Finite Group Databasebeta

← #137238

Subgroups — #137238

Order 512.

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

normal subgroups only

This group has 454 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×#2086C211yesmaximal, characteristic
2#21960C211yesmaximal, characteristic
3#12866C212yesmaximal
4C2×#1322C211yesmaximal, characteristic
5#7258C212yesmaximal
6C22×#107C2211yescharacteristic
7C22×C4⋊C8C2211yescharacteristic
8C2×#389—21no—
9#2283—21no—
10C2×C22⋊C16C2212yes—
11#2086—22no—
12#2086C414yescomplemented
13#1949—24no—
14#1808C2212yes—
15#1807—21no—
16C2×#340—21no—
17C2×#340C2211yescharacteristic
18#1380—24no—
19#1322—22no—
20C23×C8D411yescharacteristic
21C23×C8C2311yescharacteristic
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—41no—
25C22×C42D411yescharacteristic
26C22×C16—22no—
27C2×#107—22no—
28C2×#107—22no—
29C2×#107—21no—
30C2×#107C2×C412yes—
31C2×#107D414yes—
32C2×#106—22no—
33C2×C4⋊C8—22no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8C2×C412yes—
36C2×C4⋊C8D412yes—
37#389—24no—
38#389—42no—
39#383—41no—
40#382—22no—
41#381—41no—
42#380—22no—
43C4×C4⋊C4—22no—
44C4×C22⋊C4—41no—
45#376—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—
54C23×C4—41no—
55C23×C4—41no—
56C23×C4D811yescharacteristic
57C23×C4D811yescharacteristic
58C23×C4C2×D411yescharacteristic
59C22×C8—21no—
60C22×C8—21no—
61C22×C8—42no—
62C22×C8—42no—
63C22×C8C22⋊C412yes—
64C22×C8C22×C411yescharacteristic
65C22×C8C2×D412yes—
66C22×C8C2×D411yescharacteristic
67C22×C8#4112yes—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—44no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—84no—
81C2×C22⋊C4—42no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42—42no—
87C2×C42C22⋊C412yes—
88C2×C42SD1612yes—
89C2×C16—44no—
90C2×C16—42no—
91#107—24no—
92#107—42no—
93#107—44no—
94#106—82no—
95C4⋊C8—24no—
96C4⋊C8—24no—
97#95—42no—
98#95—42no—
99#95—42no—
100#95—44no—
101#95—42no—
102C24C2×D811yescharacteristic
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—42no—
112C22×C4—44no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—44no—
116C22×C4—82no—
117C22×C4—82no—
118C22×C4—81no—
119C22×C4—81no—
120C22×C4—82no—
121C22×C4—82no—
122C22×C4#10212yes—
123C22×C4#10212yes—
124C22×C4C2×C22⋊C411yescharacteristic
125C22×C4#12011yescharacteristic
126C22×C4#12311yescharacteristic
127C22×C4C2×SD1611yescharacteristic
128C22×C4AGL(1,ℤ/8)11yescharacteristic
129C22×C4#13711yescharacteristic
130C2×C8—24no—
131C2×C8—22no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—42no—
135C2×C8—42no—
136C2×C8—42no—
137C2×C8—81no—
138C2×C8C4×D412yes—
139C2×C8#12112yes—
140C4⋊C4—24no—
141C4⋊C4—44no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—44no—
146C4⋊C4—44no—
147C4⋊C4—88no—
148C4⋊C4—81no—
149C4⋊C4—81no—
150C22⋊C4—82no—
151C22⋊C4—82no—
152C42—22no—
153C42—22no—
154C42—22no—
155C42—22no—
156C42—84no—
157C16—82no—
158C23—22no—
159C23—22no—
160C23C2×#10211yescharacteristic
161C23#44711yescharacteristic
162C23#48011yescharacteristic
163C23#50812yes—
164C23#50912yes—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—44no—
181C2×C4—81no—
182C2×C4—88no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4—84no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—84no—
190C2×C4—84no—
191C2×C4—162no—
192C2×C4—162no—
193C2×C4#38611yescharacteristic
194C2×C4#41711yescharacteristic
195C2×C4#44811yescharacteristic
196C2×C4#45612yes—
197C2×C4#45812yes—
198C2×C4#48111yescharacteristic
199C8—22no—
200C8—22no—
201C8—42no—
202C22—24no—
203C22—24no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22#161012yes—
209C22#187411yescharacteristic
210C22#213811yescharacteristic
211C22#214011yescharacteristic
212C22#218612yes—
213C22#219112yes—
214C22#223412yes—
215C4—21no—
216C4—21no—
217C4—42no—
218C4—42no—
219C4—84no—
220C4—164no—
221C4—162no—
222C4#161112yes—
223C4#188611yescharacteristic
224C4#189011yescharacteristic
225C2—21no—
226C2—21no—
227C2—22no—
228C2#902711yescharacteristic
229C2#1229014yes—
230C2#1299211yescharacteristic
231C2#1300011yescharacteristic
subgroupG/H
center, socleC23#447
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2086C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20862
#219602
#128662
C2×#13222
#72582