Finite Group Databasebeta

← #139022

Subgroups — #139022

Order 512.

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

normal subgroups only

This group has 296 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#33800C211yesmaximal, characteristic
2#18091C211yesmaximal, characteristic
3#11985C211yesmaximal, characteristic
4#11935C211yesmaximal, characteristic, complemented
5#9792C211yesmaximal, characteristic, complemented
6#7293C211yesmaximal, characteristic, complemented
7#7211C211yesmaximal, characteristic, complemented
8#3169C2211yescharacteristic
9C2×#499C2211yescharacteristic
10#2979—21no—
11#2255—21no—
12#2244—21no—
13#2233C412yes—
14#2232C412yes—
15#2228—22no—
16#2163C2211yescharacteristic
17#2150C2211yescharacteristic
18#1594—21no—
19#1581—22no—
20#1580—21no—
21#1568C2211yescharacteristic
22#1564C2211yescharacteristic
23#1528—21no—
24#1484—21no—
25#1470—21no—
26#1451C2211yescharacteristic
27#1364—21no—
28#1363—21no—
29#1363—21no—
30#1361—21no—
31#532—21no—
32C2×#125—21no—
33C2×M32—21no—
34#500D411yescharacteristic
35#499—21no—
36#499—21no—
37#499—21no—
38#499—21no—
39#493D411yescharacteristic
40#486—21no—
41#466—21no—
42#466C2×C412yes—
43C4×D8—21no—
44C4×D8D412yes—
45#436—22no—
46#436—41no—
47C8×D4—22no—
48C8×D4—41no—
49#432—21no—
50#431—21no—
51#431D411yescharacteristic
52C2×#107C2×C411yescharacteristic
53C2×#107C2×C411yescharacteristic
54C2×#106—21no—
55#408—42no—
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#335—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—
81C2×M16—21no—
82C2×M16—42no—
83C22×C8—42no—
84C22×C8C22×C411yescharacteristic
85C22×C8#4111yescharacteristic
86C4⋊Q8—21no—
87#127—21no—
88#125—41no—
89#125—41no—
90#125—41no—
91#125—41no—
92#124—41no—
93#121—22no—
94#121—41no—
95C4×D4—22no—
96C4×D4—41no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C42C2×D411yescharacteristic
102M32—81no—
103C2×C16—22no—
104C2×C16—41no—
105#107—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#107—21no—
110#107C22⋊C412yes—
111#106—41no—
112#106—41no—
113C4⋊C8—21no—
114C4⋊C8—21no—
115C4⋊C8—21no—
116C4⋊C8—41no—
117#102—22no—
118#102—22no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123C22⋊C8D812yes—
124#97—21no—
125#97—41no—
126C4×C8—21no—
127C4×C8—21no—
128C4×C8—41no—
129C4×C8C2×D411yescharacteristic
130C4×C8C2×D411yescharacteristic
131C4×C8C2×D411yescharacteristic
132C4×C8#4111yescharacteristic
133#41—161no—
134#41—161no—
135C2×Q8—41no—
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—41no—
139C2×D4—41no—
140C22×C4—42no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4C4×D411yescharacteristic
144C22×C4C2×D811yescharacteristic
145C22×C4#13511yescharacteristic
146D8—42no—
147D8—41no—
148D8—41no—
149M16—81no—
150M16—162no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—22no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
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—41no—
169C4⋊C4—41no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C42#12011yescharacteristic
177C42#12111yescharacteristic
178C42AGL(1,ℤ/8)11yescharacteristic
179C42AGL(1,ℤ/8)11yescharacteristic
180C16—42no—
181C16—81no—
182C23—42no—
183C23C4×D811yescharacteristic
184Q8—161no—
185D4—42no—
186D4—42no—
187D4—82no—
188D4—161no—
189D4—162no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—161no—
201C2×C4#38611yescharacteristic
202C2×C4#44211yescharacteristic
203C2×C4#44711yescharacteristic
204C2×C4#44911yescharacteristic
205C2×C4#45911yescharacteristic
206C2×C4#46311yescharacteristic
207C8—21no—
208C8—21no—
209C8—21no—
210C8—42no—
211C8—81no—
212C8—81no—
213C8—81no—
214C8—161no—
215C8#47111yescharacteristic
216C8#47311yescharacteristic
217C22—21no—
218C22—21no—
219C22—21no—
220C22—82no—
221C22—82no—
222C22—82no—
223C22#187911yescharacteristic
224C4—21no—
225C4—21no—
226C4—21no—
227C4—21no—
228C4—21no—
229C4—82no—
230C4—161no—
231C4—161no—
232C4#190811yescharacteristic
233C4#191511yescharacteristic
234C2—41no—
235C2—162no—
236C2#927111yescharacteristic
237C2#1302411yescharacteristic
238C2#1303311yescharacteristic
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. #11935C2semidirect
5. #9792C2semidirect
6. #7293C2semidirect
7. #7211C2semidirect

Maximal subgroups

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

subgroupindex
#338002
#180912
#119852
#119352
#97922
#72932
#72112