Finite Group Databasebeta

← #138363

Subgroups — #138363

Order 512.

547 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 547 conjugacy classes of subgroups — too many to draw. Only its 90 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#32915C211yesmaximal, characteristic
2#20639C211yesmaximal, characteristic
3#11700C212yesmaximal
4#8323C211yesmaximal, characteristic
5#7253C212yesmaximal
6C2×#519C2211yescharacteristic
7C2×#462C412yes—
8C2×#451—21no—
9#2905—21no—
10C22×#101—21no—
11#2621C2211yescharacteristic
12#1878—22no—
13#1873—21no—
14#1805C412yes—
15#1801—24no—
16#1798C2211yescharacteristic
17#1742C2212yes—
18#1479—22no—
19#1458C2212yes—
20#1346—24no—
21C23×Q8D411yescharacteristic
22C22×Q16—22no—
23C22×M16—22no—
24C2×C4⋊Q8C2×C411yescharacteristic
25C2×C22⋊Q8—21no—
26#519—44no—
27C2×C4×Q8—22no—
28C2×C4×Q8C2×C412yes—
29C2×C4×Q8D412yes—
30C2×#117—22no—
31C2×#117D411yescharacteristic
32C2×#117C2311yescharacteristic
33#462—44no—
34#451—42no—
35#451—42no—
36#431—22no—
37#424—44no—
38#423—24no—
39C2×C4⋊C8C2×C412yes—
40C2×#103—22no—
41C2×#103—22no—
42C2×#101—24no—
43C2×#101—42no—
44C2×#101—42no—
45C2×C22⋊C8—21no—
46#398—22no—
47#397—22no—
48#391—21no—
49#382C2×C411yescharacteristic
50#343—22no—
51#343D414yes—
52#340—22no—
53#336—21no—
54Q8⋊C8—44no—
55#324—22no—
56#323—22no—
57C22×Q8—21no—
58C22×Q8—21no—
59C22×Q8—21no—
60C22×Q8—21no—
61C22×Q8—42no—
62C22×Q8—42no—
63C22×Q8C22⋊C412yes—
64C22×Q8D812yes—
65C22×Q8SD1612yes—
66C23×C4—21no—
67C23×C4C2×D411yescharacteristic
68C2×Q16—44no—
69C2×Q16—44no—
70C2×Q16—44no—
71C2×M16—24no—
72C2×M16—42no—
73C2×M16—42no—
74C2×M16—42no—
75C2×M16—42no—
76C2×M16—42no—
77C22×C8—22no—
78C22×C8—42no—
79C4⋊Q8—42no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82C22⋊Q8—41no—
83C22⋊Q8—41no—
84C4×Q8—24no—
85C4×Q8—24no—
86C4×Q8—42no—
87C4×Q8—42no—
88C4×Q8—48no—
89#117—24no—
90#117—42no—
91#117—48no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4C22⋊C412yes—
98C2×C4⋊C4C2×D412yes—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4#4112yes—
102C2×C42—21no—
103C2×C42—22no—
104C2×C42—22no—
105C2×C42C22×C411yescharacteristic
106C2×C42C2×D411yescharacteristic
107C4⋊C8—42no—
108C4⋊C8—48no—
109C4⋊C8—42no—
110#103—44no—
111#103—44no—
112#101—48no—
113C22⋊C8—41no—
114C22⋊C8—41no—
115C22⋊C8—42no—
116C4×C8—42no—
117#95—22no—
118#95—42no—
119C24#12311yescharacteristic
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—44no—
128C2×Q8—44no—
129C2×Q8—42no—
130C2×Q8—42no—
131C2×Q8—44no—
132C2×Q8—44no—
133C2×Q8—42no—
134C2×Q8—41no—
135C2×Q8—41no—
136C2×Q8—42no—
137C2×Q8#10214yes—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4C2×C22⋊C411yescharacteristic
149C22×C4C4×D412yes—
150C22×C4#12011yescharacteristic
151C22×C4#12112yes—
152C22×C4C2×D811yescharacteristic
153C22×C4C2×SD1611yescharacteristic
154C22×C4AGL(1,ℤ/8)11yescharacteristic
155C22×C4#13711yescharacteristic
156Q16—88no—
157M16—44no—
158M16—44no—
159M16—82no—
160M16—82no—
161C2×C8—44no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—82no—
166C2×C8—82no—
167C2×C8—82no—
168C4⋊C4—22no—
169C4⋊C4—42no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—82no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C4⋊C4#9914yes—
178C22⋊C4—24no—
179C22⋊C4—42no—
180C22⋊C4—81no—
181C22⋊C4—81no—
182C42—24no—
183C42—22no—
184C42—22no—
185C42—44no—
186C42—44no—
187C23—21no—
188C23—21no—
189C23—42no—
190C23#38611yescharacteristic
191C23#48011yescharacteristic
192C23#48111yescharacteristic
193Q8—24no—
194Q8—24no—
195Q8—44no—
196Q8—44no—
197Q8—42no—
198Q8—42no—
199Q8—82no—
200Q8—82no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4C2×#9912yes—
224C2×C4C2×#10211yescharacteristic
225C2×C4#41711yescharacteristic
226C2×C4#42112yes—
227C2×C4#44711yescharacteristic
228C2×C4#44811yescharacteristic
229C8—84no—
230C8—82no—
231C8—162no—
232C22—22no—
233C22—21no—
234C22—21no—
235C22—41no—
236C22—41no—
237C22—42no—
238C22—81no—
239C22#187411yescharacteristic
240C22#189711yescharacteristic
241C22#189811yescharacteristic
242C4—21no—
243C4—21no—
244C4—42no—
245C4—42no—
246C4—44no—
247C4—44no—
248C4—161no—
249C4—161no—
250C4#151312yes—
251C4#151912yes—
252C2—21no—
253C2—21no—
254C2—41no—
255C2—41no—
256C2#915511yescharacteristic
257C2#1309612yes—
subgroupG/H
center, socleC22#1898
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#519 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#329152
#206392
#117002
#83232
#72532