Finite Group Databasebeta

← #126362

Subgroups — #126362

Order 512.

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

normal subgroups only

This group has 247 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#32508C211yesmaximal, characteristic
2#13538C211yesmaximal, characteristic
3#13505C211yesmaximal, characteristic
4#10650C211yesmaximal, characteristic
5#7121C211yesmaximal, characteristic
6#7120C211yesmaximal, characteristic
7#7118C211yesmaximal, characteristic
8#2509—21no—
9C2×#366—21no—
10#2088—21no—
11#2087—21no—
12#2085—21no—
13#1949—21no—
14#1947—21no—
15#1942C2211yescharacteristic
16#1942C2211yescharacteristic
17#1939C2211yescharacteristic
18#1918—21no—
19#1845C2211yescharacteristic
20#1380—21no—
21#1380—21no—
22#1380—21no—
23#1374C2211yescharacteristic
24#1374C2211yescharacteristic
25#1296—21no—
26#1277C2211yescharacteristic
27C22×C16—21no—
28C22×C16—21no—
29C2×#107—21no—
30C2×#107—21no—
31C2×#107—41no—
32C2×#107D411yescharacteristic
33C2×#107D411yescharacteristic
34C2×#107D411yescharacteristic
35C2×#107D411yescharacteristic
36C2×#106—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8D411yescharacteristic
40C2×C4×C8C2311yescharacteristic
41#400—41no—
42#400—41no—
43#389—21no—
44#389—21no—
45#389—21no—
46#389—21no—
47#389—41no—
48#387—41no—
49#384—21no—
50#382—21no—
51C4×C4⋊C4—21no—
52C4×C4⋊C4D411yescharacteristic
53#366—42no—
54#340—21no—
55#340—21no—
56#340—21no—
57#340—21no—
58#340—21no—
59#340—41no—
60#336—21no—
61C22×C8—21no—
62C22×C8—41no—
63C22×C8C2×D411yescharacteristic
64C22×C8C2×D411yescharacteristic
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4D811yescharacteristic
77C2×C4⋊C4D811yescharacteristic
78C2×C42—21no—
79C2×C42—41no—
80C2×C42C2×D411yescharacteristic
81C2×C16—22no—
82C2×C16—41no—
83C2×C16—41no—
84C2×C16—41no—
85C2×C16—41no—
86C2×C16—41no—
87#107—21no—
88#107—21no—
89#107—21no—
90#107—41no—
91#107—41no—
92#107—42no—
93#107—41no—
94#107—41no—
95#107—82no—
96#107Q1612yes—
97#106—81no—
98C4⋊C8—22no—
99C4⋊C8—41no—
100C4⋊C8—41no—
101C4⋊C8—41no—
102C4⋊C8—41no—
103C4×C8—41no—
104#95—21no—
105#95—41no—
106#95—41no—
107#95—41no—
108#95—41no—
109#95—41no—
110#95—41no—
111#95—41no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—81no—
119C22×C4#12011yescharacteristic
120C22×C4C2×D811yescharacteristic
121C22×C4AGL(1,ℤ/8)11yescharacteristic
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—42no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—81no—
131C2×C8—81no—
132C2×C8C2×Q1611yescharacteristic
133C2×C8#13511yescharacteristic
134C2×C8AGL(1,ℤ/8)11yescharacteristic
135C2×C8#13711yescharacteristic
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—81no—
148C4⋊C4—81no—
149C4⋊C4—82no—
150C4⋊C4—81no—
151C4⋊C4—81no—
152C4⋊C4—81no—
153C4⋊C4—81no—
154C4⋊C4—161no—
155C4⋊C4—161no—
156C4⋊C4—161no—
157C42—21no—
158C42—21no—
159C42—41no—
160C42—41no—
161C42—161no—
162C16—42no—
163C16—81no—
164C23#44711yescharacteristic
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—161no—
189C2×C4—161no—
190C2×C4—161no—
191C2×C4#44911yescharacteristic
192C2×C4#45111yescharacteristic
193C2×C4#45411yescharacteristic
194C2×C4#45611yescharacteristic
195C2×C4#45611yescharacteristic
196C2×C4#45811yescharacteristic
197C2×C4#50811yescharacteristic
198C2×C4#51011yescharacteristic
199C8—21no—
200C8—21no—
201C8—21no—
202C8—21no—
203C8—82no—
204C8—81no—
205C22#161011yescharacteristic
206C22#162511yescharacteristic
207C22#164911yescharacteristic
208C22#218911yescharacteristic
209C22#219011yescharacteristic
210C22#219211yescharacteristic
211C22#219511yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—81no—
217C4—82no—
218C4—161no—
219C4—161no—
220C4—161no—
221C4—321no—
222C4#160911yescharacteristic
223C4#161711yescharacteristic
224C4#164811yescharacteristic
225C4#166311yescharacteristic
226C2#779211yescharacteristic
227C2#1229311yescharacteristic
228C2#1229711yescharacteristic
229C2#1234511yescharacteristic
230C2#1234811yescharacteristic
231C2#1241111yescharacteristic
232C2#1241511yescharacteristic
subgroupG/H
center, socle, prime radicalC23#447
commutator, Frattini, cosocleC2×C4×C8C23
LCSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1939 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#325082
#135382
#135052
#106502
#71212
#71202
#71182