Finite Group Databasebeta

← #125494

Subgroups — #125494

Order 512.

279 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 64 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#33269C211yesmaximal, characteristic
2#12659C211yesmaximal, characteristic, complemented
3#8800C211yesmaximal, characteristic, complemented
4#7127C211yesmaximal, characteristic, complemented
5#7126C211yesmaximal, characteristic, complemented
6#7110C211yesmaximal, characteristic
7#7110C211yesmaximal, characteristic
8#2860—21no—
9C2×#364—21no—
10C2×#347—21no—
11#1972C411yescharacteristic
12#1972C2211yescharacteristic
13#1970C411yescharacteristic
14#1944C2211yescharacteristic
15#1943C411yescharacteristic
16#1941C411yescharacteristic
17#1936—21no—
18#1936—21no—
19#1934—21no—
20C4×#97C2211yescharacteristic
21#1377C2211yescharacteristic
22#1377C2211yescharacteristic
23#1281—21no—
24#1276—21no—
25#1275C2211yescharacteristic
26#1275C2211yescharacteristic
27C22×D8C2×C411yescharacteristic
28C2×#126—41no—
29C2×M32—21no—
30C2×M32—21no—
31#431—21no—
32#431—21no—
33C2×#107C2×C411yescharacteristic
34C2×#106C2×C411yescharacteristic
35C2×#106C2×C411yescharacteristic
36C2×#102—21no—
37C2×#102—21no—
38C2×#102—21no—
39C2×#97C2×C411yescharacteristic
40C2×#97D411yescharacteristic
41C2×#97D411yescharacteristic
42C2×#97C2311yescharacteristic
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8D411yescharacteristic
45#390—21no—
46#390—21no—
47#390—21no—
48#389—21no—
49#389—21no—
50#389—21no—
51#388—41no—
52#388—41no—
53#388—41no—
54C43D411yescharacteristic
55#364—24no—
56#347—42no—
57#340—21no—
58#340—21no—
59#340—21no—
60C22×D4—21no—
61C2×D8—22no—
62C2×D8—41no—
63C2×D8—41no—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22×C411yescharacteristic
67C22×C8C2×D411yescharacteristic
68#126—84no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C22⋊C4—41no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—41no—
78C2×C22⋊C4—41no—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—41no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42C22⋊C411yescharacteristic
84C2×C42C22⋊C411yescharacteristic
85C2×C42C2×D411yescharacteristic
86M32—22no—
87M32—22no—
88M32—41no—
89M32—41no—
90C2×C16—41no—
91C2×C16—41no—
92#107—41no—
93#106—22no—
94#106—41no—
95C4⋊C8—41no—
96C4⋊C8—41no—
97#102—42no—
98#102—42no—
99#102—41no—
100#102—41no—
101C22⋊C8—41no—
102C22⋊C8—41no—
103#97—21no—
104#97—21no—
105#97—41no—
106#97—41no—
107#97—41no—
108C4×C8—22no—
109C4×C8—22no—
110C4×C8—41no—
111C4×C8—41no—
112#95—41no—
113#95—41no—
114#95—41no—
115C24—41no—
116C2×D4—41no—
117C2×D4—41no—
118C2×D4—41no—
119C2×D4—41no—
120C2×D4—41no—
121C2×D4—41no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4C2×C22⋊C411yescharacteristic
128C22×C4AGL(1,ℤ/8)11yescharacteristic
129C22×C4AGL(1,ℤ/8)11yescharacteristic
130D8—82no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—81no—
141C2×C8—81no—
142C2×C8#10412yes—
143C2×C8#13112yes—
144C2×C8AGL(1,ℤ/8)11yescharacteristic
145C2×C8#13711yescharacteristic
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—82no—
153C4⋊C4—82no—
154C4⋊C4—82no—
155C22⋊C4—82no—
156C22⋊C4—82no—
157C22⋊C4—82no—
158C22⋊C4—82no—
159C22⋊C4—82no—
160C22⋊C4—82no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42—42no—
168C42—81no—
169C16—42no—
170C16—42no—
171C23—81no—
172C23—81no—
173C23—81no—
174C23—81no—
175C23—81no—
176C23—81no—
177C23—81no—
178C23#41811yescharacteristic
179D4—81no—
180D4—81no—
181D4—81no—
182D4—81no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4#40811yescharacteristic
203C2×C4#41011yescharacteristic
204C2×C4#41311yescharacteristic
205C2×C4#41711yescharacteristic
206C2×C4C2×#10411yescharacteristic
207C2×C4#42111yescharacteristic
208C8—22no—
209C8—21no—
210C8—21no—
211C8—22no—
212C8—41no—
213C8—161no—
214C8—161no—
215C22—21no—
216C22—21no—
217C22—161no—
218C22—161no—
219C22—161no—
220C22—161no—
221C22—161no—
222C22—161no—
223C22—161no—
224C22#148411yescharacteristic
225C22#151411yescharacteristic
226C22#155711yescharacteristic
227C4—42no—
228C4—41no—
229C4—41no—
230C4—81no—
231C4—161no—
232C4—161no—
233C4—161no—
234C4#147311yescharacteristic
235C4#149511yescharacteristic
236C4#150911yescharacteristic
237C4#153211yescharacteristic
238C2—21no—
239C2—21no—
240C2—321no—
241C2#771211yescharacteristic
242C2#1183611yescharacteristic
243C2#1183711yescharacteristic
subgroupG/H
center, socleC22#1514
commutatorC22×C8C22×C4
Frattini, cosocleC2×#97C23
prime radicalC22×D8C2×C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12659C2semidirect
3. #8800C2semidirect
4. #7127C2semidirect
5. #7126C2semidirect

Maximal subgroups

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

subgroupindex
#332692
#126592
#88002
#71272
#71262
#71102