Finite Group Databasebeta

← #138387

Subgroups — #138387

Order 512.

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

normal subgroups only

This group has 420 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#33136C211yesmaximal, characteristic
2#20762C211yesmaximal, characteristic
3#12846C211yesmaximal, characteristic
4#11706C212yesmaximal
5#7262C212yesmaximal
6C2×#467C2211yescharacteristic
7C22×#97C2211yescharacteristic
8#2855—21no—
9#2285—21no—
10#2112C2212yes—
11#1971C412yescomplemented
12#1944C412yescomplemented
13#1935—22no—
14#1892—21no—
15#1836—24no—
16#1807C2211yescharacteristic
17#1735—21no—
18#1735C2212yes—
19#1377—22no—
20#1348—24no—
21C22×Q16C2×C411yescharacteristic
22C23×C8D411yescharacteristic
23C23×C8C2311yescharacteristic
24C2×C22⋊Q8—21no—
25C2×#117—22no—
26C22×C42D411yescharacteristic
27C2×M32—22no—
28#467—22no—
29#467—22no—
30C2×#107C2×C411yescharacteristic
31C2×#106C2×C412yes—
32C2×#103—22no—
33C2×#103—21no—
34C2×#97—22no—
35C2×#97—22no—
36C2×#97C2×C412yes—
37C2×#97D412yes—
38#391—22no—
39#389—22no—
40#387—42no—
41#385—41no—
42#384—22no—
43#382—22no—
44#381—41no—
45#380—21no—
46#380—41no—
47#380—41no—
48C4×C4⋊C4—41no—
49C4×C4⋊C4—41no—
50C22⋊C16—22no—
51C22⋊C16—22no—
52#340—24no—
53#340—22no—
54#340—21no—
55#340—22no—
56#340D414yes—
57C22×Q8—21no—
58C23×C4D811yescharacteristic
59C23×C4D811yescharacteristic
60C23×C4C2×D411yescharacteristic
61C2×Q16—22no—
62C2×Q16—42no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—42no—
66C22×C8—42no—
67C22×C8C22⋊C412yes—
68C22×C8C22×C411yescharacteristic
69C22×C8C2×D411yescharacteristic
70C22×C8C2×D412yes—
71C22×C8#4112yes—
72C22⋊Q8—41no—
73C22⋊Q8—41no—
74C22⋊Q8—41no—
75C22⋊Q8—41no—
76#117—44no—
77#117—44no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—41no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—42no—
93C2×C42C22⋊C412yes—
94C2×C42SD1612yes—
95M32—44no—
96C2×C16—42no—
97#107—22no—
98#106—42no—
99#103—44no—
100#103—42no—
101#97—24no—
102#97—24no—
103#95—42no—
104#95—42no—
105#95—42no—
106#95—42no—
107#95—42no—
108#95—41no—
109C24C2×D811yescharacteristic
110C2×Q8—41no—
111C2×Q8—42no—
112C2×Q8—41no—
113C2×Q8—42no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—44no—
125C22×C4—81no—
126C22×C4#10212yes—
127C22×C4#10212yes—
128C22×C4C2×C22⋊C411yescharacteristic
129C22×C4#12011yescharacteristic
130C22×C4#12311yescharacteristic
131C22×C4C2×SD1611yescharacteristic
132C22×C4AGL(1,ℤ/8)11yescharacteristic
133C22×C4#13711yescharacteristic
134Q16—82no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—42no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8C4×D412yes—
145C2×C8#12112yes—
146C4⋊C4—44no—
147C4⋊C4—44no—
148C4⋊C4—41no—
149C4⋊C4—42no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C4⋊C4—84no—
153C4⋊C4—84no—
154C4⋊C4—84no—
155C4⋊C4—81no—
156C4⋊C4—81no—
157C22⋊C4—82no—
158C22⋊C4—84no—
159C22⋊C4—81no—
160C22⋊C4—81no—
161C42—22no—
162C42—22no—
163C42—22no—
164C42—22no—
165C42—84no—
166C16—82no—
167C23—22no—
168C23—22no—
169C23—41no—
170C23C2×#10211yescharacteristic
171C23#44711yescharacteristic
172C23#48011yescharacteristic
173Q8—81no—
174Q8—81no—
175Q8—82no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—84no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—84no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—84no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4#38611yescharacteristic
202C2×C4#41012yes—
203C2×C4#41711yescharacteristic
204C2×C4#42112yes—
205C2×C4#44811yescharacteristic
206C2×C4#48111yescharacteristic
207C8—22no—
208C8—22no—
209C8—42no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—44no—
218C22#151712yes—
219C22#187411yescharacteristic
220C22#219112yes—
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—164no—
226C4—162no—
227C4—161no—
228C4—161no—
229C4#151212yes—
230C4#189711yescharacteristic
231C4#189911yescharacteristic
232C2—22no—
233C2—22no—
234C2—41no—
235C2#915611yescharacteristic
236C2#1300011yescharacteristic
237C2#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×#97 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #1971C4semidirect
12. #1944C4semidirect

Maximal subgroups

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

subgroupindex
#331362
#207622
#128462
#117062
#72622