Finite Group Databasebeta

← #137244

Subgroups — #137244

Order 512.

414 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 88 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#23666C211yesmaximal, characteristic
2#21960C211yesmaximal, characteristic
3#12870C212yesmaximal
4#11440C211yesmaximal, characteristic
5#7262C212yesmaximal
6C2×#427C2211yescharacteristic
7C22×C4⋊C8C2211yescharacteristic
8#2296—21no—
9#2283—21no—
10#2112C2212yes—
11#2088—22no—
12#2087C412yescomplemented
13#2086C412yescomplemented
14#1949—24no—
15#1808—21no—
16#1807C2212yes—
17#1735—21no—
18#1735C2211yescharacteristic
19#1386—24no—
20#1322—22no—
21C23×C8D411yescharacteristic
22C23×C8C2311yescharacteristic
23C2×#117—21no—
24C2×#117—21no—
25C22×C42D411yescharacteristic
26C2×M32—22no—
27#427—22no—
28#427—22no—
29C2×#107—22no—
30C2×#107C2×C411yescharacteristic
31C2×#107C2×C411yescharacteristic
32C2×#106—21no—
33C2×#106D414yes—
34C2×C4⋊C8—22no—
35C2×C4⋊C8—22no—
36C2×C4⋊C8C2×C412yes—
37C2×C4⋊C8D412yes—
38#389—22no—
39#389—22no—
40#385—41no—
41#385—41no—
42#383—41no—
43#382—22no—
44#381—41no—
45#380—22no—
46C4×C4⋊C4—22no—
47C4×C4⋊C4—42no—
48C4×C22⋊C4—41no—
49#376—41no—
50C22⋊C16—22no—
51C22⋊C16—22no—
52#340—22no—
53#340—22no—
54#340—22no—
55#340—22no—
56#340—21no—
57#340C2×C412yes—
58C23×C4D811yescharacteristic
59C23×C4D811yescharacteristic
60C23×C4C2×D411yescharacteristic
61C22×C8—21no—
62C22×C8—21no—
63C22×C8—42no—
64C22×C8—42no—
65C22×C8C22⋊C412yes—
66C22×C8C22×C411yescharacteristic
67C22×C8C2×D412yes—
68C22×C8C2×D411yescharacteristic
69C22×C8#4112yes—
70#117—24no—
71#117—42no—
72#117—41no—
73#117—41no—
74#117—41no—
75#117—41no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C22⋊C4—41no—
83C2×C22⋊C4—41no—
84C2×C22⋊C4—42no—
85C2×C42—21no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—41no—
90C2×C42—42no—
91C2×C42—41no—
92C2×C42C22⋊C412yes—
93C2×C42SD1612yes—
94M32—44no—
95C2×C16—42no—
96#107—22no—
97#107—22no—
98#107—82no—
99#106—42no—
100#106—44no—
101C4⋊C8—24no—
102C4⋊C8—24no—
103#95—42no—
104#95—42no—
105#95—42no—
106#95—42no—
107#95—42no—
108#95—42no—
109C24C2×D811yescharacteristic
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—44no—
121C22×C4—81no—
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—42no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—44no—
147C4⋊C4—44no—
148C4⋊C4—84no—
149C4⋊C4—84no—
150C22⋊C4—44no—
151C22⋊C4—84no—
152C22⋊C4—81no—
153C22⋊C4—81no—
154C42—22no—
155C42—22no—
156C42—22no—
157C42—22no—
158C42—44no—
159C42—84no—
160C42—81no—
161C42—81no—
162C16—82no—
163C23—22no—
164C23—22no—
165C23—41no—
166C23C2×#10211yescharacteristic
167C23#44711yescharacteristic
168C23#48011yescharacteristic
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4—81no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—84no—
189C2×C4—84no—
190C2×C4—82no—
191C2×C4—84no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—82no—
195C2×C4#38611yescharacteristic
196C2×C4#41711yescharacteristic
197C2×C4#44811yescharacteristic
198C2×C4#45612yes—
199C2×C4#45812yes—
200C2×C4#48111yescharacteristic
201C8—22no—
202C8—22no—
203C8—42no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—44no—
212C22#161012yes—
213C22#187411yescharacteristic
214C22#214512yes—
215C4—21no—
216C4—21no—
217C4—42no—
218C4—42no—
219C4—84no—
220C4—164no—
221C4—161no—
222C4—161no—
223C4#161112yes—
224C4#188611yescharacteristic
225C4#189011yescharacteristic
226C2—22no—
227C2—22no—
228C2—41no—
229C2#902711yescharacteristic
230C2#1300211yescharacteristic
231C2#1300811yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #2087C4semidirect
13. #2086C4semidirect

Maximal subgroups

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

subgroupindex
#236662
#219602
#128702
#114402
#72622