Finite Group Databasebeta

← #374479

Subgroups — #374479

Order 512.

1334 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2565C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic
3#21325C212yesmaximal
4C2×C4×#101C211yesmaximal, characteristic
5#11162C212yesmaximal
6#11162C212yesmaximal
7#10892C212yesmaximal
8#10891C212yesmaximal
9#10851C212yesmaximal
10C22×#125C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C22×#101C2212yes—
13C2×C4×M16C2212yes—
14#2565—24no—
15#2565C418yescomplemented
16C2×#398—22no—
17#2356—24no—
18#2355—24no—
19#2313—22no—
20#2310C2212yes—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#387—22no—
24#2285C2212yes—
25C2×#384C2211yescharacteristic
26#2283—22no—
27C2×#382C2211yescharacteristic
28#2274C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30#1992—24no—
31#1991—24no—
32C4×#101—24no—
33C2×#343C2212yes—
34#1736C2214yes—
35#1736C2214yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38C23×Q8C2311yescharacteristic
39C22×M16C2314yes—
40C2×#125—22no—
41C2×#125—22no—
42C2×#125—22no—
43C2×#125—24no—
44C2×#125C2×C414yes—
45C2×C4×Q8—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8—24no—
48C2×C4×Q8C2×C414yes—
49C2×C4×Q8D414yes—
50C2×#117C2314yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C42D412yes—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#101—24no—
64C2×#101—28no—
65C4×M16—24no—
66C4×M16—24no—
67C2×#97—22no—
68C2×C4×C8—22no—
69#398—216no—
70#391—22no—
71#391—22no—
72#391—28no—
73#391C2×C414yes—
74#391C2×C414yes—
75#387—216no—
76#387—216no—
77#384—22no—
78#384C2×C414yes—
79#384D414yes—
80#383—24no—
81#382—24no—
82#382—22no—
83#382C2×C414yes—
84#382D414yes—
85#381—24no—
86#380—22no—
87#380—22no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—28no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4C2×C414yes—
94C4×C22⋊C4—24no—
95#376—24no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100#343—24no—
101#340—28no—
102#340—28no—
103#339—28no—
104#339—28no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—24no—
108C22×Q8—24no—
109C22×Q8C22×C412yes—
110C22×Q8C2×D412yes—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
120C2×M16—24no—
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—24no—
124C2×M16—24no—
125C2×M16#4118yes—
126C22×C8—24no—
127#125—28no—
128#125—28no—
129#125—44no—
130#125—44no—
131C4×Q8—28no—
132C4×Q8—28no—
133C4×Q8—28no—
134C4×Q8—28no—
135#117—24no—
136#117—24no—
137#117—24no—
138#117—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4C22⋊C418yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C2×D418yes—
167C2×C4⋊C4C2×D412yes—
168C2×C22⋊C4—24no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—24no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—28no—
178C2×C42C22⋊C418yes—
179C2×C42C22×C412yes—
180C2×C42C22×C412yes—
181C2×C42C22×C412yes—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42#4114yes—
186#101—216no—
187#97—44no—
188C4×C8—44no—
189#95—28no—
190#95—28no—
191#95—28no—
192#95—24no—
193#95—28no—
194#95—28no—
195#95—28no—
196#95—24no—
197#95—24no—
198C24C22×D411yescharacteristic
199C2×Q8—24no—
200C2×Q8—24no—
201C2×Q8—24no—
202C2×Q8—24no—
203C2×Q8—24no—
204C2×Q8—24no—
205C2×Q8—44no—
206C2×Q8—44no—
207C2×Q8—44no—
208C2×Q8—44no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—28no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4C2×C22⋊C414yes—
236C22×C4C2×C22⋊C414yes—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4#12011yescharacteristic
240C22×C4#12012yes—
241C22×C4#12011yescharacteristic
242C22×C4#12312yes—
243C22×C4#12312yes—
244C22×C4C23×C411yescharacteristic
245C22×C4C22×D412yes—
246C22×C4C22×D411yescharacteristic
247C22×C4C2×#4112yes—
248C22×C4C2×#4114yes—
249C22×C42−1+414yes—
250M16—28no—
251M16—28no—
252M16—44no—
253M16—44no—
254C2×C8—28no—
255C2×C8—44no—
256C2×C8—44no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—216no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—28no—
263C4⋊C4—28no—
264C4⋊C4—48no—
265C4⋊C4—48no—
266C4⋊C4—416no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—48no—
272C4⋊C4—84no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C42—28no—
276C42—28no—
277C42—24no—
278C42—24no—
279C42—42no—
280C42—42no—
281C42—44no—
282C42—44no—
283C42—88no—
284C42C4×D414yes—
285C42C4×D414yes—
286C42#12114yes—
287C42#12114yes—
288C23—22no—
289C23—22no—
290C23C22×C22⋊C411yescharacteristic
291C23C2×#12011yescharacteristic
292C23C2×#12311yescharacteristic
293C23#53212yes—
294C23#53612yes—
295Q8—48no—
296Q8—48no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—48no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—48no—
321C2×C4—48no—
322C2×C4—48no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—82no—
327C2×C4—84no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4#38612yes—
333C2×C4#38612yes—
334C2×C4#38614yes—
335C2×C4C2×C4×D412yes—
336C2×C4C4×#4112yes—
337C2×C4#51912yes—
338C2×C4C2×#12112yes—
339C2×C4#52512yes—
340C2×C4#53612yes—
341C2×C4#54714yes—
342C2×C4#54814yes—
343C2×C4#55214yes—
344C2×C4#55614yes—
345C8—48no—
346C22—24no—
347C22—24no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22C2×#38611yescharacteristic
353C22#229711yescharacteristic
354C22#231011yescharacteristic
355C22#243212yes—
356C22#249912yes—
357C22#301812yes—
358C22#302012yes—
359C4—24no—
360C4—22no—
361C4—22no—
362C4—44no—
363C4—44no—
364C4—44no—
365C4—82no—
366C4—88no—
367C4—82no—
368C4—82no—
369C4#236214yes—
370C4#241514yes—
371C2—21no—
372C2—21no—
373C2—22no—
374C2#1413811yescharacteristic
375C2#2095211yescharacteristic
376C2#2095411yescharacteristic
377C2#2569814yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2565C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25652
#341872
#213252
C2×C4×#1012
#111622
#108922
#108912
#108512