Finite Group Databasebeta

← #194877

Subgroups — #194877

Order 512.

1358 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2#34059C211yesmaximal, characteristic
3#21840C212yesmaximal
4C2×#1467C211yesmaximal, characteristic
5#9027C214yesmaximal
6#9026C214yesmaximal
7#8021C212yesmaximal
8C22×C4⋊Q8—21no—
9C22×C4⋊Q8C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#432C2212yes—
12#2546—24no—
13#2546C418yescomplemented
14C2×#398C2212yes—
15C2×#398C2212yes—
16#2320—24no—
17#2316C2212yes—
18#2313—21no—
19#2313C2212yes—
20#2310—21no—
21#2295—24no—
22C2×#391—21no—
23C2×#389—21no—
24C2×#389—21no—
25#2283—21no—
26#2283C2212yes—
27C4×#117—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1993—24no—
30#1987—24no—
31#1823C2218yes—
32#1467—24no—
33C2×#324C2212yes—
34C2×#324C2212yes—
35#1294C2218yes—
36C23×Q8—21no—
37C23×Q8D411yescharacteristic
38C2×C4⋊Q8—28no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—48no—
44C2×C4⋊Q8C2×C414yes—
45C2×C4×Q8—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8—24no—
48C2×C4×Q8C2×C414yes—
49C2×C4×Q8D414yes—
50C2×#117—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C42—21no—
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42C2311yescharacteristic
63#432—24no—
64#432—24no—
65C2×C4⋊C8—22no—
66C2×C22⋊C8C2314yes—
67C2×#97—22no—
68#398—24no—
69#398—24no—
70#398C2×C418yes—
71#398C2×C418yes—
72#391—28no—
73#391—28no—
74#389—28no—
75#389—28no—
76#389—28no—
77#387—28no—
78#387—28no—
79#383—22no—
80#383—22no—
81#382—22no—
82#382D414yes—
83#381—22no—
84#381—22no—
85#380—22no—
86#380C2314yes—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4C2×C414yes—
92C4×C4⋊C4D414yes—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95#376—22no—
96#376—22no—
97#376—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95C2312yes—
101C43—22no—
102#340—216no—
103#324—24no—
104#324—24no—
105C22×Q8—24no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—24no—
109C22×Q8—42no—
110C22×Q8—44no—
111C22×Q8—44no—
112C22×Q8—42no—
113C22×Q8—42no—
114C22×Q8C22⋊C414yes—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2411yescharacteristic
124C22×C8—24no—
125C4⋊Q8—28no—
126C4⋊Q8—28no—
127C4⋊Q8—432no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130C4×Q8—28no—
131C4×Q8—28no—
132C4×Q8—28no—
133C4×Q8—28no—
134#117—44no—
135#117—44no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—48no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4C22⋊C414yes—
158C2×C4⋊C4C22×C414yes—
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C2×D412yes—
162C2×C4⋊C4C2×D412yes—
163C2×C4⋊C4C2×D418yes—
164C2×C22⋊C4—24no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—24no—
175C2×C42—44no—
176C2×C42C22⋊C414yes—
177C2×C42C22⋊C414yes—
178C2×C42C22×C412yes—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42#4114yes—
183C4⋊C8—44no—
184C22⋊C8—24no—
185C22⋊C8—24no—
186#97—44no—
187#95—24no—
188#95—24no—
189#95—24no—
190#95—24no—
191#95—28no—
192#95—28no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—22no—
197#95—24no—
198#95C22×C414yes—
199C24C22×D411yescharacteristic
200C2×Q8—28no—
201C2×Q8—28no—
202C2×Q8—44no—
203C2×Q8—48no—
204C2×Q8—48no—
205C2×Q8—48no—
206C2×Q8—48no—
207C2×Q8—48no—
208C2×Q8—88no—
209C2×Q8—88no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—28no—
217C22×C4—24no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—28no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4#12011yescharacteristic
244C22×C4#12012yes—
245C22×C4#12011yescharacteristic
246C22×C4#12312yes—
247C22×C4#12312yes—
248C22×C4AGL(1,ℤ/8)14yes—
249C22×C4C23×C411yescharacteristic
250C22×C4C22×D412yes—
251C22×C4C22×D411yescharacteristic
252C22×C4C2×#4112yes—
253C2×C8—48no—
254C2×C8—44no—
255C4⋊C4—216no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—28no—
261C4⋊C4—28no—
262C4⋊C4—48no—
263C4⋊C4—416no—
264C4⋊C4—48no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—416no—
269C4⋊C4—84no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C42—24no—
273C42—24no—
274C42—24no—
275C42—24no—
276C42—48no—
277C42—48no—
278C42—48no—
279C42—84no—
280C42C4×D414yes—
281C42C4×D414yes—
282C42#12114yes—
283C42#12114yes—
284C23—22no—
285C23—22no—
286C23C22×C22⋊C411yescharacteristic
287C23C2×#12011yescharacteristic
288C23C2×#12311yescharacteristic
289C23C2×AGL(1,ℤ/8)12yes—
290C23#57512yes—
291Q8—416no—
292Q8—816no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—28no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—416no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—48no—
326C2×C4—48no—
327C2×C4—48no—
328C2×C4—44no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—84no—
332C2×C4—84no—
333C2×C4#38612yes—
334C2×C4#38614yes—
335C2×C4#38612yes—
336C2×C4#41814yes—
337C2×C4#45614yes—
338C2×C4C2×C4×D412yes—
339C2×C4C2×#12112yes—
340C8—84no—
341C22—24no—
342C22—24no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22C2×#38611yescharacteristic
348C22C2×#41811yescharacteristic
349C22#289611yescharacteristic
350C22#300612yes—
351C22C2×#45612yes—
352C22#302812yes—
353C22#309412yes—
354C4—24no—
355C4—24no—
356C4—22no—
357C4—22no—
358C4—22no—
359C4—48no—
360C4—42no—
361C4—42no—
362C4—42no—
363C4—88no—
364C4—84no—
365C4#188614yes—
366C2—21no—
367C2—22no—
368C2—21no—
369C2#1834414yes—
370C2#2091211yescharacteristic
371C2C2×#188611yescharacteristic
372C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2546C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
#340592
#218402
C2×#14672
#90272
#90262
#80212