Finite Group Databasebeta

← #374149

Subgroups — #374149

Order 512.

1658 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic
2#34178C211yesmaximal, characteristic
3#20972C212yesmaximal
4C22×#343C211yesmaximal, characteristic
5#11211C212yesmaximal, complemented
6#11055C212yesmaximal, complemented
7#10957C212yesmaximal, complemented
8#10892C212yesmaximal, complemented
9#10732C212yesmaximal
10C23×M16C2212yes—
11C22×C22⋊Q8C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C2×#419C2214yes—
14#2503—24no—
15#2503C418yescomplemented
16C2×#400—22no—
17#2345—24no—
18#2344—24no—
19#2316C2212yes—
20#2313—22no—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25#2280—22no—
26#2280C2212yes—
27C2×#381—22no—
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⋊C4C2×C414yes—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4D414yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69#419—28no—
70#419—44no—
71C2×#103—28no—
72#400—216no—
73#391—22no—
74#391—22no—
75#391C2×C414yes—
76#391C2×C414yes—
77#389—28no—
78#387—216no—
79#382—22no—
80#382—22no—
81#382C2×C414yes—
82#382C2×C414yes—
83#381—24no—
84#381—216no—
85#381—22no—
86#380—24no—
87#380—24no—
88#380D414yes—
89#380D414yes—
90#379—22no—
91C4×C4⋊C4—28no—
92C4×C22⋊C4—24no—
93C4×C22⋊C4—24no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4C2×C414yes—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
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—22no—
121C23×C4—24no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—48no—
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—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4C22⋊C418yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C2×D418yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—42no—
193C2×C42—24no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—24no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—28no—
200C2×C42C22×C412yes—
201C2×C42C22×C412yes—
202#103—416no—
203#95—28no—
204#95—28no—
205#95—24no—
206#95—28no—
207#95—28no—
208#95—28no—
209#95—28no—
210#95—24no—
211#95—24no—
212C24—22no—
213C24—22no—
214C24—24no—
215C24—42no—
216C24C2×C22⋊C412yes—
217C24#12011yescharacteristic
218C24#12011yescharacteristic
219C24#12312yes—
220C24C22×D411yescharacteristic
221C2×Q8—24no—
222C2×Q8—24no—
223C2×Q8—24no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—44no—
229C2×Q8—44no—
230C2×Q8—44no—
231C2×Q8—44no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—28no—
244C22×C4—28no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—28no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—28no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—28no—
256C22×C4—48no—
257C22×C4—416no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—88no—
274C22×C4—84no—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C414yes—
277C22×C4C2×C22⋊C414yes—
278C22×C4C4×D414yes—
279C22×C4C4×D414yes—
280C22×C4#12012yes—
281C22×C4#12114yes—
282C22×C4#12114yes—
283C22×C4#12312yes—
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D412yes—
286C22×C4C22×D411yescharacteristic
287C22×C4C2×#4112yes—
288C22×C4C2×#4114yes—
289C22×C4H(5,𝔽2)12yes—
290C22×C42−1+412yes—
291M16—28no—
292M16—28no—
293M16—416no—
294M16—44no—
295M16—44no—
296C2×C8—28no—
297C2×C8—416no—
298C2×C8—44no—
299C2×C8—44no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—216no—
305C4⋊C4—24no—
306C4⋊C4—28no—
307C4⋊C4—28no—
308C4⋊C4—42no—
309C4⋊C4—48no—
310C4⋊C4—88no—
311C22⋊C4—48no—
312C22⋊C4—48no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—48no—
318C42—24no—
319C42—24no—
320C42—42no—
321C42—41no—
322C42—41no—
323C42—44no—
324C42—44no—
325C42—88no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—24no—
332C23—24no—
333C23—22no—
334C23—24no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—42no—
339C23—42no—
340C23—48no—
341C23—48no—
342C23#38612yes—
343C23#38612yes—
344C23#45514yes—
345C23C22×C22⋊C411yescharacteristic
346C23C2×#12011yescharacteristic
347C23C2×#12311yescharacteristic
348C23#53212yes—
349C23#53712yes—
350Q8—48no—
351Q8—82no—
352Q8—82no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—28no—
358C2×C4—28no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—416no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—48no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—88no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4#38614yes—
394C2×C4C2×C4×D412yes—
395C2×C4C4×#4112yes—
396C2×C4#52012yes—
397C2×C4C2×#12112yes—
398C2×C4#52512yes—
399C2×C4#53511yescharacteristic
400C2×C4#53611yescharacteristic
401C2×C4#54614yes—
402C2×C4#54714yes—
403C2×C4#55114yes—
404C2×C4#55214yes—
405C8—48no—
406C22—22no—
407C22—24no—
408C22—24no—
409C22—28no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—44no—
419C22—44no—
420C22—48no—
421C22—44no—
422C22#188514yes—
423C22C2×#38611yescharacteristic
424C22#229711yescharacteristic
425C22#231111yescharacteristic
426C22#243012yes—
427C22#249812yes—
428C22C2×#45512yes—
429C22#302012yes—
430C4—24no—
431C4—44no—
432C4—44no—
433C4—82no—
434C4—82no—
435C4—82no—
436C4—88no—
437C4—82no—
438C4#235914yes—
439C4#241112yes—
440C4#241312yes—
441C2—21no—
442C2—21no—
443C2—24no—
444C2—22no—
445C2—42no—
446C2#1413511yescharacteristic
447C2C2×#188511yescharacteristic
448C2#2095411yescharacteristic
449C2#2551514yes—
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. #11211C2semidirect
6. #11055C2semidirect
7. #10957C2semidirect
8. #10892C2semidirect
15. #2503C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
#341782
#209722
C22×#3432
#112112
#110552
#109572
#108922
#107322