Finite Group Databasebeta

← #194772

Subgroups — #194772

Order 512.

1430 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2581C211yesmaximal, characteristic
2#34182C211yesmaximal, characteristic
3#20734C212yesmaximal
4C2×#1461C211yesmaximal, characteristic
5#9023C214yesmaximal
6#9015C214yesmaximal
7#8026C212yesmaximal
8C2×#519—21no—
9C22×C4×Q8C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#431C2212yes—
12#2581—24no—
13#2581C418yescomplemented
14C2×#400C2212yes—
15C2×#395C2212yes—
16#2351—24no—
17Q8×C4⋊C4—24no—
18#2316—21no—
19#2316C2214yes—
20C2×#391—21no—
21C2×#389—21no—
22C2×#387—22no—
23#2285—22no—
24#2285C2212yes—
25C2×#382C2211yescharacteristic
26#1824C2214yes—
27#1803—24no—
28#1787C2214yes—
29C4×#103—24no—
30#1461—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C23×Q8D411yescharacteristic
34C23×Q8C2311yescharacteristic
35#519—44no—
36#519—44no—
37C2×C4×Q8—28no—
38C2×C4×Q8—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—24no—
42C2×C4×Q8—22no—
43C2×C4×Q8—22no—
44C2×C4×Q8—24no—
45C2×C4×Q8C2×C414yes—
46C2×C4×Q8C2×C414yes—
47C2×C4×Q8D414yes—
48C2×#117—22no—
49C2×#117—24no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C42—21no—
57C22×C42D412yes—
58C22×C42C2311yescharacteristic
59#431—24no—
60#431—24no—
61C2×C4⋊C8—22no—
62C2×#103—28no—
63C2×C22⋊C8C2314yes—
64C2×C4×C8—22no—
65#400—24no—
66#400C2×C418yes—
67#395—24no—
68#395C2×C418yes—
69#391—28no—
70#389—28no—
71#389—28no—
72#389—216no—
73#387—216no—
74#384—24no—
75#384D414yes—
76#382—24no—
77#382—22no—
78#382—22no—
79#382C2×C414yes—
80#380—24no—
81#380—22no—
82#380C2314yes—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4D414yes—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95C2312yes—
92#340—28no—
93#324—28no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—22no—
97C22×Q8—22no—
98C22×Q8—21no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—44no—
102C22×Q8C22⋊C414yes—
103C22×Q8SD1614yes—
104C22×Q8Q1614yes—
105C22×Q8C22×C412yes—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2411yescharacteristic
114C22×C8—24no—
115C4×Q8—28no—
116C4×Q8—28no—
117C4×Q8—28no—
118C4×Q8—28no—
119C4×Q8—28no—
120C4×Q8—28no—
121C4×Q8—432no—
122C4×Q8—44no—
123C4×Q8—44no—
124#117—48no—
125#117—48no—
126#117—44no—
127#117—44no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—28no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—48no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4C22⋊C414yes—
149C2×C4⋊C4C22×C412yes—
150C2×C4⋊C4C22×C414yes—
151C2×C4⋊C4C2×D414yes—
152C2×C4⋊C4C2×D418yes—
153C2×C22⋊C4—24no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—24no—
157C2×C42—28no—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—42no—
161C2×C42—42no—
162C2×C42C22⋊C418yes—
163C2×C42C22×C412yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42C2×D412yes—
167C2×C42#4114yes—
168C4⋊C8—44no—
169#103—416no—
170C22⋊C8—24no—
171C22⋊C8—24no—
172C4×C8—44no—
173#95—28no—
174#95—24no—
175#95—28no—
176#95—28no—
177#95—28no—
178#95—24no—
179#95—22no—
180#95—24no—
181#95C22×C414yes—
182C24C22×D411yescharacteristic
183C2×Q8—28no—
184C2×Q8—28no—
185C2×Q8—28no—
186C2×Q8—28no—
187C2×Q8—28no—
188C2×Q8—28no—
189C2×Q8—28no—
190C2×Q8—24no—
191C2×Q8—44no—
192C2×Q8—44no—
193C2×Q8—44no—
194C2×Q8—48no—
195C2×Q8—42no—
196C2×Q8#103116yescomplemented
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—28no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4C2×C22⋊C414yes—
222C22×C4C2×C22⋊C414yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4#12011yescharacteristic
226C22×C4#12011yescharacteristic
227C22×C4#12012yes—
228C22×C4#12312yes—
229C22×C4#12312yes—
230C22×C4C2×SD1614yes—
231C22×C4C2×SD1612yes—
232C22×C4C2×Q1614yes—
233C22×C4C2×Q1612yes—
234C22×C4AGL(1,ℤ/8)12yes—
235C22×C4#13712yes—
236C22×C4C23×C411yescharacteristic
237C22×C4C22×D412yes—
238C22×C4C22×D411yescharacteristic
239C22×C4C2×#4112yes—
240C2×C8—48no—
241C2×C8—44no—
242C4⋊C4—28no—
243C4⋊C4—28no—
244C4⋊C4—28no—
245C4⋊C4—24no—
246C4⋊C4—48no—
247C4⋊C4—48no—
248C4⋊C4—416no—
249C4⋊C4—416no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—48no—
254C4⋊C4—48no—
255C4⋊C4—88no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C42—28no—
259C42—28no—
260C42—416no—
261C42—48no—
262C42—48no—
263C42C4×D414yes—
264C42C4×D414yes—
265C42#12114yes—
266C42#12114yes—
267C23—22no—
268C23—22no—
269C23C22×C22⋊C411yescharacteristic
270C23C2×#12011yescharacteristic
271C23C2×#12311yescharacteristic
272C23C22×SD1611yescharacteristic
273C23C22×Q1611yescharacteristic
274C23C2×AGL(1,ℤ/8)11yescharacteristic
275C23C2×#13711yescharacteristic
276Q8—232no—
277Q8—28no—
278Q8—48no—
279Q8—44no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—48no—
307C2×C4—48no—
308C2×C4—48no—
309C2×C4—42no—
310C2×C4—82no—
311C2×C4—84no—
312C2×C4—84no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4#38612yes—
316C2×C4#38612yes—
317C2×C4#38614yes—
318C2×C4C2×#10318yes—
319C2×C4C2×#10314yes—
320C2×C4#41714yes—
321C2×C4#45014yes—
322C2×C4#45114yes—
323C2×C4#48314yes—
324C2×C4#48414yes—
325C2×C4C2×C4×D412yes—
326C2×C4C2×#12112yes—
327C8—84no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22C2×#38611yescharacteristic
337C22C22×#10311yescharacteristic
338C22C2×#41711yescharacteristic
339C22C2×#45011yescharacteristic
340C22C2×#45111yescharacteristic
341C22#302012yes—
342C22#302512yes—
343C22C2×#48311yescharacteristic
344C22C2×#48411yescharacteristic
345C4—28no—
346C4—22no—
347C4—22no—
348C4—22no—
349C4—22no—
350C4—44no—
351C4—44no—
352C4—84no—
353C4—88no—
354C4—82no—
355C4#187218yes—
356C2—21no—
357C2—21no—
358C2—21no—
359C2—21no—
360C2#1821912yes—
361C2#1822112yes—
362C2C2×#187211yescharacteristic
363C2#2095411yescharacteristic
364C2#2095911yescharacteristic
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. #2581C4semidirect
196. C2×Q8#103semidirect

Maximal subgroups

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

subgroupindex
C2×#25812
#341822
#207342
C2×#14612
#90232
#90152
#80262