Finite Group Databasebeta

← #141126

Subgroups — #141126

Order 512.

572 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 100 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#33175C211yesmaximal, characteristic
2#12785C211yesmaximal, characteristic, complemented
3C2×#1339C214yesmaximal, complemented
4C4×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
5C2×#493C411yescharacteristic, complemented
6C2×#469—21no—
7C2×#469C2212yes—
8C2×C4×M16C2211yescharacteristic
9#2856—21no—
10C4×#117—21no—
11#2150—21no—
12C2×#349C2212yes—
13#1972—21no—
14#1972C412yescomplemented
15#1957—21no—
16#1844C411yescharacteristic, complemented
17C2×H(3,ℤ/4)—21no—
18C2×H(3,ℤ/4)C2212yes—
19#1377—21no—
20#1339—28no—
21C22×D8—21no—
22C22×D8C2×C412yes—
23C22×M16D411yescharacteristic
24C22×M16C2311yescharacteristic
25C2×#127—21no—
26C2×#121—22no—
27C2×#121—42no—
28C2×#117—22no—
29C22×C42D411yescharacteristic
30C2×M32—22no—
31#493—42no—
32#469—24no—
33#469—24no—
34#469—44no—
35C2×#106—21no—
36C2×#106C2×C412yes—
37C2×#102—22no—
38C2×#102—22no—
39C4×M16—42no—
40C2×#97C2×C411yescharacteristic
41C2×#97D411yescharacteristic
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#390—22no—
45#390—42no—
46#384—21no—
47C4×C4⋊C4—42no—
48C4×C22⋊C4—42no—
49#376—22no—
50C43—22no—
51C4⋊C16—42no—
52#349—24no—
53H(3,ℤ/4)—24no—
54H(3,ℤ/4)—22no—
55H(3,ℤ/4)D414yescomplemented
56C22×D4—22no—
57C22×D4—42no—
58C22×D4—42no—
59C23×C4D812yes—
60C23×C4C2×D411yescharacteristic
61C2×D8—24no—
62C2×D8—42no—
63C2×D8—42no—
64C2×D8—42no—
65C2×D8—42no—
66C2×D8—42no—
67C2×M16—22no—
68C2×M16—22no—
69C2×M16—41no—
70C2×M16—41no—
71C2×M16—41no—
72C2×M16C2×D412yes—
73C2×M16#4112yes—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22×C411yescharacteristic
77C22×C8C2×D411yescharacteristic
78#127—42no—
79#127—41no—
80#127—41no—
81#121—44no—
82#121—44no—
83#121—88no—
84#117—24no—
85#117—24no—
86#117—42no—
87#117—42no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C22⋊C4—42no—
92C2×C22⋊C4—44no—
93C2×C22⋊C4—42no—
94C2×C42—22no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—24no—
98C2×C42—44no—
99C2×C42C22⋊C411yescharacteristic
100C2×C42C22⋊C411yescharacteristic
101C2×C42SD1612yes—
102M32—42no—
103M32—42no—
104C2×C16—42no—
105#106—24no—
106#106—42no—
107#102—44no—
108#102—42no—
109#102—42no—
110#97—41no—
111#97—41no—
112C4×C8—22no—
113C4×C8—22no—
114#95—44no—
115C24—42no—
116C24C2×D811yescharacteristic
117C2×D4—42no—
118C2×D4—42no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—84no—
124C2×D4—84no—
125C2×D4—88no—
126C2×D4—84no—
127C2×D4—82no—
128C2×D4—82no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—42no—
138C22×C4#10212yes—
139C22×C4#10212yes—
140C22×C4C2×C22⋊C411yescharacteristic
141C22×C4#12112yes—
142C22×C4C2×SD1611yescharacteristic
143C22×C4AGL(1,ℤ/8)12yes—
144D8—84no—
145D8—84no—
146M16—42no—
147M16—42no—
148M16—44no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8#10412yes—
159C2×C8C4×D412yes—
160C2×C8#12411yescharacteristic
161C2×C8#12711yescharacteristic
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—84no—
167C22⋊C4—44no—
168C22⋊C4—44no—
169C22⋊C4—88no—
170C22⋊C4—84no—
171C42—22no—
172C42—22no—
173C42—21no—
174C42—21no—
175C42—28no—
176C42—44no—
177C42—88no—
178C42#9912yes—
179C42#10411yescharacteristic
180C42#10411yescharacteristic
181C16—82no—
182C23—21no—
183C23—21no—
184C23—42no—
185C23—84no—
186C23—84no—
187C23—82no—
188C23—82no—
189C23—82no—
190C23C2×#10211yescharacteristic
191C23#45912yes—
192D4—82no—
193D4—82no—
194D4—82no—
195D4—82no—
196D4—164no—
197D4—168no—
198D4—164no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—48no—
210C2×C4—82no—
211C2×C4—88no—
212C2×C4—84no—
213C2×C4—82no—
214C2×C4#32712yes—
215C2×C4#32712yes—
216C2×C4#39011yescharacteristic
217C2×C4C2×#9911yescharacteristic
218C2×C4#41011yescharacteristic
219C2×C4#41811yescharacteristic
220C2×C4C2×#10411yescharacteristic
221C2×C4C2×#10411yescharacteristic
222C2×C4#46012yes—
223C8—21no—
224C8—22no—
225C8—21no—
226C8—22no—
227C8—41no—
228C22—22no—
229C22—22no—
230C22—42no—
231C22—42no—
232C22—81no—
233C22—162no—
234C22—162no—
235C22—164no—
236C22—164no—
237C22—162no—
238C22C2×#32711yescharacteristic
239C22#150211yescharacteristic
240C22#198211yescharacteristic
241C4—22no—
242C4—22no—
243C4—42no—
244C4—88no—
245C4—162no—
246C4#133914yescomplemented
247C4#150511yescharacteristic
248C4#150711yescharacteristic
249C4#196211yescharacteristic
250C4#196411yescharacteristic
251C2—21no—
252C2—21no—
253C2—42no—
254C2—322no—
255C2#971211yescharacteristic
256C2C2×#133912yes—
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radical#33175C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12785C2semidirect
3. C2×#1339C2semidirect
4. C4×H(3,ℤ/4)C2semidirect
5. C2×#493C4semidirect
14. #1972C4semidirect
16. #1844C4semidirect
55. H(3,ℤ/4)D4semidirect
246. C4#1339semidirect

Maximal subgroups

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

subgroupindex
#331752
#127852
C2×#13392
C4×H(3,ℤ/4)2