Finite Group Databasebeta

← #139018

Subgroups — #139018

Order 512.

315 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 76 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#33800C211yesmaximal, characteristic
2#18073C211yesmaximal, characteristic
3#11982C211yesmaximal, characteristic
4C2×#1363C211yesmaximal, characteristic, complemented
5#9792C211yesmaximal, characteristic, complemented
6#7294C211yesmaximal, characteristic, complemented
7#7210C211yesmaximal, characteristic, complemented
8#3169C2211yescharacteristic
9C2×#499C2211yescharacteristic
10#2935—21no—
11#2255—21no—
12#2244—21no—
13#2233C412yes—
14#2232C412yes—
15#2228—22no—
16#2163C2211yescharacteristic
17C2×C4⋊C16C2211yescharacteristic
18#1594—21no—
19#1586—22no—
20#1580—21no—
21#1568C2211yescharacteristic
22C2×#334C2211yescharacteristic
23#1528—21no—
24#1480—21no—
25#1470—21no—
26#1451C2211yescharacteristic
27#1365—21no—
28#1363—21no—
29#1363—21no—
30#1361—21no—
31C2×#131—41no—
32#532—21no—
33C2×#125—21no—
34C22×C16—21no—
35#500D411yescharacteristic
36#499—21no—
37#499—21no—
38#499—21no—
39#499—21no—
40#493D411yescharacteristic
41#486—21no—
42#466—21no—
43#466C2×C412yes—
44C4×D8—21no—
45C4×D8D412yes—
46#436—22no—
47#435—42no—
48C8×D4—22no—
49#432—21no—
50#431D411yescharacteristic
51C2×#107C2×C411yescharacteristic
52C2×#107C2×C411yescharacteristic
53C2×#106—21no—
54C2×C4⋊C8—21no—
55#408—41no—
56C4×M16D411yescharacteristic
57C2×C4×C8C2311yescharacteristic
58#367—22no—
59#366—22no—
60C4⋊C16—21no—
61C4⋊C16—21no—
62C4⋊C16—21no—
63#357—22no—
64#357—22no—
65C22⋊C16C2×C412yes—
66#346—21no—
67#335—21no—
68#334—21no—
69#334—21no—
70#334D411yescharacteristic
71#334D411yescharacteristic
72#330—42no—
73D4⋊C8—22no—
74D4⋊C8—22no—
75#324—42no—
76#322—21no—
77#322—21no—
78C2×#41—41no—
79C2×D8—21no—
80C2×D8C22⋊C412yes—
81#131—84no—
82C2×M16—21no—
83C2×M16—42no—
84C22×C8—42no—
85C22×C8—41no—
86C22×C8C22×C411yescharacteristic
87C22×C8#4111yescharacteristic
88C4⋊Q8—21no—
89#127—21no—
90#125—41no—
91#125—41no—
92#125—41no—
93#125—41no—
94#124—41no—
95#121—22no—
96#121—41no—
97C4×D4—22no—
98C4×D4—41no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C42C2×D411yescharacteristic
104C2×C16—22no—
105C2×C16—41no—
106C2×C16—81no—
107#107—21no—
108#107—21no—
109#107—21no—
110#107—21no—
111#107—21no—
112#107C22⋊C412yes—
113#106—41no—
114#106—41no—
115C4⋊C8—21no—
116C4⋊C8—21no—
117C4⋊C8—21no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120#102—22no—
121#102—22no—
122C22⋊C8—21no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—41no—
126#97—41no—
127#97D812yes—
128C4×C8—21no—
129C4×C8—21no—
130C4×C8C2×D411yescharacteristic
131C4×C8C2×D411yescharacteristic
132C4×C8C2×D411yescharacteristic
133C4×C8#4111yescharacteristic
134#41—82no—
135#41—82no—
136C2×Q8—41no—
137C2×D4—22no—
138C2×D4—42no—
139C2×D4—41no—
140C2×D4—41no—
141C22×C4—42no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4C4×D411yescharacteristic
145C22×C4AGL(1,ℤ/8)11yescharacteristic
146C22×C4AGL(1,ℤ/8)11yescharacteristic
147D8—42no—
148D8—41no—
149D8—41no—
150M16—42no—
151M16—84no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—22no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—84no—
160C2×C8—82no—
161C2×C8—81no—
162C2×C8C2×C22⋊C411yescharacteristic
163C2×C8C4×D411yescharacteristic
164C2×C8#12111yescharacteristic
165C2×C8#12311yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C4⋊C4—81no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C42#12011yescharacteristic
179C42#12111yescharacteristic
180C42C2×D811yescharacteristic
181C42#13511yescharacteristic
182C16—42no—
183C16—81no—
184C23—42no—
185C23#44211yescharacteristic
186Q8—82no—
187D4—42no—
188D4—42no—
189D4—82no—
190D4—84no—
191D4—161no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—82no—
202C2×C4—161no—
203C2×C4#38611yescharacteristic
204C2×C4C4×D811yescharacteristic
205C2×C4#44711yescharacteristic
206C2×C4#44911yescharacteristic
207C2×C4#45911yescharacteristic
208C2×C4#46311yescharacteristic
209C8—21no—
210C8—21no—
211C8—21no—
212C8—42no—
213C8—81no—
214C8—81no—
215C8—81no—
216C8—161no—
217C8#47111yescharacteristic
218C8#47311yescharacteristic
219C22—21no—
220C22—21no—
221C22—82no—
222C22—82no—
223C22—82no—
224C22#187911yescharacteristic
225C22AGL(1,ℤ/16)12yes—
226C4—21no—
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—82no—
232C4—161no—
233C4—161no—
234C4#190811yescharacteristic
235C4#191511yescharacteristic
236C2—22no—
237C2—162no—
238C2#927111yescharacteristic
239C2#1302111yescharacteristic
240C2#1303711yescharacteristic
subgroupG/H
center, socleC22#1879
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#3169C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1363C2semidirect
5. #9792C2semidirect
6. #7294C2semidirect
7. #7210C2semidirect

Maximal subgroups

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

subgroupindex
#338002
#180732
#119822
C2×#13632
#97922
#72942
#72102