Finite Group Databasebeta

← #137461

Subgroups — #137461

Order 512.

317 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 74 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#33663C211yesmaximal, characteristic
2#17989C211yesmaximal, characteristic
3#11948C211yesmaximal, characteristic
4#11662C211yesmaximal, characteristic, complemented
5#9671C211yesmaximal, characteristic, complemented
6#7289C211yesmaximal, characteristic, complemented
7#7198C211yesmaximal, characteristic, complemented
8#3169C2211yescharacteristic
9#2981—21no—
10C2×C4×Q16C2211yescharacteristic
11#2233C412yes—
12#2225—21no—
13#2207—21no—
14#2185C412yes—
15C4×SD32—22no—
16#2162C2211yescharacteristic
17#2162C2211yescharacteristic
18#1586—22no—
19#1581—22no—
20#1568C2211yescharacteristic
21#1564C2211yescharacteristic
22#1496—21no—
23#1479—21no—
24#1461—21no—
25#1452C2211yescharacteristic
26#1365—21no—
27#1361—21no—
28#1332—22no—
29C22×Q16C2×C411yescharacteristic
30#532—21no—
31C2×C4×Q8—21no—
32C2×SD32—22no—
33#500D411yescharacteristic
34#493D411yescharacteristic
35#486—21no—
36#466—21no—
37#466C2×C412yes—
38C4×Q16—21no—
39C4×Q16—21no—
40C4×Q16—21no—
41C4×Q16—21no—
42C4×D8—21no—
43C4×D8D412yes—
44#436—22no—
45#435—42no—
46C8×D4—22no—
47#432—21no—
48#431—21no—
49#431D411yescharacteristic
50#431D411yescharacteristic
51C2×#107C2×C411yescharacteristic
52C2×#103—21no—
53#408—42no—
54C2×C4×C8C2311yescharacteristic
55#367—22no—
56C4⋊C16—21no—
57C4⋊C16—21no—
58#358—22no—
59#357—22no—
60C22⋊C16—21no—
61C22⋊C16C2×C412yes—
62C4×C16—21no—
63C4×C16—21no—
64#335—21no—
65#335D412yes—
66#334—21no—
67#334—21no—
68Q8⋊C8—42no—
69D4⋊C8—24no—
70#324—42no—
71#322—21no—
72#322—21no—
73C2×#41—41no—
74C22×Q8—21no—
75C2×Q16—21no—
76C2×Q16—21no—
77C2×Q16—42no—
78C2×D8—21no—
79C2×D8C22⋊C412yes—
80C2×M16—42no—
81C22×C8—42no—
82C22×C8C22×C411yescharacteristic
83C22×C8#4111yescharacteristic
84C4⋊Q8—21no—
85#127—21no—
86#124—41no—
87#121—22no—
88#121—41no—
89C4×Q8—41no—
90C4×Q8—41no—
91C4×Q8—41no—
92C4×Q8—41no—
93C4×D4—22no—
94C4×D4—41no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—41no—
97C2×C42—41no—
98C2×C42C2×D411yescharacteristic
99SD32—44no—
100C2×C16—22no—
101C2×C16—41no—
102#107—21no—
103#107—21no—
104#107—21no—
105#107C22⋊C412yes—
106C4⋊C8—22no—
107C4⋊C8—21no—
108C4⋊C8—41no—
109C4⋊C8D812yes—
110#103—41no—
111#103—41no—
112#102—22no—
113#102—22no—
114C22⋊C8—21no—
115C22⋊C8—21no—
116C22⋊C8—42no—
117C22⋊C8—42no—
118C22⋊C8—41no—
119C22⋊C8—41no—
120#97—41no—
121C4×C8—21no—
122C4×C8—21no—
123C4×C8—41no—
124C4×C8C2×D411yescharacteristic
125C4×C8C2×D411yescharacteristic
126C4×C8C2×D411yescharacteristic
127C4×C8#4111yescharacteristic
128#41—162no—
129C2×Q8—22no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—41no—
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—41no—
139C2×D4—41no—
140C22×C4—41no—
141C22×C4—42no—
142C22×C4—41no—
143C22×C4C4×D411yescharacteristic
144C22×C4AGL(1,ℤ/8)11yescharacteristic
145C22×C4AGL(1,ℤ/8)11yescharacteristic
146Q16—22no—
147Q16—22no—
148Q16—41no—
149Q16—41no—
150D8—42no—
151D8—81no—
152M16—162no—
153C2×C8—22no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—42no—
157C2×C8—41no—
158C2×C8—81no—
159C2×C8—162no—
160C2×C8C2×C22⋊C411yescharacteristic
161C2×C8C4×D411yescharacteristic
162C2×C8#12111yescharacteristic
163C2×C8#12311yescharacteristic
164C4⋊C4—22no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C22⋊C4—42no—
171C22⋊C4—42no—
172C42—81no—
173C42—81no—
174C42#12011yescharacteristic
175C42#12111yescharacteristic
176C42C2×D811yescharacteristic
177C42#13511yescharacteristic
178C16—42no—
179C16—81no—
180C23—42no—
181C23#44211yescharacteristic
182Q8—42no—
183Q8—42no—
184Q8—81no—
185Q8—81no—
186Q8—161no—
187D4—42no—
188D4—42no—
189D4—161no—
190D4—161no—
191D4—162no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—81no—
203C2×C4—161no—
204C2×C4#38611yescharacteristic
205C2×C4C4×D811yescharacteristic
206C2×C4#44711yescharacteristic
207C2×C4#44911yescharacteristic
208C2×C4#45911yescharacteristic
209C2×C4#46311yescharacteristic
210C8—21no—
211C8—21no—
212C8—21no—
213C8—21no—
214C8—42no—
215C8—81no—
216C8—82no—
217C8—161no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—82no—
222C22—82no—
223C22—82no—
224C22#187911yescharacteristic
225C4—21no—
226C4—21no—
227C4—21no—
228C4—21no—
229C4—82no—
230C4—82no—
231C4—161no—
232C4#188111yescharacteristic
233C4#189111yescharacteristic
234C4#219412yes—
235C2—41no—
236C2—162no—
237C2#904911yescharacteristic
238C2#1301911yescharacteristic
239C2#1302111yescharacteristic
subgroupG/H
centerC2×C4#447
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
socleC22#1879
prime radical#3169C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1452 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11662C2semidirect
5. #9671C2semidirect
6. #7289C2semidirect
7. #7198C2semidirect

Maximal subgroups

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

subgroupindex
#336632
#179892
#119482
#116622
#96712
#72892
#71982