2280C2212yes—
28#2274C2212yes—
29C2×C4×C22⋊C4C2211yescharacteristic
30#1894—24no—
31#1892—24no—
32C2×#343—24no—
33C2×#343C2212yes—
34C2×#343C2212yes—
35#1736C2214yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38C23×Q8C2311yescharacteristic
39C24×C4D412yes—
40C24×C4C2311yescharacteristic
41C22×M16—22no—
42C22×M16—24no—
43C22×M16—28no—
44C22×M16—24no—
45C22×M16—44no—
46C22×M16C2314yes—
47C23×C8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8C2×C414yes—
53C2×#117—24no—
54C2×#117C2314yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2×C414yes—
62C22×C4⋊C4D412yes—
63C22×C4⋊C4D414yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71#419—28no—
72#419—44no—
73C2×#103—28no—
74#398—216no—
75#391—22no—
76#391—22no—
77#391C2×C414yes—
78#391C2×C414yes—
79#389—28no—
80#387—216no—
81#385—216no—
82#384—22no—
83#384—22no—
84#384C2×C414yes—
85#384C2×C414yes—
86#381—24no—
87#381—22no—
88#380—24no—
89#380—24no—
90#380D414yes—
91#380D414yes—
92#379—22no—
93C4×C4⋊C4—28no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4C2×C414yes—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103#343—24no—
104#343—24no—
105#340—28no—
106#339—28no—
107#339—44no—
108C25C2×D411yescharacteristic
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—24no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8C22×C412yes—
116C23×C4—24no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—24no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—84no—
127C23×C4C22⋊C418yes—
128C23×C4C22×C412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4#4114yes—
136C23×C4C2411yescharacteristic
137C2×M16—24no—
138C2×M16—216no—
139C2×M16—24no—
140C2×M16—24no—
141C2×M16—24no—
142C2×M16—24no—
143C2×M16—48no—
144C2×M16—416no—
145C2×M16—48no—
146C2×M16—44no—
147C2×M16#4118yes—
148C22×C8—24no—
149C22×C8—48no—
150C22×C8—44no—
151C22⋊Q8—28no—
152C22⋊Q8—28no—
153C22⋊Q8—44no—
154C22⋊Q8—44no—
155#117—24no—
156#117—24no—
157#117—24no—
158#117—24no—
159#117—88no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—84no—
180C2×C4⋊C4—84no—
181C2×C4⋊C4C22⋊C418yes—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C2×D418yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—42no—
197C2×C42—24no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—24no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—28no—
204C2×C42C22×C412yes—
205C2×C42C22×C412yes—
206#103—416no—
207#95—28no—
208#95—28no—
209#95—24no—
210#95—28no—
211#95—28no—
212#95—24no—
213#95—24no—
214C24—22no—
215C24—22no—
216C24—24no—
217C24—42no—
218C24C2×C22⋊C412yes—
219C24#12011yescharacteristic
220C24#12011yescharacteristic
221C24#12312yes—
222C24C22×D411yescharacteristic
223C2×Q8—24no—
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—44no—
233C2×Q8—44no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—28no—
246C22×C4—28no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—28no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—28no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—48no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—88no—
275C22×C4—84no—
276C22×C4—88no—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C414yes—
279C22×C4C2×C22⋊C414yes—
280C22×C4C4×D414yes—
281C22×C4C4×D414yes—
282C22×C4#12012yes—
283C22×C4#12114yes—
284C22×C4#12114yes—
285C22×C4#12312yes—
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D412yes—
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4112yes—
290C22×C4C2×#4114yes—
291C22×C4H(5,𝔽2)12yes—
292C22×C42−1+412yes—
293M16—28no—
294M16—28no—
295M16—416no—
296M16—44no—
297M16—44no—
298C2×C8—28no—
299C2×C8—416no—
300C2×C8—44no—
301C2×C8—44no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—216no—
307C4⋊C4—24no—
308C4⋊C4—28no—
309C4⋊C4—28no—
310C4⋊C4—42no—
311C4⋊C4—48no—
312C4⋊C4—88no—
313C4⋊C4—88no—
314C4⋊C4—88no—
315C22⋊C4—216no—
316C22⋊C4—48no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—48no—
322C42—24no—
323C42—24no—
324C42—42no—
325C42—41no—
326C42—41no—
327C42—44no—
328C42—44no—
329C42—88no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—24no—
336C23—24no—
337C23—22no—
338C23—24no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—42no—
343C23—42no—
344C23—48no—
345C23—48no—
346C23#38612yes—
347C23#38612yes—
348C23#45614yes—
349C23C22×C22⋊C411yescharacteristic
350C23C2×#12011yescharacteristic
351C23C2×#12311yescharacteristic
352C23#53212yes—
353C23#53712yes—
354Q8—48no—
355Q8—82no—
356Q8—82no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—28no—
362C2×C4—28no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—416no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—48no—
385C2×C4—48no—
386C2×C4—48no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—88no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4#38614yes—
398C2×C4C2×C4×D412yes—
399C2×C4C4×#4112yes—
400C2×C4#52012yes—
401C2×C4C2×#12112yes—
402C2×C4#52512yes—
403C2×C4#53511yescharacteristic
404C2×C4#53611yescharacteristic
405C2×C4#54614yes—
406C2×C4#54714yes—
407C2×C4#55114yes—
408C2×C4#55214yes—
409C8—48no—
410C22—22no—
411C22—24no—
412C22—24no—
413C22—28no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—44no—
423C22—44no—
424C22—48no—
425C22—44no—
426C22#188614yes—
427C22C2×#38611yescharacteristic
428C22#229711yescharacteristic
429C22#231111yescharacteristic
430C22#243012yes—
431C22#249812yes—
432C22C2×#45612yes—
433C22#301912yes—
434C4—24no—
435C4—44no—
436C4—44no—
437C4—44no—
438C4—44no—
439C4—82no—
440C4—88no—
441C4—82no—
442C4#235914yes—
443C4#241112yes—
444C4#241312yes—
445C2—21no—
446C2—21no—
447C2—24no—
448C2—22no—
449C2—42no—
450C2#1413511yescharacteristic
451C2C2×#188611yescharacteristic
452C2#2095311yescharacteristic
453C2#2551914yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11210C2semidirect
6. #11055C2semidirect
7. #10957C2semidirect
8. #10891C2semidirect
15. #2502C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
#341782
#209722
C22×#3432
#112102
#110552
#109572
#108912
#107322