Finite Group Databasebeta

← #138180

Subgroups — #138180

Order 512.

447 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 96 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#33419C211yesmaximal, characteristic, complemented
2#33179C211yesmaximal, characteristic
3#11931C212yesmaximal, complemented
4#8601C211yesmaximal, characteristic, complemented
5#7279C212yesmaximal, complemented
6C2×#493C2211yescharacteristic
7C2×#469C412yes—
8C2×#466—21no—
9C2×#405C2211yescharacteristic
10#2619—21no—
11#2164—24no—
12#2150C2212yes—
13#2112—21no—
14C4×M32—21no—
15#1972C412yes—
16#1970—21no—
17#1961—21no—
18#1846C2211yescharacteristic
19#1564C2212yes—
20#1458—21no—
21#1361—24no—
22#1275—21no—
23C22×D8—21no—
24C22×D8C2×C412yes—
25C22×M16C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C2×#127—21no—
28C2×#121—22no—
29C2×#117D411yescharacteristic
30C2×M32—22no—
31#493—22no—
32#493—22no—
33#469—44no—
34#466—44no—
35#431—22no—
36C2×#107—21no—
37C2×#106C2×C412yes—
38C2×#102—22no—
39C2×#102—22no—
40#405—22no—
41#405—22no—
42C2×#97C2×C411yescharacteristic
43C2×#97D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8C2311yescharacteristic
46#394—41no—
47#394—41no—
48#392—41no—
49#392—41no—
50#390—22no—
51#382—21no—
52C4⋊C16—22no—
53C4⋊C16D414yescomplemented
54C22⋊C16—24no—
55#346—22no—
56C4×C16—22no—
57#335—22no—
58#334—22no—
59#324—41no—
60#323—41no—
61C22×D4—22no—
62C22×D4—42no—
63C22×D4—41no—
64C22×D4—41no—
65C23×C4C22⋊C411yescharacteristic
66C2×D8—24no—
67C2×D8—42no—
68C2×D8—42no—
69C2×D8—41no—
70C2×D8—41no—
71C2×D8—42no—
72C2×D8—42no—
73C2×M16—22no—
74C2×M16—22no—
75C2×M16—41no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22×C411yescharacteristic
83C22×C8C2×D411yescharacteristic
84#127—42no—
85#127—42no—
86#121—44no—
87#121—44no—
88#117—22no—
89#117—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4D811yescharacteristic
92C2×C4⋊C4SD1611yescharacteristic
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4D811yescharacteristic
97C2×C22⋊C4SD1611yescharacteristic
98C2×C42C22⋊C411yescharacteristic
99C2×C42C2×D411yescharacteristic
100M32—42no—
101M32—42no—
102C2×C16—24no—
103#107—42no—
104#106—42no—
105C4⋊C8—42no—
106#102—44no—
107#102—44no—
108C22⋊C8—42no—
109#97—22no—
110#97—22no—
111C4×C8—21no—
112C4×C8—21no—
113C4×C8—42no—
114C4×C8C2×D412yes—
115C4×C8#4112yes—
116#95—41no—
117#95—41no—
118C24—42no—
119C24#10211yescharacteristic
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4—42no—
125C2×D4—42no—
126C2×D4—84no—
127C2×D4—82no—
128C2×D4—82no—
129C2×D4—162no—
130C2×D4—162no—
131C2×D4—162no—
132C2×D4—162no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—42no—
140C22×C4#10211yescharacteristic
141C22×C4#10211yescharacteristic
142C22×C4#10211yescharacteristic
143C22×C4C2×C22⋊C411yescharacteristic
144C22×C4C2×D811yescharacteristic
145C22×C4C2×SD1611yescharacteristic
146D8—84no—
147D8—82no—
148D8—82no—
149M16—44no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—82no—
160C2×C8#10012yes—
161C2×C8C4×D412yes—
162C2×C8#12011yescharacteristic
163C2×C8#12311yescharacteristic
164C2×C8#13512yes—
165C2×C8AGL(1,ℤ/8)12yes—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—84no—
173C22⋊C4—162no—
174C22⋊C4—162no—
175C42—21no—
176C42—21no—
177C42#12112yes—
178C42#13512yes—
179C16—44no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—41no—
184C23—41no—
185C23—82no—
186C23—82no—
187C23—82no—
188C23—82no—
189C23—82no—
190C23—82no—
191C23—82no—
192C23C2×#10211yescharacteristic
193D4—82no—
194D4—82no—
195D4—82no—
196D4—82no—
197D4—164no—
198D4—164no—
199D4—168no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4#38611yescharacteristic
217C2×C4C2×#10011yescharacteristic
218C2×C4#41311yescharacteristic
219C2×C4#41611yescharacteristic
220C2×C4#41611yescharacteristic
221C2×C4#41811yescharacteristic
222C2×C4#46512yes—
223C2×C4#46612yes—
224C8—22no—
225C8—22no—
226C8—41no—
227C8—162no—
228C8#44912yes—
229C8#48212yes—
230C22—21no—
231C22—21no—
232C22—42no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—81no—
237C22—162no—
238C22—162no—
239C22—162no—
240C22—162no—
241C22—162no—
242C22—162no—
243C22—162no—
244C22#154011yescharacteristic
245C22#154511yescharacteristic
246C22#189311yescharacteristic
247C4—22no—
248C4—22no—
249C4—41no—
250C4—41no—
251C4—41no—
252C4—162no—
253C4#154311yescharacteristic
254C4#155311yescharacteristic
255C4#187511yescharacteristic
256C4#190011yescharacteristic
257C2—21no—
258C2—21no—
259C2—81no—
260C2—322no—
261C2#913411yescharacteristic
262C2#1307411yescharacteristic
263C2#1307911yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#33179C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#405 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33419C2semidirect
3. #11931C2semidirect
4. #8601C2semidirect
5. #7279C2semidirect
53. C4⋊C16D4semidirect

Maximal subgroups

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

subgroupindex
#334192
#331792
#119312
#86012
#72792