Finite Group Databasebeta

← #194796

Subgroups — #194796

Order 512.

1464 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2399C211yesmaximal, characteristic
2#34182C211yesmaximal, characteristic
3#20740C212yesmaximal
4C2×#1467C211yesmaximal, characteristic
5#9027C214yesmaximal
6#9015C214yesmaximal
7#8022C212yesmaximal
8C2×#519—21no—
9C22×C4×Q8C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#432C2212yes—
12#2399—24no—
13#2399C418yescomplemented
14C2×#398—21no—
15C2×#398C2212yes—
16C2×#395—21no—
17C2×#395C2212yes—
18#2351—24no—
19Q8×C4⋊C4—24no—
20#2316—21no—
21#2316C2214yes—
22#2285—22no—
23#2285C2212yes—
24C2×C4×C4⋊C4—22no—
25C2×#376C2211yescharacteristic
26#1823C2214yes—
27#1803—24no—
28#1787C2214yes—
29#1763—24no—
30#1467—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C23×Q8D411yescharacteristic
34C23×Q8C2311yescharacteristic
35#519—48no—
36C2×C4×Q8—28no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8—24no—
41C2×C4×Q8—22no—
42C2×C4×Q8—22no—
43C2×C4×Q8—24no—
44C2×C4×Q8C2×C414yes—
45C2×C4×Q8C2×C414yes—
46C2×C4×Q8D414yes—
47C2×#117—22no—
48C2×#117—24no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C42—21no—
56C22×C42—21no—
57C22×C42D412yes—
58C22×C42C2311yescharacteristic
59#432—24no—
60#432—24no—
61C2×C4⋊C8—22no—
62C2×#103—28no—
63C2×C22⋊C8C2314yes—
64C2×#97—22no—
65#398—28no—
66#398—24no—
67#398C2×C418yes—
68#395—28no—
69#395—24no—
70#395C2×C418yes—
71#389—28no—
72#389—216no—
73#384—24no—
74#384D414yes—
75#382—24no—
76#382—22no—
77#380—24no—
78#380—22no—
79#380C2314yes—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—216no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4D414yes—
85#376—22no—
86#376C2×C414yes—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95C2312yes—
91#340—28no—
92#324—28no—
93C22×Q8—22no—
94C22×Q8—22no—
95C22×Q8—24no—
96C22×Q8—24no—
97C22×Q8—21no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—44no—
101C22×Q8C22⋊C414yes—
102C22×Q8C22×C412yes—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4C2411yescharacteristic
111C22×C8—24no—
112C4×Q8—28no—
113C4×Q8—28no—
114C4×Q8—28no—
115C4×Q8—28no—
116C4×Q8—28no—
117C4×Q8—28no—
118C4×Q8—432no—
119C4×Q8—44no—
120C4×Q8—44no—
121#117—48no—
122#117—48no—
123#117—48no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—48no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4C22⋊C414yes—
144C2×C4⋊C4C22×C412yes—
145C2×C4⋊C4C22×C414yes—
146C2×C4⋊C4C2×D414yes—
147C2×C4⋊C4C2×D418yes—
148C2×C22⋊C4—24no—
149C2×C42—28no—
150C2×C42—24no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—24no—
154C2×C42—28no—
155C2×C42—24no—
156C2×C42—24no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42—42no—
160C2×C42C22⋊C418yes—
161C2×C42C22×C412yes—
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42#4114yes—
166C4⋊C8—44no—
167#103—416no—
168C22⋊C8—24no—
169C22⋊C8—24no—
170#97—44no—
171#95—28no—
172#95—24no—
173#95—28no—
174#95—28no—
175#95—24no—
176#95—22no—
177#95—24no—
178#95C22×C414yes—
179C24C22×D411yescharacteristic
180C2×Q8—28no—
181C2×Q8—216no—
182C2×Q8—28no—
183C2×Q8—28no—
184C2×Q8—216no—
185C2×Q8—28no—
186C2×Q8—24no—
187C2×Q8—44no—
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8—48no—
191C2×Q8—42no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—28no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—28no—
207C22×C4—42no—
208C22×C4—44no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C414yes—
222C22×C4C2×C22⋊C414yes—
223C22×C4#12011yescharacteristic
224C22×C4#12012yes—
225C22×C4#12011yescharacteristic
226C22×C4#12312yes—
227C22×C4#12312yes—
228C22×C4AGL(1,ℤ/8)14yes—
229C22×C4#13714yes—
230C22×C4C23×C411yescharacteristic
231C22×C4C22×D412yes—
232C22×C4C22×D411yescharacteristic
233C22×C4C2×#4112yes—
234C2×C8—48no—
235C2×C8—44no—
236C4⋊C4—28no—
237C4⋊C4—28no—
238C4⋊C4—28no—
239C4⋊C4—24no—
240C4⋊C4—48no—
241C4⋊C4—416no—
242C4⋊C4—48no—
243C4⋊C4—416no—
244C4⋊C4—42no—
245C4⋊C4—48no—
246C4⋊C4—416no—
247C4⋊C4—48no—
248C4⋊C4—88no—
249C22⋊C4—48no—
250C42—216no—
251C42—28no—
252C42—28no—
253C42—48no—
254C42—416no—
255C42—44no—
256C42—44no—
257C42C4×D414yes—
258C42C4×D414yes—
259C42#12114yes—
260C42#12114yes—
261C23—22no—
262C23—22no—
263C23C22×C22⋊C411yescharacteristic
264C23C2×#12011yescharacteristic
265C23C2×#12311yescharacteristic
266C23C2×AGL(1,ℤ/8)12yes—
267C23C2×#13712yes—
268Q8—232no—
269Q8—28no—
270Q8—48no—
271Q8—44no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—28no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—48no—
294C2×C4—48no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—48no—
305C2×C4—48no—
306C2×C4—42no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4#38612yes—
312C2×C4#38614yes—
313C2×C4#38612yes—
314C2×C4#41718yes—
315C2×C4#45614yes—
316C2×C4C2×C4×D412yes—
317C2×C4C2×#12112yes—
318C8—84no—
319C22—24no—
320C22—24no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22C2×#38611yescharacteristic
326C22C2×#41712yes—
327C22#300311yescharacteristic
328C22#300411yescharacteristic
329C22C2×#45612yes—
330C22#302512yes—
331C22#309212yes—
332C4—24no—
333C4—28no—
334C4—22no—
335C4—22no—
336C4—22no—
337C4—48no—
338C4—44no—
339C4—44no—
340C4—88no—
341C4—82no—
342C4#188614yes—
343C2—22no—
344C2—22no—
345C2#1818712yes—
346C2#1818912yes—
347C2#2090911yescharacteristic
348C2C2×#188611yescharacteristic
349C2#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. #2399C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23992
#341822
#207402
C2×#14672
#90272
#90152
#80222