Finite Group Databasebeta

← #194928

Subgroups — #194928

Order 512.

1308 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2634C211yesmaximal, characteristic
2#34182C211yesmaximal, characteristic
3#21838C212yesmaximal
4C2×#1467C211yesmaximal, characteristic
5#9027C214yesmaximal
6#9023C214yesmaximal
7#8018C212yesmaximal
8C22×#125C2211yescharacteristic
9C2×#519—21no—
10C22×C4×Q8C2211yescharacteristic
11C2×#432C2212yes—
12#2634—24no—
13#2634C418yescomplemented
14C2×#400—21no—
15C2×#400C2212yes—
16C2×#398C2212yes—
17C2×#395—21no—
18#2351—24no—
19Q8×C4⋊C4—24no—
20#2316—21no—
21#2316C2214yes—
22C2×#387—22no—
23#2285—22no—
24#2285C2212yes—
25C2×#383C2211yescharacteristic
26#1992—24no—
27#1989—24no—
28#1824C2214yes—
29#1823C2214yes—
30#1467—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C23×Q8D411yescharacteristic
34C2×#125—22no—
35C2×#125—22no—
36C2×#125—22no—
37C2×#125—24no—
38C2×#125C2×C414yes—
39#519—48no—
40C2×C4×Q8—28no—
41C2×C4×Q8—22no—
42C2×C4×Q8—22no—
43C2×C4×Q8—24no—
44C2×C4×Q8C2×C414yes—
45C2×C4×Q8D414yes—
46C2×#117—22no—
47C2×#117—24no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C42D412yes—
56C22×C42C2311yescharacteristic
57#432—24no—
58#432—24no—
59C2×C4⋊C8—22no—
60C2×C22⋊C8C2314yes—
61C2×#97—22no—
62#400—28no—
63#400—24no—
64#400C2×C418yes—
65#398—24no—
66#398C2×C418yes—
67#395—28no—
68#389—28no—
69#389—216no—
70#387—216no—
71#384—24no—
72#384D414yes—
73#383—22no—
74#383C2×C414yes—
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—22no—
83C4×C4⋊C4D414yes—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95C2312yes—
90#340—28no—
91#340—28no—
92#324—28no—
93C22×Q8—22no—
94C22×Q8—22no—
95C22×Q8—24no—
96C22×Q8—24no—
97C22×Q8C22⋊C414yes—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D412yes—
103C23×C4C2×D412yes—
104C23×C4C2×D411yescharacteristic
105C23×C4C2411yescharacteristic
106C22×C8—24no—
107#125—28no—
108#125—28no—
109#125—44no—
110#125—44no—
111C4×Q8—28no—
112C4×Q8—28no—
113C4×Q8—28no—
114C4×Q8—28no—
115C4×Q8—432no—
116#117—48no—
117#117—48no—
118#117—48no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—42no—
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×C412yes—
146C2×C4⋊C4C22×C414yes—
147C2×C4⋊C4C2×D414yes—
148C2×C4⋊C4C2×D418yes—
149C2×C22⋊C4—24no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—24no—
153C2×C42—28no—
154C2×C42—24no—
155C2×C42C22⋊C418yes—
156C2×C42C22×C412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D412yes—
159C2×C42C2×D412yes—
160C2×C42#4114yes—
161C4⋊C8—44no—
162C22⋊C8—24no—
163C22⋊C8—24no—
164#97—44no—
165#95—28no—
166#95—24no—
167#95—24no—
168#95—28no—
169#95—28no—
170#95—28no—
171#95—24no—
172#95—22no—
173#95—24no—
174#95C22×C414yes—
175C24C22×D411yescharacteristic
176C2×Q8—28no—
177C2×Q8—216no—
178C2×Q8—28no—
179C2×Q8—28no—
180C2×Q8—216no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—28no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C414yes—
209C22×C4C2×C22⋊C414yes—
210C22×C4C2×C22⋊C412yes—
211C22×C4#12011yescharacteristic
212C22×C4#12011yescharacteristic
213C22×C4#12012yes—
214C22×C4#12312yes—
215C22×C4#12312yes—
216C22×C4AGL(1,ℤ/8)14yes—
217C22×C4#13714yes—
218C22×C4C23×C411yescharacteristic
219C22×C4C22×D412yes—
220C22×C4C22×D411yescharacteristic
221C22×C4C2×#4112yes—
222C2×C8—48no—
223C2×C8—44no—
224C4⋊C4—28no—
225C4⋊C4—28no—
226C4⋊C4—28no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—48no—
230C4⋊C4—48no—
231C4⋊C4—48no—
232C4⋊C4—416no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—48no—
238C4⋊C4—416no—
239C4⋊C4—88no—
240C22⋊C4—48no—
241C42—28no—
242C42—28no—
243C42—48no—
244C42—416no—
245C42C4×D414yes—
246C42C4×D414yes—
247C42#12114yes—
248C42#12114yes—
249C23—22no—
250C23—22no—
251C23C22×C22⋊C411yescharacteristic
252C23C2×#12011yescharacteristic
253C23C2×#12311yescharacteristic
254C23C2×AGL(1,ℤ/8)12yes—
255C23C2×#13712yes—
256Q8—232no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—48no—
284C2×C4—48no—
285C2×C4—48no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—82no—
289C2×C4—84no—
290C2×C4—84no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4#38612yes—
297C2×C4#38612yes—
298C2×C4#38614yes—
299C2×C4#41718yes—
300C2×C4C2×C4×D412yes—
301C2×C4C2×#12112yes—
302C8—84no—
303C22—24no—
304C22—24no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22C2×#38611yescharacteristic
310C22C2×#41712yes—
311C22#300311yescharacteristic
312C22#300411yescharacteristic
313C22#302012yes—
314C22#302812yes—
315C22#309212yes—
316C4—24no—
317C4—28no—
318C4—22no—
319C4—22no—
320C4—22no—
321C4—22no—
322C4—44no—
323C4—84no—
324C4—82no—
325C4—88no—
326C4—82no—
327C2—22no—
328C2—22no—
329C2#1837312yes—
330C2#1837412yes—
331C2#2090911yescharacteristic
332C2#2095411yescharacteristic
333C2#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. #2634C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26342
#341822
#218382
C2×#14672
#90272
#90232
#80182