Finite Group Databasebeta

← #126355

Subgroups — #126355

Order 512.

261 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 261 conjugacy classes of subgroups — too many to draw. Only its 66 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#32504C211yesmaximal, characteristic
2#13537C211yesmaximal, characteristic
3#13505C211yesmaximal, characteristic
4#10649C211yesmaximal, characteristic
5#7120C211yesmaximal, characteristic
6#7120C211yesmaximal, characteristic
7#7118C211yesmaximal, characteristic
8#2506—21no—
9C2×#366—21no—
10#2086—21no—
11#2086—21no—
12#2083—21no—
13#1947—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
27C2×#125—41no—
28C22×C16—21no—
29C22×C16—21no—
30C2×#107—21no—
31C2×#107—21no—
32C2×#107—21no—
33C2×#107—41no—
34C2×#107D411yescharacteristic
35C2×#107D411yescharacteristic
36C2×#107D411yescharacteristic
37C2×#107D411yescharacteristic
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8D411yescharacteristic
41C2×C4×C8C2311yescharacteristic
42#400—41no—
43#398—41no—
44#389—21no—
45#389—21no—
46#389—21no—
47#389—21no—
48#387—41no—
49#384—21no—
50#384—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
65#125—82no—
66#125—82no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4D811yescharacteristic
81C2×C4⋊C4D811yescharacteristic
82C2×C42—21no—
83C2×C42—41no—
84C2×C42C2×D411yescharacteristic
85C2×C16—22no—
86C2×C16—41no—
87C2×C16—41no—
88C2×C16—41no—
89C2×C16—41no—
90C2×C16—41no—
91#107—21no—
92#107—21no—
93#107—21no—
94#107—41no—
95#107—41no—
96#107—42no—
97#107—41no—
98#107—41no—
99#107—81no—
100#107—82no—
101#107Q1612yes—
102C4⋊C8—22no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4×C8—41no—
108#95—21no—
109#95—41no—
110#95—41no—
111#95—41no—
112#95—41no—
113#95—41no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—81no—
121C22×C4#12011yescharacteristic
122C22×C4C2×D811yescharacteristic
123C22×C4AGL(1,ℤ/8)11yescharacteristic
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—42no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8—81no—
133C2×C8—81no—
134C2×C8C2×Q1611yescharacteristic
135C2×C8#13511yescharacteristic
136C2×C8AGL(1,ℤ/8)11yescharacteristic
137C2×C8#13711yescharacteristic
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4—82no—
149C4⋊C4—82no—
150C4⋊C4—82no—
151C4⋊C4—82no—
152C4⋊C4—81no—
153C4⋊C4—81no—
154C4⋊C4—81no—
155C4⋊C4—81no—
156C4⋊C4—81no—
157C4⋊C4—81no—
158C4⋊C4—81no—
159C4⋊C4—81no—
160C4⋊C4—161no—
161C4⋊C4D1612yes—
162C42—21no—
163C42—21no—
164C42—41no—
165C42—41no—
166C42—82no—
167C16—42no—
168C16—81no—
169C23#44711yescharacteristic
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—161no—
194C2×C4—161no—
195C2×C4—161no—
196C2×C4#44911yescharacteristic
197C2×C4#45111yescharacteristic
198C2×C4#45411yescharacteristic
199C2×C4#45611yescharacteristic
200C2×C4#45611yescharacteristic
201C2×C4#45811yescharacteristic
202C2×C4C2×D1611yescharacteristic
203C2×C4#50911yescharacteristic
204C8—21no—
205C8—21no—
206C8—21no—
207C8—21no—
208C8—82no—
209C8—81no—
210C22#161011yescharacteristic
211C22#162511yescharacteristic
212C22#164911yescharacteristic
213C22#218411yescharacteristic
214C22#219111yescharacteristic
215C22#219311yescharacteristic
216C22#219511yescharacteristic
217C4—21no—
218C4—21no—
219C4—41no—
220C4—42no—
221C4—81no—
222C4—82no—
223C4—161no—
224C4—161no—
225C4—161no—
226C4—321no—
227C4#160911yescharacteristic
228C4#161711yescharacteristic
229C4#164811yescharacteristic
230C4#166311yescharacteristic
231C2#779211yescharacteristic
232C2#1228811yescharacteristic
233C2#1229611yescharacteristic
234C2#1234011yescharacteristic
235C2#1235111yescharacteristic
236C2#1240411yescharacteristic
237C2#1241311yescharacteristic
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
#325042
#135372
#135052
#106492
#71202
#71182