Finite Group Databasebeta

← #238644

Subgroups — #238644

Order 512.

996 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37641C211yesmaximal, characteristic
2#21322C212yesmaximal
3#20800C211yesmaximal, characteristic
4C2×#1526C212yesmaximal
5#10429C218yesmaximal, complemented
6#8028C211yesmaximal, characteristic
7C2×#533C2212yes—
8C2×#428—22no—
9C2×C4×M16C2212yes—
10C2×C4×M16C2211yescharacteristic
11#2822C2214yes—
12#2625—24no—
13#2621—22no—
14#2582—21no—
15#2582—21no—
16#2568—22no—
17#2546—22no—
18#2285C2211yescharacteristic
19C2×#384C2211yescharacteristic
20#1992C2218yes—
21#1990—22no—
22#1990—22no—
23#1847C2214yes—
24#1845—21no—
25#1844—21no—
26#1736—24no—
27#1526—28no—
28C2×#330C2218yes—
29#1295C2214yes—
30C22×M16—21no—
31C22×M16—22no—
32#533—24no—
33#533—24no—
34#533—24no—
35#533—24no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8D412yes—
38C2×C4⋊Q8D412yes—
39C2×#125C2314yes—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—22no—
42C2×C4×Q8—22no—
43C2×C4×Q8—22no—
44C2×#117D414yes—
45C2×#117D412yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C42C2311yescharacteristic
49#428—48no—
50C2×#107—24no—
51C2×#106—24no—
52C4×M16—22no—
53C4×M16—24no—
54C4×M16—24no—
55C4×M16D414yes—
56C2×#97C2314yes—
57C2×#97C2312yes—
58C2×C4×C8—21no—
59C2×C4×C8—22no—
60#400—28no—
61#398—24no—
62#398—24no—
63#398—28no—
64#397—24no—
65#397—24no—
66#391—24no—
67#391—24no—
68#391—24no—
69#389—24no—
70#389—24no—
71#384—21no—
72#384—22no—
73#384D411yescharacteristic
74#384D411yescharacteristic
75#384C2312yes—
76#382—22no—
77#382C2312yes—
78#380—22no—
79C4×C4⋊C4—22no—
80#340—28no—
81#340—28no—
82#330—216no—
83C22×Q8—22no—
84C22×Q8—22no—
85C23×C4—41no—
86C23×C4C2×D412yes—
87C23×C4C2×D411yescharacteristic
88C2×M16—24no—
89C2×M16—24no—
90C2×M16—24no—
91C2×M16—28no—
92C2×M16—44no—
93C2×M16—44no—
94C22×C8—22no—
95C22×C8—24no—
96C4⋊Q8—24no—
97C4⋊Q8—24no—
98C4⋊Q8—24no—
99C4⋊Q8—24no—
100C4⋊Q8—84no—
101#125—24no—
102#125—24no—
103#125—24no—
104#125—24no—
105C22⋊Q8—44no—
106C22⋊Q8—44no—
107C22⋊Q8—44no—
108C22⋊Q8—44no—
109C4×Q8—44no—
110C4×Q8—44no—
111C4×Q8—44no—
112C4×Q8—44no—
113#117—28no—
114#117—28no—
115#117—44no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—48no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4C2×D414yes—
131C2×C4⋊C4C2×D414yes—
132C2×C4⋊C4C2×D412yes—
133C2×C4⋊C4C2×D412yes—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—22no—
136C2×C42—24no—
137C2×C42—22no—
138C2×C42C2×D412yes—
139C2×C42C2×D412yes—
140C2×C42C2×D411yescharacteristic
141C2×C42#4112yes—
142C2×C42#4114yes—
143C2×C42C2411yescharacteristic
144#107—48no—
145#106—48no—
146#97—24no—
147#97—22no—
148#97—24no—
149#97C2×D414yes—
150C4×C8—24no—
151C4×C8—44no—
152#95—24no—
153#95—24no—
154#95—28no—
155#95—28no—
156#95—24no—
157C24#12011yescharacteristic
158C2×Q8—44no—
159C2×Q8—44no—
160C2×Q8—44no—
161C2×Q8—44no—
162C2×Q8—44no—
163C2×Q8—44no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—42no—
173C22×C4—82no—
174C22×C4—84no—
175C22×C4#12011yescharacteristic
176C22×C4#12012yes—
177C22×C4#12114yes—
178C22×C4#12112yes—
179C22×C4#12314yes—
180C22×C4#12412yes—
181C22×C4C22×D412yes—
182C22×C4C22×D411yescharacteristic
183M16—216no—
184M16—416no—
185C2×C8—24no—
186C2×C8—24no—
187C2×C8—24no—
188C2×C8—28no—
189C2×C8—44no—
190C2×C8—44no—
191C4⋊C4—24no—
192C4⋊C4—24no—
193C4⋊C4—24no—
194C4⋊C4—24no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C4⋊C4—48no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—84no—
208C4⋊C4—84no—
209C4⋊C4—82no—
210C4⋊C4—84no—
211C22⋊C4—48no—
212C22⋊C4—82no—
213C42—48no—
214C42—44no—
215C42#12112yes—
216C42#12114yes—
217C42#12314yes—
218C42#12412yes—
219C42C22×D411yescharacteristic
220C42C2×#4111yescharacteristic
221C42C2×#4112yes—
222C23—21no—
223C23—22no—
224C23—22no—
225C23—21no—
226C23#39412yes—
227C23C2×#12011yescharacteristic
228Q8—88no—
229Q8—88no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—28no—
235C2×C4—28no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—82no—
249C2×C4—84no—
250C2×C4—164no—
251C2×C4#39414yes—
252C2×C4#39412yes—
253C2×C4#45312yes—
254C2×C4#45414yes—
255C2×C4#45614yes—
256C2×C4#45612yes—
257C2×C4C2×#12112yes—
258C2×C4C2×#12111yescharacteristic
259C2×C4C2×#12312yes—
260C2×C4C2×#12411yescharacteristic
261C8—28no—
262C8—48no—
263C22—24no—
264C22—22no—
265C22—41no—
266C22—41no—
267C22—42no—
268C22C2×#39411yescharacteristic
269C22C2×#45311yescharacteristic
270C22C2×#45412yes—
271C22C2×#45612yes—
272C22C2×#45611yescharacteristic
273C4—22no—
274C4—22no—
275C4—21no—
276C4—21no—
277C4—82no—
278C4—82no—
279C4—84no—
280C4—84no—
281C4—162no—
282C4#203818yes—
283C4#203914yes—
284C2—42no—
285C2#1907711yescharacteristic
286C2#1907811yescharacteristic
287C2#1908412yes—
288C2C2×#203812yes—
289C2C2×#203911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10429C2semidirect

Maximal subgroups

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

subgroupindex
#376412
#213222
#208002
C2×#15262
#104292
#80282