Finite Group Databasebeta

← #137227

Subgroups — #137227

Order 512.

530 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C2×#2076C211yesmaximal, characteristic
2#21939C211yesmaximal, characteristic
3#12867C212yesmaximal, complemented
4C2×#1314C211yesmaximal, characteristic
5#7258C212yesmaximal, complemented
6C22×#107C2211yescharacteristic
7C22×C22⋊C8C2211yescharacteristic
8#2280—21no—
9C2×#380—21no—
10C2×C22⋊C16C2212yes—
11#2076—22no—
12#2076C414yescomplemented
13#1854—24no—
14#1808C2212yes—
15#1807—21no—
16C2×#340—21no—
17C2×#340C2211yescharacteristic
18#1380—24no—
19#1314—22no—
20C24×C4D411yescharacteristic
21C23×C8D411yescharacteristic
22C23×C8C2311yescharacteristic
23C2×#117—22no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C22⋊C4—41no—
27C22×C16—22no—
28C2×#107—22no—
29C2×#107—22no—
30C2×#107—21no—
31C2×#107C2×C412yes—
32C2×#107D414yes—
33C2×#106—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8C2×C412yes—
37C2×C22⋊C8D412yes—
38#381—41no—
39#380—22no—
40#380—24no—
41#380—22no—
42#380—42no—
43#379—41no—
44C4×C22⋊C4—41no—
45C4×C22⋊C4—41no—
46C2×#95—41no—
47C22⋊C16—22no—
48C22⋊C16—22no—
49#340—22no—
50#340—24no—
51#340—22no—
52#340—21no—
53#340C2×C412yes—
54C25D811yescharacteristic
55C23×C4—21no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—42no—
60C23×C4—42no—
61C23×C4—41no—
62C23×C4—41no—
63C23×C4C22⋊C412yes—
64C23×C4D811yescharacteristic
65C23×C4SD1612yes—
66C23×C4C2×D411yescharacteristic
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—42no—
70C22×C8—42no—
71C22×C8C22⋊C412yes—
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D412yes—
74C22×C8C2×D411yescharacteristic
75C22×C8#4112yes—
76#117—82no—
77#117—82no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—42no—
91C2×C22⋊C4—44no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—84no—
94C2×C42—42no—
95C2×C16—44no—
96C2×C16—42no—
97#107—24no—
98#107—42no—
99#107—44no—
100#106—82no—
101C22⋊C8—24no—
102C22⋊C8—24no—
103#95—42no—
104#95—44no—
105#95—42no—
106C24—21no—
107C24—22no—
108C24—21no—
109C24—22no—
110C24—42no—
111C24—42no—
112C24#10212yes—
113C24C2×D811yescharacteristic
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—44no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—44no—
131C22×C4—82no—
132C22×C4—81no—
133C22×C4—81no—
134C22×C4—82no—
135C22×C4—82no—
136C22×C4—82no—
137C22×C4—82no—
138C22×C4—82no—
139C22×C4—81no—
140C22×C4—81no—
141C22×C4—82no—
142C22×C4#10212yes—
143C22×C4C2×C22⋊C411yescharacteristic
144C22×C4#12011yescharacteristic
145C22×C4#12311yescharacteristic
146C22×C4C2×SD1611yescharacteristic
147C22×C4AGL(1,ℤ/8)11yescharacteristic
148C22×C4#13711yescharacteristic
149C2×C8—24no—
150C2×C8—22no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—81no—
157C2×C8C4×D412yes—
158C2×C8#12112yes—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—44no—
164C4⋊C4—44no—
165C4⋊C4—82no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C22⋊C4—84no—
169C22⋊C4—88no—
170C22⋊C4—84no—
171C22⋊C4—82no—
172C22⋊C4—82no—
173C42—84no—
174C16—82no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—42no—
181C23—42no—
182C23—48no—
183C23—42no—
184C23—42no—
185C23—42no—
186C23—42no—
187C23—81no—
188C23—81no—
189C23—84no—
190C23C2×#10211yescharacteristic
191C23#44711yescharacteristic
192C23#48011yescharacteristic
193C23#50812yes—
194C23#51012yes—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—44no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—81no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—84no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—84no—
219C2×C4—84no—
220C2×C4—164no—
221C2×C4—162no—
222C2×C4—162no—
223C2×C4#38611yescharacteristic
224C2×C4#41711yescharacteristic
225C2×C4#44811yescharacteristic
226C2×C4#45512yes—
227C2×C4#45812yes—
228C2×C4#48111yescharacteristic
229C8—22no—
230C8—22no—
231C8—42no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—24no—
237C22—24no—
238C22—44no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—82no—
243C22—84no—
244C22—84no—
245C22#160712yes—
246C22#187411yescharacteristic
247C22#213811yescharacteristic
248C22#214111yescharacteristic
249C22#218912yes—
250C22#219012yes—
251C22#223712yes—
252C4—21no—
253C4—21no—
254C4—42no—
255C4—162no—
256C4—162no—
257C4—164no—
258C4#160812yes—
259C4#188511yescharacteristic
260C4#189011yescharacteristic
261C2—21no—
262C2—21no—
263C2—22no—
264C2—82no—
265C2#902511yescharacteristic
266C2#1228314yes—
267C2#1299411yescharacteristic
268C2#1299911yescharacteristic
subgroupG/H
center, socleC23#447
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC25D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12867C2semidirect
5. #7258C2semidirect
12. #2076C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20762
#219392
#128672
C2×#13142
#72582